Optimal. Leaf size=111 \[ -\frac {\cos \left (a+b x-c x^2\right )}{x}+\sqrt {c} \sqrt {2 \pi } \cos \left (a+\frac {b^2}{4 c}\right ) S\left (\frac {b-2 c x}{\sqrt {c} \sqrt {2 \pi }}\right )-\sqrt {c} \sqrt {2 \pi } \text {FresnelC}\left (\frac {b-2 c x}{\sqrt {c} \sqrt {2 \pi }}\right ) \sin \left (a+\frac {b^2}{4 c}\right ) \]
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Rubi [A]
time = 0.06, antiderivative size = 111, normalized size of antiderivative = 1.00, 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 = {3547, 3528,
3432, 3433} \begin {gather*} -\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} \end {gather*}
Antiderivative was successfully verified.
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Rule 3432
Rule 3433
Rule 3528
Rule 3547
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*}
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Mathematica [A]
time = 3.98, size = 114, normalized size = 1.03 \begin {gather*} -\frac {\cos (a+x (b-c x))}{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 } \text {FresnelC}\left (\frac {-b+2 c x}{\sqrt {c} \sqrt {2 \pi }}\right ) \sin \left (a+\frac {b^2}{4 c}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.12, size = 0, normalized size = 0.00 \[\int \frac {\cos \left (-c \,x^{2}+b x +a \right )}{x^{2}}+\frac {b \sin \left (-c \,x^{2}+b x +a \right )}{x}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.44, size = 123, normalized size = 1.11 \begin {gather*} -\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 {gather*}
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 \begin {gather*} \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 {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {\cos \left (-c\,x^2+b\,x+a\right )}{x^2}+\frac {b\,\sin \left (-c\,x^2+b\,x+a\right )}{x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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