Optimal. Leaf size=48 \[ -\frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{x}-\frac {1}{2} b \pi S(b x)^2+\frac {1}{4} b \text {Si}\left (b^2 \pi x^2\right ) \]
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Rubi [A]
time = 0.03, 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 = {6599, 6575, 30,
3456} \begin {gather*} -\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 \end {gather*}
Antiderivative was successfully verified.
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Rule 30
Rule 3456
Rule 6575
Rule 6599
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 ) \text {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 \begin {gather*} -\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 {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.20, 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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 46, normalized size = 0.96 \begin {gather*} -\frac {2 \, \pi b x \operatorname {S}\left (b x\right )^{2} - b x \operatorname {Si}\left (\pi b^{2} x^{2}\right ) + 4 \, \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) \operatorname {S}\left (b x\right )}{4 \, x} \end {gather*}
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 \begin {gather*} \int \frac {\cos {\left (\frac {\pi b^{2} x^{2}}{2} \right )} S\left (b x\right )}{x^{2}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {\mathrm {FresnelS}\left (b\,x\right )\,\cos \left (\frac {\Pi \,b^2\,x^2}{2}\right )}{x^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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