Optimal. Leaf size=110 \[ \sqrt {2 \pi } \sqrt {c} \cos \left (a+\frac {b^2}{4 c}\right ) C\left (\frac {b-2 c x}{\sqrt {c} \sqrt {2 \pi }}\right )+\sqrt {2 \pi } \sqrt {c} \sin \left (a+\frac {b^2}{4 c}\right ) S\left (\frac {b-2 c x}{\sqrt {c} \sqrt {2 \pi }}\right )-\frac {\sin \left (a+b x-c x^2\right )}{x} \]
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Rubi [A] time = 0.08, antiderivative size = 110, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.114, Rules used = {3465, 3448, 3352, 3351} \[ \sqrt {2 \pi } \sqrt {c} \cos \left (a+\frac {b^2}{4 c}\right ) \text {FresnelC}\left (\frac {b-2 c x}{\sqrt {2 \pi } \sqrt {c}}\right )+\sqrt {2 \pi } \sqrt {c} \sin \left (a+\frac {b^2}{4 c}\right ) S\left (\frac {b-2 c x}{\sqrt {c} \sqrt {2 \pi }}\right )-\frac {\sin \left (a+b x-c x^2\right )}{x} \]
Antiderivative was successfully verified.
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Rule 3351
Rule 3352
Rule 3448
Rule 3465
Rubi steps
\begin {align*} \int \left (-\frac {b \cos \left (a+b x-c x^2\right )}{x}+\frac {\sin \left (a+b x-c x^2\right )}{x^2}\right ) \, dx &=-\left (b \int \frac {\cos \left (a+b x-c x^2\right )}{x} \, dx\right )+\int \frac {\sin \left (a+b x-c x^2\right )}{x^2} \, dx\\ &=-\frac {\sin \left (a+b x-c x^2\right )}{x}-(2 c) \int \cos \left (a+b x-c x^2\right ) \, dx\\ &=-\frac {\sin \left (a+b x-c x^2\right )}{x}-\left (2 c \cos \left (a+\frac {b^2}{4 c}\right )\right ) \int \cos \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 \sin \left (\frac {(b-2 c x)^2}{4 c}\right ) \, dx\\ &=\sqrt {c} \sqrt {2 \pi } \cos \left (a+\frac {b^2}{4 c}\right ) C\left (\frac {b-2 c x}{\sqrt {c} \sqrt {2 \pi }}\right )+\sqrt {c} \sqrt {2 \pi } S\left (\frac {b-2 c x}{\sqrt {c} \sqrt {2 \pi }}\right ) \sin \left (a+\frac {b^2}{4 c}\right )-\frac {\sin \left (a+b x-c x^2\right )}{x}\\ \end {align*}
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Mathematica [A] time = 0.81, size = 115, normalized size = 1.05 \[ -\frac {\sqrt {2 \pi } \sqrt {c} x \cos \left (a+\frac {b^2}{4 c}\right ) C\left (\frac {2 c x-b}{\sqrt {c} \sqrt {2 \pi }}\right )+\sqrt {2 \pi } \sqrt {c} x \sin \left (a+\frac {b^2}{4 c}\right ) S\left (\frac {2 c x-b}{\sqrt {c} \sqrt {2 \pi }}\right )+\sin (a+x (b-c x))}{x} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 124, normalized size = 1.13 \[ -\frac {\sqrt {2} \pi x \sqrt {\frac {c}{\pi }} \cos \left (\frac {b^{2} + 4 \, a c}{4 \, c}\right ) \operatorname {C}\left (\frac {\sqrt {2} {\left (2 \, c x - b\right )} \sqrt {\frac {c}{\pi }}}{2 \, c}\right ) + \sqrt {2} \pi x \sqrt {\frac {c}{\pi }} \operatorname {S}\left (\frac {\sqrt {2} {\left (2 \, c x - b\right )} \sqrt {\frac {c}{\pi }}}{2 \, c}\right ) \sin \left (\frac {b^{2} + 4 \, a c}{4 \, c}\right ) - \sin \left (c x^{2} - b x - a\right )}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int -\frac {b \cos \left (-c x^{2} + b x + a\right )}{x} + \frac {\sin \left (-c x^{2} + b x + a\right )}{x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.62, size = 0, normalized size = 0.00 \[ \int -\frac {b \cos \left (-c \,x^{2}+b x +a \right )}{x}+\frac {\sin \left (-c \,x^{2}+b x +a \right )}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int -\frac {b \cos \left (c x^{2} - b x - a\right )}{x} - \frac {\sin \left (c x^{2} - b x - a\right )}{x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\sin \left (-c\,x^2+b\,x+a\right )}{x^2}-\frac {b\,\cos \left (-c\,x^2+b\,x+a\right )}{x} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ - \int \left (- \frac {\sin {\left (a + b x - c x^{2} \right )}}{x^{2}}\right )\, dx - \int \frac {b \cos {\left (a + b x - c x^{2} \right )}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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