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

Optimal. Leaf size=26 \[ -4+\frac {x}{\log \left (-1+e^{-x}-x\right ) \log \left (\frac {3}{\log (x)}\right )} \]

________________________________________________________________________________________

Rubi [F]  time = 5.54, 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 (-1+e^x (1+x)\right ) \log \left (e^{-x} \left (1+e^x (-1-x)\right )\right )+\left (\left (-x-e^x x\right ) \log (x)+\left (-1+e^x (1+x)\right ) \log \left (e^{-x} \left (1+e^x (-1-x)\right )\right ) \log (x)\right ) \log \left (\frac {3}{\log (x)}\right )}{\left (-1+e^x (1+x)\right ) \log ^2\left (e^{-x} \left (1+e^x (-1-x)\right )\right ) \log (x) \log ^2\left (\frac {3}{\log (x)}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]  time = 0.94, size = 24, normalized size = 0.92 \begin {gather*} \frac {x}{\log \left (-1+e^{-x}-x\right ) \log \left (\frac {3}{\log (x)}\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [A]  time = 0.65, size = 28, normalized size = 1.08 \begin {gather*} \frac {x}{\log \left (-{\left ({\left (x + 1\right )} e^{x} - 1\right )} e^{\left (-x\right )}\right ) \log \left (\frac {3}{\log \relax (x)}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x/(log(-((x + 1)*e^x - 1)*e^(-x))*log(3/log(x)))

________________________________________________________________________________________

giac [A]  time = 0.93, size = 48, normalized size = 1.85 \begin {gather*} -\frac {x}{x \log \relax (3) - \log \relax (3) \log \left (-x e^{x} - e^{x} + 1\right ) - x \log \left (\log \relax (x)\right ) + \log \left (-x e^{x} - e^{x} + 1\right ) \log \left (\log \relax (x)\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x/(x*log(3) - log(3)*log(-x*e^x - e^x + 1) - x*log(log(x)) + log(-x*e^x - e^x + 1)*log(log(x)))

________________________________________________________________________________________

maple [C]  time = 67.79, size = 184, normalized size = 7.08

method	result	size
risch	\(-\frac {4 x}{\left (2 i \ln \relax (3)-2 i \ln \left (\ln \relax (x )\right )\right ) \left (\pi \,\mathrm {csgn}\left (i {\mathrm e}^{-x}\right ) \mathrm {csgn}\left (i \left ({\mathrm e}^{x} x +{\mathrm e}^{x}-1\right )\right ) \mathrm {csgn}\left (i {\mathrm e}^{-x} \left ({\mathrm e}^{x} x +{\mathrm e}^{x}-1\right )\right )-\pi \,\mathrm {csgn}\left (i {\mathrm e}^{-x}\right ) \mathrm {csgn}\left (i {\mathrm e}^{-x} \left ({\mathrm e}^{x} x +{\mathrm e}^{x}-1\right )\right )^{2}+2 \pi \mathrm {csgn}\left (i {\mathrm e}^{-x} \left ({\mathrm e}^{x} x +{\mathrm e}^{x}-1\right )\right )^{2}-\pi \,\mathrm {csgn}\left (i \left ({\mathrm e}^{x} x +{\mathrm e}^{x}-1\right )\right ) \mathrm {csgn}\left (i {\mathrm e}^{-x} \left ({\mathrm e}^{x} x +{\mathrm e}^{x}-1\right )\right )^{2}-\pi \mathrm {csgn}\left (i {\mathrm e}^{-x} \left ({\mathrm e}^{x} x +{\mathrm e}^{x}-1\right )\right )^{3}-2 \pi +2 i \ln \left ({\mathrm e}^{x} x +{\mathrm e}^{x}-1\right )-2 i \ln \left ({\mathrm e}^{x}\right )\right )}\)	\(184\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 0.64, size = 36, normalized size = 1.38 \begin {gather*} -\frac {x}{x \log \relax (3) - {\left (\log \relax (3) - \log \left (\log \relax (x)\right )\right )} \log \left (-{\left (x + 1\right )} e^{x} + 1\right ) - x \log \left (\log \relax (x)\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x/(x*log(3) - (log(3) - log(log(x)))*log(-(x + 1)*e^x + 1) - x*log(log(x)))

________________________________________________________________________________________

mupad [B]  time = 2.16, size = 1156, normalized size = 44.46 result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((exp(2*x) - log(x) + x*exp(2*x) + exp(2*x)*log(x) + x*exp(x) - 2*x*exp(x)*log(x) - x^2*exp(x)*log(x) - 1)/(2*
log(x)*(exp(x) + 1)^2) + (log(3/log(x))*(exp(x) - exp(3*x) - exp(2*x) - 2*x*exp(2*x) - x*exp(3*x) - x*exp(x) +
x^2*exp(2*x)*log(x) - 4*x^2*exp(x)*log(x)^2 - x^3*exp(x)*log(x)^2 + 3*x*exp(x)*log(x) + x^3*exp(2*x)*log(x)^2
+ 4*x*exp(2*x)*log(x) + x*exp(3*x)*log(x) + x^2*exp(x)*log(x) + 1))/(2*log(x)*(exp(x) + 1)^3) - (x*log(3/log(
x))^2*log(x)*(4*exp(2*x) + exp(3*x) + 3*exp(x) + x*exp(2*x) + 3*exp(x)*log(x) + 4*exp(2*x)*log(x) + exp(3*x)*l
og(x) + x*exp(x) - x^2*exp(2*x)*log(x) + 5*x*exp(x)*log(x) + x*exp(2*x)*log(x) + x^2*exp(x)*log(x)))/(2*(exp(x
) + 1)^3))/log(3/log(x)) - log(x)/6 + ((exp(x) + x*exp(x) - 1)/(log(x)*(exp(x) + 1)) + (log(3/log(x))*(exp(2*x
)*log(x) - log(x) - x*exp(2*x) - exp(2*x) - x*exp(x) + 4*x*exp(x)*log(x) + 2*x*exp(2*x)*log(x) + x^2*exp(x)*lo
g(x) + 1))/(2*log(x)*(exp(x) + 1)^2) + (x*log(3/log(x))^2*exp(x)*log(x)*(x + exp(x) + 3))/(2*(exp(x) + 1)^2))/
log(3/log(x))^2 - log(3/log(x))*((4*x + 2*x^2)/(2*x + 2*x*exp(x)) - (6*x + 3*x^2)/(2*x + 2*x*exp(x)) + log(x)*
((4*x^2*exp(x) + 2*x^2*exp(2*x) + 2*x^2)/(2*x + 2*x*exp(2*x) + 4*x*exp(x)) - (6*x^2*exp(x) + 3*x^2*exp(2*x) +
3*x^2)/(2*x + 2*x*exp(2*x) + 4*x*exp(x)) + (x*(2*x + 2*x^2) - 4*x^2 - 2*x^3 + x*exp(x)*(2*x + 2*x^2))/(2*x + 2
*x*exp(2*x) + 4*x*exp(x)) - (x*(3*x + 3*x^2) - 6*x^2 - 3*x^3 + x*exp(x)*(3*x + 3*x^2))/(2*x + 2*x*exp(2*x) + 4
*x*exp(x))) - (2*x^2*exp(x) + 2*x^2)/(2*x + 2*x*exp(x)) + (3*x^2*exp(x) + 3*x^2)/(2*x + 2*x*exp(x)) + (2*x + x
^2)/(2*x + 2*x*exp(x)) - log(x)^2*((3*x^2*exp(x) + 3*x^2*exp(2*x) + x^2*exp(3*x) + x^2)/(2*x + 6*x*exp(2*x) +
2*x*exp(3*x) + 6*x*exp(x)) + (exp(x)*(x^2*(x - x^2 + 1) + x*(3*x^2 - 2*x + 3*x^3 + x*(x - x^2 + 1))) + x*(3*x^
2 - 2*x + 3*x^3 + x*(x - x^2 + 1)) - 4*x^3 - 2*x^4 + x^2*exp(2*x)*(x - x^2 + 1))/(2*x + 6*x*exp(2*x) + 2*x*exp
(3*x) + 6*x*exp(x))) - (x^2*exp(x) + x^2)/(2*x + 2*x*exp(x))) + (x/log(3/log(x)) - (log(-exp(-x)*(exp(x)*(x +
1) - 1))*(log(3/log(x))*log(x) + 1)*(exp(x) + x*exp(x) - 1))/(log(3/log(x))^2*log(x)*(exp(x) + 1)))/log(-exp(-
x)*(exp(x)*(x + 1) - 1)) + ((exp(x) + x*exp(x) - 1)/(2*(exp(x) + 1)) - (x*exp(x)*log(x)*(x + exp(x) + 3))/(2*(
exp(x) + 1)^2))/log(x) - (log(x)*(exp(2*x)*(2*x + x^2/2 + x^3/2 - 1/2) + exp(3*x)*(x/2 - 1/6) - exp(x)*((3*x^2
)/2 - (3*x)/2 + x^3/2 + 1/2) - 1/6))/(3*exp(2*x) + exp(3*x) + 3*exp(x) + 1)

________________________________________________________________________________________

sympy [A]  time = 1.74, size = 22, normalized size = 0.85 \begin {gather*} \frac {x}{\log {\left (\left (\left (- x - 1\right ) e^{x} + 1\right ) e^{- x} \right )} \log {\left (\frac {3}{\log {\relax (x )}} \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x/(log(((-x - 1)*exp(x) + 1)*exp(-x))*log(3/log(x)))

________________________________________________________________________________________