Optimal. Leaf size=74 \[ -\frac {\tan ^2(c+d x)}{2 d (a+i a \tan (c+d x))}-\frac {3 i \tan (c+d x)}{2 a d}-\frac {\log (\cos (c+d x))}{a d}+\frac {3 i x}{2 a} \]
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Rubi [A] time = 0.07, antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {3550, 3525, 3475} \[ -\frac {\tan ^2(c+d x)}{2 d (a+i a \tan (c+d x))}-\frac {3 i \tan (c+d x)}{2 a d}-\frac {\log (\cos (c+d x))}{a d}+\frac {3 i x}{2 a} \]
Antiderivative was successfully verified.
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Rule 3475
Rule 3525
Rule 3550
Rubi steps
\begin {align*} \int \frac {\tan ^3(c+d x)}{a+i a \tan (c+d x)} \, dx &=-\frac {\tan ^2(c+d x)}{2 d (a+i a \tan (c+d x))}+\frac {\int \tan (c+d x) (2 a-3 i a \tan (c+d x)) \, dx}{2 a^2}\\ &=\frac {3 i x}{2 a}-\frac {3 i \tan (c+d x)}{2 a d}-\frac {\tan ^2(c+d x)}{2 d (a+i a \tan (c+d x))}+\frac {\int \tan (c+d x) \, dx}{a}\\ &=\frac {3 i x}{2 a}-\frac {\log (\cos (c+d x))}{a d}-\frac {3 i \tan (c+d x)}{2 a d}-\frac {\tan ^2(c+d x)}{2 d (a+i a \tan (c+d x))}\\ \end {align*}
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Mathematica [B] time = 1.14, size = 174, normalized size = 2.35 \[ -\frac {i \cos (c) \sec (c+d x) (\cos (d x)+i \sin (d x)) \left ((-4-4 i \tan (c)) \tan ^{-1}(\tan (d x))-4 d x \tan ^2(c)-2 i d x \tan (c)+4 d x \sec ^2(c)-i \tan (c) \sin (2 d x)-2 i \log \left (\cos ^2(c+d x)\right )+(\tan (c)+i) \cos (2 d x)+4 \sec (c) \sin (d x) \sec (c+d x)+2 \tan (c) \log \left (\cos ^2(c+d x)\right )+4 i \tan (c) \sec (c) \sin (d x) \sec (c+d x)-6 d x+\sin (2 d x)\right )}{4 d (a+i a \tan (c+d x))} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 93, normalized size = 1.26 \[ \frac {10 i \, d x e^{\left (4 i \, d x + 4 i \, c\right )} + {\left (10 i \, d x + 9\right )} e^{\left (2 i \, d x + 2 i \, c\right )} - 4 \, {\left (e^{\left (4 i \, d x + 4 i \, c\right )} + e^{\left (2 i \, d x + 2 i \, c\right )}\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) + 1}{4 \, {\left (a d e^{\left (4 i \, d x + 4 i \, c\right )} + a d e^{\left (2 i \, d x + 2 i \, c\right )}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 2.42, size = 70, normalized size = 0.95 \[ -\frac {\frac {\log \left (\tan \left (d x + c\right ) + i\right )}{a} - \frac {5 \, \log \left (-i \, \tan \left (d x + c\right ) - 1\right )}{a} + \frac {4 i \, \tan \left (d x + c\right )}{a} + \frac {5 \, \tan \left (d x + c\right ) - 3 i}{a {\left (\tan \left (d x + c\right ) - i\right )}}}{4 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.15, size = 73, normalized size = 0.99 \[ -\frac {i \tan \left (d x +c \right )}{d a}-\frac {\ln \left (\tan \left (d x +c \right )+i\right )}{4 d a}-\frac {i}{2 d a \left (\tan \left (d x +c \right )-i\right )}+\frac {5 \ln \left (\tan \left (d x +c \right )-i\right )}{4 d a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.00, size = 73, normalized size = 0.99 \[ \frac {5\,\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )}{4\,a\,d}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )}{4\,a\,d}-\frac {\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}}{a\,d}+\frac {1}{2\,a\,d\,\left (1+\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.40, size = 119, normalized size = 1.61 \[ \begin {cases} \frac {e^{- 2 i c} e^{- 2 i d x}}{4 a d} & \text {for}\: 4 a d e^{2 i c} \neq 0 \\x \left (- \frac {i \left (1 - 5 e^{2 i c}\right ) e^{- 2 i c}}{2 a} - \frac {5 i}{2 a}\right ) & \text {otherwise} \end {cases} - \frac {2}{- a d e^{2 i c} e^{2 i d x} - a d} + \frac {5 i x}{2 a} - \frac {\log {\left (e^{2 i d x} + e^{- 2 i c} \right )}}{a d} \]
Verification of antiderivative is not currently implemented for this CAS.
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