3.93 \(\int \frac{e^{c+d x^2} \text{Erf}(a+b x)}{x^2} \, dx\)

Optimal. Leaf size=81 \[ \frac{2 b \text{Unintegrable}\left (\frac{e^{-a^2-2 a b x+x^2 \left (d-b^2\right )+c}}{x},x\right )}{\sqrt{\pi }}+2 d \text{Unintegrable}\left (e^{c+d x^2} \text{Erf}(a+b x),x\right )-\frac{e^{c+d x^2} \text{Erf}(a+b x)}{x} \]

[Out]

-((E^(c + d*x^2)*Erf[a + b*x])/x) + (2*b*Unintegrable[E^(-a^2 + c - 2*a*b*x + (-b^2 + d)*x^2)/x, x])/Sqrt[Pi]
+ 2*d*Unintegrable[E^(c + d*x^2)*Erf[a + b*x], x]

________________________________________________________________________________________

Rubi [A]  time = 0.215272, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{e^{c+d x^2} \text{Erf}(a+b x)}{x^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 0.407365, size = 0, normalized size = 0. \[ \int \frac{e^{c+d x^2} \text{Erf}(a+b x)}{x^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.148, size = 0, normalized size = 0. \begin{align*} \int{\frac{{{\rm e}^{d{x}^{2}+c}}{\it Erf} \left ( bx+a \right ) }{{x}^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{erf}\left (b x + a\right ) e^{\left (d x^{2} + c\right )}}{x^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\operatorname{erf}\left (b x + a\right ) e^{\left (d x^{2} + c\right )}}{x^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} e^{c} \int \frac{e^{d x^{2}} \operatorname{erf}{\left (a + b x \right )}}{x^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{erf}\left (b x + a\right ) e^{\left (d x^{2} + c\right )}}{x^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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