Optimal. Leaf size=81 \[ \frac{1}{4} i \sqrt{\pi } e^{\frac{b^2}{4}-i a} \text{Erfi}\left (\frac{1}{2} (2 x-i b)\right )-\frac{1}{4} i \sqrt{\pi } e^{\frac{b^2}{4}+i a} \text{Erfi}\left (\frac{1}{2} (2 x+i b)\right ) \]
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Rubi [A] time = 0.0549439, antiderivative size = 81, 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 = {4472, 2234, 2204} \[ \frac{1}{4} i \sqrt{\pi } e^{\frac{b^2}{4}-i a} \text{Erfi}\left (\frac{1}{2} (2 x-i b)\right )-\frac{1}{4} i \sqrt{\pi } e^{\frac{b^2}{4}+i a} \text{Erfi}\left (\frac{1}{2} (2 x+i b)\right ) \]
Antiderivative was successfully verified.
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Rule 4472
Rule 2234
Rule 2204
Rubi steps
\begin{align*} \int e^{x^2} \sin (a+b x) \, dx &=\int \left (\frac{1}{2} i e^{-i a-i b x+x^2}-\frac{1}{2} i e^{i a+i b x+x^2}\right ) \, dx\\ &=\frac{1}{2} i \int e^{-i a-i b x+x^2} \, dx-\frac{1}{2} i \int e^{i a+i b x+x^2} \, dx\\ &=\frac{1}{2} \left (i e^{-i a+\frac{b^2}{4}}\right ) \int e^{\frac{1}{4} (-i b+2 x)^2} \, dx-\frac{1}{2} \left (i e^{i a+\frac{b^2}{4}}\right ) \int e^{\frac{1}{4} (i b+2 x)^2} \, dx\\ &=\frac{1}{4} i e^{-i a+\frac{b^2}{4}} \sqrt{\pi } \text{erfi}\left (\frac{1}{2} (-i b+2 x)\right )-\frac{1}{4} i e^{i a+\frac{b^2}{4}} \sqrt{\pi } \text{erfi}\left (\frac{1}{2} (i b+2 x)\right )\\ \end{align*}
Mathematica [A] time = 0.0226392, size = 81, normalized size = 1. \[ \frac{1}{4} \sqrt{\pi } e^{\frac{b^2}{4}} \left (\cos (a) \text{Erf}\left (\frac{b}{2}-i x\right )+\cos (a) \text{Erf}\left (\frac{b}{2}+i x\right )+\sin (a) \left (\text{Erfi}\left (\frac{1}{2} (2 x-i b)\right )+\text{Erfi}\left (\frac{1}{2} (2 x+i b)\right )\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0., size = 52, normalized size = 0.6 \begin{align*}{\frac{\sqrt{\pi }{{\rm e}^{ia}}}{4}{{\rm e}^{{\frac{{b}^{2}}{4}}}}{\it Erf} \left ( -ix+{\frac{b}{2}} \right ) }+{\frac{\sqrt{\pi }{{\rm e}^{-ia}}}{4}{{\rm e}^{{\frac{{b}^{2}}{4}}}}{\it Erf} \left ( ix+{\frac{b}{2}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.0317, size = 69, normalized size = 0.85 \begin{align*} \frac{1}{4} \, \sqrt{\pi }{\left ({\left (\cos \left (a\right ) - i \, \sin \left (a\right )\right )} \operatorname{erf}\left (\frac{1}{2} \, b + i \, x\right ) e^{\left (\frac{1}{4} \, b^{2}\right )} -{\left (\cos \left (a\right ) + i \, \sin \left (a\right )\right )} \operatorname{erf}\left (-\frac{1}{2} \, b + i \, x\right ) e^{\left (\frac{1}{4} \, b^{2}\right )}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.470909, size = 122, normalized size = 1.51 \begin{align*} -\frac{1}{4} \, \sqrt{\pi }{\left (\operatorname{erf}\left (-\frac{1}{2} \, b + i \, x\right ) e^{\left (\frac{1}{4} \, b^{2} + i \, a\right )} - \operatorname{erf}\left (\frac{1}{2} \, b + i \, x\right ) e^{\left (\frac{1}{4} \, b^{2} - i \, 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 e^{x^{2}} \sin{\left (a + b x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int e^{\left (x^{2}\right )} \sin \left (b x + a\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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