Integrand size = 27, antiderivative size = 188 \[ \int \frac {\left (c (d \sin (e+f x))^p\right )^n}{3+3 \sin (e+f x)} \, dx=\frac {\cos (e+f x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {n p}{2},\frac {1}{2} (2+n p),\sin ^2(e+f x)\right ) \left (c (d \sin (e+f x))^p\right )^n}{3 f \sqrt {\cos ^2(e+f x)}}-\frac {n p \cos (e+f x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2} (1+n p),\frac {1}{2} (3+n p),\sin ^2(e+f x)\right ) \sin (e+f x) \left (c (d \sin (e+f x))^p\right )^n}{3 f (1+n p) \sqrt {\cos ^2(e+f x)}}-\frac {\cos (e+f x) \left (c (d \sin (e+f x))^p\right )^n}{f (3+3 \sin (e+f x))} \]
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Time = 0.17 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.01, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {2905, 2848, 2827, 2722} \[ \int \frac {\left (c (d \sin (e+f x))^p\right )^n}{3+3 \sin (e+f x)} \, dx=\frac {\cos (e+f x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {n p}{2},\frac {1}{2} (n p+2),\sin ^2(e+f x)\right ) \left (c (d \sin (e+f x))^p\right )^n}{a f \sqrt {\cos ^2(e+f x)}}-\frac {n p \sin (e+f x) \cos (e+f x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2} (n p+1),\frac {1}{2} (n p+3),\sin ^2(e+f x)\right ) \left (c (d \sin (e+f x))^p\right )^n}{a f (n p+1) \sqrt {\cos ^2(e+f x)}}-\frac {\cos (e+f x) \left (c (d \sin (e+f x))^p\right )^n}{f (a \sin (e+f x)+a)} \]
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Rule 2722
Rule 2827
Rule 2848
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 \frac {(d \sin (e+f x))^{n p}}{a+a \sin (e+f x)} \, dx \\ & = -\frac {\cos (e+f x) \left (c (d \sin (e+f x))^p\right )^n}{f (a+a \sin (e+f x))}+\frac {\left (d n p (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} (a-a \sin (e+f x)) \, dx}{a^2} \\ & = -\frac {\cos (e+f x) \left (c (d \sin (e+f x))^p\right )^n}{f (a+a \sin (e+f x))}-\frac {\left (n p (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}{a}+\frac {\left (d n p (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}{a} \\ & = \frac {\cos (e+f x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {n p}{2},\frac {1}{2} (2+n p),\sin ^2(e+f x)\right ) \left (c (d \sin (e+f x))^p\right )^n}{a f \sqrt {\cos ^2(e+f x)}}-\frac {n p \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}{a f (1+n p) \sqrt {\cos ^2(e+f x)}}-\frac {\cos (e+f x) \left (c (d \sin (e+f x))^p\right )^n}{f (a+a \sin (e+f x))} \\ \end{align*}
Time = 0.17 (sec) , antiderivative size = 157, normalized size of antiderivative = 0.84 \[ \int \frac {\left (c (d \sin (e+f x))^p\right )^n}{3+3 \sin (e+f x)} \, dx=\frac {\cos (e+f x) \sqrt {\cos ^2(e+f x)} \sin (e+f x) \left (c (d \sin (e+f x))^p\right )^n \left (-\left ((2+n p) \operatorname {Hypergeometric2F1}\left (\frac {3}{2},\frac {1}{2} (1+n p),\frac {1}{2} (3+n p),\sin ^2(e+f x)\right )\right )+(1+n p) \operatorname {Hypergeometric2F1}\left (\frac {3}{2},1+\frac {n p}{2},2+\frac {n p}{2},\sin ^2(e+f x)\right ) \sin (e+f x)\right )}{3 f (1+n p) (2+n p) (-1+\sin (e+f x)) (1+\sin (e+f x))} \]
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\[\int \frac {\left (c \left (d \sin \left (f x +e \right )\right )^{p}\right )^{n}}{a +a \sin \left (f x +e \right )}d x\]
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\[ \int \frac {\left (c (d \sin (e+f x))^p\right )^n}{3+3 \sin (e+f x)} \, dx=\int { \frac {\left (\left (d \sin \left (f x + e\right )\right )^{p} c\right )^{n}}{a \sin \left (f x + e\right ) + a} \,d x } \]
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\[ \int \frac {\left (c (d \sin (e+f x))^p\right )^n}{3+3 \sin (e+f x)} \, dx=\frac {\int \frac {\left (c \left (d \sin {\left (e + f x \right )}\right )^{p}\right )^{n}}{\sin {\left (e + f x \right )} + 1}\, dx}{a} \]
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\[ \int \frac {\left (c (d \sin (e+f x))^p\right )^n}{3+3 \sin (e+f x)} \, dx=\int { \frac {\left (\left (d \sin \left (f x + e\right )\right )^{p} c\right )^{n}}{a \sin \left (f x + e\right ) + a} \,d x } \]
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\[ \int \frac {\left (c (d \sin (e+f x))^p\right )^n}{3+3 \sin (e+f x)} \, dx=\int { \frac {\left (\left (d \sin \left (f x + e\right )\right )^{p} c\right )^{n}}{a \sin \left (f x + e\right ) + a} \,d x } \]
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Timed out. \[ \int \frac {\left (c (d \sin (e+f x))^p\right )^n}{3+3 \sin (e+f x)} \, dx=\int \frac {{\left (c\,{\left (d\,\sin \left (e+f\,x\right )\right )}^p\right )}^n}{a+a\,\sin \left (e+f\,x\right )} \,d x \]
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