Optimal. Leaf size=65 \[ \frac{\sqrt{\pi } e^a \text{Erfi}\left (\sqrt{c+1} x\right )}{4 \sqrt{c+1}}-\frac{\sqrt{\pi } e^{-a} \text{Erfi}\left (\sqrt{1-c} x\right )}{4 \sqrt{1-c}} \]
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Rubi [A] time = 0.0832766, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {5512, 2204} \[ \frac{\sqrt{\pi } e^a \text{Erfi}\left (\sqrt{c+1} x\right )}{4 \sqrt{c+1}}-\frac{\sqrt{\pi } e^{-a} \text{Erfi}\left (\sqrt{1-c} x\right )}{4 \sqrt{1-c}} \]
Antiderivative was successfully verified.
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Rule 5512
Rule 2204
Rubi steps
\begin{align*} \int e^{x^2} \sinh \left (a+c x^2\right ) \, dx &=\int \left (-\frac{1}{2} e^{-a+(1-c) x^2}+\frac{1}{2} e^{a+(1+c) x^2}\right ) \, dx\\ &=-\left (\frac{1}{2} \int e^{-a+(1-c) x^2} \, dx\right )+\frac{1}{2} \int e^{a+(1+c) x^2} \, dx\\ &=-\frac{e^{-a} \sqrt{\pi } \text{erfi}\left (\sqrt{1-c} x\right )}{4 \sqrt{1-c}}+\frac{e^a \sqrt{\pi } \text{erfi}\left (\sqrt{1+c} x\right )}{4 \sqrt{1+c}}\\ \end{align*}
Mathematica [A] time = 0.107318, size = 72, normalized size = 1.11 \[ \frac{\sqrt{\pi } \left ((c-1) \sqrt{c+1} (\sinh (a)+\cosh (a)) \text{Erfi}\left (\sqrt{c+1} x\right )-\sqrt{c-1} (c+1) (\cosh (a)-\sinh (a)) \text{Erf}\left (\sqrt{c-1} x\right )\right )}{4 \left (c^2-1\right )} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.073, size = 48, normalized size = 0.7 \begin{align*} -{\frac{\sqrt{\pi }{{\rm e}^{-a}}}{4}{\it Erf} \left ( \sqrt{c-1}x \right ){\frac{1}{\sqrt{c-1}}}}+{\frac{\sqrt{\pi }{{\rm e}^{a}}}{4}{\it Erf} \left ( \sqrt{-1-c}x \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 = 1.08772, size = 63, normalized size = 0.97 \begin{align*} -\frac{\sqrt{\pi } \operatorname{erf}\left (\sqrt{c - 1} x\right ) e^{\left (-a\right )}}{4 \, \sqrt{c - 1}} + \frac{\sqrt{\pi } \operatorname{erf}\left (\sqrt{-c - 1} x\right ) e^{a}}{4 \, \sqrt{-c - 1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.86845, size = 235, normalized size = 3.62 \begin{align*} -\frac{\sqrt{\pi }{\left ({\left (c + 1\right )} \cosh \left (a\right ) -{\left (c + 1\right )} \sinh \left (a\right )\right )} \sqrt{c - 1} \operatorname{erf}\left (\sqrt{c - 1} x\right ) + \sqrt{\pi }{\left ({\left (c - 1\right )} \cosh \left (a\right ) +{\left (c - 1\right )} \sinh \left (a\right )\right )} \sqrt{-c - 1} \operatorname{erf}\left (\sqrt{-c - 1} x\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 + c x^{2} \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14358, size = 66, normalized size = 1.02 \begin{align*} \frac{\sqrt{\pi } \operatorname{erf}\left (-\sqrt{c - 1} x\right ) e^{\left (-a\right )}}{4 \, \sqrt{c - 1}} - \frac{\sqrt{\pi } \operatorname{erf}\left (-\sqrt{-c - 1} x\right ) e^{a}}{4 \, \sqrt{-c - 1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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