Optimal. Leaf size=87 \[ \frac{i \sqrt{\pi } e^{-i a} \text{Erfi}\left (\sqrt{1-i c} x\right )}{4 \sqrt{1-i c}}-\frac{i \sqrt{\pi } e^{i a} \text{Erfi}\left (\sqrt{1+i c} x\right )}{4 \sqrt{1+i c}} \]
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Rubi [A] time = 0.0984525, antiderivative size = 87, 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 = {4472, 2204} \[ \frac{i \sqrt{\pi } e^{-i a} \text{Erfi}\left (\sqrt{1-i c} x\right )}{4 \sqrt{1-i c}}-\frac{i \sqrt{\pi } e^{i a} \text{Erfi}\left (\sqrt{1+i c} x\right )}{4 \sqrt{1+i c}} \]
Antiderivative was successfully verified.
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Rule 4472
Rule 2204
Rubi steps
\begin{align*} \int e^{x^2} \sin \left (a+c x^2\right ) \, dx &=\int \left (\frac{1}{2} i e^{-i a+(1-i c) x^2}-\frac{1}{2} i e^{i a+(1+i c) x^2}\right ) \, dx\\ &=\frac{1}{2} i \int e^{-i a+(1-i c) x^2} \, dx-\frac{1}{2} i \int e^{i a+(1+i c) x^2} \, dx\\ &=\frac{i e^{-i a} \sqrt{\pi } \text{erfi}\left (\sqrt{1-i c} x\right )}{4 \sqrt{1-i c}}-\frac{i e^{i a} \sqrt{\pi } \text{erfi}\left (\sqrt{1+i c} x\right )}{4 \sqrt{1+i c}}\\ \end{align*}
Mathematica [A] time = 0.218311, size = 129, normalized size = 1.48 \[ -\frac{\sqrt [4]{-1} \sqrt{\pi } \left (\sqrt{c+i} \left (\sin (a) \text{Erf}\left (\frac{(1+i) \sqrt{c+i} x}{\sqrt{2}}\right )+\text{Erfi}\left ((-1)^{3/4} \sqrt{c+i} x\right ) (c \sin (a)+i c \cos (a)+\cos (a))\right )+\sqrt{c-i} (c+i) (\cos (a)+i \sin (a)) \text{Erfi}\left (\sqrt [4]{-1} \sqrt{c-i} x\right )\right )}{4 \left (c^2+1\right )} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.063, size = 62, normalized size = 0.7 \begin{align*}{-{\frac{i}{4}}\sqrt{\pi }{{\rm e}^{ia}}{\it Erf} \left ( \sqrt{-ic-1}x \right ){\frac{1}{\sqrt{-ic-1}}}}+{{\frac{i}{4}}\sqrt{\pi }{{\rm e}^{-ia}}{\it Erf} \left ( \sqrt{-1+ic}x \right ){\frac{1}{\sqrt{-1+ic}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: IndexError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.482512, size = 192, normalized size = 2.21 \begin{align*} \frac{\sqrt{\pi }{\left (c + i\right )} \sqrt{-i \, c - 1} \operatorname{erf}\left (\sqrt{-i \, c - 1} x\right ) e^{\left (i \, a\right )} + \sqrt{\pi }{\left (c - i\right )} \sqrt{i \, c - 1} \operatorname{erf}\left (\sqrt{i \, c - 1} x\right ) e^{\left (-i \, a\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}} \sin{\left (a + c x^{2} \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 (c x^{2} + a\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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