Optimal. Leaf size=44 \[ \frac {\cos (x) \, _2F_1\left (\frac {1}{2},\frac {1+p}{2};\frac {3+p}{2};\sin ^2(x)\right ) \sin ^{1+p}(x)}{(1+p) \sqrt {\cos ^2(x)}} \]
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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 = {2722}
\begin {gather*} \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)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2722
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 = 0.03, size = 44, normalized size = 1.00 \begin {gather*} -\cos (x) \, _2F_1\left (\frac {1}{2},\frac {1-p}{2};\frac {3}{2};\cos ^2(x)\right ) \sin ^{1+p}(x) \sin ^2(x)^{\frac {1}{2} (-1-p)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \sin ^{p}\left (x \right )\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \sin ^{p}{\left (x \right )}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} -\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}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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