Optimal. Leaf size=143 \[ \frac{3 \tan ^2(c+d x)}{2 d \left (a^3+i a^3 \tan (c+d x)\right )}+\frac{25 i \tan (c+d x)}{8 a^3 d}+\frac{3 \log (\cos (c+d x))}{a^3 d}-\frac{25 i x}{8 a^3}-\frac{\tan ^4(c+d x)}{6 d (a+i a \tan (c+d x))^3}+\frac{11 i \tan ^3(c+d x)}{24 a d (a+i a \tan (c+d x))^2} \]
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Rubi [A] time = 0.287057, antiderivative size = 143, 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 = {3558, 3595, 3525, 3475} \[ \frac{3 \tan ^2(c+d x)}{2 d \left (a^3+i a^3 \tan (c+d x)\right )}+\frac{25 i \tan (c+d x)}{8 a^3 d}+\frac{3 \log (\cos (c+d x))}{a^3 d}-\frac{25 i x}{8 a^3}-\frac{\tan ^4(c+d x)}{6 d (a+i a \tan (c+d x))^3}+\frac{11 i \tan ^3(c+d x)}{24 a d (a+i a \tan (c+d x))^2} \]
Antiderivative was successfully verified.
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Rule 3558
Rule 3595
Rule 3525
Rule 3475
Rubi steps
\begin{align*} \int \frac{\tan ^5(c+d x)}{(a+i a \tan (c+d x))^3} \, dx &=-\frac{\tan ^4(c+d x)}{6 d (a+i a \tan (c+d x))^3}-\frac{\int \frac{\tan ^3(c+d x) (-4 a+7 i a \tan (c+d x))}{(a+i a \tan (c+d x))^2} \, dx}{6 a^2}\\ &=-\frac{\tan ^4(c+d x)}{6 d (a+i a \tan (c+d x))^3}+\frac{11 i \tan ^3(c+d x)}{24 a d (a+i a \tan (c+d x))^2}+\frac{\int \frac{\tan ^2(c+d x) \left (-33 i a^2-39 a^2 \tan (c+d x)\right )}{a+i a \tan (c+d x)} \, dx}{24 a^4}\\ &=-\frac{\tan ^4(c+d x)}{6 d (a+i a \tan (c+d x))^3}+\frac{11 i \tan ^3(c+d x)}{24 a d (a+i a \tan (c+d x))^2}+\frac{3 \tan ^2(c+d x)}{2 d \left (a^3+i a^3 \tan (c+d x)\right )}-\frac{\int \tan (c+d x) \left (144 a^3-150 i a^3 \tan (c+d x)\right ) \, dx}{48 a^6}\\ &=-\frac{25 i x}{8 a^3}+\frac{25 i \tan (c+d x)}{8 a^3 d}-\frac{\tan ^4(c+d x)}{6 d (a+i a \tan (c+d x))^3}+\frac{11 i \tan ^3(c+d x)}{24 a d (a+i a \tan (c+d x))^2}+\frac{3 \tan ^2(c+d x)}{2 d \left (a^3+i a^3 \tan (c+d x)\right )}-\frac{3 \int \tan (c+d x) \, dx}{a^3}\\ &=-\frac{25 i x}{8 a^3}+\frac{3 \log (\cos (c+d x))}{a^3 d}+\frac{25 i \tan (c+d x)}{8 a^3 d}-\frac{\tan ^4(c+d x)}{6 d (a+i a \tan (c+d x))^3}+\frac{11 i \tan ^3(c+d x)}{24 a d (a+i a \tan (c+d x))^2}+\frac{3 \tan ^2(c+d x)}{2 d \left (a^3+i a^3 \tan (c+d x)\right )}\\ \end{align*}
Mathematica [A] time = 2.47989, size = 239, normalized size = 1.67 \[ \frac{\sec ^3(c+d x) (\cos (d x)+i \sin (d x))^3 (-138 \sin (c) \sin (2 d x)-21 \sin (c) \sin (4 d x)+300 d x \sin (3 c)+2 \sin (3 c) \sin (6 d x)-138 i \sin (c) \cos (2 d x)-21 i \sin (c) \cos (4 d x)+2 i \sin (3 c) \cos (6 d x)+\cos (c) (39 \cos (d x)+53 i \sin (d x)) (-3 \cos (3 d x)+3 i \sin (3 d x))-96 \sin (3 c) \sec (c) \sin (d x) \sec (c+d x)+288 i \sin (3 c) \log (\cos (c+d x))+\cos (3 c) (288 \log (\cos (c+d x))+96 i \sec (c) \sin (d x) \sec (c+d x)-300 i d x+2 i \sin (6 d x)-2 \cos (6 d x)))}{96 d (a+i a \tan (c+d x))^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.026, size = 112, normalized size = 0.8 \begin{align*}{\frac{i\tan \left ( dx+c \right ) }{d{a}^{3}}}+{\frac{{\frac{31\,i}{8}}}{d{a}^{3} \left ( \tan \left ( dx+c \right ) -i \right ) }}-{\frac{{\frac{i}{6}}}{d{a}^{3} \left ( \tan \left ( dx+c \right ) -i \right ) ^{3}}}-{\frac{9}{8\,d{a}^{3} \left ( \tan \left ( dx+c \right ) -i \right ) ^{2}}}-{\frac{49\,\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}}} \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 = 2.15483, size = 363, normalized size = 2.54 \begin{align*} \frac{-588 i \, d x e^{\left (8 i \, d x + 8 i \, c\right )} - 6 \,{\left (98 i \, d x + 55\right )} e^{\left (6 i \, d x + 6 i \, c\right )} + 288 \,{\left (e^{\left (8 i \, d x + 8 i \, c\right )} + e^{\left (6 i \, d x + 6 i \, c\right )}\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) - 117 \, e^{\left (4 i \, d x + 4 i \, c\right )} + 19 \, e^{\left (2 i \, d x + 2 i \, c\right )} - 2}{96 \,{\left (a^{3} d e^{\left (8 i \, d x + 8 i \, c\right )} + a^{3} d e^{\left (6 i \, d x + 6 i \, c\right )}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.5262, size = 219, normalized size = 1.53 \begin{align*} \begin{cases} \frac{\left (- 35328 a^{6} d^{2} e^{10 i c} e^{- 2 i d x} + 5376 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 (49 i e^{6 i c} - 23 i e^{4 i c} + 7 i e^{2 i c} - i\right ) e^{- 6 i c}}{8 a^{3}} + \frac{49 i}{8 a^{3}}\right ) & \text{otherwise} \end{cases} - \frac{49 i x}{8 a^{3}} + \frac{3 \log{\left (e^{2 i d x} + e^{- 2 i c} \right )}}{a^{3} d} - \frac{2 e^{- 2 i c}}{a^{3} d \left (e^{2 i d x} + e^{- 2 i c}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 3.60565, size = 123, normalized size = 0.86 \begin{align*} \frac{\frac{6 \, \log \left (\tan \left (d x + c\right ) + i\right )}{a^{3}} - \frac{294 \, \log \left (i \, \tan \left (d x + c\right ) + 1\right )}{a^{3}} + \frac{96 i \, \tan \left (d x + c\right )}{a^{3}} + \frac{539 \, \tan \left (d x + c\right )^{3} - 1245 i \, \tan \left (d x + c\right )^{2} - 981 \, \tan \left (d x + c\right ) + 259 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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