Optimal. Leaf size=12 \[ \log (\cos (x))+\frac {\tan ^2(x)}{2} \]
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Rubi [A]
time = 0.00, antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps
used = 2, 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 ^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 ^3(x) \, dx &=\frac {\tan ^2(x)}{2}-\int \tan (x) \, dx\\ &=\log (\cos (x))+\frac {\tan ^2(x)}{2}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 12, normalized size = 1.00 \begin {gather*} \log (\cos (x))+\frac {\sec ^2(x)}{2} \end {gather*}
Antiderivative was successfully verified.
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Mathics [A]
time = 1.68, size = 10, normalized size = 0.83 \begin {gather*} \text {Log}\left [\text {Cos}\left [x\right ]\right ]+\frac {1}{2 \text {Cos}\left [x\right ]^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.02, size = 17, normalized size = 1.42
method | result | size |
derivativedivides | \(\frac {\left (\tan ^{2}\left (x \right )\right )}{2}-\frac {\ln \left (1+\tan ^{2}\left (x \right )\right )}{2}\) | \(17\) |
default | \(\frac {\left (\tan ^{2}\left (x \right )\right )}{2}-\frac {\ln \left (1+\tan ^{2}\left (x \right )\right )}{2}\) | \(17\) |
norman | \(\frac {\left (\tan ^{2}\left (x \right )\right )}{2}-\frac {\ln \left (1+\tan ^{2}\left (x \right )\right )}{2}\) | \(17\) |
risch | \(-i x +\frac {2 \,{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}+1\right )^{2}}+\ln \left ({\mathrm e}^{2 i x}+1\right )\) | \(30\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 20, normalized size = 1.67 \begin {gather*} -\frac {1}{2 \, {\left (\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 = 0.34, size = 18, normalized size = 1.50 \begin {gather*} \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 = 0.05, size = 12, normalized size = 1.00 \begin {gather*} \log {\left (\cos {\left (x \right )} \right )} + \frac {1}{2 \cos ^{2}{\left (x \right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.00, size = 16, normalized size = 1.33 \begin {gather*} \frac {\tan ^{2}x-\ln \left (\tan ^{2}x+1\right )}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.02, size = 16, normalized size = 1.33 \begin {gather*} \ln \left (\cos \left (x\right )\right )-\frac {{\cos \left (x\right )}^2-1}{2\,{\cos \left (x\right )}^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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