\(\int \frac {e^{c+b^2 x^2} \text {erfi}(b x)}{x^5} \, dx\) [288]

Optimal result

Integrand size = 19, antiderivative size = 19 \[ \int \frac {e^{c+b^2 x^2} \text {erfi}(b x)}{x^5} \, dx=-\frac {b e^{c+2 b^2 x^2}}{6 \sqrt {\pi } x^3}-\frac {7 b^3 e^{c+2 b^2 x^2}}{6 \sqrt {\pi } x}-\frac {e^{c+b^2 x^2} \text {erfi}(b x)}{4 x^4}-\frac {b^2 e^{c+b^2 x^2} \text {erfi}(b x)}{4 x^2}+\frac {b^4 e^c \text {erfi}\left (\sqrt {2} b x\right )}{\sqrt {2}}+\frac {2}{3} \sqrt {2} b^4 e^c \text {erfi}\left (\sqrt {2} b x\right )+\frac {1}{2} b^4 \text {Int}\left (\frac {e^{c+b^2 x^2} \text {erfi}(b x)}{x},x\right ) \]

[Out]

-1/4*exp(b^2*x^2+c)*erfi(b*x)/x^4-1/4*b^2*exp(b^2*x^2+c)*erfi(b*x)/x^2+7/6*b^4*exp(c)*erfi(b*x*2^(1/2))*2^(1/2
)-1/6*b*exp(2*b^2*x^2+c)/x^3/Pi^(1/2)-7/6*b^3*exp(2*b^2*x^2+c)/x/Pi^(1/2)+1/2*b^4*Unintegrable(exp(b^2*x^2+c)*
erfi(b*x)/x,x)

Rubi [N/A]

Not integrable

Time = 0.17 (sec) , antiderivative size = 19, 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 = {} \[ \int \frac {e^{c+b^2 x^2} \text {erfi}(b x)}{x^5} \, dx=\int \frac {e^{c+b^2 x^2} \text {erfi}(b x)}{x^5} \, dx \]

[In]

Int[(E^(c + b^2*x^2)*Erfi[b*x])/x^5,x]

[Out]

-1/6*(b*E^(c + 2*b^2*x^2))/(Sqrt[Pi]*x^3) - (7*b^3*E^(c + 2*b^2*x^2))/(6*Sqrt[Pi]*x) - (E^(c + b^2*x^2)*Erfi[b
*x])/(4*x^4) - (b^2*E^(c + b^2*x^2)*Erfi[b*x])/(4*x^2) + (b^4*E^c*Erfi[Sqrt[2]*b*x])/Sqrt[2] + (2*Sqrt[2]*b^4*
E^c*Erfi[Sqrt[2]*b*x])/3 + (b^4*Defer[Int][(E^(c + b^2*x^2)*Erfi[b*x])/x, x])/2

Rubi steps \begin{align*} \text {integral}& = -\frac {e^{c+b^2 x^2} \text {erfi}(b x)}{4 x^4}+\frac {1}{2} b^2 \int \frac {e^{c+b^2 x^2} \text {erfi}(b x)}{x^3} \, dx+\frac {b \int \frac {e^{c+2 b^2 x^2}}{x^4} \, dx}{2 \sqrt {\pi }} \\ & = -\frac {b e^{c+2 b^2 x^2}}{6 \sqrt {\pi } x^3}-\frac {e^{c+b^2 x^2} \text {erfi}(b x)}{4 x^4}-\frac {b^2 e^{c+b^2 x^2} \text {erfi}(b x)}{4 x^2}+\frac {1}{2} b^4 \int \frac {e^{c+b^2 x^2} \text {erfi}(b x)}{x} \, dx+\frac {b^3 \int \frac {e^{c+2 b^2 x^2}}{x^2} \, dx}{2 \sqrt {\pi }}+\frac {\left (2 b^3\right ) \int \frac {e^{c+2 b^2 x^2}}{x^2} \, dx}{3 \sqrt {\pi }} \\ & = -\frac {b e^{c+2 b^2 x^2}}{6 \sqrt {\pi } x^3}-\frac {7 b^3 e^{c+2 b^2 x^2}}{6 \sqrt {\pi } x}-\frac {e^{c+b^2 x^2} \text {erfi}(b x)}{4 x^4}-\frac {b^2 e^{c+b^2 x^2} \text {erfi}(b x)}{4 x^2}+\frac {1}{2} b^4 \int \frac {e^{c+b^2 x^2} \text {erfi}(b x)}{x} \, dx+\frac {\left (2 b^5\right ) \int e^{c+2 b^2 x^2} \, dx}{\sqrt {\pi }}+\frac {\left (8 b^5\right ) \int e^{c+2 b^2 x^2} \, dx}{3 \sqrt {\pi }} \\ & = -\frac {b e^{c+2 b^2 x^2}}{6 \sqrt {\pi } x^3}-\frac {7 b^3 e^{c+2 b^2 x^2}}{6 \sqrt {\pi } x}-\frac {e^{c+b^2 x^2} \text {erfi}(b x)}{4 x^4}-\frac {b^2 e^{c+b^2 x^2} \text {erfi}(b x)}{4 x^2}+\frac {b^4 e^c \text {erfi}\left (\sqrt {2} b x\right )}{\sqrt {2}}+\frac {2}{3} \sqrt {2} b^4 e^c \text {erfi}\left (\sqrt {2} b x\right )+\frac {1}{2} b^4 \int \frac {e^{c+b^2 x^2} \text {erfi}(b x)}{x} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 0.15 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.11 \[ \int \frac {e^{c+b^2 x^2} \text {erfi}(b x)}{x^5} \, dx=\int \frac {e^{c+b^2 x^2} \text {erfi}(b x)}{x^5} \, dx \]

[In]

Integrate[(E^(c + b^2*x^2)*Erfi[b*x])/x^5,x]

[Out]

Integrate[(E^(c + b^2*x^2)*Erfi[b*x])/x^5, x]

Maple [N/A] (verified)

Not integrable

Time = 0.16 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.95

\[\int \frac {{\mathrm e}^{b^{2} x^{2}+c} \operatorname {erfi}\left (b x \right )}{x^{5}}d x\]

[In]

int(exp(b^2*x^2+c)*erfi(b*x)/x^5,x)

[Out]

int(exp(b^2*x^2+c)*erfi(b*x)/x^5,x)

Fricas [N/A]

Not integrable

Time = 0.24 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.05 \[ \int \frac {e^{c+b^2 x^2} \text {erfi}(b x)}{x^5} \, dx=\int { \frac {\operatorname {erfi}\left (b x\right ) e^{\left (b^{2} x^{2} + c\right )}}{x^{5}} \,d x } \]

[In]

integrate(exp(b^2*x^2+c)*erfi(b*x)/x^5,x, algorithm="fricas")

[Out]

integral(erfi(b*x)*e^(b^2*x^2 + c)/x^5, x)

Sympy [N/A]

Not integrable

Time = 15.44 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.05 \[ \int \frac {e^{c+b^2 x^2} \text {erfi}(b x)}{x^5} \, dx=e^{c} \int \frac {e^{b^{2} x^{2}} \operatorname {erfi}{\left (b x \right )}}{x^{5}}\, dx \]

[In]

integrate(exp(b**2*x**2+c)*erfi(b*x)/x**5,x)

[Out]

exp(c)*Integral(exp(b**2*x**2)*erfi(b*x)/x**5, x)

Maxima [N/A]

Not integrable

Time = 0.28 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.05 \[ \int \frac {e^{c+b^2 x^2} \text {erfi}(b x)}{x^5} \, dx=\int { \frac {\operatorname {erfi}\left (b x\right ) e^{\left (b^{2} x^{2} + c\right )}}{x^{5}} \,d x } \]

[In]

integrate(exp(b^2*x^2+c)*erfi(b*x)/x^5,x, algorithm="maxima")

[Out]

integrate(erfi(b*x)*e^(b^2*x^2 + c)/x^5, x)

Giac [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.05 \[ \int \frac {e^{c+b^2 x^2} \text {erfi}(b x)}{x^5} \, dx=\int { \frac {\operatorname {erfi}\left (b x\right ) e^{\left (b^{2} x^{2} + c\right )}}{x^{5}} \,d x } \]

[In]

integrate(exp(b^2*x^2+c)*erfi(b*x)/x^5,x, algorithm="giac")

[Out]

integrate(erfi(b*x)*e^(b^2*x^2 + c)/x^5, x)

Mupad [N/A]

Not integrable

Time = 5.57 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.05 \[ \int \frac {e^{c+b^2 x^2} \text {erfi}(b x)}{x^5} \, dx=\int \frac {{\mathrm {e}}^{b^2\,x^2+c}\,\mathrm {erfi}\left (b\,x\right )}{x^5} \,d x \]

[In]

int((exp(c + b^2*x^2)*erfi(b*x))/x^5,x)

[Out]

int((exp(c + b^2*x^2)*erfi(b*x))/x^5, x)