Optimal. Leaf size=49 \[ -\frac {2 \cos (b x)}{3 b^3}-\frac {2 x \sin (b x)}{3 b^2}+\frac {1}{3} x^3 \text {Si}(b x)+\frac {x^2 \cos (b x)}{3 b} \]
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Rubi [A] time = 0.05, 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 = {6503, 12, 3296, 2638} \[ -\frac {2 x \sin (b x)}{3 b^2}-\frac {2 \cos (b x)}{3 b^3}+\frac {1}{3} x^3 \text {Si}(b x)+\frac {x^2 \cos (b x)}{3 b} \]
Antiderivative was successfully verified.
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Rule 12
Rule 2638
Rule 3296
Rule 6503
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 = 0.03, size = 41, normalized size = 0.84 \[ \frac {b^3 x^3 \text {Si}(b x)+\left (b^2 x^2-2\right ) \cos (b x)-2 b x \sin (b x)}{3 b^3} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.49, size = 0, normalized size = 0.00 \[ {\rm integral}\left (x^{2} \operatorname {Si}\left (b x\right ), x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.60, size = 38, normalized size = 0.78 \[ \frac {1}{3} \, x^{3} \operatorname {Si}\left (b x\right ) - \frac {2 \, x \sin \left (b x\right )}{3 \, b^{2}} + \frac {{\left (b^{2} x^{2} - 2\right )} \cos \left (b x\right )}{3 \, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 44, normalized size = 0.90 \[ \frac {\frac {b^{3} x^{3} \Si \left (b x \right )}{3}+\frac {b^{2} x^{2} \cos \left (b x \right )}{3}-\frac {2 \cos \left (b x \right )}{3}-\frac {2 b x \sin \left (b x \right )}{3}}{b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{2} {\rm Si}\left (b x\right )\,{d x} \]
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 \[ \frac {x^3\,\mathrm {sinint}\left (b\,x\right )}{3}-\frac {\cos \left (b\,x\right )\,\left (\frac {2}{b^3}-\frac {x^2}{b}\right )}{3}-\frac {2\,x\,\sin \left (b\,x\right )}{3\,b^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.21, size = 46, normalized size = 0.94 \[ \frac {x^{3} \operatorname {Si}{\left (b x \right )}}{3} + \frac {x^{2} \cos {\left (b x \right )}}{3 b} - \frac {2 x \sin {\left (b x \right )}}{3 b^{2}} - \frac {2 \cos {\left (b x \right )}}{3 b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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