Optimal. Leaf size=112 \[ -\frac{a^2 \cot ^5(c+d x)}{5 d}-\frac{i a^2 \cot ^4(c+d x)}{2 d}+\frac{2 a^2 \cot ^3(c+d x)}{3 d}+\frac{i a^2 \cot ^2(c+d x)}{d}-\frac{2 a^2 \cot (c+d x)}{d}+\frac{2 i a^2 \log (\sin (c+d x))}{d}-2 a^2 x \]
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Rubi [A] time = 0.167012, antiderivative size = 112, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {3542, 3529, 3531, 3475} \[ -\frac{a^2 \cot ^5(c+d x)}{5 d}-\frac{i a^2 \cot ^4(c+d x)}{2 d}+\frac{2 a^2 \cot ^3(c+d x)}{3 d}+\frac{i a^2 \cot ^2(c+d x)}{d}-\frac{2 a^2 \cot (c+d x)}{d}+\frac{2 i a^2 \log (\sin (c+d x))}{d}-2 a^2 x \]
Antiderivative was successfully verified.
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Rule 3542
Rule 3529
Rule 3531
Rule 3475
Rubi steps
\begin{align*} \int \cot ^6(c+d x) (a+i a \tan (c+d x))^2 \, dx &=-\frac{a^2 \cot ^5(c+d x)}{5 d}+\int \cot ^5(c+d x) \left (2 i a^2-2 a^2 \tan (c+d x)\right ) \, dx\\ &=-\frac{i a^2 \cot ^4(c+d x)}{2 d}-\frac{a^2 \cot ^5(c+d x)}{5 d}+\int \cot ^4(c+d x) \left (-2 a^2-2 i a^2 \tan (c+d x)\right ) \, dx\\ &=\frac{2 a^2 \cot ^3(c+d x)}{3 d}-\frac{i a^2 \cot ^4(c+d x)}{2 d}-\frac{a^2 \cot ^5(c+d x)}{5 d}+\int \cot ^3(c+d x) \left (-2 i a^2+2 a^2 \tan (c+d x)\right ) \, dx\\ &=\frac{i a^2 \cot ^2(c+d x)}{d}+\frac{2 a^2 \cot ^3(c+d x)}{3 d}-\frac{i a^2 \cot ^4(c+d x)}{2 d}-\frac{a^2 \cot ^5(c+d x)}{5 d}+\int \cot ^2(c+d x) \left (2 a^2+2 i a^2 \tan (c+d x)\right ) \, dx\\ &=-\frac{2 a^2 \cot (c+d x)}{d}+\frac{i a^2 \cot ^2(c+d x)}{d}+\frac{2 a^2 \cot ^3(c+d x)}{3 d}-\frac{i a^2 \cot ^4(c+d x)}{2 d}-\frac{a^2 \cot ^5(c+d x)}{5 d}+\int \cot (c+d x) \left (2 i a^2-2 a^2 \tan (c+d x)\right ) \, dx\\ &=-2 a^2 x-\frac{2 a^2 \cot (c+d x)}{d}+\frac{i a^2 \cot ^2(c+d x)}{d}+\frac{2 a^2 \cot ^3(c+d x)}{3 d}-\frac{i a^2 \cot ^4(c+d x)}{2 d}-\frac{a^2 \cot ^5(c+d x)}{5 d}+\left (2 i a^2\right ) \int \cot (c+d x) \, dx\\ &=-2 a^2 x-\frac{2 a^2 \cot (c+d x)}{d}+\frac{i a^2 \cot ^2(c+d x)}{d}+\frac{2 a^2 \cot ^3(c+d x)}{3 d}-\frac{i a^2 \cot ^4(c+d x)}{2 d}-\frac{a^2 \cot ^5(c+d x)}{5 d}+\frac{2 i a^2 \log (\sin (c+d x))}{d}\\ \end{align*}
Mathematica [C] time = 0.799427, size = 124, normalized size = 1.11 \[ -\frac{a^2 \cot ^5(c+d x) \, _2F_1\left (-\frac{5}{2},1;-\frac{3}{2};-\tan ^2(c+d x)\right )}{5 d}+\frac{a^2 \cot ^3(c+d x) \, _2F_1\left (-\frac{3}{2},1;-\frac{1}{2};-\tan ^2(c+d x)\right )}{3 d}+\frac{i a^2 \left (-\cot ^4(c+d x)+2 \cot ^2(c+d x)+4 \log (\tan (c+d x))+4 \log (\cos (c+d x))\right )}{2 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.049, size = 113, normalized size = 1. \begin{align*}{\frac{2\,{a}^{2} \left ( \cot \left ( dx+c \right ) \right ) ^{3}}{3\,d}}-2\,{\frac{{a}^{2}\cot \left ( dx+c \right ) }{d}}-2\,{a}^{2}x-2\,{\frac{{a}^{2}c}{d}}-{\frac{{\frac{i}{2}}{a}^{2} \left ( \cot \left ( dx+c \right ) \right ) ^{4}}{d}}+{\frac{i{a}^{2} \left ( \cot \left ( dx+c \right ) \right ) ^{2}}{d}}+{\frac{2\,i{a}^{2}\ln \left ( \sin \left ( dx+c \right ) \right ) }{d}}-{\frac{{a}^{2} \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 = 2.01789, size = 147, normalized size = 1.31 \begin{align*} -\frac{60 \,{\left (d x + c\right )} a^{2} + 30 i \, a^{2} \log \left (\tan \left (d x + c\right )^{2} + 1\right ) - 60 i \, a^{2} \log \left (\tan \left (d x + c\right )\right ) + \frac{60 \, a^{2} \tan \left (d x + c\right )^{4} - 30 i \, a^{2} \tan \left (d x + c\right )^{3} - 20 \, a^{2} \tan \left (d x + c\right )^{2} + 15 i \, a^{2} \tan \left (d x + c\right ) + 6 \, a^{2}}{\tan \left (d x + c\right )^{5}}}{30 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.55803, size = 655, normalized size = 5.85 \begin{align*} \frac{-270 i \, a^{2} e^{\left (8 i \, d x + 8 i \, c\right )} + 600 i \, a^{2} e^{\left (6 i \, d x + 6 i \, c\right )} - 740 i \, a^{2} e^{\left (4 i \, d x + 4 i \, c\right )} + 400 i \, a^{2} e^{\left (2 i \, d x + 2 i \, c\right )} - 86 i \, a^{2} +{\left (30 i \, a^{2} e^{\left (10 i \, d x + 10 i \, c\right )} - 150 i \, a^{2} e^{\left (8 i \, d x + 8 i \, c\right )} + 300 i \, a^{2} e^{\left (6 i \, d x + 6 i \, c\right )} - 300 i \, a^{2} e^{\left (4 i \, d x + 4 i \, c\right )} + 150 i \, a^{2} e^{\left (2 i \, d x + 2 i \, c\right )} - 30 i \, a^{2}\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 [B] time = 10.1137, size = 226, normalized size = 2.02 \begin{align*} \frac{2 i a^{2} \log{\left (e^{2 i d x} - e^{- 2 i c} \right )}}{d} + \frac{- \frac{18 i a^{2} e^{- 2 i c} e^{8 i d x}}{d} + \frac{40 i a^{2} e^{- 4 i c} e^{6 i d x}}{d} - \frac{148 i a^{2} e^{- 6 i c} e^{4 i d x}}{3 d} + \frac{80 i a^{2} e^{- 8 i c} e^{2 i d x}}{3 d} - \frac{86 i a^{2} 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 [B] time = 1.38177, size = 288, normalized size = 2.57 \begin{align*} \frac{3 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 15 i \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 55 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 180 i \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 1920 i \, a^{2} \log \left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + i\right ) + 960 i \, a^{2} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) \right |}\right ) + 630 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + \frac{-2192 i \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 630 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 180 i \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 55 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 15 i \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 3 \, a^{2}}{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5}}}{480 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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