Optimal. Leaf size=81 \[ \frac {1}{4} i e^{-i a+\frac {b^2}{4}} \sqrt {\pi } \text {Erfi}\left (\frac {1}{2} (-i b+2 x)\right )-\frac {1}{4} i e^{i a+\frac {b^2}{4}} \sqrt {\pi } \text {Erfi}\left (\frac {1}{2} (i b+2 x)\right ) \]
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Rubi [A]
time = 0.05, 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 = {4560, 2266,
2235} \begin {gather*} \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 ) \end {gather*}
Antiderivative was successfully verified.
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Rule 2235
Rule 2266
Rule 4560
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 = 0.09, size = 81, normalized size = 1.00 \begin {gather*} \frac {1}{4} e^{\frac {b^2}{4}} \sqrt {\pi } \left (\cos (a) \text {Erf}\left (\frac {b}{2}-i x\right )+\cos (a) \text {Erf}\left (\frac {b}{2}+i x\right )+\left (\text {Erfi}\left (\frac {1}{2} (-i b+2 x)\right )+\text {Erfi}\left (\frac {1}{2} (i b+2 x)\right )\right ) \sin (a)\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 52, normalized size = 0.64
method | result | size |
risch | \(\frac {\sqrt {\pi }\, {\mathrm e}^{\frac {b^{2}}{4}} {\mathrm e}^{i a} \erf \left (-i x +\frac {b}{2}\right )}{4}+\frac {\sqrt {\pi }\, {\mathrm e}^{\frac {b^{2}}{4}} {\mathrm e}^{-i a} \erf \left (i x +\frac {b}{2}\right )}{4}\) | \(52\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 51, normalized size = 0.63 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.31, size = 45, normalized size = 0.56 \begin {gather*} -\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 {gather*}
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 \begin {gather*} \int e^{x^{2}} \sin {\left (a + b x \right )}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int {\mathrm {e}}^{x^2}\,\sin \left (a+b\,x\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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