Optimal. Leaf size=48 \[ -\frac{\text{FresnelC}(b x) \sin \left (\frac{1}{2} \pi b^2 x^2\right )}{x}+\frac{1}{4} b \text{Si}\left (b^2 \pi x^2\right )+\frac{1}{2} \pi b \text{FresnelC}(b x)^2 \]
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Rubi [A] time = 0.0427923, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {6465, 6441, 30, 3375} \[ -\frac{\text{FresnelC}(b x) \sin \left (\frac{1}{2} \pi b^2 x^2\right )}{x}+\frac{1}{4} b \text{Si}\left (b^2 \pi x^2\right )+\frac{1}{2} \pi b \text{FresnelC}(b x)^2 \]
Antiderivative was successfully verified.
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Rule 6465
Rule 6441
Rule 30
Rule 3375
Rubi steps
\begin{align*} \int \frac{C(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{x^2} \, dx &=-\frac{C(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{x}+\frac{1}{2} b \int \frac{\sin \left (b^2 \pi x^2\right )}{x} \, dx+\left (b^2 \pi \right ) \int \cos \left (\frac{1}{2} b^2 \pi x^2\right ) C(b x) \, dx\\ &=-\frac{C(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{x}+\frac{1}{4} b \text{Si}\left (b^2 \pi x^2\right )+(b \pi ) \operatorname{Subst}(\int x \, dx,x,C(b x))\\ &=\frac{1}{2} b \pi C(b x)^2-\frac{C(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{x}+\frac{1}{4} b \text{Si}\left (b^2 \pi x^2\right )\\ \end{align*}
Mathematica [A] time = 0.0050088, size = 48, normalized size = 1. \[ -\frac{\text{FresnelC}(b x) \sin \left (\frac{1}{2} \pi b^2 x^2\right )}{x}+\frac{1}{4} b \text{Si}\left (b^2 \pi x^2\right )+\frac{1}{2} \pi b \text{FresnelC}(b x)^2 \]
Antiderivative was successfully verified.
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Maple [F] time = 0.059, size = 0, normalized size = 0. \begin{align*} \int{\frac{{\it FresnelC} \left ( bx \right ) }{{x}^{2}}\sin \left ({\frac{{b}^{2}\pi \,{x}^{2}}{2}} \right ) }\, dx \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 \frac{{\rm fresnelc}\left (b x\right ) \sin \left (\frac{1}{2} \, \pi b^{2} x^{2}\right )}{x^{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 (\frac{{\rm fresnelc}\left (b x\right ) \sin \left (\frac{1}{2} \, \pi b^{2} x^{2}\right )}{x^{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 \frac{\sin{\left (\frac{\pi b^{2} x^{2}}{2} \right )} C\left (b x\right )}{x^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\rm fresnelc}\left (b x\right ) \sin \left (\frac{1}{2} \, \pi b^{2} x^{2}\right )}{x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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