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

Optimal. Leaf size=58 \[ 2 d \text{Unintegrable}\left (\text{Erfi}(b x) e^{c+d x^2},x\right )+\frac{b e^c \text{ExpIntegralEi}\left (x^2 \left (b^2+d\right )\right )}{\sqrt{\pi }}-\frac{\text{Erfi}(b x) e^{c+d x^2}}{x} \]

[Out]

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

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Rubi [A]  time = 0.113821, 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 = {} \[ \int \frac{e^{c+d x^2} \text{Erfi}(b x)}{x^2} \, dx \]

Verification is Not applicable to the result.

[In]

Int[(E^(c + d*x^2)*Erfi[b*x])/x^2,x]

[Out]

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

Rubi steps

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

Mathematica [A]  time = 0.188564, size = 0, normalized size = 0. \[ \int \frac{e^{c+d x^2} \text{Erfi}(b x)}{x^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.135, size = 0, normalized size = 0. \begin{align*} \int{\frac{{{\rm e}^{d{x}^{2}+c}}{\it erfi} \left ( bx \right ) }{{x}^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

[Out]

integral(erfi(b*x)*e^(d*x^2 + c)/x^2, 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{erfi}{\left (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)*erfi(b*x)/x**2,x)

[Out]

exp(c)*Integral(exp(d*x**2)*erfi(b*x)/x**2, 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 (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)*erfi(b*x)/x^2,x, algorithm="giac")

[Out]

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