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 = 0.07, antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, 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 2204
Rule 2234
Rule 4472
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*}
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Mathematica [A] time = 0.08, size = 81, normalized size = 1.00 \[ \frac {1}{4} \sqrt {\pi } e^{\frac {b^2}{4}} \left (\cos (a) \text {erf}\left (\frac {b}{2}-i x\right )+\cos (a) \text {erf}\left (\frac {b}{2}+i x\right )+\sin (a) \left (\text {erfi}\left (\frac {1}{2} (2 x-i b)\right )+\text {erfi}\left (\frac {1}{2} (2 x+i b)\right )\right )\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.64, size = 45, normalized size = 0.56 \[ -\frac {1}{4} \, \sqrt {\pi } {\left (\operatorname {erf}\left (-\frac {1}{2} \, b + i \, x\right ) e^{\left (\frac {1}{4} \, b^{2} + i \, a\right )} - \operatorname {erf}\left (\frac {1}{2} \, b + i \, x\right ) e^{\left (\frac {1}{4} \, b^{2} - i \, a\right )}\right )} \]
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 \[ \int e^{\left (x^{2}\right )} \sin \left (b x + a\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 52, normalized size = 0.64 \[ \frac {\sqrt {\pi }\, {\mathrm e}^{\frac {b^{2}}{4}} {\mathrm e}^{i a} \erf \left (-i x +\frac {b}{2}\right )}{4}+\frac {\sqrt {\pi }\, {\mathrm e}^{\frac {b^{2}}{4}} {\mathrm e}^{-i a} \erf \left (i x +\frac {b}{2}\right )}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 51, normalized size = 0.63 \[ \frac {1}{4} \, \sqrt {\pi } {\left ({\left (\cos \relax (a) - i \, \sin \relax (a)\right )} \operatorname {erf}\left (\frac {1}{2} \, b + i \, x\right ) e^{\left (\frac {1}{4} \, b^{2}\right )} - {\left (\cos \relax (a) + i \, \sin \relax (a)\right )} \operatorname {erf}\left (-\frac {1}{2} \, b + i \, x\right ) e^{\left (\frac {1}{4} \, b^{2}\right )}\right )} \]
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 \[ \int {\mathrm {e}}^{x^2}\,\sin \left (a+b\,x\right ) \,d x \]
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 \[ \int e^{x^{2}} \sin {\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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