Optimal. Leaf size=24 \[ \frac{x}{8}-\frac{1}{4} \sin (x) \cos ^3(x)+\frac{1}{8} \sin (x) \cos (x) \]
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Rubi [A] time = 0.0248675, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {2568, 2635, 8} \[ \frac{x}{8}-\frac{1}{4} \sin (x) \cos ^3(x)+\frac{1}{8} \sin (x) \cos (x) \]
Antiderivative was successfully verified.
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Rule 2568
Rule 2635
Rule 8
Rubi steps
\begin{align*} \int \cos ^2(x) \sin ^2(x) \, dx &=-\frac{1}{4} \cos ^3(x) \sin (x)+\frac{1}{4} \int \cos ^2(x) \, dx\\ &=\frac{1}{8} \cos (x) \sin (x)-\frac{1}{4} \cos ^3(x) \sin (x)+\frac{\int 1 \, dx}{8}\\ &=\frac{x}{8}+\frac{1}{8} \cos (x) \sin (x)-\frac{1}{4} \cos ^3(x) \sin (x)\\ \end{align*}
Mathematica [A] time = 0.0037463, size = 14, normalized size = 0.58 \[ \frac{x}{8}-\frac{1}{32} \sin (4 x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0., size = 19, normalized size = 0.8 \begin{align*}{\frac{x}{8}}+{\frac{\cos \left ( x \right ) \sin \left ( x \right ) }{8}}-{\frac{ \left ( \cos \left ( x \right ) \right ) ^{3}\sin \left ( x \right ) }{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.925287, size = 14, normalized size = 0.58 \begin{align*} \frac{1}{8} \, x - \frac{1}{32} \, \sin \left (4 \, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.03853, size = 58, normalized size = 2.42 \begin{align*} -\frac{1}{8} \,{\left (2 \, \cos \left (x\right )^{3} - \cos \left (x\right )\right )} \sin \left (x\right ) + \frac{1}{8} \, x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.065533, size = 14, normalized size = 0.58 \begin{align*} \frac{x}{8} - \frac{\sin{\left (2 x \right )} \cos{\left (2 x \right )}}{16} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.05108, size = 14, normalized size = 0.58 \begin{align*} \frac{1}{8} \, x - \frac{1}{32} \, \sin \left (4 \, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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