Optimal. Leaf size=88 \[ -\frac{16 i a^3 \sqrt{\cot (c+d x)}}{3 d}-\frac{2 \sqrt{\cot (c+d x)} \left (a^3 \cot (c+d x)+i a^3\right )}{3 d}-\frac{8 (-1)^{3/4} a^3 \tanh ^{-1}\left ((-1)^{3/4} \sqrt{\cot (c+d x)}\right )}{d} \]
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Rubi [A] time = 0.16995, antiderivative size = 88, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192, Rules used = {3673, 3556, 3592, 3533, 208} \[ -\frac{16 i a^3 \sqrt{\cot (c+d x)}}{3 d}-\frac{2 \sqrt{\cot (c+d x)} \left (a^3 \cot (c+d x)+i a^3\right )}{3 d}-\frac{8 (-1)^{3/4} a^3 \tanh ^{-1}\left ((-1)^{3/4} \sqrt{\cot (c+d x)}\right )}{d} \]
Antiderivative was successfully verified.
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Rule 3673
Rule 3556
Rule 3592
Rule 3533
Rule 208
Rubi steps
\begin{align*} \int \cot ^{\frac{5}{2}}(c+d x) (a+i a \tan (c+d x))^3 \, dx &=\int \frac{(i a+a \cot (c+d x))^3}{\sqrt{\cot (c+d x)}} \, dx\\ &=-\frac{2 \sqrt{\cot (c+d x)} \left (i a^3+a^3 \cot (c+d x)\right )}{3 d}-\frac{1}{3} (2 i a) \int \frac{(-2 i a-4 a \cot (c+d x)) (i a+a \cot (c+d x))}{\sqrt{\cot (c+d x)}} \, dx\\ &=-\frac{16 i a^3 \sqrt{\cot (c+d x)}}{3 d}-\frac{2 \sqrt{\cot (c+d x)} \left (i a^3+a^3 \cot (c+d x)\right )}{3 d}-\frac{1}{3} (2 i a) \int \frac{6 a^2-6 i a^2 \cot (c+d x)}{\sqrt{\cot (c+d x)}} \, dx\\ &=-\frac{16 i a^3 \sqrt{\cot (c+d x)}}{3 d}-\frac{2 \sqrt{\cot (c+d x)} \left (i a^3+a^3 \cot (c+d x)\right )}{3 d}-\frac{\left (48 i a^5\right ) \operatorname{Subst}\left (\int \frac{1}{-6 a^2-6 i a^2 x^2} \, dx,x,\sqrt{\cot (c+d x)}\right )}{d}\\ &=-\frac{8 (-1)^{3/4} a^3 \tanh ^{-1}\left ((-1)^{3/4} \sqrt{\cot (c+d x)}\right )}{d}-\frac{16 i a^3 \sqrt{\cot (c+d x)}}{3 d}-\frac{2 \sqrt{\cot (c+d x)} \left (i a^3+a^3 \cot (c+d x)\right )}{3 d}\\ \end{align*}
Mathematica [A] time = 2.50784, size = 125, normalized size = 1.42 \[ -\frac{2 a^3 e^{-3 i c} \sqrt{\cot (c+d x)} (\cos (3 (c+d x))+i \sin (3 (c+d x))) \left (\cot (c+d x)-12 i \sqrt{i \tan (c+d x)} \tanh ^{-1}\left (\sqrt{\frac{-1+e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}}\right )+9 i\right )}{3 d (\cos (d x)+i \sin (d x))^3} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.285, size = 791, normalized size = 9. \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.66982, size = 196, normalized size = 2.23 \begin{align*} \frac{3 \,{\left (\left (2 i + 2\right ) \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} + \frac{2}{\sqrt{\tan \left (d x + c\right )}}\right )}\right ) + \left (2 i + 2\right ) \, \sqrt{2} \arctan \left (-\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} - \frac{2}{\sqrt{\tan \left (d x + c\right )}}\right )}\right ) + \left (i - 1\right ) \, \sqrt{2} \log \left (\frac{\sqrt{2}}{\sqrt{\tan \left (d x + c\right )}} + \frac{1}{\tan \left (d x + c\right )} + 1\right ) - \left (i - 1\right ) \, \sqrt{2} \log \left (-\frac{\sqrt{2}}{\sqrt{\tan \left (d x + c\right )}} + \frac{1}{\tan \left (d x + c\right )} + 1\right )\right )} a^{3} - \frac{18 i \, a^{3}}{\sqrt{\tan \left (d x + c\right )}} - \frac{2 \, a^{3}}{\tan \left (d x + c\right )^{\frac{3}{2}}}}{3 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.42049, size = 811, normalized size = 9.22 \begin{align*} -\frac{3 \, \sqrt{-\frac{64 i \, a^{6}}{d^{2}}}{\left (d e^{\left (2 i \, d x + 2 i \, c\right )} - d\right )} \log \left (\frac{{\left (8 i \, a^{3} e^{\left (2 i \, d x + 2 i \, c\right )} + \sqrt{-\frac{64 i \, a^{6}}{d^{2}}}{\left (i \, d e^{\left (2 i \, d x + 2 i \, c\right )} - i \, d\right )} \sqrt{\frac{i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} - 1}}\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{4 \, a^{3}}\right ) - 3 \, \sqrt{-\frac{64 i \, a^{6}}{d^{2}}}{\left (d e^{\left (2 i \, d x + 2 i \, c\right )} - d\right )} \log \left (\frac{{\left (8 i \, a^{3} e^{\left (2 i \, d x + 2 i \, c\right )} + \sqrt{-\frac{64 i \, a^{6}}{d^{2}}}{\left (-i \, d e^{\left (2 i \, d x + 2 i \, c\right )} + i \, d\right )} \sqrt{\frac{i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} - 1}}\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{4 \, a^{3}}\right ) -{\left (-80 i \, a^{3} e^{\left (2 i \, d x + 2 i \, c\right )} + 64 i \, a^{3}\right )} \sqrt{\frac{i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} - 1}}}{12 \,{\left (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 [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{3} \cot \left (d x + c\right )^{\frac{5}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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