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Editor: test1 Time: 2013/07/02 00:14:41 GMT+0
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Examples from Timofeev Chapter 3 and Chapter 7

\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
ATANH ==> atanh

)boot $IOindex := 1

-- " Timofeev (1948) Ch. 3, examples 1 - 3 (p. 101) ... "

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

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

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

-- " Timofeev (1948) Ch. 3, examples 4 - 9 (p. 105) ... "

[]

[]

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

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

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

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

INT(1/(x^3*(7-6*x+2*x^2)^2),x)

INT(x^9/(x^2+3*x+2)^5,x)

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

-- " Timofeev (1948) Ch. 3, examples 13 - 14 (p. 113) ... "

INT((a-b*x^2)^3/x^7,x)

INT(x^13/(a^4+x^4)^5,x)

-- " ... end of Timofeev Ch. 3 "

)boot $IOindex := 1

-- " Timofeev (1948) Ch. 7, examples 1 - 4 (p. 334) ... "

INT(x^2*COS(x)^5,x)

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

INT(x^2*SIN(x)^6,x)

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

-- " Timofeev (1948) Ch. 7, examples 5 - 9 (p. 342-343) ... "

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

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

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

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

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

-- " Timofeev (1948) Ch. 7, examples 10 - 11 (p. 344) ... "

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

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

-- " ... end of Timofeev Ch. 7 "

\end{axiom}

Examples from Timofeev Chapter 3 and Chapter 7

fricas

(1) -> INT ==> integrate

Type: Void

fricas

pi := %pi

$\label{eq1}\pi$

(1)

Type: PiDomain?

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

ATANH ==> atanh

Type: Void

fricas

)boot $IOindex := 1

   Your user access level is compiler and this command is therefore not
      available. See the )set userlevel command for more information.

-- " Timofeev (1948) Ch. 3, examples 1 - 3 (p. 101) ... "

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

$\label{eq2}\frac{{{\left(-{2 \ {{x}^{3}}}+{6 \ {{x}^{2}}}- 8 \right)}\ {\log \left({x + 1}\right)}}+{{\left({2 \ {{x}^{3}}}-{6 \ {{x}^{2}}}+ 8 \right)}\ {\log \left({x - 2}\right)}}+{6 \ {{x}^{2}}}-{{1 5}\ x}- 3}{{{54}\ {{x}^{3}}}-{{162}\ {{x}^{2}}}+{216}}$

(2)

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

fricas

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

$\label{eq3}\frac{{{\left(-{{60}\ {{x}^{5}}}-{{780}\ {{x}^{4}}}-{{4020}\ {{x}^{3}}}-{{10260}\ {{x}^{2}}}-{{12960}\ x}-{6480}\right)}\ {\log \left({x + 3}\right)}}+{{\left({{60}\ {{x}^{5}}}+{{780}\ {{x}^{4}}}+{{4020}\ {{x}^{3}}}+{{10260}\ {{x}^{2}}}+{{12960}\ x}+{6480}\right)}\ {\log \left({x + 2}\right)}}+{{60}\ {{x}^{4}}}+{{630}\ {{x}^{3}}}+{{2450}\ {{x}^{2}}}+{{4175}\ x}+{2627}}{{6 \ {{x}^{5}}}+{{78}\ {{x}^{4}}}+{{402}\ {{x}^{3}}}+{{1026}\ {{x}^{2}}}+{{1296}\ x}+{648}}$

(3)

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

fricas

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

$\label{eq4}\frac{{{\left({{1620}\ x}+{4860}\right)}\ {\log \left({x + 3}\right)}}+{{x}^{5}}-{5 \ {{x}^{4}}}+{{30}\ {{x}^{3}}}-{{270}\ {{x}^{2}}}-{{1296}\ x}+{972}}{{4 \ x}+{12}}$

(4)

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

fricas

-- " Timofeev (1948) Ch. 3, examples 4 - 9 (p. 105) ... "

[]

$\label{eq5}\left[ \right]$

(5)

Type: List(None)

fricas

[]

$\label{eq6}\left[ \right]$

(6)

Type: List(None)

fricas

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

$\label{eq7}\frac{{{\sqrt{3}}\ {\log \left({\frac{{{\left({2 \ x}+ 3 \right)}\ {\sqrt{3}}}+{2 \ {{x}^{2}}}+{6 \ x}+ 6}{{2 \ {{x}^{2}}}+{6 \ x}+ 3}}\right)}}+{\log \left({{2 \ {{x}^{2}}}+{6 \ x}+ 3}\right)}}{4}$

(7)

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

fricas

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

$\label{eq8}\frac{{{\left({8 \ {{x}^{4}}}+{{48}\ {{x}^{3}}}+{{96}\ {{x}^{2}}}+{{72}\ x}+{18}\right)}\ {\log \left({\frac{{{\left({2 \ x}+ 3 \right)}\ {\sqrt{3}}}+{2 \ {{x}^{2}}}+{6 \ x}+ 6}{{2 \ {{x}^{2}}}+{6 \ x}+ 3}}\right)}}+{{\left(-{8 \ {{x}^{3}}}-{{36}\ {{x}^{2}}}-{{44}\ x}-{13}\right)}\ {\sqrt{3}}}}{{\left({{16}\ {{x}^{4}}}+{{96}\ {{x}^{3}}}+{{192}\ {{x}^{2}}}+{{144}\ x}+{36}\right)}\ {\sqrt{3}}}$

(8)

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

fricas

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

$\label{eq9}\frac{{{\left(-{7 \ {{x}^{2}}}-{{35}\ x}-{28}\right)}\ {\log \left({x + 4}\right)}}+{{\left({7 \ {{x}^{2}}}+{{35}\ x}+{28}\right)}\ {\log \left({x + 1}\right)}}+{{21}\ x}+{39}}{{{27}\ {{x}^{2}}}+{{1 35}\ x}+{108}}$

(9)

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

fricas

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

$\label{eq10}\frac{{{\left(-{{840}\ {{x}^{8}}}-{{10080}\ {{x}^{7}}}-{{5208 0}\ {{x}^{6}}}-{{151200}\ {{x}^{5}}}-{{269640}\ {{x}^{4}}}-{{3 02400}\ {{x}^{3}}}-{{208320}\ {{x}^{2}}}-{{80640}\ x}-{13440}\right)}\ {\log \left({x + 2}\right)}}+{{\left({{840}\ {{x}^{8}}}+{{10080}\ {{x}^{7}}}+{{52080}\ {{x}^{6}}}+{{151200}\ {{x}^{5}}}+{{269640}\ {{x}^{4}}}+{{302400}\ {{x}^{3}}}+{{208320}\ {{x}^{2}}}+{{80640}\ x}+{13440}\right)}\ {\log \left({x + 1}\right)}}+{{840}\ {{x}^{7}}}+{{8820}\ {{x}^{6}}}+{{38920}\ {{x}^{5}}}+{{93450}\ {{x}^{4}}}+{{131768}\ {{x}^{3}}}+{{109116}\ {{x}^{2}}}+{{49176}\ x}+{9315}}{{{12}\ {{x}^{8}}}+{{144}\ {{x}^{7}}}+{{744}\ {{x}^{6}}}+{{2160}\ {{x}^{5}}}+{{3852}\ {{x}^{4}}}+{{4320}\ {{x}^{3}}}+{{2976}\ {{x}^{2}}}+{{1152}\ x}+{192}}$

(10)

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

fricas

INT(1/(x^3*(7-6*x+2*x^2)^2),x)

$\label{eq11}\frac{{{\left(-{{800}\ {{x}^{4}}}+{{2400}\ {{x}^{3}}}-{{2800}\ {{x}^{2}}}\right)}\ {\sqrt{5}}\ {\log \left({{2 \ {{x}^{2}}}-{6 \ x}+ 7}\right)}}+{{\left({{1600}\ {{x}^{4}}}-{{4800}\ {{x}^{3}}}+{{5600}\ {{x}^{2}}}\right)}\ {\sqrt{5}}\ {\log \left({x}\right)}}+{{\left({{936}\ {{x}^{4}}}-{{2808}\ {{x}^{3}}}+{{3276}\ {{x}^{2}}}\right)}\ {\arctan \left({\frac{{\left({2 \ x}- 3 \right)}\ {\sqrt{5}}}{5}}\right)}}+{{\left(-{{1932}\ {{x}^{3}}}+{{5698}\ {{x}^{2}}}-{{4410}\ x}-{1715}\right)}\ {\sqrt{5}}}}{{\left({{48020}\ {{x}^{4}}}-{{14 4060}\ {{x}^{3}}}+{{168070}\ {{x}^{2}}}\right)}\ {\sqrt{5}}}$

(11)

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

fricas

INT(x^9/(x^2+3*x+2)^5,x)

$\label{eq12}\frac{{{\left({{17664}\ {{x}^{8}}}+{{211968}\ {{x}^{7}}}+{{10 95168}\ {{x}^{6}}}+{{3179520}\ {{x}^{5}}}+{{5670144}\ {{x}^{4}}}+{{6359040}\ {{x}^{3}}}+{{4380672}\ {{x}^{2}}}+{{1695744}\ x}+{282624}\right)}\ {\log \left({x + 2}\right)}}+{{\left(-{{176 52}\ {{x}^{8}}}-{{211824}\ {{x}^{7}}}-{{1094424}\ {{x}^{6}}}-{{3177360}\ {{x}^{5}}}-{{5666292}\ {{x}^{4}}}-{{6354720}\ {{x}^{3}}}-{{4377696}\ {{x}^{2}}}-{{1694592}\ x}-{282432}\right)}\ {\log \left({x + 1}\right)}}-{{17496}\ {{x}^{7}}}-{{184200}\ {{x}^{6}}}-{{813888}\ {{x}^{5}}}-{{1955853}\ {{x}^{4}}}-{{2759400}\ {{x}^{3}}}-{{2286008}\ {{x}^{2}}}-{{1030560}\ x}-{195280}}{{{12}\ {{x}^{8}}}+{{144}\ {{x}^{7}}}+{{744}\ {{x}^{6}}}+{{2160}\ {{x}^{5}}}+{{3852}\ {{x}^{4}}}+{{4320}\ {{x}^{3}}}+{{2976}\ {{x}^{2}}}+{{1 152}\ x}+{192}}$

(12)

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

fricas

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

$\label{eq13}\frac{{{\left(-{{476160}\ {{x}^{8}}}-{{4761600}\ {{x}^{7}}}-{{2 0712960}\ {{x}^{6}}}-{{51187200}\ {{x}^{5}}}-{{78596160}\ {{x}^{4}}}-{{76780800}\ {{x}^{3}}}-{{46604160}\ {{x}^{2}}}-{{16070 400}\ x}-{2410560}\right)}\ {\log \left({{2 \ x}+ 3}\right)}}+{{\left({{476160}\ {{x}^{8}}}+{{4761600}\ {{x}^{7}}}+{{207129 60}\ {{x}^{6}}}+{{51187200}\ {{x}^{5}}}+{{78596160}\ {{x}^{4}}}+{{76780800}\ {{x}^{3}}}+{{46604160}\ {{x}^{2}}}+{{16070400}\ x}+{2410560}\right)}\ {\log \left({x + 1}\right)}}+{{238080}\ {{x}^{7}}}+{{2083200}\ {{x}^{6}}}+{{7757440}\ {{x}^{5}}}+{{1593400 0}\ {{x}^{4}}}+{{19495776}\ {{x}^{3}}}+{{14209160}\ {{x}^{2}}}+{{5712464}\ x}+{977397}}{{{192}\ {{x}^{8}}}+{{1920}\ {{x}^{7}}}+{{8352}\ {{x}^{6}}}+{{20640}\ {{x}^{5}}}+{{31692}\ {{x}^{4}}}+{{30960}\ {{x}^{3}}}+{{18792}\ {{x}^{2}}}+{{6480}\ x}+{972}}$

(13)

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

fricas

-- " Timofeev (1948) Ch. 3, examples 13 - 14 (p. 113) ... "

INT((a-b*x^2)^3/x^7,x)

$\label{eq14}\frac{-{{12}\ {{b}^{3}}\ {{x}^{6}}\ {\log \left({x}\right)}}-{{18}\ a \ {{b}^{2}}\ {{x}^{4}}}+{9 \ {{a}^{2}}\ b \ {{x}^{2}}}-{2 \ {{a}^{3}}}}{{12}\ {{x}^{6}}}$

(14)

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

fricas

INT(x^13/(a^4+x^4)^5,x)

$\label{eq15}\frac{{{\left({{15}\ {{x}^{16}}}+{{60}\ {{a}^{4}}\ {{x}^{12}}}+{{90}\ {{a}^{8}}\ {{x}^{8}}}+{{60}\ {{a}^{12}}\ {{x}^{4}}}+{{1 5}\ {{a}^{16}}}\right)}\ {\arctan \left({\frac{{x}^{2}}{{a}^{2}}}\right)}}+{{15}\ {{a}^{2}}\ {{x}^{14}}}-{{73}\ {{a}^{6}}\ {{x}^{10}}}-{{5 5}\ {{a}^{10}}\ {{x}^{6}}}-{{15}\ {{a}^{14}}\ {{x}^{2}}}}{{{7 68}\ {{a}^{6}}\ {{x}^{16}}}+{{3072}\ {{a}^{10}}\ {{x}^{12}}}+{{4608}\ {{a}^{14}}\ {{x}^{8}}}+{{3072}\ {{a}^{18}}\ {{x}^{4}}}+{{768}\ {{a}^{22}}}}$

(15)

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

fricas

-- " ... end of Timofeev Ch. 3 "

fricas

)boot $IOindex := 1

   Your user access level is compiler and this command is therefore not
      available. See the )set userlevel command for more information.

-- " Timofeev (1948) Ch. 7, examples 1 - 4 (p. 334) ... "

INT(x^2*COS(x)^5,x)

$\label{eq16}\frac{{{\left({{\left({{675}\ {{x}^{2}}}-{54}\right)}\ {{\cos \left({x}\right)}^{4}}}+{{\left({{900}\ {{x}^{2}}}-{272}\right)}\ {{\cos \left({x}\right)}^{2}}}+{{1800}\ {{x}^{2}}}-{4144}\right)}\ {\sin \left({x}\right)}}+{{270}\ x \ {{\cos \left({x}\right)}^{5}}}+{{600}\ x \ {{\cos \left({x}\right)}^{3}}}+{{3600}\ x \ {\cos \left({x}\right)}}}{3375}$

(16)

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

fricas

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

$\label{eq17}\frac{{{\left({{\left(-{9 \ {{x}^{2}}}+ 2 \right)}\ {{\cos \left({x}\right)}^{2}}}+{{63}\ {{x}^{2}}}-{122}\right)}\ {\sin \left({x}\right)}}+{{\left({9 \ {{x}^{3}}}-{6 \ x}\right)}\ {{\cos \left({x}\right)}^{3}}}+{{\left(-{{27}\ {{x}^{3}}}+{{126}\ x}\right)}\ {\cos \left({x}\right)}}}{2 7}$

(17)

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

fricas

INT(x^2*SIN(x)^6,x)

$\label{eq18}\frac{{{\left({{\left(-{{576}\ {{x}^{2}}}+{32}\right)}\ {{\cos \left({x}\right)}^{5}}}+{{\left({{1872}\ {{x}^{2}}}-{194}\right)}\ {{\cos \left({x}\right)}^{3}}}+{{\left(-{{2376}\ {{x}^{2}}}+{897}\right)}\ {\cos \left({x}\right)}}\right)}\ {\sin \left({x}\right)}}-{{192}\ x \ {{\cos \left({x}\right)}^{6}}}+{{936}\ x \ {{\cos \left({x}\right)}^{4}}}-{{2376}\ x \ {{\cos \left({x}\right)}^{2}}}+{{360}\ {{x}^{3}}}+{{897}\ x}}{3456}$

(18)

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

fricas

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

$\label{eq19}\frac{{{\left({{\left(-{9 \ {{x}^{2}}}+ 2 \right)}\ {{\cos \left({x}\right)}^{2}}}+{9 \ {{x}^{2}}}-{14}\right)}\ {\sin \left({x}\right)}}-{6 \ x \ {{\cos \left({x}\right)}^{3}}}+{{18}\ x \ {\cos \left({x}\right)}}}{2 7}$

(19)

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

fricas

-- " Timofeev (1948) Ch. 7, examples 5 - 9 (p. 342-343) ... "

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

$\label{eq20}\frac{{8 \ {\sin \left({x}\right)}\ {\log \left({\frac{\sin \left({x}\right)}{2}}\right)}}+{{\left(-{2 \ {{\cos \left({x}\right)}^{2}}}-{6 \ {{x}^{2}}}+ 1 \right)}\ {\sin \left({x}\right)}}+{4 \ x \ {{\cos \left({x}\right)}^{3}}}-{{12}\ x \ {\cos \left({x}\right)}}}{8 \ {\sin \left({x}\right)}}$

(20)

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

fricas

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

$\label{eq21}\frac{{5 \ {{\cos \left({x}\right)}^{3}}\ {\log \left({{\sin \left({x}\right)}+ 1}\right)}}-{5 \ {{\cos \left({x}\right)}^{3}}\ {\log \left({-{\sin \left({x}\right)}+ 1}\right)}}-{2 \ {\cos \left({x}\right)}\ {\sin \left({x}\right)}}-{{12}\ x \ {{\cos \left({x}\right)}^{2}}}+{4 \ x}}{{12}\ {{\cos \left({x}\right)}^{3}}}$

(21)

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

fricas

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

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

(22)

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

fricas

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

$\label{eq23}\frac{-{2 \ x \ {\log \left({\frac{{{\left({\sqrt{- 1}}+ 1 \right)}\ {\sin \left({x}\right)}}+{{\left(-{\sqrt{- 1}}- 1 \right)}\ {\cos \left({x}\right)}}-{\sqrt{- 1}}- 1}{{\sin \left({x}\right)}-{{\sqrt{- 1}}\ {\cos \left({x}\right)}}-{\sqrt{- 1}}}}\right)}}-{2 \ x \ {\log \left({\frac{{{\left({\sqrt{- 1}}+ 1 \right)}\ {\sin \left({x}\right)}}+{{\left({\sqrt{- 1}}+ 1 \right)}\ {\cos \left({x}\right)}}+{\sqrt{- 1}}+ 1}{{\sin \left({x}\right)}+{{\sqrt{- 1}}\ {\cos \left({x}\right)}}+{\sqrt{- 1}}}}\right)}}-{2 \ x \ {\log \left({\frac{{{\left(-{\sqrt{- 1}}+ 1 \right)}\ {\sin \left({x}\right)}}+{{\left({\sqrt{- 1}}- 1 \right)}\ {\cos \left({x}\right)}}+{\sqrt{- 1}}- 1}{{\sin \left({x}\right)}+{{\sqrt{- 1}}\ {\cos \left({x}\right)}}+{\sqrt{- 1}}}}\right)}}-{2 \ x \ {\log \left({\frac{{{\left(-{\sqrt{- 1}}+ 1 \right)}\ {\sin \left({x}\right)}}+{{\left(-{\sqrt{- 1}}+ 1 \right)}\ {\cos \left({x}\right)}}-{\sqrt{- 1}}+ 1}{{\sin \left({x}\right)}-{{\sqrt{- 1}}\ {\cos \left({x}\right)}}-{\sqrt{- 1}}}}\right)}}-{2 \ {\sqrt{- 1}}\ {dilog \left({\frac{{{\left({\sqrt{- 1}}+ 1 \right)}\ {\sin \left({x}\right)}}+{{\left(-{\sqrt{- 1}}- 1 \right)}\ {\cos \left({x}\right)}}-{\sqrt{- 1}}- 1}{{\sin \left({x}\right)}-{{\sqrt{- 1}}\ {\cos \left({x}\right)}}-{\sqrt{- 1}}}}\right)}}+{2 \ {\sqrt{- 1}}\ {dilog \left({\frac{{{\left({\sqrt{- 1}}+ 1 \right)}\ {\sin \left({x}\right)}}+{{\left({\sqrt{- 1}}+ 1 \right)}\ {\cos \left({x}\right)}}+{\sqrt{- 1}}+ 1}{{\sin \left({x}\right)}+{{\sqrt{- 1}}\ {\cos \left({x}\right)}}+{\sqrt{- 1}}}}\right)}}+{2 \ {\sqrt{- 1}}\ {dilog \left({\frac{{{\left(-{\sqrt{- 1}}+ 1 \right)}\ {\sin \left({x}\right)}}+{{\left({\sqrt{- 1}}- 1 \right)}\ {\cos \left({x}\right)}}+{\sqrt{- 1}}- 1}{{\sin \left({x}\right)}+{{\sqrt{- 1}}\ {\cos \left({x}\right)}}+{\sqrt{- 1}}}}\right)}}-{2 \ {\sqrt{- 1}}\ {dilog \left({\frac{{{\left(-{\sqrt{- 1}}+ 1 \right)}\ {\sin \left({x}\right)}}+{{\left(-{\sqrt{- 1}}+ 1 \right)}\ {\cos \left({x}\right)}}-{\sqrt{- 1}}+ 1}{{\sin \left({x}\right)}-{{\sqrt{- 1}}\ {\cos \left({x}\right)}}-{\sqrt{- 1}}}}\right)}}-{{\cos \left({x}\right)}\ {\sin \left({x}\right)}}+{2 \ x \ {{\cos \left({x}\right)}^{2}}}- x}{4}$

(23)

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

fricas

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

$\label{eq24}\frac{{x \ {{\cos \left({x}\right)}^{2}}\ {\log \left({\frac{{{\left({\sqrt{- 1}}+ 1 \right)}\ {\sin \left({x}\right)}}+{{\left(-{\sqrt{- 1}}- 1 \right)}\ {\cos \left({x}\right)}}-{\sqrt{- 1}}- 1}{{\sin \left({x}\right)}-{{\sqrt{- 1}}\ {\cos \left({x}\right)}}-{\sqrt{- 1}}}}\right)}}+{x \ {{\cos \left({x}\right)}^{2}}\ {\log \left({\frac{{{\left({\sqrt{- 1}}+ 1 \right)}\ {\sin \left({x}\right)}}+{{\left({\sqrt{- 1}}+ 1 \right)}\ {\cos \left({x}\right)}}+{\sqrt{- 1}}+ 1}{{\sin \left({x}\right)}+{{\sqrt{- 1}}\ {\cos \left({x}\right)}}+{\sqrt{- 1}}}}\right)}}+{x \ {{\cos \left({x}\right)}^{2}}\ {\log \left({\frac{{{\left(-{\sqrt{- 1}}+ 1 \right)}\ {\sin \left({x}\right)}}+{{\left({\sqrt{- 1}}- 1 \right)}\ {\cos \left({x}\right)}}+{\sqrt{- 1}}- 1}{{\sin \left({x}\right)}+{{\sqrt{- 1}}\ {\cos \left({x}\right)}}+{\sqrt{- 1}}}}\right)}}+{x \ {{\cos \left({x}\right)}^{2}}\ {\log \left({\frac{{{\left(-{\sqrt{- 1}}+ 1 \right)}\ {\sin \left({x}\right)}}+{{\left(-{\sqrt{- 1}}+ 1 \right)}\ {\cos \left({x}\right)}}-{\sqrt{- 1}}+ 1}{{\sin \left({x}\right)}-{{\sqrt{- 1}}\ {\cos \left({x}\right)}}-{\sqrt{- 1}}}}\right)}}+{{\sqrt{- 1}}\ {{\cos \left({x}\right)}^{2}}\ {dilog \left({\frac{{{\left({\sqrt{- 1}}+ 1 \right)}\ {\sin \left({x}\right)}}+{{\left(-{\sqrt{- 1}}- 1 \right)}\ {\cos \left({x}\right)}}-{\sqrt{- 1}}- 1}{{\sin \left({x}\right)}-{{\sqrt{- 1}}\ {\cos \left({x}\right)}}-{\sqrt{- 1}}}}\right)}}-{{\sqrt{- 1}}\ {{\cos \left({x}\right)}^{2}}\ {dilog \left({\frac{{{\left({\sqrt{- 1}}+ 1 \right)}\ {\sin \left({x}\right)}}+{{\left({\sqrt{- 1}}+ 1 \right)}\ {\cos \left({x}\right)}}+{\sqrt{- 1}}+ 1}{{\sin \left({x}\right)}+{{\sqrt{- 1}}\ {\cos \left({x}\right)}}+{\sqrt{- 1}}}}\right)}}-{{\sqrt{- 1}}\ {{\cos \left({x}\right)}^{2}}\ {dilog \left({\frac{{{\left(-{\sqrt{- 1}}+ 1 \right)}\ {\sin \left({x}\right)}}+{{\left({\sqrt{- 1}}- 1 \right)}\ {\cos \left({x}\right)}}+{\sqrt{- 1}}- 1}{{\sin \left({x}\right)}+{{\sqrt{- 1}}\ {\cos \left({x}\right)}}+{\sqrt{- 1}}}}\right)}}+{{\sqrt{- 1}}\ {{\cos \left({x}\right)}^{2}}\ {dilog \left({\frac{{{\left(-{\sqrt{- 1}}+ 1 \right)}\ {\sin \left({x}\right)}}+{{\left(-{\sqrt{- 1}}+ 1 \right)}\ {\cos \left({x}\right)}}-{\sqrt{- 1}}+ 1}{{\sin \left({x}\right)}-{{\sqrt{- 1}}\ {\cos \left({x}\right)}}-{\sqrt{- 1}}}}\right)}}-{{\cos \left({x}\right)}\ {\sin \left({x}\right)}}+ x}{2 \ {{\cos \left({x}\right)}^{2}}}$

(24)

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

fricas

-- " Timofeev (1948) Ch. 7, examples 10 - 11 (p. 344) ... "

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

$\label{eq25}-{\frac{2 \ {\cos \left({x}\right)}}{{x \ {\sin \left({x}\right)}}+{\cos \left({x}\right)}}}$

(25)

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

fricas

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

$\label{eq26}\frac{-{x \ {\sin \left({x}\right)}}-{\cos \left({x}\right)}}{{\sin \left({x}\right)}-{x \ {\cos \left({x}\right)}}}$

(26)

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