Optimal. Leaf size=80 \[ -\frac {2 b^3 e^c x^2 \, _2F_2\left (1,1;\frac {3}{2},2;b^2 x^2\right )}{\sqrt {\pi }}-\frac {e^{b^2 x^2+c} \text {erfc}(b x)}{x}+\sqrt {\pi } b e^c \text {erfi}(b x)-\frac {2 b e^c \log (x)}{\sqrt {\pi }} \]
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Rubi [A] time = 0.08, antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {6392, 6377, 2204, 6376, 12, 29} \[ -\frac {2 b^3 e^c x^2 \, _2F_2\left (1,1;\frac {3}{2},2;b^2 x^2\right )}{\sqrt {\pi }}-\frac {e^{b^2 x^2+c} \text {Erfc}(b x)}{x}+\sqrt {\pi } b e^c \text {Erfi}(b x)-\frac {2 b e^c \log (x)}{\sqrt {\pi }} \]
Antiderivative was successfully verified.
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Rule 12
Rule 29
Rule 2204
Rule 6376
Rule 6377
Rule 6392
Rubi steps
\begin {align*} \int \frac {e^{c+b^2 x^2} \text {erfc}(b x)}{x^2} \, dx &=-\frac {e^{c+b^2 x^2} \text {erfc}(b x)}{x}+\left (2 b^2\right ) \int e^{c+b^2 x^2} \text {erfc}(b x) \, dx-\frac {(2 b) \int \frac {e^c}{x} \, dx}{\sqrt {\pi }}\\ &=-\frac {e^{c+b^2 x^2} \text {erfc}(b x)}{x}+\left (2 b^2\right ) \int e^{c+b^2 x^2} \, dx-\left (2 b^2\right ) \int e^{c+b^2 x^2} \text {erf}(b x) \, dx-\frac {\left (2 b e^c\right ) \int \frac {1}{x} \, dx}{\sqrt {\pi }}\\ &=-\frac {e^{c+b^2 x^2} \text {erfc}(b x)}{x}+b e^c \sqrt {\pi } \text {erfi}(b x)-\frac {2 b^3 e^c x^2 \, _2F_2\left (1,1;\frac {3}{2},2;b^2 x^2\right )}{\sqrt {\pi }}-\frac {2 b e^c \log (x)}{\sqrt {\pi }}\\ \end {align*}
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Mathematica [A] time = 0.31, size = 99, normalized size = 1.24 \[ -\frac {e^c \left (-2 b^3 x^3 \, _2F_2\left (1,1;\frac {3}{2},2;-b^2 x^2\right )+\text {erf}(b x) \left (\pi b x \text {erfi}(b x)-\sqrt {\pi } e^{b^2 x^2}\right )+\sqrt {\pi } e^{b^2 x^2}-\pi b x \text {erfi}(b x)+2 b x \log (x)\right )}{\sqrt {\pi } x} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.41, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (\operatorname {erf}\left (b x\right ) - 1\right )} e^{\left (b^{2} x^{2} + c\right )}}{x^{2}}, x\right ) \]
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 \[ \int \frac {\operatorname {erfc}\left (b x\right ) e^{\left (b^{2} x^{2} + c\right )}}{x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.09, size = 0, normalized size = 0.00 \[ \int \frac {{\mathrm e}^{b^{2} x^{2}+c} \mathrm {erfc}\left (b x \right )}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {erfc}\left (b x\right ) e^{\left (b^{2} x^{2} + c\right )}}{x^{2}}\,{d x} \]
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 \[ \int \frac {{\mathrm {e}}^{b^2\,x^2+c}\,\mathrm {erfc}\left (b\,x\right )}{x^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: AttributeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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