Optimal. Leaf size=180 \[ \frac{a^3 (47 A-45 i B) \cot ^3(c+d x)}{60 d}+\frac{2 a^3 (B+i A) \cot ^2(c+d x)}{d}-\frac{4 a^3 (A-i B) \cot (c+d x)}{d}+\frac{4 a^3 (B+i A) \log (\sin (c+d x))}{d}-\frac{(5 B+7 i A) \cot ^4(c+d x) \left (a^3+i a^3 \tan (c+d x)\right )}{20 d}-4 a^3 x (A-i B)-\frac{a A \cot ^5(c+d x) (a+i a \tan (c+d x))^2}{5 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.460285, antiderivative size = 180, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 34, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.147, Rules used = {3593, 3591, 3529, 3531, 3475} \[ \frac{a^3 (47 A-45 i B) \cot ^3(c+d x)}{60 d}+\frac{2 a^3 (B+i A) \cot ^2(c+d x)}{d}-\frac{4 a^3 (A-i B) \cot (c+d x)}{d}+\frac{4 a^3 (B+i A) \log (\sin (c+d x))}{d}-\frac{(5 B+7 i A) \cot ^4(c+d x) \left (a^3+i a^3 \tan (c+d x)\right )}{20 d}-4 a^3 x (A-i B)-\frac{a A \cot ^5(c+d x) (a+i a \tan (c+d x))^2}{5 d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3593
Rule 3591
Rule 3529
Rule 3531
Rule 3475
Rubi steps
\begin{align*} \int \cot ^6(c+d x) (a+i a \tan (c+d x))^3 (A+B \tan (c+d x)) \, dx &=-\frac{a A \cot ^5(c+d x) (a+i a \tan (c+d x))^2}{5 d}+\frac{1}{5} \int \cot ^5(c+d x) (a+i a \tan (c+d x))^2 (a (7 i A+5 B)-a (3 A-5 i B) \tan (c+d x)) \, dx\\ &=-\frac{a A \cot ^5(c+d x) (a+i a \tan (c+d x))^2}{5 d}-\frac{(7 i A+5 B) \cot ^4(c+d x) \left (a^3+i a^3 \tan (c+d x)\right )}{20 d}+\frac{1}{20} \int \cot ^4(c+d x) (a+i a \tan (c+d x)) \left (-a^2 (47 A-45 i B)-a^2 (33 i A+35 B) \tan (c+d x)\right ) \, dx\\ &=\frac{a^3 (47 A-45 i B) \cot ^3(c+d x)}{60 d}-\frac{a A \cot ^5(c+d x) (a+i a \tan (c+d x))^2}{5 d}-\frac{(7 i A+5 B) \cot ^4(c+d x) \left (a^3+i a^3 \tan (c+d x)\right )}{20 d}+\frac{1}{20} \int \cot ^3(c+d x) \left (-80 a^3 (i A+B)+80 a^3 (A-i B) \tan (c+d x)\right ) \, dx\\ &=\frac{2 a^3 (i A+B) \cot ^2(c+d x)}{d}+\frac{a^3 (47 A-45 i B) \cot ^3(c+d x)}{60 d}-\frac{a A \cot ^5(c+d x) (a+i a \tan (c+d x))^2}{5 d}-\frac{(7 i A+5 B) \cot ^4(c+d x) \left (a^3+i a^3 \tan (c+d x)\right )}{20 d}+\frac{1}{20} \int \cot ^2(c+d x) \left (80 a^3 (A-i B)+80 a^3 (i A+B) \tan (c+d x)\right ) \, dx\\ &=-\frac{4 a^3 (A-i B) \cot (c+d x)}{d}+\frac{2 a^3 (i A+B) \cot ^2(c+d x)}{d}+\frac{a^3 (47 A-45 i B) \cot ^3(c+d x)}{60 d}-\frac{a A \cot ^5(c+d x) (a+i a \tan (c+d x))^2}{5 d}-\frac{(7 i A+5 B) \cot ^4(c+d x) \left (a^3+i a^3 \tan (c+d x)\right )}{20 d}+\frac{1}{20} \int \cot (c+d x) \left (80 a^3 (i A+B)-80 a^3 (A-i B) \tan (c+d x)\right ) \, dx\\ &=-4 a^3 (A-i B) x-\frac{4 a^3 (A-i B) \cot (c+d x)}{d}+\frac{2 a^3 (i A+B) \cot ^2(c+d x)}{d}+\frac{a^3 (47 A-45 i B) \cot ^3(c+d x)}{60 d}-\frac{a A \cot ^5(c+d x) (a+i a \tan (c+d x))^2}{5 d}-\frac{(7 i A+5 B) \cot ^4(c+d x) \left (a^3+i a^3 \tan (c+d x)\right )}{20 d}+\left (4 a^3 (i A+B)\right ) \int \cot (c+d x) \, dx\\ &=-4 a^3 (A-i B) x-\frac{4 a^3 (A-i B) \cot (c+d x)}{d}+\frac{2 a^3 (i A+B) \cot ^2(c+d x)}{d}+\frac{a^3 (47 A-45 i B) \cot ^3(c+d x)}{60 d}+\frac{4 a^3 (i A+B) \log (\sin (c+d x))}{d}-\frac{a A \cot ^5(c+d x) (a+i a \tan (c+d x))^2}{5 d}-\frac{(7 i A+5 B) \cot ^4(c+d x) \left (a^3+i a^3 \tan (c+d x)\right )}{20 d}\\ \end{align*}
Mathematica [B] time = 8.77901, size = 943, normalized size = 5.24 \[ a^3 \left (\frac{(\cot (c+d x)+i)^3 (B+A \cot (c+d x)) \left (i A \cos \left (\frac{3 c}{2}\right )+B \cos \left (\frac{3 c}{2}\right )+A \sin \left (\frac{3 c}{2}\right )-i B \sin \left (\frac{3 c}{2}\right )\right ) \left (-4 i \tan ^{-1}(\tan (4 c+d x)) \cos \left (\frac{3 c}{2}\right )-4 \tan ^{-1}(\tan (4 c+d x)) \sin \left (\frac{3 c}{2}\right )\right ) \sin ^4(c+d x)}{d (\cos (d x)+i \sin (d x))^3 (A \cos (c+d x)+B \sin (c+d x))}+\frac{(\cot (c+d x)+i)^3 (B+A \cot (c+d x)) \left (i A \cos \left (\frac{3 c}{2}\right )+B \cos \left (\frac{3 c}{2}\right )+A \sin \left (\frac{3 c}{2}\right )-i B \sin \left (\frac{3 c}{2}\right )\right ) \left (2 \cos \left (\frac{3 c}{2}\right ) \log \left (\sin ^2(c+d x)\right )-2 i \log \left (\sin ^2(c+d x)\right ) \sin \left (\frac{3 c}{2}\right )\right ) \sin ^4(c+d x)}{d (\cos (d x)+i \sin (d x))^3 (A \cos (c+d x)+B \sin (c+d x))}+\frac{x (\cot (c+d x)+i)^3 (B+A \cot (c+d x)) \left (-16 A \cos ^3(c)+16 i B \cos ^3(c)-4 i A \cot (c) \cos ^3(c)-4 B \cot (c) \cos ^3(c)+24 i A \sin (c) \cos ^2(c)+24 B \sin (c) \cos ^2(c)+16 A \sin ^2(c) \cos (c)-16 i B \sin ^2(c) \cos (c)-4 i A \sin ^3(c)-4 B \sin ^3(c)+(i A+B) \cot (c) (4 \cos (3 c)-4 i \sin (3 c))\right ) \sin ^4(c+d x)}{(\cos (d x)+i \sin (d x))^3 (A \cos (c+d x)+B \sin (c+d x))}+\frac{(\cot (c+d x)+i)^3 (B+A \cot (c+d x)) \csc (c) \csc (c+d x) \left (\frac{1}{240} \cos (3 c)-\frac{1}{240} i \sin (3 c)\right ) (225 i A \cos (d x)+195 B \cos (d x)-300 A d x \cos (d x)+300 i B d x \cos (d x)-225 i A \cos (2 c+d x)-195 B \cos (2 c+d x)+300 A d x \cos (2 c+d x)-300 i B d x \cos (2 c+d x)-105 i A \cos (2 c+3 d x)-75 B \cos (2 c+3 d x)+150 A d x \cos (2 c+3 d x)-150 i B d x \cos (2 c+3 d x)+105 i A \cos (4 c+3 d x)+75 B \cos (4 c+3 d x)-150 A d x \cos (4 c+3 d x)+150 i B d x \cos (4 c+3 d x)-30 A d x \cos (4 c+5 d x)+30 i B d x \cos (4 c+5 d x)+30 A d x \cos (6 c+5 d x)-30 i B d x \cos (6 c+5 d x)+470 A \sin (d x)-420 i B \sin (d x)+360 A \sin (2 c+d x)-330 i B \sin (2 c+d x)-280 A \sin (2 c+3 d x)+270 i B \sin (2 c+3 d x)-135 A \sin (4 c+3 d x)+105 i B \sin (4 c+3 d x)+83 A \sin (4 c+5 d x)-75 i B \sin (4 c+5 d x))}{d (\cos (d x)+i \sin (d x))^3 (A \cos (c+d x)+B \sin (c+d x))}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.08, size = 224, normalized size = 1.2 \begin{align*} 4\,{\frac{B{a}^{3}\ln \left ( \sin \left ( dx+c \right ) \right ) }{d}}-4\,{\frac{A{a}^{3}c}{d}}+{\frac{4\,A{a}^{3} \left ( \cot \left ( dx+c \right ) \right ) ^{3}}{3\,d}}-4\,{\frac{A\cot \left ( dx+c \right ){a}^{3}}{d}}+2\,{\frac{B{a}^{3} \left ( \cot \left ( dx+c \right ) \right ) ^{2}}{d}}-{\frac{A{a}^{3} \left ( \cot \left ( dx+c \right ) \right ) ^{5}}{5\,d}}-{\frac{B{a}^{3} \left ( \cot \left ( dx+c \right ) \right ) ^{4}}{4\,d}}+4\,iBx{a}^{3}-{\frac{{\frac{3\,i}{4}}A{a}^{3} \left ( \cot \left ( dx+c \right ) \right ) ^{4}}{d}}+{\frac{4\,iB{a}^{3}c}{d}}+{\frac{4\,iB\cot \left ( dx+c \right ){a}^{3}}{d}}+{\frac{2\,iA{a}^{3} \left ( \cot \left ( dx+c \right ) \right ) ^{2}}{d}}-4\,A{a}^{3}x+{\frac{4\,iA{a}^{3}\ln \left ( \sin \left ( dx+c \right ) \right ) }{d}}-{\frac{iB{a}^{3} \left ( \cot \left ( dx+c \right ) \right ) ^{3}}{d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 2.24459, size = 208, normalized size = 1.16 \begin{align*} -\frac{60 \,{\left (d x + c\right )}{\left (4 \, A - 4 i \, B\right )} a^{3} + 120 \,{\left (i \, A + B\right )} a^{3} \log \left (\tan \left (d x + c\right )^{2} + 1\right ) + 240 \,{\left (-i \, A - B\right )} a^{3} \log \left (\tan \left (d x + c\right )\right ) + \frac{{\left (240 \, A - 240 i \, B\right )} a^{3} \tan \left (d x + c\right )^{4} - 120 \,{\left (i \, A + B\right )} a^{3} \tan \left (d x + c\right )^{3} -{\left (80 \, A - 60 i \, B\right )} a^{3} \tan \left (d x + c\right )^{2} - 15 \,{\left (-3 i \, A - B\right )} a^{3} \tan \left (d x + c\right ) + 12 \, A a^{3}}{\tan \left (d x + c\right )^{5}}}{60 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.64839, size = 844, normalized size = 4.69 \begin{align*} \frac{{\left (-480 i \, A - 360 \, B\right )} a^{3} e^{\left (8 i \, d x + 8 i \, c\right )} +{\left (1170 i \, A + 1050 \, B\right )} a^{3} e^{\left (6 i \, d x + 6 i \, c\right )} +{\left (-1390 i \, A - 1230 \, B\right )} a^{3} e^{\left (4 i \, d x + 4 i \, c\right )} +{\left (770 i \, A + 690 \, B\right )} a^{3} e^{\left (2 i \, d x + 2 i \, c\right )} +{\left (-166 i \, A - 150 \, B\right )} a^{3} +{\left ({\left (60 i \, A + 60 \, B\right )} a^{3} e^{\left (10 i \, d x + 10 i \, c\right )} +{\left (-300 i \, A - 300 \, B\right )} a^{3} e^{\left (8 i \, d x + 8 i \, c\right )} +{\left (600 i \, A + 600 \, B\right )} a^{3} e^{\left (6 i \, d x + 6 i \, c\right )} +{\left (-600 i \, A - 600 \, B\right )} a^{3} e^{\left (4 i \, d x + 4 i \, c\right )} +{\left (300 i \, A + 300 \, B\right )} a^{3} e^{\left (2 i \, d x + 2 i \, c\right )} +{\left (-60 i \, A - 60 \, B\right )} a^{3}\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} - 1\right )}{15 \,{\left (d e^{\left (10 i \, d x + 10 i \, c\right )} - 5 \, d e^{\left (8 i \, d x + 8 i \, c\right )} + 10 \, d e^{\left (6 i \, d x + 6 i \, c\right )} - 10 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 5 \, d e^{\left (2 i \, d x + 2 i \, c\right )} - d\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 146.15, size = 272, normalized size = 1.51 \begin{align*} \frac{4 a^{3} \left (i A + B\right ) \log{\left (e^{2 i d x} - e^{- 2 i c} \right )}}{d} + \frac{- \frac{\left (32 i A a^{3} + 24 B a^{3}\right ) e^{- 2 i c} e^{8 i d x}}{d} + \frac{\left (78 i A a^{3} + 70 B a^{3}\right ) e^{- 4 i c} e^{6 i d x}}{d} + \frac{\left (154 i A a^{3} + 138 B a^{3}\right ) e^{- 8 i c} e^{2 i d x}}{3 d} - \frac{\left (166 i A a^{3} + 150 B a^{3}\right ) e^{- 10 i c}}{15 d} - \frac{\left (278 i A a^{3} + 246 B a^{3}\right ) e^{- 6 i c} e^{4 i d x}}{3 d}}{e^{10 i d x} - 5 e^{- 2 i c} e^{8 i d x} + 10 e^{- 4 i c} e^{6 i d x} - 10 e^{- 6 i c} e^{4 i d x} + 5 e^{- 8 i c} e^{2 i d x} - e^{- 10 i c}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 1.76694, size = 529, normalized size = 2.94 \begin{align*} \frac{6 \, A a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 45 i \, A a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 15 \, B a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 190 \, A a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 120 i \, B a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 660 i \, A a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 540 \, B a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 2460 \, A a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 2280 i \, B a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 7680 \,{\left (i \, A a^{3} + B a^{3}\right )} \log \left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + i\right ) - 3840 \,{\left (-i \, A a^{3} - B a^{3}\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) \right |}\right ) + \frac{-8768 i \, A a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 8768 \, B a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 2460 \, A a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 2280 i \, B a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 660 i \, A a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 540 \, B a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 190 \, A a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 120 i \, B a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 45 i \, A a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 15 \, B a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 6 \, A a^{3}}{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5}}}{960 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]