Optimal. Leaf size=57 \[ -\frac {1}{10} \sqrt {3} \tan ^{-1}\left (\frac {1-2 \tan ^{\frac {2}{3}}(5 x)}{\sqrt {3}}\right )+\frac {3}{20} \log \left (\tan ^{\frac {2}{3}}(5 x)+1\right )-\frac {1}{20} \log \left (\tan ^2(5 x)+1\right ) \]
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Rubi [A] time = 0.06, antiderivative size = 69, normalized size of antiderivative = 1.21, number of steps used = 9, number of rules used = 9, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.125, Rules used = {3476, 329, 275, 200, 31, 634, 618, 204, 628} \[ -\frac {1}{10} \sqrt {3} \tan ^{-1}\left (\frac {1-2 \tan ^{\frac {2}{3}}(5 x)}{\sqrt {3}}\right )+\frac {1}{10} \log \left (\tan ^{\frac {2}{3}}(5 x)+1\right )-\frac {1}{20} \log \left (\tan ^{\frac {4}{3}}(5 x)-\tan ^{\frac {2}{3}}(5 x)+1\right ) \]
Antiderivative was successfully verified.
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Rule 31
Rule 200
Rule 204
Rule 275
Rule 329
Rule 618
Rule 628
Rule 634
Rule 3476
Rubi steps
\begin {align*} \int \frac {1}{\sqrt [3]{\tan (5 x)}} \, dx &=\frac {1}{5} \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{x} \left (1+x^2\right )} \, dx,x,\tan (5 x)\right )\\ &=\frac {3}{5} \operatorname {Subst}\left (\int \frac {x}{1+x^6} \, dx,x,\sqrt [3]{\tan (5 x)}\right )\\ &=\frac {3}{10} \operatorname {Subst}\left (\int \frac {1}{1+x^3} \, dx,x,\tan ^{\frac {2}{3}}(5 x)\right )\\ &=\frac {1}{10} \operatorname {Subst}\left (\int \frac {1}{1+x} \, dx,x,\tan ^{\frac {2}{3}}(5 x)\right )+\frac {1}{10} \operatorname {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} \operatorname {Subst}\left (\int \frac {-1+2 x}{1-x+x^2} \, dx,x,\tan ^{\frac {2}{3}}(5 x)\right )+\frac {3}{20} \operatorname {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} \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,-1+2 \tan ^{\frac {2}{3}}(5 x)\right )\\ &=-\frac {1}{10} \sqrt {3} \tan ^{-1}\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*}
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Mathematica [A] time = 0.05, size = 69, normalized size = 1.21 \[ \frac {1}{10} \sqrt {3} \tan ^{-1}\left (\frac {2 \tan ^{\frac {2}{3}}(5 x)-1}{\sqrt {3}}\right )+\frac {1}{10} \log \left (\tan ^{\frac {2}{3}}(5 x)+1\right )-\frac {1}{20} \log \left (\tan ^{\frac {4}{3}}(5 x)-\tan ^{\frac {2}{3}}(5 x)+1\right ) \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt [3]{\tan (5 x)}} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.95, size = 54, normalized size = 0.95 \[ \frac {1}{10} \, \sqrt {3} \arctan \left (\frac {2}{3} \, \sqrt {3} \tan \left (5 \, x\right )^{\frac {2}{3}} - \frac {1}{3} \, \sqrt {3}\right ) - \frac {1}{20} \, \log \left (\tan \left (5 \, x\right )^{\frac {4}{3}} - \tan \left (5 \, x\right )^{\frac {2}{3}} + 1\right ) + \frac {1}{10} \, \log \left (\tan \left (5 \, x\right )^{\frac {2}{3}} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\tan \left (5 \, x\right )^{\frac {1}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.10, size = 53, normalized size = 0.93
method | result | size |
derivativedivides | \(-\frac {\ln \left (1-\left (\tan ^{\frac {2}{3}}\left (5 x \right )\right )+\tan ^{\frac {4}{3}}\left (5 x \right )\right )}{20}+\frac {\sqrt {3}\, \arctan \left (\frac {\left (2 \left (\tan ^{\frac {2}{3}}\left (5 x \right )\right )-1\right ) \sqrt {3}}{3}\right )}{10}+\frac {\ln \left (1+\tan ^{\frac {2}{3}}\left (5 x \right )\right )}{10}\) | \(53\) |
default | \(-\frac {\ln \left (1-\left (\tan ^{\frac {2}{3}}\left (5 x \right )\right )+\tan ^{\frac {4}{3}}\left (5 x \right )\right )}{20}+\frac {\sqrt {3}\, \arctan \left (\frac {\left (2 \left (\tan ^{\frac {2}{3}}\left (5 x \right )\right )-1\right ) \sqrt {3}}{3}\right )}{10}+\frac {\ln \left (1+\tan ^{\frac {2}{3}}\left (5 x \right )\right )}{10}\) | \(53\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.95, size = 52, normalized size = 0.91 \[ \frac {1}{10} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, \tan \left (5 \, x\right )^{\frac {2}{3}} - 1\right )}\right ) - \frac {1}{20} \, \log \left (\tan \left (5 \, x\right )^{\frac {4}{3}} - \tan \left (5 \, x\right )^{\frac {2}{3}} + 1\right ) + \frac {1}{10} \, \log \left (\tan \left (5 \, x\right )^{\frac {2}{3}} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.69, size = 67, normalized size = 1.18 \[ \frac {\ln \left (81\,{\mathrm {tan}\left (5\,x\right )}^{2/3}+81\right )}{10}-\ln \left (81-162\,{\mathrm {tan}\left (5\,x\right )}^{2/3}+\sqrt {3}\,81{}\mathrm {i}\right )\,\left (\frac {1}{20}+\frac {\sqrt {3}\,1{}\mathrm {i}}{20}\right )+\ln \left (162\,{\mathrm {tan}\left (5\,x\right )}^{2/3}-81+\sqrt {3}\,81{}\mathrm {i}\right )\,\left (-\frac {1}{20}+\frac {\sqrt {3}\,1{}\mathrm {i}}{20}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt [3]{\tan {\left (5 x \right )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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