Integrand size = 21, antiderivative size = 58 \[ \int \cos ^2(e+f x) \sqrt [3]{b \sin (e+f x)} \, dx=\frac {3 \cos (e+f x) \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {2}{3},\frac {5}{3},\sin ^2(e+f x)\right ) (b \sin (e+f x))^{4/3}}{4 b f \sqrt {\cos ^2(e+f x)}} \]
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Time = 0.03 (sec) , antiderivative size = 58, 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 = {2657} \[ \int \cos ^2(e+f x) \sqrt [3]{b \sin (e+f x)} \, dx=\frac {3 \cos (e+f x) (b \sin (e+f x))^{4/3} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {2}{3},\frac {5}{3},\sin ^2(e+f x)\right )}{4 b f \sqrt {\cos ^2(e+f x)}} \]
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Rule 2657
Rubi steps \begin{align*} \text {integral}& = \frac {3 \cos (e+f x) \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {2}{3},\frac {5}{3},\sin ^2(e+f x)\right ) (b \sin (e+f x))^{4/3}}{4 b f \sqrt {\cos ^2(e+f x)}} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.95 \[ \int \cos ^2(e+f x) \sqrt [3]{b \sin (e+f x)} \, dx=\frac {3 \sqrt {\cos ^2(e+f x)} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {2}{3},\frac {5}{3},\sin ^2(e+f x)\right ) \sqrt [3]{b \sin (e+f x)} \tan (e+f x)}{4 f} \]
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\[\int \left (\cos ^{2}\left (f x +e \right )\right ) \left (b \sin \left (f x +e \right )\right )^{\frac {1}{3}}d x\]
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\[ \int \cos ^2(e+f x) \sqrt [3]{b \sin (e+f x)} \, dx=\int { \left (b \sin \left (f x + e\right )\right )^{\frac {1}{3}} \cos \left (f x + e\right )^{2} \,d x } \]
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\[ \int \cos ^2(e+f x) \sqrt [3]{b \sin (e+f x)} \, dx=\int \sqrt [3]{b \sin {\left (e + f x \right )}} \cos ^{2}{\left (e + f x \right )}\, dx \]
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\[ \int \cos ^2(e+f x) \sqrt [3]{b \sin (e+f x)} \, dx=\int { \left (b \sin \left (f x + e\right )\right )^{\frac {1}{3}} \cos \left (f x + e\right )^{2} \,d x } \]
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\[ \int \cos ^2(e+f x) \sqrt [3]{b \sin (e+f x)} \, dx=\int { \left (b \sin \left (f x + e\right )\right )^{\frac {1}{3}} \cos \left (f x + e\right )^{2} \,d x } \]
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Timed out. \[ \int \cos ^2(e+f x) \sqrt [3]{b \sin (e+f x)} \, dx=\int {\cos \left (e+f\,x\right )}^2\,{\left (b\,\sin \left (e+f\,x\right )\right )}^{1/3} \,d x \]
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