Optimal. Leaf size=69 \[ \frac{(a+b x) \text{FresnelC}(a+b x)^2}{b}-\frac{2 \text{FresnelC}(a+b x) \sin \left (\frac{1}{2} \pi (a+b x)^2\right )}{\pi b}+\frac{S\left (\sqrt{2} (a+b x)\right )}{\sqrt{2} \pi b} \]
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Rubi [A] time = 0.150686, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {6421, 6453, 3351} \[ \frac{(a+b x) \text{FresnelC}(a+b x)^2}{b}-\frac{2 \text{FresnelC}(a+b x) \sin \left (\frac{1}{2} \pi (a+b x)^2\right )}{\pi b}+\frac{S\left (\sqrt{2} (a+b x)\right )}{\sqrt{2} \pi b} \]
Antiderivative was successfully verified.
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Rule 6421
Rule 6453
Rule 3351
Rubi steps
\begin{align*} \int C(a+b x)^2 \, dx &=\frac{(a+b x) C(a+b x)^2}{b}-2 \int (a+b x) \cos \left (\frac{1}{2} \pi (a+b x)^2\right ) C(a+b x) \, dx\\ &=\frac{(a+b x) C(a+b x)^2}{b}-\frac{2 \operatorname{Subst}\left (\int x \cos \left (\frac{\pi x^2}{2}\right ) C(x) \, dx,x,a+b x\right )}{b}\\ &=\frac{(a+b x) C(a+b x)^2}{b}-\frac{2 C(a+b x) \sin \left (\frac{1}{2} \pi (a+b x)^2\right )}{b \pi }+\frac{\operatorname{Subst}\left (\int \sin \left (\pi x^2\right ) \, dx,x,a+b x\right )}{b \pi }\\ &=\frac{(a+b x) C(a+b x)^2}{b}+\frac{S\left (\sqrt{2} (a+b x)\right )}{\sqrt{2} b \pi }-\frac{2 C(a+b x) \sin \left (\frac{1}{2} \pi (a+b x)^2\right )}{b \pi }\\ \end{align*}
Mathematica [A] time = 0.0099422, size = 66, normalized size = 0.96 \[ \frac{2 \pi (a+b x) \text{FresnelC}(a+b x)^2-4 \text{FresnelC}(a+b x) \sin \left (\frac{1}{2} \pi (a+b x)^2\right )+\sqrt{2} S\left (\sqrt{2} (a+b x)\right )}{2 \pi b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.054, size = 60, normalized size = 0.9 \begin{align*}{\frac{1}{b} \left ( \left ( bx+a \right ) \left ({\it FresnelC} \left ( bx+a \right ) \right ) ^{2}-2\,{\frac{{\it FresnelC} \left ( bx+a \right ) \sin \left ( 1/2\,\pi \, \left ( bx+a \right ) ^{2} \right ) }{\pi }}+{\frac{\sqrt{2}{\it FresnelS} \left ( \left ( bx+a \right ) \sqrt{2} \right ) }{2\,\pi }} \right ) } \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{\rm fresnelc}\left (b x + a\right )^{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 ({\rm fresnelc}\left (b x + a\right )^{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 C^{2}\left (a + b x\right )\, 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{\rm fresnelc}\left (b x + a\right )^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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