Optimal. Leaf size=65 \[ \frac{1}{4} \sqrt{\pi } e^{a-\frac{b^2}{4}} \text{Erfi}\left (\frac{1}{2} (b+2 x)\right )-\frac{1}{4} \sqrt{\pi } e^{-a-\frac{b^2}{4}} \text{Erfi}\left (\frac{1}{2} (2 x-b)\right ) \]
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Rubi [A] time = 0.0649381, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {5512, 2234, 2204} \[ \frac{1}{4} \sqrt{\pi } e^{a-\frac{b^2}{4}} \text{Erfi}\left (\frac{1}{2} (b+2 x)\right )-\frac{1}{4} \sqrt{\pi } e^{-a-\frac{b^2}{4}} \text{Erfi}\left (\frac{1}{2} (2 x-b)\right ) \]
Antiderivative was successfully verified.
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Rule 5512
Rule 2234
Rule 2204
Rubi steps
\begin{align*} \int e^{x^2} \sinh (a+b x) \, dx &=\int \left (-\frac{1}{2} e^{-a-b x+x^2}+\frac{1}{2} e^{a+b x+x^2}\right ) \, dx\\ &=-\left (\frac{1}{2} \int e^{-a-b x+x^2} \, dx\right )+\frac{1}{2} \int e^{a+b x+x^2} \, dx\\ &=-\left (\frac{1}{2} e^{-a-\frac{b^2}{4}} \int e^{\frac{1}{4} (-b+2 x)^2} \, dx\right )+\frac{1}{2} e^{a-\frac{b^2}{4}} \int e^{\frac{1}{4} (b+2 x)^2} \, dx\\ &=-\frac{1}{4} e^{-a-\frac{b^2}{4}} \sqrt{\pi } \text{erfi}\left (\frac{1}{2} (-b+2 x)\right )+\frac{1}{4} e^{a-\frac{b^2}{4}} \sqrt{\pi } \text{erfi}\left (\frac{1}{2} (b+2 x)\right )\\ \end{align*}
Mathematica [A] time = 0.0662858, size = 51, normalized size = 0.78 \[ \frac{1}{4} \sqrt{\pi } e^{-\frac{b^2}{4}} \left ((\cosh (a)-\sinh (a)) \text{Erfi}\left (\frac{b}{2}-x\right )+(\sinh (a)+\cosh (a)) \text{Erfi}\left (\frac{b}{2}+x\right )\right ) \]
Antiderivative was successfully verified.
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Maple [C] time = 0.073, size = 52, normalized size = 0.8 \begin{align*} -{\frac{i}{4}}\sqrt{\pi }{{\rm e}^{-a-{\frac{{b}^{2}}{4}}}}{\it Erf} \left ( -ix+{\frac{i}{2}}b \right ) -{\frac{i}{4}}\sqrt{\pi }{{\rm e}^{a-{\frac{{b}^{2}}{4}}}}{\it Erf} \left ( ix+{\frac{i}{2}}b \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] time = 1.06116, size = 61, normalized size = 0.94 \begin{align*} -\frac{1}{4} i \, \sqrt{\pi } \operatorname{erf}\left (\frac{1}{2} i \, b + i \, x\right ) e^{\left (-\frac{1}{4} \, b^{2} + a\right )} + \frac{1}{4} i \, \sqrt{\pi } \operatorname{erf}\left (-\frac{1}{2} i \, b + i \, x\right ) e^{\left (-\frac{1}{4} \, b^{2} - a\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.88877, size = 130, normalized size = 2. \begin{align*} \frac{1}{4} \, \sqrt{\pi }{\left (\operatorname{erfi}\left (\frac{1}{2} \, b + x\right ) e^{\left (\frac{1}{4} \, b^{2} + a\right )} - \operatorname{erfi}\left (-\frac{1}{2} \, b + x\right ) e^{\left (\frac{1}{4} \, b^{2} - a\right )}\right )} e^{\left (-\frac{1}{2} \, b^{2}\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 \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [C] time = 1.13084, size = 61, normalized size = 0.94 \begin{align*} \frac{1}{4} i \, \sqrt{\pi } \operatorname{erf}\left (-\frac{1}{2} i \, b - i \, x\right ) e^{\left (-\frac{1}{4} \, b^{2} + a\right )} - \frac{1}{4} i \, \sqrt{\pi } \operatorname{erf}\left (\frac{1}{2} i \, b - i \, x\right ) e^{\left (-\frac{1}{4} \, b^{2} - a\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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