Integrand size = 33, antiderivative size = 277 \[ \int (d \sin (e+f x))^n (a+a \sin (e+f x))^2 (A+B \sin (e+f x)) \, dx=-\frac {a^2 (A (3+n)+B (4+n)) \cos (e+f x) (d \sin (e+f x))^{1+n}}{d f (2+n) (3+n)}+\frac {a^2 (2 B (1+n)+A (3+2 n)) \cos (e+f x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1+n}{2},\frac {3+n}{2},\sin ^2(e+f x)\right ) (d \sin (e+f x))^{1+n}}{d f (1+n) (2+n) \sqrt {\cos ^2(e+f x)}}+\frac {a^2 (2 A (3+n)+B (5+2 n)) \cos (e+f x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2+n}{2},\frac {4+n}{2},\sin ^2(e+f x)\right ) (d \sin (e+f x))^{2+n}}{d^2 f (2+n) (3+n) \sqrt {\cos ^2(e+f x)}}-\frac {B \cos (e+f x) (d \sin (e+f x))^{1+n} \left (a^2+a^2 \sin (e+f x)\right )}{d f (3+n)} \]
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Time = 0.35 (sec) , antiderivative size = 277, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.152, Rules used = {3055, 3047, 3102, 2827, 2722} \[ \int (d \sin (e+f x))^n (a+a \sin (e+f x))^2 (A+B \sin (e+f x)) \, dx=\frac {a^2 (2 A (n+3)+B (2 n+5)) \cos (e+f x) (d \sin (e+f x))^{n+2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {n+2}{2},\frac {n+4}{2},\sin ^2(e+f x)\right )}{d^2 f (n+2) (n+3) \sqrt {\cos ^2(e+f x)}}+\frac {a^2 (A (2 n+3)+2 B (n+1)) \cos (e+f x) (d \sin (e+f x))^{n+1} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {n+1}{2},\frac {n+3}{2},\sin ^2(e+f x)\right )}{d f (n+1) (n+2) \sqrt {\cos ^2(e+f x)}}-\frac {a^2 (A (n+3)+B (n+4)) \cos (e+f x) (d \sin (e+f x))^{n+1}}{d f (n+2) (n+3)}-\frac {B \cos (e+f x) \left (a^2 \sin (e+f x)+a^2\right ) (d \sin (e+f x))^{n+1}}{d f (n+3)} \]
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Rule 2722
Rule 2827
Rule 3047
Rule 3055
Rule 3102
Rubi steps \begin{align*} \text {integral}& = -\frac {B \cos (e+f x) (d \sin (e+f x))^{1+n} \left (a^2+a^2 \sin (e+f x)\right )}{d f (3+n)}+\frac {\int (d \sin (e+f x))^n (a+a \sin (e+f x)) (a d (B (1+n)+A (3+n))+a d (A (3+n)+B (4+n)) \sin (e+f x)) \, dx}{d (3+n)} \\ & = -\frac {B \cos (e+f x) (d \sin (e+f x))^{1+n} \left (a^2+a^2 \sin (e+f x)\right )}{d f (3+n)}+\frac {\int (d \sin (e+f x))^n \left (a^2 d (B (1+n)+A (3+n))+\left (a^2 d (B (1+n)+A (3+n))+a^2 d (A (3+n)+B (4+n))\right ) \sin (e+f x)+a^2 d (A (3+n)+B (4+n)) \sin ^2(e+f x)\right ) \, dx}{d (3+n)} \\ & = -\frac {a^2 (A (3+n)+B (4+n)) \cos (e+f x) (d \sin (e+f x))^{1+n}}{d f (2+n) (3+n)}-\frac {B \cos (e+f x) (d \sin (e+f x))^{1+n} \left (a^2+a^2 \sin (e+f x)\right )}{d f (3+n)}+\frac {\int (d \sin (e+f x))^n \left (a^2 d^2 (3+n) (2 B (1+n)+A (3+2 n))+a^2 d^2 (2+n) (2 A (3+n)+B (5+2 n)) \sin (e+f x)\right ) \, dx}{d^2 (2+n) (3+n)} \\ & = -\frac {a^2 (A (3+n)+B (4+n)) \cos (e+f x) (d \sin (e+f x))^{1+n}}{d f (2+n) (3+n)}-\frac {B \cos (e+f x) (d \sin (e+f x))^{1+n} \left (a^2+a^2 \sin (e+f x)\right )}{d f (3+n)}+\frac {\left (a^2 (2 B (1+n)+A (3+2 n))\right ) \int (d \sin (e+f x))^n \, dx}{2+n}+\frac {\left (a^2 (2 A (3+n)+B (5+2 n))\right ) \int (d \sin (e+f x))^{1+n} \, dx}{d (3+n)} \\ & = -\frac {a^2 (A (3+n)+B (4+n)) \cos (e+f x) (d \sin (e+f x))^{1+n}}{d f (2+n) (3+n)}+\frac {a^2 (2 B (1+n)+A (3+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^2 (2 A (3+n)+B (5+2 n)) \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) (3+n) \sqrt {\cos ^2(e+f x)}}-\frac {B \cos (e+f x) (d \sin (e+f x))^{1+n} \left (a^2+a^2 \sin (e+f x)\right )}{d f (3+n)} \\ \end{align*}
Time = 0.98 (sec) , antiderivative size = 204, normalized size of antiderivative = 0.74 \[ \int (d \sin (e+f x))^n (a+a \sin (e+f x))^2 (A+B \sin (e+f x)) \, dx=\frac {a^2 \cos (e+f x) \sin (e+f x) (d \sin (e+f x))^n \left (\frac {A \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1+n}{2},\frac {3+n}{2},\sin ^2(e+f x)\right )}{1+n}+\sin (e+f x) \left (\frac {(2 A+B) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2+n}{2},\frac {4+n}{2},\sin ^2(e+f x)\right )}{2+n}+\sin (e+f x) \left (\frac {(A+2 B) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3+n}{2},\frac {5+n}{2},\sin ^2(e+f x)\right )}{3+n}+\frac {B \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {4+n}{2},\frac {6+n}{2},\sin ^2(e+f x)\right ) \sin (e+f x)}{4+n}\right )\right )\right )}{f \sqrt {\cos ^2(e+f x)}} \]
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\[\int \left (d \sin \left (f x +e \right )\right )^{n} \left (a +a \sin \left (f x +e \right )\right )^{2} \left (A +B \sin \left (f x +e \right )\right )d x\]
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\[ \int (d \sin (e+f x))^n (a+a \sin (e+f x))^2 (A+B \sin (e+f x)) \, dx=\int { {\left (B \sin \left (f x + e\right ) + A\right )} {\left (a \sin \left (f x + e\right ) + a\right )}^{2} \left (d \sin \left (f x + e\right )\right )^{n} \,d x } \]
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Timed out. \[ \int (d \sin (e+f x))^n (a+a \sin (e+f x))^2 (A+B \sin (e+f x)) \, dx=\text {Timed out} \]
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\[ \int (d \sin (e+f x))^n (a+a \sin (e+f x))^2 (A+B \sin (e+f x)) \, dx=\int { {\left (B \sin \left (f x + e\right ) + A\right )} {\left (a \sin \left (f x + e\right ) + a\right )}^{2} \left (d \sin \left (f x + e\right )\right )^{n} \,d x } \]
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\[ \int (d \sin (e+f x))^n (a+a \sin (e+f x))^2 (A+B \sin (e+f x)) \, dx=\int { {\left (B \sin \left (f x + e\right ) + A\right )} {\left (a \sin \left (f x + e\right ) + a\right )}^{2} \left (d \sin \left (f x + e\right )\right )^{n} \,d x } \]
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Timed out. \[ \int (d \sin (e+f x))^n (a+a \sin (e+f x))^2 (A+B \sin (e+f x)) \, dx=\int {\left (d\,\sin \left (e+f\,x\right )\right )}^n\,\left (A+B\,\sin \left (e+f\,x\right )\right )\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^2 \,d x \]
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