\(\int \frac {2 e^x x+(4 x^2+e^x (-4 x+2 x^2)) \log (\frac {x}{2})+(4 x^2-4 x^2 \log (\frac {x}{2})) \log (x)+(-e^x+(e^x (1-x)-4 x) \log (\frac {x}{2})+(-4 x+4 x \log (\frac {x}{2})) \log (x)) \log (\log (\log (4)))+(\log (\frac {x}{2})+(1-\log (\frac {x}{2})) \log (x)) \log ^2(\log (\log (4)))}{4 x^4-4 x^3 \log (\log (\log (4)))+x^2 \log ^2(\log (\log (4)))} \, dx\) [1824]

Optimal result

Integrand size = 155, antiderivative size = 29 \[ \int \frac {2 e^x x+\left (4 x^2+e^x \left (-4 x+2 x^2\right )\right ) \log \left (\frac {x}{2}\right )+\left (4 x^2-4 x^2 \log \left (\frac {x}{2}\right )\right ) \log (x)+\left (-e^x+\left (e^x (1-x)-4 x\right ) \log \left (\frac {x}{2}\right )+\left (-4 x+4 x \log \left (\frac {x}{2}\right )\right ) \log (x)\right ) \log (\log (\log (4)))+\left (\log \left (\frac {x}{2}\right )+\left (1-\log \left (\frac {x}{2}\right )\right ) \log (x)\right ) \log ^2(\log (\log (4)))}{4 x^4-4 x^3 \log (\log (\log (4)))+x^2 \log ^2(\log (\log (4)))} \, dx=\frac {\log \left (\frac {x}{2}\right ) \left (\log (x)+\frac {e^x}{2 x-\log (\log (\log (4)))}\right )}{x} \] Output:

ln(1/2*x)/x*(ln(x)+exp(x)/(2*x-ln(ln(2*ln(2)))))
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00 \[ \int \frac {2 e^x x+\left (4 x^2+e^x \left (-4 x+2 x^2\right )\right ) \log \left (\frac {x}{2}\right )+\left (4 x^2-4 x^2 \log \left (\frac {x}{2}\right )\right ) \log (x)+\left (-e^x+\left (e^x (1-x)-4 x\right ) \log \left (\frac {x}{2}\right )+\left (-4 x+4 x \log \left (\frac {x}{2}\right )\right ) \log (x)\right ) \log (\log (\log (4)))+\left (\log \left (\frac {x}{2}\right )+\left (1-\log \left (\frac {x}{2}\right )\right ) \log (x)\right ) \log ^2(\log (\log (4)))}{4 x^4-4 x^3 \log (\log (\log (4)))+x^2 \log ^2(\log (\log (4)))} \, dx=\frac {\log \left (\frac {x}{2}\right ) \left (\log (x)+\frac {e^x}{2 x-\log (\log (\log (4)))}\right )}{x} \] Input:

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

Output:

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

Rubi [C] (verified)

Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.

Time = 5.18 (sec) , antiderivative size = 372, normalized size of antiderivative = 12.83, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.032, Rules used = {2026, 7277, 27, 7293, 2009}

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.

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

\(\Big \downarrow \) 2026

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

\(\Big \downarrow \) 7277

\(\displaystyle 16 \int \frac {2 e^x x+2 \left (2 x^2-e^x \left (2 x-x^2\right )\right ) \log \left (\frac {x}{2}\right )+4 \left (x^2-x^2 \log \left (\frac {x}{2}\right )\right ) \log (x)+\left (\log \left (\frac {x}{2}\right )+\left (1-\log \left (\frac {x}{2}\right )\right ) \log (x)\right ) \log ^2(\log (\log (4)))-\left (-\left (\left (e^x (1-x)-4 x\right ) \log \left (\frac {x}{2}\right )\right )+e^x+4 \left (x-x \log \left (\frac {x}{2}\right )\right ) \log (x)\right ) \log (\log (\log (4)))}{16 x^2 (2 x-\log (\log (\log (4))))^2}dx\)

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {4 \operatorname {PolyLog}\left (2,\frac {2 x}{\log (\log (\log (4)))}\right )}{\log (\log (\log (4)))}-\frac {4 \operatorname {PolyLog}\left (2,\frac {\log (\log (\log (4)))}{2 x}\right )}{\log (\log (\log (4)))}+\frac {1}{x}+\frac {2 \left (1-\log \left (\frac {x}{2}\right )\right )^3}{3 \log (\log (\log (4)))}+\frac {2 \log (x) \left (1-\log \left (\frac {x}{2}\right )\right )^2}{\log (\log (\log (4)))}-\frac {-\log (x)+1+\log (2)}{x}-\frac {\left (2-\log \left (\frac {x}{2}\right )\right ) \log (x)}{x}+\frac {\log (x)}{x}-\frac {4 \log (x) \log \left (1-\frac {2 x}{\log (\log (\log (4)))}\right )}{\log (\log (\log (4)))}+\frac {4 \log (2) \log (2 x-\log (\log (\log (4))))}{\log (\log (\log (4)))}+\frac {4 \log \left (\frac {x}{2}\right ) \log \left (1-\frac {\log (\log (\log (4)))}{2 x}\right )}{\log (\log (\log (4)))}+\frac {2 \log (2)}{2 x-\log (\log (\log (4)))}-\frac {2 \left (2-\log \left (\frac {x}{2}\right )\right )^3}{3 \log (\log (\log (4)))}-\frac {e^x \log \left (\frac {x}{2}\right )}{x \log (\log (\log (4)))}-\frac {2 \left (2-\log \left (\frac {x}{2}\right )\right )^2 \log (x)}{\log (\log (\log (4)))}-\frac {4 \log (2) \log (x)}{\log (\log (\log (4)))}+\frac {2 e^x \log \left (\frac {x}{2}\right )}{\log (\log (\log (4))) (2 x-\log (\log (\log (4))))}+\frac {4 x \log \left (\frac {x}{2}\right )}{\log (\log (\log (4))) (2 x-\log (\log (\log (4))))}-\frac {4 x \log (x)}{\log (\log (\log (4))) (2 x-\log (\log (\log (4))))}+\frac {\log (2)}{x}\)

Input:

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

Output:

x^(-1) + Log[2]/x - (1 + Log[2] - Log[x])/x + Log[x]/x - ((2 - Log[x/2])*L 
og[x])/x + (2*Log[2])/(2*x - Log[Log[Log[4]]]) + (2*(1 - Log[x/2])^3)/(3*L 
og[Log[Log[4]]]) - (2*(2 - Log[x/2])^3)/(3*Log[Log[Log[4]]]) - (E^x*Log[x/ 
2])/(x*Log[Log[Log[4]]]) - (4*Log[2]*Log[x])/Log[Log[Log[4]]] + (2*(1 - Lo 
g[x/2])^2*Log[x])/Log[Log[Log[4]]] - (2*(2 - Log[x/2])^2*Log[x])/Log[Log[L 
og[4]]] + (2*E^x*Log[x/2])/((2*x - Log[Log[Log[4]]])*Log[Log[Log[4]]]) + ( 
4*x*Log[x/2])/((2*x - Log[Log[Log[4]]])*Log[Log[Log[4]]]) - (4*x*Log[x])/( 
(2*x - Log[Log[Log[4]]])*Log[Log[Log[4]]]) - (4*Log[x]*Log[1 - (2*x)/Log[L 
og[Log[4]]]])/Log[Log[Log[4]]] + (4*Log[2]*Log[2*x - Log[Log[Log[4]]]])/Lo 
g[Log[Log[4]]] + (4*Log[x/2]*Log[1 - Log[Log[Log[4]]]/(2*x)])/Log[Log[Log[ 
4]]] - (4*PolyLog[2, (2*x)/Log[Log[Log[4]]]])/Log[Log[Log[4]]] - (4*PolyLo 
g[2, Log[Log[Log[4]]]/(2*x)])/Log[Log[Log[4]]]
 

Defintions of rubi rules used

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 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2026

Int[(Fx_.)*(Px_)^(p_.), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Int[x^(p 
*r)*ExpandToSum[Px/x^r, x]^p*Fx, x] /; IGtQ[r, 0]] /; PolyQ[Px, x] && Integ 
erQ[p] &&  !MonomialQ[Px, x] && (ILtQ[p, 0] ||  !PolyQ[u, x])
 

rule 7277

Int[(u_)*((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.))^(p_.), x_Symbol] :> 
 Simp[1/(4^p*c^p)   Int[u*(b + 2*c*x^n)^(2*p), x], x] /; FreeQ[{a, b, c, n} 
, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p] &&  !AlgebraicFu 
nctionQ[u, x]
 

rule 7293

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

Maple [A] (verified)

Time = 0.56 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.62

\[\frac {{\mathrm e}^{x} \ln \left (2\right )-{\mathrm e}^{x} \ln \left (x \right )}{x \left (\ln \left (\ln \left (2 \ln \left (2\right )\right )\right )-2 x \right )}-\frac {\ln \left (2\right ) \ln \left (x \right )}{x}+\frac {\ln \left (x \right )^{2}}{x}\]

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 63 vs. \(2 (28) = 56\).

Time = 0.09 (sec) , antiderivative size = 63, normalized size of antiderivative = 2.17 \[ \int \frac {2 e^x x+\left (4 x^2+e^x \left (-4 x+2 x^2\right )\right ) \log \left (\frac {x}{2}\right )+\left (4 x^2-4 x^2 \log \left (\frac {x}{2}\right )\right ) \log (x)+\left (-e^x+\left (e^x (1-x)-4 x\right ) \log \left (\frac {x}{2}\right )+\left (-4 x+4 x \log \left (\frac {x}{2}\right )\right ) \log (x)\right ) \log (\log (\log (4)))+\left (\log \left (\frac {x}{2}\right )+\left (1-\log \left (\frac {x}{2}\right )\right ) \log (x)\right ) \log ^2(\log (\log (4)))}{4 x^4-4 x^3 \log (\log (\log (4)))+x^2 \log ^2(\log (\log (4)))} \, dx=\frac {2 \, x \log \left (\frac {1}{2} \, x\right )^{2} + {\left (2 \, x \log \left (2\right ) + e^{x}\right )} \log \left (\frac {1}{2} \, x\right ) - {\left (\log \left (2\right ) \log \left (\frac {1}{2} \, x\right ) + \log \left (\frac {1}{2} \, x\right )^{2}\right )} \log \left (\log \left (2 \, \log \left (2\right )\right )\right )}{2 \, x^{2} - x \log \left (\log \left (2 \, \log \left (2\right )\right )\right )} \] Input:

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

Output:

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

Sympy [A] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.41 \[ \int \frac {2 e^x x+\left (4 x^2+e^x \left (-4 x+2 x^2\right )\right ) \log \left (\frac {x}{2}\right )+\left (4 x^2-4 x^2 \log \left (\frac {x}{2}\right )\right ) \log (x)+\left (-e^x+\left (e^x (1-x)-4 x\right ) \log \left (\frac {x}{2}\right )+\left (-4 x+4 x \log \left (\frac {x}{2}\right )\right ) \log (x)\right ) \log (\log (\log (4)))+\left (\log \left (\frac {x}{2}\right )+\left (1-\log \left (\frac {x}{2}\right )\right ) \log (x)\right ) \log ^2(\log (\log (4)))}{4 x^4-4 x^3 \log (\log (\log (4)))+x^2 \log ^2(\log (\log (4)))} \, dx=\frac {\left (\log {\left (x \right )} - \log {\left (2 \right )}\right ) e^{x}}{2 x^{2} - x \log {\left (\log {\left (\log {\left (2 \right )} \right )} + \log {\left (2 \right )} \right )}} + \frac {\log {\left (x \right )}^{2}}{x} - \frac {\log {\left (2 \right )} \log {\left (x \right )}}{x} \] Input:

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

Output:

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 70 vs. \(2 (28) = 56\).

Time = 0.20 (sec) , antiderivative size = 70, normalized size of antiderivative = 2.41 \[ \int \frac {2 e^x x+\left (4 x^2+e^x \left (-4 x+2 x^2\right )\right ) \log \left (\frac {x}{2}\right )+\left (4 x^2-4 x^2 \log \left (\frac {x}{2}\right )\right ) \log (x)+\left (-e^x+\left (e^x (1-x)-4 x\right ) \log \left (\frac {x}{2}\right )+\left (-4 x+4 x \log \left (\frac {x}{2}\right )\right ) \log (x)\right ) \log (\log (\log (4)))+\left (\log \left (\frac {x}{2}\right )+\left (1-\log \left (\frac {x}{2}\right )\right ) \log (x)\right ) \log ^2(\log (\log (4)))}{4 x^4-4 x^3 \log (\log (\log (4)))+x^2 \log ^2(\log (\log (4)))} \, dx=\frac {{\left (2 \, x - \log \left (\log \left (2\right ) + \log \left (\log \left (2\right )\right )\right )\right )} \log \left (x\right )^{2} - {\left (\log \left (2\right ) - \log \left (x\right )\right )} e^{x} - {\left (2 \, x \log \left (2\right ) - \log \left (2\right ) \log \left (\log \left (2\right ) + \log \left (\log \left (2\right )\right )\right )\right )} \log \left (x\right )}{2 \, x^{2} - x \log \left (\log \left (2\right ) + \log \left (\log \left (2\right )\right )\right )} \] Input:

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

Output:

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

Giac [B] (verification not implemented)

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

Time = 0.14 (sec) , antiderivative size = 75, normalized size of antiderivative = 2.59 \[ \int \frac {2 e^x x+\left (4 x^2+e^x \left (-4 x+2 x^2\right )\right ) \log \left (\frac {x}{2}\right )+\left (4 x^2-4 x^2 \log \left (\frac {x}{2}\right )\right ) \log (x)+\left (-e^x+\left (e^x (1-x)-4 x\right ) \log \left (\frac {x}{2}\right )+\left (-4 x+4 x \log \left (\frac {x}{2}\right )\right ) \log (x)\right ) \log (\log (\log (4)))+\left (\log \left (\frac {x}{2}\right )+\left (1-\log \left (\frac {x}{2}\right )\right ) \log (x)\right ) \log ^2(\log (\log (4)))}{4 x^4-4 x^3 \log (\log (\log (4)))+x^2 \log ^2(\log (\log (4)))} \, dx=\frac {2 \, x \log \left (2\right ) \log \left (\frac {1}{2} \, x\right ) + 2 \, x \log \left (\frac {1}{2} \, x\right )^{2} - \log \left (2\right ) \log \left (\frac {1}{2} \, x\right ) \log \left (\log \left (2\right ) + \log \left (\log \left (2\right )\right )\right ) - \log \left (\frac {1}{2} \, x\right )^{2} \log \left (\log \left (2\right ) + \log \left (\log \left (2\right )\right )\right ) + e^{x} \log \left (\frac {1}{2} \, x\right )}{2 \, x^{2} - x \log \left (\log \left (2\right ) + \log \left (\log \left (2\right )\right )\right )} \] Input:

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

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {2 e^x x+\left (4 x^2+e^x \left (-4 x+2 x^2\right )\right ) \log \left (\frac {x}{2}\right )+\left (4 x^2-4 x^2 \log \left (\frac {x}{2}\right )\right ) \log (x)+\left (-e^x+\left (e^x (1-x)-4 x\right ) \log \left (\frac {x}{2}\right )+\left (-4 x+4 x \log \left (\frac {x}{2}\right )\right ) \log (x)\right ) \log (\log (\log (4)))+\left (\log \left (\frac {x}{2}\right )+\left (1-\log \left (\frac {x}{2}\right )\right ) \log (x)\right ) \log ^2(\log (\log (4)))}{4 x^4-4 x^3 \log (\log (\log (4)))+x^2 \log ^2(\log (\log (4)))} \, dx=-\int \frac {\ln \left (x\right )\,\left (4\,x^2\,\ln \left (\frac {x}{2}\right )-4\,x^2\right )-2\,x\,{\mathrm {e}}^x+\ln \left (\ln \left (2\,\ln \left (2\right )\right )\right )\,\left ({\mathrm {e}}^x+\ln \left (x\right )\,\left (4\,x-4\,x\,\ln \left (\frac {x}{2}\right )\right )+\ln \left (\frac {x}{2}\right )\,\left (4\,x+{\mathrm {e}}^x\,\left (x-1\right )\right )\right )+\ln \left (\frac {x}{2}\right )\,\left ({\mathrm {e}}^x\,\left (4\,x-2\,x^2\right )-4\,x^2\right )-{\ln \left (\ln \left (2\,\ln \left (2\right )\right )\right )}^2\,\left (\ln \left (\frac {x}{2}\right )-\ln \left (x\right )\,\left (\ln \left (\frac {x}{2}\right )-1\right )\right )}{4\,x^4-4\,\ln \left (\ln \left (2\,\ln \left (2\right )\right )\right )\,x^3+{\ln \left (\ln \left (2\,\ln \left (2\right )\right )\right )}^2\,x^2} \,d x \] Input:

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.38 \[ \int \frac {2 e^x x+\left (4 x^2+e^x \left (-4 x+2 x^2\right )\right ) \log \left (\frac {x}{2}\right )+\left (4 x^2-4 x^2 \log \left (\frac {x}{2}\right )\right ) \log (x)+\left (-e^x+\left (e^x (1-x)-4 x\right ) \log \left (\frac {x}{2}\right )+\left (-4 x+4 x \log \left (\frac {x}{2}\right )\right ) \log (x)\right ) \log (\log (\log (4)))+\left (\log \left (\frac {x}{2}\right )+\left (1-\log \left (\frac {x}{2}\right )\right ) \log (x)\right ) \log ^2(\log (\log (4)))}{4 x^4-4 x^3 \log (\log (\log (4)))+x^2 \log ^2(\log (\log (4)))} \, dx=\frac {\mathrm {log}\left (\frac {x}{2}\right ) \left (-e^{x}+\mathrm {log}\left (\mathrm {log}\left (2 \,\mathrm {log}\left (2\right )\right )\right ) \mathrm {log}\left (x \right )-2 \,\mathrm {log}\left (x \right ) x \right )}{x \left (\mathrm {log}\left (\mathrm {log}\left (2 \,\mathrm {log}\left (2\right )\right )\right )-2 x \right )} \] Input:

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

Output:

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