Optimal. Leaf size=126 \[ -\frac{11 i a^3 \cot ^4(c+d x)}{20 d}+\frac{4 a^3 \cot ^3(c+d x)}{3 d}+\frac{2 i a^3 \cot ^2(c+d x)}{d}-\frac{4 a^3 \cot (c+d x)}{d}+\frac{4 i a^3 \log (\sin (c+d x))}{d}-\frac{\cot ^5(c+d x) \left (a^3+i a^3 \tan (c+d x)\right )}{5 d}-4 a^3 x \]
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Rubi [A] time = 0.210605, antiderivative size = 126, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {3553, 3591, 3529, 3531, 3475} \[ -\frac{11 i a^3 \cot ^4(c+d x)}{20 d}+\frac{4 a^3 \cot ^3(c+d x)}{3 d}+\frac{2 i a^3 \cot ^2(c+d x)}{d}-\frac{4 a^3 \cot (c+d x)}{d}+\frac{4 i a^3 \log (\sin (c+d x))}{d}-\frac{\cot ^5(c+d x) \left (a^3+i a^3 \tan (c+d x)\right )}{5 d}-4 a^3 x \]
Antiderivative was successfully verified.
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Rule 3553
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 \, dx &=-\frac{\cot ^5(c+d x) \left (a^3+i a^3 \tan (c+d x)\right )}{5 d}-\frac{1}{5} \int \cot ^5(c+d x) (a+i a \tan (c+d x)) \left (-11 i a^2+9 a^2 \tan (c+d x)\right ) \, dx\\ &=-\frac{11 i a^3 \cot ^4(c+d x)}{20 d}-\frac{\cot ^5(c+d x) \left (a^3+i a^3 \tan (c+d x)\right )}{5 d}-\frac{1}{5} \int \cot ^4(c+d x) \left (20 a^3+20 i a^3 \tan (c+d x)\right ) \, dx\\ &=\frac{4 a^3 \cot ^3(c+d x)}{3 d}-\frac{11 i a^3 \cot ^4(c+d x)}{20 d}-\frac{\cot ^5(c+d x) \left (a^3+i a^3 \tan (c+d x)\right )}{5 d}-\frac{1}{5} \int \cot ^3(c+d x) \left (20 i a^3-20 a^3 \tan (c+d x)\right ) \, dx\\ &=\frac{2 i a^3 \cot ^2(c+d x)}{d}+\frac{4 a^3 \cot ^3(c+d x)}{3 d}-\frac{11 i a^3 \cot ^4(c+d x)}{20 d}-\frac{\cot ^5(c+d x) \left (a^3+i a^3 \tan (c+d x)\right )}{5 d}-\frac{1}{5} \int \cot ^2(c+d x) \left (-20 a^3-20 i a^3 \tan (c+d x)\right ) \, dx\\ &=-\frac{4 a^3 \cot (c+d x)}{d}+\frac{2 i a^3 \cot ^2(c+d x)}{d}+\frac{4 a^3 \cot ^3(c+d x)}{3 d}-\frac{11 i a^3 \cot ^4(c+d x)}{20 d}-\frac{\cot ^5(c+d x) \left (a^3+i a^3 \tan (c+d x)\right )}{5 d}-\frac{1}{5} \int \cot (c+d x) \left (-20 i a^3+20 a^3 \tan (c+d x)\right ) \, dx\\ &=-4 a^3 x-\frac{4 a^3 \cot (c+d x)}{d}+\frac{2 i a^3 \cot ^2(c+d x)}{d}+\frac{4 a^3 \cot ^3(c+d x)}{3 d}-\frac{11 i a^3 \cot ^4(c+d x)}{20 d}-\frac{\cot ^5(c+d x) \left (a^3+i a^3 \tan (c+d x)\right )}{5 d}+\left (4 i a^3\right ) \int \cot (c+d x) \, dx\\ &=-4 a^3 x-\frac{4 a^3 \cot (c+d x)}{d}+\frac{2 i a^3 \cot ^2(c+d x)}{d}+\frac{4 a^3 \cot ^3(c+d x)}{3 d}-\frac{11 i a^3 \cot ^4(c+d x)}{20 d}+\frac{4 i a^3 \log (\sin (c+d x))}{d}-\frac{\cot ^5(c+d x) \left (a^3+i a^3 \tan (c+d x)\right )}{5 d}\\ \end{align*}
Mathematica [B] time = 2.0005, size = 359, normalized size = 2.85 \[ \frac{a^3 \csc (c) \csc ^5(c+d x) (\cos (3 d x)+i \sin (3 d x)) \left (360 \sin (2 c+d x)-280 \sin (2 c+3 d x)-135 \sin (4 c+3 d x)+83 \sin (4 c+5 d x)+600 d x \cos (2 c+d x)-225 i \cos (2 c+d x)+300 d x \cos (2 c+3 d x)-105 i \cos (2 c+3 d x)-300 d x \cos (4 c+3 d x)+105 i \cos (4 c+3 d x)-60 d x \cos (4 c+5 d x)+60 d x \cos (6 c+5 d x)+960 \sin (c) \sin ^5(c+d x) \tan ^{-1}(\tan (4 c+d x))-75 \cos (d x) \left (-2 i \log \left (\sin ^2(c+d x)\right )+8 d x-3 i\right )-150 i \cos (2 c+d x) \log \left (\sin ^2(c+d x)\right )-75 i \cos (2 c+3 d x) \log \left (\sin ^2(c+d x)\right )+75 i \cos (4 c+3 d x) \log \left (\sin ^2(c+d x)\right )+15 i \cos (4 c+5 d x) \log \left (\sin ^2(c+d x)\right )-15 i \cos (6 c+5 d x) \log \left (\sin ^2(c+d x)\right )+470 \sin (d x)\right )}{240 d (\cos (d x)+i \sin (d x))^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.052, size = 113, normalized size = 0.9 \begin{align*}{\frac{2\,i{a}^{3} \left ( \cot \left ( dx+c \right ) \right ) ^{2}}{d}}+{\frac{4\,i{a}^{3}\ln \left ( \sin \left ( dx+c \right ) \right ) }{d}}+{\frac{4\,{a}^{3} \left ( \cot \left ( dx+c \right ) \right ) ^{3}}{3\,d}}-4\,{a}^{3}x-4\,{\frac{{a}^{3}\cot \left ( dx+c \right ) }{d}}-4\,{\frac{{a}^{3}c}{d}}-{\frac{{\frac{3\,i}{4}}{a}^{3} \left ( \cot \left ( dx+c \right ) \right ) ^{4}}{d}}-{\frac{{a}^{3} \left ( \cot \left ( dx+c \right ) \right ) ^{5}}{5\,d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.68489, size = 147, normalized size = 1.17 \begin{align*} -\frac{240 \,{\left (d x + c\right )} a^{3} + 120 i \, a^{3} \log \left (\tan \left (d x + c\right )^{2} + 1\right ) - 240 i \, a^{3} \log \left (\tan \left (d x + c\right )\right ) + \frac{240 \, a^{3} \tan \left (d x + c\right )^{4} - 120 i \, a^{3} \tan \left (d x + c\right )^{3} - 80 \, a^{3} \tan \left (d x + c\right )^{2} + 45 i \, a^{3} \tan \left (d x + c\right ) + 12 \, a^{3}}{\tan \left (d x + c\right )^{5}}}{60 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.30746, size = 659, normalized size = 5.23 \begin{align*} \frac{-480 i \, a^{3} e^{\left (8 i \, d x + 8 i \, c\right )} + 1170 i \, a^{3} e^{\left (6 i \, d x + 6 i \, c\right )} - 1390 i \, a^{3} e^{\left (4 i \, d x + 4 i \, c\right )} + 770 i \, a^{3} e^{\left (2 i \, d x + 2 i \, c\right )} - 166 i \, a^{3} +{\left (60 i \, a^{3} e^{\left (10 i \, d x + 10 i \, c\right )} - 300 i \, a^{3} e^{\left (8 i \, d x + 8 i \, c\right )} + 600 i \, a^{3} e^{\left (6 i \, d x + 6 i \, c\right )} - 600 i \, a^{3} e^{\left (4 i \, d x + 4 i \, c\right )} + 300 i \, a^{3} e^{\left (2 i \, d x + 2 i \, c\right )} - 60 i \, 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.
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Sympy [A] time = 7.79321, size = 226, normalized size = 1.79 \begin{align*} \frac{4 i a^{3} \log{\left (e^{2 i d x} - e^{- 2 i c} \right )}}{d} + \frac{- \frac{32 i a^{3} e^{- 2 i c} e^{8 i d x}}{d} + \frac{78 i a^{3} e^{- 4 i c} e^{6 i d x}}{d} - \frac{278 i a^{3} e^{- 6 i c} e^{4 i d x}}{3 d} + \frac{154 i a^{3} e^{- 8 i c} e^{2 i d x}}{3 d} - \frac{166 i a^{3} e^{- 10 i c}}{15 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.
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Giac [A] time = 1.53558, size = 288, normalized size = 2.29 \begin{align*} \frac{6 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 45 i \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 190 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 660 i \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 7680 i \, a^{3} \log \left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + i\right ) + 3840 i \, a^{3} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) \right |}\right ) + 2460 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + \frac{-8768 i \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 2460 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 660 i \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 190 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 45 i \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 6 \, 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.
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