Integrand size = 88, antiderivative size = 18 \[ \int \frac {-2-20 x-20 x \log (x)}{6480 x+3600 x^2+500 x^3+\left (6480 x+3600 x^2+500 x^3\right ) \log (x)+\left (360 x+100 x^2+\left (360 x+100 x^2\right ) \log (x)\right ) \log (1+\log (x))+(5 x+5 x \log (x)) \log ^2(1+\log (x))} \, dx=\frac {2}{5 (-4+10 (4+x)+\log (1+\log (x)))} \] Output:
2/5/(ln(1+ln(x))+10*x+36)
Time = 0.02 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.89 \[ \int \frac {-2-20 x-20 x \log (x)}{6480 x+3600 x^2+500 x^3+\left (6480 x+3600 x^2+500 x^3\right ) \log (x)+\left (360 x+100 x^2+\left (360 x+100 x^2\right ) \log (x)\right ) \log (1+\log (x))+(5 x+5 x \log (x)) \log ^2(1+\log (x))} \, dx=\frac {2}{5 (36+10 x+\log (1+\log (x)))} \] Input:
Integrate[(-2 - 20*x - 20*x*Log[x])/(6480*x + 3600*x^2 + 500*x^3 + (6480*x + 3600*x^2 + 500*x^3)*Log[x] + (360*x + 100*x^2 + (360*x + 100*x^2)*Log[x ])*Log[1 + Log[x]] + (5*x + 5*x*Log[x])*Log[1 + Log[x]]^2),x]
Output:
2/(5*(36 + 10*x + Log[1 + Log[x]]))
Time = 0.47 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.89, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {7239, 27, 25, 7237}
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.
\(\displaystyle \int \frac {-20 x-20 x \log (x)-2}{500 x^3+3600 x^2+\left (100 x^2+\left (100 x^2+360 x\right ) \log (x)+360 x\right ) \log (\log (x)+1)+\left (500 x^3+3600 x^2+6480 x\right ) \log (x)+6480 x+(5 x+5 x \log (x)) \log ^2(\log (x)+1)} \, dx\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle \int \frac {2 (-10 x-10 x \log (x)-1)}{5 x (\log (x)+1) (10 x+\log (\log (x)+1)+36)^2}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {2}{5} \int -\frac {10 \log (x) x+10 x+1}{x (\log (x)+1) (10 x+\log (\log (x)+1)+36)^2}dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {2}{5} \int \frac {10 \log (x) x+10 x+1}{x (\log (x)+1) (10 x+\log (\log (x)+1)+36)^2}dx\) |
\(\Big \downarrow \) 7237 |
\(\displaystyle \frac {2}{5 (10 x+\log (\log (x)+1)+36)}\) |
Input:
Int[(-2 - 20*x - 20*x*Log[x])/(6480*x + 3600*x^2 + 500*x^3 + (6480*x + 360 0*x^2 + 500*x^3)*Log[x] + (360*x + 100*x^2 + (360*x + 100*x^2)*Log[x])*Log [1 + Log[x]] + (5*x + 5*x*Log[x])*Log[1 + Log[x]]^2),x]
Output:
2/(5*(36 + 10*x + Log[1 + Log[x]]))
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(u_)*(y_)^(m_.), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Si mp[q*(y^(m + 1)/(m + 1)), x] /; !FalseQ[q]] /; FreeQ[m, x] && NeQ[m, -1]
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Time = 0.59 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.83
method | result | size |
default | \(\frac {2}{5 \left (\ln \left (\ln \left (x \right )+1\right )+10 x +36\right )}\) | \(15\) |
risch | \(\frac {2}{5 \left (\ln \left (\ln \left (x \right )+1\right )+10 x +36\right )}\) | \(15\) |
parallelrisch | \(\frac {2}{5 \left (\ln \left (\ln \left (x \right )+1\right )+10 x +36\right )}\) | \(15\) |
Input:
int((-20*x*ln(x)-20*x-2)/((5*x*ln(x)+5*x)*ln(ln(x)+1)^2+((100*x^2+360*x)*l n(x)+100*x^2+360*x)*ln(ln(x)+1)+(500*x^3+3600*x^2+6480*x)*ln(x)+500*x^3+36 00*x^2+6480*x),x,method=_RETURNVERBOSE)
Output:
2/5/(ln(ln(x)+1)+10*x+36)
Time = 0.07 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.78 \[ \int \frac {-2-20 x-20 x \log (x)}{6480 x+3600 x^2+500 x^3+\left (6480 x+3600 x^2+500 x^3\right ) \log (x)+\left (360 x+100 x^2+\left (360 x+100 x^2\right ) \log (x)\right ) \log (1+\log (x))+(5 x+5 x \log (x)) \log ^2(1+\log (x))} \, dx=\frac {2}{5 \, {\left (10 \, x + \log \left (\log \left (x\right ) + 1\right ) + 36\right )}} \] Input:
integrate((-20*x*log(x)-20*x-2)/((5*x*log(x)+5*x)*log(1+log(x))^2+((100*x^ 2+360*x)*log(x)+100*x^2+360*x)*log(1+log(x))+(500*x^3+3600*x^2+6480*x)*log (x)+500*x^3+3600*x^2+6480*x),x, algorithm="fricas")
Output:
2/5/(10*x + log(log(x) + 1) + 36)
Time = 0.08 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.78 \[ \int \frac {-2-20 x-20 x \log (x)}{6480 x+3600 x^2+500 x^3+\left (6480 x+3600 x^2+500 x^3\right ) \log (x)+\left (360 x+100 x^2+\left (360 x+100 x^2\right ) \log (x)\right ) \log (1+\log (x))+(5 x+5 x \log (x)) \log ^2(1+\log (x))} \, dx=\frac {2}{50 x + 5 \log {\left (\log {\left (x \right )} + 1 \right )} + 180} \] Input:
integrate((-20*x*ln(x)-20*x-2)/((5*x*ln(x)+5*x)*ln(1+ln(x))**2+((100*x**2+ 360*x)*ln(x)+100*x**2+360*x)*ln(1+ln(x))+(500*x**3+3600*x**2+6480*x)*ln(x) +500*x**3+3600*x**2+6480*x),x)
Output:
2/(50*x + 5*log(log(x) + 1) + 180)
Time = 0.08 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.78 \[ \int \frac {-2-20 x-20 x \log (x)}{6480 x+3600 x^2+500 x^3+\left (6480 x+3600 x^2+500 x^3\right ) \log (x)+\left (360 x+100 x^2+\left (360 x+100 x^2\right ) \log (x)\right ) \log (1+\log (x))+(5 x+5 x \log (x)) \log ^2(1+\log (x))} \, dx=\frac {2}{5 \, {\left (10 \, x + \log \left (\log \left (x\right ) + 1\right ) + 36\right )}} \] Input:
integrate((-20*x*log(x)-20*x-2)/((5*x*log(x)+5*x)*log(1+log(x))^2+((100*x^ 2+360*x)*log(x)+100*x^2+360*x)*log(1+log(x))+(500*x^3+3600*x^2+6480*x)*log (x)+500*x^3+3600*x^2+6480*x),x, algorithm="maxima")
Output:
2/5/(10*x + log(log(x) + 1) + 36)
Time = 0.14 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.78 \[ \int \frac {-2-20 x-20 x \log (x)}{6480 x+3600 x^2+500 x^3+\left (6480 x+3600 x^2+500 x^3\right ) \log (x)+\left (360 x+100 x^2+\left (360 x+100 x^2\right ) \log (x)\right ) \log (1+\log (x))+(5 x+5 x \log (x)) \log ^2(1+\log (x))} \, dx=\frac {2}{5 \, {\left (10 \, x + \log \left (\log \left (x\right ) + 1\right ) + 36\right )}} \] Input:
integrate((-20*x*log(x)-20*x-2)/((5*x*log(x)+5*x)*log(1+log(x))^2+((100*x^ 2+360*x)*log(x)+100*x^2+360*x)*log(1+log(x))+(500*x^3+3600*x^2+6480*x)*log (x)+500*x^3+3600*x^2+6480*x),x, algorithm="giac")
Output:
2/5/(10*x + log(log(x) + 1) + 36)
Timed out. \[ \int \frac {-2-20 x-20 x \log (x)}{6480 x+3600 x^2+500 x^3+\left (6480 x+3600 x^2+500 x^3\right ) \log (x)+\left (360 x+100 x^2+\left (360 x+100 x^2\right ) \log (x)\right ) \log (1+\log (x))+(5 x+5 x \log (x)) \log ^2(1+\log (x))} \, dx=\int -\frac {20\,x+20\,x\,\ln \left (x\right )+2}{6480\,x+\ln \left (\ln \left (x\right )+1\right )\,\left (360\,x+\ln \left (x\right )\,\left (100\,x^2+360\,x\right )+100\,x^2\right )+{\ln \left (\ln \left (x\right )+1\right )}^2\,\left (5\,x+5\,x\,\ln \left (x\right )\right )+3600\,x^2+500\,x^3+\ln \left (x\right )\,\left (500\,x^3+3600\,x^2+6480\,x\right )} \,d x \] Input:
int(-(20*x + 20*x*log(x) + 2)/(6480*x + log(log(x) + 1)*(360*x + log(x)*(3 60*x + 100*x^2) + 100*x^2) + log(log(x) + 1)^2*(5*x + 5*x*log(x)) + 3600*x ^2 + 500*x^3 + log(x)*(6480*x + 3600*x^2 + 500*x^3)),x)
Output:
int(-(20*x + 20*x*log(x) + 2)/(6480*x + log(log(x) + 1)*(360*x + log(x)*(3 60*x + 100*x^2) + 100*x^2) + log(log(x) + 1)^2*(5*x + 5*x*log(x)) + 3600*x ^2 + 500*x^3 + log(x)*(6480*x + 3600*x^2 + 500*x^3)), x)
Time = 0.25 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.89 \[ \int \frac {-2-20 x-20 x \log (x)}{6480 x+3600 x^2+500 x^3+\left (6480 x+3600 x^2+500 x^3\right ) \log (x)+\left (360 x+100 x^2+\left (360 x+100 x^2\right ) \log (x)\right ) \log (1+\log (x))+(5 x+5 x \log (x)) \log ^2(1+\log (x))} \, dx=\frac {2}{5 \,\mathrm {log}\left (\mathrm {log}\left (x \right )+1\right )+50 x +180} \] Input:
int((-20*x*log(x)-20*x-2)/((5*x*log(x)+5*x)*log(1+log(x))^2+((100*x^2+360* x)*log(x)+100*x^2+360*x)*log(1+log(x))+(500*x^3+3600*x^2+6480*x)*log(x)+50 0*x^3+3600*x^2+6480*x),x)
Output:
2/(5*(log(log(x) + 1) + 10*x + 36))