Optimal. Leaf size=69 \[ \frac {1}{4} i e^{\frac {b^2}{4}} \sqrt {\pi } \text {Erfi}\left (\frac {1}{2} (-i b+2 x)\right )-\frac {1}{4} i e^{\frac {b^2}{4}} \sqrt {\pi } \text {Erfi}\left (\frac {1}{2} (i b+2 x)\right ) \]
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Rubi [A]
time = 0.04, antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {4560, 2266,
2235} \begin {gather*} \frac {1}{4} i \sqrt {\pi } e^{\frac {b^2}{4}} \text {Erfi}\left (\frac {1}{2} (2 x-i b)\right )-\frac {1}{4} i \sqrt {\pi } e^{\frac {b^2}{4}} \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 (b x) \, dx &=\int \left (\frac {1}{2} i e^{-i b x+x^2}-\frac {1}{2} i e^{i b x+x^2}\right ) \, dx\\ &=\frac {1}{2} i \int e^{-i b x+x^2} \, dx-\frac {1}{2} i \int e^{i b x+x^2} \, dx\\ &=\frac {1}{2} \left (i e^{\frac {b^2}{4}}\right ) \int e^{\frac {1}{4} (-i b+2 x)^2} \, dx-\frac {1}{2} \left (i e^{\frac {b^2}{4}}\right ) \int e^{\frac {1}{4} (i b+2 x)^2} \, dx\\ &=\frac {1}{4} i e^{\frac {b^2}{4}} \sqrt {\pi } \text {erfi}\left (\frac {1}{2} (-i b+2 x)\right )-\frac {1}{4} i e^{\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 = 0.03, size = 43, normalized size = 0.62 \begin {gather*} \frac {1}{4} e^{\frac {b^2}{4}} \sqrt {\pi } \left (\text {Erf}\left (\frac {b}{2}-i x\right )+\text {Erf}\left (\frac {b}{2}+i x\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.12, size = 42, normalized size = 0.61
method | result | size |
risch | \(\frac {\sqrt {\pi }\, {\mathrm e}^{\frac {b^{2}}{4}} \erf \left (-i x +\frac {b}{2}\right )}{4}+\frac {\sqrt {\pi }\, {\mathrm e}^{\frac {b^{2}}{4}} \erf \left (i x +\frac {b}{2}\right )}{4}\) | \(42\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 37, normalized size = 0.54 \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}\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 = 1.86, size = 30, normalized size = 0.43 \begin {gather*} \frac {1}{4} \, \sqrt {\pi } {\left (\operatorname {erf}\left (\frac {1}{2} \, b + i \, x\right ) - \operatorname {erf}\left (-\frac {1}{2} \, b + i \, x\right )\right )} e^{\left (\frac {1}{4} \, b^{2}\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 (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 (b\,x\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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