Integrand size = 10, antiderivative size = 69 \[ \int e^{x^2} \sin (b x) \, dx=\frac {1}{4} i e^{\frac {b^2}{4}} \sqrt {\pi } \text {erfi}\left (\frac {1}{2} (-i b+2 x)\right )-\frac {1}{4} i e^{\frac {b^2}{4}} \sqrt {\pi } \text {erfi}\left (\frac {1}{2} (i b+2 x)\right ) \]
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Time = 0.07 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {4560, 2266, 2235} \[ \int e^{x^2} \sin (b x) \, dx=\frac {1}{4} i \sqrt {\pi } e^{\frac {b^2}{4}} \text {erfi}\left (\frac {1}{2} (2 x-i b)\right )-\frac {1}{4} i \sqrt {\pi } e^{\frac {b^2}{4}} \text {erfi}\left (\frac {1}{2} (2 x+i b)\right ) \]
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Rule 2235
Rule 2266
Rule 4560
Rubi steps \begin{align*} \text {integral}& = \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*}
Time = 0.03 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.62 \[ \int e^{x^2} \sin (b x) \, dx=\frac {1}{4} e^{\frac {b^2}{4}} \sqrt {\pi } \left (\text {erf}\left (\frac {b}{2}-i x\right )+\text {erf}\left (\frac {b}{2}+i x\right )\right ) \]
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Time = 0.20 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.61
method | result | size |
risch | \(\frac {\sqrt {\pi }\, {\mathrm e}^{\frac {b^{2}}{4}} \operatorname {erf}\left (-i x +\frac {b}{2}\right )}{4}+\frac {\sqrt {\pi }\, {\mathrm e}^{\frac {b^{2}}{4}} \operatorname {erf}\left (i x +\frac {b}{2}\right )}{4}\) | \(42\) |
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Time = 0.25 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.43 \[ \int e^{x^2} \sin (b x) \, dx=\frac {1}{4} \, \sqrt {\pi } {\left (\operatorname {erf}\left (\frac {1}{2} \, b + i \, x\right ) - \operatorname {erf}\left (-\frac {1}{2} \, b + i \, x\right )\right )} e^{\left (\frac {1}{4} \, b^{2}\right )} \]
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\[ \int e^{x^2} \sin (b x) \, dx=\int e^{x^{2}} \sin {\left (b x \right )}\, dx \]
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Time = 0.18 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.54 \[ \int e^{x^2} \sin (b x) \, dx=\frac {1}{4} \, \sqrt {\pi } {\left (\operatorname {erf}\left (\frac {1}{2} \, b + i \, x\right ) e^{\left (\frac {1}{4} \, b^{2}\right )} - \operatorname {erf}\left (-\frac {1}{2} \, b + i \, x\right ) e^{\left (\frac {1}{4} \, b^{2}\right )}\right )} \]
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\[ \int e^{x^2} \sin (b x) \, dx=\int { e^{\left (x^{2}\right )} \sin \left (b x\right ) \,d x } \]
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Timed out. \[ \int e^{x^2} \sin (b x) \, dx=\int {\mathrm {e}}^{x^2}\,\sin \left (b\,x\right ) \,d x \]
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