Optimal. Leaf size=44 \[ \frac{\cos (x) \sin ^{p+1}(x) \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{p+1}{2},\frac{p+3}{2},\sin ^2(x)\right )}{(p+1) \sqrt{\cos ^2(x)}} \]
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Rubi [A] time = 0.0076611, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 4, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {2643} \[ \frac{\cos (x) \sin ^{p+1}(x) \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{p+1}{2},\frac{p+3}{2},\sin ^2(x)\right )}{(p+1) \sqrt{\cos ^2(x)}} \]
Antiderivative was successfully verified.
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Rule 2643
Rubi steps
\begin{align*} \int \sin ^p(x) \, dx &=\frac{\cos (x) \, _2F_1\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*}
Mathematica [A] time = 0.0395527, size = 44, normalized size = 1. \[ -\cos (x) \sin ^{p+1}(x) \sin ^2(x)^{\frac{1}{2} (-p-1)} \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{1-p}{2},\frac{3}{2},\cos ^2(x)\right ) \]
Antiderivative was successfully verified.
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Maple [F] time = 0.385, size = 0, normalized size = 0. \begin{align*} \int \left ( \sin \left ( x \right ) \right ) ^{p}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sin \left (x\right )^{p}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\sin \left (x\right )^{p}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sin ^{p}{\left (x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sin \left (x\right )^{p}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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