Integrand size = 19, antiderivative size = 71 \[ \int \frac {e^{c+b^2 x^2} \text {erf}(b x)}{x^3} \, dx=-\frac {b e^c}{\sqrt {\pi } x}-\frac {e^{c+b^2 x^2} \text {erf}(b x)}{2 x^2}+\frac {2 b^3 e^c x \, _2F_2\left (\frac {1}{2},1;\frac {3}{2},\frac {3}{2};b^2 x^2\right )}{\sqrt {\pi }} \]
[Out]
Time = 0.08 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {6526, 6523, 12, 30} \[ \int \frac {e^{c+b^2 x^2} \text {erf}(b x)}{x^3} \, dx=\frac {2 b^3 e^c x \, _2F_2\left (\frac {1}{2},1;\frac {3}{2},\frac {3}{2};b^2 x^2\right )}{\sqrt {\pi }}-\frac {e^{b^2 x^2+c} \text {erf}(b x)}{2 x^2}-\frac {b e^c}{\sqrt {\pi } x} \]
[In]
[Out]
Rule 12
Rule 30
Rule 6523
Rule 6526
Rubi steps \begin{align*} \text {integral}& = -\frac {e^{c+b^2 x^2} \text {erf}(b x)}{2 x^2}+b^2 \int \frac {e^{c+b^2 x^2} \text {erf}(b x)}{x} \, dx+\frac {b \int \frac {e^c}{x^2} \, dx}{\sqrt {\pi }} \\ & = -\frac {e^{c+b^2 x^2} \text {erf}(b x)}{2 x^2}+\frac {2 b^3 e^c x \, _2F_2\left (\frac {1}{2},1;\frac {3}{2},\frac {3}{2};b^2 x^2\right )}{\sqrt {\pi }}+\frac {\left (b e^c\right ) \int \frac {1}{x^2} \, dx}{\sqrt {\pi }} \\ & = -\frac {b e^c}{\sqrt {\pi } x}-\frac {e^{c+b^2 x^2} \text {erf}(b x)}{2 x^2}+\frac {2 b^3 e^c x \, _2F_2\left (\frac {1}{2},1;\frac {3}{2},\frac {3}{2};b^2 x^2\right )}{\sqrt {\pi }} \\ \end{align*}
Time = 0.10 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.48 \[ \int \frac {e^{c+b^2 x^2} \text {erf}(b x)}{x^3} \, dx=-\frac {2 b e^c \, _2F_2\left (-\frac {1}{2},1;\frac {1}{2},\frac {3}{2};b^2 x^2\right )}{\sqrt {\pi } x} \]
[In]
[Out]
\[\int \frac {{\mathrm e}^{b^{2} x^{2}+c} \operatorname {erf}\left (b x \right )}{x^{3}}d x\]
[In]
[Out]
\[ \int \frac {e^{c+b^2 x^2} \text {erf}(b x)}{x^3} \, dx=\int { \frac {\operatorname {erf}\left (b x\right ) e^{\left (b^{2} x^{2} + c\right )}}{x^{3}} \,d x } \]
[In]
[Out]
Time = 26.68 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.41 \[ \int \frac {e^{c+b^2 x^2} \text {erf}(b x)}{x^3} \, dx=- \frac {2 b e^{c} {{}_{2}F_{2}\left (\begin {matrix} - \frac {1}{2}, 1 \\ \frac {1}{2}, \frac {3}{2} \end {matrix}\middle | {b^{2} x^{2}} \right )}}{\sqrt {\pi } x} \]
[In]
[Out]
\[ \int \frac {e^{c+b^2 x^2} \text {erf}(b x)}{x^3} \, dx=\int { \frac {\operatorname {erf}\left (b x\right ) e^{\left (b^{2} x^{2} + c\right )}}{x^{3}} \,d x } \]
[In]
[Out]
\[ \int \frac {e^{c+b^2 x^2} \text {erf}(b x)}{x^3} \, dx=\int { \frac {\operatorname {erf}\left (b x\right ) e^{\left (b^{2} x^{2} + c\right )}}{x^{3}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {e^{c+b^2 x^2} \text {erf}(b x)}{x^3} \, dx=\int \frac {{\mathrm {e}}^{b^2\,x^2+c}\,\mathrm {erf}\left (b\,x\right )}{x^3} \,d x \]
[In]
[Out]