Optimal. Leaf size=46 \[ \frac{5 x}{16}-\frac{1}{12} \sin ^5(2 x) \cos (2 x)-\frac{5}{48} \sin ^3(2 x) \cos (2 x)-\frac{5}{32} \sin (2 x) \cos (2 x) \]
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Rubi [A] time = 0.0197261, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 6, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {2635, 8} \[ \frac{5 x}{16}-\frac{1}{12} \sin ^5(2 x) \cos (2 x)-\frac{5}{48} \sin ^3(2 x) \cos (2 x)-\frac{5}{32} \sin (2 x) \cos (2 x) \]
Antiderivative was successfully verified.
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Rule 2635
Rule 8
Rubi steps
\begin{align*} \int \sin ^6(2 x) \, dx &=-\frac{1}{12} \cos (2 x) \sin ^5(2 x)+\frac{5}{6} \int \sin ^4(2 x) \, dx\\ &=-\frac{5}{48} \cos (2 x) \sin ^3(2 x)-\frac{1}{12} \cos (2 x) \sin ^5(2 x)+\frac{5}{8} \int \sin ^2(2 x) \, dx\\ &=-\frac{5}{32} \cos (2 x) \sin (2 x)-\frac{5}{48} \cos (2 x) \sin ^3(2 x)-\frac{1}{12} \cos (2 x) \sin ^5(2 x)+\frac{5 \int 1 \, dx}{16}\\ &=\frac{5 x}{16}-\frac{5}{32} \cos (2 x) \sin (2 x)-\frac{5}{48} \cos (2 x) \sin ^3(2 x)-\frac{1}{12} \cos (2 x) \sin ^5(2 x)\\ \end{align*}
Mathematica [A] time = 0.0064757, size = 30, normalized size = 0.65 \[ \frac{5 x}{16}-\frac{15}{128} \sin (4 x)+\frac{3}{128} \sin (8 x)-\frac{1}{384} \sin (12 x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 32, normalized size = 0.7 \begin{align*} -{\frac{\cos \left ( 2\,x \right ) }{12} \left ( \left ( \sin \left ( 2\,x \right ) \right ) ^{5}+{\frac{5\, \left ( \sin \left ( 2\,x \right ) \right ) ^{3}}{4}}+{\frac{15\,\sin \left ( 2\,x \right ) }{8}} \right ) }+{\frac{5\,x}{16}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.949246, size = 32, normalized size = 0.7 \begin{align*} \frac{1}{96} \, \sin \left (4 \, x\right )^{3} + \frac{5}{16} \, x + \frac{3}{128} \, \sin \left (8 \, x\right ) - \frac{1}{8} \, \sin \left (4 \, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.9265, size = 95, normalized size = 2.07 \begin{align*} -\frac{1}{96} \,{\left (8 \, \cos \left (2 \, x\right )^{5} - 26 \, \cos \left (2 \, x\right )^{3} + 33 \, \cos \left (2 \, x\right )\right )} \sin \left (2 \, x\right ) + \frac{5}{16} \, x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.059177, size = 46, normalized size = 1. \begin{align*} \frac{5 x}{16} - \frac{\sin ^{5}{\left (2 x \right )} \cos{\left (2 x \right )}}{12} - \frac{5 \sin ^{3}{\left (2 x \right )} \cos{\left (2 x \right )}}{48} - \frac{5 \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 = 1.05715, size = 30, normalized size = 0.65 \begin{align*} \frac{5}{16} \, x - \frac{1}{384} \, \sin \left (12 \, x\right ) + \frac{3}{128} \, \sin \left (8 \, x\right ) - \frac{15}{128} \, \sin \left (4 \, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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