Optimal. Leaf size=111 \[ -\sqrt{2 \pi } \sqrt{c} \sin \left (a-\frac{b^2}{4 c}\right ) \text{FresnelC}\left (\frac{b+2 c x}{\sqrt{2 \pi } \sqrt{c}}\right )-\sqrt{2 \pi } \sqrt{c} \cos \left (a-\frac{b^2}{4 c}\right ) S\left (\frac{b+2 c x}{\sqrt{c} \sqrt{2 \pi }}\right )-\frac{\cos \left (a+b x+c x^2\right )}{x} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0935354, antiderivative size = 111, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, 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 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 = 3.74542, size = 110, normalized size = 0.99 \[ -\frac{\sqrt{2 \pi } \sqrt{c} x \sin \left (a-\frac{b^2}{4 c}\right ) \text{FresnelC}\left (\frac{b+2 c x}{\sqrt{2 \pi } \sqrt{c}}\right )+\sqrt{2 \pi } \sqrt{c} x \cos \left (a-\frac{b^2}{4 c}\right ) S\left (\frac{b+2 c x}{\sqrt{c} \sqrt{2 \pi }}\right )+\cos (a+x (b+c x))}{x} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.224, 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.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.40712, size = 312, normalized size = 2.81 \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.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]