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

Optimal. Leaf size=100 \[ d \text{Unintegrable}\left (\frac{\text{Erf}(b x) e^{c+d x^2}}{x},x\right )+b e^c \left (-\sqrt{b^2-d}\right ) \text{Erf}\left (x \sqrt{b^2-d}\right )-\frac{b e^{c-x^2 \left (b^2-d\right )}}{\sqrt{\pi } x}-\frac{\text{Erf}(b x) e^{c+d x^2}}{2 x^2} \]

[Out]

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

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Rubi [A]  time = 0.15383, 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 = {} \[ \int \frac{e^{c+d x^2} \text{Erf}(b x)}{x^3} \, dx \]

Verification is Not applicable to the result.

[In]

Int[(E^(c + d*x^2)*Erf[b*x])/x^3,x]

[Out]

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

Rubi steps

\begin{align*} \int \frac{e^{c+d x^2} \text{erf}(b x)}{x^3} \, dx &=-\frac{e^{c+d x^2} \text{erf}(b x)}{2 x^2}+d \int \frac{e^{c+d x^2} \text{erf}(b x)}{x} \, dx+\frac{b \int \frac{e^{c-\left (b^2-d\right ) x^2}}{x^2} \, dx}{\sqrt{\pi }}\\ &=-\frac{b e^{c-\left (b^2-d\right ) x^2}}{\sqrt{\pi } x}-\frac{e^{c+d x^2} \text{erf}(b x)}{2 x^2}+d \int \frac{e^{c+d x^2} \text{erf}(b x)}{x} \, dx-\frac{\left (2 b \left (b^2-d\right )\right ) \int e^{c+\left (-b^2+d\right ) x^2} \, dx}{\sqrt{\pi }}\\ &=-\frac{b e^{c-\left (b^2-d\right ) x^2}}{\sqrt{\pi } x}-\frac{e^{c+d x^2} \text{erf}(b x)}{2 x^2}-b \sqrt{b^2-d} e^c \text{erf}\left (\sqrt{b^2-d} x\right )+d \int \frac{e^{c+d x^2} \text{erf}(b x)}{x} \, dx\\ \end{align*}

Mathematica [A]  time = 0.195235, size = 0, normalized size = 0. \[ \int \frac{e^{c+d x^2} \text{Erf}(b x)}{x^3} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.293, size = 0, normalized size = 0. \begin{align*} \int{\frac{{{\rm e}^{d{x}^{2}+c}}{\it Erf} \left ( bx \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)/x^3,x)

[Out]

int(exp(d*x^2+c)*erf(b*x)/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\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)/x^3,x, algorithm="maxima")

[Out]

integrate(erf(b*x)*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\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)/x^3,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} e^{c} \int \frac{e^{d x^{2}} \operatorname{erf}{\left (b x \right )}}{x^{3}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

exp(c)*Integral(exp(d*x**2)*erf(b*x)/x**3, x)

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{erf}\left (b x\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)/x^3,x, algorithm="giac")

[Out]

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