Optimal. Leaf size=136 \[ -\frac{\sqrt{\frac{\pi }{2}} b e^{\frac{b^2}{2 c}-2 a} \text{Erf}\left (\frac{b+2 c x}{\sqrt{2} \sqrt{c}}\right )}{16 c^{3/2}}-\frac{\sqrt{\frac{\pi }{2}} b e^{2 a-\frac{b^2}{2 c}} \text{Erfi}\left (\frac{b+2 c x}{\sqrt{2} \sqrt{c}}\right )}{16 c^{3/2}}+\frac{\sinh \left (2 a+2 b x+2 c x^2\right )}{8 c}+\frac{x^2}{4} \]
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Rubi [A] time = 0.0925022, antiderivative size = 136, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {5395, 5383, 5375, 2234, 2204, 2205} \[ -\frac{\sqrt{\frac{\pi }{2}} b e^{\frac{b^2}{2 c}-2 a} \text{Erf}\left (\frac{b+2 c x}{\sqrt{2} \sqrt{c}}\right )}{16 c^{3/2}}-\frac{\sqrt{\frac{\pi }{2}} b e^{2 a-\frac{b^2}{2 c}} \text{Erfi}\left (\frac{b+2 c x}{\sqrt{2} \sqrt{c}}\right )}{16 c^{3/2}}+\frac{\sinh \left (2 a+2 b x+2 c x^2\right )}{8 c}+\frac{x^2}{4} \]
Antiderivative was successfully verified.
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Rule 5395
Rule 5383
Rule 5375
Rule 2234
Rule 2204
Rule 2205
Rubi steps
\begin{align*} \int x \cosh ^2\left (a+b x+c x^2\right ) \, dx &=\int \left (\frac{x}{2}+\frac{1}{2} x \cosh \left (2 a+2 b x+2 c x^2\right )\right ) \, dx\\ &=\frac{x^2}{4}+\frac{1}{2} \int x \cosh \left (2 a+2 b x+2 c x^2\right ) \, dx\\ &=\frac{x^2}{4}+\frac{\sinh \left (2 a+2 b x+2 c x^2\right )}{8 c}-\frac{b \int \cosh \left (2 a+2 b x+2 c x^2\right ) \, dx}{4 c}\\ &=\frac{x^2}{4}+\frac{\sinh \left (2 a+2 b x+2 c x^2\right )}{8 c}-\frac{b \int e^{-2 a-2 b x-2 c x^2} \, dx}{8 c}-\frac{b \int e^{2 a+2 b x+2 c x^2} \, dx}{8 c}\\ &=\frac{x^2}{4}+\frac{\sinh \left (2 a+2 b x+2 c x^2\right )}{8 c}-\frac{\left (b e^{2 a-\frac{b^2}{2 c}}\right ) \int e^{\frac{(2 b+4 c x)^2}{8 c}} \, dx}{8 c}-\frac{\left (b e^{-2 a+\frac{b^2}{2 c}}\right ) \int e^{-\frac{(-2 b-4 c x)^2}{8 c}} \, dx}{8 c}\\ &=\frac{x^2}{4}-\frac{b e^{-2 a+\frac{b^2}{2 c}} \sqrt{\frac{\pi }{2}} \text{erf}\left (\frac{b+2 c x}{\sqrt{2} \sqrt{c}}\right )}{16 c^{3/2}}-\frac{b e^{2 a-\frac{b^2}{2 c}} \sqrt{\frac{\pi }{2}} \text{erfi}\left (\frac{b+2 c x}{\sqrt{2} \sqrt{c}}\right )}{16 c^{3/2}}+\frac{\sinh \left (2 a+2 b x+2 c x^2\right )}{8 c}\\ \end{align*}
Mathematica [A] time = 0.393129, size = 155, normalized size = 1.14 \[ \frac{\sqrt{2 \pi } b \text{Erf}\left (\frac{b+2 c x}{\sqrt{2} \sqrt{c}}\right ) \left (\sinh \left (2 a-\frac{b^2}{2 c}\right )-\cosh \left (2 a-\frac{b^2}{2 c}\right )\right )-\sqrt{2 \pi } b \text{Erfi}\left (\frac{b+2 c x}{\sqrt{2} \sqrt{c}}\right ) \left (\sinh \left (2 a-\frac{b^2}{2 c}\right )+\cosh \left (2 a-\frac{b^2}{2 c}\right )\right )+4 \sqrt{c} \left (\sinh (2 (a+x (b+c x)))+2 c x^2\right )}{32 c^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.049, size = 141, normalized size = 1. \begin{align*}{\frac{{x}^{2}}{4}}-{\frac{{{\rm e}^{-2\,c{x}^{2}-2\,bx-2\,a}}}{16\,c}}-{\frac{b\sqrt{\pi }\sqrt{2}}{32}{{\rm e}^{-{\frac{4\,ac-{b}^{2}}{2\,c}}}}{\it Erf} \left ( \sqrt{2}\sqrt{c}x+{\frac{b\sqrt{2}}{2}{\frac{1}{\sqrt{c}}}} \right ){c}^{-{\frac{3}{2}}}}+{\frac{{{\rm e}^{2\,c{x}^{2}+2\,bx+2\,a}}}{16\,c}}+{\frac{b\sqrt{\pi }}{16\,c}{{\rm e}^{{\frac{4\,ac-{b}^{2}}{2\,c}}}}{\it Erf} \left ( -\sqrt{-2\,c}x+{b{\frac{1}{\sqrt{-2\,c}}}} \right ){\frac{1}{\sqrt{-2\,c}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.26752, size = 270, normalized size = 1.99 \begin{align*} \frac{1}{4} \, x^{2} - \frac{\sqrt{2}{\left (\frac{\sqrt{\pi }{\left (2 \, c x + b\right )} b{\left (\operatorname{erf}\left (\sqrt{\frac{1}{2}} \sqrt{-\frac{{\left (2 \, c x + b\right )}^{2}}{c}}\right ) - 1\right )}}{\sqrt{-\frac{{\left (2 \, c x + b\right )}^{2}}{c}} c^{\frac{3}{2}}} - \frac{\sqrt{2} e^{\left (\frac{{\left (2 \, c x + b\right )}^{2}}{2 \, c}\right )}}{\sqrt{c}}\right )} e^{\left (2 \, a - \frac{b^{2}}{2 \, c}\right )}}{32 \, \sqrt{c}} - \frac{\sqrt{2}{\left (\frac{\sqrt{\pi }{\left (2 \, c x + b\right )} b{\left (\operatorname{erf}\left (\sqrt{\frac{1}{2}} \sqrt{\frac{{\left (2 \, c x + b\right )}^{2}}{c}}\right ) - 1\right )}}{\sqrt{\frac{{\left (2 \, c x + b\right )}^{2}}{c}} \left (-c\right )^{\frac{3}{2}}} + \frac{\sqrt{2} c e^{\left (-\frac{{\left (2 \, c x + b\right )}^{2}}{2 \, c}\right )}}{\left (-c\right )^{\frac{3}{2}}}\right )} e^{\left (-2 \, a + \frac{b^{2}}{2 \, c}\right )}}{32 \, \sqrt{-c}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.10261, size = 1650, normalized size = 12.13 \begin{align*} \text{result too large to display} \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 x \cosh ^{2}{\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 = 1.27052, size = 192, normalized size = 1.41 \begin{align*} \frac{1}{4} \, x^{2} + \frac{\frac{\sqrt{2} \sqrt{\pi } b \operatorname{erf}\left (-\frac{1}{2} \, \sqrt{2} \sqrt{c}{\left (2 \, x + \frac{b}{c}\right )}\right ) e^{\left (\frac{b^{2} - 4 \, a c}{2 \, c}\right )}}{\sqrt{c}} - 2 \, e^{\left (-2 \, c x^{2} - 2 \, b x - 2 \, a\right )}}{32 \, c} + \frac{\frac{\sqrt{2} \sqrt{\pi } b \operatorname{erf}\left (-\frac{1}{2} \, \sqrt{2} \sqrt{-c}{\left (2 \, x + \frac{b}{c}\right )}\right ) e^{\left (-\frac{b^{2} - 4 \, a c}{2 \, c}\right )}}{\sqrt{-c}} + 2 \, e^{\left (2 \, c x^{2} + 2 \, b x + 2 \, a\right )}}{32 \, c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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