Integrand size = 15, antiderivative size = 69 \[ \int \tan (x) \left (1-7 \tan ^2(x)\right )^{2/3} \, dx=2 \sqrt {3} \arctan \left (\frac {1+\sqrt [3]{1-7 \tan ^2(x)}}{\sqrt {3}}\right )+2 \log (\cos (x))+3 \log \left (2-\sqrt [3]{1-7 \tan ^2(x)}\right )+\frac {3}{4} \left (1-7 \tan ^2(x)\right )^{2/3} \]
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Time = 0.06 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.467, Rules used = {3751, 455, 52, 57, 632, 210, 31} \[ \int \tan (x) \left (1-7 \tan ^2(x)\right )^{2/3} \, dx=2 \sqrt {3} \arctan \left (\frac {\sqrt [3]{1-7 \tan ^2(x)}+1}{\sqrt {3}}\right )+\frac {3}{4} \left (1-7 \tan ^2(x)\right )^{2/3}+3 \log \left (2-\sqrt [3]{1-7 \tan ^2(x)}\right )+2 \log (\cos (x)) \]
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Rule 31
Rule 52
Rule 57
Rule 210
Rule 455
Rule 632
Rule 3751
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {x \left (1-7 x^2\right )^{2/3}}{1+x^2} \, dx,x,\tan (x)\right ) \\ & = \frac {1}{2} \text {Subst}\left (\int \frac {(1-7 x)^{2/3}}{1+x} \, dx,x,\tan ^2(x)\right ) \\ & = \frac {3}{4} \left (1-7 \tan ^2(x)\right )^{2/3}+4 \text {Subst}\left (\int \frac {1}{\sqrt [3]{1-7 x} (1+x)} \, dx,x,\tan ^2(x)\right ) \\ & = 2 \log (\cos (x))+\frac {3}{4} \left (1-7 \tan ^2(x)\right )^{2/3}-3 \text {Subst}\left (\int \frac {1}{2-x} \, dx,x,\sqrt [3]{1-7 \tan ^2(x)}\right )+6 \text {Subst}\left (\int \frac {1}{4+2 x+x^2} \, dx,x,\sqrt [3]{1-7 \tan ^2(x)}\right ) \\ & = 2 \log (\cos (x))+3 \log \left (2-\sqrt [3]{1-7 \tan ^2(x)}\right )+\frac {3}{4} \left (1-7 \tan ^2(x)\right )^{2/3}-12 \text {Subst}\left (\int \frac {1}{-12-x^2} \, dx,x,2+2 \sqrt [3]{1-7 \tan ^2(x)}\right ) \\ & = 2 \sqrt {3} \arctan \left (\frac {1+\sqrt [3]{1-7 \tan ^2(x)}}{\sqrt {3}}\right )+2 \log (\cos (x))+3 \log \left (2-\sqrt [3]{1-7 \tan ^2(x)}\right )+\frac {3}{4} \left (1-7 \tan ^2(x)\right )^{2/3} \\ \end{align*}
Time = 0.17 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.00 \[ \int \tan (x) \left (1-7 \tan ^2(x)\right )^{2/3} \, dx=2 \sqrt {3} \arctan \left (\frac {1+\sqrt [3]{1-7 \tan ^2(x)}}{\sqrt {3}}\right )+2 \log (\cos (x))+3 \log \left (2-\sqrt [3]{1-7 \tan ^2(x)}\right )+\frac {3}{4} \left (1-7 \tan ^2(x)\right )^{2/3} \]
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\[\int \tan \left (x \right ) {\left (1-7 \left (\tan ^{2}\left (x \right )\right )\right )}^{\frac {2}{3}}d x\]
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Leaf count of result is larger than twice the leaf count of optimal. 117 vs. \(2 (58) = 116\).
Time = 0.63 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.70 \[ \int \tan (x) \left (1-7 \tan ^2(x)\right )^{2/3} \, dx=2 \, \sqrt {3} \arctan \left (\frac {7 \, \sqrt {3} \tan \left (x\right )^{2} + 4 \, \sqrt {3} {\left (-7 \, \tan \left (x\right )^{2} + 1\right )}^{\frac {2}{3}} - 16 \, \sqrt {3} {\left (-7 \, \tan \left (x\right )^{2} + 1\right )}^{\frac {1}{3}} - \sqrt {3}}{7 \, \tan \left (x\right )^{2} - 65}\right ) + \frac {3}{4} \, {\left (-7 \, \tan \left (x\right )^{2} + 1\right )}^{\frac {2}{3}} + \log \left (\frac {7 \, \tan \left (x\right )^{2} + 6 \, {\left (-7 \, \tan \left (x\right )^{2} + 1\right )}^{\frac {2}{3}} - 12 \, {\left (-7 \, \tan \left (x\right )^{2} + 1\right )}^{\frac {1}{3}} + 7}{\tan \left (x\right )^{2} + 1}\right ) \]
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\[ \int \tan (x) \left (1-7 \tan ^2(x)\right )^{2/3} \, dx=\int \left (1 - 7 \tan ^{2}{\left (x \right )}\right )^{\frac {2}{3}} \tan {\left (x \right )}\, dx \]
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\[ \int \tan (x) \left (1-7 \tan ^2(x)\right )^{2/3} \, dx=\int { {\left (-7 \, \tan \left (x\right )^{2} + 1\right )}^{\frac {2}{3}} \tan \left (x\right ) \,d x } \]
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Time = 0.28 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.14 \[ \int \tan (x) \left (1-7 \tan ^2(x)\right )^{2/3} \, dx=2 \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left ({\left (-7 \, \tan \left (x\right )^{2} + 1\right )}^{\frac {1}{3}} + 1\right )}\right ) + \frac {3}{4} \, {\left (-7 \, \tan \left (x\right )^{2} + 1\right )}^{\frac {2}{3}} - \log \left ({\left (-7 \, \tan \left (x\right )^{2} + 1\right )}^{\frac {2}{3}} + 2 \, {\left (-7 \, \tan \left (x\right )^{2} + 1\right )}^{\frac {1}{3}} + 4\right ) + 2 \, \log \left ({\left | {\left (-7 \, \tan \left (x\right )^{2} + 1\right )}^{\frac {1}{3}} - 2 \right |}\right ) \]
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Time = 0.75 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.46 \[ \int \tan (x) \left (1-7 \tan ^2(x)\right )^{2/3} \, dx=2\,\ln \left (144\,{\left (1-7\,{\mathrm {tan}\left (x\right )}^2\right )}^{1/3}-288\right )+\frac {3\,{\left (1-7\,{\mathrm {tan}\left (x\right )}^2\right )}^{2/3}}{4}+\ln \left (144\,{\left (1-7\,{\mathrm {tan}\left (x\right )}^2\right )}^{1/3}-72\,{\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}^2\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )-\ln \left (144\,{\left (1-7\,{\mathrm {tan}\left (x\right )}^2\right )}^{1/3}-72\,{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^2\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right ) \]
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