3.734 \(\int \frac{e^{x^2}}{x^2} \, dx\)

Optimal. Leaf size=19 \[ \sqrt{\pi } \text{Erfi}(x)-\frac{e^{x^2}}{x} \]

[Out]

-(E^x^2/x) + Sqrt[Pi]*Erfi[x]

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Rubi [A]  time = 0.0152378, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {2214, 2204} \[ \sqrt{\pi } \text{Erfi}(x)-\frac{e^{x^2}}{x} \]

Antiderivative was successfully verified.

[In]

Int[E^x^2/x^2,x]

[Out]

-(E^x^2/x) + Sqrt[Pi]*Erfi[x]

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]))

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

Mathematica [A]  time = 0.0043091, size = 19, normalized size = 1. \[ \sqrt{\pi } \text{Erfi}(x)-\frac{e^{x^2}}{x} \]

Antiderivative was successfully verified.

[In]

Integrate[E^x^2/x^2,x]

[Out]

-(E^x^2/x) + Sqrt[Pi]*Erfi[x]

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Maple [A]  time = 0.022, size = 17, normalized size = 0.9 \begin{align*} -{\frac{{{\rm e}^{{x}^{2}}}}{x}}+{\it erfi} \left ( x \right ) \sqrt{\pi } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(x^2)/x^2,x)

[Out]

-exp(x^2)/x+erfi(x)*Pi^(1/2)

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Maxima [A]  time = 1.05658, size = 26, normalized size = 1.37 \begin{align*} -\frac{\sqrt{-x^{2}} \Gamma \left (-\frac{1}{2}, -x^{2}\right )}{2 \, x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x^2)/x^2,x, algorithm="maxima")

[Out]

-1/2*sqrt(-x^2)*gamma(-1/2, -x^2)/x

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Fricas [A]  time = 0.801298, size = 46, normalized size = 2.42 \begin{align*} \frac{\sqrt{\pi } x \operatorname{erfi}\left (x\right ) - e^{\left (x^{2}\right )}}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x^2)/x^2,x, algorithm="fricas")

[Out]

(sqrt(pi)*x*erfi(x) - e^(x^2))/x

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Sympy [A]  time = 0.744633, size = 14, normalized size = 0.74 \begin{align*} \sqrt{\pi } \operatorname{erfi}{\left (x \right )} - \frac{e^{x^{2}}}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x**2)/x**2,x)

[Out]

sqrt(pi)*erfi(x) - exp(x**2)/x

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x^2)/x^2,x, algorithm="giac")

[Out]

integrate(e^(x^2)/x^2, x)