3.287 \(\int \frac{e^{c+b^2 x^2} \text{Erfi}(b x)}{x^3} \, dx\)

Optimal. Leaf size=92 \[ b^2 \text{Unintegrable}\left (\frac{e^{b^2 x^2+c} \text{Erfi}(b x)}{x},x\right )-\frac{e^{b^2 x^2+c} \text{Erfi}(b x)}{2 x^2}+\sqrt{2} b^2 e^c \text{Erfi}\left (\sqrt{2} b x\right )-\frac{b e^{2 b^2 x^2+c}}{\sqrt{\pi } x} \]

[Out]

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

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Rubi [A]  time = 0.111064, 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 = {} \[ \int \frac{e^{c+b^2 x^2} \text{Erfi}(b x)}{x^3} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

\begin{align*} \int \frac{e^{c+b^2 x^2} \text{erfi}(b x)}{x^3} \, dx &=-\frac{e^{c+b^2 x^2} \text{erfi}(b x)}{2 x^2}+b^2 \int \frac{e^{c+b^2 x^2} \text{erfi}(b x)}{x} \, dx+\frac{b \int \frac{e^{c+2 b^2 x^2}}{x^2} \, dx}{\sqrt{\pi }}\\ &=-\frac{b e^{c+2 b^2 x^2}}{\sqrt{\pi } x}-\frac{e^{c+b^2 x^2} \text{erfi}(b x)}{2 x^2}+b^2 \int \frac{e^{c+b^2 x^2} \text{erfi}(b x)}{x} \, dx+\frac{\left (4 b^3\right ) \int e^{c+2 b^2 x^2} \, dx}{\sqrt{\pi }}\\ &=-\frac{b e^{c+2 b^2 x^2}}{\sqrt{\pi } x}-\frac{e^{c+b^2 x^2} \text{erfi}(b x)}{2 x^2}+\sqrt{2} b^2 e^c \text{erfi}\left (\sqrt{2} b x\right )+b^2 \int \frac{e^{c+b^2 x^2} \text{erfi}(b x)}{x} \, dx\\ \end{align*}

Mathematica [A]  time = 0.146748, size = 0, normalized size = 0. \[ \int \frac{e^{c+b^2 x^2} \text{Erfi}(b x)}{x^3} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.263, size = 0, normalized size = 0. \begin{align*} \int{\frac{{{\rm e}^{{b}^{2}{x}^{2}+c}}{\it erfi} \left ( bx \right ) }{{x}^{3}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(b^2*x^2+c)*erfi(b*x)/x^3,x)

[Out]

int(exp(b^2*x^2+c)*erfi(b*x)/x^3,x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(b^2*x^2+c)*erfi(b*x)/x^3,x, algorithm="maxima")

[Out]

integrate(erfi(b*x)*e^(b^2*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{erfi}\left (b x\right ) e^{\left (b^{2} x^{2} + c\right )}}{x^{3}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(b^2*x^2+c)*erfi(b*x)/x^3,x, algorithm="fricas")

[Out]

integral(erfi(b*x)*e^(b^2*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^{b^{2} x^{2}} \operatorname{erfi}{\left (b x \right )}}{x^{3}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(b^2*x^2+c)*erfi(b*x)/x^3,x, algorithm="giac")

[Out]

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