Integrand size = 13, antiderivative size = 47 \[ \int \sqrt [3]{\sec ^{12}(x) \tan ^2(x)} \, dx=\frac {3}{5} \cos ^3(x) \sin (x) \sqrt [3]{\sec ^{12}(x) \tan ^2(x)}+\frac {3}{11} \cos (x) \sin ^3(x) \sqrt [3]{\sec ^{12}(x) \tan ^2(x)} \]
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Time = 0.05 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {1986, 15, 14} \[ \int \sqrt [3]{\sec ^{12}(x) \tan ^2(x)} \, dx=\frac {3}{5} \sin (x) \cos ^3(x) \sqrt [3]{\tan ^2(x) \sec ^{12}(x)}+\frac {3}{11} \sin ^3(x) \cos (x) \sqrt [3]{\tan ^2(x) \sec ^{12}(x)} \]
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Rule 14
Rule 15
Rule 1986
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {\sqrt [3]{x^2 \left (1+x^2\right )^6}}{1+x^2} \, dx,x,\tan (x)\right ) \\ & = \frac {\left (\cos ^4(x) \sqrt [3]{\sec ^{12}(x) \tan ^2(x)}\right ) \text {Subst}\left (\int \sqrt [3]{x^2} \left (1+x^2\right ) \, dx,x,\tan (x)\right )}{\sqrt [3]{\tan ^2(x)}} \\ & = \frac {\left (\cos ^4(x) \sqrt [3]{\sec ^{12}(x) \tan ^2(x)}\right ) \text {Subst}\left (\int x^{2/3} \left (1+x^2\right ) \, dx,x,\tan (x)\right )}{\tan ^{\frac {2}{3}}(x)} \\ & = \frac {\left (\cos ^4(x) \sqrt [3]{\sec ^{12}(x) \tan ^2(x)}\right ) \text {Subst}\left (\int \left (x^{2/3}+x^{8/3}\right ) \, dx,x,\tan (x)\right )}{\tan ^{\frac {2}{3}}(x)} \\ & = \frac {3}{5} \cos ^3(x) \sin (x) \sqrt [3]{\sec ^{12}(x) \tan ^2(x)}+\frac {3}{11} \cos (x) \sin ^3(x) \sqrt [3]{\sec ^{12}(x) \tan ^2(x)} \\ \end{align*}
Time = 0.17 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.34 \[ \int \sqrt [3]{\sec ^{12}(x) \tan ^2(x)} \, dx=\frac {3 \cos (x) \sin (x) \sqrt [3]{\sec ^{12}(x) \tan ^2(x)} \left (-3+8 \left (-\tan ^2(x)\right )^{5/6}+3 \cos (2 x) \left (-1+\left (-\tan ^2(x)\right )^{5/6}\right )\right )}{55 \left (-\tan ^2(x)\right )^{5/6}} \]
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\[\int \left (\frac {\sin ^{2}\left (x \right )}{\cos \left (x \right )^{14}}\right )^{\frac {1}{3}}d x\]
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none
Time = 0.25 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.62 \[ \int \sqrt [3]{\sec ^{12}(x) \tan ^2(x)} \, dx=\frac {3}{55} \, {\left (6 \, \cos \left (x\right )^{3} + 5 \, \cos \left (x\right )\right )} \left (-\frac {\cos \left (x\right )^{2} - 1}{\cos \left (x\right )^{14}}\right )^{\frac {1}{3}} \sin \left (x\right ) \]
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Timed out. \[ \int \sqrt [3]{\sec ^{12}(x) \tan ^2(x)} \, dx=\text {Timed out} \]
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none
Time = 0.28 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.28 \[ \int \sqrt [3]{\sec ^{12}(x) \tan ^2(x)} \, dx=\frac {3}{11} \, \tan \left (x\right )^{\frac {11}{3}} + \frac {3}{5} \, \tan \left (x\right )^{\frac {5}{3}} \]
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\[ \int \sqrt [3]{\sec ^{12}(x) \tan ^2(x)} \, dx=\int { \left (\frac {\sin \left (x\right )^{2}}{\cos \left (x\right )^{14}}\right )^{\frac {1}{3}} \,d x } \]
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Time = 4.04 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.68 \[ \int \sqrt [3]{\sec ^{12}(x) \tan ^2(x)} \, dx=\frac {6\,\sin \left (2\,x\right )\,{\left (1-\cos \left (2\,x\right )\right )}^{1/3}\,\left (3\,\cos \left (2\,x\right )+8\right )}{55\,{\left (\cos \left (2\,x\right )+1\right )}^{7/3}} \]
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