3.52.77 \(\int \frac {e^{-e^{e^3+x}-2 x+x^2 \log (x)} (-1-2 x-e^{e^3+x} x+x^2+2 x^2 \log (x))}{x^2} \, dx\)

Optimal. Leaf size=27 \[ \frac {e^{-e^{e^3+x}+x (-2+x \log (x))}}{x}+\log (5) \]

________________________________________________________________________________________

Rubi [B]  time = 0.92, antiderivative size = 67, normalized size of antiderivative = 2.48, number of steps used = 1, number of rules used = 1, integrand size = 50, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.020, Rules used = {2288} \begin {gather*} \frac {e^{-2 x-e^{x+e^3}} x^{x^2-2} \left (-x^2-2 x^2 \log (x)+e^{x+e^3} x+2 x\right )}{-x+e^{x+e^3}-2 x \log (x)+2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^(-E^(E^3 + x) - 2*x + x^2*Log[x])*(-1 - 2*x - E^(E^3 + x)*x + x^2 + 2*x^2*Log[x]))/x^2,x]

[Out]

(E^(-E^(E^3 + x) - 2*x)*x^(-2 + x^2)*(2*x + E^(E^3 + x)*x - x^2 - 2*x^2*Log[x]))/(2 + E^(E^3 + x) - x - 2*x*Lo
g[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^{-e^{e^3+x}-2 x} x^{-2+x^2} \left (2 x+e^{e^3+x} x-x^2-2 x^2 \log (x)\right )}{2+e^{e^3+x}-x-2 x \log (x)}\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.06, size = 23, normalized size = 0.85 \begin {gather*} e^{-e^{e^3+x}-2 x} x^{-1+x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(-E^(E^3 + x) - 2*x + x^2*Log[x])*(-1 - 2*x - E^(E^3 + x)*x + x^2 + 2*x^2*Log[x]))/x^2,x]

[Out]

E^(-E^(E^3 + x) - 2*x)*x^(-1 + x^2)

________________________________________________________________________________________

fricas [A]  time = 0.56, size = 22, normalized size = 0.81 \begin {gather*} \frac {e^{\left (x^{2} \log \relax (x) - 2 \, x - e^{\left (x + e^{3}\right )}\right )}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^2*log(x)-x*exp(exp(3)+x)+x^2-2*x-1)*exp(x^2*log(x)-exp(exp(3)+x)-2*x)/x^2,x, algorithm="fricas"
)

[Out]

e^(x^2*log(x) - 2*x - e^(x + e^3))/x

________________________________________________________________________________________

giac [A]  time = 0.23, size = 22, normalized size = 0.81 \begin {gather*} \frac {e^{\left (x^{2} \log \relax (x) - 2 \, x - e^{\left (x + e^{3}\right )}\right )}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^2*log(x)-x*exp(exp(3)+x)+x^2-2*x-1)*exp(x^2*log(x)-exp(exp(3)+x)-2*x)/x^2,x, algorithm="giac")

[Out]

e^(x^2*log(x) - 2*x - e^(x + e^3))/x

________________________________________________________________________________________

maple [A]  time = 0.06, size = 22, normalized size = 0.81

method	result	size
risch	\(\frac {x^{x^{2}} {\mathrm e}^{-{\mathrm e}^{{\mathrm e}^{3}+x}-2 x}}{x}\)	\(22\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x^2*ln(x)-x*exp(exp(3)+x)+x^2-2*x-1)*exp(x^2*ln(x)-exp(exp(3)+x)-2*x)/x^2,x,method=_RETURNVERBOSE)

[Out]

1/x*x^(x^2)*exp(-exp(exp(3)+x)-2*x)

________________________________________________________________________________________

maxima [A]  time = 0.45, size = 22, normalized size = 0.81 \begin {gather*} \frac {e^{\left (x^{2} \log \relax (x) - 2 \, x - e^{\left (x + e^{3}\right )}\right )}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^2*log(x)-x*exp(exp(3)+x)+x^2-2*x-1)*exp(x^2*log(x)-exp(exp(3)+x)-2*x)/x^2,x, algorithm="maxima"
)

[Out]

e^(x^2*log(x) - 2*x - e^(x + e^3))/x

________________________________________________________________________________________

mupad [B]  time = 3.26, size = 20, normalized size = 0.74 \begin {gather*} x^{x^2-1}\,{\mathrm {e}}^{-2\,x}\,{\mathrm {e}}^{-{\mathrm {e}}^{{\mathrm {e}}^3}\,{\mathrm {e}}^x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(x^2*log(x) - exp(x + exp(3)) - 2*x)*(2*x - 2*x^2*log(x) + x*exp(x + exp(3)) - x^2 + 1))/x^2,x)

[Out]

x^(x^2 - 1)*exp(-2*x)*exp(-exp(exp(3))*exp(x))

________________________________________________________________________________________

sympy [A]  time = 0.30, size = 19, normalized size = 0.70 \begin {gather*} \frac {e^{x^{2} \log {\relax (x )} - 2 x - e^{x + e^{3}}}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x**2*ln(x)-x*exp(exp(3)+x)+x**2-2*x-1)*exp(x**2*ln(x)-exp(exp(3)+x)-2*x)/x**2,x)

[Out]

exp(x**2*log(x) - 2*x - exp(x + exp(3)))/x

________________________________________________________________________________________