Integrand size = 37, antiderivative size = 220 \[ \int \frac {(A+B \sin (e+f x)) (c+d \sin (e+f x))^n}{\sqrt {a+a \sin (e+f x)}} \, dx=-\frac {(A-B) \operatorname {AppellF1}\left (1+n,\frac {1}{2},1,2+n,\frac {c+d \sin (e+f x)}{c+d},\frac {c+d \sin (e+f x)}{c-d}\right ) \cos (e+f x) \sqrt {\frac {d (1-\sin (e+f x))}{c+d}} (c+d \sin (e+f x))^{1+n}}{(c-d) f (1+n) (1-\sin (e+f x)) \sqrt {a+a \sin (e+f x)}}-\frac {2 B \cos (e+f x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-n,\frac {3}{2},\frac {d (1-\sin (e+f x))}{c+d}\right ) (c+d \sin (e+f x))^n \left (\frac {c+d \sin (e+f x)}{c+d}\right )^{-n}}{f \sqrt {a+a \sin (e+f x)}} \]
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Time = 0.28 (sec) , antiderivative size = 220, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.189, Rules used = {3066, 2867, 142, 141, 2855, 72, 71} \[ \int \frac {(A+B \sin (e+f x)) (c+d \sin (e+f x))^n}{\sqrt {a+a \sin (e+f x)}} \, dx=-\frac {(A-B) \cos (e+f x) \sqrt {\frac {d (1-\sin (e+f x))}{c+d}} (c+d \sin (e+f x))^{n+1} \operatorname {AppellF1}\left (n+1,\frac {1}{2},1,n+2,\frac {c+d \sin (e+f x)}{c+d},\frac {c+d \sin (e+f x)}{c-d}\right )}{f (n+1) (c-d) (1-\sin (e+f x)) \sqrt {a \sin (e+f x)+a}}-\frac {2 B \cos (e+f x) (c+d \sin (e+f x))^n \left (\frac {c+d \sin (e+f x)}{c+d}\right )^{-n} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-n,\frac {3}{2},\frac {d (1-\sin (e+f x))}{c+d}\right )}{f \sqrt {a \sin (e+f x)+a}} \]
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Rule 71
Rule 72
Rule 141
Rule 142
Rule 2855
Rule 2867
Rule 3066
Rubi steps \begin{align*} \text {integral}& = (A-B) \int \frac {(c+d \sin (e+f x))^n}{\sqrt {a+a \sin (e+f x)}} \, dx+\frac {B \int \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^n \, dx}{a} \\ & = \frac {\left (a^2 (A-B) \cos (e+f x)\right ) \text {Subst}\left (\int \frac {(c+d x)^n}{\sqrt {a-a x} (a+a x)} \, dx,x,\sin (e+f x)\right )}{f \sqrt {a-a \sin (e+f x)} \sqrt {a+a \sin (e+f x)}}+\frac {(a B \cos (e+f x)) \text {Subst}\left (\int \frac {(c+d x)^n}{\sqrt {a-a x}} \, dx,x,\sin (e+f x)\right )}{f \sqrt {a-a \sin (e+f x)} \sqrt {a+a \sin (e+f x)}} \\ & = \frac {\left (a^2 (A-B) \cos (e+f x) \sqrt {\frac {d (a-a \sin (e+f x))}{a c+a d}}\right ) \text {Subst}\left (\int \frac {(c+d x)^n}{(a+a x) \sqrt {\frac {a d}{a c+a d}-\frac {a d x}{a c+a d}}} \, dx,x,\sin (e+f x)\right )}{f (a-a \sin (e+f x)) \sqrt {a+a \sin (e+f x)}}+\frac {\left (a B \cos (e+f x) (c+d \sin (e+f x))^n \left (-\frac {a (c+d \sin (e+f x))}{-a c-a d}\right )^{-n}\right ) \text {Subst}\left (\int \frac {\left (\frac {c}{c+d}+\frac {d x}{c+d}\right )^n}{\sqrt {a-a x}} \, dx,x,\sin (e+f x)\right )}{f \sqrt {a-a \sin (e+f x)} \sqrt {a+a \sin (e+f x)}} \\ & = -\frac {(A-B) \operatorname {AppellF1}\left (1+n,\frac {1}{2},1,2+n,\frac {c+d \sin (e+f x)}{c+d},\frac {c+d \sin (e+f x)}{c-d}\right ) \cos (e+f x) \sqrt {\frac {d (1-\sin (e+f x))}{c+d}} (c+d \sin (e+f x))^{1+n}}{(c-d) f (1+n) (1-\sin (e+f x)) \sqrt {a+a \sin (e+f x)}}-\frac {2 B \cos (e+f x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-n,\frac {3}{2},\frac {d (1-\sin (e+f x))}{c+d}\right ) (c+d \sin (e+f x))^n \left (\frac {c+d \sin (e+f x)}{c+d}\right )^{-n}}{f \sqrt {a+a \sin (e+f x)}} \\ \end{align*}
Time = 8.63 (sec) , antiderivative size = 244, normalized size of antiderivative = 1.11 \[ \int \frac {(A+B \sin (e+f x)) (c+d \sin (e+f x))^n}{\sqrt {a+a \sin (e+f x)}} \, dx=\frac {\cos (e+f x) \sqrt {a (1+\sin (e+f x))} (c+d \sin (e+f x))^n \left (-\left ((A+B) \operatorname {AppellF1}\left (1,\frac {1}{2},-n,2,\frac {1}{2} (1+\sin (e+f x)),\frac {d (1+\sin (e+f x))}{-c+d}\right ) \sqrt {2-2 \sin (e+f x)} \left (\frac {c+d \sin (e+f x)}{c-d}\right )^{-n}\right )+\frac {4 (A-B) \operatorname {AppellF1}\left (-\frac {1}{2}-n,-\frac {1}{2},-n,\frac {1}{2}-n,\frac {2}{1+\sin (e+f x)},\frac {-c+d}{d+d \sin (e+f x)}\right ) \sqrt {\frac {-1+\sin (e+f x)}{1+\sin (e+f x)}} \left (1+\frac {c-d}{d+d \sin (e+f x)}\right )^{-n}}{1+2 n}\right )}{4 a f (-1+\sin (e+f x))} \]
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\[\int \frac {\left (A +B \sin \left (f x +e \right )\right ) \left (c +d \sin \left (f x +e \right )\right )^{n}}{\sqrt {a +a \sin \left (f x +e \right )}}d x\]
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\[ \int \frac {(A+B \sin (e+f x)) (c+d \sin (e+f x))^n}{\sqrt {a+a \sin (e+f x)}} \, dx=\int { \frac {{\left (B \sin \left (f x + e\right ) + A\right )} {\left (d \sin \left (f x + e\right ) + c\right )}^{n}}{\sqrt {a \sin \left (f x + e\right ) + a}} \,d x } \]
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\[ \int \frac {(A+B \sin (e+f x)) (c+d \sin (e+f x))^n}{\sqrt {a+a \sin (e+f x)}} \, dx=\int \frac {\left (A + B \sin {\left (e + f x \right )}\right ) \left (c + d \sin {\left (e + f x \right )}\right )^{n}}{\sqrt {a \left (\sin {\left (e + f x \right )} + 1\right )}}\, dx \]
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\[ \int \frac {(A+B \sin (e+f x)) (c+d \sin (e+f x))^n}{\sqrt {a+a \sin (e+f x)}} \, dx=\int { \frac {{\left (B \sin \left (f x + e\right ) + A\right )} {\left (d \sin \left (f x + e\right ) + c\right )}^{n}}{\sqrt {a \sin \left (f x + e\right ) + a}} \,d x } \]
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Timed out. \[ \int \frac {(A+B \sin (e+f x)) (c+d \sin (e+f x))^n}{\sqrt {a+a \sin (e+f x)}} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {(A+B \sin (e+f x)) (c+d \sin (e+f x))^n}{\sqrt {a+a \sin (e+f x)}} \, dx=\int \frac {\left (A+B\,\sin \left (e+f\,x\right )\right )\,{\left (c+d\,\sin \left (e+f\,x\right )\right )}^n}{\sqrt {a+a\,\sin \left (e+f\,x\right )}} \,d x \]
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