help links subscribe changes refresh edit

	FriCASTimofeev1
FriCASTimofeev37 ←	↑	→ FriCASTimofeev9

FriCASTimofeev5

Edit detail for FriCASTimofeev5 revision 2 of 3

	1 2 3
Editor: test1 Time: 2015/06/01 17:01:48 GMT+0
Note:

added:
Examples from Timofeev (1948) Ch. 5, divided into sections
kindly provided by clicliclic vel Martin on sci.math.symbolic


added:
-- Timofeev (

removed:
-
--- Timofeev Chapter 5 Integration Problems

added:


added:


added:


added:


added:


added:


added:


added:


removed:
-

added:

See also other chapters.

Examples from Timofeev (1948) Ch. 5, divided into sections kindly provided by clicliclic vel Martin on sci.math.symbolic

fricas

-- Timofeev (

fricas

)set break resume

-- Problems 1 - 3 (p. 202-203)
integrate(cos(x)^2, x)

$\label{eq1}{{{\cos \left({x}\right)}\ {\sin \left({x}\right)}}+ x}\over 2$

(1)

Type: Union(Expression(Integer),...)

fricas

integrate(cos(x)^3, x)

$\label{eq2}{{\left({{\cos \left({x}\right)}^{2}}+ 2 \right)}\ {\sin \left({x}\right)}}\over 3$

(2)

Type: Union(Expression(Integer),...)

fricas

integrate(sin(x)^4, x)

$\label{eq3}{{{\left({2 \ {{\cos \left({x}\right)}^{3}}}-{5 \ {\cos \left({x}\right)}}\right)}\ {\sin \left({x}\right)}}+{3 \ x}}\over 8$

(3)

Type: Union(Expression(Integer),...)

fricas

-- Problems 4 - 7 (p. 208)
integrate(cos(x)^6, x)

$\label{eq4}{{{\left({8 \ {{\cos \left({x}\right)}^{5}}}+{{10}\ {{\cos \left({x}\right)}^{3}}}+{{15}\ {\cos \left({x}\right)}}\right)}\ {\sin \left({x}\right)}}+{{15}\ x}}\over{48}$

(4)

Type: Union(Expression(Integer),...)

fricas

integrate(sin(x)^8, x)

$\label{eq5}{\left( \begin{array}{@{}l} \displaystyle {{\left({ \begin{array}{@{}l} \displaystyle {{48}\ {{\cos \left({x}\right)}^{7}}}-{{200}\ {{\cos \left({x}\right)}^{5}}}+ \ \ \displaystyle {{326}\ {{\cos \left({x}\right)}^{3}}}-{{279}\ {\cos \left({x}\right)}}$

(5)

Type: Union(Expression(Integer),...)

fricas

integrate(cos(1/4*%pi+1/2*x)^4, x)

$\label{eq6}{{{\left({4 \ {{\cos \left({{{2 \ x}+ \pi}\over 4}\right)}^{3}}}+{6 \ {\cos \left({{{2 \ x}+ \pi}\over 4}\right)}}\right)}\ {\sin \left({{{2 \ x}+ \pi}\over 4}\right)}}+{3 \ x}}\over 8$

(6)

Type: Union(Expression(Integer),...)

fricas

integrate(sin(-1/12*%pi+3*x)^3, x)

$\label{eq7}{{{\cos \left({{{{36}\ x}- \pi}\over{12}}\right)}^{3}}-{3 \ {\cos \left({{{{36}\ x}- \pi}\over{12}}\right)}}}\over 9$

(7)

Type: Union(Expression(Integer),...)

fricas

-- Problems 8 - 11 (p. 211)
integrate(1/sin(x)^6, x)

$\label{eq8}{-{8 \ {{\cos \left({x}\right)}^{5}}}+{{20}\ {{\cos \left({x}\right)}^{3}}}-{{15}\ {\cos \left({x}\right)}}}\over{{\left({{15}\ {{\cos \left({x}\right)}^{4}}}-{{30}\ {{\cos \left({x}\right)}^{2}}}+{15}\right)}\ {\sin \left({x}\right)}}$

(8)

Type: Union(Expression(Integer),...)

fricas

integrate(csc(x)^7, x)

$\label{eq9}{\left( \begin{array}{@{}l} \displaystyle {{\left({ \begin{array}{@{}l} \displaystyle {{15}\ {{\cos \left({x}\right)}^{6}}}-{{45}\ {{\cos \left({x}\right)}^{4}}}+ \ \ \displaystyle {{45}\ {{\cos \left({x}\right)}^{2}}}-{15}$

(9)

Type: Union(Expression(Integer),...)

fricas

integrate(1/cos(x)^12, x)

$\label{eq10}{\left({\left({ \begin{array}{@{}l} \displaystyle {{256}\ {{\cos \left({x}\right)}^{10}}}+{{128}\ {{\cos \left({x}\right)}^{8}}}+ \ \ \displaystyle {{96}\ {{\cos \left({x}\right)}^{6}}}+{{80}\ {{\cos \left({x}\right)}^{4}}}+{{70}\ {{\cos \left({x}\right)}^{2}}}+ \ \ \displaystyle {63}$

(10)

Type: Union(Expression(Integer),...)

fricas

integrate(1/cos(1/4*%pi+3*x)^3, x)

$\label{eq11}{\left( \begin{array}{@{}l} \displaystyle {{{\cos \left({{{{12}\ x}+ \pi}\over 4}\right)}^{2}}\ {\log{\left({{{\sin \left({{{{12}\ x}+ \pi}\over 4}\right)}+{\cos \left({{{{12}\ x}+ \pi}\over 4}\right)}+ 1}\over{{\cos \left({{{{12}\ x}+ \pi}\over 4}\right)}+ 1}}\right)}}}- \ \ \displaystyle {{{\cos \left({{{{12}\ x}+ \pi}\over 4}\right)}^{2}}\ {\log{\left({{{\sin \left({{{{12}\ x}+ \pi}\over 4}\right)}-{\cos \left({{{{12}\ x}+ \pi}\over 4}\right)}- 1}\over{{\cos \left({{{{12}\ x}+ \pi}\over 4}\right)}+ 1}}\right)}}}+ \ \ \displaystyle {\sin \left({{{{12}\ x}+ \pi}\over 4}\right)}$

(11)

Type: Union(Expression(Integer),...)

fricas

-- Problems 12 - 14 (p. 213)
integrate(tan(x)^6, x)

$\label{eq12}{{3 \ {{\tan \left({x}\right)}^{5}}}-{5 \ {{\tan \left({x}\right)}^{3}}}+{{15}\ {\tan \left({x}\right)}}-{{15}\ x}}\over{15}$

(12)

Type: Union(Expression(Integer),...)

fricas

integrate(1/tan(x)^5, x)

$\label{eq13}{\left( \begin{array}{@{}l} \displaystyle -{2 \ {{\tan \left({x}\right)}^{4}}\ {\log \left({{{\tan \left({x}\right)}^{2}}+ 1}\right)}}+ \ \ \displaystyle {4 \ {{\tan \left({x}\right)}^{4}}\ {\log \left({\tan \left({x}\right)}\right)}}+{2 \ {{\tan \left({x}\right)}^{2}}}- 1$

(13)

Type: Union(Expression(Integer),...)

fricas

integrate(cot(-3/4*%pi+1/3*x)^4, x)

$\label{eq14}{\left( \begin{array}{@{}l} \displaystyle {{\left({x \ {\cos \left({{{4 \ x}-{9 \ \pi}}\over 6}\right)}}- x \right)}\ {\sin \left({{{4 \ x}-{9 \ \pi}}\over 6}\right)}}+ \ \ \displaystyle {4 \ {{\cos \left({{{4 \ x}-{9 \ \pi}}\over 6}\right)}^{2}}}+{2 \ {\cos \left({{{4 \ x}-{9 \ \pi}}\over 6}\right)}}- 2$

(14)

Type: Union(Expression(Integer),...)

fricas

-- Problems 15 - 20 (p. 219-220)
integrate(cos(x)^6*sin(x)^4, x)

$\label{eq15}{\left( \begin{array}{@{}l} \displaystyle {{\left({ \begin{array}{@{}l} \displaystyle {{128}\ {{\cos \left({x}\right)}^{9}}}-{{176}\ {{\cos \left({x}\right)}^{7}}}+{8 \ {{\cos \left({x}\right)}^{5}}}+ \ \ \displaystyle {{10}\ {{\cos \left({x}\right)}^{3}}}+{{15}\ {\cos \left({x}\right)}}$

(15)

Type: Union(Expression(Integer),...)

fricas

integrate(cos(x)^6*sin(x)^7, x)

$\label{eq16}{{{231}\ {{\cos \left({x}\right)}^{13}}}-{{819}\ {{\cos \left({x}\right)}^{11}}}+{{1001}\ {{\cos \left({x}\right)}^{9}}}-{{429}\ {{\cos \left({x}\right)}^{7}}}}\over{3003}$

(16)

Type: Union(Expression(Integer),...)

fricas

integrate(sin(x)^11/cos(x), x)

$\label{eq17}{\left( \begin{array}{@{}l} \displaystyle {{120}\ {\log \left({2 \over{{\cos \left({x}\right)}+ 1}}\right)}}-{{120}\ {\log \left({-{{2 \ {\cos \left({x}\right)}}\over{{\cos \left({x}\right)}+ 1}}}\right)}}+ \ \ \displaystyle {{12}\ {{\cos \left({x}\right)}^{10}}}-{{75}\ {{\cos \left({x}\right)}^{8}}}+{{200}\ {{\cos \left({x}\right)}^{6}}}- \ \ \displaystyle {{300}\ {{\cos \left({x}\right)}^{4}}}+{{300}\ {{\cos \left({x}\right)}^{2}}}$

(17)

Type: Union(Expression(Integer),...)

fricas

integrate(1/(cos(x)^6*sin(x)^6), x)

$\label{eq18}{\left( \begin{array}{@{}l} \displaystyle -{{256}\ {{\cos \left({x}\right)}^{10}}}+{{640}\ {{\cos \left({x}\right)}^{8}}}-{{480}\ {{\cos \left({x}\right)}^{6}}}+ \ \ \displaystyle {{80}\ {{\cos \left({x}\right)}^{4}}}+{{10}\ {{\cos \left({x}\right)}^{2}}}+ 3$

(18)

Type: Union(Expression(Integer),...)

fricas

integrate(cos(x)^(2*m)*sin(x)^(2*m), x)

$\label{eq19}\int^{ \displaystyle x}{{{\cos \left({\%A}\right)}^{2 \ m}}\ {{\sin \left({\%A}\right)}^{2 \ m}}\ {d \%A}}$

(19)

Type: Union(Expression(Integer),...)

fricas

integrate(1/(cos(1/4*%pi+2*x)*sin(1/4*%pi+2*x)^3), x)

$\label{eq20}{\left( \begin{array}{@{}l} \displaystyle {{\left({4 \ {{\cos \left({{{8 \ x}+ \pi}\over 4}\right)}^{2}}}- 4 \right)}\ {\log \left({{\sin \left({{{8 \ x}+ \pi}\over 4}\right)}\over{{\cos \left({{{8 \ x}+ \pi}\over 4}\right)}+ 1}}\right)}}+ \ \ \displaystyle {{\left(-{4 \ {{\cos \left({{{8 \ x}+ \pi}\over 4}\right)}^{2}}}+ 4 \right)}\ {\log \left({-{{2 \ {\cos \left({{{8 \ x}+ \pi}\over 4}\right)}}\over{{\cos \left({{{8 \ x}+ \pi}\over 4}\right)}+ 1}}}\right)}}+ \ \ \displaystyle {{\cos \left({{{8 \ x}+ \pi}\over 4}\right)}^{2}}+ 1$

(20)

Type: Union(Expression(Integer),...)

fricas

-- Problems 21 - 29 (p. 223)
integrate(sec(x)^2*tan(x)^2, x)

$\label{eq21}{{\left(-{{\cos \left({x}\right)}^{2}}+ 1 \right)}\ {\sin \left({x}\right)}}\over{3 \ {{\cos \left({x}\right)}^{3}}}$

(21)

Type: Union(Expression(Integer),...)

fricas

integrate(cot(x)^3*csc(x), x)

$\label{eq22}{{3 \ {{\cos \left({x}\right)}^{2}}}- 2}\over{{\left({3 \ {{\cos \left({x}\right)}^{2}}}- 3 \right)}\ {\sin \left({x}\right)}}$

(22)

Type: Union(Expression(Integer),...)

fricas

integrate(sec(x)^3*tan(x), x)

$\label{eq23}{{{\cos \left({x}\right)}^{3}}+ 1}\over{3 \ {{\cos \left({x}\right)}^{3}}}$

(23)

Type: Union(Expression(Integer),...)

fricas

integrate(cot(x)^2*csc(x)^3, x)

$\label{eq24}{\left( \begin{array}{@{}l} \displaystyle {{\left(-{{\cos \left({x}\right)}^{4}}+{2 \ {{\cos \left({x}\right)}^{2}}}- 1 \right)}\ {\log \left({{\sin \left({x}\right)}\over{{\cos \left({x}\right)}+ 1}}\right)}}- \ \ \displaystyle {{\cos \left({x}\right)}^{3}}-{\cos \left({x}\right)}$

(24)

Type: Union(Expression(Integer),...)

fricas

integrate(cos(x)^3/sin(x)^7, x)

$\label{eq25}{-{{\cos \left({x}\right)}^{6}}+{3 \ {{\cos \left({x}\right)}^{4}}}+{3 \ {{\cos \left({x}\right)}^{2}}}- 1}\over{{{24}\ {{\cos \left({x}\right)}^{6}}}-{{72}\ {{\cos \left({x}\right)}^{4}}}+{{72}\ {{\cos \left({x}\right)}^{2}}}-{24}}$

(25)

Type: Union(Expression(Integer),...)

fricas

integrate(sqrt(sec(x)^3)*tan(x)^5, x)

$\label{eq26}{{\left({{154}\ {{\cos \left({x}\right)}^{4}}}-{{132}\ {{\cos \left({x}\right)}^{2}}}+{42}\right)}\ {\sqrt{1 \over{{\cos \left({x}\right)}^{3}}}}}\over{{231}\ {{\cos \left({x}\right)}^{4}}}$

(26)

Type: Union(Expression(Integer),...)

fricas

integrate(sec(x)^4*tan(x)^(3/2), x)

$\label{eq27}{{\left(-{8 \ {{\cos \left({x}\right)}^{4}}}-{2 \ {{\cos \left({x}\right)}^{2}}}+{10}\right)}\ {\sqrt{{\sin \left({x}\right)}\over{\cos \left({x}\right)}}}}\over{{45}\ {{\cos \left({x}\right)}^{4}}}$

(27)

Type: Union(Expression(Integer),...)

fricas

integrate(cot(x)^4*csc(x)^3, x)

$\label{eq28}{\left( \begin{array}{@{}l} \displaystyle {{\left({ \begin{array}{@{}l} \displaystyle {3 \ {{\cos \left({x}\right)}^{6}}}-{9 \ {{\cos \left({x}\right)}^{4}}}+ \ \ \displaystyle {9 \ {{\cos \left({x}\right)}^{2}}}- 3$

(28)

Type: Union(Expression(Integer),...)

fricas

)set output tex off

fricas

)set output algebra on

integrate(sec(1/4*%pi+1/2*x)^3*tan(1/4*%pi+1/2*x)^2, x)

   (29)
                               2x + %pi        2x + %pi
                           sin(--------) + cos(--------) + 1
             2x + %pi 4            4               4
       - cos(--------) log(---------------------------------)
                 4                     2x + %pi
                                   cos(--------) + 1
                                           4
     + 
                             2x + %pi        2x + %pi
                         sin(--------) - cos(--------) - 1
           2x + %pi 4            4               4
       cos(--------) log(---------------------------------)
               4                     2x + %pi
                                 cos(--------) + 1
                                         4
     + 
              2x + %pi 2         2x + %pi
       (- cos(--------)  + 2)sin(--------)
                  4                  4
  /
          2x + %pi 4
     4cos(--------)
              4

Type: Union(Expression(Integer),...)

fricas

)set output tex on

fricas

)set output algebra off

-- Problems 30 - 32 (p. 228)
integrate((1+cot(x)^3)*(a*sec(x)^2-sin(2*x)), x)

$\label{eq29}{\left( \begin{array}{@{}l} \displaystyle {{\left({4 \ {{\cos \left({x}\right)}^{4}}}+{{\left({4 \ a}-{12}\right)}\ {{\cos \left({x}\right)}^{2}}}-{4 \ a}\right)}\ {\sin \left({x}\right)}}+ \ \ \displaystyle {4 \ {{\cos \left({x}\right)}^{5}}}+{{\left({{12}\ x}+ a - 8 \right)}\ {{\cos \left({x}\right)}^{3}}}+ \ \ \displaystyle {{\left(-{{12}\ x}+ a + 4 \right)}\ {\cos \left({x}\right)}}$

(29)

Type: Union(Expression(Integer),...)

fricas

integrate((4-3*cos(x))*(1-1/2*sin(x))^4, x)

$\label{eq30}{\left( \begin{array}{@{}l} \displaystyle {{\left({ \begin{array}{@{}l} \displaystyle -{{18}\ {{\cos \left({x}\right)}^{4}}}+{{30}\ {{\cos \left({x}\right)}^{3}}}+ \ \ \displaystyle {{756}\ {{\cos \left({x}\right)}^{2}}}-{{1515}\ {\cos \left({x}\right)}}-{2178}$

(30)

Type: Union(Expression(Integer),...)

fricas

integrate((1/2-3*cot(x))*(3-2*cot(x))^3, x)

$\label{eq31}{\left( \begin{array}{@{}l} \displaystyle {{\left({8 \ {\cos \left({2 \ x}\right)}}- 8 \right)}\ {\sin \left({2 \ x}\right)}\ {\log \left({{\sin \left({2 \ x}\right)}\over{{\cos \left({2 \ x}\right)}+ 1}}\right)}}+ \ \ \displaystyle {{\left(-{4 \ {\cos \left({2 \ x}\right)}}+ 4 \right)}\ {\sin \left({2 \ x}\right)}\ {\log \left({2 \over{{\cos \left({2 \ x}\right)}+ 1}}\right)}}+ \ \ \displaystyle {{\left({{\left(-{{285}\ x}-{112}\right)}\ {\cos \left({2 \ x}\right)}}+{{285}\ x}-{112}\right)}\ {\sin \left({2 \ x}\right)}}- \ \ \displaystyle {{296}\ {{\cos \left({2 \ x}\right)}^{2}}}+{{32}\ {\cos \left({2 \ x}\right)}}+{328}$

(31)

Type: Union(Expression(Integer),...)

fricas

-- Problems 33 - 36 (p. 229)
integrate(cos(5*x)/cos(x)^5, x)

$\label{eq32}{{{\left(-{{50}\ {{\cos \left({x}\right)}^{2}}}+ 5 \right)}\ {\sin \left({x}\right)}}+{{48}\ x \ {{\cos \left({x}\right)}^{3}}}}\over{3 \ {{\cos \left({x}\right)}^{3}}}$

(32)

Type: Union(Expression(Integer),...)

fricas

integrate(cos(4*x)/cos(x), x)

$\label{eq33}{\left( \begin{array}{@{}l} \displaystyle {3 \ {\log \left({{{\sin \left({x}\right)}+{\cos \left({x}\right)}+ 1}\over{{\cos \left({x}\right)}+ 1}}\right)}}- \ \ \displaystyle {3 \ {\log \left({{{\sin \left({x}\right)}-{\cos \left({x}\right)}- 1}\over{{\cos \left({x}\right)}+ 1}}\right)}}+{{\left({8 \ {{\cos \left({x}\right)}^{2}}}- 8 \right)}\ {\sin \left({x}\right)}}$

(33)

Type: Union(Expression(Integer),...)

fricas

integrate(cos(x)*cos(4*x), x)

$\label{eq34}{{\left({{24}\ {{\cos \left({x}\right)}^{4}}}-{8 \ {{\cos \left({x}\right)}^{2}}}- 1 \right)}\ {\sin \left({x}\right)}}\over{15}$

(34)

Type: Union(Expression(Integer),...)

fricas

integrate(cos(4*x)/cos(x)^5, x)

$\label{eq35}{\left( \begin{array}{@{}l} \displaystyle {{35}\ {{\cos \left({x}\right)}^{4}}\ {\log \left({{{\sin \left({x}\right)}+{\cos \left({x}\right)}+ 1}\over{{\cos \left({x}\right)}+ 1}}\right)}}- \ \ \displaystyle {{35}\ {{\cos \left({x}\right)}^{4}}\ {\log \left({{{\sin \left({x}\right)}-{\cos \left({x}\right)}- 1}\over{{\cos \left({x}\right)}+ 1}}\right)}}+ \ \ \displaystyle {{\left(-{{29}\ {{\cos \left({x}\right)}^{2}}}+ 2 \right)}\ {\sin \left({x}\right)}}$

(35)

Type: Union(Expression(Integer),...)

fricas

-- Problems 37 - 39 (p. 233)
integrate(cos(x)^4*cos(4*x), x)

$\label{eq36}{{{\left({{48}\ {{\cos \left({x}\right)}^{7}}}-{8 \ {{\cos \left({x}\right)}^{5}}}+{2 \ {{\cos \left({x}\right)}^{3}}}+{3 \ {\cos \left({x}\right)}}\right)}\ {\sin \left({x}\right)}}+{3 \ x}}\over{48}$

(36)

Type: Union(Expression(Integer),...)

fricas

integrate(cos(5*x)/sin(x)^5, x)

$\label{eq37}{\left( \begin{array}{@{}l} \displaystyle {{\left({{512}\ {{\cos \left({x}\right)}^{4}}}-{{1024}\ {{\cos \left({x}\right)}^{2}}}+{512}\right)}\ {\log \left({{\sin \left({x}\right)}\over{{\cos \left({x}\right)}+ 1}}\right)}}+ \ \ \displaystyle {{\left({ \begin{array}{@{}l} \displaystyle -{{512}\ {{\cos \left({x}\right)}^{4}}}+ \ \ \displaystyle {{1024}\ {{\cos \left({x}\right)}^{2}}}-{512}$

(37)

Type: Union(Expression(Integer),...)

fricas

integrate(sin(4*x)/sin(x)^4, x)

$\label{eq38}{\left( \begin{array}{@{}l} \displaystyle {{\left(-{8 \ {{\cos \left({x}\right)}^{2}}}+ 8 \right)}\ {\log \left({{\sin \left({x}\right)}\over{{\cos \left({x}\right)}+ 1}}\right)}}+ \ \ \displaystyle {{\left({8 \ {{\cos \left({x}\right)}^{2}}}- 8 \right)}\ {\log \left({2 \over{{\cos \left({x}\right)}+ 1}}\right)}}+{{\cos \left({x}\right)}^{2}}+ 1$

(38)

Type: Union(Expression(Integer),...)

fricas

-- Problems 40 - 49 (p. 254-255)
integrate(cos(x)/(sin(x)*(2+sin(2*x))), x)

$\label{eq39}{\left( \begin{array}{@{}l} \displaystyle -{{\sqrt{3}}\ {\log \left({{{4 \ {\cos \left({x}\right)}\ {\sin \left({x}\right)}}+ 4}\over{{{\cos \left({x}\right)}^{2}}+{2 \ {\cos \left({x}\right)}}+ 1}}\right)}}+ \ \ \displaystyle {2 \ {\sqrt{3}}\ {\log \left({{\sin \left({x}\right)}\over{{\cos \left({x}\right)}+ 1}}\right)}}- \ \ \displaystyle {2 \ {\arctan \left({{{{\sqrt{3}}\ {\sin \left({x}\right)}}-{{\sqrt{3}}\ {\cos \left({x}\right)}}-{\sqrt{3}}}\over{{3 \ {\cos \left({x}\right)}}+ 3}}\right)}}- \ \ \displaystyle {2 \ {\arctan{\left({{{{\left({2 \ {\cos \left({x}\right)}}+ 4 \right)}\ {\sin \left({x}\right)}}+{3 \ {{\cos \left({x}\right)}^{2}}}+{2 \ {\cos \left({x}\right)}}- 1}\over{{{\sqrt{3}}\ {{\cos \left({x}\right)}^{2}}}+{2 \ {\sqrt{3}}\ {\cos \left({x}\right)}}+{\sqrt{3}}}}\right)}}}$

(39)

Type: Union(Expression(Integer),...)

fricas

integrate(cos(x)^2/(cos(3*x)*sin(x)), x)

$\label{eq40}{{2 \ {\log \left({{\sin \left({x}\right)}\over{{\cos \left({x}\right)}+ 1}}\right)}}-{\log \left({{{{16}\ {{\cos \left({x}\right)}^{2}}}-{12}}\over{{{\cos \left({x}\right)}^{2}}+{2 \ {\cos \left({x}\right)}}+ 1}}\right)}}\over 2$

(40)

Type: Union(Expression(Integer),...)

fricas

integrate(sin(2*x)/(cos(x)^4+sin(x)^4), x)

$\label{eq41}\begin{array}{@{}l} \displaystyle -{\arctan \left({{{7 \ {{\cos \left({x}\right)}^{3}}}+{5 \ {{\cos \left({x}\right)}^{2}}}-{4 \ {\cos \left({x}\right)}}- 4}\over{{{\cos \left({x}\right)}^{3}}+{3 \ {{\cos \left({x}\right)}^{2}}}+{3 \ {\cos \left({x}\right)}}+ 1}}\right)}+ \ \ \displaystyle {\arctan \left({{{3 \ {\cos \left({x}\right)}}+ 1}\over{{4 \ {\cos \left({x}\right)}}+ 4}}\right)}$

(41)

Type: Union(Expression(Integer),...)

fricas

)set output tex off

fricas

)set output algebra on

integrate(1/(4+sin(x)+cos(x)*sqrt(3)), x)

   (43)
   -
        atan
                                         +-+
               (104765547927309152411486\|3  - 181459251892870078413729)sin(x)
             + 
                                        +-+
               (18277149216643579325555\|3  - 31656951060734835109266)cos(x)
             + 
                                       +-+
               18277149216643579325555\|3  - 31656951060734835109266
          /
                                        +-+
               (63313902121469670218532\|3  - 109662895299861475953330)cos(x)
             + 
                                       +-+
               63313902121469670218532\|3  - 109662895299861475953330
     /
         +-+
        \|3

Type: Union(Expression(Integer),...)

fricas

)set output tex on

fricas

)set output algebra off

integrate(1/(3+4*cos(x)+4*sin(x)), x)

$\label{eq42}{\log{\left({{{{\left({4 \ {\sqrt{23}}}-{23}\right)}\ {\sin \left({x}\right)}}+{{\left(-{{19}\ {\sqrt{23}}}+{92}\right)}\ {\cos \left({x}\right)}}-{{20}\ {\sqrt{23}}}+{92}}\over{{4 \ {\sin \left({x}\right)}}+{4 \ {\cos \left({x}\right)}}+ 3}}\right)}}\over{\sqrt{23}}$

(42)

Type: Union(Expression(Integer),...)

fricas

integrate(1/(4-3*cos(x)^2+5*sin(x)^2), x)

$\label{eq43}{{\arctan \left({{{\left({{17}\ {\cos \left({x}\right)}}+{18}\right)}\ {\sin \left({x}\right)}}\over{{3 \ {{\cos \left({x}\right)}^{2}}}+{6 \ {\cos \left({x}\right)}}+ 3}}\right)}+{\arctan \left({{\sin \left({x}\right)}\over{{6 \ {\cos \left({x}\right)}}+ 6}}\right)}}\over 3$

(43)

Type: Union(Expression(Integer),...)

fricas

integrate(1/(4+4*cot(x)+tan(x)), x)

$\label{eq44}{\left( \begin{array}{@{}l} \displaystyle {{\left({3 \ {\tan \left({x}\right)}}+ 6 \right)}\ {\log \left({{{\tan \left({x}\right)}^{2}}+ 1}\right)}}+ \ \ \displaystyle {{\left(-{6 \ {\tan \left({x}\right)}}-{12}\right)}\ {\log \left({{\tan \left({x}\right)}+ 2}\right)}}+{8 \ x \ {\tan \left({x}\right)}}+ \ \ \displaystyle {{16}\ x}+{20}$

(44)

Type: Union(Expression(Integer),...)

fricas

integrate(1/(2*sec(x)+sin(x))^2, x)

$\label{eq45}{\left( \begin{array}{@{}l} \displaystyle {{\left({{16}\ {\cos \left({x}\right)}\ {\sin \left({x}\right)}}+{32}\right)}\ {\arctan{\left({{{2 \ {\sqrt{15}}\ {\sin \left({x}\right)}}-{{\sqrt{15}}\ {\cos \left({x}\right)}}-{\sqrt{15}}}\over{{{15}\ {\cos \left({x}\right)}}+{15}}}\right)}}}+ \ \ \displaystyle {{\left({{16}\ {\cos \left({x}\right)}\ {\sin \left({x}\right)}}+{32}\right)}\ {\arctan{\left({{{{\left({4 \ {\cos \left({x}\right)}}+ 8 \right)}\ {\sin \left({x}\right)}}+{3 \ {{\cos \left({x}\right)}^{2}}}+{2 \ {\cos \left({x}\right)}}- 1}\over{{{\sqrt{15}}\ {{\cos \left({x}\right)}^{2}}}+{2 \ {\sqrt{15}}\ {\cos \left({x}\right)}}+{\sqrt{15}}}}\right)}}}+ \ \ \displaystyle {7 \ {\sqrt{15}}\ {\cos \left({x}\right)}\ {\sin \left({x}\right)}}+{2 \ {\sqrt{15}}\ {{\cos \left({x}\right)}^{2}}}-{2 \ {\sqrt{1 5}}}$

(45)

Type: Union(Expression(Integer),...)

fricas

integrate(1/(cos(x)+2*sec(x))^2, x)

$\label{eq46}{\left( \begin{array}{@{}l} \displaystyle {{\left({{\cos \left({x}\right)}^{2}}+ 2 \right)}\ {\arctan \left({{{\sqrt{6}}\ {\sin \left({x}\right)}}\over{{4 \ {\cos \left({x}\right)}}+ 4}}\right)}}+ \ \ \displaystyle {{\left({{\cos \left({x}\right)}^{2}}+ 2 \right)}\ {\arctan{\left({{{\left({\cos \left({x}\right)}+ 4 \right)}\ {\sin \left({x}\right)}}\over{{{\sqrt{6}}\ {{\cos \left({x}\right)}^{2}}}+{2 \ {\sqrt{6}}\ {\cos \left({x}\right)}}+{\sqrt{6}}}}\right)}}}+ \ \ \displaystyle {{\sqrt{6}}\ {\cos \left({x}\right)}\ {\sin \left({x}\right)}}$

(46)

Type: Union(Expression(Integer),...)

fricas

integrate((5-tan(x)-6*tan(x)^2)/(1+3*tan(x))^3, x)

$\label{eq47}{\left( \begin{array}{@{}l} \displaystyle {{\left({{252}\ {{\tan \left({x}\right)}^{2}}}+{{168}\ {\tan \left({x}\right)}}+{28}\right)}\ {\log \left({{{\tan \left({x}\right)}^{2}}+ 1}\right)}}+ \ \ \displaystyle {{\left(-{{504}\ {{\tan \left({x}\right)}^{2}}}-{{336}\ {\tan \left({x}\right)}}-{56}\right)}\ {\log \left({{3 \ {\tan \left({x}\right)}}+ 1}\right)}}- \ \ \displaystyle {{603}\ x \ {{\tan \left({x}\right)}^{2}}}+{{\left(-{{402}\ x}-{435}\right)}\ {\tan \left({x}\right)}}-{{67}\ x}- \ \ \displaystyle {320}$

(47)

Type: Union(Expression(Integer),...)

fricas

-- Problems 50 - 56 (p. 260)
integrate(cos(x)^2/cos(3*x), x)

$\label{eq48}{{\log \left({{{4 \ {\sin \left({x}\right)}}+ 2}\over{{\cos \left({x}\right)}+ 1}}\right)}-{\log \left({{-{4 \ {\sin \left({x}\right)}}+ 2}\over{{\cos \left({x}\right)}+ 1}}\right)}}\over 4$

(48)

Type: Union(Expression(Integer),...)

fricas

integrate(sin(x)/cos(2*x), x)

$\label{eq49}{\log \left({{{{\left({6 \ {\sqrt{2}}}+ 8 \right)}\ {{\cos \left({x}\right)}^{2}}}+{{\left({8 \ {\sqrt{2}}}+{12}\right)}\ {\cos \left({x}\right)}}+{3 \ {\sqrt{2}}}+ 4}\over{{2 \ {{\cos \left({x}\right)}^{2}}}- 1}}\right)}\over{2 \ {\sqrt{2}}}$

(49)

Type: Union(Expression(Integer),...)

fricas

integrate(sin(x)^2/cos(2*x), x)

$\label{eq50}{\left( \begin{array}{@{}l} \displaystyle -{\log \left({{{2 \ {\sin \left({x}\right)}}-{2 \ {\cos \left({x}\right)}}}\over{{\cos \left({x}\right)}+ 1}}\right)}+ \ \ \displaystyle {\log \left({{-{2 \ {\sin \left({x}\right)}}-{2 \ {\cos \left({x}\right)}}}\over{{\cos \left({x}\right)}+ 1}}\right)}-{2 \ x}$

(50)

Type: Union(Expression(Integer),...)

fricas

integrate(sin(x)^3/cos(3*x), x)

$\label{eq51}{\left( \begin{array}{@{}l} \displaystyle -{\log \left({{{{16}\ {{\cos \left({x}\right)}^{2}}}-{12}}\over{{{\cos \left({x}\right)}^{2}}+{2 \ {\cos \left({x}\right)}}+ 1}}\right)}-{6 \ {\log \left({2 \over{{\cos \left({x}\right)}+ 1}}\right)}}+ \ \ \displaystyle {8 \ {\log \left({-{{2 \ {\cos \left({x}\right)}}\over{{\cos \left({x}\right)}+ 1}}}\right)}}$

(51)

Type: Union(Expression(Integer),...)

fricas

integrate(cos(x)/sin(3*x), x)

$\label{eq52}{{2 \ {\log \left({{\sin \left({x}\right)}\over{{\cos \left({x}\right)}+ 1}}\right)}}-{\log \left({{{{16}\ {{\cos \left({x}\right)}^{2}}}- 4}\over{{{\cos \left({x}\right)}^{2}}+{2 \ {\cos \left({x}\right)}}+ 1}}\right)}}\over 6$

(52)

Type: Union(Expression(Integer),...)

fricas

integrate(sin(x)/sin(4*x), x)

$\label{eq53}{\left( \begin{array}{@{}l} \displaystyle -{{\sqrt{2}}\ {\log \left({{{\sin \left({x}\right)}+{\cos \left({x}\right)}+ 1}\over{{\cos \left({x}\right)}+ 1}}\right)}}+ \ \ \displaystyle {{\sqrt{2}}\ {\log \left({{{\sin \left({x}\right)}-{\cos \left({x}\right)}- 1}\over{{\cos \left({x}\right)}+ 1}}\right)}}+ \ \ \displaystyle {\log \left({{{4 \ {\sin \left({x}\right)}}-{2 \ {\sqrt{2}}\ {{\cos \left({x}\right)}^{2}}}+{3 \ {\sqrt{2}}}}\over{{2 \ {{\cos \left({x}\right)}^{2}}}- 1}}\right)}$

(53)

Type: Union(Expression(Integer),...)

fricas

integrate(sin(x)^3/sin(4*x), x)

$\label{eq54}{\left( \begin{array}{@{}l} \displaystyle -{2 \ {\sqrt{2}}\ {\log \left({{{\sin \left({x}\right)}+{\cos \left({x}\right)}+ 1}\over{{\cos \left({x}\right)}+ 1}}\right)}}+ \ \ \displaystyle {2 \ {\sqrt{2}}\ {\log \left({{{\sin \left({x}\right)}-{\cos \left({x}\right)}- 1}\over{{\cos \left({x}\right)}+ 1}}\right)}}+ \ \ \displaystyle {\log \left({{{4 \ {\sin \left({x}\right)}}-{2 \ {\sqrt{2}}\ {{\cos \left({x}\right)}^{2}}}+{3 \ {\sqrt{2}}}}\over{{2 \ {{\cos \left({x}\right)}^{2}}}- 1}}\right)}$

(54)

Type: Union(Expression(Integer),...)

fricas

-- Problems 57 - 61 (p. 266)
integrate(sqrt(1+sin(2*x)), x)

$\label{eq55}{{\left({\sin \left({2 \ x}\right)}-{\cos \left({2 \ x}\right)}- 1 \right)}\ {\sqrt{{\sin \left({2 \ x}\right)}+ 1}}}\over{{\sin \left({2 \ x}\right)}+{\cos \left({2 \ x}\right)}+ 1}$

(55)

Type: Union(Expression(Integer),...)

fricas

integrate(1/sqrt(1+cos(2*x)), x)

$\label{eq56}{{\sqrt{2}}\ {\log \left({{{{\left(-{\cos \left({2 \ x}\right)}+ 3 \right)}\ {\sqrt{{\cos \left({2 \ x}\right)}+ 1}}}+{2 \ {\sqrt{2}}\ {\sin \left({2 \ x}\right)}}}\over{{\left({\cos \left({2 \ x}\right)}+ 1 \right)}\ {\sqrt{{\cos \left({2 \ x}\right)}+ 1}}}}\right)}}\over 4$

(56)

Type: Union(Expression(Integer),...)

fricas

integrate(1/(1-cos(3*x))^(3/2), x)

$\label{eq57}{\left( \begin{array}{@{}l} \displaystyle { \begin{array}{@{}l} \displaystyle {\left({{\sqrt{2}}\ {\cos \left({3 \ x}\right)}}-{\sqrt{2}}\right)}\ {\sin \left({3 \ x}\right)}\ \cdot \ \ \displaystyle {\log{\left({{{{\left(-{{\sqrt{2}}\ {\cos \left({3 \ x}\right)}}-{3 \ {\sqrt{2}}}\right)}\ {\sqrt{-{\cos \left({3 \ x}\right)}+ 1}}}+{4 \ {\sin \left({3 \ x}\right)}}}\over{{\left({\cos \left({3 \ x}\right)}- 1 \right)}\ {\sqrt{-{\cos \left({3 \ x}\right)}+ 1}}}}\right)}}$

(57)

Type: Union(Expression(Integer),...)

fricas

integrate((1-sin(2/3*x))^(5/2), x)

$\label{eq58}{\left({\left({ \begin{array}{@{}l} \displaystyle {{\left({ \begin{array}{@{}l} \displaystyle {3 \ {{\cos \left({{2 \ x}\over 3}\right)}^{2}}}+ \ \ \displaystyle {{14}\ {\cos{\left({{2 \ x}\over 3}\right)}}}- \ \ \displaystyle {32}$

(58)

Type: Union(Expression(Integer),...)

fricas

integrate(cos(x)*(-cos(x)^2+2*(1+2*sin(x))^(1/4))/(1+2*sin(x))^(3/2), x)

$\label{eq59}{\left( \begin{array}{@{}l} \displaystyle -{{12}\ {{\root{4}\of{{2 \ {\sin \left({x}\right)}}+ 1}}^{3}}}+ \ \ \displaystyle {{\left(-{2 \ {\sin \left({x}\right)}}-{{\cos \left({x}\right)}^{2}}+ 2 \right)}\ {{\root{4}\of{{2 \ {\sin \left({x}\right)}}+ 1}}^{2}}}$

(59)

Type: Union(Expression(Integer),...)

fricas

-- Problems 62 - 66 (p. 268)

fricas

)set output tex off

fricas

)set output algebra on

integrate(sqrt(tan(x)), x)

   (62)
       -
              +-+
            4\|2
         *
            atan
                                               1
               -----------------------------------------------------------------
                +------------------------------------------+
                |              +------+
                |    +-+       |sin(x)
                |- 2\|2 cos(x) |------  + 2sin(x) + 2cos(x)         +------+
                |             \|cos(x)                          +-+ |sin(x)
                |------------------------------------------  + \|2  |------  - 1
               \|                  cos(x)                          \|cos(x)
     + 
       -
              +-+
            4\|2
         *
                                              1
          atan(---------------------------------------------------------------)
                +----------------------------------------+
                |            +------+
                |  +-+       |sin(x)
                |2\|2 cos(x) |------  + 2sin(x) + 2cos(x)         +------+
                |           \|cos(x)                          +-+ |sin(x)
                |----------------------------------------  + \|2  |------  + 1
               \|                 cos(x)                         \|cos(x)
     + 
                             +------+
                   +-+       |sin(x)
                 2\|2 cos(x) |------  + 2sin(x) + 2cos(x)
          +-+               \|cos(x)
       - \|2 log(----------------------------------------)
                                  cos(x)
     + 
                             +------+
                   +-+       |sin(x)
               - 2\|2 cos(x) |------  + 2sin(x) + 2cos(x)
        +-+                 \|cos(x)
       \|2 log(------------------------------------------)
                                 cos(x)
  /
     4

Type: Union(Expression(Integer),...)

fricas

integrate(1/tan(5*x)^(1/3), x)

   (63)
            3+-------+2              3+-------+2          3+-------+
       2log(\|tan(5x)   + 1) - log(- \|tan(5x)   + tan(5x)\|tan(5x)  + 1)
     + 
                  3+-------+2
         +-+     2\|tan(5x)   - 1
       2\|3 atan(----------------)
                        +-+
                       \|3
  /
     20

Type: Union(Expression(Integer),...)

fricas

integrate(1/(4+3*tan(2*x))^(3/2), x)

   (64)
       -
                   +-------------------+
               +-+ |3sin(2x) + 4cos(2x)
            36\|5  |-------------------
                  \|      cos(2x)
         *
            atan
                     +-+
                 125\|5
              /
                      +--+
                     \|10
                  *
                     ROOT
                                                   +-------------------+
                                   +-+ +--+        |3sin(2x) + 4cos(2x)
                            - 9375\|5 \|10 cos(2x) |-------------------
                                                  \|      cos(2x)
                          + 
                            46875sin(2x) + 140625cos(2x)
                       /
                          cos(2x)
                 + 
                            +-------------------+
                       +--+ |3sin(2x) + 4cos(2x)        +-+
                   125\|10  |-------------------  - 375\|5
                           \|      cos(2x)
     + 
       -
                   +-------------------+
               +-+ |3sin(2x) + 4cos(2x)
            36\|5  |-------------------
                  \|      cos(2x)
         *
            atan
                     +-+
                 125\|5
              /
                      +--+
                     \|10
                  *
                     ROOT
                                                 +-------------------+
                                 +-+ +--+        |3sin(2x) + 4cos(2x)
                            9375\|5 \|10 cos(2x) |-------------------
                                                \|      cos(2x)
                          + 
                            46875sin(2x) + 140625cos(2x)
                       /
                          cos(2x)
                 + 
                            +-------------------+
                       +--+ |3sin(2x) + 4cos(2x)        +-+
                   125\|10  |-------------------  + 375\|5
                           \|      cos(2x)
     + 
                +-------------------+
            +-+ |3sin(2x) + 4cos(2x)
         13\|5  |-------------------
               \|      cos(2x)
      *
         log
                                     +-------------------+
                     +-+ +--+        |3sin(2x) + 4cos(2x)
                9375\|5 \|10 cos(2x) |-------------------  + 46875sin(2x)
                                    \|      cos(2x)
              + 
                140625cos(2x)
           /
              cos(2x)
     + 
       -
                   +-------------------+
               +-+ |3sin(2x) + 4cos(2x)
            13\|5  |-------------------
                  \|      cos(2x)
         *
            log
                                          +-------------------+
                          +-+ +--+        |3sin(2x) + 4cos(2x)
                   - 9375\|5 \|10 cos(2x) |-------------------  + 46875sin(2x)
                                         \|      cos(2x)
                 + 
                   140625cos(2x)
              /
                 cos(2x)
     + 
            +--+
       - 60\|10
  /
              +-------------------+
         +--+ |3sin(2x) + 4cos(2x)
     500\|10  |-------------------
             \|      cos(2x)

Type: Union(Expression(Integer),...)

fricas

integrate((-sqrt(4-3*tan(x))+3*tan(x))/(cos(x)^2*(4-3*tan(x))^(3/2)), x)

   (65)
                             +-------------------+
                             |- 3sin(x) + 4cos(x)
       (12sin(x) - 32cos(x)) |-------------------
                            \|       cos(x)
     + 
                                      2
       (- 3sin(x) + 4cos(x))log(cos(x) )
     + 
                                                        2
                              - 24cos(x)sin(x) + 7cos(x)  + 9
       (3sin(x) - 4cos(x))log(-------------------------------)
                                             4
  /
     18sin(x) - 24cos(x)

Type: Union(Expression(Integer),...)

fricas

integrate(tan(x)/(-1+sqrt(tan(x)))^2, x)

   (66)
                 +------+
             (- \|tan(x)  + 1)
          *
              +------------------------------------------------------+
              |        2                               2
             \|- 3%%BS1  + (- 2%%BS0 - 2)%%BS1 - 3%%BS0  - 2%%BS0 - 1
         + 
               +------+                           +------+
           (- \|tan(x)  + 1)%%BS1 + (- %%BS0 - 1)\|tan(x)  + %%BS0 + 1
      *
         log
                ((32%%BS0 + 4)%%BS1 + 4%%BS0 + 2)
             *
                 +------------------------------------------------------+
                 |        2                               2
                \|- 3%%BS1  + (- 2%%BS0 - 2)%%BS1 - 3%%BS0  - 2%%BS0 - 1
            + 
                                  2             2
              (- 32%%BS0 - 4)%%BS1  + (- 32%%BS0  - 32%%BS0 - 2)%%BS1
            + 
                +------+         2
              6\|tan(x)  - 4%%BS0  - 2%%BS0 - 3
     + 
               +------+
             (\|tan(x)  - 1)
          *
              +------------------------------------------------------+
              |        2                               2
             \|- 3%%BS1  + (- 2%%BS0 - 2)%%BS1 - 3%%BS0  - 2%%BS0 - 1
         + 
               +------+                           +------+
           (- \|tan(x)  + 1)%%BS1 + (- %%BS0 - 1)\|tan(x)  + %%BS0 + 1
      *
         log
                ((- 32%%BS0 - 4)%%BS1 - 4%%BS0 - 2)
             *
                 +------------------------------------------------------+
                 |        2                               2
                \|- 3%%BS1  + (- 2%%BS0 - 2)%%BS1 - 3%%BS0  - 2%%BS0 - 1
            + 
                                  2             2
              (- 32%%BS0 - 4)%%BS1  + (- 32%%BS0  - 32%%BS0 - 2)%%BS1
            + 
                +------+         2
              6\|tan(x)  - 4%%BS0  - 2%%BS0 - 3
     + 
            +------+
         (2\|tan(x)  - 2)%%BS1
      *
         log
                                2           2                         +------+
              (64%%BS0 + 8)%%BS1  + (64%%BS0  + 64%%BS0 + 4)%%BS1 + 6\|tan(x)
            + 
                     3          2
              64%%BS0  + 64%%BS0  + 32%%BS0 - 1
     + 
                 +------+
         (2%%BS0\|tan(x)  - 2%%BS0)
      *
               +------+          3          2
         log(6\|tan(x)  - 64%%BS0  - 56%%BS0  - 28%%BS0 + 7)
     + 
          +------+          +------+
       (2\|tan(x)  - 2)log(\|tan(x)  - 1) - 2
  /
       +------+
     2\|tan(x)  - 2

Type: Union(Expression(Integer),...)

fricas

)set output tex on

fricas

)set output algebra off

-- Problems 67 - 75 (p. 272-273)
integrate(sin(x)/sqrt(sin(2*x)), x)

$\label{eq60}{\left( \begin{array}{@{}l} \displaystyle - \ \ \displaystyle {\log{{\left({{{\left({2 \ {\sin \left({x}\right)}}+{2 \ {\cos \left({x}\right)}}\right)}\ {\sqrt{2 \ {\cos \left({x}\right)}\ {\sin \left({x}\right)}}}}+{4 \ {\cos \left({x}\right)}\ {\sin \left({x}\right)}}+ 1}\right)}}}+ \ \ \displaystyle {\log{\left({{{\left(-{2 \ {\sin \left({x}\right)}}-{2 \ {\cos \left({x}\right)}}\right)}\ {\sqrt{2 \ {\cos \left({x}\right)}\ {\sin \left({x}\right)}}}}+{4 \ {\cos \left({x}\right)}\ {\sin \left({x}\right)}}+ 1}\right)}}- \ \ \displaystyle {2 \ {\arctan \left({{{2 \ {\sqrt{2 \ {\cos \left({x}\right)}\ {\sin \left({x}\right)}}}}+{\sin \left({x}\right)}+{\cos \left({x}\right)}}\over{{\sin \left({x}\right)}-{\cos \left({x}\right)}}}\right)}}- \ \ \displaystyle {2 \ {\arctan \left({{{2 \ {\sqrt{2 \ {\cos \left({x}\right)}\ {\sin \left({x}\right)}}}}-{\sin \left({x}\right)}-{\cos \left({x}\right)}}\over{{\sin \left({x}\right)}-{\cos \left({x}\right)}}}\right)}}$

(60)

Type: Union(Expression(Integer),...)

fricas

integrate(cos(x)/sqrt(sin(2*x)), x)

$\label{eq61}{\left( \begin{array}{@{}l} \displaystyle {\log{\left({{{\left({2 \ {\sin \left({x}\right)}}+{2 \ {\cos \left({x}\right)}}\right)}\ {\sqrt{2 \ {\cos \left({x}\right)}\ {\sin \left({x}\right)}}}}+{4 \ {\cos \left({x}\right)}\ {\sin \left({x}\right)}}+ 1}\right)}}- \ \ \displaystyle {\log{\left({{{\left(-{2 \ {\sin \left({x}\right)}}-{2 \ {\cos \left({x}\right)}}\right)}\ {\sqrt{2 \ {\cos \left({x}\right)}\ {\sin \left({x}\right)}}}}+{4 \ {\cos \left({x}\right)}\ {\sin \left({x}\right)}}+ 1}\right)}}- \ \ \displaystyle {2 \ {\arctan \left({{{2 \ {\sqrt{2 \ {\cos \left({x}\right)}\ {\sin \left({x}\right)}}}}+{\sin \left({x}\right)}+{\cos \left({x}\right)}}\over{{\sin \left({x}\right)}-{\cos \left({x}\right)}}}\right)}}- \ \ \displaystyle {2 \ {\arctan \left({{{2 \ {\sqrt{2 \ {\cos \left({x}\right)}\ {\sin \left({x}\right)}}}}-{\sin \left({x}\right)}-{\cos \left({x}\right)}}\over{{\sin \left({x}\right)}-{\cos \left({x}\right)}}}\right)}}$

(61)

Type: Union(Expression(Integer),...)

fricas

integrate(sin(x)*sqrt(sin(2*x)), x)

$\label{eq62}{\left( \begin{array}{@{}l} \displaystyle {\log{\left({{{\left({2 \ {\sin \left({x}\right)}}+{2 \ {\cos \left({x}\right)}}\right)}\ {\sqrt{2 \ {\cos \left({x}\right)}\ {\sin \left({x}\right)}}}}+{4 \ {\cos \left({x}\right)}\ {\sin \left({x}\right)}}+ 1}\right)}}- \ \ \displaystyle {\log{\left({{{\left(-{2 \ {\sin \left({x}\right)}}-{2 \ {\cos \left({x}\right)}}\right)}\ {\sqrt{2 \ {\cos \left({x}\right)}\ {\sin \left({x}\right)}}}}+{4 \ {\cos \left({x}\right)}\ {\sin \left({x}\right)}}+ 1}\right)}}- \ \ \displaystyle {2 \ {\arctan \left({{{2 \ {\sqrt{2 \ {\cos \left({x}\right)}\ {\sin \left({x}\right)}}}}+{\sin \left({x}\right)}+{\cos \left({x}\right)}}\over{{\sin \left({x}\right)}-{\cos \left({x}\right)}}}\right)}}- \ \ \displaystyle {2 \ {\arctan \left({{{2 \ {\sqrt{2 \ {\cos \left({x}\right)}\ {\sin \left({x}\right)}}}}-{\sin \left({x}\right)}-{\cos \left({x}\right)}}\over{{\sin \left({x}\right)}-{\cos \left({x}\right)}}}\right)}}- \ \ \displaystyle {8 \ {\cos \left({x}\right)}\ {\sqrt{2 \ {\cos \left({x}\right)}\ {\sin \left({x}\right)}}}}$

(62)

Type: Union(Expression(Integer),...)

fricas

)set output tex off

fricas

)set output algebra on

integrate((cos(x)-sin(x))*sqrt(sin(2*x)), x)

   (70)
       -
          log
                         2                      3             +-------------+
               ((16cos(x)  + 4)sin(x) - 16cos(x)  + 20cos(x))\|2cos(x)sin(x)
             + 
                                        4           2
               16cos(x)sin(x) - 32cos(x)  + 32cos(x)  + 1
     + 
       log
                        2                      3             +-------------+
            ((- 16cos(x)  - 4)sin(x) + 16cos(x)  - 20cos(x))\|2cos(x)sin(x)
          + 
                                     4           2
            16cos(x)sin(x) - 32cos(x)  + 32cos(x)  + 1
     + 
                           +-------------+
       (8sin(x) + 8cos(x))\|2cos(x)sin(x)
  /
     16

Type: Union(Expression(Integer),...)

fricas

integrate(sin(x)^7/sin(2*x)^(7/2), x)

   (71)
              3                        +-------------+
       5cos(x) log((2sin(x) + 2cos(x))\|2cos(x)sin(x)  + 4cos(x)sin(x) + 1)
     + 
                3                          +-------------+
       - 5cos(x) log((- 2sin(x) - 2cos(x))\|2cos(x)sin(x)  + 4cos(x)sin(x) + 1)
     + 
                         +-------------+
                 3     2\|2cos(x)sin(x)  + sin(x) + cos(x)
       - 10cos(x) atan(-----------------------------------)
                                 sin(x) - cos(x)
     + 
                         +-------------+
                 3     2\|2cos(x)sin(x)  - sin(x) - cos(x)
       - 10cos(x) atan(-----------------------------------)
                                 sin(x) - cos(x)
     + 
                  2      +-------------+
       (- 48cos(x)  + 8)\|2cos(x)sin(x)
  /
              3
     320cos(x)

Type: Union(Expression(Integer),...)

fricas

)set output algebra off

fricas

)set output tex on

integrate(cos(x)^7/sin(2*x)^(7/2), x)

$\label{eq63}{\left( \begin{array}{@{}l} \displaystyle { \begin{array}{@{}l} \displaystyle {\left(-{5 \ {{\cos \left({x}\right)}^{2}}}+ 5 \right)}\ {\sin \left({x}\right)}\ \cdot \ \ \displaystyle {\log{\left({{{\left({2 \ {\sin \left({x}\right)}}+{2 \ {\cos \left({x}\right)}}\right)}\ {\sqrt{2 \ {\cos \left({x}\right)}\ {\sin \left({x}\right)}}}}+{4 \ {\cos \left({x}\right)}\ {\sin \left({x}\right)}}+ 1}\right)}}$

(63)

Type: Union(Expression(Integer),...)

fricas

integrate(sin(2*x)^(3/2)/sin(x)^5, x)

$\label{eq64}{4 \ {{\cos \left({x}\right)}^{2}}\ {\sqrt{2 \ {\cos \left({x}\right)}\ {\sin \left({x}\right)}}}}\over{{\left({5 \ {{\cos \left({x}\right)}^{2}}}- 5 \right)}\ {\sin \left({x}\right)}}$

(64)

Type: Union(Expression(Integer),...)

fricas

integrate(1/(cos(x)^3*sqrt(sin(2*x))), x)

$\label{eq65}{{\left({4 \ {{\cos \left({x}\right)}^{2}}}+ 1 \right)}\ {\sqrt{2 \ {\cos \left({x}\right)}\ {\sin \left({x}\right)}}}}\over{5 \ {{\cos \left({x}\right)}^{3}}}$

(65)

Type: Union(Expression(Integer),...)

fricas

integrate(1/(sin(x)*sin(2*x)^(3/2)), x)

$\label{eq66}{{\left({4 \ {{\cos \left({x}\right)}^{2}}}- 3 \right)}\ {\sqrt{2 \ {\cos \left({x}\right)}\ {\sin \left({x}\right)}}}}\over{{6 \ {{\cos \left({x}\right)}^{3}}}-{6 \ {\cos \left({x}\right)}}}$

(66)

Type: Union(Expression(Integer),...)

fricas

-- Fails
integrate(cos(x)^2*(cos(2*x)-3*tan(x))/((sin(x)^2-sin(2*x))*sin(2*x)^(5/2)), x)

   >> Error detected within library code:
   Imaginary part is nonzero. Cannot retract.

   Continuing to read the file...