Optimal. Leaf size=85 \[ \frac{2 x \sin ^3(k x)}{9 k^2}+\frac{4 x \sin (k x)}{3 k^2}-\frac{2 \cos ^3(k x)}{27 k^3}+\frac{14 \cos (k x)}{9 k^3}-\frac{2 x^2 \cos (k x)}{3 k}-\frac{x^2 \sin ^2(k x) \cos (k x)}{3 k} \]
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Rubi [A] time = 0.0663495, antiderivative size = 85, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {3311, 3296, 2638, 2633} \[ \frac{2 x \sin ^3(k x)}{9 k^2}+\frac{4 x \sin (k x)}{3 k^2}-\frac{2 \cos ^3(k x)}{27 k^3}+\frac{14 \cos (k x)}{9 k^3}-\frac{2 x^2 \cos (k x)}{3 k}-\frac{x^2 \sin ^2(k x) \cos (k x)}{3 k} \]
Antiderivative was successfully verified.
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Rule 3311
Rule 3296
Rule 2638
Rule 2633
Rubi steps
\begin{align*} \int x^2 \sin ^3(k x) \, dx &=-\frac{x^2 \cos (k x) \sin ^2(k x)}{3 k}+\frac{2 x \sin ^3(k x)}{9 k^2}+\frac{2}{3} \int x^2 \sin (k x) \, dx-\frac{2 \int \sin ^3(k x) \, dx}{9 k^2}\\ &=-\frac{2 x^2 \cos (k x)}{3 k}-\frac{x^2 \cos (k x) \sin ^2(k x)}{3 k}+\frac{2 x \sin ^3(k x)}{9 k^2}+\frac{2 \operatorname{Subst}\left (\int \left (1-x^2\right ) \, dx,x,\cos (k x)\right )}{9 k^3}+\frac{4 \int x \cos (k x) \, dx}{3 k}\\ &=\frac{2 \cos (k x)}{9 k^3}-\frac{2 x^2 \cos (k x)}{3 k}-\frac{2 \cos ^3(k x)}{27 k^3}+\frac{4 x \sin (k x)}{3 k^2}-\frac{x^2 \cos (k x) \sin ^2(k x)}{3 k}+\frac{2 x \sin ^3(k x)}{9 k^2}-\frac{4 \int \sin (k x) \, dx}{3 k^2}\\ &=\frac{14 \cos (k x)}{9 k^3}-\frac{2 x^2 \cos (k x)}{3 k}-\frac{2 \cos ^3(k x)}{27 k^3}+\frac{4 x \sin (k x)}{3 k^2}-\frac{x^2 \cos (k x) \sin ^2(k x)}{3 k}+\frac{2 x \sin ^3(k x)}{9 k^2}\\ \end{align*}
Mathematica [A] time = 0.0790781, size = 55, normalized size = 0.65 \[ \frac{-81 \left (k^2 x^2-2\right ) \cos (k x)+\left (9 k^2 x^2-2\right ) \cos (3 k x)-6 k x (\sin (3 k x)-27 \sin (k x))}{108 k^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.026, size = 64, normalized size = 0.8 \begin{align*}{\frac{1}{{k}^{3}} \left ( -{\frac{{k}^{2}{x}^{2} \left ( 2+ \left ( \sin \left ( kx \right ) \right ) ^{2} \right ) \cos \left ( kx \right ) }{3}}+{\frac{4\,\cos \left ( kx \right ) }{3}}+{\frac{4\,kx\sin \left ( kx \right ) }{3}}+{\frac{2\,kx \left ( \sin \left ( kx \right ) \right ) ^{3}}{9}}+{\frac{ \left ( 4+2\, \left ( \sin \left ( kx \right ) \right ) ^{2} \right ) \cos \left ( kx \right ) }{27}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.972621, size = 74, normalized size = 0.87 \begin{align*} -\frac{6 \, k x \sin \left (3 \, k x\right ) - 162 \, k x \sin \left (k x\right ) -{\left (9 \, k^{2} x^{2} - 2\right )} \cos \left (3 \, k x\right ) + 81 \,{\left (k^{2} x^{2} - 2\right )} \cos \left (k x\right )}{108 \, k^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.80226, size = 144, normalized size = 1.69 \begin{align*} \frac{{\left (9 \, k^{2} x^{2} - 2\right )} \cos \left (k x\right )^{3} - 3 \,{\left (9 \, k^{2} x^{2} - 14\right )} \cos \left (k x\right ) - 6 \,{\left (k x \cos \left (k x\right )^{2} - 7 \, k x\right )} \sin \left (k x\right )}{27 \, k^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.19431, size = 100, normalized size = 1.18 \begin{align*} \begin{cases} - \frac{x^{2} \sin ^{2}{\left (k x \right )} \cos{\left (k x \right )}}{k} - \frac{2 x^{2} \cos ^{3}{\left (k x \right )}}{3 k} + \frac{14 x \sin ^{3}{\left (k x \right )}}{9 k^{2}} + \frac{4 x \sin{\left (k x \right )} \cos ^{2}{\left (k x \right )}}{3 k^{2}} + \frac{14 \sin ^{2}{\left (k x \right )} \cos{\left (k x \right )}}{9 k^{3}} + \frac{40 \cos ^{3}{\left (k x \right )}}{27 k^{3}} & \text{for}\: k \neq 0 \\0 & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.10263, size = 81, normalized size = 0.95 \begin{align*} -\frac{x \sin \left (3 \, k x\right )}{18 \, k^{2}} + \frac{3 \, x \sin \left (k x\right )}{2 \, k^{2}} + \frac{{\left (9 \, k^{2} x^{2} - 2\right )} \cos \left (3 \, k x\right )}{108 \, k^{3}} - \frac{3 \,{\left (k^{2} x^{2} - 2\right )} \cos \left (k x\right )}{4 \, k^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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