\(\int \frac {e^{c+d x^2} \text {erfi}(a+b x)}{x^4} \, dx\) [302]

Optimal result

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

-1/3*b*exp(a^2+c+2*a*b*x+(b^2+d)*x^2)/Pi^(1/2)/x^2-2/3*a*b^2*exp(a^2+c+2*a 
*b*x+(b^2+d)*x^2)/Pi^(1/2)/x-1/3*exp(d*x^2+c)*erfi(b*x+a)/x^3-2/3*d*exp(d* 
x^2+c)*erfi(b*x+a)/x+2/3*a*b^2*(b^2+d)^(1/2)*exp(c+a^2*d/(b^2+d))*erfi((a* 
b+(b^2+d)*x)/(b^2+d)^(1/2))+4/3*a^2*b^3*Defer(Int)(exp(a^2+c+2*a*b*x+(b^2+ 
d)*x^2)/x,x)/Pi^(1/2)+4/3*b*d*Defer(Int)(exp(a^2+c+2*a*b*x+(b^2+d)*x^2)/x, 
x)/Pi^(1/2)+2/3*b*(b^2+d)*Defer(Int)(exp(a^2+c+2*a*b*x+(b^2+d)*x^2)/x,x)/P 
i^(1/2)+4/3*d^2*Defer(Int)(exp(d*x^2+c)*erfi(b*x+a),x)
 

Mathematica [N/A]

Not integrable

Time = 0.29 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.11 \[ \int \frac {e^{c+d x^2} \text {erfi}(a+b x)}{x^4} \, dx=\int \frac {e^{c+d x^2} \text {erfi}(a+b x)}{x^4} \, dx \] Input:

Integrate[(E^(c + d*x^2)*Erfi[a + b*x])/x^4,x]
 

Output:

Integrate[(E^(c + d*x^2)*Erfi[a + b*x])/x^4, x]
 

Rubi [N/A]

Not integrable

Time = 2.52 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {e^{c+d x^2} \text {erfi}(a+b x)}{x^4} \, dx\)

\(\Big \downarrow \) 6947

\(\displaystyle \frac {2 b \int \frac {e^{a^2+2 b x a+\left (b^2+d\right ) x^2+c}}{x^3}dx}{3 \sqrt {\pi }}+\frac {2}{3} d \int \frac {e^{d x^2+c} \text {erfi}(a+b x)}{x^2}dx-\frac {e^{c+d x^2} \text {erfi}(a+b x)}{3 x^3}\)

\(\Big \downarrow \) 2672

\(\displaystyle \frac {2 b \left (a b \int \frac {e^{a^2+2 b x a+\left (b^2+d\right ) x^2+c}}{x^2}dx+\left (b^2+d\right ) \int \frac {e^{a^2+2 b x a+\left (b^2+d\right ) x^2+c}}{x}dx-\frac {e^{a^2+2 a b x+x^2 \left (b^2+d\right )+c}}{2 x^2}\right )}{3 \sqrt {\pi }}+\frac {2}{3} d \int \frac {e^{d x^2+c} \text {erfi}(a+b x)}{x^2}dx-\frac {e^{c+d x^2} \text {erfi}(a+b x)}{3 x^3}\)

\(\Big \downarrow \) 2672

\(\displaystyle \frac {2 b \left (\left (b^2+d\right ) \int \frac {e^{a^2+2 b x a+\left (b^2+d\right ) x^2+c}}{x}dx+a b \left (2 \left (b^2+d\right ) \int e^{a^2+2 b x a+\left (b^2+d\right ) x^2+c}dx+2 a b \int \frac {e^{a^2+2 b x a+\left (b^2+d\right ) x^2+c}}{x}dx-\frac {e^{a^2+2 a b x+x^2 \left (b^2+d\right )+c}}{x}\right )-\frac {e^{a^2+2 a b x+x^2 \left (b^2+d\right )+c}}{2 x^2}\right )}{3 \sqrt {\pi }}+\frac {2}{3} d \int \frac {e^{d x^2+c} \text {erfi}(a+b x)}{x^2}dx-\frac {e^{c+d x^2} \text {erfi}(a+b x)}{3 x^3}\)

\(\Big \downarrow \) 2664

\(\displaystyle \frac {2 b \left (\left (b^2+d\right ) \int \frac {e^{a^2+2 b x a+\left (b^2+d\right ) x^2+c}}{x}dx+a b \left (2 a b \int \frac {e^{a^2+2 b x a+\left (b^2+d\right ) x^2+c}}{x}dx+2 \left (b^2+d\right ) e^{\frac {a^2 d}{b^2+d}+c} \int e^{\frac {\left (a b+\left (b^2+d\right ) x\right )^2}{b^2+d}}dx-\frac {e^{a^2+2 a b x+x^2 \left (b^2+d\right )+c}}{x}\right )-\frac {e^{a^2+2 a b x+x^2 \left (b^2+d\right )+c}}{2 x^2}\right )}{3 \sqrt {\pi }}+\frac {2}{3} d \int \frac {e^{d x^2+c} \text {erfi}(a+b x)}{x^2}dx-\frac {e^{c+d x^2} \text {erfi}(a+b x)}{3 x^3}\)

\(\Big \downarrow \) 2633

\(\displaystyle \frac {2 b \left (a b \left (2 a b \int \frac {e^{a^2+2 b x a+\left (b^2+d\right ) x^2+c}}{x}dx+\sqrt {\pi } \sqrt {b^2+d} e^{\frac {a^2 d}{b^2+d}+c} \text {erfi}\left (\frac {a b+x \left (b^2+d\right )}{\sqrt {b^2+d}}\right )-\frac {e^{a^2+2 a b x+x^2 \left (b^2+d\right )+c}}{x}\right )+\left (b^2+d\right ) \int \frac {e^{a^2+2 b x a+\left (b^2+d\right ) x^2+c}}{x}dx-\frac {e^{a^2+2 a b x+x^2 \left (b^2+d\right )+c}}{2 x^2}\right )}{3 \sqrt {\pi }}+\frac {2}{3} d \int \frac {e^{d x^2+c} \text {erfi}(a+b x)}{x^2}dx-\frac {e^{c+d x^2} \text {erfi}(a+b x)}{3 x^3}\)

\(\Big \downarrow \) 2673

\(\displaystyle \frac {2 b \left (a b \left (2 a b \int \frac {e^{a^2+2 b x a+\left (b^2+d\right ) x^2+c}}{x}dx+\sqrt {\pi } \sqrt {b^2+d} e^{\frac {a^2 d}{b^2+d}+c} \text {erfi}\left (\frac {a b+x \left (b^2+d\right )}{\sqrt {b^2+d}}\right )-\frac {e^{a^2+2 a b x+x^2 \left (b^2+d\right )+c}}{x}\right )+\left (b^2+d\right ) \int \frac {e^{a^2+2 b x a+\left (b^2+d\right ) x^2+c}}{x}dx-\frac {e^{a^2+2 a b x+x^2 \left (b^2+d\right )+c}}{2 x^2}\right )}{3 \sqrt {\pi }}+\frac {2}{3} d \int \frac {e^{d x^2+c} \text {erfi}(a+b x)}{x^2}dx-\frac {e^{c+d x^2} \text {erfi}(a+b x)}{3 x^3}\)

\(\Big \downarrow \) 6947

\(\displaystyle \frac {2 b \left (a b \left (2 a b \int \frac {e^{a^2+2 b x a+\left (b^2+d\right ) x^2+c}}{x}dx+\sqrt {\pi } \sqrt {b^2+d} e^{\frac {a^2 d}{b^2+d}+c} \text {erfi}\left (\frac {a b+x \left (b^2+d\right )}{\sqrt {b^2+d}}\right )-\frac {e^{a^2+2 a b x+x^2 \left (b^2+d\right )+c}}{x}\right )+\left (b^2+d\right ) \int \frac {e^{a^2+2 b x a+\left (b^2+d\right ) x^2+c}}{x}dx-\frac {e^{a^2+2 a b x+x^2 \left (b^2+d\right )+c}}{2 x^2}\right )}{3 \sqrt {\pi }}+\frac {2}{3} d \left (\frac {2 b \int \frac {e^{a^2+2 b x a+\left (b^2+d\right ) x^2+c}}{x}dx}{\sqrt {\pi }}+2 d \int e^{d x^2+c} \text {erfi}(a+b x)dx-\frac {e^{c+d x^2} \text {erfi}(a+b x)}{x}\right )-\frac {e^{c+d x^2} \text {erfi}(a+b x)}{3 x^3}\)

\(\Big \downarrow \) 2673

\(\displaystyle \frac {2 b \left (a b \left (2 a b \int \frac {e^{a^2+2 b x a+\left (b^2+d\right ) x^2+c}}{x}dx+\sqrt {\pi } \sqrt {b^2+d} e^{\frac {a^2 d}{b^2+d}+c} \text {erfi}\left (\frac {a b+x \left (b^2+d\right )}{\sqrt {b^2+d}}\right )-\frac {e^{a^2+2 a b x+x^2 \left (b^2+d\right )+c}}{x}\right )+\left (b^2+d\right ) \int \frac {e^{a^2+2 b x a+\left (b^2+d\right ) x^2+c}}{x}dx-\frac {e^{a^2+2 a b x+x^2 \left (b^2+d\right )+c}}{2 x^2}\right )}{3 \sqrt {\pi }}+\frac {2}{3} d \left (\frac {2 b \int \frac {e^{a^2+2 b x a+\left (b^2+d\right ) x^2+c}}{x}dx}{\sqrt {\pi }}+2 d \int e^{d x^2+c} \text {erfi}(a+b x)dx-\frac {e^{c+d x^2} \text {erfi}(a+b x)}{x}\right )-\frac {e^{c+d x^2} \text {erfi}(a+b x)}{3 x^3}\)

\(\Big \downarrow \) 6935

\(\displaystyle \frac {2 b \left (a b \left (2 a b \int \frac {e^{a^2+2 b x a+\left (b^2+d\right ) x^2+c}}{x}dx+\sqrt {\pi } \sqrt {b^2+d} e^{\frac {a^2 d}{b^2+d}+c} \text {erfi}\left (\frac {a b+x \left (b^2+d\right )}{\sqrt {b^2+d}}\right )-\frac {e^{a^2+2 a b x+x^2 \left (b^2+d\right )+c}}{x}\right )+\left (b^2+d\right ) \int \frac {e^{a^2+2 b x a+\left (b^2+d\right ) x^2+c}}{x}dx-\frac {e^{a^2+2 a b x+x^2 \left (b^2+d\right )+c}}{2 x^2}\right )}{3 \sqrt {\pi }}+\frac {2}{3} d \left (\frac {2 b \int \frac {e^{a^2+2 b x a+\left (b^2+d\right ) x^2+c}}{x}dx}{\sqrt {\pi }}+2 d \int e^{d x^2+c} \text {erfi}(a+b x)dx-\frac {e^{c+d x^2} \text {erfi}(a+b x)}{x}\right )-\frac {e^{c+d x^2} \text {erfi}(a+b x)}{3 x^3}\)

Input:

Int[(E^(c + d*x^2)*Erfi[a + b*x])/x^4,x]
 

Output:

$Aborted
 

Maple [N/A]

Not integrable

Time = 0.23 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.95

\[\int \frac {{\mathrm e}^{d \,x^{2}+c} \operatorname {erfi}\left (b x +a \right )}{x^{4}}d x\]

Input:

int(exp(d*x^2+c)*erfi(b*x+a)/x^4,x)
 

Output:

int(exp(d*x^2+c)*erfi(b*x+a)/x^4,x)
 

Fricas [N/A]

Not integrable

Time = 0.11 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.05 \[ \int \frac {e^{c+d x^2} \text {erfi}(a+b x)}{x^4} \, dx=\int { \frac {\operatorname {erfi}\left (b x + a\right ) e^{\left (d x^{2} + c\right )}}{x^{4}} \,d x } \] Input:

integrate(exp(d*x^2+c)*erfi(b*x+a)/x^4,x, algorithm="fricas")
 

Output:

integral(erfi(b*x + a)*e^(d*x^2 + c)/x^4, x)
 

Sympy [N/A]

Not integrable

Time = 62.33 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.05 \[ \int \frac {e^{c+d x^2} \text {erfi}(a+b x)}{x^4} \, dx=e^{c} \int \frac {e^{d x^{2}} \operatorname {erfi}{\left (a + b x \right )}}{x^{4}}\, dx \] Input:

integrate(exp(d*x**2+c)*erfi(b*x+a)/x**4,x)
 

Output:

exp(c)*Integral(exp(d*x**2)*erfi(a + b*x)/x**4, x)
 

Maxima [N/A]

Not integrable

Time = 0.08 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.05 \[ \int \frac {e^{c+d x^2} \text {erfi}(a+b x)}{x^4} \, dx=\int { \frac {\operatorname {erfi}\left (b x + a\right ) e^{\left (d x^{2} + c\right )}}{x^{4}} \,d x } \] Input:

integrate(exp(d*x^2+c)*erfi(b*x+a)/x^4,x, algorithm="maxima")
 

Output:

integrate(erfi(b*x + a)*e^(d*x^2 + c)/x^4, x)
 

Giac [N/A]

Not integrable

Time = 0.11 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.05 \[ \int \frac {e^{c+d x^2} \text {erfi}(a+b x)}{x^4} \, dx=\int { \frac {\operatorname {erfi}\left (b x + a\right ) e^{\left (d x^{2} + c\right )}}{x^{4}} \,d x } \] Input:

integrate(exp(d*x^2+c)*erfi(b*x+a)/x^4,x, algorithm="giac")
 

Output:

integrate(erfi(b*x + a)*e^(d*x^2 + c)/x^4, x)
 

Mupad [N/A]

Not integrable

Time = 4.51 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.05 \[ \int \frac {e^{c+d x^2} \text {erfi}(a+b x)}{x^4} \, dx=\int \frac {\mathrm {erfi}\left (a+b\,x\right )\,{\mathrm {e}}^{d\,x^2+c}}{x^4} \,d x \] Input:

int((erfi(a + b*x)*exp(c + d*x^2))/x^4,x)
 

Output:

int((erfi(a + b*x)*exp(c + d*x^2))/x^4, x)
 

Reduce [N/A]

Not integrable

Time = 0.16 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.47 \[ \int \frac {e^{c+d x^2} \text {erfi}(a+b x)}{x^4} \, dx=-e^{c} \left (\int \frac {e^{d \,x^{2}} \mathrm {erf}\left (b i x +a i \right )}{x^{4}}d x \right ) i \] Input:

int(exp(d*x^2+c)*erfi(b*x+a)/x^4,x)
 

Output:

 - e**c*int((e**(d*x**2)*erf(a*i + b*i*x))/x**4,x)*i