\(\int \frac {4+4 \log (\frac {5 \log (3)}{e^2 x})+(-4-8 x) \log ^2(\frac {5 \log (3)}{e^2 x})+(-2-2 \log (\frac {5 \log (3)}{e^2 x})+(2+4 x) \log ^2(\frac {5 \log (3)}{e^2 x})) \log (\frac {x+(-x-x^2) \log (\frac {5 \log (3)}{e^2 x})}{\log (\frac {5 \log (3)}{e^2 x})})}{-x \log (\frac {5 \log (3)}{e^2 x})+(x+x^2) \log ^2(\frac {5 \log (3)}{e^2 x})} \, dx\) [1116]

Optimal result

Integrand size = 144, antiderivative size = 31 \[ \int \frac {4+4 \log \left (\frac {5 \log (3)}{e^2 x}\right )+(-4-8 x) \log ^2\left (\frac {5 \log (3)}{e^2 x}\right )+\left (-2-2 \log \left (\frac {5 \log (3)}{e^2 x}\right )+(2+4 x) \log ^2\left (\frac {5 \log (3)}{e^2 x}\right )\right ) \log \left (\frac {x+\left (-x-x^2\right ) \log \left (\frac {5 \log (3)}{e^2 x}\right )}{\log \left (\frac {5 \log (3)}{e^2 x}\right )}\right )}{-x \log \left (\frac {5 \log (3)}{e^2 x}\right )+\left (x+x^2\right ) \log ^2\left (\frac {5 \log (3)}{e^2 x}\right )} \, dx=\left (2-\log \left (-x-x^2+\frac {x}{\log \left (\frac {5 \log (3)}{e^2 x}\right )}\right )\right )^2 \]

[Out]

(2-ln(x/ln(5*ln(3)/exp(2)/x)-x^2-x))^2

Rubi [A] (verified)

Time = 0.28 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.84, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.021, Rules used = {6820, 12, 6818} \[ \int \frac {4+4 \log \left (\frac {5 \log (3)}{e^2 x}\right )+(-4-8 x) \log ^2\left (\frac {5 \log (3)}{e^2 x}\right )+\left (-2-2 \log \left (\frac {5 \log (3)}{e^2 x}\right )+(2+4 x) \log ^2\left (\frac {5 \log (3)}{e^2 x}\right )\right ) \log \left (\frac {x+\left (-x-x^2\right ) \log \left (\frac {5 \log (3)}{e^2 x}\right )}{\log \left (\frac {5 \log (3)}{e^2 x}\right )}\right )}{-x \log \left (\frac {5 \log (3)}{e^2 x}\right )+\left (x+x^2\right ) \log ^2\left (\frac {5 \log (3)}{e^2 x}\right )} \, dx=\left (2-\log \left (-x \left (x+\frac {1}{2-\log \left (\frac {\log (243)}{x}\right )}+1\right )\right )\right )^2 \]

[In]

Int[(4 + 4*Log[(5*Log[3])/(E^2*x)] + (-4 - 8*x)*Log[(5*Log[3])/(E^2*x)]^2 + (-2 - 2*Log[(5*Log[3])/(E^2*x)] +
(2 + 4*x)*Log[(5*Log[3])/(E^2*x)]^2)*Log[(x + (-x - x^2)*Log[(5*Log[3])/(E^2*x)])/Log[(5*Log[3])/(E^2*x)]])/(-
(x*Log[(5*Log[3])/(E^2*x)]) + (x + x^2)*Log[(5*Log[3])/(E^2*x)]^2),x]

[Out]

(2 - Log[-(x*(1 + x + (2 - Log[Log[243]/x])^(-1)))])^2

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 6818

Int[(u_)*(y_)^(m_.), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[q*(y^(m + 1)/(m + 1)), x] /;  !F
alseQ[q]] /; FreeQ[m, x] && NeQ[m, -1]

Rule 6820

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

Rubi steps \begin{align*} \text {integral}& = \int \frac {2 \left (1+\left (-2+\log \left (\frac {1}{x}\right )+\log (\log (243))\right )^2+2 x \left (-2+\log \left (\frac {1}{x}\right )+\log (\log (243))\right )^2-\log \left (\frac {\log (243)}{x}\right )\right ) \left (-2+\log \left (x \left (-1-x+\frac {1}{-2+\log \left (\frac {\log (243)}{x}\right )}\right )\right )\right )}{x \left (1-(1+x) \left (-2+\log \left (\frac {1}{x}\right )+\log (\log (243))\right )\right ) \left (2-\log \left (\frac {\log (243)}{x}\right )\right )} \, dx \\ & = 2 \int \frac {\left (1+\left (-2+\log \left (\frac {1}{x}\right )+\log (\log (243))\right )^2+2 x \left (-2+\log \left (\frac {1}{x}\right )+\log (\log (243))\right )^2-\log \left (\frac {\log (243)}{x}\right )\right ) \left (-2+\log \left (x \left (-1-x+\frac {1}{-2+\log \left (\frac {\log (243)}{x}\right )}\right )\right )\right )}{x \left (1-(1+x) \left (-2+\log \left (\frac {1}{x}\right )+\log (\log (243))\right )\right ) \left (2-\log \left (\frac {\log (243)}{x}\right )\right )} \, dx \\ & = \left (2-\log \left (-x \left (1+x+\frac {1}{2-\log \left (\frac {\log (243)}{x}\right )}\right )\right )\right )^2 \\ \end{align*}

Mathematica [F]

\[ \int \frac {4+4 \log \left (\frac {5 \log (3)}{e^2 x}\right )+(-4-8 x) \log ^2\left (\frac {5 \log (3)}{e^2 x}\right )+\left (-2-2 \log \left (\frac {5 \log (3)}{e^2 x}\right )+(2+4 x) \log ^2\left (\frac {5 \log (3)}{e^2 x}\right )\right ) \log \left (\frac {x+\left (-x-x^2\right ) \log \left (\frac {5 \log (3)}{e^2 x}\right )}{\log \left (\frac {5 \log (3)}{e^2 x}\right )}\right )}{-x \log \left (\frac {5 \log (3)}{e^2 x}\right )+\left (x+x^2\right ) \log ^2\left (\frac {5 \log (3)}{e^2 x}\right )} \, dx=\int \frac {4+4 \log \left (\frac {5 \log (3)}{e^2 x}\right )+(-4-8 x) \log ^2\left (\frac {5 \log (3)}{e^2 x}\right )+\left (-2-2 \log \left (\frac {5 \log (3)}{e^2 x}\right )+(2+4 x) \log ^2\left (\frac {5 \log (3)}{e^2 x}\right )\right ) \log \left (\frac {x+\left (-x-x^2\right ) \log \left (\frac {5 \log (3)}{e^2 x}\right )}{\log \left (\frac {5 \log (3)}{e^2 x}\right )}\right )}{-x \log \left (\frac {5 \log (3)}{e^2 x}\right )+\left (x+x^2\right ) \log ^2\left (\frac {5 \log (3)}{e^2 x}\right )} \, dx \]

[In]

Integrate[(4 + 4*Log[(5*Log[3])/(E^2*x)] + (-4 - 8*x)*Log[(5*Log[3])/(E^2*x)]^2 + (-2 - 2*Log[(5*Log[3])/(E^2*
x)] + (2 + 4*x)*Log[(5*Log[3])/(E^2*x)]^2)*Log[(x + (-x - x^2)*Log[(5*Log[3])/(E^2*x)])/Log[(5*Log[3])/(E^2*x)
]])/(-(x*Log[(5*Log[3])/(E^2*x)]) + (x + x^2)*Log[(5*Log[3])/(E^2*x)]^2),x]

[Out]

Integrate[(4 + 4*Log[(5*Log[3])/(E^2*x)] + (-4 - 8*x)*Log[(5*Log[3])/(E^2*x)]^2 + (-2 - 2*Log[(5*Log[3])/(E^2*
x)] + (2 + 4*x)*Log[(5*Log[3])/(E^2*x)]^2)*Log[(x + (-x - x^2)*Log[(5*Log[3])/(E^2*x)])/Log[(5*Log[3])/(E^2*x)
]])/(-(x*Log[(5*Log[3])/(E^2*x)]) + (x + x^2)*Log[(5*Log[3])/(E^2*x)]^2), x]

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(117\) vs. \(2(32)=64\).

Time = 0.49 (sec) , antiderivative size = 118, normalized size of antiderivative = 3.81

method	result	size
default	\(8 \ln \left (\frac {1}{x}\right )-4 \ln \left (\frac {\ln \left (5\right )}{x}+\frac {\ln \left (\ln \left (3\right )\right )}{x}+\frac {\ln \left (\frac {1}{x}\right )}{x}+\ln \left (5\right )+\ln \left (\ln \left (3\right )\right )+\ln \left (\frac {1}{x}\right )-\frac {3}{x}-2\right )+4 \ln \left (\ln \left (5\right )+\ln \left (\ln \left (3\right )\right )-2+\ln \left (\frac {1}{x}\right )\right )+\ln \left (\frac {x \left (-x \ln \left (5\right )-\ln \left (\ln \left (3\right )\right ) x +2 x -x \ln \left (\frac {1}{x}\right )-\ln \left (5\right )-\ln \left (\ln \left (3\right )\right )+3-\ln \left (\frac {1}{x}\right )\right )}{\ln \left (5\right )+\ln \left (\ln \left (3\right )\right )-2+\ln \left (\frac {1}{x}\right )}\right )^{2}\)	\(118\)
parts	\(8 \ln \left (\frac {5 \ln \left (3\right ) {\mathrm e}^{-2}}{x}\right )-4 \ln \left (\frac {5 \ln \left (\frac {5 \ln \left (3\right ) {\mathrm e}^{-2}}{x}\right ) \ln \left (3\right )}{x}-\frac {5 \ln \left (3\right )}{x}+5 \ln \left (3\right ) \ln \left (\frac {5 \ln \left (3\right ) {\mathrm e}^{-2}}{x}\right )\right )+4 \ln \left (\ln \left (\frac {5 \ln \left (3\right ) {\mathrm e}^{-2}}{x}\right )\right )+\ln \left (\frac {x \left (-x \ln \left (5\right )-\ln \left (\ln \left (3\right )\right ) x +2 x -x \ln \left (\frac {1}{x}\right )-\ln \left (5\right )-\ln \left (\ln \left (3\right )\right )+3-\ln \left (\frac {1}{x}\right )\right )}{\ln \left (5\right )+\ln \left (\ln \left (3\right )\right )-2+\ln \left (\frac {1}{x}\right )}\right )^{2}\)	\(133\)

[In]

int((((4*x+2)*ln(5*ln(3)/exp(2)/x)^2-2*ln(5*ln(3)/exp(2)/x)-2)*ln(((-x^2-x)*ln(5*ln(3)/exp(2)/x)+x)/ln(5*ln(3)
/exp(2)/x))+(-8*x-4)*ln(5*ln(3)/exp(2)/x)^2+4*ln(5*ln(3)/exp(2)/x)+4)/((x^2+x)*ln(5*ln(3)/exp(2)/x)^2-x*ln(5*l
n(3)/exp(2)/x)),x,method=_RETURNVERBOSE)

[Out]

8*ln(1/x)-4*ln(ln(5)/x+ln(ln(3))/x+ln(1/x)/x+ln(5)+ln(ln(3))+ln(1/x)-3/x-2)+4*ln(ln(5)+ln(ln(3))-2+ln(1/x))+ln
(x*(-x*ln(5)-ln(ln(3))*x+2*x-x*ln(1/x)-ln(5)-ln(ln(3))+3-ln(1/x))/(ln(5)+ln(ln(3))-2+ln(1/x)))^2

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 75 vs. \(2 (28) = 56\).

Time = 0.27 (sec) , antiderivative size = 75, normalized size of antiderivative = 2.42 \[ \int \frac {4+4 \log \left (\frac {5 \log (3)}{e^2 x}\right )+(-4-8 x) \log ^2\left (\frac {5 \log (3)}{e^2 x}\right )+\left (-2-2 \log \left (\frac {5 \log (3)}{e^2 x}\right )+(2+4 x) \log ^2\left (\frac {5 \log (3)}{e^2 x}\right )\right ) \log \left (\frac {x+\left (-x-x^2\right ) \log \left (\frac {5 \log (3)}{e^2 x}\right )}{\log \left (\frac {5 \log (3)}{e^2 x}\right )}\right )}{-x \log \left (\frac {5 \log (3)}{e^2 x}\right )+\left (x+x^2\right ) \log ^2\left (\frac {5 \log (3)}{e^2 x}\right )} \, dx=\log \left (-\frac {{\left (x^{2} + x\right )} \log \left (\frac {5 \, e^{\left (-2\right )} \log \left (3\right )}{x}\right ) - x}{\log \left (\frac {5 \, e^{\left (-2\right )} \log \left (3\right )}{x}\right )}\right )^{2} - 4 \, \log \left (-\frac {{\left (x^{2} + x\right )} \log \left (\frac {5 \, e^{\left (-2\right )} \log \left (3\right )}{x}\right ) - x}{\log \left (\frac {5 \, e^{\left (-2\right )} \log \left (3\right )}{x}\right )}\right ) \]

[In]

integrate((((4*x+2)*log(5*log(3)/exp(2)/x)^2-2*log(5*log(3)/exp(2)/x)-2)*log(((-x^2-x)*log(5*log(3)/exp(2)/x)+
x)/log(5*log(3)/exp(2)/x))+(-8*x-4)*log(5*log(3)/exp(2)/x)^2+4*log(5*log(3)/exp(2)/x)+4)/((x^2+x)*log(5*log(3)
/exp(2)/x)^2-x*log(5*log(3)/exp(2)/x)),x, algorithm="fricas")

[Out]

log(-((x^2 + x)*log(5*e^(-2)*log(3)/x) - x)/log(5*e^(-2)*log(3)/x))^2 - 4*log(-((x^2 + x)*log(5*e^(-2)*log(3)/
x) - x)/log(5*e^(-2)*log(3)/x))

Sympy [F(-2)]

Exception generated. \[ \int \frac {4+4 \log \left (\frac {5 \log (3)}{e^2 x}\right )+(-4-8 x) \log ^2\left (\frac {5 \log (3)}{e^2 x}\right )+\left (-2-2 \log \left (\frac {5 \log (3)}{e^2 x}\right )+(2+4 x) \log ^2\left (\frac {5 \log (3)}{e^2 x}\right )\right ) \log \left (\frac {x+\left (-x-x^2\right ) \log \left (\frac {5 \log (3)}{e^2 x}\right )}{\log \left (\frac {5 \log (3)}{e^2 x}\right )}\right )}{-x \log \left (\frac {5 \log (3)}{e^2 x}\right )+\left (x+x^2\right ) \log ^2\left (\frac {5 \log (3)}{e^2 x}\right )} \, dx=\text {Exception raised: PolynomialError} \]

[In]

integrate((((4*x+2)*ln(5*ln(3)/exp(2)/x)**2-2*ln(5*ln(3)/exp(2)/x)-2)*ln(((-x**2-x)*ln(5*ln(3)/exp(2)/x)+x)/ln
(5*ln(3)/exp(2)/x))+(-8*x-4)*ln(5*ln(3)/exp(2)/x)**2+4*ln(5*ln(3)/exp(2)/x)+4)/((x**2+x)*ln(5*ln(3)/exp(2)/x)*
*2-x*ln(5*ln(3)/exp(2)/x)),x)

[Out]

Exception raised: PolynomialError >> 1/(x**3 + 2*x**2 + x) contains an element of the set of generators.

Maxima [F]

\[ \int \frac {4+4 \log \left (\frac {5 \log (3)}{e^2 x}\right )+(-4-8 x) \log ^2\left (\frac {5 \log (3)}{e^2 x}\right )+\left (-2-2 \log \left (\frac {5 \log (3)}{e^2 x}\right )+(2+4 x) \log ^2\left (\frac {5 \log (3)}{e^2 x}\right )\right ) \log \left (\frac {x+\left (-x-x^2\right ) \log \left (\frac {5 \log (3)}{e^2 x}\right )}{\log \left (\frac {5 \log (3)}{e^2 x}\right )}\right )}{-x \log \left (\frac {5 \log (3)}{e^2 x}\right )+\left (x+x^2\right ) \log ^2\left (\frac {5 \log (3)}{e^2 x}\right )} \, dx=\int { -\frac {2 \, {\left (2 \, {\left (2 \, x + 1\right )} \log \left (\frac {5 \, e^{\left (-2\right )} \log \left (3\right )}{x}\right )^{2} - {\left ({\left (2 \, x + 1\right )} \log \left (\frac {5 \, e^{\left (-2\right )} \log \left (3\right )}{x}\right )^{2} - \log \left (\frac {5 \, e^{\left (-2\right )} \log \left (3\right )}{x}\right ) - 1\right )} \log \left (-\frac {{\left (x^{2} + x\right )} \log \left (\frac {5 \, e^{\left (-2\right )} \log \left (3\right )}{x}\right ) - x}{\log \left (\frac {5 \, e^{\left (-2\right )} \log \left (3\right )}{x}\right )}\right ) - 2 \, \log \left (\frac {5 \, e^{\left (-2\right )} \log \left (3\right )}{x}\right ) - 2\right )}}{{\left (x^{2} + x\right )} \log \left (\frac {5 \, e^{\left (-2\right )} \log \left (3\right )}{x}\right )^{2} - x \log \left (\frac {5 \, e^{\left (-2\right )} \log \left (3\right )}{x}\right )} \,d x } \]

[In]

integrate((((4*x+2)*log(5*log(3)/exp(2)/x)^2-2*log(5*log(3)/exp(2)/x)-2)*log(((-x^2-x)*log(5*log(3)/exp(2)/x)+
x)/log(5*log(3)/exp(2)/x))+(-8*x-4)*log(5*log(3)/exp(2)/x)^2+4*log(5*log(3)/exp(2)/x)+4)/((x^2+x)*log(5*log(3)
/exp(2)/x)^2-x*log(5*log(3)/exp(2)/x)),x, algorithm="maxima")

[Out]

-2*integrate((2*(2*x + 1)*log(5*e^(-2)*log(3)/x)^2 - ((2*x + 1)*log(5*e^(-2)*log(3)/x)^2 - log(5*e^(-2)*log(3)
/x) - 1)*log(-((x^2 + x)*log(5*e^(-2)*log(3)/x) - x)/log(5*e^(-2)*log(3)/x)) - 2*log(5*e^(-2)*log(3)/x) - 2)/(
(x^2 + x)*log(5*e^(-2)*log(3)/x)^2 - x*log(5*e^(-2)*log(3)/x)), x)

Giac [F]

\[ \int \frac {4+4 \log \left (\frac {5 \log (3)}{e^2 x}\right )+(-4-8 x) \log ^2\left (\frac {5 \log (3)}{e^2 x}\right )+\left (-2-2 \log \left (\frac {5 \log (3)}{e^2 x}\right )+(2+4 x) \log ^2\left (\frac {5 \log (3)}{e^2 x}\right )\right ) \log \left (\frac {x+\left (-x-x^2\right ) \log \left (\frac {5 \log (3)}{e^2 x}\right )}{\log \left (\frac {5 \log (3)}{e^2 x}\right )}\right )}{-x \log \left (\frac {5 \log (3)}{e^2 x}\right )+\left (x+x^2\right ) \log ^2\left (\frac {5 \log (3)}{e^2 x}\right )} \, dx=\int { -\frac {2 \, {\left (2 \, {\left (2 \, x + 1\right )} \log \left (\frac {5 \, e^{\left (-2\right )} \log \left (3\right )}{x}\right )^{2} - {\left ({\left (2 \, x + 1\right )} \log \left (\frac {5 \, e^{\left (-2\right )} \log \left (3\right )}{x}\right )^{2} - \log \left (\frac {5 \, e^{\left (-2\right )} \log \left (3\right )}{x}\right ) - 1\right )} \log \left (-\frac {{\left (x^{2} + x\right )} \log \left (\frac {5 \, e^{\left (-2\right )} \log \left (3\right )}{x}\right ) - x}{\log \left (\frac {5 \, e^{\left (-2\right )} \log \left (3\right )}{x}\right )}\right ) - 2 \, \log \left (\frac {5 \, e^{\left (-2\right )} \log \left (3\right )}{x}\right ) - 2\right )}}{{\left (x^{2} + x\right )} \log \left (\frac {5 \, e^{\left (-2\right )} \log \left (3\right )}{x}\right )^{2} - x \log \left (\frac {5 \, e^{\left (-2\right )} \log \left (3\right )}{x}\right )} \,d x } \]

[In]

integrate((((4*x+2)*log(5*log(3)/exp(2)/x)^2-2*log(5*log(3)/exp(2)/x)-2)*log(((-x^2-x)*log(5*log(3)/exp(2)/x)+
x)/log(5*log(3)/exp(2)/x))+(-8*x-4)*log(5*log(3)/exp(2)/x)^2+4*log(5*log(3)/exp(2)/x)+4)/((x^2+x)*log(5*log(3)
/exp(2)/x)^2-x*log(5*log(3)/exp(2)/x)),x, algorithm="giac")

[Out]

integrate(-2*(2*(2*x + 1)*log(5*e^(-2)*log(3)/x)^2 - ((2*x + 1)*log(5*e^(-2)*log(3)/x)^2 - log(5*e^(-2)*log(3)
/x) - 1)*log(-((x^2 + x)*log(5*e^(-2)*log(3)/x) - x)/log(5*e^(-2)*log(3)/x)) - 2*log(5*e^(-2)*log(3)/x) - 2)/(
(x^2 + x)*log(5*e^(-2)*log(3)/x)^2 - x*log(5*e^(-2)*log(3)/x)), x)

Mupad [B] (verification not implemented)

Time = 9.86 (sec) , antiderivative size = 223, normalized size of antiderivative = 7.19 \[ \int \frac {4+4 \log \left (\frac {5 \log (3)}{e^2 x}\right )+(-4-8 x) \log ^2\left (\frac {5 \log (3)}{e^2 x}\right )+\left (-2-2 \log \left (\frac {5 \log (3)}{e^2 x}\right )+(2+4 x) \log ^2\left (\frac {5 \log (3)}{e^2 x}\right )\right ) \log \left (\frac {x+\left (-x-x^2\right ) \log \left (\frac {5 \log (3)}{e^2 x}\right )}{\log \left (\frac {5 \log (3)}{e^2 x}\right )}\right )}{-x \log \left (\frac {5 \log (3)}{e^2 x}\right )+\left (x+x^2\right ) \log ^2\left (\frac {5 \log (3)}{e^2 x}\right )} \, dx={\ln \left (\frac {x-\ln \left (\frac {5\,{\mathrm {e}}^{-2}\,\ln \left (3\right )}{x}\right )\,\left (x^2+x\right )}{\ln \left (\frac {5\,{\mathrm {e}}^{-2}\,\ln \left (3\right )}{x}\right )}\right )}^2+4\,\ln \left (x+1\right )-4\,\ln \left (\frac {4\,\left (x-x\,\ln \left (\frac {5\,{\mathrm {e}}^{-2}\,\ln \left (3\right )}{x}\right )+1\right )}{x\,{\left (x+1\right )}^2}-\frac {4\,\left (2\,\ln \left (\frac {5\,{\mathrm {e}}^{-2}\,\ln \left (3\right )}{x}\right )-x+3\,x\,\ln \left (\frac {5\,{\mathrm {e}}^{-2}\,\ln \left (3\right )}{x}\right )+2\,x^2\,\ln \left (\frac {5\,{\mathrm {e}}^{-2}\,\ln \left (3\right )}{x}\right )-1\right )}{x\,{\left (x+1\right )}^2}\right )+4\,\ln \left (\frac {4\,\left (2\,\ln \left (\frac {5\,{\mathrm {e}}^{-2}\,\ln \left (3\right )}{x}\right )-x+3\,x\,\ln \left (\frac {5\,{\mathrm {e}}^{-2}\,\ln \left (3\right )}{x}\right )+2\,x^2\,\ln \left (\frac {5\,{\mathrm {e}}^{-2}\,\ln \left (3\right )}{x}\right )-1\right )}{x\,{\left (x+1\right )}^2}+\frac {4\,\left (x-x\,\ln \left (\frac {5\,{\mathrm {e}}^{-2}\,\ln \left (3\right )}{x}\right )+1\right )}{x\,{\left (x+1\right )}^2}\right )-4\,\ln \left (x\,\left (x^2+x+1\right )\right ) \]

[In]

int(-(4*log((5*exp(-2)*log(3))/x) - log((5*exp(-2)*log(3))/x)^2*(8*x + 4) - log((x - log((5*exp(-2)*log(3))/x)
*(x + x^2))/log((5*exp(-2)*log(3))/x))*(2*log((5*exp(-2)*log(3))/x) - log((5*exp(-2)*log(3))/x)^2*(4*x + 2) +
2) + 4)/(x*log((5*exp(-2)*log(3))/x) - log((5*exp(-2)*log(3))/x)^2*(x + x^2)),x)

[Out]

4*log(x + 1) - 4*log((4*(x - x*log((5*exp(-2)*log(3))/x) + 1))/(x*(x + 1)^2) - (4*(2*log((5*exp(-2)*log(3))/x)
- x + 3*x*log((5*exp(-2)*log(3))/x) + 2*x^2*log((5*exp(-2)*log(3))/x) - 1))/(x*(x + 1)^2)) + 4*log((4*(2*log(
(5*exp(-2)*log(3))/x) - x + 3*x*log((5*exp(-2)*log(3))/x) + 2*x^2*log((5*exp(-2)*log(3))/x) - 1))/(x*(x + 1)^2
) + (4*(x - x*log((5*exp(-2)*log(3))/x) + 1))/(x*(x + 1)^2)) - 4*log(x*(x + x^2 + 1)) + log((x - log((5*exp(-2
)*log(3))/x)*(x + x^2))/log((5*exp(-2)*log(3))/x))^2