\(\int \frac {(3 \log (3)+\log (\frac {4+4 x}{x}))^{500+200 x+20 x^2} (-500-200 x-20 x^2+((600 x+720 x^2+120 x^3) \log (3)+(200 x+240 x^2+40 x^3) \log (\frac {4+4 x}{x})) \log (3 \log (3)+\log (\frac {4+4 x}{x})))}{(3 x+3 x^2) \log (3)+(x+x^2) \log (\frac {4+4 x}{x})} \, dx\) [1086]

Optimal result

Integrand size = 128, antiderivative size = 23 \[ \int \frac {\left (3 \log (3)+\log \left (\frac {4+4 x}{x}\right )\right )^{500+200 x+20 x^2} \left (-500-200 x-20 x^2+\left (\left (600 x+720 x^2+120 x^3\right ) \log (3)+\left (200 x+240 x^2+40 x^3\right ) \log \left (\frac {4+4 x}{x}\right )\right ) \log \left (3 \log (3)+\log \left (\frac {4+4 x}{x}\right )\right )\right )}{\left (3 x+3 x^2\right ) \log (3)+\left (x+x^2\right ) \log \left (\frac {4+4 x}{x}\right )} \, dx=\left (3 \log (3)+\log \left (4+\frac {4}{x}\right )\right )^{5 (10+2 x)^2} \] Output:

exp(5*ln(3*ln(3)+ln(4+4/x))*(2*x+10)^2)
 

Mathematica [A] (verified)

Time = 5.14 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.78 \[ \int \frac {\left (3 \log (3)+\log \left (\frac {4+4 x}{x}\right )\right )^{500+200 x+20 x^2} \left (-500-200 x-20 x^2+\left (\left (600 x+720 x^2+120 x^3\right ) \log (3)+\left (200 x+240 x^2+40 x^3\right ) \log \left (\frac {4+4 x}{x}\right )\right ) \log \left (3 \log (3)+\log \left (\frac {4+4 x}{x}\right )\right )\right )}{\left (3 x+3 x^2\right ) \log (3)+\left (x+x^2\right ) \log \left (\frac {4+4 x}{x}\right )} \, dx=\log ^{500+20 x (10+x)}\left (\frac {108 (1+x)}{x}\right ) \] Input:

Integrate[((3*Log[3] + Log[(4 + 4*x)/x])^(500 + 200*x + 20*x^2)*(-500 - 20 
0*x - 20*x^2 + ((600*x + 720*x^2 + 120*x^3)*Log[3] + (200*x + 240*x^2 + 40 
*x^3)*Log[(4 + 4*x)/x])*Log[3*Log[3] + Log[(4 + 4*x)/x]]))/((3*x + 3*x^2)* 
Log[3] + (x + x^2)*Log[(4 + 4*x)/x]),x]
 

Output:

Log[(108*(1 + x))/x]^(500 + 20*x*(10 + x))
 

Rubi [F]

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.

\(\displaystyle \int \frac {\left (\log \left (\frac {4 x+4}{x}\right )+3 \log (3)\right )^{20 x^2+200 x+500} \left (-20 x^2+\left (\left (120 x^3+720 x^2+600 x\right ) \log (3)+\left (40 x^3+240 x^2+200 x\right ) \log \left (\frac {4 x+4}{x}\right )\right ) \log \left (\log \left (\frac {4 x+4}{x}\right )+3 \log (3)\right )-200 x-500\right )}{\left (3 x^2+3 x\right ) \log (3)+\left (x^2+x\right ) \log \left (\frac {4 x+4}{x}\right )} \, dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {20 (x+5) \log ^{20 x^2+200 x+499}\left (\frac {108}{x}+108\right ) \left (-x+2 (x+1) x \log \left (\frac {108 (x+1)}{x}\right ) \log \left (\log \left (\frac {108 (x+1)}{x}\right )\right )-5\right )}{x (x+1)}dx\)

\(\Big \downarrow \) 27

\(\displaystyle 20 \int -\frac {(x+5) \log ^{20 x^2+200 x+499}\left (108+\frac {108}{x}\right ) \left (-2 (x+1) \log \left (\frac {108 (x+1)}{x}\right ) \log \left (\log \left (\frac {108 (x+1)}{x}\right )\right ) x+x+5\right )}{x (x+1)}dx\)

\(\Big \downarrow \) 25

\(\displaystyle -20 \int \frac {(x+5) \log ^{20 x^2+200 x+499}\left (108+\frac {108}{x}\right ) \left (-2 (x+1) \log \left (\frac {108 (x+1)}{x}\right ) \log \left (\log \left (\frac {108 (x+1)}{x}\right )\right ) x+x+5\right )}{x (x+1)}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle -20 \int \left (\frac {(x+5)^2 \log ^{20 x^2+200 x+499}\left (108+\frac {108}{x}\right )}{x (x+1)}+2 (-x-5) \log \left (\log \left (108+\frac {108}{x}\right )\right ) \log ^{20 x^2+200 x+500}\left (108+\frac {108}{x}\right )\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -20 \left (\int \log ^{20 x^2+200 x+499}\left (108+\frac {108}{x}\right )dx+25 \int \frac {\log ^{20 x^2+200 x+499}\left (108+\frac {108}{x}\right )}{x}dx-16 \int \frac {\log ^{20 x^2+200 x+499}\left (108+\frac {108}{x}\right )}{x+1}dx-10 \int \log ^{20 x^2+200 x+500}\left (108+\frac {108}{x}\right ) \log \left (\log \left (108+\frac {108}{x}\right )\right )dx-2 \int x \log ^{20 x^2+200 x+500}\left (108+\frac {108}{x}\right ) \log \left (\log \left (108+\frac {108}{x}\right )\right )dx\right )\)

Input:

Int[((3*Log[3] + Log[(4 + 4*x)/x])^(500 + 200*x + 20*x^2)*(-500 - 200*x - 
20*x^2 + ((600*x + 720*x^2 + 120*x^3)*Log[3] + (200*x + 240*x^2 + 40*x^3)* 
Log[(4 + 4*x)/x])*Log[3*Log[3] + Log[(4 + 4*x)/x]]))/((3*x + 3*x^2)*Log[3] 
 + (x + x^2)*Log[(4 + 4*x)/x]),x]
 

Output:

$Aborted
 

Maple [F(-1)]

Timed out.

\[\int \frac {\left (\left (\left (40 x^{3}+240 x^{2}+200 x \right ) \ln \left (\frac {4+4 x}{x}\right )+\left (120 x^{3}+720 x^{2}+600 x \right ) \ln \left (3\right )\right ) \ln \left (\ln \left (\frac {4+4 x}{x}\right )+3 \ln \left (3\right )\right )-20 x^{2}-200 x -500\right ) {\mathrm e}^{\left (20 x^{2}+200 x +500\right ) \ln \left (\ln \left (\frac {4+4 x}{x}\right )+3 \ln \left (3\right )\right )}}{\left (x^{2}+x \right ) \ln \left (\frac {4+4 x}{x}\right )+\left (3 x^{2}+3 x \right ) \ln \left (3\right )}d x\]

Input:

int((((40*x^3+240*x^2+200*x)*ln((4+4*x)/x)+(120*x^3+720*x^2+600*x)*ln(3))* 
ln(ln((4+4*x)/x)+3*ln(3))-20*x^2-200*x-500)*exp((20*x^2+200*x+500)*ln(ln(( 
4+4*x)/x)+3*ln(3)))/((x^2+x)*ln((4+4*x)/x)+(3*x^2+3*x)*ln(3)),x)
 

Output:

int((((40*x^3+240*x^2+200*x)*ln((4+4*x)/x)+(120*x^3+720*x^2+600*x)*ln(3))* 
ln(ln((4+4*x)/x)+3*ln(3))-20*x^2-200*x-500)*exp((20*x^2+200*x+500)*ln(ln(( 
4+4*x)/x)+3*ln(3)))/((x^2+x)*ln((4+4*x)/x)+(3*x^2+3*x)*ln(3)),x)
 

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \frac {\left (3 \log (3)+\log \left (\frac {4+4 x}{x}\right )\right )^{500+200 x+20 x^2} \left (-500-200 x-20 x^2+\left (\left (600 x+720 x^2+120 x^3\right ) \log (3)+\left (200 x+240 x^2+40 x^3\right ) \log \left (\frac {4+4 x}{x}\right )\right ) \log \left (3 \log (3)+\log \left (\frac {4+4 x}{x}\right )\right )\right )}{\left (3 x+3 x^2\right ) \log (3)+\left (x+x^2\right ) \log \left (\frac {4+4 x}{x}\right )} \, dx={\left (3 \, \log \left (3\right ) + \log \left (\frac {4 \, {\left (x + 1\right )}}{x}\right )\right )}^{20 \, x^{2} + 200 \, x + 500} \] Input:

integrate((((40*x^3+240*x^2+200*x)*log((4+4*x)/x)+(120*x^3+720*x^2+600*x)* 
log(3))*log(log((4+4*x)/x)+3*log(3))-20*x^2-200*x-500)*exp((20*x^2+200*x+5 
00)*log(log((4+4*x)/x)+3*log(3)))/((x^2+x)*log((4+4*x)/x)+(3*x^2+3*x)*log( 
3)),x, algorithm="fricas")
 

Output:

(3*log(3) + log(4*(x + 1)/x))^(20*x^2 + 200*x + 500)
 

Sympy [A] (verification not implemented)

Time = 0.57 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.13 \[ \int \frac {\left (3 \log (3)+\log \left (\frac {4+4 x}{x}\right )\right )^{500+200 x+20 x^2} \left (-500-200 x-20 x^2+\left (\left (600 x+720 x^2+120 x^3\right ) \log (3)+\left (200 x+240 x^2+40 x^3\right ) \log \left (\frac {4+4 x}{x}\right )\right ) \log \left (3 \log (3)+\log \left (\frac {4+4 x}{x}\right )\right )\right )}{\left (3 x+3 x^2\right ) \log (3)+\left (x+x^2\right ) \log \left (\frac {4+4 x}{x}\right )} \, dx=e^{\left (20 x^{2} + 200 x + 500\right ) \log {\left (\log {\left (\frac {4 x + 4}{x} \right )} + 3 \log {\left (3 \right )} \right )}} \] Input:

integrate((((40*x**3+240*x**2+200*x)*ln((4+4*x)/x)+(120*x**3+720*x**2+600* 
x)*ln(3))*ln(ln((4+4*x)/x)+3*ln(3))-20*x**2-200*x-500)*exp((20*x**2+200*x+ 
500)*ln(ln((4+4*x)/x)+3*ln(3)))/((x**2+x)*ln((4+4*x)/x)+(3*x**2+3*x)*ln(3) 
),x)
 

Output:

exp((20*x**2 + 200*x + 500)*log(log((4*x + 4)/x) + 3*log(3)))
 

Maxima [F(-1)]

Timed out. \[ \int \frac {\left (3 \log (3)+\log \left (\frac {4+4 x}{x}\right )\right )^{500+200 x+20 x^2} \left (-500-200 x-20 x^2+\left (\left (600 x+720 x^2+120 x^3\right ) \log (3)+\left (200 x+240 x^2+40 x^3\right ) \log \left (\frac {4+4 x}{x}\right )\right ) \log \left (3 \log (3)+\log \left (\frac {4+4 x}{x}\right )\right )\right )}{\left (3 x+3 x^2\right ) \log (3)+\left (x+x^2\right ) \log \left (\frac {4+4 x}{x}\right )} \, dx=\text {Timed out} \] Input:

integrate((((40*x^3+240*x^2+200*x)*log((4+4*x)/x)+(120*x^3+720*x^2+600*x)* 
log(3))*log(log((4+4*x)/x)+3*log(3))-20*x^2-200*x-500)*exp((20*x^2+200*x+5 
00)*log(log((4+4*x)/x)+3*log(3)))/((x^2+x)*log((4+4*x)/x)+(3*x^2+3*x)*log( 
3)),x, algorithm="maxima")
 

Output:

Timed out
 

Giac [F(-1)]

Timed out. \[ \int \frac {\left (3 \log (3)+\log \left (\frac {4+4 x}{x}\right )\right )^{500+200 x+20 x^2} \left (-500-200 x-20 x^2+\left (\left (600 x+720 x^2+120 x^3\right ) \log (3)+\left (200 x+240 x^2+40 x^3\right ) \log \left (\frac {4+4 x}{x}\right )\right ) \log \left (3 \log (3)+\log \left (\frac {4+4 x}{x}\right )\right )\right )}{\left (3 x+3 x^2\right ) \log (3)+\left (x+x^2\right ) \log \left (\frac {4+4 x}{x}\right )} \, dx=\text {Timed out} \] Input:

integrate((((40*x^3+240*x^2+200*x)*log((4+4*x)/x)+(120*x^3+720*x^2+600*x)* 
log(3))*log(log((4+4*x)/x)+3*log(3))-20*x^2-200*x-500)*exp((20*x^2+200*x+5 
00)*log(log((4+4*x)/x)+3*log(3)))/((x^2+x)*log((4+4*x)/x)+(3*x^2+3*x)*log( 
3)),x, algorithm="giac")
 

Output:

Timed out
 

Mupad [B] (verification not implemented)

Time = 5.16 (sec) , antiderivative size = 29038, normalized size of antiderivative = 1262.52 \[ \int \frac {\left (3 \log (3)+\log \left (\frac {4+4 x}{x}\right )\right )^{500+200 x+20 x^2} \left (-500-200 x-20 x^2+\left (\left (600 x+720 x^2+120 x^3\right ) \log (3)+\left (200 x+240 x^2+40 x^3\right ) \log \left (\frac {4+4 x}{x}\right )\right ) \log \left (3 \log (3)+\log \left (\frac {4+4 x}{x}\right )\right )\right )}{\left (3 x+3 x^2\right ) \log (3)+\left (x+x^2\right ) \log \left (\frac {4+4 x}{x}\right )} \, dx=\text {Too large to display} \] Input:

int(-(exp(log(3*log(3) + log((4*x + 4)/x))*(200*x + 20*x^2 + 500))*(200*x 
- log(3*log(3) + log((4*x + 4)/x))*(log(3)*(600*x + 720*x^2 + 120*x^3) + l 
og((4*x + 4)/x)*(200*x + 240*x^2 + 40*x^3)) + 20*x^2 + 500))/(log(3)*(3*x 
+ 3*x^2) + log((4*x + 4)/x)*(x + x^2)),x)
 

Output:

36360291795869936842385267079543319118023385026001623040346035832580600191 
58389548419850826297938878330817970253440385575285593151701306614299243091 
65620257800217712478476434501253428365658132099725903715901525787280083859 
90139795377610001*log(3)^500*(3*log(3) + log((4*x + 4)/x))^(200*x)*(3*log( 
3) + log((4*x + 4)/x))^(20*x^2) + log((4*x + 4)/x)^500*(3*log(3) + log((4* 
x + 4)/x))^(200*x)*(3*log(3) + log((4*x + 4)/x))^(20*x^2) + 1122750*log(3) 
^2*log((4*x + 4)/x)^498*(3*log(3) + log((4*x + 4)/x))^(200*x)*(3*log(3) + 
log((4*x + 4)/x))^(20*x^2) + 559129500*log(3)^3*log((4*x + 4)/x)^497*(3*lo 
g(3) + log((4*x + 4)/x))^(200*x)*(3*log(3) + log((4*x + 4)/x))^(20*x^2) + 
208415521125*log(3)^4*log((4*x + 4)/x)^496*(3*log(3) + log((4*x + 4)/x))^( 
200*x)*(3*log(3) + log((4*x + 4)/x))^(20*x^2) + 62024459086800*log(3)^5*lo 
g((4*x + 4)/x)^495*(3*log(3) + log((4*x + 4)/x))^(200*x)*(3*log(3) + log(( 
4*x + 4)/x))^(20*x^2) + 15351053623983000*log(3)^6*log((4*x + 4)/x)^494*(3 
*log(3) + log((4*x + 4)/x))^(200*x)*(3*log(3) + log((4*x + 4)/x))^(20*x^2) 
 + 3250037352963258000*log(3)^7*log((4*x + 4)/x)^493*(3*log(3) + log((4*x 
+ 4)/x))^(200*x)*(3*log(3) + log((4*x + 4)/x))^(20*x^2) + 6008506556290823 
22750*log(3)^8*log((4*x + 4)/x)^492*(3*log(3) + log((4*x + 4)/x))^(200*x)* 
(3*log(3) + log((4*x + 4)/x))^(20*x^2) + 98539507523169500931000*log(3)^9* 
log((4*x + 4)/x)^491*(3*log(3) + log((4*x + 4)/x))^(200*x)*(3*log(3) + log 
((4*x + 4)/x))^(20*x^2) + 14514869458162867487136300*log(3)^10*log((4*x...
 

Reduce [B] (verification not implemented)

Time = 16.37 (sec) , antiderivative size = 9023, normalized size of antiderivative = 392.30 \[ \int \frac {\left (3 \log (3)+\log \left (\frac {4+4 x}{x}\right )\right )^{500+200 x+20 x^2} \left (-500-200 x-20 x^2+\left (\left (600 x+720 x^2+120 x^3\right ) \log (3)+\left (200 x+240 x^2+40 x^3\right ) \log \left (\frac {4+4 x}{x}\right )\right ) \log \left (3 \log (3)+\log \left (\frac {4+4 x}{x}\right )\right )\right )}{\left (3 x+3 x^2\right ) \log (3)+\left (x+x^2\right ) \log \left (\frac {4+4 x}{x}\right )} \, dx =\text {Too large to display} \] Input:

int((((40*x^3+240*x^2+200*x)*log((4+4*x)/x)+(120*x^3+720*x^2+600*x)*log(3) 
)*log(log((4+4*x)/x)+3*log(3))-20*x^2-200*x-500)*exp((20*x^2+200*x+500)*lo 
g(log((4+4*x)/x)+3*log(3)))/((x^2+x)*log((4+4*x)/x)+(3*x^2+3*x)*log(3)),x)
 

Output:

(log((4*x + 4)/x) + 3*log(3))**(20*x**2 + 200*x)*(log((4*x + 4)/x)**500 + 
1500*log((4*x + 4)/x)**499*log(3) + 1122750*log((4*x + 4)/x)**498*log(3)** 
2 + 559129500*log((4*x + 4)/x)**497*log(3)**3 + 208415521125*log((4*x + 4) 
/x)**496*log(3)**4 + 62024459086800*log((4*x + 4)/x)**495*log(3)**5 + 1535 
1053623983000*log((4*x + 4)/x)**494*log(3)**6 + 3250037352963258000*log((4 
*x + 4)/x)**493*log(3)**7 + 600850655629082322750*log((4*x + 4)/x)**492*lo 
g(3)**8 + 98539507523169500931000*log((4*x + 4)/x)**491*log(3)**9 + 145148 
69458162867487136300*log((4*x + 4)/x)**490*log(3)**10 + 193971437304540138 
2371851000*log((4*x + 4)/x)**489*log(3)**11 + 2371300821048003189949587847 
50*log((4*x + 4)/x)**488*log(3)**12 + 26704495400109820539124589298000*log 
((4*x + 4)/x)**487*log(3)**13 + 2786804841397174843404358926027000*log((4* 
x + 4)/x)**486*log(3)**14 + 270877430583805394778903687609824400*log((4*x 
+ 4)/x)**485*log(3)**15 + 24632916343714803087706554092018406375*log((4*x 
+ 4)/x)**484*log(3)**16 + 2103940854769052593138230384800630944500*log((4* 
x + 4)/x)**483*log(3)**17 + 169367238808908733747627545976450791032250*log 
((4*x + 4)/x)**482*log(3)**18 + 128897382798780015262668121832604128332965 
00*log((4*x + 4)/x)**481*log(3)**19 + 929994616893197810120150499022238785 
922342475*log((4*x + 4)/x)**480*log(3)**20 + 63771059444104992693953177075 
810659606103484000*log((4*x + 4)/x)**479*log(3)**21 + 41654096555081306591 
45941611724541720635032114000*log((4*x + 4)/x)**478*log(3)**22 + 259704...