Integrand size = 19, antiderivative size = 115 \[ \int \frac {e^{c+b^2 x^2} \text {erf}(b x)}{x^4} \, dx=-\frac {b e^c}{3 \sqrt {\pi } x^2}-\frac {e^{c+b^2 x^2} \text {erf}(b x)}{3 x^3}-\frac {2 b^2 e^{c+b^2 x^2} \text {erf}(b x)}{3 x}+\frac {4 b^5 e^c x^2 \, _2F_2\left (1,1;\frac {3}{2},2;b^2 x^2\right )}{3 \sqrt {\pi }}+\frac {4 b^3 e^c \log (x)}{3 \sqrt {\pi }} \] Output:
-1/3*b*exp(c)/Pi^(1/2)/x^2-1/3*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+4/3*b^5*exp(c)*x^2*hypergeom([1, 1],[3/2, 2],b^2*x^2)/P i^(1/2)+4/3*b^3*exp(c)*ln(x)/Pi^(1/2)
Time = 0.23 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.87 \[ \int \frac {e^{c+b^2 x^2} \text {erf}(b x)}{x^4} \, dx=-\frac {e^c \left (b x+e^{b^2 x^2} \sqrt {\pi } \left (1+2 b^2 x^2\right ) \text {erf}(b x)-2 b^3 \pi x^3 \text {erf}(b x) \text {erfi}(b x)+4 b^5 x^5 \, _2F_2\left (1,1;\frac {3}{2},2;-b^2 x^2\right )-4 b^3 x^3 \log (x)\right )}{3 \sqrt {\pi } x^3} \] Input:
Integrate[(E^(c + b^2*x^2)*Erf[b*x])/x^4,x]
Output:
-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*HypergeometricPFQ[{1, 1}, {3/2, 2}, -(b ^2*x^2)] - 4*b^3*x^3*Log[x]))/(Sqrt[Pi]*x^3)
Time = 0.46 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.97, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {6945, 15, 6945, 14, 6930}
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.
\(\displaystyle \int \frac {e^{b^2 x^2+c} \text {erf}(b x)}{x^4} \, dx\) |
\(\Big \downarrow \) 6945 |
\(\displaystyle \frac {2}{3} b^2 \int \frac {e^{b^2 x^2+c} \text {erf}(b x)}{x^2}dx+\frac {2 b \int \frac {e^c}{x^3}dx}{3 \sqrt {\pi }}-\frac {e^{b^2 x^2+c} \text {erf}(b x)}{3 x^3}\) |
\(\Big \downarrow \) 15 |
\(\displaystyle \frac {2}{3} b^2 \int \frac {e^{b^2 x^2+c} \text {erf}(b x)}{x^2}dx-\frac {e^{b^2 x^2+c} \text {erf}(b x)}{3 x^3}-\frac {b e^c}{3 \sqrt {\pi } x^2}\) |
\(\Big \downarrow \) 6945 |
\(\displaystyle \frac {2}{3} b^2 \left (2 b^2 \int e^{b^2 x^2+c} \text {erf}(b x)dx+\frac {2 b \int \frac {e^c}{x}dx}{\sqrt {\pi }}-\frac {e^{b^2 x^2+c} \text {erf}(b x)}{x}\right )-\frac {e^{b^2 x^2+c} \text {erf}(b x)}{3 x^3}-\frac {b e^c}{3 \sqrt {\pi } x^2}\) |
\(\Big \downarrow \) 14 |
\(\displaystyle \frac {2}{3} b^2 \left (2 b^2 \int e^{b^2 x^2+c} \text {erf}(b x)dx-\frac {e^{b^2 x^2+c} \text {erf}(b x)}{x}+\frac {2 b e^c \log (x)}{\sqrt {\pi }}\right )-\frac {e^{b^2 x^2+c} \text {erf}(b x)}{3 x^3}-\frac {b e^c}{3 \sqrt {\pi } x^2}\) |
\(\Big \downarrow \) 6930 |
\(\displaystyle \frac {2}{3} b^2 \left (\frac {2 b^3 e^c x^2 \, _2F_2\left (1,1;\frac {3}{2},2;b^2 x^2\right )}{\sqrt {\pi }}-\frac {e^{b^2 x^2+c} \text {erf}(b x)}{x}+\frac {2 b e^c \log (x)}{\sqrt {\pi }}\right )-\frac {e^{b^2 x^2+c} \text {erf}(b x)}{3 x^3}-\frac {b e^c}{3 \sqrt {\pi } x^2}\) |
Input:
Int[(E^(c + b^2*x^2)*Erf[b*x])/x^4,x]
Output:
-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])/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]))/3
Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ {a, m}, x] && NeQ[m, -1]
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]
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] + (-Simp[2*(d/(m + 1)) Int[x^(m + 2)*E^(c + d*x^2)*Erf[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]
\[\int \frac {{\mathrm e}^{b^{2} x^{2}+c} \operatorname {erf}\left (b x \right )}{x^{4}}d x\]
Input:
int(exp(b^2*x^2+c)*erf(b*x)/x^4,x)
Output:
int(exp(b^2*x^2+c)*erf(b*x)/x^4,x)
\[ \int \frac {e^{c+b^2 x^2} \text {erf}(b x)}{x^4} \, dx=\int { \frac {\operatorname {erf}\left (b x\right ) e^{\left (b^{2} x^{2} + c\right )}}{x^{4}} \,d x } \] Input:
integrate(exp(b^2*x^2+c)*erf(b*x)/x^4,x, algorithm="fricas")
Output:
integral(erf(b*x)*e^(b^2*x^2 + c)/x^4, x)
Result contains complex when optimal does not.
Time = 81.04 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.21 \[ \int \frac {e^{c+b^2 x^2} \text {erf}(b x)}{x^4} \, dx=\frac {b^{3} {G_{3, 2}^{1, 2}\left (\begin {matrix} 2, 1 & \frac {5}{2} \\2 & 0 \end {matrix} \middle | {\frac {e^{- i \pi }}{b^{2} x^{2}}} \right )} e^{c}}{2} \] Input:
integrate(exp(b**2*x**2+c)*erf(b*x)/x**4,x)
Output:
b**3*meijerg(((2, 1), (5/2,)), ((2,), (0,)), exp_polar(-I*pi)/(b**2*x**2)) *exp(c)/2
\[ \int \frac {e^{c+b^2 x^2} \text {erf}(b x)}{x^4} \, dx=\int { \frac {\operatorname {erf}\left (b x\right ) e^{\left (b^{2} x^{2} + c\right )}}{x^{4}} \,d x } \] Input:
integrate(exp(b^2*x^2+c)*erf(b*x)/x^4,x, algorithm="maxima")
Output:
integrate(erf(b*x)*e^(b^2*x^2 + c)/x^4, x)
\[ \int \frac {e^{c+b^2 x^2} \text {erf}(b x)}{x^4} \, dx=\int { \frac {\operatorname {erf}\left (b x\right ) e^{\left (b^{2} x^{2} + c\right )}}{x^{4}} \,d x } \] Input:
integrate(exp(b^2*x^2+c)*erf(b*x)/x^4,x, algorithm="giac")
Output:
integrate(erf(b*x)*e^(b^2*x^2 + c)/x^4, x)
Timed out. \[ \int \frac {e^{c+b^2 x^2} \text {erf}(b x)}{x^4} \, dx=\int \frac {{\mathrm {e}}^{b^2\,x^2+c}\,\mathrm {erf}\left (b\,x\right )}{x^4} \,d x \] Input:
int((exp(c + b^2*x^2)*erf(b*x))/x^4,x)
Output:
int((exp(c + b^2*x^2)*erf(b*x))/x^4, x)
\[ \int \frac {e^{c+b^2 x^2} \text {erf}(b x)}{x^4} \, dx=e^{c} \left (\int \frac {e^{b^{2} x^{2}} \mathrm {erf}\left (b x \right )}{x^{4}}d x \right ) \] Input:
int(exp(b^2*x^2+c)*erf(b*x)/x^4,x)
Output:
e**c*int((e**(b**2*x**2)*erf(b*x))/x**4,x)