Optimal. Leaf size=36 \[ \frac {x}{16}+\frac {1}{16} \cos (x) \sin (x)-\frac {1}{8} \cos ^3(x) \sin (x)-\frac {1}{6} \cos ^3(x) \sin ^3(x) \]
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Rubi [A]
time = 0.03, 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 = {2648, 2715, 8}
\begin {gather*} \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) \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 2648
Rule 2715
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 \begin {gather*} \frac {x}{16}-\frac {1}{64} \sin (2 x)-\frac {1}{64} \sin (4 x)+\frac {1}{192} \sin (6 x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.03, size = 29, normalized size = 0.81
method | result | size |
risch | \(\frac {x}{16}+\frac {\sin \left (6 x \right )}{192}-\frac {\sin \left (4 x \right )}{64}-\frac {\sin \left (2 x \right )}{64}\) | \(23\) |
default | \(\frac {x}{16}+\frac {\cos \left (x \right ) \sin \left (x \right )}{16}-\frac {\left (\cos ^{3}\left (x \right )\right ) \sin \left (x \right )}{8}-\frac {\left (\cos ^{3}\left (x \right )\right ) \left (\sin ^{3}\left (x \right )\right )}{6}\) | \(29\) |
norman | \(\frac {\frac {x}{16}-\frac {17 \left (\tan ^{3}\left (\frac {x}{2}\right )\right )}{24}+\frac {19 \left (\tan ^{5}\left (\frac {x}{2}\right )\right )}{4}-\frac {19 \left (\tan ^{7}\left (\frac {x}{2}\right )\right )}{4}+\frac {17 \left (\tan ^{9}\left (\frac {x}{2}\right )\right )}{24}+\frac {\left (\tan ^{11}\left (\frac {x}{2}\right )\right )}{8}+\frac {3 x \left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{8}+\frac {15 x \left (\tan ^{4}\left (\frac {x}{2}\right )\right )}{16}+\frac {5 x \left (\tan ^{6}\left (\frac {x}{2}\right )\right )}{4}+\frac {15 x \left (\tan ^{8}\left (\frac {x}{2}\right )\right )}{16}+\frac {3 x \left (\tan ^{10}\left (\frac {x}{2}\right )\right )}{8}+\frac {x \left (\tan ^{12}\left (\frac {x}{2}\right )\right )}{16}-\frac {\tan \left (\frac {x}{2}\right )}{8}}{\left (1+\tan ^{2}\left (\frac {x}{2}\right )\right )^{6}}\) | \(116\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 1.12, size = 18, normalized size = 0.50 \begin {gather*} -\frac {1}{48} \, \sin \left (2 \, x\right )^{3} + \frac {1}{16} \, x - \frac {1}{64} \, \sin \left (4 \, x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.66, size = 25, normalized size = 0.69 \begin {gather*} \frac {1}{48} \, {\left (8 \, \cos \left (x\right )^{5} - 14 \, \cos \left (x\right )^{3} + 3 \, \cos \left (x\right )\right )} \sin \left (x\right ) + \frac {1}{16} \, x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.01, size = 31, normalized size = 0.86 \begin {gather*} \frac {x}{16} + \frac {\sin ^{5}{\left (x \right )} \cos {\left (x \right )}}{6} - \frac {\sin ^{3}{\left (x \right )} \cos {\left (x \right )}}{24} - \frac {\sin {\left (x \right )} \cos {\left (x \right )}}{16} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.98, size = 22, normalized size = 0.61 \begin {gather*} \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 {gather*}
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 \begin {gather*} \frac {\cos \left (x\right )\,{\sin \left (x\right )}^5}{6}+\frac {x}{16}-\frac {\sin \left (2\,x\right )}{24}+\frac {\sin \left (4\,x\right )}{192} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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