Optimal. Leaf size=46 \[ \frac {x}{16}+\frac {1}{32} \cos (2 x) \sin (2 x)+\frac {1}{48} \cos ^3(2 x) \sin (2 x)-\frac {1}{12} \cos ^5(2 x) \sin (2 x) \]
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Rubi [A]
time = 0.03, antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {2648, 2715, 8}
\begin {gather*} \frac {x}{16}-\frac {1}{12} \sin (2 x) \cos ^5(2 x)+\frac {1}{48} \sin (2 x) \cos ^3(2 x)+\frac {1}{32} \sin (2 x) \cos (2 x) \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 2648
Rule 2715
Rubi steps
\begin {align*} \int \cos ^4(2 x) \sin ^2(2 x) \, dx &=-\frac {1}{12} \cos ^5(2 x) \sin (2 x)+\frac {1}{6} \int \cos ^4(2 x) \, dx\\ &=\frac {1}{48} \cos ^3(2 x) \sin (2 x)-\frac {1}{12} \cos ^5(2 x) \sin (2 x)+\frac {1}{8} \int \cos ^2(2 x) \, dx\\ &=\frac {1}{32} \cos (2 x) \sin (2 x)+\frac {1}{48} \cos ^3(2 x) \sin (2 x)-\frac {1}{12} \cos ^5(2 x) \sin (2 x)+\frac {\int 1 \, dx}{16}\\ &=\frac {x}{16}+\frac {1}{32} \cos (2 x) \sin (2 x)+\frac {1}{48} \cos ^3(2 x) \sin (2 x)-\frac {1}{12} \cos ^5(2 x) \sin (2 x)\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 30, normalized size = 0.65 \begin {gather*} \frac {x}{16}+\frac {1}{128} \sin (4 x)-\frac {1}{128} \sin (8 x)-\frac {1}{384} \sin (12 x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.03, size = 36, normalized size = 0.78
method | result | size |
risch | \(\frac {x}{16}-\frac {\sin \left (12 x \right )}{384}-\frac {\sin \left (8 x \right )}{128}+\frac {\sin \left (4 x \right )}{128}\) | \(23\) |
derivativedivides | \(-\frac {\left (\cos ^{5}\left (2 x \right )\right ) \sin \left (2 x \right )}{12}+\frac {\left (\cos ^{3}\left (2 x \right )+\frac {3 \cos \left (2 x \right )}{2}\right ) \sin \left (2 x \right )}{48}+\frac {x}{16}\) | \(36\) |
default | \(-\frac {\left (\cos ^{5}\left (2 x \right )\right ) \sin \left (2 x \right )}{12}+\frac {\left (\cos ^{3}\left (2 x \right )+\frac {3 \cos \left (2 x \right )}{2}\right ) \sin \left (2 x \right )}{48}+\frac {x}{16}\) | \(36\) |
norman | \(\frac {\frac {x}{16}+\frac {47 \left (\tan ^{3}\left (x \right )\right )}{48}-\frac {13 \left (\tan ^{5}\left (x \right )\right )}{8}+\frac {13 \left (\tan ^{7}\left (x \right )\right )}{8}-\frac {47 \left (\tan ^{9}\left (x \right )\right )}{48}+\frac {\left (\tan ^{11}\left (x \right )\right )}{16}+\frac {3 x \left (\tan ^{2}\left (x \right )\right )}{8}+\frac {15 x \left (\tan ^{4}\left (x \right )\right )}{16}+\frac {5 x \left (\tan ^{6}\left (x \right )\right )}{4}+\frac {15 x \left (\tan ^{8}\left (x \right )\right )}{16}+\frac {3 x \left (\tan ^{10}\left (x \right )\right )}{8}+\frac {x \left (\tan ^{12}\left (x \right )\right )}{16}-\frac {\tan \left (x \right )}{16}}{\left (1+\tan ^{2}\left (x \right )\right )^{6}}\) | \(90\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 1.02, size = 18, normalized size = 0.39 \begin {gather*} \frac {1}{96} \, \sin \left (4 \, x\right )^{3} + \frac {1}{16} \, x - \frac {1}{128} \, \sin \left (8 \, x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.10, size = 33, normalized size = 0.72 \begin {gather*} -\frac {1}{96} \, {\left (8 \, \cos \left (2 \, x\right )^{5} - 2 \, \cos \left (2 \, x\right )^{3} - 3 \, \cos \left (2 \, x\right )\right )} \sin \left (2 \, 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 = 41, normalized size = 0.89 \begin {gather*} \frac {x}{16} - \frac {\sin {\left (2 x \right )} \cos ^{5}{\left (2 x \right )}}{12} + \frac {\sin {\left (2 x \right )} \cos ^{3}{\left (2 x \right )}}{48} + \frac {\sin {\left (2 x \right )} \cos {\left (2 x \right )}}{32} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.64, size = 22, normalized size = 0.48 \begin {gather*} \frac {1}{16} \, x - \frac {1}{384} \, \sin \left (12 \, x\right ) - \frac {1}{128} \, \sin \left (8 \, x\right ) + \frac {1}{128} \, \sin \left (4 \, x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.07, size = 37, normalized size = 0.80 \begin {gather*} \frac {x}{16}-\frac {\cos \left (2\,x\right )\,\sin \left (2\,x\right )}{32}+\frac {{\sin \left (2\,x\right )}^3\,\left (\frac {{\cos \left (2\,x\right )}^3}{6}+\frac {\cos \left (2\,x\right )}{8}\right )}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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