Integrand size = 8, antiderivative size = 57 \[ \int \frac {1}{\sqrt [3]{\tan (5 x)}} \, dx=-\frac {1}{10} \sqrt {3} \arctan \left (\frac {1-2 \tan ^{\frac {2}{3}}(5 x)}{\sqrt {3}}\right )+\frac {3}{20} \log \left (1+\tan ^{\frac {2}{3}}(5 x)\right )-\frac {1}{20} \log \left (1+\tan ^2(5 x)\right ) \]
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Time = 0.05 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.21, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.125, Rules used = {3557, 335, 281, 206, 31, 648, 632, 210, 642} \[ \int \frac {1}{\sqrt [3]{\tan (5 x)}} \, dx=-\frac {1}{10} \sqrt {3} \arctan \left (\frac {1-2 \tan ^{\frac {2}{3}}(5 x)}{\sqrt {3}}\right )+\frac {1}{10} \log \left (\tan ^{\frac {2}{3}}(5 x)+1\right )-\frac {1}{20} \log \left (\tan ^{\frac {4}{3}}(5 x)-\tan ^{\frac {2}{3}}(5 x)+1\right ) \]
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Rule 31
Rule 206
Rule 210
Rule 281
Rule 335
Rule 632
Rule 642
Rule 648
Rule 3557
Rubi steps \begin{align*} \text {integral}& = \frac {1}{5} \text {Subst}\left (\int \frac {1}{\sqrt [3]{x} \left (1+x^2\right )} \, dx,x,\tan (5 x)\right ) \\ & = \frac {3}{5} \text {Subst}\left (\int \frac {x}{1+x^6} \, dx,x,\sqrt [3]{\tan (5 x)}\right ) \\ & = \frac {3}{10} \text {Subst}\left (\int \frac {1}{1+x^3} \, dx,x,\tan ^{\frac {2}{3}}(5 x)\right ) \\ & = \frac {1}{10} \text {Subst}\left (\int \frac {1}{1+x} \, dx,x,\tan ^{\frac {2}{3}}(5 x)\right )+\frac {1}{10} \text {Subst}\left (\int \frac {2-x}{1-x+x^2} \, dx,x,\tan ^{\frac {2}{3}}(5 x)\right ) \\ & = \frac {1}{10} \log \left (1+\tan ^{\frac {2}{3}}(5 x)\right )-\frac {1}{20} \text {Subst}\left (\int \frac {-1+2 x}{1-x+x^2} \, dx,x,\tan ^{\frac {2}{3}}(5 x)\right )+\frac {3}{20} \text {Subst}\left (\int \frac {1}{1-x+x^2} \, dx,x,\tan ^{\frac {2}{3}}(5 x)\right ) \\ & = \frac {1}{10} \log \left (1+\tan ^{\frac {2}{3}}(5 x)\right )-\frac {1}{20} \log \left (1-\tan ^{\frac {2}{3}}(5 x)+\tan ^{\frac {4}{3}}(5 x)\right )-\frac {3}{10} \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,-1+2 \tan ^{\frac {2}{3}}(5 x)\right ) \\ & = -\frac {1}{10} \sqrt {3} \arctan \left (\frac {1-2 \tan ^{\frac {2}{3}}(5 x)}{\sqrt {3}}\right )+\frac {1}{10} \log \left (1+\tan ^{\frac {2}{3}}(5 x)\right )-\frac {1}{20} \log \left (1-\tan ^{\frac {2}{3}}(5 x)+\tan ^{\frac {4}{3}}(5 x)\right ) \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.21 \[ \int \frac {1}{\sqrt [3]{\tan (5 x)}} \, dx=\frac {1}{10} \sqrt {3} \arctan \left (\frac {-1+2 \tan ^{\frac {2}{3}}(5 x)}{\sqrt {3}}\right )+\frac {1}{10} \log \left (1+\tan ^{\frac {2}{3}}(5 x)\right )-\frac {1}{20} \log \left (1-\tan ^{\frac {2}{3}}(5 x)+\tan ^{\frac {4}{3}}(5 x)\right ) \]
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Time = 0.10 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.93
method | result | size |
derivativedivides | \(\frac {\ln \left (1+\tan ^{\frac {2}{3}}\left (5 x \right )\right )}{10}-\frac {\ln \left (\tan ^{\frac {4}{3}}\left (5 x \right )-\left (\tan ^{\frac {2}{3}}\left (5 x \right )\right )+1\right )}{20}+\frac {\sqrt {3}\, \arctan \left (\frac {\left (2 \left (\tan ^{\frac {2}{3}}\left (5 x \right )\right )-1\right ) \sqrt {3}}{3}\right )}{10}\) | \(53\) |
default | \(\frac {\ln \left (1+\tan ^{\frac {2}{3}}\left (5 x \right )\right )}{10}-\frac {\ln \left (\tan ^{\frac {4}{3}}\left (5 x \right )-\left (\tan ^{\frac {2}{3}}\left (5 x \right )\right )+1\right )}{20}+\frac {\sqrt {3}\, \arctan \left (\frac {\left (2 \left (\tan ^{\frac {2}{3}}\left (5 x \right )\right )-1\right ) \sqrt {3}}{3}\right )}{10}\) | \(53\) |
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Time = 0.26 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.95 \[ \int \frac {1}{\sqrt [3]{\tan (5 x)}} \, dx=\frac {1}{10} \, \sqrt {3} \arctan \left (\frac {2}{3} \, \sqrt {3} \tan \left (5 \, x\right )^{\frac {2}{3}} - \frac {1}{3} \, \sqrt {3}\right ) - \frac {1}{20} \, \log \left (\tan \left (5 \, x\right )^{\frac {4}{3}} - \tan \left (5 \, x\right )^{\frac {2}{3}} + 1\right ) + \frac {1}{10} \, \log \left (\tan \left (5 \, x\right )^{\frac {2}{3}} + 1\right ) \]
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\[ \int \frac {1}{\sqrt [3]{\tan (5 x)}} \, dx=\int \frac {1}{\sqrt [3]{\tan {\left (5 x \right )}}}\, dx \]
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Time = 0.27 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.91 \[ \int \frac {1}{\sqrt [3]{\tan (5 x)}} \, dx=\frac {1}{10} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, \tan \left (5 \, x\right )^{\frac {2}{3}} - 1\right )}\right ) - \frac {1}{20} \, \log \left (\tan \left (5 \, x\right )^{\frac {4}{3}} - \tan \left (5 \, x\right )^{\frac {2}{3}} + 1\right ) + \frac {1}{10} \, \log \left (\tan \left (5 \, x\right )^{\frac {2}{3}} + 1\right ) \]
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Time = 0.33 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.91 \[ \int \frac {1}{\sqrt [3]{\tan (5 x)}} \, dx=\frac {1}{10} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, \tan \left (5 \, x\right )^{\frac {2}{3}} - 1\right )}\right ) - \frac {1}{20} \, \log \left (\tan \left (5 \, x\right )^{\frac {4}{3}} - \tan \left (5 \, x\right )^{\frac {2}{3}} + 1\right ) + \frac {1}{10} \, \log \left (\tan \left (5 \, x\right )^{\frac {2}{3}} + 1\right ) \]
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Time = 0.75 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.18 \[ \int \frac {1}{\sqrt [3]{\tan (5 x)}} \, dx=\frac {\ln \left (81\,{\mathrm {tan}\left (5\,x\right )}^{2/3}+81\right )}{10}-\ln \left (81-162\,{\mathrm {tan}\left (5\,x\right )}^{2/3}+\sqrt {3}\,81{}\mathrm {i}\right )\,\left (\frac {1}{20}+\frac {\sqrt {3}\,1{}\mathrm {i}}{20}\right )+\ln \left (162\,{\mathrm {tan}\left (5\,x\right )}^{2/3}-81+\sqrt {3}\,81{}\mathrm {i}\right )\,\left (-\frac {1}{20}+\frac {\sqrt {3}\,1{}\mathrm {i}}{20}\right ) \]
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