3.18.67 \(\int \frac {e^{x^2} (1-2 x^2)}{x^2} \, dx\)

Optimal. Leaf size=18 \[ \frac {37}{3}-\frac {e^{x^2}}{x}-\log (3) \]

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Rubi [A]  time = 0.03, antiderivative size = 10, normalized size of antiderivative = 0.56, number of steps used = 1, number of rules used = 1, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {2288} \begin {gather*} -\frac {e^{x^2}}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^x^2*(1 - 2*x^2))/x^2,x]

[Out]

-(E^x^2/x)

Rule 2288

Int[(y_.)*(F_)^(u_)*((v_) + (w_)), x_Symbol] :> With[{z = (v*y)/(Log[F]*D[u, x])}, Simp[F^u*z, x] /; EqQ[D[z,
x], w*y]] /; FreeQ[F, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=-\frac {e^{x^2}}{x}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 10, normalized size = 0.56 \begin {gather*} -\frac {e^{x^2}}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^x^2*(1 - 2*x^2))/x^2,x]

[Out]

-(E^x^2/x)

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fricas [A]  time = 0.83, size = 9, normalized size = 0.50 \begin {gather*} -\frac {e^{\left (x^{2}\right )}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x^2+1)*exp(x^2)/x^2,x, algorithm="fricas")

[Out]

-e^(x^2)/x

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giac [A]  time = 0.25, size = 9, normalized size = 0.50 \begin {gather*} -\frac {e^{\left (x^{2}\right )}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x^2+1)*exp(x^2)/x^2,x, algorithm="giac")

[Out]

-e^(x^2)/x

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maple [A]  time = 0.03, size = 10, normalized size = 0.56

method	result	size
gosper	\(-\frac {{\mathrm e}^{x^{2}}}{x}\)	\(10\)
default	\(-\frac {{\mathrm e}^{x^{2}}}{x}\)	\(10\)
norman	\(-\frac {{\mathrm e}^{x^{2}}}{x}\)	\(10\)
risch	\(-\frac {{\mathrm e}^{x^{2}}}{x}\)	\(10\)
meijerg	\(-\sqrt {\pi }\, \erfi \relax (x )+\frac {i \left (\frac {2 i {\mathrm e}^{x^{2}}}{x}-2 i \sqrt {\pi }\, \erfi \relax (x )\right )}{2}\)	\(31\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-2*x^2+1)*exp(x^2)/x^2,x,method=_RETURNVERBOSE)

[Out]

-exp(x^2)/x

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maxima [C]  time = 0.52, size = 29, normalized size = 1.61 \begin {gather*} i \, \sqrt {\pi } \operatorname {erf}\left (i \, x\right ) - \frac {\sqrt {-x^{2}} \Gamma \left (-\frac {1}{2}, -x^{2}\right )}{2 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x^2+1)*exp(x^2)/x^2,x, algorithm="maxima")

[Out]

I*sqrt(pi)*erf(I*x) - 1/2*sqrt(-x^2)*gamma(-1/2, -x^2)/x

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mupad [B]  time = 0.05, size = 9, normalized size = 0.50 \begin {gather*} -\frac {{\mathrm {e}}^{x^2}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(x^2)*(2*x^2 - 1))/x^2,x)

[Out]

-exp(x^2)/x

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sympy [A]  time = 0.08, size = 7, normalized size = 0.39 \begin {gather*} - \frac {e^{x^{2}}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x**2+1)*exp(x**2)/x**2,x)

[Out]

-exp(x**2)/x

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