\(\int \frac {e^{x^2}}{x^2} \, dx\) [734]

Optimal result

Integrand size = 9, antiderivative size = 19 \[ \int \frac {e^{x^2}}{x^2} \, dx=-\frac {e^{x^2}}{x}+\sqrt {\pi } \text {erfi}(x) \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2245, 2235} \[ \int \frac {e^{x^2}}{x^2} \, dx=\sqrt {\pi } \text {erfi}(x)-\frac {e^{x^2}}{x} \]

[In]

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

[Out]

-(E^x^2/x) + Sqrt[Pi]*Erfi[x]

Rule 2235

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]

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]))

Rubi steps \begin{align*} \text {integral}& = -\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] (verified)

Time = 0.02 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00 \[ \int \frac {e^{x^2}}{x^2} \, dx=-\frac {e^{x^2}}{x}+\sqrt {\pi } \text {erfi}(x) \]

[In]

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

[Out]

-(E^x^2/x) + Sqrt[Pi]*Erfi[x]

Maple [A] (verified)

Time = 0.02 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.89

method	result	size
default	\(-\frac {{\mathrm e}^{x^{2}}}{x}+\operatorname {erfi}\left (x \right ) \sqrt {\pi }\)	\(17\)
risch	\(-\frac {{\mathrm e}^{x^{2}}}{x}+\operatorname {erfi}\left (x \right ) \sqrt {\pi }\)	\(17\)
meijerg	\(\frac {i \left (\frac {2 i {\mathrm e}^{x^{2}}}{x}-2 i \operatorname {erfi}\left (x \right ) \sqrt {\pi }\right )}{2}\)	\(23\)

[In]

int(exp(x^2)/x^2,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.95 \[ \int \frac {e^{x^2}}{x^2} \, dx=\frac {\sqrt {\pi } x \operatorname {erfi}\left (x\right ) - e^{\left (x^{2}\right )}}{x} \]

[In]

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

[Out]

(sqrt(pi)*x*erfi(x) - e^(x^2))/x

Sympy [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.74 \[ \int \frac {e^{x^2}}{x^2} \, dx=\sqrt {\pi } \operatorname {erfi}{\left (x \right )} - \frac {e^{x^{2}}}{x} \]

[In]

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

[Out]

sqrt(pi)*erfi(x) - exp(x**2)/x

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00 \[ \int \frac {e^{x^2}}{x^2} \, dx=-\frac {\sqrt {-x^{2}} \Gamma \left (-\frac {1}{2}, -x^{2}\right )}{2 \, x} \]

[In]

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

[Out]

-1/2*sqrt(-x^2)*gamma(-1/2, -x^2)/x

Giac [F]

\[ \int \frac {e^{x^2}}{x^2} \, dx=\int { \frac {e^{\left (x^{2}\right )}}{x^{2}} \,d x } \]

[In]

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

[Out]

integrate(e^(x^2)/x^2, x)

Mupad [B] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.11 \[ \int \frac {e^{x^2}}{x^2} \, dx=-\frac {{\mathrm {e}}^{x^2}}{x}+\sqrt {\pi }\,\mathrm {erfc}\left (x\,1{}\mathrm {i}\right )\,1{}\mathrm {i} \]

[In]

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

[Out]

pi^(1/2)*erfc(x*1i)*1i - exp(x^2)/x