Integrand size = 95, antiderivative size = 27 \[ \int \frac {-2 x+x^2+(-24-6 x) \log (2)+\left (-24-2 x+x^2+(-24-6 x) \log (2)\right ) \log \left (\frac {4+x}{x}\right )+\left (-24-6 x+(-24-6 x) \log \left (\frac {4+x}{x}\right )\right ) \log \left (x+x \log \left (\frac {4+x}{x}\right )\right )}{24+6 x+(24+6 x) \log \left (\frac {4+x}{x}\right )} \, dx=x \left (\frac {x}{12}-\log (2)-\log \left (x+x \log \left (\frac {4+x}{x}\right )\right )\right ) \]
Time = 0.05 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.22 \[ \int \frac {-2 x+x^2+(-24-6 x) \log (2)+\left (-24-2 x+x^2+(-24-6 x) \log (2)\right ) \log \left (\frac {4+x}{x}\right )+\left (-24-6 x+(-24-6 x) \log \left (\frac {4+x}{x}\right )\right ) \log \left (x+x \log \left (\frac {4+x}{x}\right )\right )}{24+6 x+(24+6 x) \log \left (\frac {4+x}{x}\right )} \, dx=\frac {1}{6} \left (\frac {x^2}{2}-6 x \log (2)-6 x \log \left (x \left (1+\log \left (\frac {4+x}{x}\right )\right )\right )\right ) \]
Integrate[(-2*x + x^2 + (-24 - 6*x)*Log[2] + (-24 - 2*x + x^2 + (-24 - 6*x )*Log[2])*Log[(4 + x)/x] + (-24 - 6*x + (-24 - 6*x)*Log[(4 + x)/x])*Log[x + x*Log[(4 + x)/x]])/(24 + 6*x + (24 + 6*x)*Log[(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.
\(\displaystyle \int \frac {x^2+\left (x^2-2 x+(-6 x-24) \log (2)-24\right ) \log \left (\frac {x+4}{x}\right )-2 x+\left (-6 x+(-6 x-24) \log \left (\frac {x+4}{x}\right )-24\right ) \log \left (x+x \log \left (\frac {x+4}{x}\right )\right )+(-6 x-24) \log (2)}{6 x+(6 x+24) \log \left (\frac {x+4}{x}\right )+24} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {x^2+\left (x^2-2 x+(-6 x-24) \log (2)-24\right ) \log \left (\frac {x+4}{x}\right )-2 x+\left (-6 x+(-6 x-24) \log \left (\frac {x+4}{x}\right )-24\right ) \log \left (x+x \log \left (\frac {x+4}{x}\right )\right )+(-6 x-24) \log (2)}{6 (x+4) \left (\log \left (\frac {x+4}{x}\right )+1\right )}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{6} \int -\frac {-x^2+2 x+\left (-x^2+2 x+6 (x+4) \log (2)+24\right ) \log \left (\frac {x+4}{x}\right )+6 \left (x+(x+4) \log \left (\frac {x+4}{x}\right )+4\right ) \log \left (\log \left (\frac {x+4}{x}\right ) x+x\right )+6 (x+4) \log (2)}{(x+4) \left (\log \left (\frac {x+4}{x}\right )+1\right )}dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {1}{6} \int \frac {-x^2+2 x+\left (-x^2+2 x+6 (x+4) \log (2)+24\right ) \log \left (\frac {x+4}{x}\right )+6 \left (x+(x+4) \log \left (\frac {x+4}{x}\right )+4\right ) \log \left (\log \left (\frac {x+4}{x}\right ) x+x\right )+6 (x+4) \log (2)}{(x+4) \left (\log \left (\frac {x+4}{x}\right )+1\right )}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle -\frac {1}{6} \int \left (\frac {-\log \left (\frac {x+4}{x}\right ) x^2-x^2+2 (1+\log (8)) \log \left (\frac {x+4}{x}\right ) x+2 (1+\log (8)) x+24 (1+\log (2)) \log \left (\frac {x+4}{x}\right )+24 \log (2)}{(x+4) \left (\log \left (\frac {x+4}{x}\right )+1\right )}+6 \log \left (\log \left (\frac {x+4}{x}\right ) x+x\right )\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{6} \left (24 \int \frac {1}{(x+4) \left (\log \left (1+\frac {4}{x}\right )+1\right )}dx-6 \int \log \left (\log \left (\frac {x+4}{x}\right ) x+x\right )dx+\frac {1}{2} (-x+6+\log (64))^2\right )\) |
Int[(-2*x + x^2 + (-24 - 6*x)*Log[2] + (-24 - 2*x + x^2 + (-24 - 6*x)*Log[ 2])*Log[(4 + x)/x] + (-24 - 6*x + (-24 - 6*x)*Log[(4 + x)/x])*Log[x + x*Lo g[(4 + x)/x]])/(24 + 6*x + (24 + 6*x)*Log[(4 + x)/x]),x]
3.28.77.3.1 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]]
Time = 3.32 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.04
method | result | size |
norman | \(\frac {x^{2}}{12}-x \ln \left (2\right )-\ln \left (\ln \left (\frac {4+x}{x}\right ) x +x \right ) x\) | \(28\) |
parallelrisch | \(-x \ln \left (2\right )-\frac {4}{3}+\frac {x^{2}}{12}-\ln \left (x \left (\ln \left (\frac {4+x}{x}\right )+1\right )\right ) x +8 \ln \left (2\right )\) | \(33\) |
default | \(-\frac {4 \left (-4+\frac {3 x +12}{x}\right )}{3 \left (\frac {4+x}{x}-1\right )^{2}}-x \ln \left (2\right )-\ln \left (\ln \left (\frac {4+x}{x}\right ) x +x \right ) x +x\) | \(47\) |
int((((-6*x-24)*ln((4+x)/x)-6*x-24)*ln(ln((4+x)/x)*x+x)+((-6*x-24)*ln(2)+x ^2-2*x-24)*ln((4+x)/x)+(-6*x-24)*ln(2)+x^2-2*x)/((24+6*x)*ln((4+x)/x)+24+6 *x),x,method=_RETURNVERBOSE)
Time = 0.24 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00 \[ \int \frac {-2 x+x^2+(-24-6 x) \log (2)+\left (-24-2 x+x^2+(-24-6 x) \log (2)\right ) \log \left (\frac {4+x}{x}\right )+\left (-24-6 x+(-24-6 x) \log \left (\frac {4+x}{x}\right )\right ) \log \left (x+x \log \left (\frac {4+x}{x}\right )\right )}{24+6 x+(24+6 x) \log \left (\frac {4+x}{x}\right )} \, dx=\frac {1}{12} \, x^{2} - x \log \left (2\right ) - x \log \left (x \log \left (\frac {x + 4}{x}\right ) + x\right ) \]
integrate((((-6*x-24)*log((4+x)/x)-6*x-24)*log(log((4+x)/x)*x+x)+((-6*x-24 )*log(2)+x^2-2*x-24)*log((4+x)/x)+(-6*x-24)*log(2)+x^2-2*x)/((24+6*x)*log( (4+x)/x)+24+6*x),x, algorithm=\
Leaf count of result is larger than twice the leaf count of optimal. 48 vs. \(2 (19) = 38\).
Time = 0.35 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.78 \[ \int \frac {-2 x+x^2+(-24-6 x) \log (2)+\left (-24-2 x+x^2+(-24-6 x) \log (2)\right ) \log \left (\frac {4+x}{x}\right )+\left (-24-6 x+(-24-6 x) \log \left (\frac {4+x}{x}\right )\right ) \log \left (x+x \log \left (\frac {4+x}{x}\right )\right )}{24+6 x+(24+6 x) \log \left (\frac {4+x}{x}\right )} \, dx=\frac {x^{2}}{12} - x \log {\left (2 \right )} + \left (- x - \frac {2}{3}\right ) \log {\left (x \log {\left (\frac {x + 4}{x} \right )} + x \right )} + \frac {2 \log {\left (x \right )}}{3} + \frac {2 \log {\left (\log {\left (\frac {x + 4}{x} \right )} + 1 \right )}}{3} \]
integrate((((-6*x-24)*ln((4+x)/x)-6*x-24)*ln(ln((4+x)/x)*x+x)+((-6*x-24)*l n(2)+x**2-2*x-24)*ln((4+x)/x)+(-6*x-24)*ln(2)+x**2-2*x)/((24+6*x)*ln((4+x) /x)+24+6*x),x)
x**2/12 - x*log(2) + (-x - 2/3)*log(x*log((x + 4)/x) + x) + 2*log(x)/3 + 2 *log(log((x + 4)/x) + 1)/3
Time = 0.33 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.11 \[ \int \frac {-2 x+x^2+(-24-6 x) \log (2)+\left (-24-2 x+x^2+(-24-6 x) \log (2)\right ) \log \left (\frac {4+x}{x}\right )+\left (-24-6 x+(-24-6 x) \log \left (\frac {4+x}{x}\right )\right ) \log \left (x+x \log \left (\frac {4+x}{x}\right )\right )}{24+6 x+(24+6 x) \log \left (\frac {4+x}{x}\right )} \, dx=\frac {1}{12} \, x^{2} - x \log \left (2\right ) - x \log \left (x\right ) - x \log \left (\log \left (x + 4\right ) - \log \left (x\right ) + 1\right ) \]
integrate((((-6*x-24)*log((4+x)/x)-6*x-24)*log(log((4+x)/x)*x+x)+((-6*x-24 )*log(2)+x^2-2*x-24)*log((4+x)/x)+(-6*x-24)*log(2)+x^2-2*x)/((24+6*x)*log( (4+x)/x)+24+6*x),x, algorithm=\
Time = 0.30 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.11 \[ \int \frac {-2 x+x^2+(-24-6 x) \log (2)+\left (-24-2 x+x^2+(-24-6 x) \log (2)\right ) \log \left (\frac {4+x}{x}\right )+\left (-24-6 x+(-24-6 x) \log \left (\frac {4+x}{x}\right )\right ) \log \left (x+x \log \left (\frac {4+x}{x}\right )\right )}{24+6 x+(24+6 x) \log \left (\frac {4+x}{x}\right )} \, dx=\frac {1}{12} \, x^{2} - x \log \left (2\right ) - x \log \left (x\right ) - x \log \left (\log \left (\frac {x + 4}{x}\right ) + 1\right ) \]
integrate((((-6*x-24)*log((4+x)/x)-6*x-24)*log(log((4+x)/x)*x+x)+((-6*x-24 )*log(2)+x^2-2*x-24)*log((4+x)/x)+(-6*x-24)*log(2)+x^2-2*x)/((24+6*x)*log( (4+x)/x)+24+6*x),x, algorithm=\
Time = 9.66 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00 \[ \int \frac {-2 x+x^2+(-24-6 x) \log (2)+\left (-24-2 x+x^2+(-24-6 x) \log (2)\right ) \log \left (\frac {4+x}{x}\right )+\left (-24-6 x+(-24-6 x) \log \left (\frac {4+x}{x}\right )\right ) \log \left (x+x \log \left (\frac {4+x}{x}\right )\right )}{24+6 x+(24+6 x) \log \left (\frac {4+x}{x}\right )} \, dx=\frac {x^2}{12}-x\,\ln \left (x+x\,\ln \left (\frac {x+4}{x}\right )\right )-x\,\ln \left (2\right ) \]