Integrand size = 21, antiderivative size = 57 \[ \int \frac {\tan ^2(e+f x)}{\sqrt [3]{d \sec (e+f x)}} \, dx=\frac {\cos ^2(e+f x)^{4/3} \operatorname {Hypergeometric2F1}\left (\frac {4}{3},\frac {3}{2},\frac {5}{2},\sin ^2(e+f x)\right ) \tan ^3(e+f x)}{3 f \sqrt [3]{d \sec (e+f x)}} \]
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Time = 0.04 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {2697} \[ \int \frac {\tan ^2(e+f x)}{\sqrt [3]{d \sec (e+f x)}} \, dx=\frac {\cos ^2(e+f x)^{4/3} \tan ^3(e+f x) \operatorname {Hypergeometric2F1}\left (\frac {4}{3},\frac {3}{2},\frac {5}{2},\sin ^2(e+f x)\right )}{3 f \sqrt [3]{d \sec (e+f x)}} \]
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Rule 2697
Rubi steps \begin{align*} \text {integral}& = \frac {\cos ^2(e+f x)^{4/3} \operatorname {Hypergeometric2F1}\left (\frac {4}{3},\frac {3}{2},\frac {5}{2},\sin ^2(e+f x)\right ) \tan ^3(e+f x)}{3 f \sqrt [3]{d \sec (e+f x)}} \\ \end{align*}
Time = 0.11 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.96 \[ \int \frac {\tan ^2(e+f x)}{\sqrt [3]{d \sec (e+f x)}} \, dx=-\frac {3 \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},-\frac {1}{6},\frac {5}{6},\sec ^2(e+f x)\right ) \tan (e+f x)}{f \sqrt [3]{d \sec (e+f x)} \sqrt {-\tan ^2(e+f x)}} \]
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\[\int \frac {\tan ^{2}\left (f x +e \right )}{\left (d \sec \left (f x +e \right )\right )^{\frac {1}{3}}}d x\]
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\[ \int \frac {\tan ^2(e+f x)}{\sqrt [3]{d \sec (e+f x)}} \, dx=\int { \frac {\tan \left (f x + e\right )^{2}}{\left (d \sec \left (f x + e\right )\right )^{\frac {1}{3}}} \,d x } \]
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\[ \int \frac {\tan ^2(e+f x)}{\sqrt [3]{d \sec (e+f x)}} \, dx=\int \frac {\tan ^{2}{\left (e + f x \right )}}{\sqrt [3]{d \sec {\left (e + f x \right )}}}\, dx \]
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\[ \int \frac {\tan ^2(e+f x)}{\sqrt [3]{d \sec (e+f x)}} \, dx=\int { \frac {\tan \left (f x + e\right )^{2}}{\left (d \sec \left (f x + e\right )\right )^{\frac {1}{3}}} \,d x } \]
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\[ \int \frac {\tan ^2(e+f x)}{\sqrt [3]{d \sec (e+f x)}} \, dx=\int { \frac {\tan \left (f x + e\right )^{2}}{\left (d \sec \left (f x + e\right )\right )^{\frac {1}{3}}} \,d x } \]
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Timed out. \[ \int \frac {\tan ^2(e+f x)}{\sqrt [3]{d \sec (e+f x)}} \, dx=\int \frac {{\mathrm {tan}\left (e+f\,x\right )}^2}{{\left (\frac {d}{\cos \left (e+f\,x\right )}\right )}^{1/3}} \,d x \]
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