\(\int \frac {e^{c+d x^2} \text {erf}(a+b x)}{x^4} \, dx\) [94]

Optimal result

Integrand size = 19, antiderivative size = 19 \[ \int \frac {e^{c+d x^2} \text {erf}(a+b x)}{x^4} \, dx=-\frac {b e^{-a^2+c-2 a b x-\left (b^2-d\right ) x^2}}{3 \sqrt {\pi } x^2}+\frac {2 a b^2 e^{-a^2+c-2 a b x-\left (b^2-d\right ) x^2}}{3 \sqrt {\pi } x}-\frac {e^{c+d x^2} \text {erf}(a+b x)}{3 x^3}-\frac {2 d e^{c+d x^2} \text {erf}(a+b x)}{3 x}+\frac {2}{3} a b^2 \sqrt {b^2-d} e^{c+\frac {a^2 d}{b^2-d}} \text {erf}\left (\frac {a b+\left (b^2-d\right ) x}{\sqrt {b^2-d}}\right )+\frac {4 a^2 b^3 \text {Int}\left (\frac {e^{-a^2+c-2 a b x+\left (-b^2+d\right ) x^2}}{x},x\right )}{3 \sqrt {\pi }}-\frac {2 b \left (b^2-d\right ) \text {Int}\left (\frac {e^{-a^2+c-2 a b x+\left (-b^2+d\right ) x^2}}{x},x\right )}{3 \sqrt {\pi }}+\frac {4 b d \text {Int}\left (\frac {e^{-a^2+c-2 a b x+\left (-b^2+d\right ) x^2}}{x},x\right )}{3 \sqrt {\pi }}+\frac {4}{3} d^2 \text {Int}\left (e^{c+d x^2} \text {erf}(a+b x),x\right ) \] 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)*erf(b*x+a)/x^3-2/3*d*exp(d 
*x^2+c)*erf(b*x+a)/x+2/3*a*b^2*(b^2-d)^(1/2)*exp(c+a^2*d/(b^2-d))*erf((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)-2/3*b*(b^2-d)*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)+4/3*d^2*Defer(Int)(exp(d*x^2+c)*erf(b*x+a),x)
 

Mathematica [N/A]

Not integrable

Time = 0.32 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.11 \[ \int \frac {e^{c+d x^2} \text {erf}(a+b x)}{x^4} \, dx=\int \frac {e^{c+d x^2} \text {erf}(a+b x)}{x^4} \, dx \] Input:

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

Output:

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

Rubi [N/A]

Not integrable

Time = 2.58 (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.

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

\(\Big \downarrow \) 6945

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

\(\Big \downarrow \) 2672

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

\(\Big \downarrow \) 2672

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

\(\Big \downarrow \) 2664

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

\(\Big \downarrow \) 2634

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

\(\Big \downarrow \) 2673

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

\(\Big \downarrow \) 6945

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

\(\Big \downarrow \) 2673

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

\(\Big \downarrow \) 6933

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

Input:

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

Output:

$Aborted
 

Maple [N/A]

Not integrable

Time = 0.21 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.95

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

Input:

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

Output:

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

Fricas [N/A]

Not integrable

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

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

Output:

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

Sympy [N/A]

Not integrable

Time = 76.81 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.05 \[ \int \frac {e^{c+d x^2} \text {erf}(a+b x)}{x^4} \, dx=e^{c} \int \frac {e^{d x^{2}} \operatorname {erf}{\left (a + b x \right )}}{x^{4}}\, dx \] Input:

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

Output:

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

Maxima [N/A]

Not integrable

Time = 0.09 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.05 \[ \int \frac {e^{c+d x^2} \text {erf}(a+b x)}{x^4} \, dx=\int { \frac {\operatorname {erf}\left (b x + a\right ) e^{\left (d x^{2} + c\right )}}{x^{4}} \,d x } \] Input:

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

Output:

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

Giac [N/A]

Not integrable

Time = 0.13 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.05 \[ \int \frac {e^{c+d x^2} \text {erf}(a+b x)}{x^4} \, dx=\int { \frac {\operatorname {erf}\left (b x + a\right ) e^{\left (d x^{2} + c\right )}}{x^{4}} \,d x } \] Input:

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

Output:

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

Mupad [N/A]

Not integrable

Time = 4.66 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.05 \[ \int \frac {e^{c+d x^2} \text {erf}(a+b x)}{x^4} \, dx=\int \frac {\mathrm {erf}\left (a+b\,x\right )\,{\mathrm {e}}^{d\,x^2+c}}{x^4} \,d x \] Input:

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

Output:

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

Reduce [N/A]

Not integrable

Time = 0.16 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.21 \[ \int \frac {e^{c+d x^2} \text {erf}(a+b x)}{x^4} \, dx=e^{c} \left (\int \frac {e^{d \,x^{2}} \mathrm {erf}\left (b x +a \right )}{x^{4}}d x \right ) \] Input:

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

Output:

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