Optimal. Leaf size=14 \[ x+\frac {\tan ^3(x)}{3}-\tan (x) \]
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Rubi [A] time = 0.01, antiderivative size = 14, 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 = {3473, 8} \[ x+\frac {\tan ^3(x)}{3}-\tan (x) \]
Antiderivative was successfully verified.
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Rule 8
Rule 3473
Rubi steps
\begin {align*} \int \tan ^4(x) \, dx &=\frac {\tan ^3(x)}{3}-\int \tan ^2(x) \, dx\\ &=-\tan (x)+\frac {\tan ^3(x)}{3}+\int 1 \, dx\\ &=x-\tan (x)+\frac {\tan ^3(x)}{3}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 18, normalized size = 1.29 \[ x-\frac {4 \tan (x)}{3}+\frac {1}{3} \tan (x) \sec ^2(x) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 12, normalized size = 0.86 \[ \frac {1}{3} \, \tan \relax (x)^{3} + x - \tan \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.01, size = 12, normalized size = 0.86 \[ \frac {1}{3} \, \tan \relax (x)^{3} + x - \tan \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 13, normalized size = 0.93 \[ \frac {\left (\tan ^{3}\relax (x )\right )}{3}+x -\tan \relax (x ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.27, size = 12, normalized size = 0.86 \[ \frac {1}{3} \, \tan \relax (x)^{3} + x - \tan \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.03, size = 12, normalized size = 0.86 \[ \frac {{\mathrm {tan}\relax (x)}^3}{3}-\mathrm {tan}\relax (x)+x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.07, size = 19, normalized size = 1.36 \[ x + \frac {\sin ^{3}{\relax (x )}}{3 \cos ^{3}{\relax (x )}} - \frac {\sin {\relax (x )}}{\cos {\relax (x )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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