Optimal. Leaf size=49 \[ -\frac {2 \cos (b x)}{3 b^3}+\frac {x^2 \cos (b x)}{3 b}-\frac {2 x \sin (b x)}{3 b^2}+\frac {1}{3} x^3 \text {Si}(b x) \]
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Rubi [A]
time = 0.03, antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6638, 12, 3377,
2718} \begin {gather*} -\frac {2 \cos (b x)}{3 b^3}-\frac {2 x \sin (b x)}{3 b^2}+\frac {1}{3} x^3 \text {Si}(b x)+\frac {x^2 \cos (b x)}{3 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 2718
Rule 3377
Rule 6638
Rubi steps
\begin {align*} \int x^2 \text {Si}(b x) \, dx &=\frac {1}{3} x^3 \text {Si}(b x)-\frac {1}{3} b \int \frac {x^2 \sin (b x)}{b} \, dx\\ &=\frac {1}{3} x^3 \text {Si}(b x)-\frac {1}{3} \int x^2 \sin (b x) \, dx\\ &=\frac {x^2 \cos (b x)}{3 b}+\frac {1}{3} x^3 \text {Si}(b x)-\frac {2 \int x \cos (b x) \, dx}{3 b}\\ &=\frac {x^2 \cos (b x)}{3 b}-\frac {2 x \sin (b x)}{3 b^2}+\frac {1}{3} x^3 \text {Si}(b x)+\frac {2 \int \sin (b x) \, dx}{3 b^2}\\ &=-\frac {2 \cos (b x)}{3 b^3}+\frac {x^2 \cos (b x)}{3 b}-\frac {2 x \sin (b x)}{3 b^2}+\frac {1}{3} x^3 \text {Si}(b x)\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 41, normalized size = 0.84 \begin {gather*} \frac {\left (-2+b^2 x^2\right ) \cos (b x)-2 b x \sin (b x)+b^3 x^3 \text {Si}(b x)}{3 b^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.26, size = 44, normalized size = 0.90
method | result | size |
derivativedivides | \(\frac {\frac {b^{3} x^{3} \sinIntegral \left (b x \right )}{3}+\frac {b^{2} x^{2} \cos \left (b x \right )}{3}-\frac {2 \cos \left (b x \right )}{3}-\frac {2 b x \sin \left (b x \right )}{3}}{b^{3}}\) | \(44\) |
default | \(\frac {\frac {b^{3} x^{3} \sinIntegral \left (b x \right )}{3}+\frac {b^{2} x^{2} \cos \left (b x \right )}{3}-\frac {2 \cos \left (b x \right )}{3}-\frac {2 b x \sin \left (b x \right )}{3}}{b^{3}}\) | \(44\) |
meijerg | \(\frac {2 \sqrt {\pi }\, \left (\frac {1}{3 \sqrt {\pi }}-\frac {\left (-\frac {b^{2} x^{2}}{2}+1\right ) \cos \left (b x \right )}{3 \sqrt {\pi }}-\frac {b x \sin \left (b x \right )}{3 \sqrt {\pi }}+\frac {b^{3} x^{3} \sinIntegral \left (b x \right )}{6 \sqrt {\pi }}\right )}{b^{3}}\) | \(60\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 39, normalized size = 0.80 \begin {gather*} \frac {1}{3} \, x^{3} \operatorname {Si}\left (b x\right ) - \frac {2 \, b x \sin \left (b x\right ) - {\left (b^{2} x^{2} - 2\right )} \cos \left (b x\right )}{3 \, b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, size = 39, normalized size = 0.80 \begin {gather*} \frac {b^{3} x^{3} \operatorname {Si}\left (b x\right ) - 2 \, b x \sin \left (b x\right ) + {\left (b^{2} x^{2} - 2\right )} \cos \left (b x\right )}{3 \, b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.79, size = 46, normalized size = 0.94 \begin {gather*} \frac {x^{3} \operatorname {Si}{\left (b x \right )}}{3} + \frac {x^{2} \cos {\left (b x \right )}}{3 b} - \frac {2 x \sin {\left (b x \right )}}{3 b^{2}} - \frac {2 \cos {\left (b x \right )}}{3 b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.40, size = 38, normalized size = 0.78 \begin {gather*} \frac {1}{3} \, x^{3} \operatorname {Si}\left (b x\right ) - \frac {2 \, x \sin \left (b x\right )}{3 \, b^{2}} + \frac {{\left (b^{2} x^{2} - 2\right )} \cos \left (b x\right )}{3 \, b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \frac {x^3\,\mathrm {sinint}\left (b\,x\right )}{3}-\frac {\cos \left (b\,x\right )\,\left (\frac {2}{b^3}-\frac {x^2}{b}\right )}{3}-\frac {2\,x\,\sin \left (b\,x\right )}{3\,b^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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