Optimal. Leaf size=70 \[ \frac{(a+b x) S(a+b x)^2}{b}-\frac{S\left (\sqrt{2} (a+b x)\right )}{\sqrt{2} \pi b}+\frac{2 S(a+b x) \cos \left (\frac{1}{2} \pi (a+b x)^2\right )}{\pi b} \]
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Rubi [A] time = 0.167726, antiderivative size = 70, 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 = {6420, 6452, 3351} \[ \frac{(a+b x) S(a+b x)^2}{b}-\frac{S\left (\sqrt{2} (a+b x)\right )}{\sqrt{2} \pi b}+\frac{2 S(a+b x) \cos \left (\frac{1}{2} \pi (a+b x)^2\right )}{\pi b} \]
Antiderivative was successfully verified.
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Rule 6420
Rule 6452
Rule 3351
Rubi steps
\begin{align*} \int S(a+b x)^2 \, dx &=\frac{(a+b x) S(a+b x)^2}{b}-2 \int (a+b x) S(a+b x) \sin \left (\frac{1}{2} \pi (a+b x)^2\right ) \, dx\\ &=\frac{(a+b x) S(a+b x)^2}{b}-\frac{2 \operatorname{Subst}\left (\int x S(x) \sin \left (\frac{\pi x^2}{2}\right ) \, dx,x,a+b x\right )}{b}\\ &=\frac{2 \cos \left (\frac{1}{2} \pi (a+b x)^2\right ) S(a+b x)}{b \pi }+\frac{(a+b x) S(a+b x)^2}{b}-\frac{\operatorname{Subst}\left (\int \sin \left (\pi x^2\right ) \, dx,x,a+b x\right )}{b \pi }\\ &=\frac{2 \cos \left (\frac{1}{2} \pi (a+b x)^2\right ) S(a+b x)}{b \pi }+\frac{(a+b x) S(a+b x)^2}{b}-\frac{S\left (\sqrt{2} (a+b x)\right )}{\sqrt{2} b \pi }\\ \end{align*}
Mathematica [A] time = 0.0104769, size = 67, normalized size = 0.96 \[ \frac{2 \pi (a+b x) S(a+b x)^2-\sqrt{2} S\left (\sqrt{2} (a+b x)\right )+4 S(a+b x) \cos \left (\frac{1}{2} \pi (a+b x)^2\right )}{2 \pi b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.056, size = 60, normalized size = 0.9 \begin{align*}{\frac{1}{b} \left ( \left ( bx+a \right ) \left ({\it FresnelS} \left ( bx+a \right ) \right ) ^{2}+2\,{\frac{{\it FresnelS} \left ( bx+a \right ) \cos \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 fresnels}\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 fresnels}\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 S^{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 fresnels}\left (b x + a\right )^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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