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 = 0.0360118, antiderivative size = 46, normalized size of antiderivative = 1., 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 2568
Rule 2635
Rule 8
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*}
Mathematica [A] time = 0.0138663, 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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Maple [A] time = 0.011, size = 36, normalized size = 0.8 \begin{align*} -{\frac{ \left ( \cos \left ( 2\,x \right ) \right ) ^{5}\sin \left ( 2\,x \right ) }{12}}+{\frac{\sin \left ( 2\,x \right ) }{48} \left ( \left ( \cos \left ( 2\,x \right ) \right ) ^{3}+{\frac{3\,\cos \left ( 2\,x \right ) }{2}} \right ) }+{\frac{x}{16}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.938778, size = 24, normalized size = 0.52 \begin{align*} \frac{1}{96} \, \sin \left (4 \, x\right )^{3} + \frac{1}{16} \, x - \frac{1}{128} \, \sin \left (8 \, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.99528, size = 92, normalized size = 2. \begin{align*} -\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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.061795, size = 41, normalized size = 0.89 \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.05166, size = 30, normalized size = 0.65 \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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