Optimal. Leaf size=44 \[ \frac {\cos (x) \sin ^{p+1}(x) \, _2F_1\left (\frac {1}{2},\frac {p+1}{2};\frac {p+3}{2};\sin ^2(x)\right )}{(p+1) \sqrt {\cos ^2(x)}} \]
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Rubi [A] time = 0.01, antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, 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*}
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Mathematica [A] time = 0.04, size = 44, normalized size = 1.00 \[ -\cos (x) \sin ^{p+1}(x) \sin ^2(x)^{\frac {1}{2} (-p-1)} \, _2F_1\left (\frac {1}{2},\frac {1-p}{2};\frac {3}{2};\cos ^2(x)\right ) \]
Antiderivative was successfully verified.
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fricas [F] time = 0.46, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sin \relax (x)^{p}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sin \relax (x)^{p}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.53, size = 0, normalized size = 0.00 \[ \int \sin ^{p}\relax (x )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sin \relax (x)^{p}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.32, size = 35, normalized size = 0.80 \[ -\frac {\cos \relax (x)\,{\sin \relax (x)}^{p+1}\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {1}{2}-\frac {p}{2};\ \frac {3}{2};\ {\cos \relax (x)}^2\right )}{{\left ({\sin \relax (x)}^2\right )}^{\frac {p}{2}+\frac {1}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sin ^{p}{\relax (x )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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