Integrand size = 24, antiderivative size = 109 \[ \int \frac {\tan ^5(c+d x)}{a+i a \tan (c+d x)} \, dx=-\frac {5 i x}{2 a}+\frac {2 \log (\cos (c+d x))}{a d}+\frac {5 i \tan (c+d x)}{2 a d}+\frac {\tan ^2(c+d x)}{a d}-\frac {5 i \tan ^3(c+d x)}{6 a d}-\frac {\tan ^4(c+d x)}{2 d (a+i a \tan (c+d x))} \]
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Time = 0.14 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3631, 3609, 3606, 3556} \[ \int \frac {\tan ^5(c+d x)}{a+i a \tan (c+d x)} \, dx=-\frac {\tan ^4(c+d x)}{2 d (a+i a \tan (c+d x))}-\frac {5 i \tan ^3(c+d x)}{6 a d}+\frac {\tan ^2(c+d x)}{a d}+\frac {5 i \tan (c+d x)}{2 a d}+\frac {2 \log (\cos (c+d x))}{a d}-\frac {5 i x}{2 a} \]
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Rule 3556
Rule 3606
Rule 3609
Rule 3631
Rubi steps \begin{align*} \text {integral}& = -\frac {\tan ^4(c+d x)}{2 d (a+i a \tan (c+d x))}+\frac {\int \tan ^3(c+d x) (4 a-5 i a \tan (c+d x)) \, dx}{2 a^2} \\ & = -\frac {5 i \tan ^3(c+d x)}{6 a d}-\frac {\tan ^4(c+d x)}{2 d (a+i a \tan (c+d x))}+\frac {\int \tan ^2(c+d x) (5 i a+4 a \tan (c+d x)) \, dx}{2 a^2} \\ & = \frac {\tan ^2(c+d x)}{a d}-\frac {5 i \tan ^3(c+d x)}{6 a d}-\frac {\tan ^4(c+d x)}{2 d (a+i a \tan (c+d x))}+\frac {\int \tan (c+d x) (-4 a+5 i a \tan (c+d x)) \, dx}{2 a^2} \\ & = -\frac {5 i x}{2 a}+\frac {5 i \tan (c+d x)}{2 a d}+\frac {\tan ^2(c+d x)}{a d}-\frac {5 i \tan ^3(c+d x)}{6 a d}-\frac {\tan ^4(c+d x)}{2 d (a+i a \tan (c+d x))}-\frac {2 \int \tan (c+d x) \, dx}{a} \\ & = -\frac {5 i x}{2 a}+\frac {2 \log (\cos (c+d x))}{a d}+\frac {5 i \tan (c+d x)}{2 a d}+\frac {\tan ^2(c+d x)}{a d}-\frac {5 i \tan ^3(c+d x)}{6 a d}-\frac {\tan ^4(c+d x)}{2 d (a+i a \tan (c+d x))} \\ \end{align*}
Time = 0.54 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.97 \[ \int \frac {\tan ^5(c+d x)}{a+i a \tan (c+d x)} \, dx=\frac {-12 i \log (\cos (c+d x))+3 (5+4 \log (\cos (c+d x))) \tan (c+d x)+9 i \tan ^2(c+d x)+\tan ^3(c+d x)-2 i \tan ^4(c+d x)-15 i \arctan (\tan (c+d x)) (-i+\tan (c+d x))}{6 a d (-i+\tan (c+d x))} \]
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Time = 0.30 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.93
method | result | size |
risch | \(-\frac {9 i x}{2 a}-\frac {{\mathrm e}^{-2 i \left (d x +c \right )}}{4 a d}-\frac {4 i c}{a d}-\frac {2 \left (6 \,{\mathrm e}^{4 i \left (d x +c \right )}+9 \,{\mathrm e}^{2 i \left (d x +c \right )}+7\right )}{3 d a \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{3}}+\frac {2 \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{a d}\) | \(101\) |
derivativedivides | \(\frac {2 i \tan \left (d x +c \right )}{d a}-\frac {i \left (\tan ^{3}\left (d x +c \right )\right )}{3 d a}+\frac {\tan ^{2}\left (d x +c \right )}{2 a d}-\frac {\ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{d a}-\frac {5 i \arctan \left (\tan \left (d x +c \right )\right )}{2 d a}+\frac {i}{2 d a \left (\tan \left (d x +c \right )-i\right )}\) | \(105\) |
default | \(\frac {2 i \tan \left (d x +c \right )}{d a}-\frac {i \left (\tan ^{3}\left (d x +c \right )\right )}{3 d a}+\frac {\tan ^{2}\left (d x +c \right )}{2 a d}-\frac {\ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{d a}-\frac {5 i \arctan \left (\tan \left (d x +c \right )\right )}{2 d a}+\frac {i}{2 d a \left (\tan \left (d x +c \right )-i\right )}\) | \(105\) |
norman | \(\frac {-\frac {1}{a d}+\frac {\tan ^{4}\left (d x +c \right )}{2 a d}-\frac {5 i x}{2 a}+\frac {5 i \tan \left (d x +c \right )}{2 d a}+\frac {5 i \left (\tan ^{3}\left (d x +c \right )\right )}{3 d a}-\frac {i \left (\tan ^{5}\left (d x +c \right )\right )}{3 a d}-\frac {5 i x \left (\tan ^{2}\left (d x +c \right )\right )}{2 a}}{1+\tan ^{2}\left (d x +c \right )}-\frac {\ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{d a}\) | \(130\) |
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Time = 0.25 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.61 \[ \int \frac {\tan ^5(c+d x)}{a+i a \tan (c+d x)} \, dx=\frac {-54 i \, d x e^{\left (8 i \, d x + 8 i \, c\right )} - 3 \, {\left (54 i \, d x + 17\right )} e^{\left (6 i \, d x + 6 i \, c\right )} - 81 \, {\left (2 i \, d x + 1\right )} e^{\left (4 i \, d x + 4 i \, c\right )} + {\left (-54 i \, d x - 65\right )} e^{\left (2 i \, d x + 2 i \, c\right )} + 24 \, {\left (e^{\left (8 i \, d x + 8 i \, c\right )} + 3 \, e^{\left (6 i \, d x + 6 i \, c\right )} + 3 \, e^{\left (4 i \, d x + 4 i \, c\right )} + e^{\left (2 i \, d x + 2 i \, c\right )}\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) - 3}{12 \, {\left (a d e^{\left (8 i \, d x + 8 i \, c\right )} + 3 \, a d e^{\left (6 i \, d x + 6 i \, c\right )} + 3 \, a d e^{\left (4 i \, d x + 4 i \, c\right )} + a d e^{\left (2 i \, d x + 2 i \, c\right )}\right )}} \]
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Time = 0.28 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.81 \[ \int \frac {\tan ^5(c+d x)}{a+i a \tan (c+d x)} \, dx=\frac {- 12 e^{4 i c} e^{4 i d x} - 18 e^{2 i c} e^{2 i d x} - 14}{3 a d e^{6 i c} e^{6 i d x} + 9 a d e^{4 i c} e^{4 i d x} + 9 a d e^{2 i c} e^{2 i d x} + 3 a d} + \begin {cases} - \frac {e^{- 2 i c} e^{- 2 i d x}}{4 a d} & \text {for}\: a d e^{2 i c} \neq 0 \\x \left (\frac {\left (- 9 i e^{2 i c} + i\right ) e^{- 2 i c}}{2 a} + \frac {9 i}{2 a}\right ) & \text {otherwise} \end {cases} - \frac {9 i x}{2 a} + \frac {2 \log {\left (e^{2 i d x} + e^{- 2 i c} \right )}}{a d} \]
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Exception generated. \[ \int \frac {\tan ^5(c+d x)}{a+i a \tan (c+d x)} \, dx=\text {Exception raised: RuntimeError} \]
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Time = 1.61 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.94 \[ \int \frac {\tan ^5(c+d x)}{a+i a \tan (c+d x)} \, dx=\frac {\frac {3 \, \log \left (\tan \left (d x + c\right ) + i\right )}{a} - \frac {27 \, \log \left (\tan \left (d x + c\right ) - i\right )}{a} + \frac {3 \, {\left (9 \, \tan \left (d x + c\right ) - 7 i\right )}}{a {\left (\tan \left (d x + c\right ) - i\right )}} - \frac {2 \, {\left (2 i \, a^{2} \tan \left (d x + c\right )^{3} - 3 \, a^{2} \tan \left (d x + c\right )^{2} - 12 i \, a^{2} \tan \left (d x + c\right )\right )}}{a^{3}}}{12 \, d} \]
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Time = 4.27 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.97 \[ \int \frac {\tan ^5(c+d x)}{a+i a \tan (c+d x)} \, dx=\frac {{\mathrm {tan}\left (c+d\,x\right )}^2}{2\,a\,d}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )}{4\,a\,d}+\frac {\mathrm {tan}\left (c+d\,x\right )\,2{}\mathrm {i}}{a\,d}-\frac {1}{2\,a\,d\,\left (1+\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}-\frac {9\,\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )}{4\,a\,d}-\frac {{\mathrm {tan}\left (c+d\,x\right )}^3\,1{}\mathrm {i}}{3\,a\,d} \]
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