Optimal. Leaf size=111 \[ -\sqrt{2 \pi } \sqrt{c} \sin \left (a+\frac{b^2}{4 c}\right ) \text{FresnelC}\left (\frac{b-2 c x}{\sqrt{2 \pi } \sqrt{c}}\right )+\sqrt{2 \pi } \sqrt{c} \cos \left (a+\frac{b^2}{4 c}\right ) S\left (\frac{b-2 c x}{\sqrt{c} \sqrt{2 \pi }}\right )-\frac{\cos \left (a+b x-c x^2\right )}{x} \]
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Rubi [A] time = 0.0923342, antiderivative size = 111, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 34, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {3466, 3447, 3351, 3352} \[ -\sqrt{2 \pi } \sqrt{c} \sin \left (a+\frac{b^2}{4 c}\right ) \text{FresnelC}\left (\frac{b-2 c x}{\sqrt{2 \pi } \sqrt{c}}\right )+\sqrt{2 \pi } \sqrt{c} \cos \left (a+\frac{b^2}{4 c}\right ) S\left (\frac{b-2 c x}{\sqrt{c} \sqrt{2 \pi }}\right )-\frac{\cos \left (a+b x-c x^2\right )}{x} \]
Antiderivative was successfully verified.
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Rule 3466
Rule 3447
Rule 3351
Rule 3352
Rubi steps
\begin{align*} \int \left (\frac{\cos \left (a+b x-c x^2\right )}{x^2}+\frac{b \sin \left (a+b x-c x^2\right )}{x}\right ) \, dx &=b \int \frac{\sin \left (a+b x-c x^2\right )}{x} \, dx+\int \frac{\cos \left (a+b x-c x^2\right )}{x^2} \, dx\\ &=-\frac{\cos \left (a+b x-c x^2\right )}{x}+(2 c) \int \sin \left (a+b x-c x^2\right ) \, dx\\ &=-\frac{\cos \left (a+b x-c x^2\right )}{x}-\left (2 c \cos \left (a+\frac{b^2}{4 c}\right )\right ) \int \sin \left (\frac{(b-2 c x)^2}{4 c}\right ) \, dx+\left (2 c \sin \left (a+\frac{b^2}{4 c}\right )\right ) \int \cos \left (\frac{(b-2 c x)^2}{4 c}\right ) \, dx\\ &=-\frac{\cos \left (a+b x-c x^2\right )}{x}+\sqrt{c} \sqrt{2 \pi } \cos \left (a+\frac{b^2}{4 c}\right ) S\left (\frac{b-2 c x}{\sqrt{c} \sqrt{2 \pi }}\right )-\sqrt{c} \sqrt{2 \pi } C\left (\frac{b-2 c x}{\sqrt{c} \sqrt{2 \pi }}\right ) \sin \left (a+\frac{b^2}{4 c}\right )\\ \end{align*}
Mathematica [A] time = 4.62468, size = 114, normalized size = 1.03 \[ \sqrt{2 \pi } \sqrt{c} \sin \left (a+\frac{b^2}{4 c}\right ) \text{FresnelC}\left (\frac{2 c x-b}{\sqrt{2 \pi } \sqrt{c}}\right )-\sqrt{2 \pi } \sqrt{c} \cos \left (a+\frac{b^2}{4 c}\right ) S\left (\frac{2 c x-b}{\sqrt{c} \sqrt{2 \pi }}\right )-\frac{\cos (a+x (b-c x))}{x} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.223, size = 0, normalized size = 0. \begin{align*} \int{\frac{\cos \left ( -c{x}^{2}+bx+a \right ) }{{x}^{2}}}+{\frac{b\sin \left ( -c{x}^{2}+bx+a \right ) }{x}}\, dx \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 -\frac{b \sin \left (c x^{2} - b x - a\right )}{x} + \frac{\cos \left (c x^{2} - b x - a\right )}{x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.46302, size = 309, normalized size = 2.78 \begin{align*} -\frac{\sqrt{2} \pi x \sqrt{\frac{c}{\pi }} \cos \left (\frac{b^{2} + 4 \, a c}{4 \, c}\right ) \operatorname{S}\left (\frac{\sqrt{2}{\left (2 \, c x - b\right )} \sqrt{\frac{c}{\pi }}}{2 \, c}\right ) - \sqrt{2} \pi x \sqrt{\frac{c}{\pi }} \operatorname{C}\left (\frac{\sqrt{2}{\left (2 \, c x - b\right )} \sqrt{\frac{c}{\pi }}}{2 \, c}\right ) \sin \left (\frac{b^{2} + 4 \, a c}{4 \, c}\right ) + \cos \left (c x^{2} - b x - a\right )}{x} \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 \frac{b x \sin{\left (a + b x - c x^{2} \right )} + \cos{\left (a + b x - c x^{2} \right )}}{x^{2}}\, 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 \frac{b \sin \left (-c x^{2} + b x + a\right )}{x} + \frac{\cos \left (-c x^{2} + b x + a\right )}{x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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