Optimal. Leaf size=22 \[ -\log (\cos (x))-\frac {\tan ^2(x)}{2}+\frac {\tan ^4(x)}{4} \]
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Rubi [A]
time = 0.01, antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3554, 3556}
\begin {gather*} \frac {\tan ^4(x)}{4}-\frac {\tan ^2(x)}{2}-\log (\cos (x)) \end {gather*}
Antiderivative was successfully verified.
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Rule 3554
Rule 3556
Rubi steps
\begin {align*} \int \tan ^5(x) \, dx &=\frac {\tan ^4(x)}{4}-\int \tan ^3(x) \, dx\\ &=-\frac {1}{2} \tan ^2(x)+\frac {\tan ^4(x)}{4}+\int \tan (x) \, dx\\ &=-\log (\cos (x))-\frac {\tan ^2(x)}{2}+\frac {\tan ^4(x)}{4}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 20, normalized size = 0.91 \begin {gather*} -\log (\cos (x))-\sec ^2(x)+\frac {\sec ^4(x)}{4} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.02, size = 23, normalized size = 1.05
method | result | size |
derivativedivides | \(\frac {\left (\tan ^{4}\left (x \right )\right )}{4}-\frac {\left (\tan ^{2}\left (x \right )\right )}{2}+\frac {\ln \left (1+\tan ^{2}\left (x \right )\right )}{2}\) | \(23\) |
default | \(\frac {\left (\tan ^{4}\left (x \right )\right )}{4}-\frac {\left (\tan ^{2}\left (x \right )\right )}{2}+\frac {\ln \left (1+\tan ^{2}\left (x \right )\right )}{2}\) | \(23\) |
norman | \(\frac {\left (\tan ^{4}\left (x \right )\right )}{4}-\frac {\left (\tan ^{2}\left (x \right )\right )}{2}+\frac {\ln \left (1+\tan ^{2}\left (x \right )\right )}{2}\) | \(23\) |
risch | \(i x -\frac {4 \left ({\mathrm e}^{6 i x}+{\mathrm e}^{4 i x}+{\mathrm e}^{2 i x}\right )}{\left ({\mathrm e}^{2 i x}+1\right )^{4}}-\ln \left ({\mathrm e}^{2 i x}+1\right )\) | \(43\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 1.97, size = 34, normalized size = 1.55 \begin {gather*} \frac {4 \, \sin \left (x\right )^{2} - 3}{4 \, {\left (\sin \left (x\right )^{4} - 2 \, \sin \left (x\right )^{2} + 1\right )}} - \frac {1}{2} \, \log \left (\sin \left (x\right )^{2} - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.61, size = 24, normalized size = 1.09 \begin {gather*} \frac {1}{4} \, \tan \left (x\right )^{4} - \frac {1}{2} \, \tan \left (x\right )^{2} - \frac {1}{2} \, \log \left (\frac {1}{\tan \left (x\right )^{2} + 1}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.04, size = 20, normalized size = 0.91 \begin {gather*} - \frac {4 \cos ^{2}{\left (x \right )} - 1}{4 \cos ^{4}{\left (x \right )}} - \log {\left (\cos {\left (x \right )} \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.79, size = 22, normalized size = 1.00 \begin {gather*} \frac {1}{4} \, \tan \left (x\right )^{4} - \frac {1}{2} \, \tan \left (x\right )^{2} + \frac {1}{2} \, \log \left (\tan \left (x\right )^{2} + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.03, size = 18, normalized size = 0.82 \begin {gather*} \frac {{\mathrm {tan}\left (x\right )}^4}{4}-\frac {{\mathrm {tan}\left (x\right )}^2}{2}-\ln \left (\cos \left (x\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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