Optimal. Leaf size=25 \[ -\frac{1}{7} \cos ^7(x)+\frac{2 \cos ^5(x)}{5}-\frac{\cos ^3(x)}{3} \]
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Rubi [A] time = 0.0265598, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {2565, 270} \[ -\frac{1}{7} \cos ^7(x)+\frac{2 \cos ^5(x)}{5}-\frac{\cos ^3(x)}{3} \]
Antiderivative was successfully verified.
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Rule 2565
Rule 270
Rubi steps
\begin{align*} \int \cos ^2(x) \sin ^5(x) \, dx &=-\operatorname{Subst}\left (\int x^2 \left (1-x^2\right )^2 \, dx,x,\cos (x)\right )\\ &=-\operatorname{Subst}\left (\int \left (x^2-2 x^4+x^6\right ) \, dx,x,\cos (x)\right )\\ &=-\frac{1}{3} \cos ^3(x)+\frac{2 \cos ^5(x)}{5}-\frac{\cos ^7(x)}{7}\\ \end{align*}
Mathematica [A] time = 0.0099994, size = 31, normalized size = 1.24 \[ -\frac{5 \cos (x)}{64}-\frac{1}{192} \cos (3 x)+\frac{3}{320} \cos (5 x)-\frac{1}{448} \cos (7 x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 28, normalized size = 1.1 \begin{align*} -{\frac{ \left ( \cos \left ( x \right ) \right ) ^{3} \left ( \sin \left ( x \right ) \right ) ^{4}}{7}}-{\frac{4\, \left ( \sin \left ( x \right ) \right ) ^{2} \left ( \cos \left ( x \right ) \right ) ^{3}}{35}}-{\frac{8\, \left ( \cos \left ( x \right ) \right ) ^{3}}{105}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.949246, size = 26, normalized size = 1.04 \begin{align*} -\frac{1}{7} \, \cos \left (x\right )^{7} + \frac{2}{5} \, \cos \left (x\right )^{5} - \frac{1}{3} \, \cos \left (x\right )^{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.07527, size = 61, normalized size = 2.44 \begin{align*} -\frac{1}{7} \, \cos \left (x\right )^{7} + \frac{2}{5} \, \cos \left (x\right )^{5} - \frac{1}{3} \, \cos \left (x\right )^{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.062005, size = 20, normalized size = 0.8 \begin{align*} - \frac{\cos ^{7}{\left (x \right )}}{7} + \frac{2 \cos ^{5}{\left (x \right )}}{5} - \frac{\cos ^{3}{\left (x \right )}}{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.0583, size = 26, normalized size = 1.04 \begin{align*} -\frac{1}{7} \, \cos \left (x\right )^{7} + \frac{2}{5} \, \cos \left (x\right )^{5} - \frac{1}{3} \, \cos \left (x\right )^{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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