Optimal. Leaf size=46 \[ \frac {x}{16}-\frac {1}{12} \sin (2 x) \cos ^5(2 x)+\frac {1}{48} \sin (2 x) \cos ^3(2 x)+\frac {1}{32} \sin (2 x) \cos (2 x) \]
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Rubi [A] time = 0.04, 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 = {2568, 2635, 8} \[ \frac {x}{16}-\frac {1}{12} \sin (2 x) \cos ^5(2 x)+\frac {1}{48} \sin (2 x) \cos ^3(2 x)+\frac {1}{32} \sin (2 x) \cos (2 x) \]
Antiderivative was successfully verified.
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Rule 8
Rule 2568
Rule 2635
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 = 0.01, size = 30, normalized size = 0.65 \[ \frac {x}{16}+\frac {1}{128} \sin (4 x)-\frac {1}{128} \sin (8 x)-\frac {1}{384} \sin (12 x) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 33, normalized size = 0.72 \[ -\frac {1}{96} \, {\left (8 \, \cos \left (2 \, x\right )^{5} - 2 \, \cos \left (2 \, x\right )^{3} - 3 \, \cos \left (2 \, x\right )\right )} \sin \left (2 \, x\right ) + \frac {1}{16} \, x \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.83, size = 22, normalized size = 0.48 \[ \frac {1}{16} \, x - \frac {1}{384} \, \sin \left (12 \, x\right ) - \frac {1}{128} \, \sin \left (8 \, x\right ) + \frac {1}{128} \, \sin \left (4 \, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 36, normalized size = 0.78 \[ -\frac {\left (\cos ^{5}\left (2 x \right )\right ) \sin \left (2 x \right )}{12}+\frac {x}{16}+\frac {\left (\cos ^{3}\left (2 x \right )+\frac {3 \cos \left (2 x \right )}{2}\right ) \sin \left (2 x \right )}{48} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 18, normalized size = 0.39 \[ \frac {1}{96} \, \sin \left (4 \, x\right )^{3} + \frac {1}{16} \, x - \frac {1}{128} \, \sin \left (8 \, x\right ) \]
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 \[ \frac {x}{16}-\frac {\cos \left (2\,x\right )\,\sin \left (2\,x\right )}{32}+\frac {{\sin \left (2\,x\right )}^3\,\left (\frac {{\cos \left (2\,x\right )}^3}{6}+\frac {\cos \left (2\,x\right )}{8}\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.07, size = 41, normalized size = 0.89 \[ \frac {x}{16} - \frac {\sin {\left (2 x \right )} \cos ^{5}{\left (2 x \right )}}{12} + \frac {\sin {\left (2 x \right )} \cos ^{3}{\left (2 x \right )}}{48} + \frac {\sin {\left (2 x \right )} \cos {\left (2 x \right )}}{32} \]
Verification of antiderivative is not currently implemented for this CAS.
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