\(\int \frac {e^{c+b^2 x^2} \text {erf}(b x)}{x^4} \, dx\) [74]

Optimal result

Integrand size = 19, antiderivative size = 115 \[ \int \frac {e^{c+b^2 x^2} \text {erf}(b x)}{x^4} \, dx=-\frac {b e^c}{3 \sqrt {\pi } x^2}-\frac {e^{c+b^2 x^2} \text {erf}(b x)}{3 x^3}-\frac {2 b^2 e^{c+b^2 x^2} \text {erf}(b x)}{3 x}+\frac {4 b^5 e^c x^2 \, _2F_2\left (1,1;\frac {3}{2},2;b^2 x^2\right )}{3 \sqrt {\pi }}+\frac {4 b^3 e^c \log (x)}{3 \sqrt {\pi }} \]

[Out]

-1/3*exp(b^2*x^2+c)*erf(b*x)/x^3-2/3*b^2*exp(b^2*x^2+c)*erf(b*x)/x-1/3*b*exp(c)/x^2/Pi^(1/2)+4/3*b^5*exp(c)*x^
2*hypergeom([1, 1],[3/2, 2],b^2*x^2)/Pi^(1/2)+4/3*b^3*exp(c)*ln(x)/Pi^(1/2)

Rubi [A] (verified)

Time = 0.09 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {6526, 6511, 12, 29, 30} \[ \int \frac {e^{c+b^2 x^2} \text {erf}(b x)}{x^4} \, dx=\frac {4 b^5 e^c x^2 \, _2F_2\left (1,1;\frac {3}{2},2;b^2 x^2\right )}{3 \sqrt {\pi }}+\frac {4 b^3 e^c \log (x)}{3 \sqrt {\pi }}-\frac {2 b^2 e^{b^2 x^2+c} \text {erf}(b x)}{3 x}-\frac {e^{b^2 x^2+c} \text {erf}(b x)}{3 x^3}-\frac {b e^c}{3 \sqrt {\pi } x^2} \]

[In]

Int[(E^(c + b^2*x^2)*Erf[b*x])/x^4,x]

[Out]

-1/3*(b*E^c)/(Sqrt[Pi]*x^2) - (E^(c + b^2*x^2)*Erf[b*x])/(3*x^3) - (2*b^2*E^(c + b^2*x^2)*Erf[b*x])/(3*x) + (4
*b^5*E^c*x^2*HypergeometricPFQ[{1, 1}, {3/2, 2}, b^2*x^2])/(3*Sqrt[Pi]) + (4*b^3*E^c*Log[x])/(3*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 30

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

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)}{3 x^3}+\frac {1}{3} \left (2 b^2\right ) \int \frac {e^{c+b^2 x^2} \text {erf}(b x)}{x^2} \, dx+\frac {(2 b) \int \frac {e^c}{x^3} \, dx}{3 \sqrt {\pi }} \\ & = -\frac {e^{c+b^2 x^2} \text {erf}(b x)}{3 x^3}-\frac {2 b^2 e^{c+b^2 x^2} \text {erf}(b x)}{3 x}+\frac {1}{3} \left (4 b^4\right ) \int e^{c+b^2 x^2} \text {erf}(b x) \, dx+\frac {\left (4 b^3\right ) \int \frac {e^c}{x} \, dx}{3 \sqrt {\pi }}+\frac {\left (2 b e^c\right ) \int \frac {1}{x^3} \, dx}{3 \sqrt {\pi }} \\ & = -\frac {b e^c}{3 \sqrt {\pi } x^2}-\frac {e^{c+b^2 x^2} \text {erf}(b x)}{3 x^3}-\frac {2 b^2 e^{c+b^2 x^2} \text {erf}(b x)}{3 x}+\frac {4 b^5 e^c x^2 \, _2F_2\left (1,1;\frac {3}{2},2;b^2 x^2\right )}{3 \sqrt {\pi }}+\frac {\left (4 b^3 e^c\right ) \int \frac {1}{x} \, dx}{3 \sqrt {\pi }} \\ & = -\frac {b e^c}{3 \sqrt {\pi } x^2}-\frac {e^{c+b^2 x^2} \text {erf}(b x)}{3 x^3}-\frac {2 b^2 e^{c+b^2 x^2} \text {erf}(b x)}{3 x}+\frac {4 b^5 e^c x^2 \, _2F_2\left (1,1;\frac {3}{2},2;b^2 x^2\right )}{3 \sqrt {\pi }}+\frac {4 b^3 e^c \log (x)}{3 \sqrt {\pi }} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.23 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.87 \[ \int \frac {e^{c+b^2 x^2} \text {erf}(b x)}{x^4} \, dx=-\frac {e^c \left (b x+e^{b^2 x^2} \sqrt {\pi } \left (1+2 b^2 x^2\right ) \text {erf}(b x)-2 b^3 \pi x^3 \text {erf}(b x) \text {erfi}(b x)+4 b^5 x^5 \, _2F_2\left (1,1;\frac {3}{2},2;-b^2 x^2\right )-4 b^3 x^3 \log (x)\right )}{3 \sqrt {\pi } x^3} \]

[In]

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

[Out]

-1/3*(E^c*(b*x + E^(b^2*x^2)*Sqrt[Pi]*(1 + 2*b^2*x^2)*Erf[b*x] - 2*b^3*Pi*x^3*Erf[b*x]*Erfi[b*x] + 4*b^5*x^5*H
ypergeometricPFQ[{1, 1}, {3/2, 2}, -(b^2*x^2)] - 4*b^3*x^3*Log[x]))/(Sqrt[Pi]*x^3)

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 75.75 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.21 \[ \int \frac {e^{c+b^2 x^2} \text {erf}(b x)}{x^4} \, dx=\frac {b^{3} {G_{3, 2}^{1, 2}\left (\begin {matrix} 2, 1 & \frac {5}{2} \\2 & 0 \end {matrix} \middle | {\frac {e^{- i \pi }}{b^{2} x^{2}}} \right )} e^{c}}{2} \]

[In]

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

[Out]

b**3*meijerg(((2, 1), (5/2,)), ((2,), (0,)), exp_polar(-I*pi)/(b**2*x**2))*exp(c)/2

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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