Integrand size = 107, antiderivative size = 29 \[ \int \frac {-10 x+5 \log (\log (2))+\left (\left (-20 x-4 x^2\right ) \log \left (\frac {5+x}{x}\right )+(5+x) \log \left (\frac {5+x}{x}\right ) \log (\log (2))\right ) \log \left (\log \left (\frac {5+x}{x}\right )\right )+(150+30 x) \log \left (\frac {5+x}{x}\right ) \log ^2\left (\log \left (\frac {5+x}{x}\right )\right )}{(25+5 x) \log \left (\frac {5+x}{x}\right ) \log ^2\left (\log \left (\frac {5+x}{x}\right )\right )} \, dx=6 x-\frac {x (2 x-\log (\log (2)))}{5 \log \left (\log \left (1+\frac {5}{x}\right )\right )} \]
Time = 0.64 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.90 \[ \int \frac {-10 x+5 \log (\log (2))+\left (\left (-20 x-4 x^2\right ) \log \left (\frac {5+x}{x}\right )+(5+x) \log \left (\frac {5+x}{x}\right ) \log (\log (2))\right ) \log \left (\log \left (\frac {5+x}{x}\right )\right )+(150+30 x) \log \left (\frac {5+x}{x}\right ) \log ^2\left (\log \left (\frac {5+x}{x}\right )\right )}{(25+5 x) \log \left (\frac {5+x}{x}\right ) \log ^2\left (\log \left (\frac {5+x}{x}\right )\right )} \, dx=\frac {1}{5} x \left (30+\frac {-2 x+\log (\log (2))}{\log \left (\log \left (\frac {5+x}{x}\right )\right )}\right ) \]
Integrate[(-10*x + 5*Log[Log[2]] + ((-20*x - 4*x^2)*Log[(5 + x)/x] + (5 + x)*Log[(5 + x)/x]*Log[Log[2]])*Log[Log[(5 + x)/x]] + (150 + 30*x)*Log[(5 + x)/x]*Log[Log[(5 + x)/x]]^2)/((25 + 5*x)*Log[(5 + x)/x]*Log[Log[(5 + x)/x ]]^2),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.
\(\displaystyle \int \frac {\left (\left (-4 x^2-20 x\right ) \log \left (\frac {x+5}{x}\right )+(x+5) \log (\log (2)) \log \left (\frac {x+5}{x}\right )\right ) \log \left (\log \left (\frac {x+5}{x}\right )\right )-10 x+(30 x+150) \log \left (\frac {x+5}{x}\right ) \log ^2\left (\log \left (\frac {x+5}{x}\right )\right )+5 \log (\log (2))}{(5 x+25) \log \left (\frac {x+5}{x}\right ) \log ^2\left (\log \left (\frac {x+5}{x}\right )\right )} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {\left (\left (-4 x^2-20 x\right ) \log \left (\frac {x+5}{x}\right )+(x+5) \log (\log (2)) \log \left (\frac {x+5}{x}\right )\right ) \log \left (\log \left (\frac {x+5}{x}\right )\right )-10 x+(30 x+150) \log \left (\frac {x+5}{x}\right ) \log ^2\left (\log \left (\frac {x+5}{x}\right )\right )+5 \log (\log (2))}{(5 x+25) \log \left (\frac {5}{x}+1\right ) \log ^2\left (\log \left (\frac {5}{x}+1\right )\right )}dx\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle \int \frac {\frac {5 (\log (\log (2))-2 x)}{(x+5) \log \left (\frac {x+5}{x}\right )}+\log \left (\log \left (\frac {x+5}{x}\right )\right ) \left (-4 x+30 \log \left (\log \left (\frac {x+5}{x}\right )\right )+\log (\log (2))\right )}{5 \log ^2\left (\log \left (\frac {5}{x}+1\right )\right )}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{5} \int -\frac {\frac {5 (2 x-\log (\log (2)))}{(x+5) \log \left (\frac {x+5}{x}\right )}+\left (4 x-30 \log \left (\log \left (\frac {x+5}{x}\right )\right )-\log (\log (2))\right ) \log \left (\log \left (\frac {x+5}{x}\right )\right )}{\log ^2\left (\log \left (1+\frac {5}{x}\right )\right )}dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {1}{5} \int \frac {\frac {5 (2 x-\log (\log (2)))}{(x+5) \log \left (\frac {x+5}{x}\right )}+\left (4 x-30 \log \left (\log \left (\frac {x+5}{x}\right )\right )-\log (\log (2))\right ) \log \left (\log \left (\frac {x+5}{x}\right )\right )}{\log ^2\left (\log \left (1+\frac {5}{x}\right )\right )}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle -\frac {1}{5} \int \left (\frac {5 (2 x-\log (\log (2)))}{(x+5) \log \left (1+\frac {5}{x}\right ) \log ^2\left (\log \left (1+\frac {5}{x}\right )\right )}+\frac {4 x-\log (\log (2))}{\log \left (\log \left (1+\frac {5}{x}\right )\right )}-30\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{5} \left (-10 \int \frac {1}{\log \left (1+\frac {5}{x}\right ) \log ^2\left (\log \left (1+\frac {5}{x}\right )\right )}dx+5 (10+\log (\log (2))) \int \frac {1}{(x+5) \log \left (1+\frac {5}{x}\right ) \log ^2\left (\log \left (1+\frac {5}{x}\right )\right )}dx+\log (\log (2)) \int \frac {1}{\log \left (\log \left (1+\frac {5}{x}\right )\right )}dx-4 \int \frac {x}{\log \left (\log \left (1+\frac {5}{x}\right )\right )}dx+30 x\right )\) |
Int[(-10*x + 5*Log[Log[2]] + ((-20*x - 4*x^2)*Log[(5 + x)/x] + (5 + x)*Log [(5 + x)/x]*Log[Log[2]])*Log[Log[(5 + x)/x]] + (150 + 30*x)*Log[(5 + x)/x] *Log[Log[(5 + x)/x]]^2)/((25 + 5*x)*Log[(5 + x)/x]*Log[Log[(5 + x)/x]]^2), x]
3.26.91.3.1 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]]
Time = 1.08 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.66
method | result | size |
parallelrisch | \(\frac {x \ln \left (\ln \left (2\right )\right )-2 x^{2}+30 \ln \left (\ln \left (\frac {5+x}{x}\right )\right ) x -300 \ln \left (\ln \left (\frac {5+x}{x}\right )\right )}{5 \ln \left (\ln \left (\frac {5+x}{x}\right )\right )}\) | \(48\) |
int(((30*x+150)*ln(1/x*(5+x))*ln(ln(1/x*(5+x)))^2+((5+x)*ln(1/x*(5+x))*ln( ln(2))+(-4*x^2-20*x)*ln(1/x*(5+x)))*ln(ln(1/x*(5+x)))+5*ln(ln(2))-10*x)/(2 5+5*x)/ln(1/x*(5+x))/ln(ln(1/x*(5+x)))^2,x,method=_RETURNVERBOSE)
Time = 0.26 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.28 \[ \int \frac {-10 x+5 \log (\log (2))+\left (\left (-20 x-4 x^2\right ) \log \left (\frac {5+x}{x}\right )+(5+x) \log \left (\frac {5+x}{x}\right ) \log (\log (2))\right ) \log \left (\log \left (\frac {5+x}{x}\right )\right )+(150+30 x) \log \left (\frac {5+x}{x}\right ) \log ^2\left (\log \left (\frac {5+x}{x}\right )\right )}{(25+5 x) \log \left (\frac {5+x}{x}\right ) \log ^2\left (\log \left (\frac {5+x}{x}\right )\right )} \, dx=-\frac {2 \, x^{2} - x \log \left (\log \left (2\right )\right ) - 30 \, x \log \left (\log \left (\frac {x + 5}{x}\right )\right )}{5 \, \log \left (\log \left (\frac {x + 5}{x}\right )\right )} \]
integrate(((30*x+150)*log(1/x*(5+x))*log(log(1/x*(5+x)))^2+((5+x)*log(1/x* (5+x))*log(log(2))+(-4*x^2-20*x)*log(1/x*(5+x)))*log(log(1/x*(5+x)))+5*log (log(2))-10*x)/(25+5*x)/log(1/x*(5+x))/log(log(1/x*(5+x)))^2,x, algorithm= \
Time = 0.08 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.83 \[ \int \frac {-10 x+5 \log (\log (2))+\left (\left (-20 x-4 x^2\right ) \log \left (\frac {5+x}{x}\right )+(5+x) \log \left (\frac {5+x}{x}\right ) \log (\log (2))\right ) \log \left (\log \left (\frac {5+x}{x}\right )\right )+(150+30 x) \log \left (\frac {5+x}{x}\right ) \log ^2\left (\log \left (\frac {5+x}{x}\right )\right )}{(25+5 x) \log \left (\frac {5+x}{x}\right ) \log ^2\left (\log \left (\frac {5+x}{x}\right )\right )} \, dx=6 x + \frac {- 2 x^{2} + x \log {\left (\log {\left (2 \right )} \right )}}{5 \log {\left (\log {\left (\frac {x + 5}{x} \right )} \right )}} \]
integrate(((30*x+150)*ln(1/x*(5+x))*ln(ln(1/x*(5+x)))**2+((5+x)*ln(1/x*(5+ x))*ln(ln(2))+(-4*x**2-20*x)*ln(1/x*(5+x)))*ln(ln(1/x*(5+x)))+5*ln(ln(2))- 10*x)/(25+5*x)/ln(1/x*(5+x))/ln(ln(1/x*(5+x)))**2,x)
Time = 0.30 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.34 \[ \int \frac {-10 x+5 \log (\log (2))+\left (\left (-20 x-4 x^2\right ) \log \left (\frac {5+x}{x}\right )+(5+x) \log \left (\frac {5+x}{x}\right ) \log (\log (2))\right ) \log \left (\log \left (\frac {5+x}{x}\right )\right )+(150+30 x) \log \left (\frac {5+x}{x}\right ) \log ^2\left (\log \left (\frac {5+x}{x}\right )\right )}{(25+5 x) \log \left (\frac {5+x}{x}\right ) \log ^2\left (\log \left (\frac {5+x}{x}\right )\right )} \, dx=-\frac {2 \, x^{2} - 30 \, x \log \left (\log \left (x + 5\right ) - \log \left (x\right )\right ) - x \log \left (\log \left (2\right )\right )}{5 \, \log \left (\log \left (x + 5\right ) - \log \left (x\right )\right )} \]
integrate(((30*x+150)*log(1/x*(5+x))*log(log(1/x*(5+x)))^2+((5+x)*log(1/x* (5+x))*log(log(2))+(-4*x^2-20*x)*log(1/x*(5+x)))*log(log(1/x*(5+x)))+5*log (log(2))-10*x)/(25+5*x)/log(1/x*(5+x))/log(log(1/x*(5+x)))^2,x, algorithm= \
Leaf count of result is larger than twice the leaf count of optimal. 64 vs. \(2 (27) = 54\).
Time = 0.40 (sec) , antiderivative size = 64, normalized size of antiderivative = 2.21 \[ \int \frac {-10 x+5 \log (\log (2))+\left (\left (-20 x-4 x^2\right ) \log \left (\frac {5+x}{x}\right )+(5+x) \log \left (\frac {5+x}{x}\right ) \log (\log (2))\right ) \log \left (\log \left (\frac {5+x}{x}\right )\right )+(150+30 x) \log \left (\frac {5+x}{x}\right ) \log ^2\left (\log \left (\frac {5+x}{x}\right )\right )}{(25+5 x) \log \left (\frac {5+x}{x}\right ) \log ^2\left (\log \left (\frac {5+x}{x}\right )\right )} \, dx=6 \, x - \frac {2 \, x^{2} \log \left (\frac {x + 5}{x}\right ) - x \log \left (\frac {x + 5}{x}\right ) \log \left (\log \left (2\right )\right )}{5 \, {\left (\log \left (x + 5\right ) \log \left (\log \left (\frac {x + 5}{x}\right )\right ) - \log \left (x\right ) \log \left (\log \left (\frac {x + 5}{x}\right )\right )\right )}} \]
integrate(((30*x+150)*log(1/x*(5+x))*log(log(1/x*(5+x)))^2+((5+x)*log(1/x* (5+x))*log(log(2))+(-4*x^2-20*x)*log(1/x*(5+x)))*log(log(1/x*(5+x)))+5*log (log(2))-10*x)/(25+5*x)/log(1/x*(5+x))/log(log(1/x*(5+x)))^2,x, algorithm= \
6*x - 1/5*(2*x^2*log((x + 5)/x) - x*log((x + 5)/x)*log(log(2)))/(log(x + 5 )*log(log((x + 5)/x)) - log(x)*log(log((x + 5)/x)))
Time = 12.65 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.10 \[ \int \frac {-10 x+5 \log (\log (2))+\left (\left (-20 x-4 x^2\right ) \log \left (\frac {5+x}{x}\right )+(5+x) \log \left (\frac {5+x}{x}\right ) \log (\log (2))\right ) \log \left (\log \left (\frac {5+x}{x}\right )\right )+(150+30 x) \log \left (\frac {5+x}{x}\right ) \log ^2\left (\log \left (\frac {5+x}{x}\right )\right )}{(25+5 x) \log \left (\frac {5+x}{x}\right ) \log ^2\left (\log \left (\frac {5+x}{x}\right )\right )} \, dx=\frac {x\,\left (\ln \left (\ln \left (2\right )\right )-2\,x+30\,\ln \left (\ln \left (\frac {x+5}{x}\right )\right )\right )}{5\,\ln \left (\ln \left (\frac {x+5}{x}\right )\right )} \]