\(\int \frac {e^{c+b^2 x^2} \text {erfi}(b x)}{x^4} \, dx\) [293]

Optimal result

Integrand size = 19, antiderivative size = 118 \[ \int \frac {e^{c+b^2 x^2} \text {erfi}(b x)}{x^4} \, dx=-\frac {b e^{c+2 b^2 x^2}}{3 \sqrt {\pi } x^2}-\frac {e^{c+b^2 x^2} \text {erfi}(b x)}{3 x^3}-\frac {2 b^2 e^{c+b^2 x^2} \text {erfi}(b x)}{3 x}+\frac {1}{3} b^3 e^c \sqrt {\pi } \text {erfi}(b x)^2+\frac {4 b^3 e^c \operatorname {ExpIntegralEi}\left (2 b^2 x^2\right )}{3 \sqrt {\pi }} \]

[Out]

-1/3*exp(b^2*x^2+c)*erfi(b*x)/x^3-2/3*b^2*exp(b^2*x^2+c)*erfi(b*x)/x-1/3*b*exp(2*b^2*x^2+c)/x^2/Pi^(1/2)+4/3*b
^3*exp(c)*Ei(2*b^2*x^2)/Pi^(1/2)+1/3*b^3*exp(c)*erfi(b*x)^2*Pi^(1/2)

Rubi [A] (verified)

Time = 0.13 (sec) , antiderivative size = 118, 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 = {6528, 6510, 30, 2241, 2245} \[ \int \frac {e^{c+b^2 x^2} \text {erfi}(b x)}{x^4} \, dx=\frac {1}{3} \sqrt {\pi } b^3 e^c \text {erfi}(b x)^2-\frac {2 b^2 e^{b^2 x^2+c} \text {erfi}(b x)}{3 x}-\frac {e^{b^2 x^2+c} \text {erfi}(b x)}{3 x^3}-\frac {b e^{2 b^2 x^2+c}}{3 \sqrt {\pi } x^2}+\frac {4 b^3 e^c \operatorname {ExpIntegralEi}\left (2 b^2 x^2\right )}{3 \sqrt {\pi }} \]

[In]

Int[(E^(c + b^2*x^2)*Erfi[b*x])/x^4,x]

[Out]

-1/3*(b*E^(c + 2*b^2*x^2))/(Sqrt[Pi]*x^2) - (E^(c + b^2*x^2)*Erfi[b*x])/(3*x^3) - (2*b^2*E^(c + b^2*x^2)*Erfi[
b*x])/(3*x) + (b^3*E^c*Sqrt[Pi]*Erfi[b*x]^2)/3 + (4*b^3*E^c*ExpIntegralEi[2*b^2*x^2])/(3*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 2245

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[(c + d*x)^(m
+ 1)*(F^(a + b*(c + d*x)^n)/(d*(m + 1))), x] - Dist[b*n*(Log[F]/(m + 1)), Int[(c + d*x)^(m + n)*F^(a + b*(c +
d*x)^n), x], x] /; FreeQ[{F, a, b, c, d}, x] && IntegerQ[2*((m + 1)/n)] && LtQ[-4, (m + 1)/n, 5] && IntegerQ[n
] && ((GtQ[n, 0] && LtQ[m, -1]) || (GtQ[-n, 0] && LeQ[-n, m + 1]))

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)}{3 x^3}+\frac {1}{3} \left (2 b^2\right ) \int \frac {e^{c+b^2 x^2} \text {erfi}(b x)}{x^2} \, dx+\frac {(2 b) \int \frac {e^{c+2 b^2 x^2}}{x^3} \, dx}{3 \sqrt {\pi }} \\ & = -\frac {b e^{c+2 b^2 x^2}}{3 \sqrt {\pi } x^2}-\frac {e^{c+b^2 x^2} \text {erfi}(b x)}{3 x^3}-\frac {2 b^2 e^{c+b^2 x^2} \text {erfi}(b x)}{3 x}+\frac {1}{3} \left (4 b^4\right ) \int e^{c+b^2 x^2} \text {erfi}(b x) \, dx+2 \frac {\left (4 b^3\right ) \int \frac {e^{c+2 b^2 x^2}}{x} \, dx}{3 \sqrt {\pi }} \\ & = -\frac {b e^{c+2 b^2 x^2}}{3 \sqrt {\pi } x^2}-\frac {e^{c+b^2 x^2} \text {erfi}(b x)}{3 x^3}-\frac {2 b^2 e^{c+b^2 x^2} \text {erfi}(b x)}{3 x}+\frac {4 b^3 e^c \operatorname {ExpIntegralEi}\left (2 b^2 x^2\right )}{3 \sqrt {\pi }}+\frac {1}{3} \left (2 b^3 e^c \sqrt {\pi }\right ) \text {Subst}(\int x \, dx,x,\text {erfi}(b x)) \\ & = -\frac {b e^{c+2 b^2 x^2}}{3 \sqrt {\pi } x^2}-\frac {e^{c+b^2 x^2} \text {erfi}(b x)}{3 x^3}-\frac {2 b^2 e^{c+b^2 x^2} \text {erfi}(b x)}{3 x}+\frac {1}{3} b^3 e^c \sqrt {\pi } \text {erfi}(b x)^2+\frac {4 b^3 e^c \operatorname {ExpIntegralEi}\left (2 b^2 x^2\right )}{3 \sqrt {\pi }} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.77 \[ \int \frac {e^{c+b^2 x^2} \text {erfi}(b x)}{x^4} \, dx=-\frac {e^c \left (e^{b^2 x^2} \sqrt {\pi } \left (1+2 b^2 x^2\right ) \text {erfi}(b x)-b^3 \pi x^3 \text {erfi}(b x)^2+b x \left (e^{2 b^2 x^2}-4 b^2 x^2 \operatorname {ExpIntegralEi}\left (2 b^2 x^2\right )\right )\right )}{3 \sqrt {\pi } x^3} \]

[In]

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

[Out]

-1/3*(E^c*(E^(b^2*x^2)*Sqrt[Pi]*(1 + 2*b^2*x^2)*Erfi[b*x] - b^3*Pi*x^3*Erfi[b*x]^2 + b*x*(E^(2*b^2*x^2) - 4*b^
2*x^2*ExpIntegralEi[2*b^2*x^2])))/(Sqrt[Pi]*x^3)

Maple [F]

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

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.72 \[ \int \frac {e^{c+b^2 x^2} \text {erfi}(b x)}{x^4} \, dx=-\frac {{\left ({\left (\pi + 2 \, \pi b^{2} x^{2}\right )} \operatorname {erfi}\left (b x\right ) e^{\left (b^{2} x^{2}\right )} - \sqrt {\pi } {\left (\pi b^{3} x^{3} \operatorname {erfi}\left (b x\right )^{2} + 4 \, b^{3} x^{3} {\rm Ei}\left (2 \, b^{2} x^{2}\right ) - b x e^{\left (2 \, b^{2} x^{2}\right )}\right )}\right )} e^{c}}{3 \, \pi x^{3}} \]

[In]

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

[Out]

-1/3*((pi + 2*pi*b^2*x^2)*erfi(b*x)*e^(b^2*x^2) - sqrt(pi)*(pi*b^3*x^3*erfi(b*x)^2 + 4*b^3*x^3*Ei(2*b^2*x^2) -
b*x*e^(2*b^2*x^2)))*e^c/(pi*x^3)

Sympy [F]

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

[In]

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

[Out]

exp(c)*Integral(exp(b**2*x**2)*erfi(b*x)/x**4, x)

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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