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FriCASTimofeev1

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Editor: test1 Time: 2013/09/30 21:02:22 GMT+0
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Examples from Timofeev (1948) Ch. 1, divided into sections
kindly provided by clicliclic vel Martin on sci.math.symbolic

\begin{axiom}

INT ==> integrate
pi ==> %pi
SIN ==> sin
COS ==> cos
TAN ==> tan
COT ==> cot
SEC ==> sec
CSC ==> csc

LN ==> log
SQRT ==> sqrt

ATAN ==> atan
ACOS ==> acos
ASIN ==> asin

-- Timofeev (1948) Ch. 1, examples 1 - 21
INT(1/(a^2-b^2*x^2),x)

INT(1/(a^2+b^2*x^2),x)

INT(SEC(2*a*x),x)

INT(1/(4*SIN(x/3)),x)

INT(1/COS(3/4*pi-2*x),x)

INT(SEC(x)*TAN(x),x)

INT(CSC(x)*COT(x),x)

INT(TAN(x)/SIN(2*x),x)

INT(1/(1+COS(x)),x)

INT(1/(1-COS(x)),x)
INT(SIN(x)/(a-b*COS(x)),x)

INT(COS(x)/(a^2+b^2*SIN(x)^2),x)

INT(COS(x)/(a^2-b^2*SIN(x)^2),x)

INT(SIN(2*x)/(b^2*SIN(x)^2+a^2),x)
INT(SIN(2*x)/(b^2*SIN(x)^2-a^2),x)

INT(SIN(2*x)/(b^2*COS(x)^2+a^2),x)
INT(SIN(2*x)/(b^2*COS(x)^2-a^2),x)

INT(1/(4-COS(x)^2),x)

INT(%e^x/(%e^(2*x)-1),x)

INT(1/(x*LN(x)),x)

INT(1/(x*(1+LN(x)^2)),x)

INT(1/(x*(1-LN(x))),x)

INT(1/(x*(1+LN(x/a))),x)

-- Timofeev (1948) Ch. 1, examples 22 - 40 (p. 27-28)
INT(((1-SQRT(x)+x)/x)^2,x)

INT((2-x^(2/3))*(x+SQRT(x))/x^(3/2),x)

INT((2*x-1)/(2*x+3),x)

INT((2*x-5)/(3*x^2-2),x)

INT((2*x-5)/(3*x^2+2),x)

INT(SIN(x)*SIN(x/4),x)

INT(COS(3*x)*COS(4*x),x)

INT(TAN(x)*TAN(x-a),x)

INT(SIN(x)^2,x)
INT(COS(x)^2,x)

INT(SIN(x)*COS(x)^3,x)

INT(COS(x)^3/SIN(x)^4,x)

INT(1/(SIN(x)^2*COS(x)^2),x)

INT(COT(3/4*x)^2,x)

INT((1+TAN(2*x))^2,x)

INT((TAN(x)-COT(x))^2,x)

INT((TAN(x)-SEC(x))^2,x)

INT(SIN(x)/(1+SIN(x)),x)

INT(COS(x)/(1-COS(x)),x)

INT((%e^(x/2)-1)^3*%e^(-x/2),x)

-- Timofeev (1948) Ch. 1, examples 41 - 65 (p. 35-37)

INT(1/(x^2-6*x+5),x)

INT(x^2/(13-6*x^3+x^6),x)

INT((x+2)/(x^2-4*x-1),x)

INT(1/(1+(x+1)^(1/3)),x)

INT(1/((a*x+b)*SQRT(x)),x)

INT(x^3*SQRT(1+x^2),x)

INT(x/SQRT(a^4-x^4),x)

INT(1/(x*SQRT(x^2-a^2)),x)
INT(1/(x*SQRT(a^2-x^2)),x)
INT(1/(x*SQRT(x^2+a^2)),x)

INT(1/SQRT(2+x-x^2),x)

INT(1/SQRT(3*x^2-4*x+5),x)

INT(1/SQRT(x-x^2),x)

INT((2*x+1)/SQRT(2+x-x^2),x)

INT(1/(x*SQRT(2+x-x^2)),x)

INT(1/((x-2)*SQRT(2+x-x^2)),x)

INT((2+3*SIN(x))/(SIN(x)*(1-COS(x))),x)

INT(1/(2+3*COS(x)^2),x)

INT((1-TAN(x))/SIN(2*x),x)

INT((1+TAN(x)^2)/(1-TAN(x)^2),x)

INT((a^2-4*COS(x)^2)^(3/4)*SIN(2*x),x)

INT(SIN(2*x)/(a^2-4*SIN(x)^2)^(1/3),x)

INT(1/SQRT(a^(2*x)-1),x)

INT(%e^(x/2)/SQRT(%e^x-1),x)

INT(ATAN(x)^n/(1+x^2),x)

INT(ASIN(x/a)^(3/2)/SQRT(a^2-x^2),x)

INT(1/(ACOS(x)^3*SQRT(1-x^2)),x)

-- Timofeev (1948) Ch. 1, examples 66 - 81 (p. 41-42)

INT(LN(x)^2*x,x)

INT(LN(x)/x^5,x)

INT(x^2*LN((x-1)/x),x)

INT(COS(x)^5,x)

INT(SIN(x)^2*COS(x)^4,x)

INT(1/SIN(x)^5,x)

INT(SIN(x)/%e^x,x)

INT(%e^(2*x)*SIN(3*x),x)

INT(a^x*COS(x),x)

INT(COS(LN(x)),x)

INT(SEC(x)^2*LN(COS(x)),x)

INT(x*TAN(x)^2,x)

INT(ASIN(x)/x^2,x)

INT(ASIN(x)^2,x)

INT(x^2*ATAN(x)/(1+x^2),x)

INT(ACOS(SQRT(x/(x+1))),x)

\end{axiom}

See also SandboxFriCASTimofeev37, SandboxFriCASTimofeev5, SandboxFriCASTimofeev9

Examples from Timofeev (1948) Ch. 1, divided into sections kindly provided by clicliclic vel Martin on sci.math.symbolic

fricas

INT ==> integrate

Type: Void

fricas

pi ==> %pi

Type: Void

fricas

SIN ==> sin

Type: Void

fricas

COS ==> cos

Type: Void

fricas

TAN ==> tan

Type: Void

fricas

COT ==> cot

Type: Void

fricas

SEC ==> sec

Type: Void

fricas

CSC ==> csc

Type: Void

fricas

LN ==> log

Type: Void

fricas

SQRT ==> sqrt

Type: Void

fricas

ATAN ==> atan

Type: Void

fricas

ACOS ==> acos

Type: Void

fricas

ASIN ==> asin

Type: Void

fricas

-- Timofeev (1948) Ch. 1, examples 1 - 21
INT(1/(a^2-b^2*x^2),x)

$\label{eq1}{{\log \left({{b \ x}+ a}\right)}-{\log \left({{b \ x}- a}\right)}}\over{2 \ a \ b}$

(1)

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

fricas

INT(1/(a^2+b^2*x^2),x)

$\label{eq2}{\arctan \left({{b \ x}\over a}\right)}\over{a \ b}$

(2)

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

fricas

INT(SEC(2*a*x),x)

$\label{eq3}{\left( \begin{array}{@{}l} \displaystyle {\log \left({{{\sin \left({2 \ a \ x}\right)}+{\cos \left({2 \ a \ x}\right)}+ 1}\over{{\cos \left({2 \ a \ x}\right)}+ 1}}\right)}- \ \ \displaystyle {\log \left({{{\sin \left({2 \ a \ x}\right)}-{\cos \left({2 \ a \ x}\right)}- 1}\over{{\cos \left({2 \ a \ x}\right)}+ 1}}\right)}$

(3)

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

fricas

INT(1/(4*SIN(x/3)),x)

$\label{eq4}{3 \ {\log \left({{\sin \left({x \over 3}\right)}\over{{\cos \left({x \over 3}\right)}+ 1}}\right)}}\over 4$

(4)

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

fricas

INT(1/COS(3/4*pi-2*x),x)

$\label{eq5}{\left( \begin{array}{@{}l} \displaystyle {\log \left({{{\sin \left({{{8 \ x}-{3 \ \pi}}\over 4}\right)}+{\cos \left({{{8 \ x}-{3 \ \pi}}\over 4}\right)}+ 1}\over{{\cos \left({{{8 \ x}-{3 \ \pi}}\over 4}\right)}+ 1}}\right)}- \ \ \displaystyle {\log \left({{{\sin \left({{{8 \ x}-{3 \ \pi}}\over 4}\right)}-{\cos \left({{{8 \ x}-{3 \ \pi}}\over 4}\right)}- 1}\over{{\cos \left({{{8 \ x}-{3 \ \pi}}\over 4}\right)}+ 1}}\right)}$

(5)

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

fricas

INT(SEC(x)*TAN(x),x)

$\label{eq6}{{\cos \left({x}\right)}+ 1}\over{\cos \left({x}\right)}$

(6)

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

fricas

INT(CSC(x)*COT(x),x)

$\label{eq7}-{1 \over{\sin \left({x}\right)}}$

(7)

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

fricas

INT(TAN(x)/SIN(2*x),x)

$\label{eq8}{\tan \left({x}\right)}\over 2$

(8)

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

fricas

INT(1/(1+COS(x)),x)

$\label{eq9}{\sin \left({x}\right)}\over{{\cos \left({x}\right)}+ 1}$

(9)

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

fricas

INT(1/(1-COS(x)),x)

$\label{eq10}{-{\cos \left({x}\right)}- 1}\over{\sin \left({x}\right)}$

(10)

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

fricas

INT(SIN(x)/(a-b*COS(x)),x)

$\label{eq11}{-{\log \left({2 \over{{\cos \left({x}\right)}+ 1}}\right)}+{\log \left({{-{2 \ b \ {\cos \left({x}\right)}}+{2 \ a}}\over{{\cos \left({x}\right)}+ 1}}\right)}}\over b$

(11)

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

fricas

INT(COS(x)/(a^2+b^2*SIN(x)^2),x)

$\label{eq12}{\left( \begin{array}{@{}l} \displaystyle {\arctan \left({{{\left({2 \ {{b}^{2}}\ {\cos \left({x}\right)}}+{2 \ {{b}^{2}}}+{{a}^{2}}\right)}\ {\sin \left({x}\right)}}\over{{a \ b \ {{\cos \left({x}\right)}^{2}}}+{2 \ a \ b \ {\cos \left({x}\right)}}+{a \ b}}}\right)}- \ \ \displaystyle {\arctan \left({{a \ {\sin \left({x}\right)}}\over{{2 \ b \ {\cos \left({x}\right)}}+{2 \ b}}}\right)}$

(12)

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

fricas

INT(COS(x)/(a^2-b^2*SIN(x)^2),x)

$\label{eq13}{{\log \left({{{2 \ b \ {\sin \left({x}\right)}}+{2 \ a}}\over{{\cos \left({x}\right)}+ 1}}\right)}-{\log \left({{-{2 \ b \ {\sin \left({x}\right)}}+{2 \ a}}\over{{\cos \left({x}\right)}+ 1}}\right)}}\over{2 \ a \ b}$

(13)

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

fricas

INT(SIN(2*x)/(b^2*SIN(x)^2+a^2),x)

$\label{eq14}{-{2 \ {\log \left({2 \over{{\cos \left({x}\right)}+ 1}}\right)}}+{\log \left({{-{4 \ {{b}^{2}}\ {{\cos \left({x}\right)}^{2}}}+{4 \ {{b}^{2}}}+{4 \ {{a}^{2}}}}\over{{{\cos \left({x}\right)}^{2}}+{2 \ {\cos \left({x}\right)}}+ 1}}\right)}}\over{{b}^{2}}$

(14)

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

fricas

INT(SIN(2*x)/(b^2*SIN(x)^2-a^2),x)

$\label{eq15}{{\log \left({{{4 \ {{b}^{2}}\ {{\cos \left({x}\right)}^{2}}}-{4 \ {{b}^{2}}}+{4 \ {{a}^{2}}}}\over{{{\cos \left({x}\right)}^{2}}+{2 \ {\cos \left({x}\right)}}+ 1}}\right)}-{2 \ {\log \left({2 \over{{\cos \left({x}\right)}+ 1}}\right)}}}\over{{b}^{2}}$

(15)

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

fricas

INT(SIN(2*x)/(b^2*COS(x)^2+a^2),x)

$\label{eq16}{-{\log \left({{{4 \ {{b}^{2}}\ {{\cos \left({x}\right)}^{2}}}+{4 \ {{a}^{2}}}}\over{{{\cos \left({x}\right)}^{2}}+{2 \ {\cos \left({x}\right)}}+ 1}}\right)}+{2 \ {\log \left({2 \over{{\cos \left({x}\right)}+ 1}}\right)}}}\over{{b}^{2}}$

(16)

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

fricas

INT(SIN(2*x)/(b^2*COS(x)^2-a^2),x)

$\label{eq17}{-{\log \left({{{4 \ {{b}^{2}}\ {{\cos \left({x}\right)}^{2}}}-{4 \ {{a}^{2}}}}\over{{{\cos \left({x}\right)}^{2}}+{2 \ {\cos \left({x}\right)}}+ 1}}\right)}+{2 \ {\log \left({2 \over{{\cos \left({x}\right)}+ 1}}\right)}}}\over{{b}^{2}}$

(17)

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

fricas

INT(1/(4-COS(x)^2),x)

$\label{eq18}{\left( \begin{array}{@{}l} \displaystyle {\arctan \left({{{\sqrt{3}}\ {\sin \left({x}\right)}}\over{{4 \ {\cos \left({x}\right)}}+ 4}}\right)}+ \ \ \displaystyle {\arctan \left({{{\left({5 \ {\cos \left({x}\right)}}+ 8 \right)}\ {\sin \left({x}\right)}}\over{{2 \ {\sqrt{3}}\ {{\cos \left({x}\right)}^{2}}}+{4 \ {\sqrt{3}}\ {\cos \left({x}\right)}}+{2 \ {\sqrt{3}}}}}\right)}$

(18)

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

fricas

INT(%e^x/(%e^(2*x)-1),x)

$\label{eq19}{-{\log \left({{{e}^{x}}+ 1}\right)}+{\log \left({{{e}^{x}}- 1}\right)}}\over 2$

(19)

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

fricas

INT(1/(x*LN(x)),x)

$\label{eq20}\log \left({\log \left({x}\right)}\right)$

(20)

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

fricas

INT(1/(x*(1+LN(x)^2)),x)

$\label{eq21}\arctan \left({\log \left({x}\right)}\right)$

(21)

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

fricas

INT(1/(x*(1-LN(x))),x)

$\label{eq22}-{\log \left({{\log \left({x}\right)}- 1}\right)}$

(22)

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

fricas

INT(1/(x*(1+LN(x/a))),x)

$\label{eq23}\log \left({{\log \left({x \over a}\right)}+ 1}\right)$

(23)

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

fricas

-- Timofeev (1948) Ch. 1, examples 22 - 40 (p. 27-28)
INT(((1-SQRT(x)+x)/x)^2,x)

$\label{eq24}{{6 \ x \ {\log \left({\sqrt{x}}\right)}}+{{\left(-{4 \ x}+ 4 \right)}\ {\sqrt{x}}}+{{x}^{2}}- 1}\over x$

(24)

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

fricas

INT((2-x^(2/3))*(x+SQRT(x))/x^(3/2),x)

$\label{eq25}{{{168}\ {\log \left({\root{6}\of{x}}\right)}}-{{21}\ {{\root{6}\of{x}}^{4}}}+{{56}\ {{\root{6}\of{x}}^{3}}}-{{12}\ x \ {\root{6}\of{x}}}}\over{1 4}$

(25)

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

fricas

INT((2*x-1)/(2*x+3),x)

$\label{eq26}-{2 \ {\log \left({{2 \ x}+ 3}\right)}}+ x$

(26)

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

fricas

INT((2*x-5)/(3*x^2-2),x)

$\label{eq27}{{{15}\ {\log \left({{{{\left({3 \ {{x}^{2}}}+ 2 \right)}\ {\sqrt{6}}}+{{12}\ x}}\over{{3 \ {{x}^{2}}}- 2}}\right)}}+{2 \ {\sqrt{6}}\ {\log \left({{3 \ {{x}^{2}}}- 2}\right)}}}\over{6 \ {\sqrt{6}}}$

(27)

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

fricas

INT((2*x-5)/(3*x^2+2),x)

$\label{eq28}{{{\sqrt{6}}\ {\log \left({{3 \ {{x}^{2}}}+ 2}\right)}}-{{15}\ {\arctan \left({{x \ {\sqrt{6}}}\over 2}\right)}}}\over{3 \ {\sqrt{6}}}$

(28)

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

fricas

INT(SIN(x)*SIN(x/4),x)

$\label{eq29}{{\left(-{{96}\ {{\cos \left({x \over 4}\right)}^{4}}}+{{112}\ {{\cos \left({x \over 4}\right)}^{2}}}-{16}\right)}\ {\sin \left({x \over 4}\right)}}\over{15}$

(29)

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

fricas

INT(COS(3*x)*COS(4*x),x)

$\label{eq30}{{\left({{32}\ {{\cos \left({x}\right)}^{6}}}-{{40}\ {{\cos \left({x}\right)}^{4}}}+{{12}\ {{\cos \left({x}\right)}^{2}}}+ 3 \right)}\ {\sin \left({x}\right)}}\over 7$

(30)

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

fricas

INT(TAN(x)*TAN(x-a),x)

$\label{eq31}{{\log \left({{{\tan \left({a}\right)}\ {\tan \left({x}\right)}}+ 1}\right)}-{x \ {\tan \left({a}\right)}}}\over{\tan \left({a}\right)}$

(31)

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

fricas

INT(SIN(x)^2,x)

$\label{eq32}{-{{\cos \left({x}\right)}\ {\sin \left({x}\right)}}+ x}\over 2$

(32)

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

fricas

INT(COS(x)^2,x)

$\label{eq33}{{{\cos \left({x}\right)}\ {\sin \left({x}\right)}}+ x}\over 2$

(33)

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

fricas

INT(SIN(x)*COS(x)^3,x)

$\label{eq34}-{{{\cos \left({x}\right)}^{4}}\over 4}$

(34)

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

fricas

INT(COS(x)^3/SIN(x)^4,x)

$\label{eq35}{{3 \ {{\cos \left({x}\right)}^{2}}}- 2}\over{{\left({3 \ {{\cos \left({x}\right)}^{2}}}- 3 \right)}\ {\sin \left({x}\right)}}$

(35)

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

fricas

INT(1/(SIN(x)^2*COS(x)^2),x)

$\label{eq36}{-{2 \ {{\cos \left({x}\right)}^{2}}}+ 1}\over{{\cos \left({x}\right)}\ {\sin \left({x}\right)}}$

(36)

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

fricas

INT(COT(3/4*x)^2,x)

$\label{eq37}{-{3 \ x \ {\sin \left({{3 \ x}\over 2}\right)}}-{4 \ {\cos \left({{3 \ x}\over 2}\right)}}- 4}\over{3 \ {\sin \left({{3 \ x}\over 2}\right)}}$

(37)

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

fricas

INT((1+TAN(2*x))^2,x)

$\label{eq38}{{\log \left({{{\tan \left({2 \ x}\right)}^{2}}+ 1}\right)}+{\tan \left({2 \ x}\right)}}\over 2$

(38)

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

fricas

INT((TAN(x)-COT(x))^2,x)

$\label{eq39}{{{\tan \left({x}\right)}^{2}}-{4 \ x \ {\tan \left({x}\right)}}- 1}\over{\tan \left({x}\right)}$

(39)

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

fricas

INT((TAN(x)-SEC(x))^2,x)

$\label{eq40}{-{x \ {\sin \left({x}\right)}}+{{\left(- x - 4 \right)}\ {\cos \left({x}\right)}}- x - 4}\over{{\sin \left({x}\right)}+{\cos \left({x}\right)}+ 1}$

(40)

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

fricas

INT(SIN(x)/(1+SIN(x)),x)

$\label{eq41}{{x \ {\sin \left({x}\right)}}+{{\left(x + 2 \right)}\ {\cos \left({x}\right)}}+ x + 2}\over{{\sin \left({x}\right)}+{\cos \left({x}\right)}+ 1}$

(41)

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

fricas

INT(COS(x)/(1-COS(x)),x)

$\label{eq42}{-{x \ {\sin \left({x}\right)}}-{\cos \left({x}\right)}- 1}\over{\sin \left({x}\right)}$

(42)

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

fricas

INT((%e^(x/2)-1)^3*%e^(-x/2),x)

$\label{eq43}{{{{e}^{x \over 2}}^{3}}-{6 \ {{{e}^{x \over 2}}^{2}}}+{3 \ x \ {{e}^{x \over 2}}}+ 2}\over{{e}^{x \over 2}}$

(43)

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

fricas

-- Timofeev (1948) Ch. 1, examples 41 - 65 (p. 35-37)

INT(1/(x^2-6*x+5),x)

$\label{eq44}{-{\log \left({x - 1}\right)}+{\log \left({x - 5}\right)}}\over 4$

(44)

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

fricas

INT(x^2/(13-6*x^3+x^6),x)

$\label{eq45}{\arctan \left({{{{x}^{3}}- 3}\over 2}\right)}\over 6$

(45)

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

fricas

INT((x+2)/(x^2-4*x-1),x)

$\label{eq46}{\left( \begin{array}{@{}l} \displaystyle {4 \ {\log \left({{{{\left({{x}^{2}}-{4 \ x}+ 9 \right)}\ {\sqrt{5}}}-{{10}\ x}+{20}}\over{{{x}^{2}}-{4 \ x}- 1}}\right)}}+ \ \ \displaystyle {{\sqrt{5}}\ {\log \left({{{x}^{2}}-{4 \ x}- 1}\right)}}$

(46)

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

fricas

INT(1/(1+(x+1)^(1/3)),x)

$\label{eq47}{{6 \ {\log \left({{\root{3}\of{x + 1}}+ 1}\right)}}+{3 \ {{\root{3}\of{x + 1}}^{2}}}-{6 \ {\root{3}\of{x + 1}}}}\over 2$

(47)

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

fricas

INT(1/((a*x+b)*SQRT(x)),x)

$\label{eq48}\begin{array}{@{}l} \displaystyle \left[{{\log \left({{{2 \ a \ b \ {\sqrt{x}}}+{{\left({a \ x}- b \right)}\ {\sqrt{-{a \ b}}}}}\over{{a \ x}+ b}}\right)}\over{\sqrt{-{a \ b}}}}, \:{{2 \ {\arctan \left({{{\sqrt{a \ b}}\ {\sqrt{x}}}\over b}\right)}}\over{\sqrt{a \ b}}}\right]$

(48)

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

fricas

INT(x^3*SQRT(1+x^2),x)

$\label{eq49}{\left( \begin{array}{@{}l} \displaystyle {{\left(-{{48}\ {{x}^{9}}}-{{76}\ {{x}^{7}}}-{3 \ {{x}^{5}}}+{{35}\ {{x}^{3}}}+{{10}\ x}\right)}\ {\sqrt{{{x}^{2}}+ 1}}}+ \ \ \displaystyle {{48}\ {{x}^{10}}}+{{100}\ {{x}^{8}}}+{{35}\ {{x}^{6}}}-{{40}\ {{x}^{4}}}-{{25}\ {{x}^{2}}}- 2$

(49)

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

fricas

INT(x/SQRT(a^4-x^4),x)

$\label{eq50}-{\arctan \left({{{\sqrt{-{{x}^{4}}+{{a}^{4}}}}-{{a}^{2}}}\over{{x}^{2}}}\right)}$

(50)

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

fricas

INT(1/(x*SQRT(x^2-a^2)),x)

$\label{eq51}{2 \ {\arctan \left({{{\sqrt{{{x}^{2}}-{{a}^{2}}}}- x}\over a}\right)}}\over a$

(51)

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

fricas

INT(1/(x*SQRT(a^2-x^2)),x)

$\label{eq52}{\log \left({{{\sqrt{-{{x}^{2}}+{{a}^{2}}}}- a}\over x}\right)}\over a$

(52)

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

fricas

INT(1/(x*SQRT(x^2+a^2)),x)

$\label{eq53}{-{\log \left({{\sqrt{{{x}^{2}}+{{a}^{2}}}}- x + a}\right)}+{\log \left({{\sqrt{{{x}^{2}}+{{a}^{2}}}}- x - a}\right)}}\over a$

(53)

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

fricas

INT(1/SQRT(2+x-x^2),x)

$\label{eq54}-{\arctan \left({{2 \ {\sqrt{-{{x}^{2}}+ x + 2}}}\over{{2 \ x}- 1}}\right)}$

(54)

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

fricas

INT(1/SQRT(3*x^2-4*x+5),x)

$\label{eq55}{\log{\left({{{\left(-{{18}\ x}+{12}\right)}\ {\sqrt{{3 \ {{x}^{2}}}-{4 \ x}+ 5}}}+{{\left(-{{18}\ {{x}^{2}}}+{{24}\ x}-{1 9}\right)}\ {\sqrt{3}}}}\right)}}\over{2 \ {\sqrt{3}}}$

(55)

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

fricas

INT(1/SQRT(x-x^2),x)

$\label{eq56}-{2 \ {\arctan \left({{\sqrt{-{{x}^{2}}+ x}}\over x}\right)}}$

(56)

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

fricas

INT((2*x+1)/SQRT(2+x-x^2),x)

$\label{eq57}-{2 \ {\arctan \left({{2 \ {\sqrt{-{{x}^{2}}+ x + 2}}}\over{{2 \ x}- 1}}\right)}}-{2 \ {\sqrt{-{{x}^{2}}+ x + 2}}}$

(57)

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

fricas

INT(1/(x*SQRT(2+x-x^2)),x)

$\label{eq58}{\left({\sqrt{2}}\ {\log{\left({{{{\left(-{4 \ x}-{16}\right)}\ {\sqrt{2}}\ {\sqrt{-{{x}^{2}}+ x + 2}}}-{7 \ {{x}^{2}}}+{{16}\ x}+{32}}\over{{x}^{2}}}\right)}}\right)}\over 4$

(58)

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

fricas

INT(1/((x-2)*SQRT(2+x-x^2)),x)

$\label{eq59}{2 \ {\sqrt{-{{x}^{2}}+ x + 2}}}\over{{3 \ x}- 6}$

(59)

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

fricas

INT((2+3*SIN(x))/(SIN(x)*(1-COS(x))),x)

$\label{eq60}{{{\left({2 \ {\cos \left({x}\right)}}- 2 \right)}\ {\log \left({{\sin \left({x}\right)}\over{{\cos \left({x}\right)}+ 1}}\right)}}+{6 \ {\sin \left({x}\right)}}+{\cos \left({x}\right)}+ 1}\over{{2 \ {\cos \left({x}\right)}}- 2}$

(60)

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

fricas

INT(1/(2+3*COS(x)^2),x)

$\label{eq61}{\left( \begin{array}{@{}l} \displaystyle {\arctan \left({{{\sqrt{10}}\ {\sin \left({x}\right)}}\over{{4 \ {\cos \left({x}\right)}}+ 4}}\right)}- \ \ \displaystyle {\arctan \left({{{\left({\cos \left({x}\right)}- 4 \right)}\ {\sin \left({x}\right)}}\over{{{\sqrt{10}}\ {{\cos \left({x}\right)}^{2}}}+{2 \ {\sqrt{10}}\ {\cos \left({x}\right)}}+{\sqrt{10}}}}\right)}$

(61)

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

fricas

INT((1-TAN(x))/SIN(2*x),x)

$\label{eq62}{{\log \left({\tan \left({x}\right)}\right)}-{\tan \left({x}\right)}}\over 2$

(62)

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

fricas

INT((1+TAN(x)^2)/(1-TAN(x)^2),x)

$\label{eq63}{{\log \left({{\tan \left({x}\right)}+ 1}\right)}-{\log \left({{\tan \left({x}\right)}- 1}\right)}}\over 2$

(63)

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

fricas

INT((a^2-4*COS(x)^2)^(3/4)*SIN(2*x),x)

$\label{eq64}{{\left(-{4 \ {{\cos \left({x}\right)}^{2}}}+{{a}^{2}}\right)}\ {{\root{4}\of{-{4 \ {{\cos \left({x}\right)}^{2}}}+{{a}^{2}}}}^{3}}}\over 7$

(64)

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

fricas

INT(SIN(2*x)/(a^2-4*SIN(x)^2)^(1/3),x)

$\label{eq65}-{{3 \ {{\root{3}\of{{4 \ {{\cos \left({x}\right)}^{2}}}+{{a}^{2}}- 4}}^{2}}}\over 8}$

(65)

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

fricas

INT(1/SQRT(a^(2*x)-1),x)

$\label{eq66}{\arctan \left({\sqrt{{{e}^{2 \ x \ {\log \left({a}\right)}}}- 1}}\right)}\over{\log \left({a}\right)}$

(66)

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

fricas

INT(%e^(x/2)/SQRT(%e^x-1),x)

$\label{eq67}-{2 \ {\log \left({{\sqrt{{{{e}^{x \over 2}}^{2}}- 1}}-{{e}^{x \over 2}}}\right)}}$

(67)

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

fricas

INT(ATAN(x)^n/(1+x^2),x)

$\label{eq68}{{\arctan \left({x}\right)}\ {{e}^{n \ {\log \left({\arctan \left({x}\right)}\right)}}}}\over{n + 1}$

(68)

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

fricas

INT(ASIN(x/a)^(3/2)/SQRT(a^2-x^2),x)

$\label{eq69}{2 \ {{\arctan \left({x \over{\sqrt{-{{x}^{2}}+{{a}^{2}}}}}\right)}^{2}}\ {\sqrt{2 \ {\arctan \left({x \over{\sqrt{-{{x}^{2}}+{{a}^{2}}}}}\right)}}}}\over{5 \ {\sqrt{2}}}$

(69)

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

fricas

INT(1/(ACOS(x)^3*SQRT(1-x^2)),x)

$\label{eq70}1 \over{2 \ {{\arccos \left({x}\right)}^{2}}}$

(70)

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

fricas

-- Timofeev (1948) Ch. 1, examples 66 - 81 (p. 41-42)

INT(LN(x)^2*x,x)

$\label{eq71}{{2 \ {{x}^{2}}\ {{\log \left({x}\right)}^{2}}}-{2 \ {{x}^{2}}\ {\log \left({x}\right)}}+{{x}^{2}}}\over 4$

(71)

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

fricas

INT(LN(x)/x^5,x)

$\label{eq72}{-{4 \ {\log \left({x}\right)}}- 1}\over{{16}\ {{x}^{4}}}$

(72)

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

fricas

INT(x^2*LN((x-1)/x),x)

$\label{eq73}{-{2 \ {\log \left({x - 1}\right)}}+{2 \ {{x}^{3}}\ {\log \left({{x - 1}\over x}\right)}}-{{x}^{2}}-{2 \ x}}\over 6$

(73)

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

fricas

INT(COS(x)^5,x)

$\label{eq74}{{\left({3 \ {{\cos \left({x}\right)}^{4}}}+{4 \ {{\cos \left({x}\right)}^{2}}}+ 8 \right)}\ {\sin \left({x}\right)}}\over{15}$

(74)

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

fricas

INT(SIN(x)^2*COS(x)^4,x)

$\label{eq75}{{{\left(-{8 \ {{\cos \left({x}\right)}^{5}}}+{2 \ {{\cos \left({x}\right)}^{3}}}+{3 \ {\cos \left({x}\right)}}\right)}\ {\sin \left({x}\right)}}+{3 \ x}}\over{48}$

(75)

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

fricas

INT(1/SIN(x)^5,x)

$\label{eq76}{\left( \begin{array}{@{}l} \displaystyle {{\left({3 \ {{\cos \left({x}\right)}^{4}}}-{6 \ {{\cos \left({x}\right)}^{2}}}+ 3 \right)}\ {\log \left({{\sin \left({x}\right)}\over{{\cos \left({x}\right)}+ 1}}\right)}}+ \ \ \displaystyle {3 \ {{\cos \left({x}\right)}^{3}}}-{5 \ {\cos \left({x}\right)}}$

(76)

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

fricas

INT(SIN(x)/%e^x,x)

$\label{eq77}{-{\sin \left({x}\right)}-{\cos \left({x}\right)}}\over{2 \ {{e}^{x}}}$

(77)

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

fricas

INT(%e^(2*x)*SIN(3*x),x)

$\label{eq78}{{2 \ {{e}^{2 \ x}}\ {\sin \left({3 \ x}\right)}}-{3 \ {\cos \left({3 \ x}\right)}\ {{e}^{2 \ x}}}}\over{13}$

(78)

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

fricas

INT(a^x*COS(x),x)

$\label{eq79}{{\left({\sin \left({x}\right)}+{{\cos \left({x}\right)}\ {\log \left({a}\right)}}\right)}\ {{e}^{x \ {\log \left({a}\right)}}}}\over{{{\log \left({a}\right)}^{2}}+ 1}$

(79)

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

fricas

INT(COS(LN(x)),x)

$\label{eq80}{{x \ {\sin \left({\log \left({x}\right)}\right)}}+{x \ {\cos \left({\log \left({x}\right)}\right)}}}\over 2$

(80)

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

fricas

INT(SEC(x)^2*LN(COS(x)),x)

$\label{eq81}{{{\sin \left({x}\right)}\ {\log \left({\cos \left({x}\right)}\right)}}+{\sin \left({x}\right)}-{x \ {\cos \left({x}\right)}}}\over{\cos \left({x}\right)}$

(81)

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

fricas

INT(x*TAN(x)^2,x)

$\label{eq82}{-{\log \left({{{\tan \left({x}\right)}^{2}}+ 1}\right)}+{2 \ x \ {\tan \left({x}\right)}}-{{x}^{2}}}\over 2$

(82)

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

fricas

INT(ASIN(x)/x^2,x)

$\label{eq83}{\left( \begin{array}{@{}l} \displaystyle -{x \ {\log \left({{\sqrt{-{{x}^{2}}+ 1}}+ 1}\right)}}+{x \ {\log \left({{\sqrt{-{{x}^{2}}+ 1}}- 1}\right)}}- \ \ \displaystyle {2 \ {\arcsin \left({x}\right)}}$

(83)

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

fricas

INT(ASIN(x)^2,x)

$\label{eq84}{2 \ {\arcsin \left({x}\right)}\ {\sqrt{-{{x}^{2}}+ 1}}}+{x \ {{\arcsin \left({x}\right)}^{2}}}-{2 \ x}$

(84)

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

fricas

INT(x^2*ATAN(x)/(1+x^2),x)

$\label{eq85}{-{\log \left({{{x}^{2}}+ 1}\right)}-{{\arctan \left({x}\right)}^{2}}+{2 \ x \ {\arctan \left({x}\right)}}}\over 2$

(85)

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

fricas

INT(ACOS(SQRT(x/(x+1))),x)

$\label{eq86}{{\left(x + 1 \right)}\ {\arctan \left({1 \over{{\sqrt{x \over{x + 1}}}\ {\sqrt{x + 1}}}}\right)}}+{{\sqrt{x \over{x + 1}}}\ {\sqrt{x + 1}}}$

(86)

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

See also SandboxFriCASTimofeev37?, SandboxFriCASTimofeev5?, SandboxFriCASTimofeev9?