Integrand size = 31, antiderivative size = 191 \[ \int (d \sin (e+f x))^n (a+a \sin (e+f x)) (A+B \sin (e+f x)) \, dx=-\frac {a B \cos (e+f x) (d \sin (e+f x))^{1+n}}{d f (2+n)}+\frac {a (B (1+n)+A (2+n)) \cos (e+f x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1+n}{2},\frac {3+n}{2},\sin ^2(e+f x)\right ) (d \sin (e+f x))^{1+n}}{d f (1+n) (2+n) \sqrt {\cos ^2(e+f x)}}+\frac {a (A+B) \cos (e+f x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2+n}{2},\frac {4+n}{2},\sin ^2(e+f x)\right ) (d \sin (e+f x))^{2+n}}{d^2 f (2+n) \sqrt {\cos ^2(e+f x)}} \]
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Time = 0.16 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.129, Rules used = {3047, 3102, 2827, 2722} \[ \int (d \sin (e+f x))^n (a+a \sin (e+f x)) (A+B \sin (e+f x)) \, dx=\frac {a (A+B) \cos (e+f x) (d \sin (e+f x))^{n+2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {n+2}{2},\frac {n+4}{2},\sin ^2(e+f x)\right )}{d^2 f (n+2) \sqrt {\cos ^2(e+f x)}}+\frac {a (A (n+2)+B (n+1)) \cos (e+f x) (d \sin (e+f x))^{n+1} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {n+1}{2},\frac {n+3}{2},\sin ^2(e+f x)\right )}{d f (n+1) (n+2) \sqrt {\cos ^2(e+f x)}}-\frac {a B \cos (e+f x) (d \sin (e+f x))^{n+1}}{d f (n+2)} \]
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Rule 2722
Rule 2827
Rule 3047
Rule 3102
Rubi steps \begin{align*} \text {integral}& = \int (d \sin (e+f x))^n \left (a A+(a A+a B) \sin (e+f x)+a B \sin ^2(e+f x)\right ) \, dx \\ & = -\frac {a B \cos (e+f x) (d \sin (e+f x))^{1+n}}{d f (2+n)}+\frac {\int (d \sin (e+f x))^n (a d (B (1+n)+A (2+n))+a (A+B) d (2+n) \sin (e+f x)) \, dx}{d (2+n)} \\ & = -\frac {a B \cos (e+f x) (d \sin (e+f x))^{1+n}}{d f (2+n)}+\frac {(a (A+B)) \int (d \sin (e+f x))^{1+n} \, dx}{d}+\frac {(a (B (1+n)+A (2+n))) \int (d \sin (e+f x))^n \, dx}{2+n} \\ & = -\frac {a B \cos (e+f x) (d \sin (e+f x))^{1+n}}{d f (2+n)}+\frac {a (B (1+n)+A (2+n)) \cos (e+f x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1+n}{2},\frac {3+n}{2},\sin ^2(e+f x)\right ) (d \sin (e+f x))^{1+n}}{d f (1+n) (2+n) \sqrt {\cos ^2(e+f x)}}+\frac {a (A+B) \cos (e+f x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2+n}{2},\frac {4+n}{2},\sin ^2(e+f x)\right ) (d \sin (e+f x))^{2+n}}{d^2 f (2+n) \sqrt {\cos ^2(e+f x)}} \\ \end{align*}
Time = 0.66 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.76 \[ \int (d \sin (e+f x))^n (a+a \sin (e+f x)) (A+B \sin (e+f x)) \, dx=\frac {a \cos (e+f x) \sin (e+f x) (d \sin (e+f x))^n \left ((B (1+n)+A (2+n)) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1+n}{2},\frac {3+n}{2},\sin ^2(e+f x)\right )-(1+n) \left (B \sqrt {\cos ^2(e+f x)}-(A+B) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2+n}{2},\frac {4+n}{2},\sin ^2(e+f x)\right ) \sin (e+f x)\right )\right )}{f (1+n) (2+n) \sqrt {\cos ^2(e+f x)}} \]
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\[\int \left (d \sin \left (f x +e \right )\right )^{n} \left (a +a \sin \left (f x +e \right )\right ) \left (A +B \sin \left (f x +e \right )\right )d x\]
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\[ \int (d \sin (e+f x))^n (a+a \sin (e+f x)) (A+B \sin (e+f x)) \, dx=\int { {\left (B \sin \left (f x + e\right ) + A\right )} {\left (a \sin \left (f x + e\right ) + a\right )} \left (d \sin \left (f x + e\right )\right )^{n} \,d x } \]
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Timed out. \[ \int (d \sin (e+f x))^n (a+a \sin (e+f x)) (A+B \sin (e+f x)) \, dx=\text {Timed out} \]
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\[ \int (d \sin (e+f x))^n (a+a \sin (e+f x)) (A+B \sin (e+f x)) \, dx=\int { {\left (B \sin \left (f x + e\right ) + A\right )} {\left (a \sin \left (f x + e\right ) + a\right )} \left (d \sin \left (f x + e\right )\right )^{n} \,d x } \]
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\[ \int (d \sin (e+f x))^n (a+a \sin (e+f x)) (A+B \sin (e+f x)) \, dx=\int { {\left (B \sin \left (f x + e\right ) + A\right )} {\left (a \sin \left (f x + e\right ) + a\right )} \left (d \sin \left (f x + e\right )\right )^{n} \,d x } \]
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Timed out. \[ \int (d \sin (e+f x))^n (a+a \sin (e+f x)) (A+B \sin (e+f x)) \, dx=\int {\left (d\,\sin \left (e+f\,x\right )\right )}^n\,\left (A+B\,\sin \left (e+f\,x\right )\right )\,\left (a+a\,\sin \left (e+f\,x\right )\right ) \,d x \]
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