Integrand size = 25, antiderivative size = 23 \[ \int \left (\frac {e^{(a+x)^2}}{x^2}-\frac {2 a e^{(a+x)^2}}{x}\right ) \, dx=-\frac {e^{(a+x)^2}}{x}+\sqrt {\pi } \text {erfi}(a+x) \]
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Time = 0.03 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {2252, 2235} \[ \int \left (\frac {e^{(a+x)^2}}{x^2}-\frac {2 a e^{(a+x)^2}}{x}\right ) \, dx=\sqrt {\pi } \text {erfi}(a+x)-\frac {e^{(a+x)^2}}{x} \]
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Rule 2235
Rule 2252
Rubi steps \begin{align*} \text {integral}& = -\left ((2 a) \int \frac {e^{(a+x)^2}}{x} \, dx\right )+\int \frac {e^{(a+x)^2}}{x^2} \, dx \\ & = -\frac {e^{(a+x)^2}}{x}+2 \int e^{(a+x)^2} \, dx \\ & = -\frac {e^{(a+x)^2}}{x}+\sqrt {\pi } \text {erfi}(a+x) \\ \end{align*}
Time = 0.21 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int \left (\frac {e^{(a+x)^2}}{x^2}-\frac {2 a e^{(a+x)^2}}{x}\right ) \, dx=-\frac {e^{(a+x)^2}}{x}+\sqrt {\pi } \text {erfi}(a+x) \]
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\[\int \left (\frac {{\mathrm e}^{\left (a +x \right )^{2}}}{x^{2}}-\frac {2 a \,{\mathrm e}^{\left (a +x \right )^{2}}}{x}\right )d x\]
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none
Time = 0.29 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.22 \[ \int \left (\frac {e^{(a+x)^2}}{x^2}-\frac {2 a e^{(a+x)^2}}{x}\right ) \, dx=\frac {\sqrt {\pi } x \operatorname {erfi}\left (a + x\right ) - e^{\left (a^{2} + 2 \, a x + x^{2}\right )}}{x} \]
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\[ \int \left (\frac {e^{(a+x)^2}}{x^2}-\frac {2 a e^{(a+x)^2}}{x}\right ) \, dx=- \left (\int \left (- \frac {e^{x^{2}} e^{2 a x}}{x^{2}}\right )\, dx + \int \frac {2 a e^{x^{2}} e^{2 a x}}{x}\, dx\right ) e^{a^{2}} \]
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\[ \int \left (\frac {e^{(a+x)^2}}{x^2}-\frac {2 a e^{(a+x)^2}}{x}\right ) \, dx=\int { -\frac {2 \, a e^{\left ({\left (a + x\right )}^{2}\right )}}{x} + \frac {e^{\left ({\left (a + x\right )}^{2}\right )}}{x^{2}} \,d x } \]
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\[ \int \left (\frac {e^{(a+x)^2}}{x^2}-\frac {2 a e^{(a+x)^2}}{x}\right ) \, dx=\int { -\frac {2 \, a e^{\left ({\left (a + x\right )}^{2}\right )}}{x} + \frac {e^{\left ({\left (a + x\right )}^{2}\right )}}{x^{2}} \,d x } \]
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Time = 0.29 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.17 \[ \int \left (\frac {e^{(a+x)^2}}{x^2}-\frac {2 a e^{(a+x)^2}}{x}\right ) \, dx=\sqrt {\pi }\,\mathrm {erfi}\left (a+x\right )-\frac {{\mathrm {e}}^{a^2}\,{\mathrm {e}}^{x^2}\,{\mathrm {e}}^{2\,a\,x}}{x} \]
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