\(\int \frac {e^{c+d x^2} \text {erfi}(b x)}{x^3} \, dx\) [263]

Optimal result

Integrand size = 17, antiderivative size = 17 \[ \int \frac {e^{c+d x^2} \text {erfi}(b x)}{x^3} \, dx=-\frac {b e^{c+\left (b^2+d\right ) x^2}}{\sqrt {\pi } x}-\frac {e^{c+d x^2} \text {erfi}(b x)}{2 x^2}+b \sqrt {b^2+d} e^c \text {erfi}\left (\sqrt {b^2+d} x\right )+d \text {Int}\left (\frac {e^{c+d x^2} \text {erfi}(b x)}{x},x\right ) \]

[Out]

-1/2*exp(d*x^2+c)*erfi(b*x)/x^2+b*exp(c)*erfi(x*(b^2+d)^(1/2))*(b^2+d)^(1/2)-b*exp(c+(b^2+d)*x^2)/x/Pi^(1/2)+d
*Unintegrable(exp(d*x^2+c)*erfi(b*x)/x,x)

Rubi [N/A]

Not integrable

Time = 0.10 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {e^{c+d x^2} \text {erfi}(b x)}{x^3} \, dx=\int \frac {e^{c+d x^2} \text {erfi}(b x)}{x^3} \, dx \]

[In]

Int[(E^(c + d*x^2)*Erfi[b*x])/x^3,x]

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 0.13 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.12 \[ \int \frac {e^{c+d x^2} \text {erfi}(b x)}{x^3} \, dx=\int \frac {e^{c+d x^2} \text {erfi}(b x)}{x^3} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.16 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.94

\[\int \frac {{\mathrm e}^{d \,x^{2}+c} \operatorname {erfi}\left (b x \right )}{x^{3}}d x\]

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.24 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.06 \[ \int \frac {e^{c+d x^2} \text {erfi}(b x)}{x^3} \, dx=\int { \frac {\operatorname {erfi}\left (b x\right ) e^{\left (d x^{2} + c\right )}}{x^{3}} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 5.81 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.12 \[ \int \frac {e^{c+d x^2} \text {erfi}(b x)}{x^3} \, dx=e^{c} \int \frac {e^{d x^{2}} \operatorname {erfi}{\left (b x \right )}}{x^{3}}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.06 \[ \int \frac {e^{c+d x^2} \text {erfi}(b x)}{x^3} \, dx=\int { \frac {\operatorname {erfi}\left (b x\right ) e^{\left (d x^{2} + c\right )}}{x^{3}} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.27 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.06 \[ \int \frac {e^{c+d x^2} \text {erfi}(b x)}{x^3} \, dx=\int { \frac {\operatorname {erfi}\left (b x\right ) e^{\left (d x^{2} + c\right )}}{x^{3}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 5.84 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.06 \[ \int \frac {e^{c+d x^2} \text {erfi}(b x)}{x^3} \, dx=\int \frac {{\mathrm {e}}^{d\,x^2+c}\,\mathrm {erfi}\left (b\,x\right )}{x^3} \,d x \]

[In]

int((exp(c + d*x^2)*erfi(b*x))/x^3,x)

[Out]

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