Optimal. Leaf size=43 \[ \frac {\tan ^4(c+d x)}{4 d}-\frac {\tan ^2(c+d x)}{2 d}-\frac {\log (\cos (c+d x))}{d} \]
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Rubi [A] time = 0.02, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3473, 3475} \[ \frac {\tan ^4(c+d x)}{4 d}-\frac {\tan ^2(c+d x)}{2 d}-\frac {\log (\cos (c+d x))}{d} \]
Antiderivative was successfully verified.
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Rule 3473
Rule 3475
Rubi steps
\begin {align*} \int \tan ^5(c+d x) \, dx &=\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*}
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Mathematica [A] time = 0.05, size = 37, normalized size = 0.86 \[ -\frac {-\tan ^4(c+d x)+2 \tan ^2(c+d x)+4 \log (\cos (c+d x))}{4 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.55, size = 39, normalized size = 0.91 \[ \frac {\tan \left (d x + c\right )^{4} - 2 \, \tan \left (d x + c\right )^{2} - 2 \, \log \left (\frac {1}{\tan \left (d x + c\right )^{2} + 1}\right )}{4 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 6.28, size = 512, normalized size = 11.91 \[ -\frac {2 \, \log \left (\frac {4 \, {\left (\tan \left (d x\right )^{4} \tan \relax (c)^{2} - 2 \, \tan \left (d x\right )^{3} \tan \relax (c) + \tan \left (d x\right )^{2} \tan \relax (c)^{2} + \tan \left (d x\right )^{2} - 2 \, \tan \left (d x\right ) \tan \relax (c) + 1\right )}}{\tan \relax (c)^{2} + 1}\right ) \tan \left (d x\right )^{4} \tan \relax (c)^{4} + 3 \, \tan \left (d x\right )^{4} \tan \relax (c)^{4} - 8 \, \log \left (\frac {4 \, {\left (\tan \left (d x\right )^{4} \tan \relax (c)^{2} - 2 \, \tan \left (d x\right )^{3} \tan \relax (c) + \tan \left (d x\right )^{2} \tan \relax (c)^{2} + \tan \left (d x\right )^{2} - 2 \, \tan \left (d x\right ) \tan \relax (c) + 1\right )}}{\tan \relax (c)^{2} + 1}\right ) \tan \left (d x\right )^{3} \tan \relax (c)^{3} + 2 \, \tan \left (d x\right )^{4} \tan \relax (c)^{2} - 8 \, \tan \left (d x\right )^{3} \tan \relax (c)^{3} + 2 \, \tan \left (d x\right )^{2} \tan \relax (c)^{4} + 12 \, \log \left (\frac {4 \, {\left (\tan \left (d x\right )^{4} \tan \relax (c)^{2} - 2 \, \tan \left (d x\right )^{3} \tan \relax (c) + \tan \left (d x\right )^{2} \tan \relax (c)^{2} + \tan \left (d x\right )^{2} - 2 \, \tan \left (d x\right ) \tan \relax (c) + 1\right )}}{\tan \relax (c)^{2} + 1}\right ) \tan \left (d x\right )^{2} \tan \relax (c)^{2} - \tan \left (d x\right )^{4} - 8 \, \tan \left (d x\right )^{3} \tan \relax (c) + 4 \, \tan \left (d x\right )^{2} \tan \relax (c)^{2} - 8 \, \tan \left (d x\right ) \tan \relax (c)^{3} - \tan \relax (c)^{4} - 8 \, \log \left (\frac {4 \, {\left (\tan \left (d x\right )^{4} \tan \relax (c)^{2} - 2 \, \tan \left (d x\right )^{3} \tan \relax (c) + \tan \left (d x\right )^{2} \tan \relax (c)^{2} + \tan \left (d x\right )^{2} - 2 \, \tan \left (d x\right ) \tan \relax (c) + 1\right )}}{\tan \relax (c)^{2} + 1}\right ) \tan \left (d x\right ) \tan \relax (c) + 2 \, \tan \left (d x\right )^{2} - 8 \, \tan \left (d x\right ) \tan \relax (c) + 2 \, \tan \relax (c)^{2} + 2 \, \log \left (\frac {4 \, {\left (\tan \left (d x\right )^{4} \tan \relax (c)^{2} - 2 \, \tan \left (d x\right )^{3} \tan \relax (c) + \tan \left (d x\right )^{2} \tan \relax (c)^{2} + \tan \left (d x\right )^{2} - 2 \, \tan \left (d x\right ) \tan \relax (c) + 1\right )}}{\tan \relax (c)^{2} + 1}\right ) + 3}{4 \, {\left (d \tan \left (d x\right )^{4} \tan \relax (c)^{4} - 4 \, d \tan \left (d x\right )^{3} \tan \relax (c)^{3} + 6 \, d \tan \left (d x\right )^{2} \tan \relax (c)^{2} - 4 \, d \tan \left (d x\right ) \tan \relax (c) + d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 44, normalized size = 1.02 \[ \frac {\tan ^{4}\left (d x +c \right )}{4 d}-\frac {\tan ^{2}\left (d x +c \right )}{2 d}+\frac {\ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.48, size = 54, normalized size = 1.26 \[ \frac {\frac {4 \, \sin \left (d x + c\right )^{2} - 3}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - 2 \, \log \left (\sin \left (d x + c\right )^{2} - 1\right )}{4 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.50, size = 38, normalized size = 0.88 \[ \frac {\frac {\ln \left ({\mathrm {tan}\left (c+d\,x\right )}^2+1\right )}{2}-\frac {{\mathrm {tan}\left (c+d\,x\right )}^2}{2}+\frac {{\mathrm {tan}\left (c+d\,x\right )}^4}{4}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.34, size = 44, normalized size = 1.02 \[ \begin {cases} \frac {\log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} + \frac {\tan ^{4}{\left (c + d x \right )}}{4 d} - \frac {\tan ^{2}{\left (c + d x \right )}}{2 d} & \text {for}\: d \neq 0 \\x \tan ^{5}{\relax (c )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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