Optimal. Leaf size=75 \[ \frac{\text{CosIntegral}(2 b x)}{2 b^2}-\frac{\text{CosIntegral}(b x) \cos (b x)}{b^2}+\frac{\log (x)}{2 b^2}+\frac{\sin ^2(b x)}{2 b^2}+\frac{1}{2} x^2 \text{CosIntegral}(b x)^2-\frac{x \text{CosIntegral}(b x) \sin (b x)}{b} \]
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Rubi [A] time = 0.0906658, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 8, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 1., 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 6508
Rule 6514
Rule 12
Rule 2564
Rule 30
Rule 6518
Rule 3312
Rule 3302
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*}
Mathematica [A] time = 0.0489505, size = 58, normalized size = 0.77 \[ \frac{2 b^2 x^2 \text{CosIntegral}(b x)^2+2 \text{CosIntegral}(2 b x)-4 \text{CosIntegral}(b x) (b x \sin (b x)+\cos (b x))-\cos (2 b x)+2 \log (x)}{4 b^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.059, size = 70, normalized size = 0.9 \begin{align*}{\frac{{x}^{2} \left ({\it Ci} \left ( bx \right ) \right ) ^{2}}{2}}-{\frac{x{\it Ci} \left ( bx \right ) \sin \left ( bx \right ) }{b}}-{\frac{{\it Ci} \left ( bx \right ) \cos \left ( bx \right ) }{{b}^{2}}}-{\frac{ \left ( \cos \left ( bx \right ) \right ) ^{2}}{2\,{b}^{2}}}+{\frac{\ln \left ( bx \right ) }{2\,{b}^{2}}}+{\frac{{\it Ci} \left ( 2\,bx \right ) }{2\,{b}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x{\rm Ci}\left (b x\right )^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (x \operatorname{Ci}\left (b x\right )^{2}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x \operatorname{Ci}^{2}{\left (b x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.29807, size = 76, normalized size = 1.01 \begin{align*} \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{\operatorname{Ci}\left (2 \, b x\right ) + \operatorname{Ci}\left (-2 \, b x\right ) + 2 \, \log \left (x\right )}{4 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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