3.726 \(\int (\frac{e^{(a+x)^2}}{x^2}-\frac{2 a e^{(a+x)^2}}{x}) \, dx\)

Optimal. Leaf size=23 \[ \sqrt{\pi } \text{Erfi}(a+x)-\frac{e^{(a+x)^2}}{x} \]

[Out]

-(E^(a + x)^2/x) + Sqrt[Pi]*Erfi[a + x]

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Rubi [A]  time = 0.0481148, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08, Rules used = {2220, 2204} \[ \sqrt{\pi } \text{Erfi}(a+x)-\frac{e^{(a+x)^2}}{x} \]

Antiderivative was successfully verified.

[In]

Int[E^(a + x)^2/x^2 - (2*a*E^(a + x)^2)/x,x]

[Out]

-(E^(a + x)^2/x) + Sqrt[Pi]*Erfi[a + x]

Rule 2220

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2)*((e_.) + (f_.)*(x_))^(m_), x_Symbol] :> Simp[(f*(e + f*x)^(m +
1)*F^(a + b*(c + d*x)^2))/((m + 1)*f^2), x] + (-Dist[(2*b*d^2*Log[F])/(f^2*(m + 1)), Int[(e + f*x)^(m + 2)*F^
(a + b*(c + d*x)^2), x], x] + Dist[(2*b*d*(d*e - c*f)*Log[F])/(f^2*(m + 1)), Int[(e + f*x)^(m + 1)*F^(a + b*(c
+ d*x)^2), x], x]) /; FreeQ[{F, a, b, c, d, e, f}, x] && NeQ[d*e - c*f, 0] && LtQ[m, -1]

Rule 2204

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[(F^a*Sqrt[Pi]*Erfi[(c + d*x)*Rt[b*Log[F], 2
]])/(2*d*Rt[b*Log[F], 2]), x] /; FreeQ[{F, a, b, c, d}, x] && PosQ[b]

Rubi steps

\begin{align*} \int \left (\frac{e^{(a+x)^2}}{x^2}-\frac{2 a e^{(a+x)^2}}{x}\right ) \, dx &=-\left ((2 a) \int \frac{e^{(a+x)^2}}{x} \, dx\right )+\int \frac{e^{(a+x)^2}}{x^2} \, dx\\ &=-\frac{e^{(a+x)^2}}{x}+2 \int e^{(a+x)^2} \, dx\\ &=-\frac{e^{(a+x)^2}}{x}+\sqrt{\pi } \text{erfi}(a+x)\\ \end{align*}

Mathematica [A]  time = 0.0752959, size = 23, normalized size = 1. \[ \sqrt{\pi } \text{Erfi}(a+x)-\frac{e^{(a+x)^2}}{x} \]

Antiderivative was successfully verified.

[In]

Integrate[E^(a + x)^2/x^2 - (2*a*E^(a + x)^2)/x,x]

[Out]

-(E^(a + x)^2/x) + Sqrt[Pi]*Erfi[a + x]

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Maple [F]  time = 0.137, size = 0, normalized size = 0. \begin{align*} \int{\frac{{{\rm e}^{ \left ( a+x \right ) ^{2}}}}{{x}^{2}}}-2\,{\frac{a{{\rm e}^{ \left ( a+x \right ) ^{2}}}}{x}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp((a+x)^2)/x^2-2*a*exp((a+x)^2)/x,x)

[Out]

int(exp((a+x)^2)/x^2-2*a*exp((a+x)^2)/x,x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp((a+x)^2)/x^2-2*a*exp((a+x)^2)/x,x, algorithm="maxima")

[Out]

integrate(-2*a*e^((a + x)^2)/x + e^((a + x)^2)/x^2, x)

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Fricas [A]  time = 0.733647, size = 70, normalized size = 3.04 \begin{align*} \frac{\sqrt{\pi } x \operatorname{erfi}\left (a + x\right ) - e^{\left (a^{2} + 2 \, a x + x^{2}\right )}}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp((a+x)^2)/x^2-2*a*exp((a+x)^2)/x,x, algorithm="fricas")

[Out]

(sqrt(pi)*x*erfi(a + x) - e^(a^2 + 2*a*x + x^2))/x

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} - \left (\int - \frac{e^{x^{2}} e^{2 a x}}{x^{2}}\, dx + \int \frac{2 a e^{x^{2}} e^{2 a x}}{x}\, dx\right ) e^{a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp((a+x)**2)/x**2-2*a*exp((a+x)**2)/x,x)

[Out]

-(Integral(-exp(x**2)*exp(2*a*x)/x**2, x) + Integral(2*a*exp(x**2)*exp(2*a*x)/x, x))*exp(a**2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp((a+x)^2)/x^2-2*a*exp((a+x)^2)/x,x, algorithm="giac")

[Out]

integrate(-2*a*e^((a + x)^2)/x + e^((a + x)^2)/x^2, x)