Optimal. Leaf size=75 \[ \frac {\text {Ci}(2 b x)}{2 b^2}-\frac {\text {Ci}(b x) \cos (b x)}{b^2}+\frac {\log (x)}{2 b^2}+\frac {\sin ^2(b x)}{2 b^2}+\frac {1}{2} x^2 \text {Ci}(b x)^2-\frac {x \text {Ci}(b x) \sin (b x)}{b} \]
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Rubi [A] time = 0.09, antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {6508, 6514, 12, 2564, 30, 6518, 3312, 3302} \[ \frac {\text {CosIntegral}(2 b x)}{2 b^2}-\frac {\text {CosIntegral}(b x) \cos (b x)}{b^2}+\frac {\log (x)}{2 b^2}+\frac {\sin ^2(b x)}{2 b^2}+\frac {1}{2} x^2 \text {CosIntegral}(b x)^2-\frac {x \text {CosIntegral}(b x) \sin (b x)}{b} \]
Antiderivative was successfully verified.
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Rule 12
Rule 30
Rule 2564
Rule 3302
Rule 3312
Rule 6508
Rule 6514
Rule 6518
Rubi steps
\begin {align*} \int x \text {Ci}(b x)^2 \, dx &=\frac {1}{2} x^2 \text {Ci}(b x)^2-\int x \cos (b x) \text {Ci}(b x) \, dx\\ &=\frac {1}{2} x^2 \text {Ci}(b x)^2-\frac {x \text {Ci}(b x) \sin (b x)}{b}+\frac {\int \text {Ci}(b x) \sin (b x) \, dx}{b}+\int \frac {\cos (b x) \sin (b x)}{b} \, dx\\ &=-\frac {\cos (b x) \text {Ci}(b x)}{b^2}+\frac {1}{2} x^2 \text {Ci}(b x)^2-\frac {x \text {Ci}(b x) \sin (b x)}{b}+\frac {\int \frac {\cos ^2(b x)}{b x} \, dx}{b}+\frac {\int \cos (b x) \sin (b x) \, dx}{b}\\ &=-\frac {\cos (b x) \text {Ci}(b x)}{b^2}+\frac {1}{2} x^2 \text {Ci}(b x)^2-\frac {x \text {Ci}(b x) \sin (b x)}{b}+\frac {\int \frac {\cos ^2(b x)}{x} \, dx}{b^2}+\frac {\operatorname {Subst}(\int x \, dx,x,\sin (b x))}{b^2}\\ &=-\frac {\cos (b x) \text {Ci}(b x)}{b^2}+\frac {1}{2} x^2 \text {Ci}(b x)^2-\frac {x \text {Ci}(b x) \sin (b x)}{b}+\frac {\sin ^2(b x)}{2 b^2}+\frac {\int \left (\frac {1}{2 x}+\frac {\cos (2 b x)}{2 x}\right ) \, dx}{b^2}\\ &=-\frac {\cos (b x) \text {Ci}(b x)}{b^2}+\frac {1}{2} x^2 \text {Ci}(b x)^2+\frac {\log (x)}{2 b^2}-\frac {x \text {Ci}(b x) \sin (b x)}{b}+\frac {\sin ^2(b x)}{2 b^2}+\frac {\int \frac {\cos (2 b x)}{x} \, dx}{2 b^2}\\ &=-\frac {\cos (b x) \text {Ci}(b x)}{b^2}+\frac {1}{2} x^2 \text {Ci}(b x)^2+\frac {\text {Ci}(2 b x)}{2 b^2}+\frac {\log (x)}{2 b^2}-\frac {x \text {Ci}(b x) \sin (b x)}{b}+\frac {\sin ^2(b x)}{2 b^2}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 58, normalized size = 0.77 \[ \frac {2 b^2 x^2 \text {Ci}(b x)^2+2 \text {Ci}(2 b x)-4 \text {Ci}(b x) (b x \sin (b x)+\cos (b x))-\cos (2 b x)+2 \log (x)}{4 b^2} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.47, size = 0, normalized size = 0.00 \[ {\rm integral}\left (x \operatorname {Ci}\left (b x\right )^{2}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.22, size = 65, normalized size = 0.87 \[ \frac {1}{2} \, x^{2} \operatorname {Ci}\left (b x\right )^{2} - {\left (\frac {x \sin \left (b x\right )}{b} + \frac {\cos \left (b x\right )}{b^{2}}\right )} \operatorname {Ci}\left (b x\right ) - \frac {\cos \left (2 \, b x\right ) - \operatorname {Ci}\left (2 \, b x\right ) - \operatorname {Ci}\left (-2 \, b x\right ) - 2 \, \log \relax (x)}{4 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 70, normalized size = 0.93 \[ \frac {x^{2} \Ci \left (b x \right )^{2}}{2}-\frac {\Ci \left (b x \right ) \cos \left (b x \right )}{b^{2}}-\frac {x \Ci \left (b x \right ) \sin \left (b x \right )}{b}+\frac {\ln \left (b x \right )}{2 b^{2}}+\frac {\Ci \left (2 b x \right )}{2 b^{2}}-\frac {\cos ^{2}\left (b x \right )}{2 b^{2}} \]
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 \[ \int x {\rm Ci}\left (b x\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int x\,{\mathrm {cosint}\left (b\,x\right )}^2 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x \operatorname {Ci}^{2}{\left (b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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