\(\int \frac {(1-x-x^2) \log (4 e^{e^x})+\log ^2(4 e^{e^x}) (-2 x^2+4 x^3-2 x^4+(-2 x+4 x^2-2 x^3) \log (-x+x^2))+(e^x (-x^2+x^3)+e^x (-x+x^2) \log (-x+x^2)) \log (x+\log (-x+x^2))}{\log ^2(4 e^{e^x}) (-x^2+x^3+(-x+x^2) \log (-x+x^2))} \, dx\) [2937]

Optimal result

Integrand size = 153, antiderivative size = 35 \[ \int \frac {\left (1-x-x^2\right ) \log \left (4 e^{e^x}\right )+\log ^2\left (4 e^{e^x}\right ) \left (-2 x^2+4 x^3-2 x^4+\left (-2 x+4 x^2-2 x^3\right ) \log \left (-x+x^2\right )\right )+\left (e^x \left (-x^2+x^3\right )+e^x \left (-x+x^2\right ) \log \left (-x+x^2\right )\right ) \log \left (x+\log \left (-x+x^2\right )\right )}{\log ^2\left (4 e^{e^x}\right ) \left (-x^2+x^3+\left (-x+x^2\right ) \log \left (-x+x^2\right )\right )} \, dx=2 x-x^2-\frac {\log (x+\log (x-(2-x) x))}{\log \left (4 e^{e^x}\right )} \]

[Out]

2*x-ln(ln(x-(2-x)*x)+x)/ln(4*exp(exp(x)))-x^2

Rubi [F]

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

[In]

Int[((1 - x - x^2)*Log[4*E^E^x] + Log[4*E^E^x]^2*(-2*x^2 + 4*x^3 - 2*x^4 + (-2*x + 4*x^2 - 2*x^3)*Log[-x + x^2
]) + (E^x*(-x^2 + x^3) + E^x*(-x + x^2)*Log[-x + x^2])*Log[x + Log[-x + x^2]])/(Log[4*E^E^x]^2*(-x^2 + x^3 + (
-x + x^2)*Log[-x + x^2])),x]

[Out]

2*x - x^2 - Defer[Int][1/(Log[4*E^E^x]*(x + Log[(-1 + x)*x])), x] - Defer[Int][1/((-1 + x)*Log[4*E^E^x]*(x + L
og[(-1 + x)*x])), x] - Defer[Int][1/(x*Log[4*E^E^x]*(x + Log[(-1 + x)*x])), x] + Defer[Int][(E^x*Log[x + Log[(
-1 + x)*x]])/Log[4*E^E^x]^2, x]

Rubi steps \begin{align*} \text {integral}& = \int \left (2-2 x-\frac {-1+x+x^2}{(-1+x) x \log \left (4 e^{e^x}\right ) (x+\log ((-1+x) x))}+\frac {e^x \log (x+\log ((-1+x) x))}{\log ^2\left (4 e^{e^x}\right )}\right ) \, dx \\ & = 2 x-x^2-\int \frac {-1+x+x^2}{(-1+x) x \log \left (4 e^{e^x}\right ) (x+\log ((-1+x) x))} \, dx+\int \frac {e^x \log (x+\log ((-1+x) x))}{\log ^2\left (4 e^{e^x}\right )} \, dx \\ & = 2 x-x^2-\int \left (\frac {1}{\log \left (4 e^{e^x}\right ) (x+\log ((-1+x) x))}+\frac {1}{(-1+x) \log \left (4 e^{e^x}\right ) (x+\log ((-1+x) x))}+\frac {1}{x \log \left (4 e^{e^x}\right ) (x+\log ((-1+x) x))}\right ) \, dx+\int \frac {e^x \log (x+\log ((-1+x) x))}{\log ^2\left (4 e^{e^x}\right )} \, dx \\ & = 2 x-x^2-\int \frac {1}{\log \left (4 e^{e^x}\right ) (x+\log ((-1+x) x))} \, dx-\int \frac {1}{(-1+x) \log \left (4 e^{e^x}\right ) (x+\log ((-1+x) x))} \, dx-\int \frac {1}{x \log \left (4 e^{e^x}\right ) (x+\log ((-1+x) x))} \, dx+\int \frac {e^x \log (x+\log ((-1+x) x))}{\log ^2\left (4 e^{e^x}\right )} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.48 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.86 \[ \int \frac {\left (1-x-x^2\right ) \log \left (4 e^{e^x}\right )+\log ^2\left (4 e^{e^x}\right ) \left (-2 x^2+4 x^3-2 x^4+\left (-2 x+4 x^2-2 x^3\right ) \log \left (-x+x^2\right )\right )+\left (e^x \left (-x^2+x^3\right )+e^x \left (-x+x^2\right ) \log \left (-x+x^2\right )\right ) \log \left (x+\log \left (-x+x^2\right )\right )}{\log ^2\left (4 e^{e^x}\right ) \left (-x^2+x^3+\left (-x+x^2\right ) \log \left (-x+x^2\right )\right )} \, dx=2 x-x^2-\frac {\log (x+\log ((-1+x) x))}{\log \left (4 e^{e^x}\right )} \]

[In]

Integrate[((1 - x - x^2)*Log[4*E^E^x] + Log[4*E^E^x]^2*(-2*x^2 + 4*x^3 - 2*x^4 + (-2*x + 4*x^2 - 2*x^3)*Log[-x
+ x^2]) + (E^x*(-x^2 + x^3) + E^x*(-x + x^2)*Log[-x + x^2])*Log[x + Log[-x + x^2]])/(Log[4*E^E^x]^2*(-x^2 + x
^3 + (-x + x^2)*Log[-x + x^2])),x]

[Out]

2*x - x^2 - Log[x + Log[(-1 + x)*x]]/Log[4*E^E^x]

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 0.51 (sec) , antiderivative size = 83, normalized size of antiderivative = 2.37

\[-x^{2}+2 x -\frac {2 i \ln \left (\ln \left (-1+x \right )+\ln \left (x \right )-\frac {i \pi \,\operatorname {csgn}\left (i x \left (-1+x \right )\right ) \left (-\operatorname {csgn}\left (i x \left (-1+x \right )\right )+\operatorname {csgn}\left (i x \right )\right ) \left (-\operatorname {csgn}\left (i x \left (-1+x \right )\right )+\operatorname {csgn}\left (i \left (-1+x \right )\right )\right )}{2}+x \right )}{4 i \ln \left (2\right )+2 i \ln \left ({\mathrm e}^{{\mathrm e}^{x}}\right )}\]

[In]

int((((-2*x^3+4*x^2-2*x)*ln(x^2-x)-2*x^4+4*x^3-2*x^2)*ln(4*exp(exp(x)))^2+(-x^2-x+1)*ln(4*exp(exp(x)))+((x^2-x
)*exp(x)*ln(x^2-x)+(x^3-x^2)*exp(x))*ln(ln(x^2-x)+x))/((x^2-x)*ln(x^2-x)+x^3-x^2)/ln(4*exp(exp(x)))^2,x)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.26 \[ \int \frac {\left (1-x-x^2\right ) \log \left (4 e^{e^x}\right )+\log ^2\left (4 e^{e^x}\right ) \left (-2 x^2+4 x^3-2 x^4+\left (-2 x+4 x^2-2 x^3\right ) \log \left (-x+x^2\right )\right )+\left (e^x \left (-x^2+x^3\right )+e^x \left (-x+x^2\right ) \log \left (-x+x^2\right )\right ) \log \left (x+\log \left (-x+x^2\right )\right )}{\log ^2\left (4 e^{e^x}\right ) \left (-x^2+x^3+\left (-x+x^2\right ) \log \left (-x+x^2\right )\right )} \, dx=-\frac {{\left (x^{2} - 2 \, x\right )} e^{x} + 2 \, {\left (x^{2} - 2 \, x\right )} \log \left (2\right ) + \log \left (x + \log \left (x^{2} - x\right )\right )}{e^{x} + 2 \, \log \left (2\right )} \]

[In]

integrate((((-2*x^3+4*x^2-2*x)*log(x^2-x)-2*x^4+4*x^3-2*x^2)*log(4*exp(exp(x)))^2+(-x^2-x+1)*log(4*exp(exp(x))
)+((x^2-x)*exp(x)*log(x^2-x)+(x^3-x^2)*exp(x))*log(log(x^2-x)+x))/((x^2-x)*log(x^2-x)+x^3-x^2)/log(4*exp(exp(x
)))^2,x, algorithm="fricas")

[Out]

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

Sympy [A] (verification not implemented)

Time = 3.15 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.69 \[ \int \frac {\left (1-x-x^2\right ) \log \left (4 e^{e^x}\right )+\log ^2\left (4 e^{e^x}\right ) \left (-2 x^2+4 x^3-2 x^4+\left (-2 x+4 x^2-2 x^3\right ) \log \left (-x+x^2\right )\right )+\left (e^x \left (-x^2+x^3\right )+e^x \left (-x+x^2\right ) \log \left (-x+x^2\right )\right ) \log \left (x+\log \left (-x+x^2\right )\right )}{\log ^2\left (4 e^{e^x}\right ) \left (-x^2+x^3+\left (-x+x^2\right ) \log \left (-x+x^2\right )\right )} \, dx=- x^{2} + 2 x - \frac {\log {\left (x + \log {\left (x^{2} - x \right )} \right )}}{e^{x} + 2 \log {\left (2 \right )}} \]

[In]

integrate((((-2*x**3+4*x**2-2*x)*ln(x**2-x)-2*x**4+4*x**3-2*x**2)*ln(4*exp(exp(x)))**2+(-x**2-x+1)*ln(4*exp(ex
p(x)))+((x**2-x)*exp(x)*ln(x**2-x)+(x**3-x**2)*exp(x))*ln(ln(x**2-x)+x))/((x**2-x)*ln(x**2-x)+x**3-x**2)/ln(4*
exp(exp(x)))**2,x)

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.38 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.23 \[ \int \frac {\left (1-x-x^2\right ) \log \left (4 e^{e^x}\right )+\log ^2\left (4 e^{e^x}\right ) \left (-2 x^2+4 x^3-2 x^4+\left (-2 x+4 x^2-2 x^3\right ) \log \left (-x+x^2\right )\right )+\left (e^x \left (-x^2+x^3\right )+e^x \left (-x+x^2\right ) \log \left (-x+x^2\right )\right ) \log \left (x+\log \left (-x+x^2\right )\right )}{\log ^2\left (4 e^{e^x}\right ) \left (-x^2+x^3+\left (-x+x^2\right ) \log \left (-x+x^2\right )\right )} \, dx=-\frac {2 \, x^{2} \log \left (2\right ) + {\left (x^{2} - 2 \, x\right )} e^{x} - 4 \, x \log \left (2\right ) + \log \left (x + \log \left (x - 1\right ) + \log \left (x\right )\right )}{e^{x} + 2 \, \log \left (2\right )} \]

[In]

integrate((((-2*x^3+4*x^2-2*x)*log(x^2-x)-2*x^4+4*x^3-2*x^2)*log(4*exp(exp(x)))^2+(-x^2-x+1)*log(4*exp(exp(x))
)+((x^2-x)*exp(x)*log(x^2-x)+(x^3-x^2)*exp(x))*log(log(x^2-x)+x))/((x^2-x)*log(x^2-x)+x^3-x^2)/log(4*exp(exp(x
)))^2,x, algorithm="maxima")

[Out]

-(2*x^2*log(2) + (x^2 - 2*x)*e^x - 4*x*log(2) + log(x + log(x - 1) + log(x)))/(e^x + 2*log(2))

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 96 vs. \(2 (30) = 60\).

Time = 0.38 (sec) , antiderivative size = 96, normalized size of antiderivative = 2.74 \[ \int \frac {\left (1-x-x^2\right ) \log \left (4 e^{e^x}\right )+\log ^2\left (4 e^{e^x}\right ) \left (-2 x^2+4 x^3-2 x^4+\left (-2 x+4 x^2-2 x^3\right ) \log \left (-x+x^2\right )\right )+\left (e^x \left (-x^2+x^3\right )+e^x \left (-x+x^2\right ) \log \left (-x+x^2\right )\right ) \log \left (x+\log \left (-x+x^2\right )\right )}{\log ^2\left (4 e^{e^x}\right ) \left (-x^2+x^3+\left (-x+x^2\right ) \log \left (-x+x^2\right )\right )} \, dx=-\frac {x^{2} e^{x} + 2 \, x^{2} \log \left (2\right ) - 2 \, x e^{x} - 4 \, x \log \left (2\right ) + 4 \, e^{x} \log \left (e^{x} + 2 \, \log \left (2\right )\right ) + 8 \, \log \left (2\right ) \log \left (e^{x} + 2 \, \log \left (2\right )\right ) - 4 \, e^{x} \log \left (-e^{x} - 2 \, \log \left (2\right )\right ) - 8 \, \log \left (2\right ) \log \left (-e^{x} - 2 \, \log \left (2\right )\right ) + \log \left (x + \log \left (x - 1\right ) + \log \left (x\right )\right )}{e^{x} + 2 \, \log \left (2\right )} \]

[In]

integrate((((-2*x^3+4*x^2-2*x)*log(x^2-x)-2*x^4+4*x^3-2*x^2)*log(4*exp(exp(x)))^2+(-x^2-x+1)*log(4*exp(exp(x))
)+((x^2-x)*exp(x)*log(x^2-x)+(x^3-x^2)*exp(x))*log(log(x^2-x)+x))/((x^2-x)*log(x^2-x)+x^3-x^2)/log(4*exp(exp(x
)))^2,x, algorithm="giac")

[Out]

-(x^2*e^x + 2*x^2*log(2) - 2*x*e^x - 4*x*log(2) + 4*e^x*log(e^x + 2*log(2)) + 8*log(2)*log(e^x + 2*log(2)) - 4
*e^x*log(-e^x - 2*log(2)) - 8*log(2)*log(-e^x - 2*log(2)) + log(x + log(x - 1) + log(x)))/(e^x + 2*log(2))

Mupad [B] (verification not implemented)

Time = 9.14 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.83 \[ \int \frac {\left (1-x-x^2\right ) \log \left (4 e^{e^x}\right )+\log ^2\left (4 e^{e^x}\right ) \left (-2 x^2+4 x^3-2 x^4+\left (-2 x+4 x^2-2 x^3\right ) \log \left (-x+x^2\right )\right )+\left (e^x \left (-x^2+x^3\right )+e^x \left (-x+x^2\right ) \log \left (-x+x^2\right )\right ) \log \left (x+\log \left (-x+x^2\right )\right )}{\log ^2\left (4 e^{e^x}\right ) \left (-x^2+x^3+\left (-x+x^2\right ) \log \left (-x+x^2\right )\right )} \, dx=2\,x-\frac {\ln \left (x+\ln \left (x^2-x\right )\right )}{\ln \left (4\right )+{\mathrm {e}}^x}-x^2 \]

[In]

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

[Out]

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