Optimal. Leaf size=33 \[ -\frac{1}{8} \cos (3 x)+\frac{1}{72} \cos (9 x)-\frac{1}{40} \cos (15 x)-\frac{1}{216} \cos (27 x) \]
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Rubi [A] time = 0.0328019, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {4354, 2638} \[ -\frac{1}{8} \cos (3 x)+\frac{1}{72} \cos (9 x)-\frac{1}{40} \cos (15 x)-\frac{1}{216} \cos (27 x) \]
Antiderivative was successfully verified.
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Rule 4354
Rule 2638
Rubi steps
\begin{align*} \int \cos ^3(6 x) \sin (9 x) \, dx &=\int \left (\frac{3}{8} \sin (3 x)-\frac{1}{8} \sin (9 x)+\frac{3}{8} \sin (15 x)+\frac{1}{8} \sin (27 x)\right ) \, dx\\ &=-\left (\frac{1}{8} \int \sin (9 x) \, dx\right )+\frac{1}{8} \int \sin (27 x) \, dx+\frac{3}{8} \int \sin (3 x) \, dx+\frac{3}{8} \int \sin (15 x) \, dx\\ &=-\frac{1}{8} \cos (3 x)+\frac{1}{72} \cos (9 x)-\frac{1}{40} \cos (15 x)-\frac{1}{216} \cos (27 x)\\ \end{align*}
Mathematica [A] time = 0.0173943, size = 33, normalized size = 1. \[ -\frac{1}{8} \cos (3 x)+\frac{1}{72} \cos (9 x)-\frac{1}{40} \cos (15 x)-\frac{1}{216} \cos (27 x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.038, size = 26, normalized size = 0.8 \begin{align*} -{\frac{\cos \left ( 3\,x \right ) }{8}}+{\frac{\cos \left ( 9\,x \right ) }{72}}-{\frac{\cos \left ( 15\,x \right ) }{40}}-{\frac{\cos \left ( 27\,x \right ) }{216}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.01214, size = 34, normalized size = 1.03 \begin{align*} -\frac{1}{216} \, \cos \left (27 \, x\right ) - \frac{1}{40} \, \cos \left (15 \, x\right ) + \frac{1}{72} \, \cos \left (9 \, x\right ) - \frac{1}{8} \, \cos \left (3 \, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.44613, size = 117, normalized size = 3.55 \begin{align*} -\frac{32}{27} \, \cos \left (3 \, x\right )^{9} + \frac{8}{3} \, \cos \left (3 \, x\right )^{7} - \frac{12}{5} \, \cos \left (3 \, x\right )^{5} + \frac{10}{9} \, \cos \left (3 \, x\right )^{3} - \frac{1}{3} \, \cos \left (3 \, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 23.3911, size = 71, normalized size = 2.15 \begin{align*} - \frac{16 \sin ^{3}{\left (6 x \right )} \sin{\left (9 x \right )}}{135} - \frac{8 \sin ^{2}{\left (6 x \right )} \cos{\left (6 x \right )} \cos{\left (9 x \right )}}{45} - \frac{2 \sin{\left (6 x \right )} \sin{\left (9 x \right )} \cos ^{2}{\left (6 x \right )}}{45} - \frac{19 \cos ^{3}{\left (6 x \right )} \cos{\left (9 x \right )}}{135} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12231, size = 34, normalized size = 1.03 \begin{align*} -\frac{1}{216} \, \cos \left (27 \, x\right ) - \frac{1}{40} \, \cos \left (15 \, x\right ) + \frac{1}{72} \, \cos \left (9 \, x\right ) - \frac{1}{8} \, \cos \left (3 \, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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