Optimal. Leaf size=111 \[ -\sqrt {2 \pi } \sqrt {c} \sin \left (a+\frac {b^2}{4 c}\right ) C\left (\frac {b-2 c x}{\sqrt {c} \sqrt {2 \pi }}\right )+\sqrt {2 \pi } \sqrt {c} \cos \left (a+\frac {b^2}{4 c}\right ) S\left (\frac {b-2 c x}{\sqrt {c} \sqrt {2 \pi }}\right )-\frac {\cos \left (a+b x-c x^2\right )}{x} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.09, 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 = {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.
[In]
[Out]
Rule 3351
Rule 3352
Rule 3447
Rule 3466
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 = 4.12, size = 114, normalized size = 1.03 \[ \sqrt {2 \pi } \sqrt {c} \sin \left (a+\frac {b^2}{4 c}\right ) C\left (\frac {2 c x-b}{\sqrt {c} \sqrt {2 \pi }}\right )-\sqrt {2 \pi } \sqrt {c} \cos \left (a+\frac {b^2}{4 c}\right ) S\left (\frac {2 c x-b}{\sqrt {c} \sqrt {2 \pi }}\right )-\frac {\cos (a+x (b-c x))}{x} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.53, size = 123, normalized size = 1.11 \[ -\frac {\sqrt {2} \pi x \sqrt {\frac {c}{\pi }} \cos \left (\frac {b^{2} + 4 \, a c}{4 \, c}\right ) \operatorname {S}\left (\frac {\sqrt {2} {\left (2 \, c x - b\right )} \sqrt {\frac {c}{\pi }}}{2 \, c}\right ) - \sqrt {2} \pi x \sqrt {\frac {c}{\pi }} \operatorname {C}\left (\frac {\sqrt {2} {\left (2 \, c x - b\right )} \sqrt {\frac {c}{\pi }}}{2 \, c}\right ) \sin \left (\frac {b^{2} + 4 \, a c}{4 \, c}\right ) + \cos \left (c x^{2} - b x - a\right )}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {b \sin \left (-c x^{2} + b x + a\right )}{x} + \frac {\cos \left (-c x^{2} + b x + a\right )}{x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.32, 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.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int -\frac {b \sin \left (c x^{2} - b x - a\right )}{x} + \frac {\cos \left (c x^{2} - b x - a\right )}{x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\cos \left (-c\,x^2+b\,x+a\right )}{x^2}+\frac {b\,\sin \left (-c\,x^2+b\,x+a\right )}{x} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {b x \sin {\left (a + b x - c x^{2} \right )} + \cos {\left (a + b x - c x^{2} \right )}}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________