Integrand size = 18, antiderivative size = 18 \[ \int \frac {e^{-b^2 x^2} \text {erf}(b x)}{x^3} \, dx=-\frac {b e^{-2 b^2 x^2}}{\sqrt {\pi } x}-\frac {e^{-b^2 x^2} \text {erf}(b x)}{2 x^2}-\sqrt {2} b^2 \text {erf}\left (\sqrt {2} b x\right )-b^2 \text {Int}\left (\frac {e^{-b^2 x^2} \text {erf}(b x)}{x},x\right ) \]
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Not integrable
Time = 0.06 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {e^{-b^2 x^2} \text {erf}(b x)}{x^3} \, dx=\int \frac {e^{-b^2 x^2} \text {erf}(b x)}{x^3} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = -\frac {e^{-b^2 x^2} \text {erf}(b x)}{2 x^2}-b^2 \int \frac {e^{-b^2 x^2} \text {erf}(b x)}{x} \, dx+\frac {b \int \frac {e^{-2 b^2 x^2}}{x^2} \, dx}{\sqrt {\pi }} \\ & = -\frac {b e^{-2 b^2 x^2}}{\sqrt {\pi } x}-\frac {e^{-b^2 x^2} \text {erf}(b x)}{2 x^2}-b^2 \int \frac {e^{-b^2 x^2} \text {erf}(b x)}{x} \, dx-\frac {\left (4 b^3\right ) \int e^{-2 b^2 x^2} \, dx}{\sqrt {\pi }} \\ & = -\frac {b e^{-2 b^2 x^2}}{\sqrt {\pi } x}-\frac {e^{-b^2 x^2} \text {erf}(b x)}{2 x^2}-\sqrt {2} b^2 \text {erf}\left (\sqrt {2} b x\right )-b^2 \int \frac {e^{-b^2 x^2} \text {erf}(b x)}{x} \, dx \\ \end{align*}
Not integrable
Time = 0.11 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int \frac {e^{-b^2 x^2} \text {erf}(b x)}{x^3} \, dx=\int \frac {e^{-b^2 x^2} \text {erf}(b x)}{x^3} \, dx \]
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Not integrable
Time = 0.16 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00
\[\int \frac {\operatorname {erf}\left (b x \right ) {\mathrm e}^{-b^{2} x^{2}}}{x^{3}}d x\]
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Not integrable
Time = 0.25 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.06 \[ \int \frac {e^{-b^2 x^2} \text {erf}(b x)}{x^3} \, dx=\int { \frac {\operatorname {erf}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )}}{x^{3}} \,d x } \]
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Not integrable
Time = 3.51 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.94 \[ \int \frac {e^{-b^2 x^2} \text {erf}(b x)}{x^3} \, dx=\int \frac {e^{- b^{2} x^{2}} \operatorname {erf}{\left (b x \right )}}{x^{3}}\, dx \]
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Not integrable
Time = 0.21 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.06 \[ \int \frac {e^{-b^2 x^2} \text {erf}(b x)}{x^3} \, dx=\int { \frac {\operatorname {erf}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )}}{x^{3}} \,d x } \]
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Not integrable
Time = 0.27 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.06 \[ \int \frac {e^{-b^2 x^2} \text {erf}(b x)}{x^3} \, dx=\int { \frac {\operatorname {erf}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )}}{x^{3}} \,d x } \]
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Not integrable
Time = 5.74 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.06 \[ \int \frac {e^{-b^2 x^2} \text {erf}(b x)}{x^3} \, dx=\int \frac {{\mathrm {e}}^{-b^2\,x^2}\,\mathrm {erf}\left (b\,x\right )}{x^3} \,d x \]
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