Optimal. Leaf size=37 \[ \frac {x}{2}-\frac {1}{2} \log (\cos (x))+\frac {1}{6} \log (1+\tan (x))-\frac {1}{3} \log \left (1-\tan (x)+\tan ^2(x)\right ) \]
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Rubi [A]
time = 0.04, antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 6, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {3742, 2099, 649,
209, 266, 642} \begin {gather*} \frac {x}{2}-\frac {1}{3} \log \left (\tan ^2(x)-\tan (x)+1\right )+\frac {1}{6} \log (\tan (x)+1)-\frac {1}{2} \log (\cos (x)) \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 266
Rule 642
Rule 649
Rule 2099
Rule 3742
Rubi steps
\begin {align*} \int \frac {1}{1+\tan ^3(x)} \, dx &=\text {Subst}\left (\int \frac {1}{\left (1+x^2\right ) \left (1+x^3\right )} \, dx,x,\tan (x)\right )\\ &=\text {Subst}\left (\int \left (\frac {1}{6 (1+x)}+\frac {1+x}{2 \left (1+x^2\right )}+\frac {1-2 x}{3 \left (1-x+x^2\right )}\right ) \, dx,x,\tan (x)\right )\\ &=\frac {1}{6} \log (1+\tan (x))+\frac {1}{3} \text {Subst}\left (\int \frac {1-2 x}{1-x+x^2} \, dx,x,\tan (x)\right )+\frac {1}{2} \text {Subst}\left (\int \frac {1+x}{1+x^2} \, dx,x,\tan (x)\right )\\ &=\frac {1}{6} \log (1+\tan (x))-\frac {1}{3} \log \left (1-\tan (x)+\tan ^2(x)\right )+\frac {1}{2} \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\tan (x)\right )+\frac {1}{2} \text {Subst}\left (\int \frac {x}{1+x^2} \, dx,x,\tan (x)\right )\\ &=\frac {x}{2}-\frac {1}{2} \log (\cos (x))+\frac {1}{6} \log (1+\tan (x))-\frac {1}{3} \log \left (1-\tan (x)+\tan ^2(x)\right )\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 43, normalized size = 1.16 \begin {gather*} \frac {1}{2} \text {ArcTan}(\tan (x))+\frac {1}{6} \log (1+\tan (x))+\frac {1}{4} \log \left (1+\tan ^2(x)\right )-\frac {1}{3} \log \left (1-\tan (x)+\tan ^2(x)\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.07, size = 36, normalized size = 0.97
method | result | size |
norman | \(\frac {x}{2}+\frac {\ln \left (1+\tan \left (x \right )\right )}{6}+\frac {\ln \left (1+\tan ^{2}\left (x \right )\right )}{4}-\frac {\ln \left (1-\tan \left (x \right )+\tan ^{2}\left (x \right )\right )}{3}\) | \(34\) |
derivativedivides | \(-\frac {\ln \left (1-\tan \left (x \right )+\tan ^{2}\left (x \right )\right )}{3}+\frac {\ln \left (1+\tan ^{2}\left (x \right )\right )}{4}+\frac {\arctan \left (\tan \left (x \right )\right )}{2}+\frac {\ln \left (1+\tan \left (x \right )\right )}{6}\) | \(36\) |
default | \(-\frac {\ln \left (1-\tan \left (x \right )+\tan ^{2}\left (x \right )\right )}{3}+\frac {\ln \left (1+\tan ^{2}\left (x \right )\right )}{4}+\frac {\arctan \left (\tan \left (x \right )\right )}{2}+\frac {\ln \left (1+\tan \left (x \right )\right )}{6}\) | \(36\) |
risch | \(\frac {x}{2}+\frac {i x}{2}+\frac {\ln \left ({\mathrm e}^{2 i x}+i\right )}{6}-\frac {\ln \left ({\mathrm e}^{4 i x}-4 i {\mathrm e}^{2 i x}-1\right )}{3}\) | \(38\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 33, normalized size = 0.89 \begin {gather*} \frac {1}{2} \, x - \frac {1}{3} \, \log \left (\tan \left (x\right )^{2} - \tan \left (x\right ) + 1\right ) + \frac {1}{4} \, \log \left (\tan \left (x\right )^{2} + 1\right ) + \frac {1}{6} \, \log \left (\tan \left (x\right ) + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.00, size = 48, normalized size = 1.30 \begin {gather*} \frac {1}{2} \, x + \frac {1}{12} \, \log \left (\frac {\tan \left (x\right )^{2} + 2 \, \tan \left (x\right ) + 1}{\tan \left (x\right )^{2} + 1}\right ) - \frac {1}{3} \, \log \left (\frac {\tan \left (x\right )^{2} - \tan \left (x\right ) + 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 = 0.06, size = 34, normalized size = 0.92 \begin {gather*} \frac {x}{2} + \frac {\log {\left (\tan {\left (x \right )} + 1 \right )}}{6} + \frac {\log {\left (\tan ^{2}{\left (x \right )} + 1 \right )}}{4} - \frac {\log {\left (\tan ^{2}{\left (x \right )} - \tan {\left (x \right )} + 1 \right )}}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.42, size = 34, normalized size = 0.92 \begin {gather*} \frac {1}{2} \, x - \frac {1}{3} \, \log \left (\tan \left (x\right )^{2} - \tan \left (x\right ) + 1\right ) + \frac {1}{4} \, \log \left (\tan \left (x\right )^{2} + 1\right ) + \frac {1}{6} \, \log \left ({\left | \tan \left (x\right ) + 1 \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 11.62, size = 41, normalized size = 1.11 \begin {gather*} \frac {\ln \left (\mathrm {tan}\left (x\right )+1\right )}{6}-\frac {\ln \left ({\mathrm {tan}\left (x\right )}^2-\mathrm {tan}\left (x\right )+1\right )}{3}+\ln \left (\mathrm {tan}\left (x\right )-\mathrm {i}\right )\,\left (\frac {1}{4}-\frac {1}{4}{}\mathrm {i}\right )+\ln \left (\mathrm {tan}\left (x\right )+1{}\mathrm {i}\right )\,\left (\frac {1}{4}+\frac {1}{4}{}\mathrm {i}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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