Integrand size = 23, antiderivative size = 89 \[ \int (a \sin (e+f x))^{3/2} (b \tan (e+f x))^n \, dx=\frac {2 \cos ^2(e+f x)^{\frac {1+n}{2}} \operatorname {Hypergeometric2F1}\left (\frac {1+n}{2},\frac {1}{4} (5+2 n),\frac {1}{4} (9+2 n),\sin ^2(e+f x)\right ) (a \sin (e+f x))^{3/2} (b \tan (e+f x))^{1+n}}{b f (5+2 n)} \]
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Time = 0.12 (sec) , antiderivative size = 89, 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} \[ \int (a \sin (e+f x))^{3/2} (b \tan (e+f x))^n \, dx=\frac {2 (a \sin (e+f x))^{3/2} \cos ^2(e+f x)^{\frac {n+1}{2}} (b \tan (e+f x))^{n+1} \operatorname {Hypergeometric2F1}\left (\frac {n+1}{2},\frac {1}{4} (2 n+5),\frac {1}{4} (2 n+9),\sin ^2(e+f x)\right )}{b f (2 n+5)} \]
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Rule 2657
Rule 2682
Rubi steps \begin{align*} \text {integral}& = \frac {\left (a \cos ^{1+n}(e+f x) (a \sin (e+f x))^{-1-n} (b \tan (e+f x))^{1+n}\right ) \int \cos ^{-n}(e+f x) (a \sin (e+f x))^{\frac {3}{2}+n} \, dx}{b} \\ & = \frac {2 \cos ^2(e+f x)^{\frac {1+n}{2}} \operatorname {Hypergeometric2F1}\left (\frac {1+n}{2},\frac {1}{4} (5+2 n),\frac {1}{4} (9+2 n),\sin ^2(e+f x)\right ) (a \sin (e+f x))^{3/2} (b \tan (e+f x))^{1+n}}{b f (5+2 n)} \\ \end{align*}
Result contains higher order function than in optimal. Order 6 vs. order 5 in optimal.
Time = 32.84 (sec) , antiderivative size = 297, normalized size of antiderivative = 3.34 \[ \int (a \sin (e+f x))^{3/2} (b \tan (e+f x))^n \, dx=\frac {8 (9+2 n) \operatorname {AppellF1}\left (\frac {5}{4}+\frac {n}{2},n,\frac {5}{2},\frac {9}{4}+\frac {n}{2},\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right ) \cos ^3\left (\frac {1}{2} (e+f x)\right ) \sin \left (\frac {1}{2} (e+f x)\right ) (a \sin (e+f x))^{3/2} (b \tan (e+f x))^n}{f (5+2 n) \left (2 (9+2 n) \operatorname {AppellF1}\left (\frac {5}{4}+\frac {n}{2},n,\frac {5}{2},\frac {9}{4}+\frac {n}{2},\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right ) \cos ^2\left (\frac {1}{2} (e+f x)\right )+2 \left (5 \operatorname {AppellF1}\left (\frac {9}{4}+\frac {n}{2},n,\frac {7}{2},\frac {13}{4}+\frac {n}{2},\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )-2 n \operatorname {AppellF1}\left (\frac {9}{4}+\frac {n}{2},1+n,\frac {5}{2},\frac {13}{4}+\frac {n}{2},\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )\right ) (-1+\cos (e+f x))\right )} \]
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\[\int \left (\sin \left (f x +e \right ) a \right )^{\frac {3}{2}} \left (b \tan \left (f x +e \right )\right )^{n}d x\]
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\[ \int (a \sin (e+f x))^{3/2} (b \tan (e+f x))^n \, dx=\int { \left (a \sin \left (f x + e\right )\right )^{\frac {3}{2}} \left (b \tan \left (f x + e\right )\right )^{n} \,d x } \]
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Timed out. \[ \int (a \sin (e+f x))^{3/2} (b \tan (e+f x))^n \, dx=\text {Timed out} \]
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\[ \int (a \sin (e+f x))^{3/2} (b \tan (e+f x))^n \, dx=\int { \left (a \sin \left (f x + e\right )\right )^{\frac {3}{2}} \left (b \tan \left (f x + e\right )\right )^{n} \,d x } \]
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\[ \int (a \sin (e+f x))^{3/2} (b \tan (e+f x))^n \, dx=\int { \left (a \sin \left (f x + e\right )\right )^{\frac {3}{2}} \left (b \tan \left (f x + e\right )\right )^{n} \,d x } \]
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Timed out. \[ \int (a \sin (e+f x))^{3/2} (b \tan (e+f x))^n \, dx=\int {\left (a\,\sin \left (e+f\,x\right )\right )}^{3/2}\,{\left (b\,\mathrm {tan}\left (e+f\,x\right )\right )}^n \,d x \]
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