Optimal. Leaf size=83 \[ \frac{\sqrt{\pi } e^{-i a} \text{Erfi}\left (\sqrt{1-i c} x\right )}{4 \sqrt{1-i c}}+\frac{\sqrt{\pi } e^{i a} \text{Erfi}\left (\sqrt{1+i c} x\right )}{4 \sqrt{1+i c}} \]
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Rubi [A] time = 0.078642, antiderivative size = 83, 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 = {4473, 2204} \[ \frac{\sqrt{\pi } e^{-i a} \text{Erfi}\left (\sqrt{1-i c} x\right )}{4 \sqrt{1-i c}}+\frac{\sqrt{\pi } e^{i a} \text{Erfi}\left (\sqrt{1+i c} x\right )}{4 \sqrt{1+i c}} \]
Antiderivative was successfully verified.
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Rule 4473
Rule 2204
Rubi steps
\begin{align*} \int e^{x^2} \cos \left (a+c x^2\right ) \, dx &=\int \left (\frac{1}{2} e^{-i a+(1-i c) x^2}+\frac{1}{2} e^{i a+(1+i c) x^2}\right ) \, dx\\ &=\frac{1}{2} \int e^{-i a+(1-i c) x^2} \, dx+\frac{1}{2} \int e^{i a+(1+i c) x^2} \, dx\\ &=\frac{e^{-i a} \sqrt{\pi } \text{erfi}\left (\sqrt{1-i c} x\right )}{4 \sqrt{1-i c}}+\frac{e^{i a} \sqrt{\pi } \text{erfi}\left (\sqrt{1+i c} x\right )}{4 \sqrt{1+i c}}\\ \end{align*}
Mathematica [A] time = 0.193847, size = 107, normalized size = 1.29 \[ \frac{\sqrt [4]{-1} \sqrt{\pi } \left ((1-i c) \sqrt{c-i} (\cos (a)+i \sin (a)) \text{Erfi}\left (\sqrt [4]{-1} \sqrt{c-i} x\right )-(c-i) \sqrt{c+i} (\cos (a)-i \sin (a)) \text{Erfi}\left ((-1)^{3/4} \sqrt{c+i} x\right )\right )}{4 \left (c^2+1\right )} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.043, size = 60, normalized size = 0.7 \begin{align*}{\frac{\sqrt{\pi }{{\rm e}^{-ia}}}{4}{\it Erf} \left ( \sqrt{-1+ic}x \right ){\frac{1}{\sqrt{-1+ic}}}}+{\frac{\sqrt{\pi }{{\rm e}^{ia}}}{4}{\it Erf} \left ( \sqrt{-ic-1}x \right ){\frac{1}{\sqrt{-ic-1}}}} \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 = 0.479143, size = 198, normalized size = 2.39 \begin{align*} \frac{\sqrt{\pi }{\left (i \, c - 1\right )} \sqrt{-i \, c - 1} \operatorname{erf}\left (\sqrt{-i \, c - 1} x\right ) e^{\left (i \, a\right )} + \sqrt{\pi } \sqrt{i \, c - 1}{\left (-i \, c - 1\right )} \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}} \cos{\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 \cos \left (c x^{2} + a\right ) e^{\left (x^{2}\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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