Integrand size = 19, antiderivative size = 59 \[ \int \frac {e^{c+b^2 x^2} \text {erfi}(b x)}{x^2} \, dx=-\frac {e^{c+b^2 x^2} \text {erfi}(b x)}{x}+\frac {1}{2} b e^c \sqrt {\pi } \text {erfi}(b x)^2+\frac {b e^c \operatorname {ExpIntegralEi}\left (2 b^2 x^2\right )}{\sqrt {\pi }} \] Output:
-exp(b^2*x^2+c)*erfi(b*x)/x+1/2*b*exp(c)*Pi^(1/2)*erfi(b*x)^2+b*exp(c)*Ei( 2*b^2*x^2)/Pi^(1/2)
Time = 0.02 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.95 \[ \int \frac {e^{c+b^2 x^2} \text {erfi}(b x)}{x^2} \, dx=\frac {1}{2} e^c \left (-\frac {2 e^{b^2 x^2} \text {erfi}(b x)}{x}+b \sqrt {\pi } \text {erfi}(b x)^2+\frac {2 b \operatorname {ExpIntegralEi}\left (2 b^2 x^2\right )}{\sqrt {\pi }}\right ) \] Input:
Integrate[(E^(c + b^2*x^2)*Erfi[b*x])/x^2,x]
Output:
(E^c*((-2*E^(b^2*x^2)*Erfi[b*x])/x + b*Sqrt[Pi]*Erfi[b*x]^2 + (2*b*ExpInte gralEi[2*b^2*x^2])/Sqrt[Pi]))/2
Time = 0.37 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {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.
\(\displaystyle \int \frac {e^{b^2 x^2+c} \text {erfi}(b x)}{x^2} \, dx\) |
\(\Big \downarrow \) 6947 |
\(\displaystyle 2 b^2 \int e^{b^2 x^2+c} \text {erfi}(b x)dx+\frac {2 b \int \frac {e^{2 b^2 x^2+c}}{x}dx}{\sqrt {\pi }}-\frac {e^{b^2 x^2+c} \text {erfi}(b x)}{x}\) |
\(\Big \downarrow \) 2639 |
\(\displaystyle 2 b^2 \int e^{b^2 x^2+c} \text {erfi}(b x)dx-\frac {e^{b^2 x^2+c} \text {erfi}(b x)}{x}+\frac {b e^c \operatorname {ExpIntegralEi}\left (2 b^2 x^2\right )}{\sqrt {\pi }}\) |
\(\Big \downarrow \) 6929 |
\(\displaystyle \sqrt {\pi } b e^c \int \text {erfi}(b x)d\text {erfi}(b x)-\frac {e^{b^2 x^2+c} \text {erfi}(b x)}{x}+\frac {b e^c \operatorname {ExpIntegralEi}\left (2 b^2 x^2\right )}{\sqrt {\pi }}\) |
\(\Big \downarrow \) 15 |
\(\displaystyle -\frac {e^{b^2 x^2+c} \text {erfi}(b x)}{x}+\frac {b e^c \operatorname {ExpIntegralEi}\left (2 b^2 x^2\right )}{\sqrt {\pi }}+\frac {1}{2} \sqrt {\pi } b e^c \text {erfi}(b x)^2\) |
Input:
Int[(E^(c + b^2*x^2)*Erfi[b*x])/x^2,x]
Output:
-((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]
Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ {a, m}, x] && NeQ[m, -1]
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]
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]
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]
\[\int \frac {{\mathrm e}^{b^{2} x^{2}+c} \operatorname {erfi}\left (b x \right )}{x^{2}}d x\]
Input:
int(exp(b^2*x^2+c)*erfi(b*x)/x^2,x)
Output:
int(exp(b^2*x^2+c)*erfi(b*x)/x^2,x)
Time = 0.09 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.93 \[ \int \frac {e^{c+b^2 x^2} \text {erfi}(b x)}{x^2} \, dx=-\frac {{\left (2 \, \pi \operatorname {erfi}\left (b x\right ) e^{\left (b^{2} x^{2}\right )} - \sqrt {\pi } {\left (\pi b x \operatorname {erfi}\left (b x\right )^{2} + 2 \, b x {\rm Ei}\left (2 \, b^{2} x^{2}\right )\right )}\right )} e^{c}}{2 \, \pi x} \] Input:
integrate(exp(b^2*x^2+c)*erfi(b*x)/x^2,x, algorithm="fricas")
Output:
-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)
\[ \int \frac {e^{c+b^2 x^2} \text {erfi}(b x)}{x^2} \, dx=e^{c} \int \frac {e^{b^{2} x^{2}} \operatorname {erfi}{\left (b x \right )}}{x^{2}}\, dx \] Input:
integrate(exp(b**2*x**2+c)*erfi(b*x)/x**2,x)
Output:
exp(c)*Integral(exp(b**2*x**2)*erfi(b*x)/x**2, x)
\[ \int \frac {e^{c+b^2 x^2} \text {erfi}(b x)}{x^2} \, dx=\int { \frac {\operatorname {erfi}\left (b x\right ) e^{\left (b^{2} x^{2} + c\right )}}{x^{2}} \,d x } \] Input:
integrate(exp(b^2*x^2+c)*erfi(b*x)/x^2,x, algorithm="maxima")
Output:
integrate(erfi(b*x)*e^(b^2*x^2 + c)/x^2, x)
\[ \int \frac {e^{c+b^2 x^2} \text {erfi}(b x)}{x^2} \, dx=\int { \frac {\operatorname {erfi}\left (b x\right ) e^{\left (b^{2} x^{2} + c\right )}}{x^{2}} \,d x } \] Input:
integrate(exp(b^2*x^2+c)*erfi(b*x)/x^2,x, algorithm="giac")
Output:
integrate(erfi(b*x)*e^(b^2*x^2 + c)/x^2, x)
Timed out. \[ \int \frac {e^{c+b^2 x^2} \text {erfi}(b x)}{x^2} \, dx=\int \frac {{\mathrm {e}}^{b^2\,x^2+c}\,\mathrm {erfi}\left (b\,x\right )}{x^2} \,d x \] Input:
int((exp(c + b^2*x^2)*erfi(b*x))/x^2,x)
Output:
int((exp(c + b^2*x^2)*erfi(b*x))/x^2, x)
\[ \int \frac {e^{c+b^2 x^2} \text {erfi}(b x)}{x^2} \, dx=-e^{c} \left (\int \frac {e^{b^{2} x^{2}} \mathrm {erf}\left (b i x \right )}{x^{2}}d x \right ) i \] Input:
int(exp(b^2*x^2+c)*erfi(b*x)/x^2,x)
Output:
- e**c*int((e**(b**2*x**2)*erf(b*i*x))/x**2,x)*i