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 = 0.08, antiderivative size = 83, normalized size of antiderivative = 1.00, 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 2204
Rule 4473
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*}
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Mathematica [A] time = 0.20, 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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fricas [A] time = 0.90, size = 70, normalized size = 0.84 \[ \frac {\sqrt {\pi } {\left (i \, c - 1\right )} \sqrt {-i \, c - 1} \operatorname {erf}\left (\sqrt {-i \, c - 1} x\right ) e^{\left (i \, a\right )} + \sqrt {\pi } \sqrt {i \, c - 1} {\left (-i \, c - 1\right )} \operatorname {erf}\left (\sqrt {i \, c - 1} x\right ) e^{\left (-i \, a\right )}}{4 \, {\left (c^{2} + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \cos \left (c x^{2} + a\right ) e^{\left (x^{2}\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.10, size = 60, normalized size = 0.72 \[ \frac {\sqrt {\pi }\, {\mathrm e}^{-i a} \erf \left (\sqrt {i c -1}\, x \right )}{4 \sqrt {i c -1}}+\frac {\sqrt {\pi }\, {\mathrm e}^{i a} \erf \left (\sqrt {-i c -1}\, x \right )}{4 \sqrt {-i c -1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.32, size = 133, normalized size = 1.60 \[ -\frac {\sqrt {\pi } \sqrt {2 \, c^{2} + 2} {\left ({\left (i \, \cos \relax (a) + \sin \relax (a)\right )} \operatorname {erf}\left (\sqrt {i \, c - 1} x\right ) + {\left (-i \, \cos \relax (a) + \sin \relax (a)\right )} \operatorname {erf}\left (\sqrt {-i \, c - 1} x\right )\right )} \sqrt {\sqrt {c^{2} + 1} + 1} - \sqrt {\pi } \sqrt {2 \, c^{2} + 2} {\left ({\left (\cos \relax (a) - i \, \sin \relax (a)\right )} \operatorname {erf}\left (\sqrt {i \, c - 1} x\right ) + {\left (\cos \relax (a) + i \, \sin \relax (a)\right )} \operatorname {erf}\left (\sqrt {-i \, c - 1} x\right )\right )} \sqrt {\sqrt {c^{2} + 1} - 1}}{8 \, {\left (c^{2} + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\mathrm {e}}^{x^2}\,\cos \left (c\,x^2+a\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int e^{x^{2}} \cos {\left (a + c x^{2} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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