Integrand size = 4, antiderivative size = 44 \[ \int \sin ^p(x) \, dx=\frac {\cos (x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1+p}{2},\frac {3+p}{2},\sin ^2(x)\right ) \sin ^{1+p}(x)}{(1+p) \sqrt {\cos ^2(x)}} \]
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Time = 0.01 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2722} \[ \int \sin ^p(x) \, dx=\frac {\cos (x) \sin ^{p+1}(x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {p+1}{2},\frac {p+3}{2},\sin ^2(x)\right )}{(p+1) \sqrt {\cos ^2(x)}} \]
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Rule 2722
Rubi steps \begin{align*} \text {integral}& = \frac {\cos (x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1+p}{2},\frac {3+p}{2},\sin ^2(x)\right ) \sin ^{1+p}(x)}{(1+p) \sqrt {\cos ^2(x)}} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.00 \[ \int \sin ^p(x) \, dx=\frac {\sqrt {\cos ^2(x)} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1+p}{2},\frac {3+p}{2},\sin ^2(x)\right ) \sec (x) \sin ^{1+p}(x)}{1+p} \]
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\[\int \left (\sin ^{p}\left (x \right )\right )d x\]
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\[ \int \sin ^p(x) \, dx=\int { \sin \left (x\right )^{p} \,d x } \]
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\[ \int \sin ^p(x) \, dx=\int \sin ^{p}{\left (x \right )}\, dx \]
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\[ \int \sin ^p(x) \, dx=\int { \sin \left (x\right )^{p} \,d x } \]
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\[ \int \sin ^p(x) \, dx=\int { \sin \left (x\right )^{p} \,d x } \]
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Time = 0.31 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.80 \[ \int \sin ^p(x) \, dx=-\frac {\cos \left (x\right )\,{\sin \left (x\right )}^{p+1}\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {1}{2}-\frac {p}{2};\ \frac {3}{2};\ {\cos \left (x\right )}^2\right )}{{\left ({\sin \left (x\right )}^2\right )}^{\frac {p}{2}+\frac {1}{2}}} \]
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