Optimal. Leaf size=48 \[ -\frac{S(b x) \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{x}+\frac{1}{4} b \text{Si}\left (b^2 \pi x^2\right )-\frac{1}{2} \pi b S(b x)^2 \]
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Rubi [A] time = 0.0413899, 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 = {6464, 6440, 30, 3375} \[ -\frac{S(b x) \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{x}+\frac{1}{4} b \text{Si}\left (b^2 \pi x^2\right )-\frac{1}{2} \pi b S(b x)^2 \]
Antiderivative was successfully verified.
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Rule 6464
Rule 6440
Rule 30
Rule 3375
Rubi steps
\begin{align*} \int \frac{\cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{x^2} \, dx &=-\frac{\cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{x}+\frac{1}{2} b \int \frac{\sin \left (b^2 \pi x^2\right )}{x} \, dx-\left (b^2 \pi \right ) \int S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right ) \, dx\\ &=-\frac{\cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{x}+\frac{1}{4} b \text{Si}\left (b^2 \pi x^2\right )-(b \pi ) \operatorname{Subst}(\int x \, dx,x,S(b x))\\ &=-\frac{\cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{x}-\frac{1}{2} b \pi S(b x)^2+\frac{1}{4} b \text{Si}\left (b^2 \pi x^2\right )\\ \end{align*}
Mathematica [A] time = 0.0045022, size = 48, normalized size = 1. \[ -\frac{S(b x) \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{x}+\frac{1}{4} b \text{Si}\left (b^2 \pi x^2\right )-\frac{1}{2} \pi b S(b x)^2 \]
Antiderivative was successfully verified.
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Maple [F] time = 0.063, size = 0, normalized size = 0. \begin{align*} \int{\frac{{\it FresnelS} \left ( bx \right ) }{{x}^{2}}\cos \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{\cos \left (\frac{1}{2} \, \pi b^{2} x^{2}\right ){\rm fresnels}\left (b x\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{\cos \left (\frac{1}{2} \, \pi b^{2} x^{2}\right ){\rm fresnels}\left (b x\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{\cos{\left (\frac{\pi b^{2} x^{2}}{2} \right )} S\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{\cos \left (\frac{1}{2} \, \pi b^{2} x^{2}\right ){\rm fresnels}\left (b x\right )}{x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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