Integrand size = 8, antiderivative size = 37 \[ \int \frac {1}{1+\tan ^3(x)} \, dx=\frac {x}{2}-\frac {1}{2} \log (\cos (x))+\frac {1}{6} \log (1+\tan (x))-\frac {1}{3} \log \left (1-\tan (x)+\tan ^2(x)\right ) \]
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Time = 0.06 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {3742, 2099, 649, 209, 266, 642} \[ \int \frac {1}{1+\tan ^3(x)} \, dx=\frac {x}{2}-\frac {1}{3} \log \left (\tan ^2(x)-\tan (x)+1\right )+\frac {1}{6} \log (\tan (x)+1)-\frac {1}{2} \log (\cos (x)) \]
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Rule 209
Rule 266
Rule 642
Rule 649
Rule 2099
Rule 3742
Rubi steps \begin{align*} \text {integral}& = \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*}
Result contains complex when optimal does not.
Time = 0.03 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.54 \[ \int \frac {1}{1+\tan ^3(x)} \, dx=\left (\frac {1}{4}-\frac {i}{4}\right ) \log (i-\tan (x))+\left (\frac {1}{4}+\frac {i}{4}\right ) \log (i+\tan (x))+\frac {1}{6} \log (1+\tan (x))-\frac {1}{3} \log \left (1-\tan (x)+\tan ^2(x)\right ) \]
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Time = 0.07 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.92
method | result | size |
norman | \(\frac {x}{2}+\frac {\ln \left (1+\tan \left (x \right )\right )}{6}+\frac {\ln \left (1+\tan \left (x \right )^{2}\right )}{4}-\frac {\ln \left (1-\tan \left (x \right )+\tan \left (x \right )^{2}\right )}{3}\) | \(34\) |
parallelrisch | \(\frac {x}{2}+\frac {\ln \left (1+\tan \left (x \right )\right )}{6}+\frac {\ln \left (1+\tan \left (x \right )^{2}\right )}{4}-\frac {\ln \left (1-\tan \left (x \right )+\tan \left (x \right )^{2}\right )}{3}\) | \(34\) |
derivativedivides | \(\frac {\ln \left (1+\tan \left (x \right )^{2}\right )}{4}+\frac {\arctan \left (\tan \left (x \right )\right )}{2}-\frac {\ln \left (1-\tan \left (x \right )+\tan \left (x \right )^{2}\right )}{3}+\frac {\ln \left (1+\tan \left (x \right )\right )}{6}\) | \(36\) |
default | \(\frac {\ln \left (1+\tan \left (x \right )^{2}\right )}{4}+\frac {\arctan \left (\tan \left (x \right )\right )}{2}-\frac {\ln \left (1-\tan \left (x \right )+\tan \left (x \right )^{2}\right )}{3}+\frac {\ln \left (1+\tan \left (x \right )\right )}{6}\) | \(36\) |
risch | \(\frac {x}{2}+\frac {i x}{2}+\frac {\ln \left ({\mathrm e}^{2 i x}+i\right )}{6}-\frac {\ln \left ({\mathrm e}^{4 i x}-4 i {\mathrm e}^{2 i x}-1\right )}{3}\) | \(38\) |
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Time = 0.26 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.30 \[ \int \frac {1}{1+\tan ^3(x)} \, dx=\frac {1}{2} \, x + \frac {1}{12} \, \log \left (\frac {\tan \left (x\right )^{2} + 2 \, \tan \left (x\right ) + 1}{\tan \left (x\right )^{2} + 1}\right ) - \frac {1}{3} \, \log \left (\frac {\tan \left (x\right )^{2} - \tan \left (x\right ) + 1}{\tan \left (x\right )^{2} + 1}\right ) \]
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Time = 0.09 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.92 \[ \int \frac {1}{1+\tan ^3(x)} \, dx=\frac {x}{2} + \frac {\log {\left (\tan {\left (x \right )} + 1 \right )}}{6} + \frac {\log {\left (\tan ^{2}{\left (x \right )} + 1 \right )}}{4} - \frac {\log {\left (\tan ^{2}{\left (x \right )} - \tan {\left (x \right )} + 1 \right )}}{3} \]
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Time = 0.29 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.89 \[ \int \frac {1}{1+\tan ^3(x)} \, dx=\frac {1}{2} \, x - \frac {1}{3} \, \log \left (\tan \left (x\right )^{2} - \tan \left (x\right ) + 1\right ) + \frac {1}{4} \, \log \left (\tan \left (x\right )^{2} + 1\right ) + \frac {1}{6} \, \log \left (\tan \left (x\right ) + 1\right ) \]
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Time = 0.26 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.92 \[ \int \frac {1}{1+\tan ^3(x)} \, dx=\frac {1}{2} \, x - \frac {1}{3} \, \log \left (\tan \left (x\right )^{2} - \tan \left (x\right ) + 1\right ) + \frac {1}{4} \, \log \left (\tan \left (x\right )^{2} + 1\right ) + \frac {1}{6} \, \log \left ({\left | \tan \left (x\right ) + 1 \right |}\right ) \]
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Time = 11.99 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.11 \[ \int \frac {1}{1+\tan ^3(x)} \, dx=\frac {\ln \left (\mathrm {tan}\left (x\right )+1\right )}{6}-\frac {\ln \left ({\mathrm {tan}\left (x\right )}^2-\mathrm {tan}\left (x\right )+1\right )}{3}+\ln \left (\mathrm {tan}\left (x\right )-\mathrm {i}\right )\,\left (\frac {1}{4}-\frac {1}{4}{}\mathrm {i}\right )+\ln \left (\mathrm {tan}\left (x\right )+1{}\mathrm {i}\right )\,\left (\frac {1}{4}+\frac {1}{4}{}\mathrm {i}\right ) \]
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