Optimal. Leaf size=98 \[ \frac{7}{8 d \left (a^3+i a^3 \tan (c+d x)\right )}+\frac{\log (\sin (c+d x))}{a^3 d}-\frac{7 i x}{8 a^3}+\frac{3}{8 a d (a+i a \tan (c+d x))^2}+\frac{1}{6 d (a+i a \tan (c+d x))^3} \]
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Rubi [A] time = 0.230147, antiderivative size = 98, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {3559, 3596, 3531, 3475} \[ \frac{7}{8 d \left (a^3+i a^3 \tan (c+d x)\right )}+\frac{\log (\sin (c+d x))}{a^3 d}-\frac{7 i x}{8 a^3}+\frac{3}{8 a d (a+i a \tan (c+d x))^2}+\frac{1}{6 d (a+i a \tan (c+d x))^3} \]
Antiderivative was successfully verified.
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Rule 3559
Rule 3596
Rule 3531
Rule 3475
Rubi steps
\begin{align*} \int \frac{\cot (c+d x)}{(a+i a \tan (c+d x))^3} \, dx &=\frac{1}{6 d (a+i a \tan (c+d x))^3}+\frac{\int \frac{\cot (c+d x) (6 a-3 i a \tan (c+d x))}{(a+i a \tan (c+d x))^2} \, dx}{6 a^2}\\ &=\frac{1}{6 d (a+i a \tan (c+d x))^3}+\frac{3}{8 a d (a+i a \tan (c+d x))^2}+\frac{\int \frac{\cot (c+d x) \left (24 a^2-18 i a^2 \tan (c+d x)\right )}{a+i a \tan (c+d x)} \, dx}{24 a^4}\\ &=\frac{1}{6 d (a+i a \tan (c+d x))^3}+\frac{3}{8 a d (a+i a \tan (c+d x))^2}+\frac{7}{8 d \left (a^3+i a^3 \tan (c+d x)\right )}+\frac{\int \cot (c+d x) \left (48 a^3-42 i a^3 \tan (c+d x)\right ) \, dx}{48 a^6}\\ &=-\frac{7 i x}{8 a^3}+\frac{1}{6 d (a+i a \tan (c+d x))^3}+\frac{3}{8 a d (a+i a \tan (c+d x))^2}+\frac{7}{8 d \left (a^3+i a^3 \tan (c+d x)\right )}+\frac{\int \cot (c+d x) \, dx}{a^3}\\ &=-\frac{7 i x}{8 a^3}+\frac{\log (\sin (c+d x))}{a^3 d}+\frac{1}{6 d (a+i a \tan (c+d x))^3}+\frac{3}{8 a d (a+i a \tan (c+d x))^2}+\frac{7}{8 d \left (a^3+i a^3 \tan (c+d x)\right )}\\ \end{align*}
Mathematica [A] time = 0.381996, size = 118, normalized size = 1.2 \[ \frac{\sec ^3(c+d x) (-51 \sin (c+d x)+84 i d x \sin (3 (c+d x))+2 \sin (3 (c+d x))+81 i \cos (c+d x)-96 \sin (3 (c+d x)) \log (\sin (c+d x))+\cos (3 (c+d x)) (96 i \log (\sin (c+d x))+84 d x+2 i))}{96 a^3 d (\tan (c+d x)-i)^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.084, size = 111, normalized size = 1.1 \begin{align*}{\frac{{\frac{i}{6}}}{d{a}^{3} \left ( \tan \left ( dx+c \right ) -i \right ) ^{3}}}-{\frac{{\frac{7\,i}{8}}}{d{a}^{3} \left ( \tan \left ( dx+c \right ) -i \right ) }}-{\frac{3}{8\,d{a}^{3} \left ( \tan \left ( dx+c \right ) -i \right ) ^{2}}}-{\frac{15\,\ln \left ( \tan \left ( dx+c \right ) -i \right ) }{16\,d{a}^{3}}}-{\frac{\ln \left ( \tan \left ( dx+c \right ) +i \right ) }{16\,d{a}^{3}}}+{\frac{\ln \left ( \tan \left ( dx+c \right ) \right ) }{d{a}^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.3378, size = 238, normalized size = 2.43 \begin{align*} \frac{{\left (-180 i \, d x e^{\left (6 i \, d x + 6 i \, c\right )} + 96 \, e^{\left (6 i \, d x + 6 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} - 1\right ) + 66 \, e^{\left (4 i \, d x + 4 i \, c\right )} + 15 \, e^{\left (2 i \, d x + 2 i \, c\right )} + 2\right )} e^{\left (-6 i \, d x - 6 i \, c\right )}}{96 \, a^{3} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 3.20898, size = 187, normalized size = 1.91 \begin{align*} \begin{cases} \frac{\left (16896 a^{6} d^{2} e^{10 i c} e^{- 2 i d x} + 3840 a^{6} d^{2} e^{8 i c} e^{- 4 i d x} + 512 a^{6} d^{2} e^{6 i c} e^{- 6 i d x}\right ) e^{- 12 i c}}{24576 a^{9} d^{3}} & \text{for}\: 24576 a^{9} d^{3} e^{12 i c} \neq 0 \\x \left (- \frac{\left (15 i e^{6 i c} + 11 i e^{4 i c} + 5 i e^{2 i c} + i\right ) e^{- 6 i c}}{8 a^{3}} + \frac{15 i}{8 a^{3}}\right ) & \text{otherwise} \end{cases} - \frac{15 i x}{8 a^{3}} + \frac{\log{\left (e^{2 i d x} - e^{- 2 i c} \right )}}{a^{3} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.32418, size = 127, normalized size = 1.3 \begin{align*} -\frac{\frac{90 \, \log \left (\tan \left (d x + c\right ) - i\right )}{a^{3}} + \frac{6 \, \log \left (i \, \tan \left (d x + c\right ) - 1\right )}{a^{3}} - \frac{96 \, \log \left ({\left | \tan \left (d x + c\right ) \right |}\right )}{a^{3}} - \frac{165 \, \tan \left (d x + c\right )^{3} - 579 i \, \tan \left (d x + c\right )^{2} - 699 \, \tan \left (d x + c\right ) + 301 i}{a^{3}{\left (\tan \left (d x + c\right ) - i\right )}^{3}}}{96 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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