Integrand size = 19, antiderivative size = 53 \[ \int \sec ^{\frac {7}{3}}(e+f x) \sin ^2(e+f x) \, dx=\frac {3 \operatorname {Hypergeometric2F1}\left (-\frac {2}{3},-\frac {1}{2},\frac {1}{3},\cos ^2(e+f x)\right ) \sec ^{\frac {4}{3}}(e+f x) \sin (e+f x)}{4 f \sqrt {\sin ^2(e+f x)}} \]
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Time = 0.07 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {2712, 2656} \[ \int \sec ^{\frac {7}{3}}(e+f x) \sin ^2(e+f x) \, dx=\frac {3 \sin (e+f x) \sec ^{\frac {4}{3}}(e+f x) \operatorname {Hypergeometric2F1}\left (-\frac {2}{3},-\frac {1}{2},\frac {1}{3},\cos ^2(e+f x)\right )}{4 f \sqrt {\sin ^2(e+f x)}} \]
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Rule 2656
Rule 2712
Rubi steps \begin{align*} \text {integral}& = \left (\sqrt [3]{\cos (e+f x)} \sqrt [3]{\sec (e+f x)}\right ) \int \frac {\sin ^2(e+f x)}{\cos ^{\frac {7}{3}}(e+f x)} \, dx \\ & = \frac {3 \operatorname {Hypergeometric2F1}\left (-\frac {2}{3},-\frac {1}{2},\frac {1}{3},\cos ^2(e+f x)\right ) \sec ^{\frac {4}{3}}(e+f x) \sin (e+f x)}{4 f \sqrt {\sin ^2(e+f x)}} \\ \end{align*}
Time = 0.14 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.06 \[ \int \sec ^{\frac {7}{3}}(e+f x) \sin ^2(e+f x) \, dx=-\frac {3 \left (-1+\cos ^2(e+f x)^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2}{3},\frac {3}{2},\sin ^2(e+f x)\right )\right ) \sec ^{\frac {4}{3}}(e+f x) \sin (e+f x)}{4 f} \]
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\[\int \left (\sec ^{\frac {1}{3}}\left (f x +e \right )\right ) \left (\tan ^{2}\left (f x +e \right )\right )d x\]
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\[ \int \sec ^{\frac {7}{3}}(e+f x) \sin ^2(e+f x) \, dx=\int { \sec \left (f x + e\right )^{\frac {1}{3}} \tan \left (f x + e\right )^{2} \,d x } \]
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\[ \int \sec ^{\frac {7}{3}}(e+f x) \sin ^2(e+f x) \, dx=\int \tan ^{2}{\left (e + f x \right )} \sqrt [3]{\sec {\left (e + f x \right )}}\, dx \]
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\[ \int \sec ^{\frac {7}{3}}(e+f x) \sin ^2(e+f x) \, dx=\int { \sec \left (f x + e\right )^{\frac {1}{3}} \tan \left (f x + e\right )^{2} \,d x } \]
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\[ \int \sec ^{\frac {7}{3}}(e+f x) \sin ^2(e+f x) \, dx=\int { \sec \left (f x + e\right )^{\frac {1}{3}} \tan \left (f x + e\right )^{2} \,d x } \]
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Timed out. \[ \int \sec ^{\frac {7}{3}}(e+f x) \sin ^2(e+f x) \, dx=\int {\mathrm {tan}\left (e+f\,x\right )}^2\,{\left (\frac {1}{\cos \left (e+f\,x\right )}\right )}^{1/3} \,d x \]
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