Optimal. Leaf size=25 \[ \frac {\sin ^6(x)}{6}-\frac {\sin ^8(x)}{4}+\frac {\sin ^{10}(x)}{10} \]
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Rubi [A]
time = 0.03, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2644, 272, 45}
\begin {gather*} \frac {\sin ^{10}(x)}{10}-\frac {\sin ^8(x)}{4}+\frac {\sin ^6(x)}{6} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 272
Rule 2644
Rubi steps
\begin {align*} \int \cos ^5(x) \sin ^5(x) \, dx &=\text {Subst}\left (\int x^5 \left (1-x^2\right )^2 \, dx,x,\sin (x)\right )\\ &=\frac {1}{2} \text {Subst}\left (\int (1-x)^2 x^2 \, dx,x,\sin ^2(x)\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \left (x^2-2 x^3+x^4\right ) \, dx,x,\sin ^2(x)\right )\\ &=\frac {\sin ^6(x)}{6}-\frac {\sin ^8(x)}{4}+\frac {\sin ^{10}(x)}{10}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 25, normalized size = 1.00 \begin {gather*} -\frac {5}{512} \cos (2 x)+\frac {5 \cos (6 x)}{3072}-\frac {\cos (10 x)}{5120} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.02, size = 28, normalized size = 1.12
method | result | size |
risch | \(-\frac {\cos \left (10 x \right )}{5120}+\frac {5 \cos \left (6 x \right )}{3072}-\frac {5 \cos \left (2 x \right )}{512}\) | \(20\) |
default | \(-\frac {\left (\cos ^{6}\left (x \right )\right ) \left (\sin ^{4}\left (x \right )\right )}{10}-\frac {\left (\sin ^{2}\left (x \right )\right ) \left (\cos ^{6}\left (x \right )\right )}{20}-\frac {\left (\cos ^{6}\left (x \right )\right )}{60}\) | \(28\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.93, size = 19, normalized size = 0.76 \begin {gather*} \frac {1}{10} \, \sin \left (x\right )^{10} - \frac {1}{4} \, \sin \left (x\right )^{8} + \frac {1}{6} \, \sin \left (x\right )^{6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.79, size = 19, normalized size = 0.76 \begin {gather*} -\frac {1}{10} \, \cos \left (x\right )^{10} + \frac {1}{4} \, \cos \left (x\right )^{8} - \frac {1}{6} \, \cos \left (x\right )^{6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.01, size = 19, normalized size = 0.76 \begin {gather*} \frac {\sin ^{10}{\left (x \right )}}{10} - \frac {\sin ^{8}{\left (x \right )}}{4} + \frac {\sin ^{6}{\left (x \right )}}{6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.55, size = 19, normalized size = 0.76 \begin {gather*} -\frac {1}{10} \, \cos \left (x\right )^{10} + \frac {1}{4} \, \cos \left (x\right )^{8} - \frac {1}{6} \, \cos \left (x\right )^{6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.05, size = 19, normalized size = 0.76 \begin {gather*} \frac {{\sin \left (x\right )}^{10}}{10}-\frac {{\sin \left (x\right )}^8}{4}+\frac {{\sin \left (x\right )}^6}{6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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