Optimal. Leaf size=91 \[ \frac{\sqrt{\pi } e^{\frac{b^2}{4 c}-a} \text{Erf}\left (\frac{b+2 c x}{2 \sqrt{c}}\right )}{4 \sqrt{c}}+\frac{\sqrt{\pi } e^{a-\frac{b^2}{4 c}} \text{Erfi}\left (\frac{b+2 c x}{2 \sqrt{c}}\right )}{4 \sqrt{c}} \]
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Rubi [A] time = 0.0339899, antiderivative size = 91, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364, Rules used = {5375, 2234, 2204, 2205} \[ \frac{\sqrt{\pi } e^{\frac{b^2}{4 c}-a} \text{Erf}\left (\frac{b+2 c x}{2 \sqrt{c}}\right )}{4 \sqrt{c}}+\frac{\sqrt{\pi } e^{a-\frac{b^2}{4 c}} \text{Erfi}\left (\frac{b+2 c x}{2 \sqrt{c}}\right )}{4 \sqrt{c}} \]
Antiderivative was successfully verified.
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Rule 5375
Rule 2234
Rule 2204
Rule 2205
Rubi steps
\begin{align*} \int \cosh \left (a+b x+c x^2\right ) \, dx &=\frac{1}{2} \int e^{-a-b x-c x^2} \, dx+\frac{1}{2} \int e^{a+b x+c x^2} \, dx\\ &=\frac{1}{2} e^{a-\frac{b^2}{4 c}} \int e^{\frac{(b+2 c x)^2}{4 c}} \, dx+\frac{1}{2} e^{-a+\frac{b^2}{4 c}} \int e^{-\frac{(-b-2 c x)^2}{4 c}} \, dx\\ &=\frac{e^{-a+\frac{b^2}{4 c}} \sqrt{\pi } \text{erf}\left (\frac{b+2 c x}{2 \sqrt{c}}\right )}{4 \sqrt{c}}+\frac{e^{a-\frac{b^2}{4 c}} \sqrt{\pi } \text{erfi}\left (\frac{b+2 c x}{2 \sqrt{c}}\right )}{4 \sqrt{c}}\\ \end{align*}
Mathematica [A] time = 0.0629771, size = 105, normalized size = 1.15 \[ \frac{\sqrt{\pi } \left (\text{Erf}\left (\frac{b+2 c x}{2 \sqrt{c}}\right ) \left (\cosh \left (a-\frac{b^2}{4 c}\right )-\sinh \left (a-\frac{b^2}{4 c}\right )\right )+\text{Erfi}\left (\frac{b+2 c x}{2 \sqrt{c}}\right ) \left (\sinh \left (a-\frac{b^2}{4 c}\right )+\cosh \left (a-\frac{b^2}{4 c}\right )\right )\right )}{4 \sqrt{c}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.03, size = 83, normalized size = 0.9 \begin{align*}{\frac{\sqrt{\pi }}{4}{{\rm e}^{-{\frac{4\,ac-{b}^{2}}{4\,c}}}}{\it Erf} \left ( \sqrt{c}x+{\frac{b}{2}{\frac{1}{\sqrt{c}}}} \right ){\frac{1}{\sqrt{c}}}}-{\frac{\sqrt{\pi }}{4}{{\rm e}^{{\frac{4\,ac-{b}^{2}}{4\,c}}}}{\it Erf} \left ( -\sqrt{-c}x+{\frac{b}{2}{\frac{1}{\sqrt{-c}}}} \right ){\frac{1}{\sqrt{-c}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.51132, size = 626, normalized size = 6.88 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.77096, size = 302, normalized size = 3.32 \begin{align*} -\frac{\sqrt{\pi } \sqrt{-c}{\left (\cosh \left (-\frac{b^{2} - 4 \, a c}{4 \, c}\right ) + \sinh \left (-\frac{b^{2} - 4 \, a c}{4 \, c}\right )\right )} \operatorname{erf}\left (\frac{{\left (2 \, c x + b\right )} \sqrt{-c}}{2 \, c}\right ) - \sqrt{\pi } \sqrt{c}{\left (\cosh \left (-\frac{b^{2} - 4 \, a c}{4 \, c}\right ) - \sinh \left (-\frac{b^{2} - 4 \, a c}{4 \, c}\right )\right )} \operatorname{erf}\left (\frac{2 \, c x + b}{2 \, \sqrt{c}}\right )}{4 \, c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \cosh{\left (a + b x + c x^{2} \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.3526, size = 107, normalized size = 1.18 \begin{align*} -\frac{\sqrt{\pi } \operatorname{erf}\left (-\frac{1}{2} \, \sqrt{c}{\left (2 \, x + \frac{b}{c}\right )}\right ) e^{\left (\frac{b^{2} - 4 \, a c}{4 \, c}\right )}}{4 \, \sqrt{c}} - \frac{\sqrt{\pi } \operatorname{erf}\left (-\frac{1}{2} \, \sqrt{-c}{\left (2 \, x + \frac{b}{c}\right )}\right ) e^{\left (-\frac{b^{2} - 4 \, a c}{4 \, c}\right )}}{4 \, \sqrt{-c}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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