Optimal. Leaf size=41 \[ -\frac{3}{16} \cos (2 x)+\frac{3}{64} \cos (4 x)+\frac{1}{48} \cos (6 x)-\frac{3}{128} \cos (8 x)+\frac{1}{192} \cos (12 x) \]
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Rubi [A] time = 0.0433548, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {4354, 2638} \[ -\frac{3}{16} \cos (2 x)+\frac{3}{64} \cos (4 x)+\frac{1}{48} \cos (6 x)-\frac{3}{128} \cos (8 x)+\frac{1}{192} \cos (12 x) \]
Antiderivative was successfully verified.
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Rule 4354
Rule 2638
Rubi steps
\begin{align*} \int \cos ^2(3 x) \sin ^3(2 x) \, dx &=\int \left (\frac{3}{8} \sin (2 x)-\frac{3}{16} \sin (4 x)-\frac{1}{8} \sin (6 x)+\frac{3}{16} \sin (8 x)-\frac{1}{16} \sin (12 x)\right ) \, dx\\ &=-\left (\frac{1}{16} \int \sin (12 x) \, dx\right )-\frac{1}{8} \int \sin (6 x) \, dx-\frac{3}{16} \int \sin (4 x) \, dx+\frac{3}{16} \int \sin (8 x) \, dx+\frac{3}{8} \int \sin (2 x) \, dx\\ &=-\frac{3}{16} \cos (2 x)+\frac{3}{64} \cos (4 x)+\frac{1}{48} \cos (6 x)-\frac{3}{128} \cos (8 x)+\frac{1}{192} \cos (12 x)\\ \end{align*}
Mathematica [A] time = 0.0185465, size = 41, normalized size = 1. \[ -\frac{3}{16} \cos (2 x)+\frac{3}{64} \cos (4 x)+\frac{1}{48} \cos (6 x)-\frac{3}{128} \cos (8 x)+\frac{1}{192} \cos (12 x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.066, size = 32, normalized size = 0.8 \begin{align*} -{\frac{3\,\cos \left ( 2\,x \right ) }{16}}+{\frac{3\,\cos \left ( 4\,x \right ) }{64}}+{\frac{\cos \left ( 6\,x \right ) }{48}}-{\frac{3\,\cos \left ( 8\,x \right ) }{128}}+{\frac{\cos \left ( 12\,x \right ) }{192}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.0121, size = 42, normalized size = 1.02 \begin{align*} \frac{1}{192} \, \cos \left (12 \, x\right ) - \frac{3}{128} \, \cos \left (8 \, x\right ) + \frac{1}{48} \, \cos \left (6 \, x\right ) + \frac{3}{64} \, \cos \left (4 \, x\right ) - \frac{3}{16} \, \cos \left (2 \, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.39628, size = 80, normalized size = 1.95 \begin{align*} \frac{32}{3} \, \cos \left (x\right )^{12} - 32 \, \cos \left (x\right )^{10} + 33 \, \cos \left (x\right )^{8} - 12 \, \cos \left (x\right )^{6} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 52.9172, size = 228, normalized size = 5.56 \begin{align*} - \frac{x \sin ^{3}{\left (2 x \right )} \sin ^{2}{\left (3 x \right )}}{16} + \frac{x \sin ^{3}{\left (2 x \right )} \cos ^{2}{\left (3 x \right )}}{16} - \frac{3 x \sin ^{2}{\left (2 x \right )} \sin{\left (3 x \right )} \cos{\left (2 x \right )} \cos{\left (3 x \right )}}{8} + \frac{3 x \sin{\left (2 x \right )} \sin ^{2}{\left (3 x \right )} \cos ^{2}{\left (2 x \right )}}{16} - \frac{3 x \sin{\left (2 x \right )} \cos ^{2}{\left (2 x \right )} \cos ^{2}{\left (3 x \right )}}{16} + \frac{x \sin{\left (3 x \right )} \cos ^{3}{\left (2 x \right )} \cos{\left (3 x \right )}}{8} - \frac{\sin ^{2}{\left (2 x \right )} \sin ^{2}{\left (3 x \right )} \cos{\left (2 x \right )}}{32} - \frac{15 \sin ^{2}{\left (2 x \right )} \cos{\left (2 x \right )} \cos ^{2}{\left (3 x \right )}}{32} + \frac{9 \sin{\left (2 x \right )} \sin{\left (3 x \right )} \cos ^{2}{\left (2 x \right )} \cos{\left (3 x \right )}}{16} - \frac{13 \sin ^{2}{\left (3 x \right )} \cos ^{3}{\left (2 x \right )}}{48} - \frac{\cos ^{3}{\left (2 x \right )} \cos ^{2}{\left (3 x \right )}}{16} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16051, size = 42, normalized size = 1.02 \begin{align*} \frac{1}{192} \, \cos \left (12 \, x\right ) - \frac{3}{128} \, \cos \left (8 \, x\right ) + \frac{1}{48} \, \cos \left (6 \, x\right ) + \frac{3}{64} \, \cos \left (4 \, x\right ) - \frac{3}{16} \, \cos \left (2 \, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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