Optimal. Leaf size=14 \[ x-\tan (x)+\frac {\tan ^3(x)}{3} \]
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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 = {3554, 8}
\begin {gather*} x+\frac {\tan ^3(x)}{3}-\tan (x) \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 3554
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 \begin {gather*} x-\frac {4 \tan (x)}{3}+\frac {1}{3} \sec ^2(x) \tan (x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.01, size = 15, normalized size = 1.07
method | result | size |
norman | \(x -\tan \left (x \right )+\frac {\left (\tan ^{3}\left (x \right )\right )}{3}\) | \(13\) |
derivativedivides | \(\frac {\left (\tan ^{3}\left (x \right )\right )}{3}-\tan \left (x \right )+\arctan \left (\tan \left (x \right )\right )\) | \(15\) |
default | \(\frac {\left (\tan ^{3}\left (x \right )\right )}{3}-\tan \left (x \right )+\arctan \left (\tan \left (x \right )\right )\) | \(15\) |
risch | \(x -\frac {4 i \left (3 \,{\mathrm e}^{4 i x}+3 \,{\mathrm e}^{2 i x}+2\right )}{3 \left ({\mathrm e}^{2 i x}+1\right )^{3}}\) | \(31\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 3.18, size = 12, normalized size = 0.86 \begin {gather*} \frac {1}{3} \, \tan \left (x\right )^{3} + x - \tan \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.08, size = 12, normalized size = 0.86 \begin {gather*} \frac {1}{3} \, \tan \left (x\right )^{3} + x - \tan \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.01, size = 19, normalized size = 1.36 \begin {gather*} x + \frac {\sin ^{3}{\left (x \right )}}{3 \cos ^{3}{\left (x \right )}} - \frac {\sin {\left (x \right )}}{\cos {\left (x \right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.44, size = 12, normalized size = 0.86 \begin {gather*} \frac {1}{3} \, \tan \left (x\right )^{3} + x - \tan \left (x\right ) \end {gather*}
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 \begin {gather*} \frac {{\mathrm {tan}\left (x\right )}^3}{3}-\mathrm {tan}\left (x\right )+x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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