\(\int \frac {81-\log (x)+e^{-5+x \log (\frac {81-\log (x)}{x})} (82-\log (x)+(-81+\log (x)) \log (\frac {81-\log (x)}{x}))}{\log (4+e^{-5+x \log (\frac {81-\log (x)}{x})}-x) (-1296+324 x+(16-4 x) \log (x)+e^{-5+x \log (\frac {81-\log (x)}{x})} (-324+4 \log (x)))} \, dx\) [2303]

Optimal result

Integrand size = 111, antiderivative size = 28 \[ \int \frac {81-\log (x)+e^{-5+x \log \left (\frac {81-\log (x)}{x}\right )} \left (82-\log (x)+(-81+\log (x)) \log \left (\frac {81-\log (x)}{x}\right )\right )}{\log \left (4+e^{-5+x \log \left (\frac {81-\log (x)}{x}\right )}-x\right ) \left (-1296+324 x+(16-4 x) \log (x)+e^{-5+x \log \left (\frac {81-\log (x)}{x}\right )} (-324+4 \log (x))\right )} \, dx=\frac {1}{4} \log \left (\log \left (4+e^{-5+x \log \left (\frac {81-\log (x)}{x}\right )}-x\right )\right ) \] Output:

1/4*ln(ln(exp(x*ln((-ln(x)+81)/x)-5)-x+4))
 

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.96 \[ \int \frac {81-\log (x)+e^{-5+x \log \left (\frac {81-\log (x)}{x}\right )} \left (82-\log (x)+(-81+\log (x)) \log \left (\frac {81-\log (x)}{x}\right )\right )}{\log \left (4+e^{-5+x \log \left (\frac {81-\log (x)}{x}\right )}-x\right ) \left (-1296+324 x+(16-4 x) \log (x)+e^{-5+x \log \left (\frac {81-\log (x)}{x}\right )} (-324+4 \log (x))\right )} \, dx=\frac {1}{4} \log \left (\log \left (4-x+\frac {\left (\frac {81-\log (x)}{x}\right )^x}{e^5}\right )\right ) \] Input:

Integrate[(81 - Log[x] + E^(-5 + x*Log[(81 - Log[x])/x])*(82 - Log[x] + (- 
81 + Log[x])*Log[(81 - Log[x])/x]))/(Log[4 + E^(-5 + x*Log[(81 - Log[x])/x 
]) - x]*(-1296 + 324*x + (16 - 4*x)*Log[x] + E^(-5 + x*Log[(81 - Log[x])/x 
])*(-324 + 4*Log[x]))),x]
 

Output:

Log[Log[4 - x + ((81 - Log[x])/x)^x/E^5]]/4
 

Rubi [A] (verified)

Time = 0.46 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.96, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.009, Rules used = {7235}

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.

\(\displaystyle \int \frac {-\log (x)+e^{x \log \left (\frac {81-\log (x)}{x}\right )-5} \left (-\log (x)+(\log (x)-81) \log \left (\frac {81-\log (x)}{x}\right )+82\right )+81}{\log \left (-x+e^{x \log \left (\frac {81-\log (x)}{x}\right )-5}+4\right ) \left (324 x+(16-4 x) \log (x)+e^{x \log \left (\frac {81-\log (x)}{x}\right )-5} (4 \log (x)-324)-1296\right )} \, dx\)

\(\Big \downarrow \) 7235

\(\displaystyle \frac {1}{4} \log \left (\log \left (-x+\frac {\left (\frac {81-\log (x)}{x}\right )^x}{e^5}+4\right )\right )\)

Input:

Int[(81 - Log[x] + E^(-5 + x*Log[(81 - Log[x])/x])*(82 - Log[x] + (-81 + L 
og[x])*Log[(81 - Log[x])/x]))/(Log[4 + E^(-5 + x*Log[(81 - Log[x])/x]) - x 
]*(-1296 + 324*x + (16 - 4*x)*Log[x] + E^(-5 + x*Log[(81 - Log[x])/x])*(-3 
24 + 4*Log[x]))),x]
 

Output:

Log[Log[4 - x + ((81 - Log[x])/x)^x/E^5]]/4
 

Defintions of rubi rules used

rule 7235

Int[(u_)/(y_), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[q*L 
og[RemoveContent[y, x]], x] /;  !FalseQ[q]]
 

Maple [A] (verified)

Time = 14.93 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.89

method	result	size
parallelrisch	\(\frac {\ln \left (\ln \left ({\mathrm e}^{x \ln \left (-\frac {\ln \left (x \right )-81}{x}\right )-5}-x +4\right )\right )}{4}\)	\(25\)
risch	\(\frac {\ln \left (\ln \left (x^{-x} \left (\ln \left (x \right )-81\right )^{x} {\mathrm e}^{-5} {\mathrm e}^{i \pi x} {\mathrm e}^{\frac {i x \pi \operatorname {csgn}\left (\frac {i \left (\ln \left (x \right )-81\right )}{x}\right )^{3}}{2}} {\mathrm e}^{\frac {i x \pi \operatorname {csgn}\left (\frac {i \left (\ln \left (x \right )-81\right )}{x}\right )^{2} \operatorname {csgn}\left (\frac {i}{x}\right )}{2}} {\mathrm e}^{\frac {i x \pi \operatorname {csgn}\left (\frac {i \left (\ln \left (x \right )-81\right )}{x}\right )^{2} \operatorname {csgn}\left (i \left (\ln \left (x \right )-81\right )\right )}{2}} {\mathrm e}^{-\frac {i x \pi \,\operatorname {csgn}\left (\frac {i \left (\ln \left (x \right )-81\right )}{x}\right ) \operatorname {csgn}\left (\frac {i}{x}\right ) \operatorname {csgn}\left (i \left (\ln \left (x \right )-81\right )\right )}{2}} {\mathrm e}^{-i x \pi \operatorname {csgn}\left (\frac {i \left (\ln \left (x \right )-81\right )}{x}\right )^{2}}-x +4\right )\right )}{4}\)	\(153\)

Input:

int((((ln(x)-81)*ln((-ln(x)+81)/x)-ln(x)+82)*exp(x*ln((-ln(x)+81)/x)-5)-ln 
(x)+81)/((4*ln(x)-324)*exp(x*ln((-ln(x)+81)/x)-5)+(-4*x+16)*ln(x)+324*x-12 
96)/ln(exp(x*ln((-ln(x)+81)/x)-5)-x+4),x,method=_RETURNVERBOSE)
 

Output:

1/4*ln(ln(exp(x*ln(-(ln(x)-81)/x)-5)-x+4))
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.86 \[ \int \frac {81-\log (x)+e^{-5+x \log \left (\frac {81-\log (x)}{x}\right )} \left (82-\log (x)+(-81+\log (x)) \log \left (\frac {81-\log (x)}{x}\right )\right )}{\log \left (4+e^{-5+x \log \left (\frac {81-\log (x)}{x}\right )}-x\right ) \left (-1296+324 x+(16-4 x) \log (x)+e^{-5+x \log \left (\frac {81-\log (x)}{x}\right )} (-324+4 \log (x))\right )} \, dx=\frac {1}{4} \, \log \left (\log \left (-x + e^{\left (x \log \left (-\frac {\log \left (x\right ) - 81}{x}\right ) - 5\right )} + 4\right )\right ) \] Input:

integrate((((log(x)-81)*log((-log(x)+81)/x)-log(x)+82)*exp(x*log((-log(x)+ 
81)/x)-5)-log(x)+81)/((4*log(x)-324)*exp(x*log((-log(x)+81)/x)-5)+(-4*x+16 
)*log(x)+324*x-1296)/log(exp(x*log((-log(x)+81)/x)-5)-x+4),x, algorithm="f 
ricas")
 

Output:

1/4*log(log(-x + e^(x*log(-(log(x) - 81)/x) - 5) + 4))
 

Sympy [F(-1)]

Timed out. \[ \int \frac {81-\log (x)+e^{-5+x \log \left (\frac {81-\log (x)}{x}\right )} \left (82-\log (x)+(-81+\log (x)) \log \left (\frac {81-\log (x)}{x}\right )\right )}{\log \left (4+e^{-5+x \log \left (\frac {81-\log (x)}{x}\right )}-x\right ) \left (-1296+324 x+(16-4 x) \log (x)+e^{-5+x \log \left (\frac {81-\log (x)}{x}\right )} (-324+4 \log (x))\right )} \, dx=\text {Timed out} \] Input:

integrate((((ln(x)-81)*ln((-ln(x)+81)/x)-ln(x)+82)*exp(x*ln((-ln(x)+81)/x) 
-5)-ln(x)+81)/((4*ln(x)-324)*exp(x*ln((-ln(x)+81)/x)-5)+(-4*x+16)*ln(x)+32 
4*x-1296)/ln(exp(x*ln((-ln(x)+81)/x)-5)-x+4),x)
 

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.25 \[ \int \frac {81-\log (x)+e^{-5+x \log \left (\frac {81-\log (x)}{x}\right )} \left (82-\log (x)+(-81+\log (x)) \log \left (\frac {81-\log (x)}{x}\right )\right )}{\log \left (4+e^{-5+x \log \left (\frac {81-\log (x)}{x}\right )}-x\right ) \left (-1296+324 x+(16-4 x) \log (x)+e^{-5+x \log \left (\frac {81-\log (x)}{x}\right )} (-324+4 \log (x))\right )} \, dx=\frac {1}{4} \, \log \left (\log \left (-{\left (x e^{5} - 4 \, e^{5}\right )} x^{x} + {\left (-\log \left (x\right ) + 81\right )}^{x}\right ) - \log \left (x^{x}\right ) - 5\right ) \] Input:

integrate((((log(x)-81)*log((-log(x)+81)/x)-log(x)+82)*exp(x*log((-log(x)+ 
81)/x)-5)-log(x)+81)/((4*log(x)-324)*exp(x*log((-log(x)+81)/x)-5)+(-4*x+16 
)*log(x)+324*x-1296)/log(exp(x*log((-log(x)+81)/x)-5)-x+4),x, algorithm="m 
axima")
 

Output:

1/4*log(log(-(x*e^5 - 4*e^5)*x^x + (-log(x) + 81)^x) - log(x^x) - 5)
 

Giac [A] (verification not implemented)

Time = 0.85 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.14 \[ \int \frac {81-\log (x)+e^{-5+x \log \left (\frac {81-\log (x)}{x}\right )} \left (82-\log (x)+(-81+\log (x)) \log \left (\frac {81-\log (x)}{x}\right )\right )}{\log \left (4+e^{-5+x \log \left (\frac {81-\log (x)}{x}\right )}-x\right ) \left (-1296+324 x+(16-4 x) \log (x)+e^{-5+x \log \left (\frac {81-\log (x)}{x}\right )} (-324+4 \log (x))\right )} \, dx=\frac {1}{4} \, \log \left (\log \left (-x e^{5} + 4 \, e^{5} + e^{\left (-x \log \left (x\right ) + x \log \left (-\log \left (x\right ) + 81\right )\right )}\right ) - 5\right ) \] Input:

integrate((((log(x)-81)*log((-log(x)+81)/x)-log(x)+82)*exp(x*log((-log(x)+ 
81)/x)-5)-log(x)+81)/((4*log(x)-324)*exp(x*log((-log(x)+81)/x)-5)+(-4*x+16 
)*log(x)+324*x-1296)/log(exp(x*log((-log(x)+81)/x)-5)-x+4),x, algorithm="g 
iac")
 

Output:

1/4*log(log(-x*e^5 + 4*e^5 + e^(-x*log(x) + x*log(-log(x) + 81))) - 5)
 

Mupad [B] (verification not implemented)

Time = 3.86 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.82 \[ \int \frac {81-\log (x)+e^{-5+x \log \left (\frac {81-\log (x)}{x}\right )} \left (82-\log (x)+(-81+\log (x)) \log \left (\frac {81-\log (x)}{x}\right )\right )}{\log \left (4+e^{-5+x \log \left (\frac {81-\log (x)}{x}\right )}-x\right ) \left (-1296+324 x+(16-4 x) \log (x)+e^{-5+x \log \left (\frac {81-\log (x)}{x}\right )} (-324+4 \log (x))\right )} \, dx=\frac {\ln \left (\ln \left ({\mathrm {e}}^{-5}\,{\left (-\frac {\ln \left (x\right )-81}{x}\right )}^x-x+4\right )\right )}{4} \] Input:

int((exp(x*log(-(log(x) - 81)/x) - 5)*(log(-(log(x) - 81)/x)*(log(x) - 81) 
 - log(x) + 82) - log(x) + 81)/(log(exp(x*log(-(log(x) - 81)/x) - 5) - x + 
 4)*(324*x - log(x)*(4*x - 16) + exp(x*log(-(log(x) - 81)/x) - 5)*(4*log(x 
) - 324) - 1296)),x)
 

Output:

log(log(exp(-5)*(-(log(x) - 81)/x)^x - x + 4))/4
 

Reduce [F]

\[ \int \frac {81-\log (x)+e^{-5+x \log \left (\frac {81-\log (x)}{x}\right )} \left (82-\log (x)+(-81+\log (x)) \log \left (\frac {81-\log (x)}{x}\right )\right )}{\log \left (4+e^{-5+x \log \left (\frac {81-\log (x)}{x}\right )}-x\right ) \left (-1296+324 x+(16-4 x) \log (x)+e^{-5+x \log \left (\frac {81-\log (x)}{x}\right )} (-324+4 \log (x))\right )} \, dx=\text {too large to display} \] Input:

int((((log(x)-81)*log((-log(x)+81)/x)-log(x)+82)*exp(x*log((-log(x)+81)/x) 
-5)-log(x)+81)/((4*log(x)-324)*exp(x*log((-log(x)+81)/x)-5)+(-4*x+16)*log( 
x)+324*x-1296)/log(exp(x*log((-log(x)+81)/x)-5)-x+4),x)
 

Output:

(82*int(( - log(x) + 81)**x/(( - log(x) + 81)**x*log((( - log(x) + 81)**x 
- x**x*e**5*x + 4*x**x*e**5)/(x**x*e**5))*log(x) - 81*( - log(x) + 81)**x* 
log((( - log(x) + 81)**x - x**x*e**5*x + 4*x**x*e**5)/(x**x*e**5)) - x**x* 
log((( - log(x) + 81)**x - x**x*e**5*x + 4*x**x*e**5)/(x**x*e**5))*log(x)* 
e**5*x + 4*x**x*log((( - log(x) + 81)**x - x**x*e**5*x + 4*x**x*e**5)/(x** 
x*e**5))*log(x)*e**5 + 81*x**x*log((( - log(x) + 81)**x - x**x*e**5*x + 4* 
x**x*e**5)/(x**x*e**5))*e**5*x - 324*x**x*log((( - log(x) + 81)**x - x**x* 
e**5*x + 4*x**x*e**5)/(x**x*e**5))*e**5),x) + 81*int(x**x/(( - log(x) + 81 
)**x*log((( - log(x) + 81)**x - x**x*e**5*x + 4*x**x*e**5)/(x**x*e**5))*lo 
g(x) - 81*( - log(x) + 81)**x*log((( - log(x) + 81)**x - x**x*e**5*x + 4*x 
**x*e**5)/(x**x*e**5)) - x**x*log((( - log(x) + 81)**x - x**x*e**5*x + 4*x 
**x*e**5)/(x**x*e**5))*log(x)*e**5*x + 4*x**x*log((( - log(x) + 81)**x - x 
**x*e**5*x + 4*x**x*e**5)/(x**x*e**5))*log(x)*e**5 + 81*x**x*log((( - log( 
x) + 81)**x - x**x*e**5*x + 4*x**x*e**5)/(x**x*e**5))*e**5*x - 324*x**x*lo 
g((( - log(x) + 81)**x - x**x*e**5*x + 4*x**x*e**5)/(x**x*e**5))*e**5),x)* 
e**5 + int((( - log(x) + 81)**x*log(( - log(x) + 81)/x)*log(x))/(( - log(x 
) + 81)**x*log((( - log(x) + 81)**x - x**x*e**5*x + 4*x**x*e**5)/(x**x*e** 
5))*log(x) - 81*( - log(x) + 81)**x*log((( - log(x) + 81)**x - x**x*e**5*x 
 + 4*x**x*e**5)/(x**x*e**5)) - x**x*log((( - log(x) + 81)**x - x**x*e**5*x 
 + 4*x**x*e**5)/(x**x*e**5))*log(x)*e**5*x + 4*x**x*log((( - log(x) + 8...