Integrand size = 8, antiderivative size = 43 \[ \int \tan ^5(c+d x) \, dx=-\frac {\log (\cos (c+d x))}{d}-\frac {\tan ^2(c+d x)}{2 d}+\frac {\tan ^4(c+d x)}{4 d} \]
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Time = 0.03 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3554, 3556} \[ \int \tan ^5(c+d x) \, dx=\frac {\tan ^4(c+d x)}{4 d}-\frac {\tan ^2(c+d x)}{2 d}-\frac {\log (\cos (c+d x))}{d} \]
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Rule 3554
Rule 3556
Rubi steps \begin{align*} \text {integral}& = \frac {\tan ^4(c+d x)}{4 d}-\int \tan ^3(c+d x) \, dx \\ & = -\frac {\tan ^2(c+d x)}{2 d}+\frac {\tan ^4(c+d x)}{4 d}+\int \tan (c+d x) \, dx \\ & = -\frac {\log (\cos (c+d x))}{d}-\frac {\tan ^2(c+d x)}{2 d}+\frac {\tan ^4(c+d x)}{4 d} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.86 \[ \int \tan ^5(c+d x) \, dx=-\frac {4 \log (\cos (c+d x))+2 \tan ^2(c+d x)-\tan ^4(c+d x)}{4 d} \]
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Time = 0.09 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.88
method | result | size |
parallelrisch | \(\frac {\tan ^{4}\left (d x +c \right )-2 \left (\tan ^{2}\left (d x +c \right )\right )+2 \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{4 d}\) | \(38\) |
derivativedivides | \(\frac {\frac {\left (\tan ^{4}\left (d x +c \right )\right )}{4}-\frac {\left (\tan ^{2}\left (d x +c \right )\right )}{2}+\frac {\ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}}{d}\) | \(39\) |
default | \(\frac {\frac {\left (\tan ^{4}\left (d x +c \right )\right )}{4}-\frac {\left (\tan ^{2}\left (d x +c \right )\right )}{2}+\frac {\ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}}{d}\) | \(39\) |
norman | \(-\frac {\tan ^{2}\left (d x +c \right )}{2 d}+\frac {\tan ^{4}\left (d x +c \right )}{4 d}+\frac {\ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2 d}\) | \(44\) |
risch | \(i x +\frac {2 i c}{d}-\frac {4 \left ({\mathrm e}^{6 i \left (d x +c \right )}+{\mathrm e}^{4 i \left (d x +c \right )}+{\mathrm e}^{2 i \left (d x +c \right )}\right )}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{4}}-\frac {\ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{d}\) | \(76\) |
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Time = 0.23 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.91 \[ \int \tan ^5(c+d x) \, dx=\frac {\tan \left (d x + c\right )^{4} - 2 \, \tan \left (d x + c\right )^{2} - 2 \, \log \left (\frac {1}{\tan \left (d x + c\right )^{2} + 1}\right )}{4 \, d} \]
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Time = 0.12 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.02 \[ \int \tan ^5(c+d x) \, dx=\begin {cases} \frac {\log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} + \frac {\tan ^{4}{\left (c + d x \right )}}{4 d} - \frac {\tan ^{2}{\left (c + d x \right )}}{2 d} & \text {for}\: d \neq 0 \\x \tan ^{5}{\left (c \right )} & \text {otherwise} \end {cases} \]
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Time = 0.32 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.26 \[ \int \tan ^5(c+d x) \, dx=\frac {\frac {4 \, \sin \left (d x + c\right )^{2} - 3}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - 2 \, \log \left (\sin \left (d x + c\right )^{2} - 1\right )}{4 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 462 vs. \(2 (39) = 78\).
Time = 1.37 (sec) , antiderivative size = 462, normalized size of antiderivative = 10.74 \[ \int \tan ^5(c+d x) \, dx=-\frac {2 \, \log \left (\frac {4 \, {\left (\tan \left (d x\right )^{2} \tan \left (c\right )^{2} - 2 \, \tan \left (d x\right ) \tan \left (c\right ) + 1\right )}}{\tan \left (d x\right )^{2} \tan \left (c\right )^{2} + \tan \left (d x\right )^{2} + \tan \left (c\right )^{2} + 1}\right ) \tan \left (d x\right )^{4} \tan \left (c\right )^{4} + 3 \, \tan \left (d x\right )^{4} \tan \left (c\right )^{4} - 8 \, \log \left (\frac {4 \, {\left (\tan \left (d x\right )^{2} \tan \left (c\right )^{2} - 2 \, \tan \left (d x\right ) \tan \left (c\right ) + 1\right )}}{\tan \left (d x\right )^{2} \tan \left (c\right )^{2} + \tan \left (d x\right )^{2} + \tan \left (c\right )^{2} + 1}\right ) \tan \left (d x\right )^{3} \tan \left (c\right )^{3} + 2 \, \tan \left (d x\right )^{4} \tan \left (c\right )^{2} - 8 \, \tan \left (d x\right )^{3} \tan \left (c\right )^{3} + 2 \, \tan \left (d x\right )^{2} \tan \left (c\right )^{4} + 12 \, \log \left (\frac {4 \, {\left (\tan \left (d x\right )^{2} \tan \left (c\right )^{2} - 2 \, \tan \left (d x\right ) \tan \left (c\right ) + 1\right )}}{\tan \left (d x\right )^{2} \tan \left (c\right )^{2} + \tan \left (d x\right )^{2} + \tan \left (c\right )^{2} + 1}\right ) \tan \left (d x\right )^{2} \tan \left (c\right )^{2} - \tan \left (d x\right )^{4} - 8 \, \tan \left (d x\right )^{3} \tan \left (c\right ) + 4 \, \tan \left (d x\right )^{2} \tan \left (c\right )^{2} - 8 \, \tan \left (d x\right ) \tan \left (c\right )^{3} - \tan \left (c\right )^{4} - 8 \, \log \left (\frac {4 \, {\left (\tan \left (d x\right )^{2} \tan \left (c\right )^{2} - 2 \, \tan \left (d x\right ) \tan \left (c\right ) + 1\right )}}{\tan \left (d x\right )^{2} \tan \left (c\right )^{2} + \tan \left (d x\right )^{2} + \tan \left (c\right )^{2} + 1}\right ) \tan \left (d x\right ) \tan \left (c\right ) + 2 \, \tan \left (d x\right )^{2} - 8 \, \tan \left (d x\right ) \tan \left (c\right ) + 2 \, \tan \left (c\right )^{2} + 2 \, \log \left (\frac {4 \, {\left (\tan \left (d x\right )^{2} \tan \left (c\right )^{2} - 2 \, \tan \left (d x\right ) \tan \left (c\right ) + 1\right )}}{\tan \left (d x\right )^{2} \tan \left (c\right )^{2} + \tan \left (d x\right )^{2} + \tan \left (c\right )^{2} + 1}\right ) + 3}{4 \, {\left (d \tan \left (d x\right )^{4} \tan \left (c\right )^{4} - 4 \, d \tan \left (d x\right )^{3} \tan \left (c\right )^{3} + 6 \, d \tan \left (d x\right )^{2} \tan \left (c\right )^{2} - 4 \, d \tan \left (d x\right ) \tan \left (c\right ) + d\right )}} \]
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Time = 2.74 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.88 \[ \int \tan ^5(c+d x) \, dx=\frac {\frac {\ln \left ({\mathrm {tan}\left (c+d\,x\right )}^2+1\right )}{2}-\frac {{\mathrm {tan}\left (c+d\,x\right )}^2}{2}+\frac {{\mathrm {tan}\left (c+d\,x\right )}^4}{4}}{d} \]
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