Optimal. Leaf size=34 \[ \frac {x}{2}-\frac {4}{3} \sin ^3(x) \cos ^3(x)-\sin (x) \cos ^3(x)+\frac {1}{2} \sin (x) \cos (x) \]
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Rubi [A] time = 0.05, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {12, 2568, 2635, 8} \[ \frac {x}{2}-\frac {4}{3} \sin ^3(x) \cos ^3(x)-\sin (x) \cos ^3(x)+\frac {1}{2} \sin (x) \cos (x) \]
Antiderivative was successfully verified.
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Rule 8
Rule 12
Rule 2568
Rule 2635
Rubi steps
\begin {align*} \int 8 \cos ^2(x) \sin ^4(x) \, dx &=8 \int \cos ^2(x) \sin ^4(x) \, dx\\ &=-\frac {4}{3} \cos ^3(x) \sin ^3(x)+4 \int \cos ^2(x) \sin ^2(x) \, dx\\ &=-\cos ^3(x) \sin (x)-\frac {4}{3} \cos ^3(x) \sin ^3(x)+\int \cos ^2(x) \, dx\\ &=\frac {1}{2} \cos (x) \sin (x)-\cos ^3(x) \sin (x)-\frac {4}{3} \cos ^3(x) \sin ^3(x)+\frac {\int 1 \, dx}{2}\\ &=\frac {x}{2}+\frac {1}{2} \cos (x) \sin (x)-\cos ^3(x) \sin (x)-\frac {4}{3} \cos ^3(x) \sin ^3(x)\\ \end {align*}
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Mathematica [A] time = 0.01, size = 32, normalized size = 0.94 \[ 8 \left (\frac {x}{16}-\frac {1}{64} \sin (2 x)-\frac {1}{64} \sin (4 x)+\frac {1}{192} \sin (6 x)\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 2.39, size = 25, normalized size = 0.74 \[ \frac {1}{6} \, {\left (8 \, \cos \relax (x)^{5} - 14 \, \cos \relax (x)^{3} + 3 \, \cos \relax (x)\right )} \sin \relax (x) + \frac {1}{2} \, x \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.12, size = 22, normalized size = 0.65 \[ \frac {1}{2} \, x + \frac {1}{24} \, \sin \left (6 \, x\right ) - \frac {1}{8} \, \sin \left (4 \, x\right ) - \frac {1}{8} \, \sin \left (2 \, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 29, normalized size = 0.85 \[ \frac {x}{2}+\frac {\cos \relax (x ) \sin \relax (x )}{2}-\left (\cos ^{3}\relax (x )\right ) \sin \relax (x )-\frac {4 \left (\cos ^{3}\relax (x )\right ) \left (\sin ^{3}\relax (x )\right )}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 18, normalized size = 0.53 \[ -\frac {1}{6} \, \sin \left (2 \, x\right )^{3} + \frac {1}{2} \, x - \frac {1}{8} \, \sin \left (4 \, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 24, normalized size = 0.71 \[ \frac {4\,\cos \relax (x)\,{\sin \relax (x)}^5}{3}+\frac {x}{2}-\frac {\sin \left (2\,x\right )}{3}+\frac {\sin \left (4\,x\right )}{24} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.05, size = 32, normalized size = 0.94 \[ \frac {x}{2} + \frac {4 \sin ^{5}{\relax (x )} \cos {\relax (x )}}{3} - \frac {\sin ^{3}{\relax (x )} \cos {\relax (x )}}{3} - \frac {\sin {\relax (x )} \cos {\relax (x )}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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