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 = 0.03, antiderivative size = 24, normalized size of antiderivative = 1.00, 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 8
Rule 2568
Rule 2635
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*}
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Mathematica [A] time = 0.00, size = 14, normalized size = 0.58 \[ \frac {x}{8}-\frac {1}{32} \sin (4 x) \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \cos ^2(x) \sin ^2(x) \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.70, size = 19, normalized size = 0.79 \[ -\frac {1}{8} \, {\left (2 \, \cos \relax (x)^{3} - \cos \relax (x)\right )} \sin \relax (x) + \frac {1}{8} \, x \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.00, size = 10, normalized size = 0.42 \[ \frac {1}{8} \, x - \frac {1}{32} \, \sin \left (4 \, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 11, normalized size = 0.46
method | result | size |
risch | \(\frac {x}{8}-\frac {\sin \left (4 x \right )}{32}\) | \(11\) |
default | \(\frac {x}{8}+\frac {\cos \relax (x ) \sin \relax (x )}{8}-\frac {\left (\cos ^{3}\relax (x )\right ) \sin \relax (x )}{4}\) | \(19\) |
norman | \(\frac {\frac {x}{8}+\frac {7 \left (\tan ^{3}\left (\frac {x}{2}\right )\right )}{4}-\frac {7 \left (\tan ^{5}\left (\frac {x}{2}\right )\right )}{4}+\frac {\left (\tan ^{7}\left (\frac {x}{2}\right )\right )}{4}+\frac {x \left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{2}+\frac {3 x \left (\tan ^{4}\left (\frac {x}{2}\right )\right )}{4}+\frac {x \left (\tan ^{6}\left (\frac {x}{2}\right )\right )}{2}+\frac {x \left (\tan ^{8}\left (\frac {x}{2}\right )\right )}{8}-\frac {\tan \left (\frac {x}{2}\right )}{4}}{\left (1+\tan ^{2}\left (\frac {x}{2}\right )\right )^{4}}\) | \(82\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.57, size = 10, normalized size = 0.42 \[ \frac {1}{8} \, x - \frac {1}{32} \, \sin \left (4 \, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 18, normalized size = 0.75 \[ \frac {\cos \relax (x)\,{\sin \relax (x)}^3}{4}-\frac {\cos \relax (x)\,\sin \relax (x)}{8}+\frac {x}{8} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.07, size = 14, normalized size = 0.58 \[ \frac {x}{8} - \frac {\sin {\left (2 x \right )} \cos {\left (2 x \right )}}{16} \]
Verification of antiderivative is not currently implemented for this CAS.
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