Optimal. Leaf size=82 \[ b \text {Int}\left (\frac {e^{a+b x-c x^2}}{x},x\right )+\sqrt {\pi } \sqrt {c} e^{a+\frac {b^2}{4 c}} \text {erf}\left (\frac {b-2 c x}{2 \sqrt {c}}\right )-\frac {e^{a+b x-c x^2}}{x} \]
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Rubi [A] time = 0.07, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {e^{a+b x-c x^2}}{x^2} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {e^{a+b x-c x^2}}{x^2} \, dx &=-\frac {e^{a+b x-c x^2}}{x}+b \int \frac {e^{a+b x-c x^2}}{x} \, dx-(2 c) \int e^{a+b x-c x^2} \, dx\\ &=-\frac {e^{a+b x-c x^2}}{x}+b \int \frac {e^{a+b x-c x^2}}{x} \, dx-\left (2 c e^{a+\frac {b^2}{4 c}}\right ) \int e^{-\frac {(b-2 c x)^2}{4 c}} \, dx\\ &=-\frac {e^{a+b x-c x^2}}{x}+\sqrt {c} e^{a+\frac {b^2}{4 c}} \sqrt {\pi } \text {erf}\left (\frac {b-2 c x}{2 \sqrt {c}}\right )+b \int \frac {e^{a+b x-c x^2}}{x} \, dx\\ \end {align*}
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Mathematica [A] time = 0.23, size = 0, normalized size = 0.00 \[ \int \frac {e^{a+b x-c x^2}}{x^2} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.40, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {e^{\left (-c x^{2} + b x + a\right )}}{x^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {e^{\left (-c x^{2} + b x + a\right )}}{x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 0, normalized size = 0.00 \[ \int \frac {{\mathrm e}^{-c \,x^{2}+b x +a}}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {e^{\left (-c x^{2} + b x + a\right )}}{x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [A] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\mathrm {e}}^{-c\,x^2+b\,x+a}}{x^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.00, size = 0, normalized size = 0.00 \[ e^{a} \int \frac {e^{b x} e^{- c x^{2}}}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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