Optimal. Leaf size=25 \[ \frac{\sin ^{10}(x)}{10}-\frac{\sin ^8(x)}{4}+\frac{\sin ^6(x)}{6} \]
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Rubi [A] time = 0.0299897, antiderivative size = 25, normalized size of antiderivative = 1., 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 = {2564, 266, 43} \[ \frac{\sin ^{10}(x)}{10}-\frac{\sin ^8(x)}{4}+\frac{\sin ^6(x)}{6} \]
Antiderivative was successfully verified.
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Rule 2564
Rule 266
Rule 43
Rubi steps
\begin{align*} \int \cos ^5(x) \sin ^5(x) \, dx &=\operatorname{Subst}\left (\int x^5 \left (1-x^2\right )^2 \, dx,x,\sin (x)\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int (1-x)^2 x^2 \, dx,x,\sin ^2(x)\right )\\ &=\frac{1}{2} \operatorname{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*}
Mathematica [A] time = 0.0102496, size = 25, normalized size = 1. \[ -\frac{5}{512} \cos (2 x)+\frac{5 \cos (6 x)}{3072}-\frac{\cos (10 x)}{5120} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 28, normalized size = 1.1 \begin{align*} -{\frac{ \left ( \cos \left ( x \right ) \right ) ^{6} \left ( \sin \left ( x \right ) \right ) ^{4}}{10}}-{\frac{ \left ( \sin \left ( x \right ) \right ) ^{2} \left ( \cos \left ( x \right ) \right ) ^{6}}{20}}-{\frac{ \left ( \cos \left ( x \right ) \right ) ^{6}}{60}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.927547, size = 26, normalized size = 1.04 \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.00251, size = 63, normalized size = 2.52 \begin{align*} -\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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.067664, size = 19, normalized size = 0.76 \begin{align*} \frac{\sin ^{10}{\left (x \right )}}{10} - \frac{\sin ^{8}{\left (x \right )}}{4} + \frac{\sin ^{6}{\left (x \right )}}{6} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.05139, size = 26, normalized size = 1.04 \begin{align*} -\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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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