Optimal. Leaf size=35 \[ -\frac {\cos (b x)}{2 b^2}+\frac {1}{2} x^2 \text {Ci}(b x)-\frac {x \sin (b x)}{2 b} \]
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Rubi [A] time = 0.02, antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {6504, 12, 3296, 2638} \[ -\frac {\cos (b x)}{2 b^2}+\frac {1}{2} x^2 \text {CosIntegral}(b x)-\frac {x \sin (b x)}{2 b} \]
Antiderivative was successfully verified.
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Rule 12
Rule 2638
Rule 3296
Rule 6504
Rubi steps
\begin {align*} \int x \text {Ci}(b x) \, dx &=\frac {1}{2} x^2 \text {Ci}(b x)-\frac {1}{2} b \int \frac {x \cos (b x)}{b} \, dx\\ &=\frac {1}{2} x^2 \text {Ci}(b x)-\frac {1}{2} \int x \cos (b x) \, dx\\ &=\frac {1}{2} x^2 \text {Ci}(b x)-\frac {x \sin (b x)}{2 b}+\frac {\int \sin (b x) \, dx}{2 b}\\ &=-\frac {\cos (b x)}{2 b^2}+\frac {1}{2} x^2 \text {Ci}(b x)-\frac {x \sin (b x)}{2 b}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 35, normalized size = 1.00 \[ -\frac {\cos (b x)}{2 b^2}+\frac {1}{2} x^2 \text {Ci}(b x)-\frac {x \sin (b x)}{2 b} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.71, size = 0, normalized size = 0.00 \[ {\rm integral}\left (x \operatorname {Ci}\left (b x\right ), x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 29, normalized size = 0.83 \[ \frac {1}{2} \, x^{2} \operatorname {Ci}\left (b x\right ) - \frac {x \sin \left (b x\right )}{2 \, b} - \frac {\cos \left (b x\right )}{2 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 32, normalized size = 0.91 \[ \frac {\frac {b^{2} x^{2} \Ci \left (b x \right )}{2}-\frac {\cos \left (b x \right )}{2}-\frac {b x \sin \left (b x \right )}{2}}{b^{2}} \]
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 \[ \int x {\rm Ci}\left (b x\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \frac {x^2\,\mathrm {cosint}\left (b\,x\right )}{2}-\frac {\cos \left (b\,x\right )+b\,x\,\sin \left (b\,x\right )}{2\,b^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.12, size = 53, normalized size = 1.51 \[ - \frac {x^{2} \log {\left (b x \right )}}{2} + \frac {x^{2} \log {\left (b^{2} x^{2} \right )}}{4} + \frac {x^{2} \operatorname {Ci}{\left (b x \right )}}{2} - \frac {x \sin {\left (b x \right )}}{2 b} - \frac {\cos {\left (b x \right )}}{2 b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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