\(\int \frac {e^{c+b^2 x^2} \text {erfi}(b x)}{x^2} \, dx\) [292]

Optimal result

Integrand size = 19, antiderivative size = 59 \[ \int \frac {e^{c+b^2 x^2} \text {erfi}(b x)}{x^2} \, dx=-\frac {e^{c+b^2 x^2} \text {erfi}(b x)}{x}+\frac {1}{2} b e^c \sqrt {\pi } \text {erfi}(b x)^2+\frac {b e^c \operatorname {ExpIntegralEi}\left (2 b^2 x^2\right )}{\sqrt {\pi }} \]

[Out]

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

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {6528, 6510, 30, 2241} \[ \int \frac {e^{c+b^2 x^2} \text {erfi}(b x)}{x^2} \, dx=-\frac {e^{b^2 x^2+c} \text {erfi}(b x)}{x}+\frac {b e^c \operatorname {ExpIntegralEi}\left (2 b^2 x^2\right )}{\sqrt {\pi }}+\frac {1}{2} \sqrt {\pi } b e^c \text {erfi}(b x)^2 \]

[In]

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

[Out]

-((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]

Rule 30

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

Rule 2241

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] /; FreeQ[{F, a, b, c, d, e, f, n}, x] && EqQ[d*e - c*f, 0]

Rule 6510

Int[E^((c_.) + (d_.)*(x_)^2)*Erfi[(b_.)*(x_)]^(n_.), x_Symbol] :> Dist[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 6528

Int[E^((c_.) + (d_.)*(x_)^2)*Erfi[(a_.) + (b_.)*(x_)]*(x_)^(m_), x_Symbol] :> Simp[x^(m + 1)*E^(c + d*x^2)*(Er
fi[a + b*x]/(m + 1)), x] + (-Dist[2*(d/(m + 1)), Int[x^(m + 2)*E^(c + d*x^2)*Erfi[a + b*x], x], x] - Dist[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]

Rubi steps \begin{align*} \text {integral}& = -\frac {e^{c+b^2 x^2} \text {erfi}(b x)}{x}+\left (2 b^2\right ) \int e^{c+b^2 x^2} \text {erfi}(b x) \, dx+\frac {(2 b) \int \frac {e^{c+2 b^2 x^2}}{x} \, dx}{\sqrt {\pi }} \\ & = -\frac {e^{c+b^2 x^2} \text {erfi}(b x)}{x}+\frac {b e^c \operatorname {ExpIntegralEi}\left (2 b^2 x^2\right )}{\sqrt {\pi }}+\left (b e^c \sqrt {\pi }\right ) \text {Subst}(\int x \, dx,x,\text {erfi}(b x)) \\ & = -\frac {e^{c+b^2 x^2} \text {erfi}(b x)}{x}+\frac {1}{2} b e^c \sqrt {\pi } \text {erfi}(b x)^2+\frac {b e^c \operatorname {ExpIntegralEi}\left (2 b^2 x^2\right )}{\sqrt {\pi }} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.95 \[ \int \frac {e^{c+b^2 x^2} \text {erfi}(b x)}{x^2} \, dx=\frac {1}{2} e^c \left (-\frac {2 e^{b^2 x^2} \text {erfi}(b x)}{x}+b \sqrt {\pi } \text {erfi}(b x)^2+\frac {2 b \operatorname {ExpIntegralEi}\left (2 b^2 x^2\right )}{\sqrt {\pi }}\right ) \]

[In]

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

[Out]

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

Maple [F]

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

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.93 \[ \int \frac {e^{c+b^2 x^2} \text {erfi}(b x)}{x^2} \, dx=-\frac {{\left (2 \, \pi \operatorname {erfi}\left (b x\right ) e^{\left (b^{2} x^{2}\right )} - \sqrt {\pi } {\left (\pi b x \operatorname {erfi}\left (b x\right )^{2} + 2 \, b x {\rm Ei}\left (2 \, b^{2} x^{2}\right )\right )}\right )} e^{c}}{2 \, \pi x} \]

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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