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

Optimal. Leaf size=26 \[ \frac {1}{2} \left (-x+\log \left (x \left (e^x+x\right )\right )\right ) \left (-1-x+\log ^2(\log (x))\right ) \]

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Rubi [F]  time = 3.32, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {\left (-2 x-x^2+2 x^3+e^x \left (-1-x+x^2\right )\right ) \log (x)+\left (-e^x x-x^2\right ) \log (x) \log \left (e^x x+x^2\right )+\left (-2 e^x x-2 x^2+\left (2 e^x+2 x\right ) \log \left (e^x x+x^2\right )\right ) \log (\log (x))+\left (e^x+2 x-x^2\right ) \log (x) \log ^2(\log (x))}{\left (2 e^x x+2 x^2\right ) \log (x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {\left (-2 x-x^2+2 x^3+e^x \left (-1-x+x^2\right )\right ) \log (x)+\left (-e^x x-x^2\right ) \log (x) \log \left (e^x x+x^2\right )+\left (-2 e^x x-2 x^2+\left (2 e^x+2 x\right ) \log \left (e^x x+x^2\right )\right ) \log (\log (x))+\left (e^x+2 x-x^2\right ) \log (x) \log ^2(\log (x))}{2 x \left (e^x+x\right ) \log (x)} \, dx\\ &=\frac {1}{2} \int \frac {\left (-2 x-x^2+2 x^3+e^x \left (-1-x+x^2\right )\right ) \log (x)+\left (-e^x x-x^2\right ) \log (x) \log \left (e^x x+x^2\right )+\left (-2 e^x x-2 x^2+\left (2 e^x+2 x\right ) \log \left (e^x x+x^2\right )\right ) \log (\log (x))+\left (e^x+2 x-x^2\right ) \log (x) \log ^2(\log (x))}{x \left (e^x+x\right ) \log (x)} \, dx\\ &=\frac {1}{2} \int \frac {-\frac {2 \left (x-\log \left (x \left (e^x+x\right )\right )\right ) \log (\log (x))}{\log (x)}-\frac {e^x+2 x+e^x x+x^2-e^x x^2-2 x^3+x \left (e^x+x\right ) \log \left (x \left (e^x+x\right )\right )+\left (-e^x+(-2+x) x\right ) \log ^2(\log (x))}{e^x+x}}{x} \, dx\\ &=\frac {1}{2} \int \left (\frac {(-1+x) \left (1+x-\log ^2(\log (x))\right )}{e^x+x}+\frac {-\log (x)-x \log (x)+x^2 \log (x)-x \log (x) \log \left (x \left (e^x+x\right )\right )-2 x \log (\log (x))+2 \log \left (x \left (e^x+x\right )\right ) \log (\log (x))+\log (x) \log ^2(\log (x))}{x \log (x)}\right ) \, dx\\ &=\frac {1}{2} \int \frac {(-1+x) \left (1+x-\log ^2(\log (x))\right )}{e^x+x} \, dx+\frac {1}{2} \int \frac {-\log (x)-x \log (x)+x^2 \log (x)-x \log (x) \log \left (x \left (e^x+x\right )\right )-2 x \log (\log (x))+2 \log \left (x \left (e^x+x\right )\right ) \log (\log (x))+\log (x) \log ^2(\log (x))}{x \log (x)} \, dx\\ &=\frac {1}{2} \int \left (\frac {-1-x+x^2-x \log \left (x \left (e^x+x\right )\right )}{x}-\frac {2 \left (x-\log \left (x \left (e^x+x\right )\right )\right ) \log (\log (x))}{x \log (x)}+\frac {\log ^2(\log (x))}{x}\right ) \, dx+\frac {1}{2} \int \left (-\frac {1+x-\log ^2(\log (x))}{e^x+x}+\frac {x \left (1+x-\log ^2(\log (x))\right )}{e^x+x}\right ) \, dx\\ &=\frac {1}{2} \int \frac {-1-x+x^2-x \log \left (x \left (e^x+x\right )\right )}{x} \, dx+\frac {1}{2} \int \frac {\log ^2(\log (x))}{x} \, dx-\frac {1}{2} \int \frac {1+x-\log ^2(\log (x))}{e^x+x} \, dx+\frac {1}{2} \int \frac {x \left (1+x-\log ^2(\log (x))\right )}{e^x+x} \, dx-\int \frac {\left (x-\log \left (x \left (e^x+x\right )\right )\right ) \log (\log (x))}{x \log (x)} \, dx\\ &=\frac {1}{2} \int \left (\frac {-1-x+x^2}{x}-\log \left (e^x x+x^2\right )\right ) \, dx-\frac {1}{2} \int \left (\frac {1}{e^x+x}+\frac {x}{e^x+x}-\frac {\log ^2(\log (x))}{e^x+x}\right ) \, dx+\frac {1}{2} \int \left (\frac {x}{e^x+x}+\frac {x^2}{e^x+x}-\frac {x \log ^2(\log (x))}{e^x+x}\right ) \, dx+\frac {1}{2} \operatorname {Subst}\left (\int \log ^2(x) \, dx,x,\log (x)\right )-\int \left (\frac {\log (\log (x))}{\log (x)}-\frac {\log \left (e^x x+x^2\right ) \log (\log (x))}{x \log (x)}\right ) \, dx\\ &=\frac {1}{2} \log (x) \log ^2(\log (x))-\frac {1}{2} \int \frac {1}{e^x+x} \, dx+\frac {1}{2} \int \frac {x^2}{e^x+x} \, dx+\frac {1}{2} \int \frac {-1-x+x^2}{x} \, dx-\frac {1}{2} \int \log \left (e^x x+x^2\right ) \, dx+\frac {1}{2} \int \frac {\log ^2(\log (x))}{e^x+x} \, dx-\frac {1}{2} \int \frac {x \log ^2(\log (x))}{e^x+x} \, dx-\int \frac {\log (\log (x))}{\log (x)} \, dx+\int \frac {\log \left (e^x x+x^2\right ) \log (\log (x))}{x \log (x)} \, dx-\operatorname {Subst}(\int \log (x) \, dx,x,\log (x))\\ &=\log (x)-\frac {1}{2} x \log \left (e^x x+x^2\right )-\log (x) \log (\log (x))+\frac {1}{2} \log (x) \log ^2(\log (x))-\frac {1}{2} \int \frac {1}{e^x+x} \, dx+\frac {1}{2} \int \frac {x^2}{e^x+x} \, dx+\frac {1}{2} \int \left (-1-\frac {1}{x}+x\right ) \, dx+\frac {1}{2} \int \frac {2 x+e^x (1+x)}{e^x+x} \, dx+\frac {1}{2} \int \frac {\log ^2(\log (x))}{e^x+x} \, dx-\frac {1}{2} \int \frac {x \log ^2(\log (x))}{e^x+x} \, dx-\int \frac {\log (\log (x))}{\log (x)} \, dx+\int \frac {\log \left (e^x x+x^2\right ) \log (\log (x))}{x \log (x)} \, dx\\ &=-\frac {x}{2}+\frac {x^2}{4}+\frac {\log (x)}{2}-\frac {1}{2} x \log \left (e^x x+x^2\right )-\log (x) \log (\log (x))+\frac {1}{2} \log (x) \log ^2(\log (x))-\frac {1}{2} \int \frac {1}{e^x+x} \, dx+\frac {1}{2} \int \frac {x^2}{e^x+x} \, dx+\frac {1}{2} \int \left (1+x-\frac {(-1+x) x}{e^x+x}\right ) \, dx+\frac {1}{2} \int \frac {\log ^2(\log (x))}{e^x+x} \, dx-\frac {1}{2} \int \frac {x \log ^2(\log (x))}{e^x+x} \, dx-\int \frac {\log (\log (x))}{\log (x)} \, dx+\int \frac {\log \left (e^x x+x^2\right ) \log (\log (x))}{x \log (x)} \, dx\\ &=\frac {x^2}{2}+\frac {\log (x)}{2}-\frac {1}{2} x \log \left (e^x x+x^2\right )-\log (x) \log (\log (x))+\frac {1}{2} \log (x) \log ^2(\log (x))-\frac {1}{2} \int \frac {1}{e^x+x} \, dx-\frac {1}{2} \int \frac {(-1+x) x}{e^x+x} \, dx+\frac {1}{2} \int \frac {x^2}{e^x+x} \, dx+\frac {1}{2} \int \frac {\log ^2(\log (x))}{e^x+x} \, dx-\frac {1}{2} \int \frac {x \log ^2(\log (x))}{e^x+x} \, dx-\int \frac {\log (\log (x))}{\log (x)} \, dx+\int \frac {\log \left (e^x x+x^2\right ) \log (\log (x))}{x \log (x)} \, dx\\ &=\frac {x^2}{2}+\frac {\log (x)}{2}-\frac {1}{2} x \log \left (e^x x+x^2\right )-\log (x) \log (\log (x))+\frac {1}{2} \log (x) \log ^2(\log (x))-\frac {1}{2} \int \frac {1}{e^x+x} \, dx+\frac {1}{2} \int \frac {x^2}{e^x+x} \, dx-\frac {1}{2} \int \left (-\frac {x}{e^x+x}+\frac {x^2}{e^x+x}\right ) \, dx+\frac {1}{2} \int \frac {\log ^2(\log (x))}{e^x+x} \, dx-\frac {1}{2} \int \frac {x \log ^2(\log (x))}{e^x+x} \, dx-\int \frac {\log (\log (x))}{\log (x)} \, dx+\int \frac {\log \left (e^x x+x^2\right ) \log (\log (x))}{x \log (x)} \, dx\\ &=\frac {x^2}{2}+\frac {\log (x)}{2}-\frac {1}{2} x \log \left (e^x x+x^2\right )-\log (x) \log (\log (x))+\frac {1}{2} \log (x) \log ^2(\log (x))-\frac {1}{2} \int \frac {1}{e^x+x} \, dx+\frac {1}{2} \int \frac {x}{e^x+x} \, dx+\frac {1}{2} \int \frac {\log ^2(\log (x))}{e^x+x} \, dx-\frac {1}{2} \int \frac {x \log ^2(\log (x))}{e^x+x} \, dx-\int \frac {\log (\log (x))}{\log (x)} \, dx+\int \frac {\log \left (e^x x+x^2\right ) \log (\log (x))}{x \log (x)} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.25, size = 51, normalized size = 1.96 \begin {gather*} \frac {1}{2} \left (x+x^2-\log (x)-\log \left (e^x+x\right )-x \log \left (x \left (e^x+x\right )\right )-\left (x-\log \left (x \left (e^x+x\right )\right )\right ) \log ^2(\log (x))\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(x + x^2 - Log[x] - Log[E^x + x] - x*Log[x*(E^x + x)] - (x - Log[x*(E^x + x)])*Log[Log[x]]^2)/2

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fricas [A]  time = 1.01, size = 43, normalized size = 1.65 \begin {gather*} -\frac {1}{2} \, {\left (x - \log \left (x^{2} + x e^{x}\right )\right )} \log \left (\log \relax (x)\right )^{2} + \frac {1}{2} \, x^{2} - \frac {1}{2} \, {\left (x + 1\right )} \log \left (x^{2} + x e^{x}\right ) + \frac {1}{2} \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(x - log(x^2 + x*e^x))*log(log(x))^2 + 1/2*x^2 - 1/2*(x + 1)*log(x^2 + x*e^x) + 1/2*x

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giac [B]  time = 0.24, size = 62, normalized size = 2.38 \begin {gather*} -\frac {1}{2} \, x \log \left (\log \relax (x)\right )^{2} + \frac {1}{2} \, \log \left (x + e^{x}\right ) \log \left (\log \relax (x)\right )^{2} + \frac {1}{2} \, \log \relax (x) \log \left (\log \relax (x)\right )^{2} + \frac {1}{2} \, x^{2} - \frac {1}{2} \, x \log \left (x + e^{x}\right ) - \frac {1}{2} \, x \log \relax (x) + \frac {1}{2} \, x - \frac {1}{2} \, \log \left (x + e^{x}\right ) - \frac {1}{2} \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*x*log(log(x))^2 + 1/2*log(x + e^x)*log(log(x))^2 + 1/2*log(x)*log(log(x))^2 + 1/2*x^2 - 1/2*x*log(x + e^x
) - 1/2*x*log(x) + 1/2*x - 1/2*log(x + e^x) - 1/2*log(x)

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maple [C]  time = 0.13, size = 252, normalized size = 9.69

method	result	size
risch	\(\left (\frac {\ln \left (\ln \relax (x )\right )^{2}}{2}-\frac {x}{2}\right ) \ln \left ({\mathrm e}^{x}+x \right )-\frac {x \ln \left (\ln \relax (x )\right )^{2}}{2}+\frac {\ln \relax (x ) \ln \left (\ln \relax (x )\right )^{2}}{2}+\frac {i x \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i \left ({\mathrm e}^{x}+x \right )\right ) \mathrm {csgn}\left (i x \left ({\mathrm e}^{x}+x \right )\right )}{4}+\frac {i \ln \left (\ln \relax (x )\right )^{2} \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x \left ({\mathrm e}^{x}+x \right )\right )^{2}}{4}+\frac {i \ln \left (\ln \relax (x )\right )^{2} \pi \,\mathrm {csgn}\left (i \left ({\mathrm e}^{x}+x \right )\right ) \mathrm {csgn}\left (i x \left ({\mathrm e}^{x}+x \right )\right )^{2}}{4}-\frac {i \ln \left (\ln \relax (x )\right )^{2} \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i \left ({\mathrm e}^{x}+x \right )\right ) \mathrm {csgn}\left (i x \left ({\mathrm e}^{x}+x \right )\right )}{4}-\frac {x \ln \relax (x )}{2}+\frac {i x \pi \mathrm {csgn}\left (i x \left ({\mathrm e}^{x}+x \right )\right )^{3}}{4}-\frac {i x \pi \,\mathrm {csgn}\left (i \left ({\mathrm e}^{x}+x \right )\right ) \mathrm {csgn}\left (i x \left ({\mathrm e}^{x}+x \right )\right )^{2}}{4}-\frac {i x \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x \left ({\mathrm e}^{x}+x \right )\right )^{2}}{4}-\frac {i \ln \left (\ln \relax (x )\right )^{2} \pi \mathrm {csgn}\left (i x \left ({\mathrm e}^{x}+x \right )\right )^{3}}{4}+\frac {x^{2}}{2}-\frac {\ln \relax (x )}{2}+\frac {x}{2}-\frac {\ln \left ({\mathrm e}^{x}+x \right )}{2}\)	\(252\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(1/2*ln(ln(x))^2-1/2*x)*ln(exp(x)+x)-1/2*x*ln(ln(x))^2+1/2*ln(x)*ln(ln(x))^2+1/4*I*x*Pi*csgn(I*x)*csgn(I*(exp(
x)+x))*csgn(I*x*(exp(x)+x))+1/4*I*ln(ln(x))^2*Pi*csgn(I*x)*csgn(I*x*(exp(x)+x))^2+1/4*I*ln(ln(x))^2*Pi*csgn(I*
(exp(x)+x))*csgn(I*x*(exp(x)+x))^2-1/4*I*ln(ln(x))^2*Pi*csgn(I*x)*csgn(I*(exp(x)+x))*csgn(I*x*(exp(x)+x))-1/2*
x*ln(x)+1/4*I*x*Pi*csgn(I*x*(exp(x)+x))^3-1/4*I*x*Pi*csgn(I*(exp(x)+x))*csgn(I*x*(exp(x)+x))^2-1/4*I*x*Pi*csgn
(I*x)*csgn(I*x*(exp(x)+x))^2-1/4*I*ln(ln(x))^2*Pi*csgn(I*x*(exp(x)+x))^3+1/2*x^2-1/2*ln(x)+1/2*x-1/2*ln(exp(x)
+x)

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maxima [A]  time = 0.42, size = 46, normalized size = 1.77 \begin {gather*} -\frac {1}{2} \, {\left (x - \log \relax (x)\right )} \log \left (\log \relax (x)\right )^{2} + \frac {1}{2} \, x^{2} + \frac {1}{2} \, {\left (\log \left (\log \relax (x)\right )^{2} - x - 1\right )} \log \left (x + e^{x}\right ) - \frac {1}{2} \, {\left (x + 1\right )} \log \relax (x) + \frac {1}{2} \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(x - log(x))*log(log(x))^2 + 1/2*x^2 + 1/2*(log(log(x))^2 - x - 1)*log(x + e^x) - 1/2*(x + 1)*log(x) + 1/
2*x

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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int -\frac {-\ln \relax (x)\,\left (2\,x+{\mathrm {e}}^x-x^2\right )\,{\ln \left (\ln \relax (x)\right )}^2+\left (2\,x\,{\mathrm {e}}^x-\ln \left (x\,{\mathrm {e}}^x+x^2\right )\,\left (2\,x+2\,{\mathrm {e}}^x\right )+2\,x^2\right )\,\ln \left (\ln \relax (x)\right )+\ln \relax (x)\,\left (2\,x+{\mathrm {e}}^x\,\left (-x^2+x+1\right )+x^2-2\,x^3\right )+\ln \left (x\,{\mathrm {e}}^x+x^2\right )\,\ln \relax (x)\,\left (x\,{\mathrm {e}}^x+x^2\right )}{\ln \relax (x)\,\left (2\,x\,{\mathrm {e}}^x+2\,x^2\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [B]  time = 1.74, size = 53, normalized size = 2.04 \begin {gather*} \frac {x^{2}}{2} - \frac {x \log {\left (\log {\relax (x )} \right )}^{2}}{2} + \frac {x}{2} + \left (- \frac {x}{2} + \frac {\log {\left (\log {\relax (x )} \right )}^{2}}{2}\right ) \log {\left (x^{2} + x e^{x} \right )} - \frac {\log {\relax (x )}}{2} - \frac {\log {\left (x + e^{x} \right )}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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