Optimal. Leaf size=74 \[ -\frac{a^2 \cot ^3(c+d x)}{3 d}-\frac{i a^2 \cot ^2(c+d x)}{d}+\frac{2 a^2 \cot (c+d x)}{d}-\frac{2 i a^2 \log (\sin (c+d x))}{d}+2 a^2 x \]
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Rubi [A] time = 0.113194, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {3542, 3529, 3531, 3475} \[ -\frac{a^2 \cot ^3(c+d x)}{3 d}-\frac{i a^2 \cot ^2(c+d x)}{d}+\frac{2 a^2 \cot (c+d x)}{d}-\frac{2 i a^2 \log (\sin (c+d x))}{d}+2 a^2 x \]
Antiderivative was successfully verified.
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Rule 3542
Rule 3529
Rule 3531
Rule 3475
Rubi steps
\begin{align*} \int \cot ^4(c+d x) (a+i a \tan (c+d x))^2 \, dx &=-\frac{a^2 \cot ^3(c+d x)}{3 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{a^2 \cot ^3(c+d x)}{3 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{a^2 \cot ^3(c+d x)}{3 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{a^2 \cot ^3(c+d x)}{3 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{a^2 \cot ^3(c+d x)}{3 d}-\frac{2 i a^2 \log (\sin (c+d x))}{d}\\ \end{align*}
Mathematica [C] time = 0.4579, size = 105, normalized size = 1.42 \[ -\frac{a^2 \cot ^3(c+d x) \, _2F_1\left (-\frac{3}{2},1;-\frac{1}{2};-\tan ^2(c+d x)\right )}{3 d}+\frac{a^2 \cot (c+d x) \, _2F_1\left (-\frac{1}{2},1;\frac{1}{2};-\tan ^2(c+d x)\right )}{d}-\frac{i a^2 \left (\cot ^2(c+d x)+2 \log (\tan (c+d x))+2 \log (\cos (c+d x))\right )}{d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.046, size = 80, normalized size = 1.1 \begin{align*} 2\,{a}^{2}x+2\,{\frac{{a}^{2}\cot \left ( dx+c \right ) }{d}}+2\,{\frac{{a}^{2}c}{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 ) ^{3}}{3\,d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 2.36703, size = 112, normalized size = 1.51 \begin{align*} \frac{6 \,{\left (d x + c\right )} a^{2} + 3 i \, a^{2} \log \left (\tan \left (d x + c\right )^{2} + 1\right ) - 6 i \, a^{2} \log \left (\tan \left (d x + c\right )\right ) + \frac{6 \, a^{2} \tan \left (d x + c\right )^{2} - 3 i \, a^{2} \tan \left (d x + c\right ) - a^{2}}{\tan \left (d x + c\right )^{3}}}{3 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.45229, size = 394, normalized size = 5.32 \begin{align*} \frac{30 i \, a^{2} e^{\left (4 i \, d x + 4 i \, c\right )} - 36 i \, a^{2} e^{\left (2 i \, d x + 2 i \, c\right )} + 14 i \, a^{2} +{\left (-6 i \, a^{2} e^{\left (6 i \, d x + 6 i \, c\right )} + 18 i \, a^{2} e^{\left (4 i \, d x + 4 i \, c\right )} - 18 i \, a^{2} e^{\left (2 i \, d x + 2 i \, c\right )} + 6 i \, a^{2}\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} - 1\right )}{3 \,{\left (d e^{\left (6 i \, d x + 6 i \, c\right )} - 3 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 3 \, 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 = 3.32621, size = 141, normalized size = 1.91 \begin{align*} - \frac{2 i a^{2} \log{\left (e^{2 i d x} - e^{- 2 i c} \right )}}{d} + \frac{\frac{10 i a^{2} e^{- 2 i c} e^{4 i d x}}{d} - \frac{12 i a^{2} e^{- 4 i c} e^{2 i d x}}{d} + \frac{14 i a^{2} e^{- 6 i c}}{3 d}}{e^{6 i d x} - 3 e^{- 2 i c} e^{4 i d x} + 3 e^{- 4 i c} e^{2 i d x} - e^{- 6 i c}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.36316, size = 198, normalized size = 2.68 \begin{align*} \frac{a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 6 i \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 96 i \, a^{2} \log \left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + i\right ) - 48 i \, a^{2} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) \right |}\right ) - 27 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - \frac{-88 i \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 27 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 6 i \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + a^{2}}{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3}}}{24 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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