Optimal. Leaf size=36 \[ \frac {x}{16}-\frac {1}{6} \sin ^3(x) \cos ^3(x)-\frac {1}{8} \sin (x) \cos ^3(x)+\frac {1}{16} \sin (x) \cos (x) \]
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Rubi [A] time = 0.04, antiderivative size = 36, normalized size of antiderivative = 1.00, 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} \[ \frac {x}{16}-\frac {1}{6} \sin ^3(x) \cos ^3(x)-\frac {1}{8} \sin (x) \cos ^3(x)+\frac {1}{16} \sin (x) \cos (x) \]
Antiderivative was successfully verified.
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Rule 8
Rule 2568
Rule 2635
Rubi steps
\begin {align*} \int \cos ^2(x) \sin ^4(x) \, dx &=-\frac {1}{6} \cos ^3(x) \sin ^3(x)+\frac {1}{2} \int \cos ^2(x) \sin ^2(x) \, dx\\ &=-\frac {1}{8} \cos ^3(x) \sin (x)-\frac {1}{6} \cos ^3(x) \sin ^3(x)+\frac {1}{8} \int \cos ^2(x) \, dx\\ &=\frac {1}{16} \cos (x) \sin (x)-\frac {1}{8} \cos ^3(x) \sin (x)-\frac {1}{6} \cos ^3(x) \sin ^3(x)+\frac {\int 1 \, dx}{16}\\ &=\frac {x}{16}+\frac {1}{16} \cos (x) \sin (x)-\frac {1}{8} \cos ^3(x) \sin (x)-\frac {1}{6} \cos ^3(x) \sin ^3(x)\\ \end {align*}
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Mathematica [A] time = 0.01, size = 30, normalized size = 0.83 \[ \frac {x}{16}-\frac {1}{64} \sin (2 x)-\frac {1}{64} \sin (4 x)+\frac {1}{192} \sin (6 x) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 25, normalized size = 0.69 \[ \frac {1}{48} \, {\left (8 \, \cos \relax (x)^{5} - 14 \, \cos \relax (x)^{3} + 3 \, \cos \relax (x)\right )} \sin \relax (x) + \frac {1}{16} \, x \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.92, size = 22, normalized size = 0.61 \[ \frac {1}{16} \, x + \frac {1}{192} \, \sin \left (6 \, x\right ) - \frac {1}{64} \, \sin \left (4 \, x\right ) - \frac {1}{64} \, \sin \left (2 \, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 29, normalized size = 0.81 \[ -\frac {\left (\cos ^{3}\relax (x )\right ) \left (\sin ^{3}\relax (x )\right )}{6}-\frac {\left (\cos ^{3}\relax (x )\right ) \sin \relax (x )}{8}+\frac {\cos \relax (x ) \sin \relax (x )}{16}+\frac {x}{16} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.44, size = 18, normalized size = 0.50 \[ -\frac {1}{48} \, \sin \left (2 \, x\right )^{3} + \frac {1}{16} \, x - \frac {1}{64} \, \sin \left (4 \, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 24, normalized size = 0.67 \[ \frac {\cos \relax (x)\,{\sin \relax (x)}^5}{6}+\frac {x}{16}-\frac {\sin \left (2\,x\right )}{24}+\frac {\sin \left (4\,x\right )}{192} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.06, size = 31, normalized size = 0.86 \[ \frac {x}{16} + \frac {\sin ^{5}{\relax (x )} \cos {\relax (x )}}{6} - \frac {\sin ^{3}{\relax (x )} \cos {\relax (x )}}{24} - \frac {\sin {\relax (x )} \cos {\relax (x )}}{16} \]
Verification of antiderivative is not currently implemented for this CAS.
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