Integrand size = 24, antiderivative size = 74 \[ \int \frac {\tan ^3(c+d x)}{a+i a \tan (c+d x)} \, dx=\frac {3 i x}{2 a}-\frac {\log (\cos (c+d x))}{a d}-\frac {3 i \tan (c+d x)}{2 a d}-\frac {\tan ^2(c+d x)}{2 d (a+i a \tan (c+d x))} \]
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Time = 0.08 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {3631, 3606, 3556} \[ \int \frac {\tan ^3(c+d x)}{a+i a \tan (c+d x)} \, dx=-\frac {\tan ^2(c+d x)}{2 d (a+i a \tan (c+d x))}-\frac {3 i \tan (c+d x)}{2 a d}-\frac {\log (\cos (c+d x))}{a d}+\frac {3 i x}{2 a} \]
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Rule 3556
Rule 3606
Rule 3631
Rubi steps \begin{align*} \text {integral}& = -\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*}
Time = 0.33 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.15 \[ \int \frac {\tan ^3(c+d x)}{a+i a \tan (c+d x)} \, dx=\frac {2 i \log (\cos (c+d x))+\arctan (\tan (c+d x)) (3+3 i \tan (c+d x))-(3+2 \log (\cos (c+d x))) \tan (c+d x)-2 i \tan ^2(c+d x)}{2 a d (-i+\tan (c+d x))} \]
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Time = 0.27 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.97
method | result | size |
derivativedivides | \(-\frac {i \tan \left (d x +c \right )}{d a}-\frac {i}{2 d a \left (\tan \left (d x +c \right )-i\right )}+\frac {\ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2 d a}+\frac {3 i \arctan \left (\tan \left (d x +c \right )\right )}{2 d a}\) | \(72\) |
default | \(-\frac {i \tan \left (d x +c \right )}{d a}-\frac {i}{2 d a \left (\tan \left (d x +c \right )-i\right )}+\frac {\ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2 d a}+\frac {3 i \arctan \left (\tan \left (d x +c \right )\right )}{2 d a}\) | \(72\) |
risch | \(\frac {5 i x}{2 a}+\frac {{\mathrm e}^{-2 i \left (d x +c \right )}}{4 a d}+\frac {2 i c}{a d}+\frac {2}{d a \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}-\frac {\ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{a d}\) | \(77\) |
norman | \(\frac {\frac {1}{2 a d}+\frac {3 i x}{2 a}-\frac {3 i \tan \left (d x +c \right )}{2 d a}-\frac {i \left (\tan ^{3}\left (d x +c \right )\right )}{d a}+\frac {3 i x \left (\tan ^{2}\left (d x +c \right )\right )}{2 a}}{1+\tan ^{2}\left (d x +c \right )}+\frac {\ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2 d a}\) | \(97\) |
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Time = 0.23 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.26 \[ \int \frac {\tan ^3(c+d x)}{a+i a \tan (c+d x)} \, dx=\frac {10 i \, d x e^{\left (4 i \, d x + 4 i \, c\right )} + {\left (10 i \, d x + 9\right )} e^{\left (2 i \, d x + 2 i \, c\right )} - 4 \, {\left (e^{\left (4 i \, d x + 4 i \, c\right )} + e^{\left (2 i \, d x + 2 i \, c\right )}\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) + 1}{4 \, {\left (a d e^{\left (4 i \, d x + 4 i \, c\right )} + a d e^{\left (2 i \, d x + 2 i \, c\right )}\right )}} \]
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Time = 0.19 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.54 \[ \int \frac {\tan ^3(c+d x)}{a+i a \tan (c+d x)} \, dx=\begin {cases} \frac {e^{- 2 i c} e^{- 2 i d x}}{4 a d} & \text {for}\: a d e^{2 i c} \neq 0 \\x \left (\frac {\left (5 i e^{2 i c} - i\right ) e^{- 2 i c}}{2 a} - \frac {5 i}{2 a}\right ) & \text {otherwise} \end {cases} + \frac {2}{a d e^{2 i c} e^{2 i d x} + a d} + \frac {5 i x}{2 a} - \frac {\log {\left (e^{2 i d x} + e^{- 2 i c} \right )}}{a d} \]
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Exception generated. \[ \int \frac {\tan ^3(c+d x)}{a+i a \tan (c+d x)} \, dx=\text {Exception raised: RuntimeError} \]
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Time = 0.68 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.92 \[ \int \frac {\tan ^3(c+d x)}{a+i a \tan (c+d x)} \, dx=-\frac {\frac {\log \left (\tan \left (d x + c\right ) + i\right )}{a} - \frac {5 \, \log \left (\tan \left (d x + c\right ) - i\right )}{a} + \frac {4 i \, \tan \left (d x + c\right )}{a} + \frac {5 \, \tan \left (d x + c\right ) - 3 i}{a {\left (\tan \left (d x + c\right ) - i\right )}}}{4 \, d} \]
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Time = 4.62 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.99 \[ \int \frac {\tan ^3(c+d x)}{a+i a \tan (c+d x)} \, dx=\frac {5\,\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )}{4\,a\,d}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )}{4\,a\,d}-\frac {\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}}{a\,d}+\frac {1}{2\,a\,d\,\left (1+\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )} \]
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