Integrand size = 23, antiderivative size = 79 \[ \int \frac {(a \sin (e+f x))^m}{\sqrt {b \tan (e+f x)}} \, dx=\frac {2 \sqrt [4]{\cos ^2(e+f x)} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{4} (1+2 m),\frac {1}{4} (5+2 m),\sin ^2(e+f x)\right ) (a \sin (e+f x))^m \sqrt {b \tan (e+f x)}}{b f (1+2 m)} \]
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Time = 0.11 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {2682, 2657} \[ \int \frac {(a \sin (e+f x))^m}{\sqrt {b \tan (e+f x)}} \, dx=\frac {2 \sqrt [4]{\cos ^2(e+f x)} \sqrt {b \tan (e+f x)} (a \sin (e+f x))^m \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{4} (2 m+1),\frac {1}{4} (2 m+5),\sin ^2(e+f x)\right )}{b f (2 m+1)} \]
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Rule 2657
Rule 2682
Rubi steps \begin{align*} \text {integral}& = \frac {\left (a \sqrt {\cos (e+f x)} \sqrt {b \tan (e+f x)}\right ) \int \sqrt {\cos (e+f x)} (a \sin (e+f x))^{-\frac {1}{2}+m} \, dx}{b \sqrt {a \sin (e+f x)}} \\ & = \frac {2 \sqrt [4]{\cos ^2(e+f x)} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{4} (1+2 m),\frac {1}{4} (5+2 m),\sin ^2(e+f x)\right ) (a \sin (e+f x))^m \sqrt {b \tan (e+f x)}}{b f (1+2 m)} \\ \end{align*}
Time = 0.54 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.10 \[ \int \frac {(a \sin (e+f x))^m}{\sqrt {b \tan (e+f x)}} \, dx=\frac {2 \operatorname {Hypergeometric2F1}\left (\frac {2+m}{2},\frac {1}{4} (1+2 m),\frac {1}{4} (5+2 m),-\tan ^2(e+f x)\right ) \sec ^2(e+f x)^{m/2} (a \sin (e+f x))^m \sqrt {b \tan (e+f x)}}{b f (1+2 m)} \]
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\[\int \frac {\left (\sin \left (f x +e \right ) a \right )^{m}}{\sqrt {b \tan \left (f x +e \right )}}d x\]
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\[ \int \frac {(a \sin (e+f x))^m}{\sqrt {b \tan (e+f x)}} \, dx=\int { \frac {\left (a \sin \left (f x + e\right )\right )^{m}}{\sqrt {b \tan \left (f x + e\right )}} \,d x } \]
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\[ \int \frac {(a \sin (e+f x))^m}{\sqrt {b \tan (e+f x)}} \, dx=\int \frac {\left (a \sin {\left (e + f x \right )}\right )^{m}}{\sqrt {b \tan {\left (e + f x \right )}}}\, dx \]
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\[ \int \frac {(a \sin (e+f x))^m}{\sqrt {b \tan (e+f x)}} \, dx=\int { \frac {\left (a \sin \left (f x + e\right )\right )^{m}}{\sqrt {b \tan \left (f x + e\right )}} \,d x } \]
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\[ \int \frac {(a \sin (e+f x))^m}{\sqrt {b \tan (e+f x)}} \, dx=\int { \frac {\left (a \sin \left (f x + e\right )\right )^{m}}{\sqrt {b \tan \left (f x + e\right )}} \,d x } \]
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Timed out. \[ \int \frac {(a \sin (e+f x))^m}{\sqrt {b \tan (e+f x)}} \, dx=\int \frac {{\left (a\,\sin \left (e+f\,x\right )\right )}^m}{\sqrt {b\,\mathrm {tan}\left (e+f\,x\right )}} \,d x \]
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