Optimal. Leaf size=54 \[ -\frac {2 C(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b}+x C(b x)^2+\frac {S\left (\sqrt {2} b x\right )}{\sqrt {2} \pi b} \]
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Rubi [A] time = 0.04, antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {6421, 12, 6453, 3351} \[ -\frac {2 \text {FresnelC}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b}+x \text {FresnelC}(b x)^2+\frac {S\left (\sqrt {2} b x\right )}{\sqrt {2} \pi b} \]
Antiderivative was successfully verified.
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Rule 12
Rule 3351
Rule 6421
Rule 6453
Rubi steps
\begin {align*} \int C(b x)^2 \, dx &=x C(b x)^2-2 \int b x \cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x) \, dx\\ &=x C(b x)^2-(2 b) \int x \cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x) \, dx\\ &=x C(b x)^2-\frac {2 C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b \pi }+\frac {\int \sin \left (b^2 \pi x^2\right ) \, dx}{\pi }\\ &=x C(b x)^2+\frac {S\left (\sqrt {2} b x\right )}{\sqrt {2} b \pi }-\frac {2 C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b \pi }\\ \end {align*}
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Mathematica [A] time = 0.01, size = 54, normalized size = 1.00 \[ -\frac {2 C(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b}+x C(b x)^2+\frac {S\left (\sqrt {2} b x\right )}{\sqrt {2} \pi b} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.47, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\rm fresnelc}\left (b x\right )^{2}, x\right ) \]
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 \[ \int {\rm fresnelc}\left (b x\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 49, normalized size = 0.91 \[ \frac {b x \FresnelC \left (b x \right )^{2}-\frac {2 \FresnelC \left (b x \right ) \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{\pi }+\frac {\sqrt {2}\, \mathrm {S}\left (b x \sqrt {2}\right )}{2 \pi }}{b} \]
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 \[ \int {\rm fresnelc}\left (b x\right )^{2}\,{d x} \]
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 \[ \int {\mathrm {FresnelC}\left (b\,x\right )}^2 \,d x \]
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 \[ \int C^{2}\left (b x\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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