Optimal. Leaf size=115 \[ \frac{\sqrt{\pi } e^{-a-\frac{b^2}{4 (1-c)}} \text{Erfi}\left (\frac{b-2 (1-c) x}{2 \sqrt{1-c}}\right )}{4 \sqrt{1-c}}+\frac{\sqrt{\pi } e^{a-\frac{b^2}{4 (c+1)}} \text{Erfi}\left (\frac{b+2 (c+1) x}{2 \sqrt{c+1}}\right )}{4 \sqrt{c+1}} \]
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Rubi [A] time = 0.167993, antiderivative size = 115, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {5512, 2234, 2204} \[ \frac{\sqrt{\pi } e^{-a-\frac{b^2}{4 (1-c)}} \text{Erfi}\left (\frac{b-2 (1-c) x}{2 \sqrt{1-c}}\right )}{4 \sqrt{1-c}}+\frac{\sqrt{\pi } e^{a-\frac{b^2}{4 (c+1)}} \text{Erfi}\left (\frac{b+2 (c+1) x}{2 \sqrt{c+1}}\right )}{4 \sqrt{c+1}} \]
Antiderivative was successfully verified.
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Rule 5512
Rule 2234
Rule 2204
Rubi steps
\begin{align*} \int e^{x^2} \sinh \left (a+b x+c x^2\right ) \, dx &=\int \left (-\frac{1}{2} e^{-a-b x+(1-c) x^2}+\frac{1}{2} e^{a+b x+(1+c) x^2}\right ) \, dx\\ &=-\left (\frac{1}{2} \int e^{-a-b x+(1-c) x^2} \, dx\right )+\frac{1}{2} \int e^{a+b x+(1+c) x^2} \, dx\\ &=-\left (\frac{1}{2} e^{-a-\frac{b^2}{4 (1-c)}} \int e^{\frac{(-b+2 (1-c) x)^2}{4 (1-c)}} \, dx\right )+\frac{1}{2} e^{a-\frac{b^2}{4 (1+c)}} \int e^{\frac{(b+2 (1+c) x)^2}{4 (1+c)}} \, dx\\ &=\frac{e^{-a-\frac{b^2}{4 (1-c)}} \sqrt{\pi } \text{erfi}\left (\frac{b-2 (1-c) x}{2 \sqrt{1-c}}\right )}{4 \sqrt{1-c}}+\frac{e^{a-\frac{b^2}{4 (1+c)}} \sqrt{\pi } \text{erfi}\left (\frac{b+2 (1+c) x}{2 \sqrt{1+c}}\right )}{4 \sqrt{1+c}}\\ \end{align*}
Mathematica [A] time = 0.396314, size = 123, normalized size = 1.07 \[ \frac{\sqrt{\pi } e^{-\frac{b^2}{4 c+4}} \left ((c-1) \sqrt{c+1} (\sinh (a)+\cosh (a)) \text{Erfi}\left (\frac{b+2 (c+1) x}{2 \sqrt{c+1}}\right )-\sqrt{c-1} (c+1) e^{\frac{b^2 c}{2 \left (c^2-1\right )}} (\cosh (a)-\sinh (a)) \text{Erf}\left (\frac{b+2 (c-1) x}{2 \sqrt{c-1}}\right )\right )}{4 \left (c^2-1\right )} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.145, size = 105, normalized size = 0.9 \begin{align*} -{\frac{\sqrt{\pi }}{4}{{\rm e}^{-{\frac{4\,ac-{b}^{2}-4\,a}{4\,c-4}}}}{\it Erf} \left ( \sqrt{c-1}x+{\frac{b}{2}{\frac{1}{\sqrt{c-1}}}} \right ){\frac{1}{\sqrt{c-1}}}}-{\frac{\sqrt{\pi }}{4}{{\rm e}^{{\frac{4\,ac-{b}^{2}+4\,a}{4+4\,c}}}}{\it Erf} \left ( -\sqrt{-1-c}x+{\frac{b}{2}{\frac{1}{\sqrt{-1-c}}}} \right ){\frac{1}{\sqrt{-1-c}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.06368, size = 120, normalized size = 1.04 \begin{align*} \frac{\sqrt{\pi } \operatorname{erf}\left (\sqrt{-c - 1} x - \frac{b}{2 \, \sqrt{-c - 1}}\right ) e^{\left (a - \frac{b^{2}}{4 \,{\left (c + 1\right )}}\right )}}{4 \, \sqrt{-c - 1}} - \frac{\sqrt{\pi } \operatorname{erf}\left (\sqrt{c - 1} x + \frac{b}{2 \, \sqrt{c - 1}}\right ) e^{\left (-a + \frac{b^{2}}{4 \,{\left (c - 1\right )}}\right )}}{4 \, \sqrt{c - 1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.82426, size = 467, normalized size = 4.06 \begin{align*} -\frac{\sqrt{\pi }{\left ({\left (c + 1\right )} \cosh \left (-\frac{b^{2} - 4 \, a c + 4 \, a}{4 \,{\left (c - 1\right )}}\right ) -{\left (c + 1\right )} \sinh \left (-\frac{b^{2} - 4 \, a c + 4 \, a}{4 \,{\left (c - 1\right )}}\right )\right )} \sqrt{c - 1} \operatorname{erf}\left (\frac{2 \,{\left (c - 1\right )} x + b}{2 \, \sqrt{c - 1}}\right ) + \sqrt{\pi }{\left ({\left (c - 1\right )} \cosh \left (-\frac{b^{2} - 4 \, a c - 4 \, a}{4 \,{\left (c + 1\right )}}\right ) +{\left (c - 1\right )} \sinh \left (-\frac{b^{2} - 4 \, a c - 4 \, a}{4 \,{\left (c + 1\right )}}\right )\right )} \sqrt{-c - 1} \operatorname{erf}\left (\frac{{\left (2 \,{\left (c + 1\right )} x + b\right )} \sqrt{-c - 1}}{2 \,{\left (c + 1\right )}}\right )}{4 \,{\left (c^{2} - 1\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 e^{x^{2}} \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 = 1.24533, size = 136, normalized size = 1.18 \begin{align*} -\frac{\sqrt{\pi } \operatorname{erf}\left (-\frac{1}{2} \, \sqrt{-c - 1}{\left (2 \, x + \frac{b}{c + 1}\right )}\right ) e^{\left (-\frac{b^{2} - 4 \, a c - 4 \, a}{4 \,{\left (c + 1\right )}}\right )}}{4 \, \sqrt{-c - 1}} + \frac{\sqrt{\pi } \operatorname{erf}\left (-\frac{1}{2} \, \sqrt{c - 1}{\left (2 \, x + \frac{b}{c - 1}\right )}\right ) e^{\left (\frac{b^{2} - 4 \, a c + 4 \, a}{4 \,{\left (c - 1\right )}}\right )}}{4 \, \sqrt{c - 1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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