\(\int \frac {e^{c+b^2 x^2} \text {erfi}(b x)}{x^4} \, dx\) [293]

Optimal result

Integrand size = 19, antiderivative size = 118 \[ \int \frac {e^{c+b^2 x^2} \text {erfi}(b x)}{x^4} \, dx=-\frac {b e^{c+2 b^2 x^2}}{3 \sqrt {\pi } x^2}-\frac {e^{c+b^2 x^2} \text {erfi}(b x)}{3 x^3}-\frac {2 b^2 e^{c+b^2 x^2} \text {erfi}(b x)}{3 x}+\frac {1}{3} b^3 e^c \sqrt {\pi } \text {erfi}(b x)^2+\frac {4 b^3 e^c \operatorname {ExpIntegralEi}\left (2 b^2 x^2\right )}{3 \sqrt {\pi }} \] Output:

-1/3*b*exp(2*b^2*x^2+c)/Pi^(1/2)/x^2-1/3*exp(b^2*x^2+c)*erfi(b*x)/x^3-2/3* 
b^2*exp(b^2*x^2+c)*erfi(b*x)/x+1/3*b^3*exp(c)*Pi^(1/2)*erfi(b*x)^2+4/3*b^3 
*exp(c)*Ei(2*b^2*x^2)/Pi^(1/2)
 

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.77 \[ \int \frac {e^{c+b^2 x^2} \text {erfi}(b x)}{x^4} \, dx=-\frac {e^c \left (e^{b^2 x^2} \sqrt {\pi } \left (1+2 b^2 x^2\right ) \text {erfi}(b x)-b^3 \pi x^3 \text {erfi}(b x)^2+b x \left (e^{2 b^2 x^2}-4 b^2 x^2 \operatorname {ExpIntegralEi}\left (2 b^2 x^2\right )\right )\right )}{3 \sqrt {\pi } x^3} \] Input:

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

Output:

-1/3*(E^c*(E^(b^2*x^2)*Sqrt[Pi]*(1 + 2*b^2*x^2)*Erfi[b*x] - b^3*Pi*x^3*Erf 
i[b*x]^2 + b*x*(E^(2*b^2*x^2) - 4*b^2*x^2*ExpIntegralEi[2*b^2*x^2])))/(Sqr 
t[Pi]*x^3)
 

Rubi [A] (verified)

Time = 0.70 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.14, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.368, Rules used = {6947, 2643, 2639, 6947, 2639, 6929, 15}

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.

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

\(\Big \downarrow \) 6947

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

\(\Big \downarrow \) 2643

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

\(\Big \downarrow \) 2639

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

\(\Big \downarrow \) 6947

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

\(\Big \downarrow \) 2639

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

\(\Big \downarrow \) 6929

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

\(\Big \downarrow \) 15

\(\displaystyle \frac {2}{3} b^2 \left (-\frac {e^{b^2 x^2+c} \text {erfi}(b x)}{x}+\frac {b e^c \operatorname {ExpIntegralEi}\left (2 b^2 x^2\right )}{\sqrt {\pi }}+\frac {1}{2} \sqrt {\pi } b e^c \text {erfi}(b x)^2\right )-\frac {e^{b^2 x^2+c} \text {erfi}(b x)}{3 x^3}+\frac {2 b \left (b^2 e^c \operatorname {ExpIntegralEi}\left (2 b^2 x^2\right )-\frac {e^{2 b^2 x^2+c}}{2 x^2}\right )}{3 \sqrt {\pi }}\)

Input:

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

Output:

-1/3*(E^(c + b^2*x^2)*Erfi[b*x])/x^3 + (2*b*(-1/2*E^(c + 2*b^2*x^2)/x^2 + 
b^2*E^c*ExpIntegralEi[2*b^2*x^2]))/(3*Sqrt[Pi]) + (2*b^2*(-((E^(c + b^2*x^ 
2)*Erfi[b*x])/x) + (b*E^c*Sqrt[Pi]*Erfi[b*x]^2)/2 + (b*E^c*ExpIntegralEi[2 
*b^2*x^2])/Sqrt[Pi]))/3
 

Defintions of rubi rules used

rule 15

Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ 
{a, m}, x] && NeQ[m, -1]
 

rule 2639

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_ 
Symbol] :> Simp[F^a*(ExpIntegralEi[b*(c + d*x)^n*Log[F]]/(f*n)), x] /; Free 
Q[{F, a, b, c, d, e, f, n}, x] && EqQ[d*e - c*f, 0]
 

rule 2643

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((c_.) + (d_.)*(x_))^(m_ 
.), x_Symbol] :> Simp[(c + d*x)^(m + 1)*(F^(a + b*(c + d*x)^n)/(d*(m + 1))) 
, x] - Simp[b*n*(Log[F]/(m + 1))   Int[(c + d*x)^(m + n)*F^(a + b*(c + d*x) 
^n), x], x] /; FreeQ[{F, a, b, c, d}, x] && IntegerQ[2*((m + 1)/n)] && LtQ[ 
-4, (m + 1)/n, 5] && IntegerQ[n] && ((GtQ[n, 0] && LtQ[m, -1]) || (GtQ[-n, 
0] && LeQ[-n, m + 1]))
 

rule 6929

Int[E^((c_.) + (d_.)*(x_)^2)*Erfi[(b_.)*(x_)]^(n_.), x_Symbol] :> Simp[E^c* 
(Sqrt[Pi]/(2*b))   Subst[Int[x^n, x], x, Erfi[b*x]], x] /; FreeQ[{b, c, d, 
n}, x] && EqQ[d, b^2]
 

rule 6947

Int[E^((c_.) + (d_.)*(x_)^2)*Erfi[(a_.) + (b_.)*(x_)]*(x_)^(m_), x_Symbol] 
:> Simp[x^(m + 1)*E^(c + d*x^2)*(Erfi[a + b*x]/(m + 1)), x] + (-Simp[2*(d/( 
m + 1))   Int[x^(m + 2)*E^(c + d*x^2)*Erfi[a + b*x], x], x] - Simp[2*(b/((m 
 + 1)*Sqrt[Pi]))   Int[x^(m + 1)*E^(a^2 + c + 2*a*b*x + (b^2 + d)*x^2), x], 
 x]) /; FreeQ[{a, b, c, d}, x] && ILtQ[m, -1]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.72 \[ \int \frac {e^{c+b^2 x^2} \text {erfi}(b x)}{x^4} \, dx=-\frac {{\left ({\left (\pi + 2 \, \pi b^{2} x^{2}\right )} \operatorname {erfi}\left (b x\right ) e^{\left (b^{2} x^{2}\right )} - \sqrt {\pi } {\left (\pi b^{3} x^{3} \operatorname {erfi}\left (b x\right )^{2} + 4 \, b^{3} x^{3} {\rm Ei}\left (2 \, b^{2} x^{2}\right ) - b x e^{\left (2 \, b^{2} x^{2}\right )}\right )}\right )} e^{c}}{3 \, \pi x^{3}} \] Input:

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

Output:

-1/3*((pi + 2*pi*b^2*x^2)*erfi(b*x)*e^(b^2*x^2) - sqrt(pi)*(pi*b^3*x^3*erf 
i(b*x)^2 + 4*b^3*x^3*Ei(2*b^2*x^2) - b*x*e^(2*b^2*x^2)))*e^c/(pi*x^3)
 

Sympy [F]

\[ \int \frac {e^{c+b^2 x^2} \text {erfi}(b x)}{x^4} \, dx=e^{c} \int \frac {e^{b^{2} x^{2}} \operatorname {erfi}{\left (b x \right )}}{x^{4}}\, dx \] Input:

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

Output:

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

Maxima [F]

\[ \int \frac {e^{c+b^2 x^2} \text {erfi}(b x)}{x^4} \, dx=\int { \frac {\operatorname {erfi}\left (b x\right ) e^{\left (b^{2} x^{2} + c\right )}}{x^{4}} \,d x } \] Input:

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

Output:

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

Giac [F]

\[ \int \frac {e^{c+b^2 x^2} \text {erfi}(b x)}{x^4} \, dx=\int { \frac {\operatorname {erfi}\left (b x\right ) e^{\left (b^{2} x^{2} + c\right )}}{x^{4}} \,d x } \] Input:

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

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {e^{c+b^2 x^2} \text {erfi}(b x)}{x^4} \, dx=\int \frac {{\mathrm {e}}^{b^2\,x^2+c}\,\mathrm {erfi}\left (b\,x\right )}{x^4} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \frac {e^{c+b^2 x^2} \text {erfi}(b x)}{x^4} \, dx=-e^{c} \left (\int \frac {e^{b^{2} x^{2}} \mathrm {erf}\left (b i x \right )}{x^{4}}d x \right ) i \] Input:

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

Output:

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