3.293 \(\int \frac{e^{c+b^2 x^2} \text{Erfi}(b x)}{x^4} \, dx\)

Optimal. Leaf size=118 \[ -\frac{2 b^2 e^{b^2 x^2+c} \text{Erfi}(b x)}{3 x}-\frac{e^{b^2 x^2+c} \text{Erfi}(b x)}{3 x^3}+\frac{1}{3} \sqrt{\pi } b^3 e^c \text{Erfi}(b x)^2+\frac{4 b^3 e^c \text{ExpIntegralEi}\left (2 b^2 x^2\right )}{3 \sqrt{\pi }}-\frac{b e^{2 b^2 x^2+c}}{3 \sqrt{\pi } x^2} \]

[Out]

-(b*E^(c + 2*b^2*x^2))/(3*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])

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Rubi [A]  time = 0.172764, antiderivative size = 118, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263, Rules used = {6393, 6375, 30, 2210, 2214} \[ -\frac{2 b^2 e^{b^2 x^2+c} \text{Erfi}(b x)}{3 x}-\frac{e^{b^2 x^2+c} \text{Erfi}(b x)}{3 x^3}+\frac{1}{3} \sqrt{\pi } b^3 e^c \text{Erfi}(b x)^2+\frac{4 b^3 e^c \text{Ei}\left (2 b^2 x^2\right )}{3 \sqrt{\pi }}-\frac{b e^{2 b^2 x^2+c}}{3 \sqrt{\pi } x^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(b*E^(c + 2*b^2*x^2))/(3*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 6393

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]

Rule 6375

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 30

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

Rule 2210

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 2214

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]))

Rubi steps

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

Mathematica [A]  time = 0.0357399, size = 91, normalized size = 0.77 \[ -\frac{e^c \left (\pi \left (-b^3\right ) x^3 \text{Erfi}(b x)^2+\sqrt{\pi } e^{b^2 x^2} \left (2 b^2 x^2+1\right ) \text{Erfi}(b x)+b x \left (e^{2 b^2 x^2}-4 b^2 x^2 \text{ExpIntegralEi}\left (2 b^2 x^2\right )\right )\right )}{3 \sqrt{\pi } x^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(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])))/(3*Sqrt[Pi]*x^3)

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Maple [F]  time = 0.279, size = 0, normalized size = 0. \begin{align*} \int{\frac{{{\rm e}^{{b}^{2}{x}^{2}+c}}{\it erfi} \left ( bx \right ) }{{x}^{4}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{erfi}\left (b x\right ) e^{\left (b^{2} x^{2} + c\right )}}{x^{4}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [A]  time = 2.65977, size = 196, normalized size = 1.66 \begin{align*} -\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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} e^{c} \int \frac{e^{b^{2} x^{2}} \operatorname{erfi}{\left (b x \right )}}{x^{4}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{erfi}\left (b x\right ) e^{\left (b^{2} x^{2} + c\right )}}{x^{4}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)