Optimal. Leaf size=49 \[ \frac {2 \sin (b x)}{3 b^3}-\frac {2 x \cos (b x)}{3 b^2}+\frac {1}{3} x^3 \text {Ci}(b x)-\frac {x^2 \sin (b x)}{3 b} \]
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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 = {6504, 12, 3296, 2637} \[ \frac {2 \sin (b x)}{3 b^3}-\frac {2 x \cos (b x)}{3 b^2}+\frac {1}{3} x^3 \text {CosIntegral}(b x)-\frac {x^2 \sin (b x)}{3 b} \]
Antiderivative was successfully verified.
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Rule 12
Rule 2637
Rule 3296
Rule 6504
Rubi steps
\begin {align*} \int x^2 \text {Ci}(b x) \, dx &=\frac {1}{3} x^3 \text {Ci}(b x)-\frac {1}{3} b \int \frac {x^2 \cos (b x)}{b} \, dx\\ &=\frac {1}{3} x^3 \text {Ci}(b x)-\frac {1}{3} \int x^2 \cos (b x) \, dx\\ &=\frac {1}{3} x^3 \text {Ci}(b x)-\frac {x^2 \sin (b x)}{3 b}+\frac {2 \int x \sin (b x) \, dx}{3 b}\\ &=-\frac {2 x \cos (b x)}{3 b^2}+\frac {1}{3} x^3 \text {Ci}(b x)-\frac {x^2 \sin (b x)}{3 b}+\frac {2 \int \cos (b x) \, dx}{3 b^2}\\ &=-\frac {2 x \cos (b x)}{3 b^2}+\frac {1}{3} x^3 \text {Ci}(b x)+\frac {2 \sin (b x)}{3 b^3}-\frac {x^2 \sin (b x)}{3 b}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 44, normalized size = 0.90 \[ -\frac {2 x \cos (b x)}{3 b^2}-\frac {\left (b^2 x^2-2\right ) \sin (b x)}{3 b^3}+\frac {1}{3} x^3 \text {Ci}(b x) \]
Antiderivative was successfully verified.
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fricas [F] time = 0.60, size = 0, normalized size = 0.00 \[ {\rm integral}\left (x^{2} \operatorname {Ci}\left (b x\right ), x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 38, normalized size = 0.78 \[ \frac {1}{3} \, x^{3} \operatorname {Ci}\left (b x\right ) - \frac {2 \, x \cos \left (b x\right )}{3 \, b^{2}} - \frac {{\left (b^{2} x^{2} - 2\right )} \sin \left (b x\right )}{3 \, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 44, normalized size = 0.90 \[ \frac {\frac {b^{3} x^{3} \Ci \left (b x \right )}{3}-\frac {b^{2} x^{2} \sin \left (b x \right )}{3}+\frac {2 \sin \left (b x \right )}{3}-\frac {2 b x \cos \left (b x \right )}{3}}{b^{3}} \]
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 \[ \int x^{2} {\rm Ci}\left (b x\right )\,{d x} \]
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 \[ \frac {x^3\,\mathrm {cosint}\left (b\,x\right )}{3}-\frac {b^2\,x^2\,\sin \left (b\,x\right )-2\,\sin \left (b\,x\right )+2\,b\,x\,\cos \left (b\,x\right )}{3\,b^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.79, size = 70, normalized size = 1.43 \[ - \frac {x^{3} \log {\left (b x \right )}}{3} + \frac {x^{3} \log {\left (b^{2} x^{2} \right )}}{6} + \frac {x^{3} \operatorname {Ci}{\left (b x \right )}}{3} - \frac {x^{2} \sin {\left (b x \right )}}{3 b} - \frac {2 x \cos {\left (b x \right )}}{3 b^{2}} + \frac {2 \sin {\left (b x \right )}}{3 b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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