3.89 \(\int \frac{e^{c+d x^2} \text{Erf}(a+b x)}{x^3} \, dx\)

Optimal. Leaf size=183 \[ -\frac{2 a b^2 \text{Unintegrable}\left (\frac{e^{-a^2-2 a b x+x^2 \left (d-b^2\right )+c}}{x},x\right )}{\sqrt{\pi }}+d \text{Unintegrable}\left (\frac{e^{c+d x^2} \text{Erf}(a+b x)}{x},x\right )-b \sqrt{b^2-d} e^{\frac{a^2 d}{b^2-d}+c} \text{Erf}\left (\frac{a b+x \left (b^2-d\right )}{\sqrt{b^2-d}}\right )-\frac{b e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{\sqrt{\pi } x}-\frac{e^{c+d x^2} \text{Erf}(a+b x)}{2 x^2} \]

[Out]

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

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Rubi [A]  time = 0.416555, 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 = {} \[ \int \frac{e^{c+d x^2} \text{Erf}(a+b x)}{x^3} \, dx \]

Verification is Not applicable to the result.

[In]

Int[(E^(c + d*x^2)*Erf[a + b*x])/x^3,x]

[Out]

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

Rubi steps

\begin{align*} \int \frac{e^{c+d x^2} \text{erf}(a+b x)}{x^3} \, dx &=-\frac{e^{c+d x^2} \text{erf}(a+b x)}{2 x^2}+d \int \frac{e^{c+d x^2} \text{erf}(a+b x)}{x} \, dx+\frac{b \int \frac{e^{-a^2+c-2 a b x+\left (-b^2+d\right ) x^2}}{x^2} \, dx}{\sqrt{\pi }}\\ &=-\frac{b e^{-a^2+c-2 a b x-\left (b^2-d\right ) x^2}}{\sqrt{\pi } x}-\frac{e^{c+d x^2} \text{erf}(a+b x)}{2 x^2}+d \int \frac{e^{c+d x^2} \text{erf}(a+b x)}{x} \, dx-\frac{\left (2 a b^2\right ) \int \frac{e^{-a^2+c-2 a b x+\left (-b^2+d\right ) x^2}}{x} \, dx}{\sqrt{\pi }}-\frac{\left (2 b \left (b^2-d\right )\right ) \int e^{-a^2+c-2 a b x+\left (-b^2+d\right ) x^2} \, dx}{\sqrt{\pi }}\\ &=-\frac{b e^{-a^2+c-2 a b x-\left (b^2-d\right ) x^2}}{\sqrt{\pi } x}-\frac{e^{c+d x^2} \text{erf}(a+b x)}{2 x^2}+d \int \frac{e^{c+d x^2} \text{erf}(a+b x)}{x} \, dx-\frac{\left (2 a b^2\right ) \int \frac{e^{-a^2+c-2 a b x+\left (-b^2+d\right ) x^2}}{x} \, dx}{\sqrt{\pi }}-\frac{\left (2 b \left (b^2-d\right ) e^{\frac{b^2 c+a^2 d-c d}{b^2-d}}\right ) \int \exp \left (\frac{\left (-2 a b+2 \left (-b^2+d\right ) x\right )^2}{4 \left (-b^2+d\right )}\right ) \, dx}{\sqrt{\pi }}\\ &=-\frac{b e^{-a^2+c-2 a b x-\left (b^2-d\right ) x^2}}{\sqrt{\pi } x}-\frac{e^{c+d x^2} \text{erf}(a+b x)}{2 x^2}-b \sqrt{b^2-d} e^{\frac{b^2 c+a^2 d-c d}{b^2-d}} \text{erf}\left (\frac{a b+\left (b^2-d\right ) x}{\sqrt{b^2-d}}\right )+d \int \frac{e^{c+d x^2} \text{erf}(a+b x)}{x} \, dx-\frac{\left (2 a b^2\right ) \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 = 0.36426, size = 0, normalized size = 0. \[ \int \frac{e^{c+d x^2} \text{Erf}(a+b x)}{x^3} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[(E^(c + d*x^2)*Erf[a + b*x])/x^3,x]

[Out]

Integrate[(E^(c + d*x^2)*Erf[a + b*x])/x^3, x]

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Maple [A]  time = 0.337, size = 0, normalized size = 0. \begin{align*} \int{\frac{{{\rm e}^{d{x}^{2}+c}}{\it Erf} \left ( bx+a \right ) }{{x}^{3}}}\, 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^3,x)

[Out]

int(exp(d*x^2+c)*erf(b*x+a)/x^3,x)

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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^{3}}\,{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^3,x, algorithm="maxima")

[Out]

integrate(erf(b*x + a)*e^(d*x^2 + c)/x^3, x)

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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^{3}}, 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^3,x, algorithm="fricas")

[Out]

integral(erf(b*x + a)*e^(d*x^2 + c)/x^3, x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(d*x**2+c)*erf(b*x+a)/x**3,x)

[Out]

Timed out

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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^{3}}\,{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^3,x, algorithm="giac")

[Out]

integrate(erf(b*x + a)*e^(d*x^2 + c)/x^3, x)