Optimal. Leaf size=95 \[ -\frac{a^2 \text{FresnelC}(a+b x)}{2 b^2}+\frac{S(a+b x)}{2 \pi b^2}+\frac{a \sin \left (\frac{1}{2} \pi (a+b x)^2\right )}{\pi b^2}-\frac{(a+b x) \sin \left (\frac{1}{2} \pi (a+b x)^2\right )}{2 \pi b^2}+\frac{1}{2} x^2 \text{FresnelC}(a+b x) \]
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Rubi [A] time = 0.0707863, antiderivative size = 95, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.875, Rules used = {6429, 3434, 3352, 3380, 2637, 3386, 3351} \[ -\frac{a^2 \text{FresnelC}(a+b x)}{2 b^2}+\frac{S(a+b x)}{2 \pi b^2}+\frac{a \sin \left (\frac{1}{2} \pi (a+b x)^2\right )}{\pi b^2}-\frac{(a+b x) \sin \left (\frac{1}{2} \pi (a+b x)^2\right )}{2 \pi b^2}+\frac{1}{2} x^2 \text{FresnelC}(a+b x) \]
Antiderivative was successfully verified.
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Rule 6429
Rule 3434
Rule 3352
Rule 3380
Rule 2637
Rule 3386
Rule 3351
Rubi steps
\begin{align*} \int x C(a+b x) \, dx &=\frac{1}{2} x^2 C(a+b x)-\frac{1}{2} b \int x^2 \cos \left (\frac{1}{2} \pi (a+b x)^2\right ) \, dx\\ &=\frac{1}{2} x^2 C(a+b x)-\frac{\operatorname{Subst}\left (\int \left (a^2 \cos \left (\frac{\pi x^2}{2}\right )-2 a x \cos \left (\frac{\pi x^2}{2}\right )+x^2 \cos \left (\frac{\pi x^2}{2}\right )\right ) \, dx,x,a+b x\right )}{2 b^2}\\ &=\frac{1}{2} x^2 C(a+b x)-\frac{\operatorname{Subst}\left (\int x^2 \cos \left (\frac{\pi x^2}{2}\right ) \, dx,x,a+b x\right )}{2 b^2}+\frac{a \operatorname{Subst}\left (\int x \cos \left (\frac{\pi x^2}{2}\right ) \, dx,x,a+b x\right )}{b^2}-\frac{a^2 \operatorname{Subst}\left (\int \cos \left (\frac{\pi x^2}{2}\right ) \, dx,x,a+b x\right )}{2 b^2}\\ &=-\frac{a^2 C(a+b x)}{2 b^2}+\frac{1}{2} x^2 C(a+b x)-\frac{(a+b x) \sin \left (\frac{1}{2} \pi (a+b x)^2\right )}{2 b^2 \pi }+\frac{a \operatorname{Subst}\left (\int \cos \left (\frac{\pi x}{2}\right ) \, dx,x,(a+b x)^2\right )}{2 b^2}+\frac{\operatorname{Subst}\left (\int \sin \left (\frac{\pi x^2}{2}\right ) \, dx,x,a+b x\right )}{2 b^2 \pi }\\ &=-\frac{a^2 C(a+b x)}{2 b^2}+\frac{1}{2} x^2 C(a+b x)+\frac{S(a+b x)}{2 b^2 \pi }+\frac{a \sin \left (\frac{1}{2} \pi (a+b x)^2\right )}{b^2 \pi }-\frac{(a+b x) \sin \left (\frac{1}{2} \pi (a+b x)^2\right )}{2 b^2 \pi }\\ \end{align*}
Mathematica [A] time = 0.148444, size = 59, normalized size = 0.62 \[ \frac{\left (\pi b^2 x^2-\pi a^2\right ) \text{FresnelC}(a+b x)+S(a+b x)+(a-b x) \sin \left (\frac{1}{2} \pi (a+b x)^2\right )}{2 \pi b^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.052, size = 79, normalized size = 0.8 \begin{align*}{\frac{1}{{b}^{2}} \left ({\it FresnelC} \left ( bx+a \right ) \left ({\frac{ \left ( bx+a \right ) ^{2}}{2}}-a \left ( bx+a \right ) \right ) -{\frac{bx+a}{2\,\pi }\sin \left ({\frac{\pi \, \left ( bx+a \right ) ^{2}}{2}} \right ) }+{\frac{{\it FresnelS} \left ( bx+a \right ) }{2\,\pi }}+{\frac{a}{\pi }\sin \left ({\frac{\pi \, \left ( bx+a \right ) ^{2}}{2}} \right ) } \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 x{\rm fresnelc}\left (b x + a\right )\,{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 (x{\rm fresnelc}\left (b x + a\right ), 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 x C\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 x{\rm fresnelc}\left (b x + a\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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