Optimal. Leaf size=8 \[ \frac {\tan ^4(x)}{4} \]
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Rubi [A] time = 0.02, antiderivative size = 8, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 2, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {3473, 3475} \[ \frac {\tan ^4(x)}{4} \]
Antiderivative was successfully verified.
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Rule 3473
Rule 3475
Rubi steps
\begin {align*} \int \left (\tan ^3(x)+\tan ^5(x)\right ) \, dx &=\int \tan ^3(x) \, dx+\int \tan ^5(x) \, dx\\ &=\frac {\tan ^2(x)}{2}+\frac {\tan ^4(x)}{4}-\int \tan (x) \, dx-\int \tan ^3(x) \, dx\\ &=\log (\cos (x))+\frac {\tan ^4(x)}{4}+\int \tan (x) \, dx\\ &=\frac {\tan ^4(x)}{4}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 8, normalized size = 1.00 \[ \frac {\tan ^4(x)}{4} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.47, size = 6, normalized size = 0.75 \[ \frac {1}{4} \, \tan \relax (x)^{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 6, normalized size = 0.75 \[ \frac {1}{4} \, \tan \relax (x)^{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 7, normalized size = 0.88 \[ \frac {\left (\tan ^{4}\relax (x )\right )}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.32, size = 35, normalized size = 4.38 \[ \frac {4 \, \sin \relax (x)^{2} - 3}{4 \, {\left (\sin \relax (x)^{4} - 2 \, \sin \relax (x)^{2} + 1\right )}} - \frac {1}{2 \, {\left (\sin \relax (x)^{2} - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.95, size = 6, normalized size = 0.75 \[ \frac {{\mathrm {tan}\relax (x)}^4}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.12, size = 22, normalized size = 2.75 \[ - \frac {4 \cos ^{2}{\relax (x )} - 1}{4 \cos ^{4}{\relax (x )}} + \frac {1}{2 \cos ^{2}{\relax (x )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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