Integrand size = 12, antiderivative size = 77 \[ \int \left (b \sin ^n(c+d x)\right )^p \, dx=\frac {\cos (c+d x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2} (1+n p),\frac {1}{2} (3+n p),\sin ^2(c+d x)\right ) \sin (c+d x) \left (b \sin ^n(c+d x)\right )^p}{d (1+n p) \sqrt {\cos ^2(c+d x)}} \]
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Time = 0.02 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3287, 2722} \[ \int \left (b \sin ^n(c+d x)\right )^p \, dx=\frac {\sin (c+d x) \cos (c+d x) \left (b \sin ^n(c+d x)\right )^p \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2} (n p+1),\frac {1}{2} (n p+3),\sin ^2(c+d x)\right )}{d (n p+1) \sqrt {\cos ^2(c+d x)}} \]
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Rule 2722
Rule 3287
Rubi steps \begin{align*} \text {integral}& = \left (\sin ^{-n p}(c+d x) \left (b \sin ^n(c+d x)\right )^p\right ) \int \sin ^{n p}(c+d x) \, dx \\ & = \frac {\cos (c+d x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2} (1+n p),\frac {1}{2} (3+n p),\sin ^2(c+d x)\right ) \sin (c+d x) \left (b \sin ^n(c+d x)\right )^p}{d (1+n p) \sqrt {\cos ^2(c+d x)}} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.92 \[ \int \left (b \sin ^n(c+d x)\right )^p \, dx=\frac {\sqrt {\cos ^2(c+d x)} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2} (1+n p),\frac {1}{2} (3+n p),\sin ^2(c+d x)\right ) \left (b \sin ^n(c+d x)\right )^p \tan (c+d x)}{d (1+n p)} \]
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\[\int {\left (b \left (\sin ^{n}\left (d x +c \right )\right )\right )}^{p}d x\]
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\[ \int \left (b \sin ^n(c+d x)\right )^p \, dx=\int { \left (b \sin \left (d x + c\right )^{n}\right )^{p} \,d x } \]
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\[ \int \left (b \sin ^n(c+d x)\right )^p \, dx=\int \left (b \sin ^{n}{\left (c + d x \right )}\right )^{p}\, dx \]
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\[ \int \left (b \sin ^n(c+d x)\right )^p \, dx=\int { \left (b \sin \left (d x + c\right )^{n}\right )^{p} \,d x } \]
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\[ \int \left (b \sin ^n(c+d x)\right )^p \, dx=\int { \left (b \sin \left (d x + c\right )^{n}\right )^{p} \,d x } \]
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Timed out. \[ \int \left (b \sin ^n(c+d x)\right )^p \, dx=\int {\left (b\,{\sin \left (c+d\,x\right )}^n\right )}^p \,d x \]
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