Optimal. Leaf size=108 \[ \frac{1}{2} \sqrt{\pi } \sqrt{c} e^{a+\frac{b^2}{4 c}} \text{Erf}\left (\frac{b-2 c x}{2 \sqrt{c}}\right )-\frac{1}{2} \sqrt{\pi } \sqrt{c} e^{-a-\frac{b^2}{4 c}} \text{Erfi}\left (\frac{b-2 c x}{2 \sqrt{c}}\right )-\frac{\cosh \left (a+b x-c x^2\right )}{x} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0891469, antiderivative size = 108, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {5391, 5374, 2234, 2205, 2204} \[ \frac{1}{2} \sqrt{\pi } \sqrt{c} e^{a+\frac{b^2}{4 c}} \text{Erf}\left (\frac{b-2 c x}{2 \sqrt{c}}\right )-\frac{1}{2} \sqrt{\pi } \sqrt{c} e^{-a-\frac{b^2}{4 c}} \text{Erfi}\left (\frac{b-2 c x}{2 \sqrt{c}}\right )-\frac{\cosh \left (a+b x-c x^2\right )}{x} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 5391
Rule 5374
Rule 2234
Rule 2205
Rule 2204
Rubi steps
\begin{align*} \int \left (\frac{\cosh \left (a+b x-c x^2\right )}{x^2}-\frac{b \sinh \left (a+b x-c x^2\right )}{x}\right ) \, dx &=-\left (b \int \frac{\sinh \left (a+b x-c x^2\right )}{x} \, dx\right )+\int \frac{\cosh \left (a+b x-c x^2\right )}{x^2} \, dx\\ &=-\frac{\cosh \left (a+b x-c x^2\right )}{x}-(2 c) \int \sinh \left (a+b x-c x^2\right ) \, dx\\ &=-\frac{\cosh \left (a+b x-c x^2\right )}{x}-c \int e^{a+b x-c x^2} \, dx+c \int e^{-a-b x+c x^2} \, dx\\ &=-\frac{\cosh \left (a+b x-c x^2\right )}{x}+\left (c e^{-a-\frac{b^2}{4 c}}\right ) \int e^{\frac{(-b+2 c x)^2}{4 c}} \, dx-\left (c e^{a+\frac{b^2}{4 c}}\right ) \int e^{-\frac{(b-2 c x)^2}{4 c}} \, dx\\ &=-\frac{\cosh \left (a+b x-c x^2\right )}{x}+\frac{1}{2} \sqrt{c} e^{a+\frac{b^2}{4 c}} \sqrt{\pi } \text{erf}\left (\frac{b-2 c x}{2 \sqrt{c}}\right )-\frac{1}{2} \sqrt{c} e^{-a-\frac{b^2}{4 c}} \sqrt{\pi } \text{erfi}\left (\frac{b-2 c x}{2 \sqrt{c}}\right )\\ \end{align*}
Mathematica [A] time = 9.15112, size = 136, normalized size = 1.26 \[ \frac{1}{2} \left (-\sqrt{\pi } \sqrt{c} \text{Erf}\left (\frac{2 c x-b}{2 \sqrt{c}}\right ) \left (\sinh \left (a+\frac{b^2}{4 c}\right )+\cosh \left (a+\frac{b^2}{4 c}\right )\right )+\sqrt{\pi } \sqrt{c} \text{Erfi}\left (\frac{2 c x-b}{2 \sqrt{c}}\right ) \left (\cosh \left (a+\frac{b^2}{4 c}\right )-\sinh \left (a+\frac{b^2}{4 c}\right )\right )-\frac{2 \cosh (a+x (b-c x))}{x}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.111, size = 0, normalized size = 0. \begin{align*} \int{\frac{\cosh \left ( -c{x}^{2}+bx+a \right ) }{{x}^{2}}}-{\frac{b\sinh \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 \sinh \left (c x^{2} - b x - a\right )}{x} + \frac{\cosh \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 [B] time = 2.09882, size = 861, normalized size = 7.97 \begin{align*} -\frac{\sqrt{\pi }{\left (x \cosh \left (c x^{2} - b x - a\right ) \cosh \left (\frac{b^{2} + 4 \, a c}{4 \, c}\right ) - x \cosh \left (c x^{2} - b x - a\right ) \sinh \left (\frac{b^{2} + 4 \, a c}{4 \, c}\right ) +{\left (x \cosh \left (\frac{b^{2} + 4 \, a c}{4 \, c}\right ) - x \sinh \left (\frac{b^{2} + 4 \, a c}{4 \, c}\right )\right )} \sinh \left (c x^{2} - b x - a\right )\right )} \sqrt{-c} \operatorname{erf}\left (\frac{{\left (2 \, c x - b\right )} \sqrt{-c}}{2 \, c}\right ) + \sqrt{\pi }{\left (x \cosh \left (c x^{2} - b x - a\right ) \cosh \left (\frac{b^{2} + 4 \, a c}{4 \, c}\right ) + x \cosh \left (c x^{2} - b x - a\right ) \sinh \left (\frac{b^{2} + 4 \, a c}{4 \, c}\right ) +{\left (x \cosh \left (\frac{b^{2} + 4 \, a c}{4 \, c}\right ) + x \sinh \left (\frac{b^{2} + 4 \, a c}{4 \, c}\right )\right )} \sinh \left (c x^{2} - b x - a\right )\right )} \sqrt{c} \operatorname{erf}\left (\frac{2 \, c x - b}{2 \, \sqrt{c}}\right ) + \cosh \left (c x^{2} - b x - a\right )^{2} + 2 \, \cosh \left (c x^{2} - b x - a\right ) \sinh \left (c x^{2} - b x - a\right ) + \sinh \left (c x^{2} - b x - a\right )^{2} + 1}{2 \,{\left (x \cosh \left (c x^{2} - b x - a\right ) + x \sinh \left (c x^{2} - b x - a\right )\right )}} \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{\cosh{\left (a + b x - c x^{2} \right )}}{x^{2}}\, dx - \int \frac{b \sinh{\left (a + b x - c x^{2} \right )}}{x}\, 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 \sinh \left (-c x^{2} + b x + a\right )}{x} + \frac{\cosh \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]