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 = 0.04, antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, 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 30
Rule 3375
Rule 6440
Rule 6464
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*}
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Mathematica [A] time = 0.01, size = 48, normalized size = 1.00 \[ -\frac {S(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{x}+\frac {1}{4} b \text {Si}\left (b^2 \pi x^2\right )-\frac {1}{2} \pi b S(b x)^2 \]
Antiderivative was successfully verified.
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fricas [F] time = 0.40, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) {\rm fresnels}\left (b x\right )}{x^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) {\rm fresnels}\left (b x\right )}{x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.03, size = 0, normalized size = 0.00 \[ \int \frac {\cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right ) \mathrm {S}\left (b x \right )}{x^{2}}\, dx \]
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 \[ \int \frac {\cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) {\rm fresnels}\left (b x\right )}{x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {\mathrm {FresnelS}\left (b\,x\right )\,\cos \left (\frac {\Pi \,b^2\,x^2}{2}\right )}{x^2} \,d x \]
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 \[ \int \frac {\cos {\left (\frac {\pi b^{2} x^{2}}{2} \right )} S\left (b x\right )}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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