Integrand size = 128, antiderivative size = 23 \[ \int \frac {\left (3 \log (3)+\log \left (\frac {4+4 x}{x}\right )\right )^{500+200 x+20 x^2} \left (-500-200 x-20 x^2+\left (\left (600 x+720 x^2+120 x^3\right ) \log (3)+\left (200 x+240 x^2+40 x^3\right ) \log \left (\frac {4+4 x}{x}\right )\right ) \log \left (3 \log (3)+\log \left (\frac {4+4 x}{x}\right )\right )\right )}{\left (3 x+3 x^2\right ) \log (3)+\left (x+x^2\right ) \log \left (\frac {4+4 x}{x}\right )} \, dx=\left (3 \log (3)+\log \left (4+\frac {4}{x}\right )\right )^{5 (10+2 x)^2} \]
Time = 5.23 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.78 \[ \int \frac {\left (3 \log (3)+\log \left (\frac {4+4 x}{x}\right )\right )^{500+200 x+20 x^2} \left (-500-200 x-20 x^2+\left (\left (600 x+720 x^2+120 x^3\right ) \log (3)+\left (200 x+240 x^2+40 x^3\right ) \log \left (\frac {4+4 x}{x}\right )\right ) \log \left (3 \log (3)+\log \left (\frac {4+4 x}{x}\right )\right )\right )}{\left (3 x+3 x^2\right ) \log (3)+\left (x+x^2\right ) \log \left (\frac {4+4 x}{x}\right )} \, dx=\log ^{500+20 x (10+x)}\left (\frac {108 (1+x)}{x}\right ) \]
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]
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 |
| \(\displaystyle -20 \left (\int \log ^{20 x^2+200 x+499}\left (108+\frac {108}{x}\right )dx+25 \int \frac {\log ^{20 x^2+200 x+499}\left (108+\frac {108}{x}\right )}{x}dx-16 \int \frac {\log ^{20 x^2+200 x+499}\left (108+\frac {108}{x}\right )}{x+1}dx-10 \int \log ^{20 x^2+200 x+500}\left (108+\frac {108}{x}\right ) \log \left (\log \left (108+\frac {108}{x}\right )\right )dx-2 \int x \log ^{20 x^2+200 x+500}\left (108+\frac {108}{x}\right ) \log \left (\log \left (108+\frac {108}{x}\right )\right )dx\right )\) |
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]
3.11.86.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Timed out.
\[\int \frac {\left (\left (\left (40 x^{3}+240 x^{2}+200 x \right ) \ln \left (\frac {4+4 x}{x}\right )+\left (120 x^{3}+720 x^{2}+600 x \right ) \ln \left (3\right )\right ) \ln \left (\ln \left (\frac {4+4 x}{x}\right )+3 \ln \left (3\right )\right )-20 x^{2}-200 x -500\right ) {\mathrm e}^{\left (20 x^{2}+200 x +500\right ) \ln \left (\ln \left (\frac {4+4 x}{x}\right )+3 \ln \left (3\right )\right )}}{\left (x^{2}+x \right ) \ln \left (\frac {4+4 x}{x}\right )+\left (3 x^{2}+3 x \right ) \ln \left (3\right )}d x\]
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)
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)
Time = 0.28 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \frac {\left (3 \log (3)+\log \left (\frac {4+4 x}{x}\right )\right )^{500+200 x+20 x^2} \left (-500-200 x-20 x^2+\left (\left (600 x+720 x^2+120 x^3\right ) \log (3)+\left (200 x+240 x^2+40 x^3\right ) \log \left (\frac {4+4 x}{x}\right )\right ) \log \left (3 \log (3)+\log \left (\frac {4+4 x}{x}\right )\right )\right )}{\left (3 x+3 x^2\right ) \log (3)+\left (x+x^2\right ) \log \left (\frac {4+4 x}{x}\right )} \, dx={\left (3 \, \log \left (3\right ) + \log \left (\frac {4 \, {\left (x + 1\right )}}{x}\right )\right )}^{20 \, x^{2} + 200 \, x + 500} \]
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=\
Time = 0.56 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.13 \[ \int \frac {\left (3 \log (3)+\log \left (\frac {4+4 x}{x}\right )\right )^{500+200 x+20 x^2} \left (-500-200 x-20 x^2+\left (\left (600 x+720 x^2+120 x^3\right ) \log (3)+\left (200 x+240 x^2+40 x^3\right ) \log \left (\frac {4+4 x}{x}\right )\right ) \log \left (3 \log (3)+\log \left (\frac {4+4 x}{x}\right )\right )\right )}{\left (3 x+3 x^2\right ) \log (3)+\left (x+x^2\right ) \log \left (\frac {4+4 x}{x}\right )} \, dx=e^{\left (20 x^{2} + 200 x + 500\right ) \log {\left (\log {\left (\frac {4 x + 4}{x} \right )} + 3 \log {\left (3 \right )} \right )}} \]
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)
Timed out. \[ \int \frac {\left (3 \log (3)+\log \left (\frac {4+4 x}{x}\right )\right )^{500+200 x+20 x^2} \left (-500-200 x-20 x^2+\left (\left (600 x+720 x^2+120 x^3\right ) \log (3)+\left (200 x+240 x^2+40 x^3\right ) \log \left (\frac {4+4 x}{x}\right )\right ) \log \left (3 \log (3)+\log \left (\frac {4+4 x}{x}\right )\right )\right )}{\left (3 x+3 x^2\right ) \log (3)+\left (x+x^2\right ) \log \left (\frac {4+4 x}{x}\right )} \, dx=\text {Timed out} \]
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=\
Timed out. \[ \int \frac {\left (3 \log (3)+\log \left (\frac {4+4 x}{x}\right )\right )^{500+200 x+20 x^2} \left (-500-200 x-20 x^2+\left (\left (600 x+720 x^2+120 x^3\right ) \log (3)+\left (200 x+240 x^2+40 x^3\right ) \log \left (\frac {4+4 x}{x}\right )\right ) \log \left (3 \log (3)+\log \left (\frac {4+4 x}{x}\right )\right )\right )}{\left (3 x+3 x^2\right ) \log (3)+\left (x+x^2\right ) \log \left (\frac {4+4 x}{x}\right )} \, dx=\text {Timed out} \]
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=\
Time = 13.80 (sec) , antiderivative size = 29038, normalized size of antiderivative = 1262.52 \[ \int \frac {\left (3 \log (3)+\log \left (\frac {4+4 x}{x}\right )\right )^{500+200 x+20 x^2} \left (-500-200 x-20 x^2+\left (\left (600 x+720 x^2+120 x^3\right ) \log (3)+\left (200 x+240 x^2+40 x^3\right ) \log \left (\frac {4+4 x}{x}\right )\right ) \log \left (3 \log (3)+\log \left (\frac {4+4 x}{x}\right )\right )\right )}{\left (3 x+3 x^2\right ) \log (3)+\left (x+x^2\right ) \log \left (\frac {4+4 x}{x}\right )} \, dx=\text {Too large to display} \]
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)
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...