Optimal. Leaf size=17 \[ \frac{\cos ^7(x)}{7}-\frac{\cos ^5(x)}{5} \]
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Rubi [A] time = 0.022827, antiderivative size = 17, 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, 14} \[ \frac{\cos ^7(x)}{7}-\frac{\cos ^5(x)}{5} \]
Antiderivative was successfully verified.
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Rule 2565
Rule 14
Rubi steps
\begin{align*} \int \cos ^4(x) \sin ^3(x) \, dx &=-\operatorname{Subst}\left (\int x^4 \left (1-x^2\right ) \, dx,x,\cos (x)\right )\\ &=-\operatorname{Subst}\left (\int \left (x^4-x^6\right ) \, dx,x,\cos (x)\right )\\ &=-\frac{1}{5} \cos ^5(x)+\frac{\cos ^7(x)}{7}\\ \end{align*}
Mathematica [A] time = 0.009907, size = 31, normalized size = 1.82 \[ -\frac{3 \cos (x)}{64}-\frac{1}{64} \cos (3 x)+\frac{1}{320} \cos (5 x)+\frac{1}{448} \cos (7 x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 18, normalized size = 1.1 \begin{align*} -{\frac{ \left ( \cos \left ( x \right ) \right ) ^{5} \left ( \sin \left ( x \right ) \right ) ^{2}}{7}}-{\frac{2\, \left ( \cos \left ( x \right ) \right ) ^{5}}{35}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.95728, size = 18, normalized size = 1.06 \begin{align*} \frac{1}{7} \, \cos \left (x\right )^{7} - \frac{1}{5} \, \cos \left (x\right )^{5} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.21902, size = 39, normalized size = 2.29 \begin{align*} \frac{1}{7} \, \cos \left (x\right )^{7} - \frac{1}{5} \, \cos \left (x\right )^{5} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.061924, size = 12, normalized size = 0.71 \begin{align*} \frac{\cos ^{7}{\left (x \right )}}{7} - \frac{\cos ^{5}{\left (x \right )}}{5} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.0605, size = 18, normalized size = 1.06 \begin{align*} \frac{1}{7} \, \cos \left (x\right )^{7} - \frac{1}{5} \, \cos \left (x\right )^{5} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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