\(\int \frac {11+e^{2 x} x^2+e^x (-2 x+10 x^2)+(-11+12 e^x x-e^{2 x} x^2) \log (\frac {-44 x+4 e^x x^2}{-1+e^x x}) \log (\log (\frac {-44 x+4 e^x x^2}{-1+e^x x}))}{(55 x^2-60 e^x x^3+5 e^{2 x} x^4) \log (\frac {-44 x+4 e^x x^2}{-1+e^x x})+(-110 x+120 e^x x^2-10 e^{2 x} x^3) \log (\frac {-44 x+4 e^x x^2}{-1+e^x x}) \log (\log (\frac {-44 x+4 e^x x^2}{-1+e^x x}))+(55-60 e^x x+5 e^{2 x} x^2) \log (\frac {-44 x+4 e^x x^2}{-1+e^x x}) \log ^2(\log (\frac {-44 x+4 e^x x^2}{-1+e^x x}))} \, dx\) [1373]

Optimal result

Integrand size = 280, antiderivative size = 30 \[ \int \frac {11+e^{2 x} x^2+e^x \left (-2 x+10 x^2\right )+\left (-11+12 e^x x-e^{2 x} x^2\right ) \log \left (\frac {-44 x+4 e^x x^2}{-1+e^x x}\right ) \log \left (\log \left (\frac {-44 x+4 e^x x^2}{-1+e^x x}\right )\right )}{\left (55 x^2-60 e^x x^3+5 e^{2 x} x^4\right ) \log \left (\frac {-44 x+4 e^x x^2}{-1+e^x x}\right )+\left (-110 x+120 e^x x^2-10 e^{2 x} x^3\right ) \log \left (\frac {-44 x+4 e^x x^2}{-1+e^x x}\right ) \log \left (\log \left (\frac {-44 x+4 e^x x^2}{-1+e^x x}\right )\right )+\left (55-60 e^x x+5 e^{2 x} x^2\right ) \log \left (\frac {-44 x+4 e^x x^2}{-1+e^x x}\right ) \log ^2\left (\log \left (\frac {-44 x+4 e^x x^2}{-1+e^x x}\right )\right )} \, dx=\frac {x}{5 \left (x-\log \left (\log \left (\frac {40}{-e^x+\frac {1}{x}}+4 x\right )\right )\right )} \] Output:

x/(5*x-5*ln(ln(4*x+40/(1/x-exp(x)))))
 

Mathematica [A] (verified)

Time = 0.13 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.07 \[ \int \frac {11+e^{2 x} x^2+e^x \left (-2 x+10 x^2\right )+\left (-11+12 e^x x-e^{2 x} x^2\right ) \log \left (\frac {-44 x+4 e^x x^2}{-1+e^x x}\right ) \log \left (\log \left (\frac {-44 x+4 e^x x^2}{-1+e^x x}\right )\right )}{\left (55 x^2-60 e^x x^3+5 e^{2 x} x^4\right ) \log \left (\frac {-44 x+4 e^x x^2}{-1+e^x x}\right )+\left (-110 x+120 e^x x^2-10 e^{2 x} x^3\right ) \log \left (\frac {-44 x+4 e^x x^2}{-1+e^x x}\right ) \log \left (\log \left (\frac {-44 x+4 e^x x^2}{-1+e^x x}\right )\right )+\left (55-60 e^x x+5 e^{2 x} x^2\right ) \log \left (\frac {-44 x+4 e^x x^2}{-1+e^x x}\right ) \log ^2\left (\log \left (\frac {-44 x+4 e^x x^2}{-1+e^x x}\right )\right )} \, dx=-\frac {x}{5 \left (-x+\log \left (\log \left (\frac {4 x \left (-11+e^x x\right )}{-1+e^x x}\right )\right )\right )} \] Input:

Integrate[(11 + E^(2*x)*x^2 + E^x*(-2*x + 10*x^2) + (-11 + 12*E^x*x - E^(2 
*x)*x^2)*Log[(-44*x + 4*E^x*x^2)/(-1 + E^x*x)]*Log[Log[(-44*x + 4*E^x*x^2) 
/(-1 + E^x*x)]])/((55*x^2 - 60*E^x*x^3 + 5*E^(2*x)*x^4)*Log[(-44*x + 4*E^x 
*x^2)/(-1 + E^x*x)] + (-110*x + 120*E^x*x^2 - 10*E^(2*x)*x^3)*Log[(-44*x + 
 4*E^x*x^2)/(-1 + E^x*x)]*Log[Log[(-44*x + 4*E^x*x^2)/(-1 + E^x*x)]] + (55 
 - 60*E^x*x + 5*E^(2*x)*x^2)*Log[(-44*x + 4*E^x*x^2)/(-1 + E^x*x)]*Log[Log 
[(-44*x + 4*E^x*x^2)/(-1 + E^x*x)]]^2),x]
 

Output:

-1/5*x/(-x + Log[Log[(4*x*(-11 + E^x*x))/(-1 + E^x*x)]])
 

Rubi [A] (verified)

Time = 1.74 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.20, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.014, Rules used = {7239, 27, 7262, 17}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {e^{2 x} x^2+e^x \left (10 x^2-2 x\right )+\left (-e^{2 x} x^2+12 e^x x-11\right ) \log \left (\frac {4 e^x x^2-44 x}{e^x x-1}\right ) \log \left (\log \left (\frac {4 e^x x^2-44 x}{e^x x-1}\right )\right )+11}{\left (5 e^{2 x} x^2-60 e^x x+55\right ) \log \left (\frac {4 e^x x^2-44 x}{e^x x-1}\right ) \log ^2\left (\log \left (\frac {4 e^x x^2-44 x}{e^x x-1}\right )\right )+\left (-10 e^{2 x} x^3+120 e^x x^2-110 x\right ) \log \left (\frac {4 e^x x^2-44 x}{e^x x-1}\right ) \log \left (\log \left (\frac {4 e^x x^2-44 x}{e^x x-1}\right )\right )+\left (5 e^{2 x} x^4-60 e^x x^3+55 x^2\right ) \log \left (\frac {4 e^x x^2-44 x}{e^x x-1}\right )} \, dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {e^{2 x} x^2-\left (e^{2 x} x^2-12 e^x x+11\right ) \log \left (\frac {4 x \left (e^x x-11\right )}{e^x x-1}\right ) \log \left (\log \left (\frac {4 x \left (e^x x-11\right )}{e^x x-1}\right )\right )+2 e^x (5 x-1) x+11}{5 \left (e^{2 x} x^2-12 e^x x+11\right ) \log \left (\frac {4 x \left (e^x x-11\right )}{e^x x-1}\right ) \left (x-\log \left (\log \left (\frac {4 x \left (e^x x-11\right )}{e^x x-1}\right )\right )\right )^2}dx\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{5} \int \frac {e^{2 x} x^2-2 e^x (1-5 x) x-\left (e^{2 x} x^2-12 e^x x+11\right ) \log \left (\frac {4 x \left (11-e^x x\right )}{1-e^x x}\right ) \log \left (\log \left (\frac {4 x \left (11-e^x x\right )}{1-e^x x}\right )\right )+11}{\left (e^{2 x} x^2-12 e^x x+11\right ) \log \left (\frac {4 x \left (11-e^x x\right )}{1-e^x x}\right ) \left (x-\log \left (\log \left (\frac {4 x \left (11-e^x x\right )}{1-e^x x}\right )\right )\right )^2}dx\)

\(\Big \downarrow \) 7262

\(\displaystyle -\frac {1}{5} \int \frac {1}{\left (\frac {x}{\log \left (\log \left (\frac {4 x \left (11-e^x x\right )}{1-e^x x}\right )\right )}-1\right )^2}d\frac {x}{\log \left (\log \left (\frac {4 x \left (11-e^x x\right )}{1-e^x x}\right )\right )}\)

\(\Big \downarrow \) 17

\(\displaystyle -\frac {1}{5 \left (1-\frac {x}{\log \left (\log \left (\frac {4 x \left (11-e^x x\right )}{1-e^x x}\right )\right )}\right )}\)

Input:

Int[(11 + E^(2*x)*x^2 + E^x*(-2*x + 10*x^2) + (-11 + 12*E^x*x - E^(2*x)*x^ 
2)*Log[(-44*x + 4*E^x*x^2)/(-1 + E^x*x)]*Log[Log[(-44*x + 4*E^x*x^2)/(-1 + 
 E^x*x)]])/((55*x^2 - 60*E^x*x^3 + 5*E^(2*x)*x^4)*Log[(-44*x + 4*E^x*x^2)/ 
(-1 + E^x*x)] + (-110*x + 120*E^x*x^2 - 10*E^(2*x)*x^3)*Log[(-44*x + 4*E^x 
*x^2)/(-1 + E^x*x)]*Log[Log[(-44*x + 4*E^x*x^2)/(-1 + E^x*x)]] + (55 - 60* 
E^x*x + 5*E^(2*x)*x^2)*Log[(-44*x + 4*E^x*x^2)/(-1 + E^x*x)]*Log[Log[(-44* 
x + 4*E^x*x^2)/(-1 + E^x*x)]]^2),x]
 

Output:

-1/5*1/(1 - x/Log[Log[(4*x*(11 - E^x*x))/(1 - E^x*x)]])
 

Defintions of rubi rules used

rule 17

Int[(c_.)*((a_.) + (b_.)*(x_))^(m_.), x_Symbol] :> Simp[c*((a + b*x)^(m + 1 
)/(b*(m + 1))), x] /; FreeQ[{a, b, c, m}, x] && NeQ[m, -1]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 7239

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

rule 7262

Int[(u_)*((a_.)*(v_)^(p_.) + (b_.)*(w_)^(q_.))^(m_.), x_Symbol] :> With[{c 
= Simplify[u/(p*w*D[v, x] - q*v*D[w, x])]}, Simp[c*p   Subst[Int[(b + a*x^p 
)^m, x], x, v*w^(m*q + 1)], x] /; FreeQ[c, x]] /; FreeQ[{a, b, m, p, q}, x] 
 && EqQ[p + q*(m*p + 1), 0] && IntegerQ[p] && IntegerQ[m]
 

Maple [C] (warning: unable to verify)

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

Time = 0.09 (sec) , antiderivative size = 210, normalized size of antiderivative = 7.00

\[\frac {x}{5 x -5 \ln \left (2 \ln \left (2\right )+\ln \left (x \right )+\ln \left ({\mathrm e}^{x} x -11\right )-\ln \left ({\mathrm e}^{x} x -1\right )-\frac {i \pi \,\operatorname {csgn}\left (\frac {i \left ({\mathrm e}^{x} x -11\right )}{{\mathrm e}^{x} x -1}\right ) \left (-\operatorname {csgn}\left (\frac {i \left ({\mathrm e}^{x} x -11\right )}{{\mathrm e}^{x} x -1}\right )+\operatorname {csgn}\left (i \left ({\mathrm e}^{x} x -11\right )\right )\right ) \left (-\operatorname {csgn}\left (\frac {i \left ({\mathrm e}^{x} x -11\right )}{{\mathrm e}^{x} x -1}\right )+\operatorname {csgn}\left (\frac {i}{{\mathrm e}^{x} x -1}\right )\right )}{2}-\frac {i \pi \,\operatorname {csgn}\left (\frac {i x \left ({\mathrm e}^{x} x -11\right )}{{\mathrm e}^{x} x -1}\right ) \left (-\operatorname {csgn}\left (\frac {i x \left ({\mathrm e}^{x} x -11\right )}{{\mathrm e}^{x} x -1}\right )+\operatorname {csgn}\left (i x \right )\right ) \left (-\operatorname {csgn}\left (\frac {i x \left ({\mathrm e}^{x} x -11\right )}{{\mathrm e}^{x} x -1}\right )+\operatorname {csgn}\left (\frac {i \left ({\mathrm e}^{x} x -11\right )}{{\mathrm e}^{x} x -1}\right )\right )}{2}\right )}\]

Input:

int(((-exp(x)^2*x^2+12*exp(x)*x-11)*ln((4*exp(x)*x^2-44*x)/(exp(x)*x-1))*l 
n(ln((4*exp(x)*x^2-44*x)/(exp(x)*x-1)))+exp(x)^2*x^2+(10*x^2-2*x)*exp(x)+1 
1)/((5*exp(x)^2*x^2-60*exp(x)*x+55)*ln((4*exp(x)*x^2-44*x)/(exp(x)*x-1))*l 
n(ln((4*exp(x)*x^2-44*x)/(exp(x)*x-1)))^2+(-10*exp(x)^2*x^3+120*exp(x)*x^2 
-110*x)*ln((4*exp(x)*x^2-44*x)/(exp(x)*x-1))*ln(ln((4*exp(x)*x^2-44*x)/(ex 
p(x)*x-1)))+(5*exp(x)^2*x^4-60*exp(x)*x^3+55*x^2)*ln((4*exp(x)*x^2-44*x)/( 
exp(x)*x-1))),x)
 

Output:

1/5*x/(x-ln(2*ln(2)+ln(x)+ln(exp(x)*x-11)-ln(exp(x)*x-1)-1/2*I*Pi*csgn(I*( 
exp(x)*x-11)/(exp(x)*x-1))*(-csgn(I*(exp(x)*x-11)/(exp(x)*x-1))+csgn(I*(ex 
p(x)*x-11)))*(-csgn(I*(exp(x)*x-11)/(exp(x)*x-1))+csgn(I/(exp(x)*x-1)))-1/ 
2*I*Pi*csgn(I*x/(exp(x)*x-1)*(exp(x)*x-11))*(-csgn(I*x/(exp(x)*x-1)*(exp(x 
)*x-11))+csgn(I*x))*(-csgn(I*x/(exp(x)*x-1)*(exp(x)*x-11))+csgn(I*(exp(x)* 
x-11)/(exp(x)*x-1)))))
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.03 \[ \int \frac {11+e^{2 x} x^2+e^x \left (-2 x+10 x^2\right )+\left (-11+12 e^x x-e^{2 x} x^2\right ) \log \left (\frac {-44 x+4 e^x x^2}{-1+e^x x}\right ) \log \left (\log \left (\frac {-44 x+4 e^x x^2}{-1+e^x x}\right )\right )}{\left (55 x^2-60 e^x x^3+5 e^{2 x} x^4\right ) \log \left (\frac {-44 x+4 e^x x^2}{-1+e^x x}\right )+\left (-110 x+120 e^x x^2-10 e^{2 x} x^3\right ) \log \left (\frac {-44 x+4 e^x x^2}{-1+e^x x}\right ) \log \left (\log \left (\frac {-44 x+4 e^x x^2}{-1+e^x x}\right )\right )+\left (55-60 e^x x+5 e^{2 x} x^2\right ) \log \left (\frac {-44 x+4 e^x x^2}{-1+e^x x}\right ) \log ^2\left (\log \left (\frac {-44 x+4 e^x x^2}{-1+e^x x}\right )\right )} \, dx=\frac {x}{5 \, {\left (x - \log \left (\log \left (\frac {4 \, {\left (x^{2} e^{x} - 11 \, x\right )}}{x e^{x} - 1}\right )\right )\right )}} \] Input:

integrate(((-exp(x)^2*x^2+12*exp(x)*x-11)*log((4*exp(x)*x^2-44*x)/(exp(x)* 
x-1))*log(log((4*exp(x)*x^2-44*x)/(exp(x)*x-1)))+exp(x)^2*x^2+(10*x^2-2*x) 
*exp(x)+11)/((5*exp(x)^2*x^2-60*exp(x)*x+55)*log((4*exp(x)*x^2-44*x)/(exp( 
x)*x-1))*log(log((4*exp(x)*x^2-44*x)/(exp(x)*x-1)))^2+(-10*exp(x)^2*x^3+12 
0*exp(x)*x^2-110*x)*log((4*exp(x)*x^2-44*x)/(exp(x)*x-1))*log(log((4*exp(x 
)*x^2-44*x)/(exp(x)*x-1)))+(5*exp(x)^2*x^4-60*exp(x)*x^3+55*x^2)*log((4*ex 
p(x)*x^2-44*x)/(exp(x)*x-1))),x, algorithm="fricas")
 

Output:

1/5*x/(x - log(log(4*(x^2*e^x - 11*x)/(x*e^x - 1))))
 

Sympy [A] (verification not implemented)

Time = 1.46 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.97 \[ \int \frac {11+e^{2 x} x^2+e^x \left (-2 x+10 x^2\right )+\left (-11+12 e^x x-e^{2 x} x^2\right ) \log \left (\frac {-44 x+4 e^x x^2}{-1+e^x x}\right ) \log \left (\log \left (\frac {-44 x+4 e^x x^2}{-1+e^x x}\right )\right )}{\left (55 x^2-60 e^x x^3+5 e^{2 x} x^4\right ) \log \left (\frac {-44 x+4 e^x x^2}{-1+e^x x}\right )+\left (-110 x+120 e^x x^2-10 e^{2 x} x^3\right ) \log \left (\frac {-44 x+4 e^x x^2}{-1+e^x x}\right ) \log \left (\log \left (\frac {-44 x+4 e^x x^2}{-1+e^x x}\right )\right )+\left (55-60 e^x x+5 e^{2 x} x^2\right ) \log \left (\frac {-44 x+4 e^x x^2}{-1+e^x x}\right ) \log ^2\left (\log \left (\frac {-44 x+4 e^x x^2}{-1+e^x x}\right )\right )} \, dx=- \frac {x}{- 5 x + 5 \log {\left (\log {\left (\frac {4 x^{2} e^{x} - 44 x}{x e^{x} - 1} \right )} \right )}} \] Input:

integrate(((-exp(x)**2*x**2+12*exp(x)*x-11)*ln((4*exp(x)*x**2-44*x)/(exp(x 
)*x-1))*ln(ln((4*exp(x)*x**2-44*x)/(exp(x)*x-1)))+exp(x)**2*x**2+(10*x**2- 
2*x)*exp(x)+11)/((5*exp(x)**2*x**2-60*exp(x)*x+55)*ln((4*exp(x)*x**2-44*x) 
/(exp(x)*x-1))*ln(ln((4*exp(x)*x**2-44*x)/(exp(x)*x-1)))**2+(-10*exp(x)**2 
*x**3+120*exp(x)*x**2-110*x)*ln((4*exp(x)*x**2-44*x)/(exp(x)*x-1))*ln(ln(( 
4*exp(x)*x**2-44*x)/(exp(x)*x-1)))+(5*exp(x)**2*x**4-60*exp(x)*x**3+55*x** 
2)*ln((4*exp(x)*x**2-44*x)/(exp(x)*x-1))),x)
 

Output:

-x/(-5*x + 5*log(log((4*x**2*exp(x) - 44*x)/(x*exp(x) - 1))))
 

Maxima [A] (verification not implemented)

Time = 1.00 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.10 \[ \int \frac {11+e^{2 x} x^2+e^x \left (-2 x+10 x^2\right )+\left (-11+12 e^x x-e^{2 x} x^2\right ) \log \left (\frac {-44 x+4 e^x x^2}{-1+e^x x}\right ) \log \left (\log \left (\frac {-44 x+4 e^x x^2}{-1+e^x x}\right )\right )}{\left (55 x^2-60 e^x x^3+5 e^{2 x} x^4\right ) \log \left (\frac {-44 x+4 e^x x^2}{-1+e^x x}\right )+\left (-110 x+120 e^x x^2-10 e^{2 x} x^3\right ) \log \left (\frac {-44 x+4 e^x x^2}{-1+e^x x}\right ) \log \left (\log \left (\frac {-44 x+4 e^x x^2}{-1+e^x x}\right )\right )+\left (55-60 e^x x+5 e^{2 x} x^2\right ) \log \left (\frac {-44 x+4 e^x x^2}{-1+e^x x}\right ) \log ^2\left (\log \left (\frac {-44 x+4 e^x x^2}{-1+e^x x}\right )\right )} \, dx=\frac {x}{5 \, {\left (x - \log \left (2 \, \log \left (2\right ) - \log \left (x e^{x} - 1\right ) + \log \left (x e^{x} - 11\right ) + \log \left (x\right )\right )\right )}} \] Input:

integrate(((-exp(x)^2*x^2+12*exp(x)*x-11)*log((4*exp(x)*x^2-44*x)/(exp(x)* 
x-1))*log(log((4*exp(x)*x^2-44*x)/(exp(x)*x-1)))+exp(x)^2*x^2+(10*x^2-2*x) 
*exp(x)+11)/((5*exp(x)^2*x^2-60*exp(x)*x+55)*log((4*exp(x)*x^2-44*x)/(exp( 
x)*x-1))*log(log((4*exp(x)*x^2-44*x)/(exp(x)*x-1)))^2+(-10*exp(x)^2*x^3+12 
0*exp(x)*x^2-110*x)*log((4*exp(x)*x^2-44*x)/(exp(x)*x-1))*log(log((4*exp(x 
)*x^2-44*x)/(exp(x)*x-1)))+(5*exp(x)^2*x^4-60*exp(x)*x^3+55*x^2)*log((4*ex 
p(x)*x^2-44*x)/(exp(x)*x-1))),x, algorithm="maxima")
 

Output:

1/5*x/(x - log(2*log(2) - log(x*e^x - 1) + log(x*e^x - 11) + log(x)))
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1340 vs. \(2 (27) = 54\).

Time = 3.44 (sec) , antiderivative size = 1340, normalized size of antiderivative = 44.67 \[ \int \frac {11+e^{2 x} x^2+e^x \left (-2 x+10 x^2\right )+\left (-11+12 e^x x-e^{2 x} x^2\right ) \log \left (\frac {-44 x+4 e^x x^2}{-1+e^x x}\right ) \log \left (\log \left (\frac {-44 x+4 e^x x^2}{-1+e^x x}\right )\right )}{\left (55 x^2-60 e^x x^3+5 e^{2 x} x^4\right ) \log \left (\frac {-44 x+4 e^x x^2}{-1+e^x x}\right )+\left (-110 x+120 e^x x^2-10 e^{2 x} x^3\right ) \log \left (\frac {-44 x+4 e^x x^2}{-1+e^x x}\right ) \log \left (\log \left (\frac {-44 x+4 e^x x^2}{-1+e^x x}\right )\right )+\left (55-60 e^x x+5 e^{2 x} x^2\right ) \log \left (\frac {-44 x+4 e^x x^2}{-1+e^x x}\right ) \log ^2\left (\log \left (\frac {-44 x+4 e^x x^2}{-1+e^x x}\right )\right )} \, dx=\text {Too large to display} \] Input:

integrate(((-exp(x)^2*x^2+12*exp(x)*x-11)*log((4*exp(x)*x^2-44*x)/(exp(x)* 
x-1))*log(log((4*exp(x)*x^2-44*x)/(exp(x)*x-1)))+exp(x)^2*x^2+(10*x^2-2*x) 
*exp(x)+11)/((5*exp(x)^2*x^2-60*exp(x)*x+55)*log((4*exp(x)*x^2-44*x)/(exp( 
x)*x-1))*log(log((4*exp(x)*x^2-44*x)/(exp(x)*x-1)))^2+(-10*exp(x)^2*x^3+12 
0*exp(x)*x^2-110*x)*log((4*exp(x)*x^2-44*x)/(exp(x)*x-1))*log(log((4*exp(x 
)*x^2-44*x)/(exp(x)*x-1)))+(5*exp(x)^2*x^4-60*exp(x)*x^3+55*x^2)*log((4*ex 
p(x)*x^2-44*x)/(exp(x)*x-1))),x, algorithm="giac")
 

Output:

1/5*(x^4*e^(2*x)*log(4*x^2*e^x - 44*x)*log(4*(x^2*e^x - 11*x)/(x*e^x - 1)) 
 - x^4*e^(2*x)*log(x*e^x - 1)*log(4*(x^2*e^x - 11*x)/(x*e^x - 1)) - 12*x^3 
*e^x*log(4*x^2*e^x - 44*x)*log(4*(x^2*e^x - 11*x)/(x*e^x - 1)) + 12*x^3*e^ 
x*log(x*e^x - 1)*log(4*(x^2*e^x - 11*x)/(x*e^x - 1)) - x^3*e^(2*x)*log(4*( 
x^2*e^x - 11*x)/(x*e^x - 1)) - 10*x^3*e^x*log(4*(x^2*e^x - 11*x)/(x*e^x - 
1)) + 2*x^2*e^x*log(4*(x^2*e^x - 11*x)/(x*e^x - 1)) + 11*x^2*log(4*x^2*e^x 
 - 44*x)*log(4*(x^2*e^x - 11*x)/(x*e^x - 1)) - 11*x^2*log(x*e^x - 1)*log(4 
*(x^2*e^x - 11*x)/(x*e^x - 1)) - 11*x*log(4*(x^2*e^x - 11*x)/(x*e^x - 1))) 
/(x^4*e^(2*x)*log(4*x^2*e^x - 44*x)*log(4*(x^2*e^x - 11*x)/(x*e^x - 1)) - 
x^4*e^(2*x)*log(x*e^x - 1)*log(4*(x^2*e^x - 11*x)/(x*e^x - 1)) - x^3*e^(2* 
x)*log(4*x^2*e^x - 44*x)*log(4*(x^2*e^x - 11*x)/(x*e^x - 1))*log(log(4*(x^ 
2*e^x - 11*x)/(x*e^x - 1))) + x^3*e^(2*x)*log(x*e^x - 1)*log(4*(x^2*e^x - 
11*x)/(x*e^x - 1))*log(log(4*(x^2*e^x - 11*x)/(x*e^x - 1))) - 12*x^3*e^x*l 
og(4*x^2*e^x - 44*x)*log(4*(x^2*e^x - 11*x)/(x*e^x - 1)) + 12*x^3*e^x*log( 
x*e^x - 1)*log(4*(x^2*e^x - 11*x)/(x*e^x - 1)) + 12*x^2*e^x*log(4*x^2*e^x 
- 44*x)*log(4*(x^2*e^x - 11*x)/(x*e^x - 1))*log(log(4*(x^2*e^x - 11*x)/(x* 
e^x - 1))) - 12*x^2*e^x*log(x*e^x - 1)*log(4*(x^2*e^x - 11*x)/(x*e^x - 1)) 
*log(log(4*(x^2*e^x - 11*x)/(x*e^x - 1))) - x^3*e^(2*x)*log(4*x^2*e^x - 44 
*x) - 10*x^3*e^x*log(4*x^2*e^x - 44*x) + x^3*e^(2*x)*log(x*e^x - 1) + 10*x 
^3*e^x*log(x*e^x - 1) + x^2*e^(2*x)*log(4*x^2*e^x - 44*x)*log(log(4*(x^...
 

Mupad [F(-1)]

Timed out. \[ \int \frac {11+e^{2 x} x^2+e^x \left (-2 x+10 x^2\right )+\left (-11+12 e^x x-e^{2 x} x^2\right ) \log \left (\frac {-44 x+4 e^x x^2}{-1+e^x x}\right ) \log \left (\log \left (\frac {-44 x+4 e^x x^2}{-1+e^x x}\right )\right )}{\left (55 x^2-60 e^x x^3+5 e^{2 x} x^4\right ) \log \left (\frac {-44 x+4 e^x x^2}{-1+e^x x}\right )+\left (-110 x+120 e^x x^2-10 e^{2 x} x^3\right ) \log \left (\frac {-44 x+4 e^x x^2}{-1+e^x x}\right ) \log \left (\log \left (\frac {-44 x+4 e^x x^2}{-1+e^x x}\right )\right )+\left (55-60 e^x x+5 e^{2 x} x^2\right ) \log \left (\frac {-44 x+4 e^x x^2}{-1+e^x x}\right ) \log ^2\left (\log \left (\frac {-44 x+4 e^x x^2}{-1+e^x x}\right )\right )} \, dx=\int \frac {x^2\,{\mathrm {e}}^{2\,x}-{\mathrm {e}}^x\,\left (2\,x-10\,x^2\right )-\ln \left (-\frac {44\,x-4\,x^2\,{\mathrm {e}}^x}{x\,{\mathrm {e}}^x-1}\right )\,\ln \left (\ln \left (-\frac {44\,x-4\,x^2\,{\mathrm {e}}^x}{x\,{\mathrm {e}}^x-1}\right )\right )\,\left (x^2\,{\mathrm {e}}^{2\,x}-12\,x\,{\mathrm {e}}^x+11\right )+11}{\ln \left (-\frac {44\,x-4\,x^2\,{\mathrm {e}}^x}{x\,{\mathrm {e}}^x-1}\right )\,\left (5\,x^2\,{\mathrm {e}}^{2\,x}-60\,x\,{\mathrm {e}}^x+55\right )\,{\ln \left (\ln \left (-\frac {44\,x-4\,x^2\,{\mathrm {e}}^x}{x\,{\mathrm {e}}^x-1}\right )\right )}^2-\ln \left (-\frac {44\,x-4\,x^2\,{\mathrm {e}}^x}{x\,{\mathrm {e}}^x-1}\right )\,\left (110\,x-120\,x^2\,{\mathrm {e}}^x+10\,x^3\,{\mathrm {e}}^{2\,x}\right )\,\ln \left (\ln \left (-\frac {44\,x-4\,x^2\,{\mathrm {e}}^x}{x\,{\mathrm {e}}^x-1}\right )\right )+\ln \left (-\frac {44\,x-4\,x^2\,{\mathrm {e}}^x}{x\,{\mathrm {e}}^x-1}\right )\,\left (5\,x^4\,{\mathrm {e}}^{2\,x}-60\,x^3\,{\mathrm {e}}^x+55\,x^2\right )} \,d x \] Input:

int((x^2*exp(2*x) - exp(x)*(2*x - 10*x^2) - log(-(44*x - 4*x^2*exp(x))/(x* 
exp(x) - 1))*log(log(-(44*x - 4*x^2*exp(x))/(x*exp(x) - 1)))*(x^2*exp(2*x) 
 - 12*x*exp(x) + 11) + 11)/(log(-(44*x - 4*x^2*exp(x))/(x*exp(x) - 1))*(5* 
x^4*exp(2*x) - 60*x^3*exp(x) + 55*x^2) - log(-(44*x - 4*x^2*exp(x))/(x*exp 
(x) - 1))*log(log(-(44*x - 4*x^2*exp(x))/(x*exp(x) - 1)))*(110*x - 120*x^2 
*exp(x) + 10*x^3*exp(2*x)) + log(-(44*x - 4*x^2*exp(x))/(x*exp(x) - 1))*lo 
g(log(-(44*x - 4*x^2*exp(x))/(x*exp(x) - 1)))^2*(5*x^2*exp(2*x) - 60*x*exp 
(x) + 55)),x)
 

Output:

int((x^2*exp(2*x) - exp(x)*(2*x - 10*x^2) - log(-(44*x - 4*x^2*exp(x))/(x* 
exp(x) - 1))*log(log(-(44*x - 4*x^2*exp(x))/(x*exp(x) - 1)))*(x^2*exp(2*x) 
 - 12*x*exp(x) + 11) + 11)/(log(-(44*x - 4*x^2*exp(x))/(x*exp(x) - 1))*(5* 
x^4*exp(2*x) - 60*x^3*exp(x) + 55*x^2) - log(-(44*x - 4*x^2*exp(x))/(x*exp 
(x) - 1))*log(log(-(44*x - 4*x^2*exp(x))/(x*exp(x) - 1)))*(110*x - 120*x^2 
*exp(x) + 10*x^3*exp(2*x)) + log(-(44*x - 4*x^2*exp(x))/(x*exp(x) - 1))*lo 
g(log(-(44*x - 4*x^2*exp(x))/(x*exp(x) - 1)))^2*(5*x^2*exp(2*x) - 60*x*exp 
(x) + 55)), x)
 

Reduce [B] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.93 \[ \int \frac {11+e^{2 x} x^2+e^x \left (-2 x+10 x^2\right )+\left (-11+12 e^x x-e^{2 x} x^2\right ) \log \left (\frac {-44 x+4 e^x x^2}{-1+e^x x}\right ) \log \left (\log \left (\frac {-44 x+4 e^x x^2}{-1+e^x x}\right )\right )}{\left (55 x^2-60 e^x x^3+5 e^{2 x} x^4\right ) \log \left (\frac {-44 x+4 e^x x^2}{-1+e^x x}\right )+\left (-110 x+120 e^x x^2-10 e^{2 x} x^3\right ) \log \left (\frac {-44 x+4 e^x x^2}{-1+e^x x}\right ) \log \left (\log \left (\frac {-44 x+4 e^x x^2}{-1+e^x x}\right )\right )+\left (55-60 e^x x+5 e^{2 x} x^2\right ) \log \left (\frac {-44 x+4 e^x x^2}{-1+e^x x}\right ) \log ^2\left (\log \left (\frac {-44 x+4 e^x x^2}{-1+e^x x}\right )\right )} \, dx=-\frac {\mathrm {log}\left (\mathrm {log}\left (\frac {4 e^{x} x^{2}-44 x}{e^{x} x -1}\right )\right )}{5 \,\mathrm {log}\left (\mathrm {log}\left (\frac {4 e^{x} x^{2}-44 x}{e^{x} x -1}\right )\right )-5 x} \] Input:

int(((-exp(x)^2*x^2+12*exp(x)*x-11)*log((4*exp(x)*x^2-44*x)/(exp(x)*x-1))* 
log(log((4*exp(x)*x^2-44*x)/(exp(x)*x-1)))+exp(x)^2*x^2+(10*x^2-2*x)*exp(x 
)+11)/((5*exp(x)^2*x^2-60*exp(x)*x+55)*log((4*exp(x)*x^2-44*x)/(exp(x)*x-1 
))*log(log((4*exp(x)*x^2-44*x)/(exp(x)*x-1)))^2+(-10*exp(x)^2*x^3+120*exp( 
x)*x^2-110*x)*log((4*exp(x)*x^2-44*x)/(exp(x)*x-1))*log(log((4*exp(x)*x^2- 
44*x)/(exp(x)*x-1)))+(5*exp(x)^2*x^4-60*exp(x)*x^3+55*x^2)*log((4*exp(x)*x 
^2-44*x)/(exp(x)*x-1))),x)
 

Output:

( - log(log((4*e**x*x**2 - 44*x)/(e**x*x - 1))))/(5*(log(log((4*e**x*x**2 
- 44*x)/(e**x*x - 1))) - x))