\(\int \frac {e^{c+d x^2} \text {erf}(b x)}{x^2} \, dx\) [62]

Optimal result

Integrand size = 17, antiderivative size = 17 \[ \int \frac {e^{c+d x^2} \text {erf}(b x)}{x^2} \, dx=-\frac {e^{c+d x^2} \text {erf}(b x)}{x}+\frac {b e^c \operatorname {ExpIntegralEi}\left (-\left (\left (b^2-d\right ) x^2\right )\right )}{\sqrt {\pi }}+2 d \text {Int}\left (e^{c+d x^2} \text {erf}(b x),x\right ) \]

[Out]

-exp(d*x^2+c)*erf(b*x)/x+b*exp(c)*Ei(-(b^2-d)*x^2)/Pi^(1/2)+2*d*Unintegrable(exp(d*x^2+c)*erf(b*x),x)

Rubi [N/A]

Not integrable

Time = 0.09 (sec) , antiderivative size = 17, 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 = {} \[ \int \frac {e^{c+d x^2} \text {erf}(b x)}{x^2} \, dx=\int \frac {e^{c+d x^2} \text {erf}(b x)}{x^2} \, dx \]

[In]

Int[(E^(c + d*x^2)*Erf[b*x])/x^2,x]

[Out]

-((E^(c + d*x^2)*Erf[b*x])/x) + (b*E^c*ExpIntegralEi[-((b^2 - d)*x^2)])/Sqrt[Pi] + 2*d*Defer[Int][E^(c + d*x^2
)*Erf[b*x], x]

Rubi steps \begin{align*} \text {integral}& = -\frac {e^{c+d x^2} \text {erf}(b x)}{x}+(2 d) \int e^{c+d x^2} \text {erf}(b x) \, dx+\frac {(2 b) \int \frac {e^{c-\left (b^2-d\right ) x^2}}{x} \, dx}{\sqrt {\pi }} \\ & = -\frac {e^{c+d x^2} \text {erf}(b x)}{x}+\frac {b e^c \operatorname {ExpIntegralEi}\left (-\left (\left (b^2-d\right ) x^2\right )\right )}{\sqrt {\pi }}+(2 d) \int e^{c+d x^2} \text {erf}(b x) \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 0.15 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.12 \[ \int \frac {e^{c+d x^2} \text {erf}(b x)}{x^2} \, dx=\int \frac {e^{c+d x^2} \text {erf}(b x)}{x^2} \, dx \]

[In]

Integrate[(E^(c + d*x^2)*Erf[b*x])/x^2,x]

[Out]

Integrate[(E^(c + d*x^2)*Erf[b*x])/x^2, x]

Maple [N/A] (verified)

Not integrable

Time = 0.08 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.94

\[\int \frac {{\mathrm e}^{d \,x^{2}+c} \operatorname {erf}\left (b x \right )}{x^{2}}d x\]

[In]

int(exp(d*x^2+c)*erf(b*x)/x^2,x)

[Out]

int(exp(d*x^2+c)*erf(b*x)/x^2,x)

Fricas [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.06 \[ \int \frac {e^{c+d x^2} \text {erf}(b x)}{x^2} \, dx=\int { \frac {\operatorname {erf}\left (b x\right ) e^{\left (d x^{2} + c\right )}}{x^{2}} \,d x } \]

[In]

integrate(exp(d*x^2+c)*erf(b*x)/x^2,x, algorithm="fricas")

[Out]

integral(erf(b*x)*e^(d*x^2 + c)/x^2, x)

Sympy [N/A]

Not integrable

Time = 2.86 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.12 \[ \int \frac {e^{c+d x^2} \text {erf}(b x)}{x^2} \, dx=e^{c} \int \frac {e^{d x^{2}} \operatorname {erf}{\left (b x \right )}}{x^{2}}\, dx \]

[In]

integrate(exp(d*x**2+c)*erf(b*x)/x**2,x)

[Out]

exp(c)*Integral(exp(d*x**2)*erf(b*x)/x**2, x)

Maxima [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.06 \[ \int \frac {e^{c+d x^2} \text {erf}(b x)}{x^2} \, dx=\int { \frac {\operatorname {erf}\left (b x\right ) e^{\left (d x^{2} + c\right )}}{x^{2}} \,d x } \]

[In]

integrate(exp(d*x^2+c)*erf(b*x)/x^2,x, algorithm="maxima")

[Out]

integrate(erf(b*x)*e^(d*x^2 + c)/x^2, x)

Giac [N/A]

Not integrable

Time = 0.27 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.06 \[ \int \frac {e^{c+d x^2} \text {erf}(b x)}{x^2} \, dx=\int { \frac {\operatorname {erf}\left (b x\right ) e^{\left (d x^{2} + c\right )}}{x^{2}} \,d x } \]

[In]

integrate(exp(d*x^2+c)*erf(b*x)/x^2,x, algorithm="giac")

[Out]

integrate(erf(b*x)*e^(d*x^2 + c)/x^2, x)

Mupad [N/A]

Not integrable

Time = 5.76 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.06 \[ \int \frac {e^{c+d x^2} \text {erf}(b x)}{x^2} \, dx=\int \frac {{\mathrm {e}}^{d\,x^2+c}\,\mathrm {erf}\left (b\,x\right )}{x^2} \,d x \]

[In]

int((exp(c + d*x^2)*erf(b*x))/x^2,x)

[Out]

int((exp(c + d*x^2)*erf(b*x))/x^2, x)