Optimal. Leaf size=121 \[ -\frac{(b c-a d)^2 S(a+b x)}{2 b^2 d}+\frac{(b c-a d) \cos \left (\frac{1}{2} \pi (a+b x)^2\right )}{\pi b^2}-\frac{d \text{FresnelC}(a+b x)}{2 \pi b^2}+\frac{d (a+b x) \cos \left (\frac{1}{2} \pi (a+b x)^2\right )}{2 \pi b^2}+\frac{(c+d x)^2 S(a+b x)}{2 d} \]
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Rubi [A] time = 0.1145, antiderivative size = 121, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.583, Rules used = {6428, 3433, 3351, 3379, 2638, 3385, 3352} \[ -\frac{(b c-a d)^2 S(a+b x)}{2 b^2 d}+\frac{(b c-a d) \cos \left (\frac{1}{2} \pi (a+b x)^2\right )}{\pi b^2}-\frac{d \text{FresnelC}(a+b x)}{2 \pi b^2}+\frac{d (a+b x) \cos \left (\frac{1}{2} \pi (a+b x)^2\right )}{2 \pi b^2}+\frac{(c+d x)^2 S(a+b x)}{2 d} \]
Antiderivative was successfully verified.
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Rule 6428
Rule 3433
Rule 3351
Rule 3379
Rule 2638
Rule 3385
Rule 3352
Rubi steps
\begin{align*} \int (c+d x) S(a+b x) \, dx &=\frac{(c+d x)^2 S(a+b x)}{2 d}-\frac{b \int (c+d x)^2 \sin \left (\frac{1}{2} \pi (a+b x)^2\right ) \, dx}{2 d}\\ &=\frac{(c+d x)^2 S(a+b x)}{2 d}-\frac{\operatorname{Subst}\left (\int \left (b^2 c^2 \left (1+\frac{a d (-2 b c+a d)}{b^2 c^2}\right ) \sin \left (\frac{\pi x^2}{2}\right )+2 b c d \left (1-\frac{a d}{b c}\right ) x \sin \left (\frac{\pi x^2}{2}\right )+d^2 x^2 \sin \left (\frac{\pi x^2}{2}\right )\right ) \, dx,x,a+b x\right )}{2 b^2 d}\\ &=\frac{(c+d x)^2 S(a+b x)}{2 d}-\frac{d \operatorname{Subst}\left (\int x^2 \sin \left (\frac{\pi x^2}{2}\right ) \, dx,x,a+b x\right )}{2 b^2}-\frac{(b c-a d) \operatorname{Subst}\left (\int x \sin \left (\frac{\pi x^2}{2}\right ) \, dx,x,a+b x\right )}{b^2}-\frac{(b c-a d)^2 \operatorname{Subst}\left (\int \sin \left (\frac{\pi x^2}{2}\right ) \, dx,x,a+b x\right )}{2 b^2 d}\\ &=\frac{d (a+b x) \cos \left (\frac{1}{2} \pi (a+b x)^2\right )}{2 b^2 \pi }-\frac{(b c-a d)^2 S(a+b x)}{2 b^2 d}+\frac{(c+d x)^2 S(a+b x)}{2 d}-\frac{(b c-a d) \operatorname{Subst}\left (\int \sin \left (\frac{\pi x}{2}\right ) \, dx,x,(a+b x)^2\right )}{2 b^2}-\frac{d \operatorname{Subst}\left (\int \cos \left (\frac{\pi x^2}{2}\right ) \, dx,x,a+b x\right )}{2 b^2 \pi }\\ &=\frac{(b c-a d) \cos \left (\frac{1}{2} \pi (a+b x)^2\right )}{b^2 \pi }+\frac{d (a+b x) \cos \left (\frac{1}{2} \pi (a+b x)^2\right )}{2 b^2 \pi }-\frac{d C(a+b x)}{2 b^2 \pi }-\frac{(b c-a d)^2 S(a+b x)}{2 b^2 d}+\frac{(c+d x)^2 S(a+b x)}{2 d}\\ \end{align*}
Mathematica [A] time = 0.208895, size = 61, normalized size = 0.5 \[ \frac{(-a d+2 b c+b d x) \left (\pi (a+b x) S(a+b x)+\cos \left (\frac{1}{2} \pi (a+b x)^2\right )\right )-d \text{FresnelC}(a+b x)}{2 \pi b^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.051, size = 108, normalized size = 0.9 \begin{align*}{\frac{1}{b} \left ({\frac{{\it FresnelS} \left ( bx+a \right ) }{b} \left ({\frac{d \left ( bx+a \right ) ^{2}}{2}}-ad \left ( bx+a \right ) +bc \left ( bx+a \right ) \right ) }-{\frac{1}{2\,b} \left ( -{\frac{d \left ( bx+a \right ) }{\pi }\cos \left ({\frac{\pi \, \left ( bx+a \right ) ^{2}}{2}} \right ) }+{\frac{d{\it FresnelC} \left ( bx+a \right ) }{\pi }}-{\frac{-2\,ad+2\,bc}{\pi }\cos \left ({\frac{\pi \, \left ( bx+a \right ) ^{2}}{2}} \right ) } \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{\left (d x + c\right )}{\rm fresnels}\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 ({\left (d x + c\right )}{\rm fresnels}\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 \left (c + d x\right ) S\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{\left (d x + c\right )}{\rm fresnels}\left (b x + a\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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