Integrand size = 25, antiderivative size = 163 \[ \int \left (c (d \sin (e+f x))^p\right )^n (3+3 \sin (e+f x)) \, dx=\frac {3 \cos (e+f x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2} (1+n p),\frac {1}{2} (3+n p),\sin ^2(e+f x)\right ) \sin (e+f x) \left (c (d \sin (e+f x))^p\right )^n}{f (1+n p) \sqrt {\cos ^2(e+f x)}}+\frac {3 \cos (e+f x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2} (2+n p),\frac {1}{2} (4+n p),\sin ^2(e+f x)\right ) \sin ^2(e+f x) \left (c (d \sin (e+f x))^p\right )^n}{f (2+n p) \sqrt {\cos ^2(e+f x)}} \]
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Time = 0.08 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {2905, 2827, 2722} \[ \int \left (c (d \sin (e+f x))^p\right )^n (3+3 \sin (e+f x)) \, dx=\frac {a \sin ^2(e+f x) \cos (e+f x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2} (n p+2),\frac {1}{2} (n p+4),\sin ^2(e+f x)\right ) \left (c (d \sin (e+f x))^p\right )^n}{f (n p+2) \sqrt {\cos ^2(e+f x)}}+\frac {a \sin (e+f x) \cos (e+f x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2} (n p+1),\frac {1}{2} (n p+3),\sin ^2(e+f x)\right ) \left (c (d \sin (e+f x))^p\right )^n}{f (n p+1) \sqrt {\cos ^2(e+f x)}} \]
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Rule 2722
Rule 2827
Rule 2905
Rubi steps \begin{align*} \text {integral}& = \left ((d \sin (e+f x))^{-n p} \left (c (d \sin (e+f x))^p\right )^n\right ) \int (d \sin (e+f x))^{n p} (a+a \sin (e+f x)) \, dx \\ & = \left (a (d \sin (e+f x))^{-n p} \left (c (d \sin (e+f x))^p\right )^n\right ) \int (d \sin (e+f x))^{n p} \, dx+\frac {\left (a (d \sin (e+f x))^{-n p} \left (c (d \sin (e+f x))^p\right )^n\right ) \int (d \sin (e+f x))^{1+n p} \, dx}{d} \\ & = \frac {a \cos (e+f x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2} (1+n p),\frac {1}{2} (3+n p),\sin ^2(e+f x)\right ) \sin (e+f x) \left (c (d \sin (e+f x))^p\right )^n}{f (1+n p) \sqrt {\cos ^2(e+f x)}}+\frac {a \cos (e+f x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2} (2+n p),\frac {1}{2} (4+n p),\sin ^2(e+f x)\right ) \sin ^2(e+f x) \left (c (d \sin (e+f x))^p\right )^n}{f (2+n p) \sqrt {\cos ^2(e+f x)}} \\ \end{align*}
Time = 0.41 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.79 \[ \int \left (c (d \sin (e+f x))^p\right )^n (3+3 \sin (e+f x)) \, dx=\frac {3 \sqrt {\cos ^2(e+f x)} \left (c (d \sin (e+f x))^p\right )^n \left ((2+n p) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2} (1+n p),\frac {1}{2} (3+n p),\sin ^2(e+f x)\right )+(1+n p) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},1+\frac {n p}{2},2+\frac {n p}{2},\sin ^2(e+f x)\right ) \sin (e+f x)\right ) \tan (e+f x)}{f (1+n p) (2+n p)} \]
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\[\int \left (c \left (d \sin \left (f x +e \right )\right )^{p}\right )^{n} \left (a +a \sin \left (f x +e \right )\right )d x\]
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\[ \int \left (c (d \sin (e+f x))^p\right )^n (3+3 \sin (e+f x)) \, dx=\int { {\left (a \sin \left (f x + e\right ) + a\right )} \left (\left (d \sin \left (f x + e\right )\right )^{p} c\right )^{n} \,d x } \]
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\[ \int \left (c (d \sin (e+f x))^p\right )^n (3+3 \sin (e+f x)) \, dx=a \left (\int \left (c \left (d \sin {\left (e + f x \right )}\right )^{p}\right )^{n}\, dx + \int \left (c \left (d \sin {\left (e + f x \right )}\right )^{p}\right )^{n} \sin {\left (e + f x \right )}\, dx\right ) \]
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\[ \int \left (c (d \sin (e+f x))^p\right )^n (3+3 \sin (e+f x)) \, dx=\int { {\left (a \sin \left (f x + e\right ) + a\right )} \left (\left (d \sin \left (f x + e\right )\right )^{p} c\right )^{n} \,d x } \]
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\[ \int \left (c (d \sin (e+f x))^p\right )^n (3+3 \sin (e+f x)) \, dx=\int { {\left (a \sin \left (f x + e\right ) + a\right )} \left (\left (d \sin \left (f x + e\right )\right )^{p} c\right )^{n} \,d x } \]
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Timed out. \[ \int \left (c (d \sin (e+f x))^p\right )^n (3+3 \sin (e+f x)) \, dx=\int {\left (c\,{\left (d\,\sin \left (e+f\,x\right )\right )}^p\right )}^n\,\left (a+a\,\sin \left (e+f\,x\right )\right ) \,d x \]
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