Optimal. Leaf size=34 \[ \frac{x}{16}-\frac{1}{6} \sin (x) \cos ^5(x)+\frac{1}{24} \sin (x) \cos ^3(x)+\frac{1}{16} \sin (x) \cos (x) \]
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Rubi [A] time = 0.0326611, antiderivative size = 34, 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 = {2568, 2635, 8} \[ \frac{x}{16}-\frac{1}{6} \sin (x) \cos ^5(x)+\frac{1}{24} \sin (x) \cos ^3(x)+\frac{1}{16} \sin (x) \cos (x) \]
Antiderivative was successfully verified.
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Rule 2568
Rule 2635
Rule 8
Rubi steps
\begin{align*} \int \cos ^4(x) \sin ^2(x) \, dx &=-\frac{1}{6} \cos ^5(x) \sin (x)+\frac{1}{6} \int \cos ^4(x) \, dx\\ &=\frac{1}{24} \cos ^3(x) \sin (x)-\frac{1}{6} \cos ^5(x) \sin (x)+\frac{1}{8} \int \cos ^2(x) \, dx\\ &=\frac{1}{16} \cos (x) \sin (x)+\frac{1}{24} \cos ^3(x) \sin (x)-\frac{1}{6} \cos ^5(x) \sin (x)+\frac{\int 1 \, dx}{16}\\ &=\frac{x}{16}+\frac{1}{16} \cos (x) \sin (x)+\frac{1}{24} \cos ^3(x) \sin (x)-\frac{1}{6} \cos ^5(x) \sin (x)\\ \end{align*}
Mathematica [A] time = 0.0078148, size = 30, normalized size = 0.88 \[ \frac{x}{16}+\frac{1}{64} \sin (2 x)-\frac{1}{64} \sin (4 x)-\frac{1}{192} \sin (6 x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 26, normalized size = 0.8 \begin{align*} -{\frac{ \left ( \cos \left ( x \right ) \right ) ^{5}\sin \left ( x \right ) }{6}}+{\frac{\sin \left ( x \right ) }{24} \left ( \left ( \cos \left ( x \right ) \right ) ^{3}+{\frac{3\,\cos \left ( x \right ) }{2}} \right ) }+{\frac{x}{16}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.937827, size = 24, normalized size = 0.71 \begin{align*} \frac{1}{48} \, \sin \left (2 \, x\right )^{3} + \frac{1}{16} \, x - \frac{1}{64} \, \sin \left (4 \, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.30492, size = 81, normalized size = 2.38 \begin{align*} -\frac{1}{48} \,{\left (8 \, \cos \left (x\right )^{5} - 2 \, \cos \left (x\right )^{3} - 3 \, \cos \left (x\right )\right )} \sin \left (x\right ) + \frac{1}{16} \, x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.060285, size = 31, normalized size = 0.91 \begin{align*} \frac{x}{16} - \frac{\sin{\left (x \right )} \cos ^{5}{\left (x \right )}}{6} + \frac{\sin{\left (x \right )} \cos ^{3}{\left (x \right )}}{24} + \frac{\sin{\left (x \right )} \cos{\left (x \right )}}{16} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.06386, size = 30, normalized size = 0.88 \begin{align*} \frac{1}{16} \, x - \frac{1}{192} \, \sin \left (6 \, x\right ) - \frac{1}{64} \, \sin \left (4 \, x\right ) + \frac{1}{64} \, \sin \left (2 \, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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