Optimal. Leaf size=85 \[ -\frac {2 \cos ^3(k x)}{27 k^3}+\frac {14 \cos (k x)}{9 k^3}+\frac {2 x \sin ^3(k x)}{9 k^2}+\frac {4 x \sin (k x)}{3 k^2}-\frac {2 x^2 \cos (k x)}{3 k}-\frac {x^2 \sin ^2(k x) \cos (k x)}{3 k} \]
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Rubi [A] time = 0.07, antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, 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 2633
Rule 2638
Rule 3296
Rule 3311
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*}
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Mathematica [A] time = 0.08, 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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fricas [A] time = 0.43, size = 59, normalized size = 0.69 \[ \frac {{\left (9 \, k^{2} x^{2} - 2\right )} \cos \left (k x\right )^{3} - 3 \, {\left (9 \, k^{2} x^{2} - 14\right )} \cos \left (k x\right ) - 6 \, {\left (k x \cos \left (k x\right )^{2} - 7 \, k x\right )} \sin \left (k x\right )}{27 \, k^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.14, size = 60, normalized size = 0.71 \[ -\frac {x \sin \left (3 \, k x\right )}{18 \, k^{2}} + \frac {3 \, x \sin \left (k x\right )}{2 \, k^{2}} + \frac {{\left (9 \, k^{2} x^{2} - 2\right )} \cos \left (3 \, k x\right )}{108 \, k^{3}} - \frac {3 \, {\left (k^{2} x^{2} - 2\right )} \cos \left (k x\right )}{4 \, k^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 64, normalized size = 0.75 \[ \frac {-\frac {\left (\sin ^{2}\left (k x \right )+2\right ) k^{2} x^{2} \cos \left (k x \right )}{3}+\frac {2 k x \left (\sin ^{3}\left (k x \right )\right )}{9}+\frac {4 k x \sin \left (k x \right )}{3}+\frac {4 \cos \left (k x \right )}{3}+\frac {2 \left (\sin ^{2}\left (k x \right )+2\right ) \cos \left (k x \right )}{27}}{k^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 55, normalized size = 0.65 \[ -\frac {6 \, k x \sin \left (3 \, k x\right ) - 162 \, k x \sin \left (k x\right ) - {\left (9 \, k^{2} x^{2} - 2\right )} \cos \left (3 \, k x\right ) + 81 \, {\left (k^{2} x^{2} - 2\right )} \cos \left (k x\right )}{108 \, k^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.23, size = 67, normalized size = 0.79 \[ \frac {\frac {14\,\cos \left (k\,x\right )}{9}-\frac {2\,{\cos \left (k\,x\right )}^3}{27}+k\,\left (\frac {14\,x\,\sin \left (k\,x\right )}{9}-\frac {2\,x\,{\cos \left (k\,x\right )}^2\,\sin \left (k\,x\right )}{9}\right )+k^2\,\left (\frac {x^2\,{\cos \left (k\,x\right )}^3}{3}-x^2\,\cos \left (k\,x\right )\right )}{k^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.10, size = 100, normalized size = 1.18 \[ \begin {cases} - \frac {x^{2} \sin ^{2}{\left (k x \right )} \cos {\left (k x \right )}}{k} - \frac {2 x^{2} \cos ^{3}{\left (k x \right )}}{3 k} + \frac {14 x \sin ^{3}{\left (k x \right )}}{9 k^{2}} + \frac {4 x \sin {\left (k x \right )} \cos ^{2}{\left (k x \right )}}{3 k^{2}} + \frac {14 \sin ^{2}{\left (k x \right )} \cos {\left (k x \right )}}{9 k^{3}} + \frac {40 \cos ^{3}{\left (k x \right )}}{27 k^{3}} & \text {for}\: k \neq 0 \\0 & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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