Integrand size = 35, antiderivative size = 223 \[ \int \frac {(A+B \sin (e+f x)) (c+d \sin (e+f x))^n}{(a+a \sin (e+f x))^2} \, dx=-\frac {B \operatorname {AppellF1}\left (\frac {1}{2},\frac {3}{2},-n,\frac {3}{2},\frac {1}{2} (1-\sin (e+f x)),\frac {d (1-\sin (e+f x))}{c+d}\right ) \cos (e+f x) (c+d \sin (e+f x))^n \left (\frac {c+d \sin (e+f x)}{c+d}\right )^{-n}}{\sqrt {2} a^2 f \sqrt {1+\sin (e+f x)}}-\frac {(A-B) \operatorname {AppellF1}\left (\frac {1}{2},\frac {5}{2},-n,\frac {3}{2},\frac {1}{2} (1-\sin (e+f x)),\frac {d (1-\sin (e+f x))}{c+d}\right ) \cos (e+f x) (c+d \sin (e+f x))^n \left (\frac {c+d \sin (e+f x)}{c+d}\right )^{-n}}{2 \sqrt {2} a^2 f \sqrt {1+\sin (e+f x)}} \]
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Time = 0.22 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.114, Rules used = {3066, 2863, 144, 143} \[ \int \frac {(A+B \sin (e+f x)) (c+d \sin (e+f x))^n}{(a+a \sin (e+f x))^2} \, dx=-\frac {(A-B) \cos (e+f x) (c+d \sin (e+f x))^n \left (\frac {c+d \sin (e+f x)}{c+d}\right )^{-n} \operatorname {AppellF1}\left (\frac {1}{2},\frac {5}{2},-n,\frac {3}{2},\frac {1}{2} (1-\sin (e+f x)),\frac {d (1-\sin (e+f x))}{c+d}\right )}{2 \sqrt {2} a^2 f \sqrt {\sin (e+f x)+1}}-\frac {B \cos (e+f x) (c+d \sin (e+f x))^n \left (\frac {c+d \sin (e+f x)}{c+d}\right )^{-n} \operatorname {AppellF1}\left (\frac {1}{2},\frac {3}{2},-n,\frac {3}{2},\frac {1}{2} (1-\sin (e+f x)),\frac {d (1-\sin (e+f x))}{c+d}\right )}{\sqrt {2} a^2 f \sqrt {\sin (e+f x)+1}} \]
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Rule 143
Rule 144
Rule 2863
Rule 3066
Rubi steps \begin{align*} \text {integral}& = (A-B) \int \frac {(c+d \sin (e+f x))^n}{(a+a \sin (e+f x))^2} \, dx+\frac {B \int \frac {(c+d \sin (e+f x))^n}{a+a \sin (e+f x)} \, dx}{a} \\ & = \frac {((A-B) \cos (e+f x)) \text {Subst}\left (\int \frac {(c+d x)^n}{\sqrt {1-x} (1+x)^{5/2}} \, dx,x,\sin (e+f x)\right )}{a^2 f \sqrt {1-\sin (e+f x)} \sqrt {1+\sin (e+f x)}}+\frac {(B \cos (e+f x)) \text {Subst}\left (\int \frac {(c+d x)^n}{\sqrt {1-x} (1+x)^{3/2}} \, dx,x,\sin (e+f x)\right )}{a^2 f \sqrt {1-\sin (e+f x)} \sqrt {1+\sin (e+f x)}} \\ & = \frac {\left ((A-B) \cos (e+f x) (c+d \sin (e+f x))^n \left (-\frac {c+d \sin (e+f x)}{-c-d}\right )^{-n}\right ) \text {Subst}\left (\int \frac {\left (-\frac {c}{-c-d}-\frac {d x}{-c-d}\right )^n}{\sqrt {1-x} (1+x)^{5/2}} \, dx,x,\sin (e+f x)\right )}{a^2 f \sqrt {1-\sin (e+f x)} \sqrt {1+\sin (e+f x)}}+\frac {\left (B \cos (e+f x) (c+d \sin (e+f x))^n \left (-\frac {c+d \sin (e+f x)}{-c-d}\right )^{-n}\right ) \text {Subst}\left (\int \frac {\left (-\frac {c}{-c-d}-\frac {d x}{-c-d}\right )^n}{\sqrt {1-x} (1+x)^{3/2}} \, dx,x,\sin (e+f x)\right )}{a^2 f \sqrt {1-\sin (e+f x)} \sqrt {1+\sin (e+f x)}} \\ & = -\frac {B \operatorname {AppellF1}\left (\frac {1}{2},\frac {3}{2},-n,\frac {3}{2},\frac {1}{2} (1-\sin (e+f x)),\frac {d (1-\sin (e+f x))}{c+d}\right ) \cos (e+f x) (c+d \sin (e+f x))^n \left (\frac {c+d \sin (e+f x)}{c+d}\right )^{-n}}{\sqrt {2} a^2 f \sqrt {1+\sin (e+f x)}}-\frac {(A-B) \operatorname {AppellF1}\left (\frac {1}{2},\frac {5}{2},-n,\frac {3}{2},\frac {1}{2} (1-\sin (e+f x)),\frac {d (1-\sin (e+f x))}{c+d}\right ) \cos (e+f x) (c+d \sin (e+f x))^n \left (\frac {c+d \sin (e+f x)}{c+d}\right )^{-n}}{2 \sqrt {2} a^2 f \sqrt {1+\sin (e+f x)}} \\ \end{align*}
\[ \int \frac {(A+B \sin (e+f x)) (c+d \sin (e+f x))^n}{(a+a \sin (e+f x))^2} \, dx=\int \frac {(A+B \sin (e+f x)) (c+d \sin (e+f x))^n}{(a+a \sin (e+f x))^2} \, dx \]
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\[\int \frac {\left (A +B \sin \left (f x +e \right )\right ) \left (c +d \sin \left (f x +e \right )\right )^{n}}{\left (a +a \sin \left (f x +e \right )\right )^{2}}d x\]
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\[ \int \frac {(A+B \sin (e+f x)) (c+d \sin (e+f x))^n}{(a+a \sin (e+f x))^2} \, dx=\int { \frac {{\left (B \sin \left (f x + e\right ) + A\right )} {\left (d \sin \left (f x + e\right ) + c\right )}^{n}}{{\left (a \sin \left (f x + e\right ) + a\right )}^{2}} \,d x } \]
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Timed out. \[ \int \frac {(A+B \sin (e+f x)) (c+d \sin (e+f x))^n}{(a+a \sin (e+f x))^2} \, dx=\text {Timed out} \]
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\[ \int \frac {(A+B \sin (e+f x)) (c+d \sin (e+f x))^n}{(a+a \sin (e+f x))^2} \, dx=\int { \frac {{\left (B \sin \left (f x + e\right ) + A\right )} {\left (d \sin \left (f x + e\right ) + c\right )}^{n}}{{\left (a \sin \left (f x + e\right ) + a\right )}^{2}} \,d x } \]
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\[ \int \frac {(A+B \sin (e+f x)) (c+d \sin (e+f x))^n}{(a+a \sin (e+f x))^2} \, dx=\int { \frac {{\left (B \sin \left (f x + e\right ) + A\right )} {\left (d \sin \left (f x + e\right ) + c\right )}^{n}}{{\left (a \sin \left (f x + e\right ) + a\right )}^{2}} \,d x } \]
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Timed out. \[ \int \frac {(A+B \sin (e+f x)) (c+d \sin (e+f x))^n}{(a+a \sin (e+f x))^2} \, dx=\int \frac {\left (A+B\,\sin \left (e+f\,x\right )\right )\,{\left (c+d\,\sin \left (e+f\,x\right )\right )}^n}{{\left (a+a\,\sin \left (e+f\,x\right )\right )}^2} \,d x \]
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