Integrand size = 19, antiderivative size = 89 \[ \int \sec ^n(e+f x) (a \sin (e+f x))^m \, dx=-\frac {a \operatorname {Hypergeometric2F1}\left (\frac {1-m}{2},\frac {1-n}{2},\frac {3-n}{2},\cos ^2(e+f x)\right ) \sec ^{-1+n}(e+f x) (a \sin (e+f x))^{-1+m} \sin ^2(e+f x)^{\frac {1-m}{2}}}{f (1-n)} \]
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Time = 0.06 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {2667, 2656} \[ \int \sec ^n(e+f x) (a \sin (e+f x))^m \, dx=-\frac {a \sin ^2(e+f x)^{\frac {1-m}{2}} \sec ^{n-1}(e+f x) (a \sin (e+f x))^{m-1} \operatorname {Hypergeometric2F1}\left (\frac {1-m}{2},\frac {1-n}{2},\frac {3-n}{2},\cos ^2(e+f x)\right )}{f (1-n)} \]
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Rule 2656
Rule 2667
Rubi steps \begin{align*} \text {integral}& = \left (\cos ^n(e+f x) \sec ^n(e+f x)\right ) \int \cos ^{-n}(e+f x) (a \sin (e+f x))^m \, dx \\ & = -\frac {a \operatorname {Hypergeometric2F1}\left (\frac {1-m}{2},\frac {1-n}{2},\frac {3-n}{2},\cos ^2(e+f x)\right ) \sec ^{-1+n}(e+f x) (a \sin (e+f x))^{-1+m} \sin ^2(e+f x)^{\frac {1-m}{2}}}{f (1-n)} \\ \end{align*}
Result contains higher order function than in optimal. Order 6 vs. order 5 in optimal.
Time = 0.35 (sec) , antiderivative size = 287, normalized size of antiderivative = 3.22 \[ \int \sec ^n(e+f x) (a \sin (e+f x))^m \, dx=\frac {4 (3+m) \operatorname {AppellF1}\left (\frac {1+m}{2},n,1+m-n,\frac {3+m}{2},\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right ) \cos ^3\left (\frac {1}{2} (e+f x)\right ) \sec ^n(e+f x) \sin \left (\frac {1}{2} (e+f x)\right ) (a \sin (e+f x))^m}{f (1+m) \left ((3+m) \operatorname {AppellF1}\left (\frac {1+m}{2},n,1+m-n,\frac {3+m}{2},\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right ) (1+\cos (e+f x))-4 \left ((1+m-n) \operatorname {AppellF1}\left (\frac {3+m}{2},n,2+m-n,\frac {5+m}{2},\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )-n \operatorname {AppellF1}\left (\frac {3+m}{2},1+n,1+m-n,\frac {5+m}{2},\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )\right ) \sin ^2\left (\frac {1}{2} (e+f x)\right )\right )} \]
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\[\int \left (\sec ^{n}\left (f x +e \right )\right ) \left (a \sin \left (f x +e \right )\right )^{m}d x\]
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\[ \int \sec ^n(e+f x) (a \sin (e+f x))^m \, dx=\int { \left (a \sin \left (f x + e\right )\right )^{m} \sec \left (f x + e\right )^{n} \,d x } \]
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\[ \int \sec ^n(e+f x) (a \sin (e+f x))^m \, dx=\int \left (a \sin {\left (e + f x \right )}\right )^{m} \sec ^{n}{\left (e + f x \right )}\, dx \]
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\[ \int \sec ^n(e+f x) (a \sin (e+f x))^m \, dx=\int { \left (a \sin \left (f x + e\right )\right )^{m} \sec \left (f x + e\right )^{n} \,d x } \]
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\[ \int \sec ^n(e+f x) (a \sin (e+f x))^m \, dx=\int { \left (a \sin \left (f x + e\right )\right )^{m} \sec \left (f x + e\right )^{n} \,d x } \]
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Timed out. \[ \int \sec ^n(e+f x) (a \sin (e+f x))^m \, dx=\int {\left (a\,\sin \left (e+f\,x\right )\right )}^m\,{\left (\frac {1}{\cos \left (e+f\,x\right )}\right )}^n \,d x \]
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