\(\int \frac {e^{c+b^2 x^2} \text {erf}(b x)}{x^2} \, dx\) [73]

Optimal result

Integrand size = 19, antiderivative size = 66 \[ \int \frac {e^{c+b^2 x^2} \text {erf}(b x)}{x^2} \, dx=-\frac {e^{c+b^2 x^2} \text {erf}(b x)}{x}+\frac {2 b^3 e^c x^2 \, _2F_2\left (1,1;\frac {3}{2},2;b^2 x^2\right )}{\sqrt {\pi }}+\frac {2 b e^c \log (x)}{\sqrt {\pi }} \]

[Out]

-exp(b^2*x^2+c)*erf(b*x)/x+2*b^3*exp(c)*x^2*hypergeom([1, 1],[3/2, 2],b^2*x^2)/Pi^(1/2)+2*b*exp(c)*ln(x)/Pi^(1
/2)

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 66, 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 = {6526, 6511, 12, 29} \[ \int \frac {e^{c+b^2 x^2} \text {erf}(b x)}{x^2} \, dx=\frac {2 b^3 e^c x^2 \, _2F_2\left (1,1;\frac {3}{2},2;b^2 x^2\right )}{\sqrt {\pi }}-\frac {e^{b^2 x^2+c} \text {erf}(b x)}{x}+\frac {2 b e^c \log (x)}{\sqrt {\pi }} \]

[In]

Int[(E^(c + b^2*x^2)*Erf[b*x])/x^2,x]

[Out]

-((E^(c + b^2*x^2)*Erf[b*x])/x) + (2*b^3*E^c*x^2*HypergeometricPFQ[{1, 1}, {3/2, 2}, b^2*x^2])/Sqrt[Pi] + (2*b
*E^c*Log[x])/Sqrt[Pi]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 6511

Int[E^((c_.) + (d_.)*(x_)^2)*Erf[(b_.)*(x_)], x_Symbol] :> Simp[b*E^c*(x^2/Sqrt[Pi])*HypergeometricPFQ[{1, 1},
{3/2, 2}, b^2*x^2], x] /; FreeQ[{b, c, d}, x] && EqQ[d, b^2]

Rule 6526

Int[E^((c_.) + (d_.)*(x_)^2)*Erf[(a_.) + (b_.)*(x_)]*(x_)^(m_), x_Symbol] :> Simp[x^(m + 1)*E^(c + d*x^2)*(Erf
[a + b*x]/(m + 1)), x] + (-Dist[2*(d/(m + 1)), Int[x^(m + 2)*E^(c + d*x^2)*Erf[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 {erf}(b x)}{x}+\left (2 b^2\right ) \int e^{c+b^2 x^2} \text {erf}(b x) \, dx+\frac {(2 b) \int \frac {e^c}{x} \, dx}{\sqrt {\pi }} \\ & = -\frac {e^{c+b^2 x^2} \text {erf}(b x)}{x}+\frac {2 b^3 e^c x^2 \, _2F_2\left (1,1;\frac {3}{2},2;b^2 x^2\right )}{\sqrt {\pi }}+\frac {\left (2 b e^c\right ) \int \frac {1}{x} \, dx}{\sqrt {\pi }} \\ & = -\frac {e^{c+b^2 x^2} \text {erf}(b x)}{x}+\frac {2 b^3 e^c x^2 \, _2F_2\left (1,1;\frac {3}{2},2;b^2 x^2\right )}{\sqrt {\pi }}+\frac {2 b e^c \log (x)}{\sqrt {\pi }} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.15 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.12 \[ \int \frac {e^{c+b^2 x^2} \text {erf}(b x)}{x^2} \, dx=\frac {e^c \left (\text {erf}(b x) \left (-e^{b^2 x^2} \sqrt {\pi }+b \pi x \text {erfi}(b x)\right )-2 b^3 x^3 \, _2F_2\left (1,1;\frac {3}{2},2;-b^2 x^2\right )+2 b x \log (x)\right )}{\sqrt {\pi } x} \]

[In]

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

[Out]

(E^c*(Erf[b*x]*(-(E^(b^2*x^2)*Sqrt[Pi]) + b*Pi*x*Erfi[b*x]) - 2*b^3*x^3*HypergeometricPFQ[{1, 1}, {3/2, 2}, -(
b^2*x^2)] + 2*b*x*Log[x]))/(Sqrt[Pi]*x)

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 9.92 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.70 \[ \int \frac {e^{c+b^2 x^2} \text {erf}(b x)}{x^2} \, dx=\frac {2 b^{3} x^{2} e^{c} {{}_{2}F_{2}\left (\begin {matrix} 1, 1 \\ 2, \frac {5}{2} \end {matrix}\middle | {b^{2} x^{2}} \right )}}{3 \sqrt {\pi }} + \frac {b e^{c} \log {\left (b^{2} x^{2} \right )}}{\sqrt {\pi }} \]

[In]

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

[Out]

2*b**3*x**2*exp(c)*hyper((1, 1), (2, 5/2), b**2*x**2)/(3*sqrt(pi)) + b*exp(c)*log(b**2*x**2)/sqrt(pi)

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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