Optimal. Leaf size=68 \[ \frac {\sin ^{2 m+1}(x) \cos ^{2 m-1}(x) \cos ^2(x)^{\frac {1}{2}-m} \, _2F_1\left (\frac {1}{2} (1-2 m),\frac {1}{2} (2 m+1);\frac {1}{2} (2 m+3);\sin ^2(x)\right )}{2 m+1} \]
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Rubi [A] time = 0.04, antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {2577} \[ \frac {\sin ^{2 m+1}(x) \cos ^{2 m-1}(x) \cos ^2(x)^{\frac {1}{2}-m} \text {Hypergeometric2F1}\left (\frac {1}{2} (1-2 m),\frac {1}{2} (2 m+1),\frac {1}{2} (2 m+3),\sin ^2(x)\right )}{2 m+1} \]
Antiderivative was successfully verified.
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Rule 2577
Rubi steps
\begin {align*} \int \cos ^{2 m}(x) \sin ^{2 m}(x) \, dx &=\frac {\cos ^{-1+2 m}(x) \cos ^2(x)^{\frac {1}{2}-m} \, _2F_1\left (\frac {1}{2} (1-2 m),\frac {1}{2} (1+2 m);\frac {1}{2} (3+2 m);\sin ^2(x)\right ) \sin ^{1+2 m}(x)}{1+2 m}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 58, normalized size = 0.85 \[ \frac {\sin ^{2 m+1}(x) \cos ^{2 m-1}(x) \cos ^2(x)^{\frac {1}{2}-m} \, _2F_1\left (\frac {1}{2}-m,m+\frac {1}{2};m+\frac {3}{2};\sin ^2(x)\right )}{2 m+1} \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \cos ^{2 m}(x) \sin ^{2 m}(x) \, dx \]
Verification is Not applicable to the result.
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fricas [F] time = 0.89, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\cos \relax (x)^{2 \, m} \sin \relax (x)^{2 \, m}, x\right ) \]
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 \[ \int \cos \relax (x)^{2 \, m} \sin \relax (x)^{2 \, m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.03, size = 0, normalized size = 0.00 \[\int \left (\cos ^{2 m}\relax (x )\right ) \left (\sin ^{2 m}\relax (x )\right )\, dx\]
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 \[ \int \cos \relax (x)^{2 \, m} \sin \relax (x)^{2 \, m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.75, size = 52, normalized size = 0.76 \[ -\frac {{\cos \relax (x)}^{2\,m+1}\,{\sin \relax (x)}^{2\,m+1}\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2}-m,m+\frac {1}{2};\ m+\frac {3}{2};\ {\cos \relax (x)}^2\right )}{\left (2\,m+1\right )\,{\left ({\sin \relax (x)}^2\right )}^{m+\frac {1}{2}}} \]
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 \[ \int \sin ^{2 m}{\relax (x )} \cos ^{2 m}{\relax (x )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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