Optimal. Leaf size=111 \[ \frac{\sqrt{\pi } b e^{\frac{b^2}{4 c}-a} \text{Erf}\left (\frac{b+2 c x}{2 \sqrt{c}}\right )}{8 c^{3/2}}-\frac{\sqrt{\pi } b e^{a-\frac{b^2}{4 c}} \text{Erfi}\left (\frac{b+2 c x}{2 \sqrt{c}}\right )}{8 c^{3/2}}+\frac{\cosh \left (a+b x+c x^2\right )}{2 c} \]
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Rubi [A] time = 0.0462197, antiderivative size = 111, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.385, Rules used = {5382, 5374, 2234, 2204, 2205} \[ \frac{\sqrt{\pi } b e^{\frac{b^2}{4 c}-a} \text{Erf}\left (\frac{b+2 c x}{2 \sqrt{c}}\right )}{8 c^{3/2}}-\frac{\sqrt{\pi } b e^{a-\frac{b^2}{4 c}} \text{Erfi}\left (\frac{b+2 c x}{2 \sqrt{c}}\right )}{8 c^{3/2}}+\frac{\cosh \left (a+b x+c x^2\right )}{2 c} \]
Antiderivative was successfully verified.
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Rule 5382
Rule 5374
Rule 2234
Rule 2204
Rule 2205
Rubi steps
\begin{align*} \int x \sinh \left (a+b x+c x^2\right ) \, dx &=\frac{\cosh \left (a+b x+c x^2\right )}{2 c}-\frac{b \int \sinh \left (a+b x+c x^2\right ) \, dx}{2 c}\\ &=\frac{\cosh \left (a+b x+c x^2\right )}{2 c}+\frac{b \int e^{-a-b x-c x^2} \, dx}{4 c}-\frac{b \int e^{a+b x+c x^2} \, dx}{4 c}\\ &=\frac{\cosh \left (a+b x+c x^2\right )}{2 c}-\frac{\left (b e^{a-\frac{b^2}{4 c}}\right ) \int e^{\frac{(b+2 c x)^2}{4 c}} \, dx}{4 c}+\frac{\left (b e^{-a+\frac{b^2}{4 c}}\right ) \int e^{-\frac{(-b-2 c x)^2}{4 c}} \, dx}{4 c}\\ &=\frac{\cosh \left (a+b x+c x^2\right )}{2 c}+\frac{b e^{-a+\frac{b^2}{4 c}} \sqrt{\pi } \text{erf}\left (\frac{b+2 c x}{2 \sqrt{c}}\right )}{8 c^{3/2}}-\frac{b e^{a-\frac{b^2}{4 c}} \sqrt{\pi } \text{erfi}\left (\frac{b+2 c x}{2 \sqrt{c}}\right )}{8 c^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.194711, size = 130, normalized size = 1.17 \[ \frac{\sqrt{\pi } b \text{Erf}\left (\frac{b+2 c x}{2 \sqrt{c}}\right ) \left (\cosh \left (a-\frac{b^2}{4 c}\right )-\sinh \left (a-\frac{b^2}{4 c}\right )\right )-\sqrt{\pi } b \text{Erfi}\left (\frac{b+2 c x}{2 \sqrt{c}}\right ) \left (\sinh \left (a-\frac{b^2}{4 c}\right )+\cosh \left (a-\frac{b^2}{4 c}\right )\right )+4 \sqrt{c} \cosh (a+x (b+c x))}{8 c^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.031, size = 124, normalized size = 1.1 \begin{align*}{\frac{{{\rm e}^{-c{x}^{2}-bx-a}}}{4\,c}}+{\frac{b\sqrt{\pi }}{8}{{\rm e}^{-{\frac{4\,ac-{b}^{2}}{4\,c}}}}{\it Erf} \left ( \sqrt{c}x+{\frac{b}{2}{\frac{1}{\sqrt{c}}}} \right ){c}^{-{\frac{3}{2}}}}+{\frac{{{\rm e}^{c{x}^{2}+bx+a}}}{4\,c}}+{\frac{b\sqrt{\pi }}{8\,c}{{\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 = 1.59159, size = 825, normalized size = 7.43 \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 [B] time = 2.04929, size = 892, normalized size = 8.04 \begin{align*} \frac{2 \, c \cosh \left (c x^{2} + b x + a\right )^{2} + \sqrt{\pi }{\left (b \cosh \left (c x^{2} + b x + a\right ) \cosh \left (-\frac{b^{2} - 4 \, a c}{4 \, c}\right ) + b \cosh \left (c x^{2} + b x + a\right ) \sinh \left (-\frac{b^{2} - 4 \, a c}{4 \, c}\right ) +{\left (b \cosh \left (-\frac{b^{2} - 4 \, a c}{4 \, c}\right ) + b \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 (b \cosh \left (c x^{2} + b x + a\right ) \cosh \left (-\frac{b^{2} - 4 \, a c}{4 \, c}\right ) - b \cosh \left (c x^{2} + b x + a\right ) \sinh \left (-\frac{b^{2} - 4 \, a c}{4 \, c}\right ) +{\left (b \cosh \left (-\frac{b^{2} - 4 \, a c}{4 \, c}\right ) - b \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 ) + 4 \, c \cosh \left (c x^{2} + b x + a\right ) \sinh \left (c x^{2} + b x + a\right ) + 2 \, c \sinh \left (c x^{2} + b x + a\right )^{2} + 2 \, c}{8 \,{\left (c^{2} \cosh \left (c x^{2} + b x + a\right ) + c^{2} \sinh \left (c x^{2} + b x + a\right )\right )}} \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 \sinh{\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 = 1.36486, size = 163, normalized size = 1.47 \begin{align*} -\frac{\frac{\sqrt{\pi } b \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 )}}{\sqrt{c}} - 2 \, e^{\left (-c x^{2} - b x - a\right )}}{8 \, c} + \frac{\frac{\sqrt{\pi } b \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 )}}{\sqrt{-c}} + 2 \, e^{\left (c x^{2} + b x + a\right )}}{8 \, c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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