Integrand size = 95, antiderivative size = 25 \[ \int \frac {5+e^{4-x} (-1+x)-5 x^2+(-1+x) \log (-1+x)+\left (e^{4-x} (-1+x)+x+\left (-1+x^2\right ) \log (-1+x)\right ) \log (x)}{5 x-5 x^2+\left (e^{4-x} \left (-x+x^2\right )+\left (-x+x^2\right ) \log (-1+x)\right ) \log (x)} \, dx=x+\log \left (x \left (-1+\frac {1}{5} \left (e^{4-x}+\log (-1+x)\right ) \log (x)\right )\right ) \] Output:
x+ln(x*(1/5*ln(x)*(exp(4-x)+ln(-1+x))-1))
Time = 0.10 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.04 \[ \int \frac {5+e^{4-x} (-1+x)-5 x^2+(-1+x) \log (-1+x)+\left (e^{4-x} (-1+x)+x+\left (-1+x^2\right ) \log (-1+x)\right ) \log (x)}{5 x-5 x^2+\left (e^{4-x} \left (-x+x^2\right )+\left (-x+x^2\right ) \log (-1+x)\right ) \log (x)} \, dx=\log (x)+\log \left (-5 e^x+e^4 \log (x)+e^x \log (-1+x) \log (x)\right ) \] Input:
Integrate[(5 + E^(4 - x)*(-1 + x) - 5*x^2 + (-1 + x)*Log[-1 + x] + (E^(4 - x)*(-1 + x) + x + (-1 + x^2)*Log[-1 + x])*Log[x])/(5*x - 5*x^2 + (E^(4 - x)*(-x + x^2) + (-x + x^2)*Log[-1 + x])*Log[x]),x]
Output:
Log[x] + Log[-5*E^x + E^4*Log[x] + E^x*Log[-1 + x]*Log[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.
\(\displaystyle \int \frac {-5 x^2+\left (\left (x^2-1\right ) \log (x-1)+e^{4-x} (x-1)+x\right ) \log (x)+e^{4-x} (x-1)+(x-1) \log (x-1)+5}{-5 x^2+\left (e^{4-x} \left (x^2-x\right )+\left (x^2-x\right ) \log (x-1)\right ) \log (x)+5 x} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {e^x \left (-5 x^2+\left (\left (x^2-1\right ) \log (x-1)+e^{4-x} (x-1)+x\right ) \log (x)+e^{4-x} (x-1)+(x-1) \log (x-1)+5\right )}{(1-x) x \left (5 e^x-e^x \log (x-1) \log (x)-e^4 \log (x)\right )}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {e^x \left (x^2 \log (x-1) \log ^2(x)-5 x^2 \log (x)+5 x-x \log (x-1) \log ^2(x)+x \log ^2(x)+5 x \log (x)-5\right )}{(x-1) x \log (x) \left (-5 e^x+e^x \log (x-1) \log (x)+e^4 \log (x)\right )}+\frac {\log (x)+1}{x \log (x)}\right )dx\) |
\(\Big \downarrow \) 7299 |
\(\displaystyle \int \left (\frac {e^x \left (x^2 \log (x-1) \log ^2(x)-5 x^2 \log (x)+5 x-x \log (x-1) \log ^2(x)+x \log ^2(x)+5 x \log (x)-5\right )}{(x-1) x \log (x) \left (-5 e^x+e^x \log (x-1) \log (x)+e^4 \log (x)\right )}+\frac {\log (x)+1}{x \log (x)}\right )dx\) |
Input:
Int[(5 + E^(4 - x)*(-1 + x) - 5*x^2 + (-1 + x)*Log[-1 + x] + (E^(4 - x)*(- 1 + x) + x + (-1 + x^2)*Log[-1 + x])*Log[x])/(5*x - 5*x^2 + (E^(4 - x)*(-x + x^2) + (-x + x^2)*Log[-1 + x])*Log[x]),x]
Output:
$Aborted
Time = 37.24 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00
method | result | size |
parallelrisch | \(\ln \left (x \right )+\ln \left (\ln \left (x \right ) \ln \left (-1+x \right )+{\mathrm e}^{-x +4} \ln \left (x \right )-5\right )+x +\frac {1}{2}\) | \(25\) |
risch | \(x +\ln \left (x \right )+\ln \left (\ln \left (x \right )\right )+\ln \left (\ln \left (-1+x \right )+\frac {{\mathrm e}^{-x +4} \ln \left (x \right )-5}{\ln \left (x \right )}\right )\) | \(30\) |
Input:
int((((x^2-1)*ln(-1+x)+(-1+x)*exp(-x+4)+x)*ln(x)+(-1+x)*ln(-1+x)+(-1+x)*ex p(-x+4)-5*x^2+5)/(((x^2-x)*ln(-1+x)+(x^2-x)*exp(-x+4))*ln(x)-5*x^2+5*x),x, method=_RETURNVERBOSE)
Output:
ln(x)+ln(ln(x)*ln(-1+x)+exp(-x+4)*ln(x)-5)+x+1/2
Leaf count of result is larger than twice the leaf count of optimal. 47 vs. \(2 (22) = 44\).
Time = 0.16 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.88 \[ \int \frac {5+e^{4-x} (-1+x)-5 x^2+(-1+x) \log (-1+x)+\left (e^{4-x} (-1+x)+x+\left (-1+x^2\right ) \log (-1+x)\right ) \log (x)}{5 x-5 x^2+\left (e^{4-x} \left (-x+x^2\right )+\left (-x+x^2\right ) \log (-1+x)\right ) \log (x)} \, dx=x + \log \left (x\right ) + \log \left (\frac {{\left (e^{\left (-x + 4\right )} + \log \left (x - 1\right )\right )} \log \left (x\right ) - 5}{e^{\left (-x + 4\right )} + \log \left (x - 1\right )}\right ) + \log \left (e^{\left (-x + 4\right )} + \log \left (x - 1\right )\right ) \] Input:
integrate((((x^2-1)*log(-1+x)+(-1+x)*exp(-x+4)+x)*log(x)+(-1+x)*log(-1+x)+ (-1+x)*exp(-x+4)-5*x^2+5)/(((x^2-x)*log(-1+x)+(x^2-x)*exp(-x+4))*log(x)-5* x^2+5*x),x, algorithm="fricas")
Output:
x + log(x) + log(((e^(-x + 4) + log(x - 1))*log(x) - 5)/(e^(-x + 4) + log( x - 1))) + log(e^(-x + 4) + log(x - 1))
Time = 0.36 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.16 \[ \int \frac {5+e^{4-x} (-1+x)-5 x^2+(-1+x) \log (-1+x)+\left (e^{4-x} (-1+x)+x+\left (-1+x^2\right ) \log (-1+x)\right ) \log (x)}{5 x-5 x^2+\left (e^{4-x} \left (-x+x^2\right )+\left (-x+x^2\right ) \log (-1+x)\right ) \log (x)} \, dx=x + \log {\left (x \right )} + \log {\left (\frac {\log {\left (x \right )} \log {\left (x - 1 \right )} - 5}{\log {\left (x \right )}} + e^{4 - x} \right )} + \log {\left (\log {\left (x \right )} \right )} \] Input:
integrate((((x**2-1)*ln(-1+x)+(-1+x)*exp(-x+4)+x)*ln(x)+(-1+x)*ln(-1+x)+(- 1+x)*exp(-x+4)-5*x**2+5)/(((x**2-x)*ln(-1+x)+(x**2-x)*exp(-x+4))*ln(x)-5*x **2+5*x),x)
Output:
x + log(x) + log((log(x)*log(x - 1) - 5)/log(x) + exp(4 - x)) + log(log(x) )
Time = 0.11 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.44 \[ \int \frac {5+e^{4-x} (-1+x)-5 x^2+(-1+x) \log (-1+x)+\left (e^{4-x} (-1+x)+x+\left (-1+x^2\right ) \log (-1+x)\right ) \log (x)}{5 x-5 x^2+\left (e^{4-x} \left (-x+x^2\right )+\left (-x+x^2\right ) \log (-1+x)\right ) \log (x)} \, dx=x + \log \left (x\right ) + \log \left (\frac {{\left (e^{x} \log \left (x - 1\right ) \log \left (x\right ) + e^{4} \log \left (x\right ) - 5 \, e^{x}\right )} e^{\left (-x\right )}}{\log \left (x\right )}\right ) + \log \left (\log \left (x\right )\right ) \] Input:
integrate((((x^2-1)*log(-1+x)+(-1+x)*exp(-x+4)+x)*log(x)+(-1+x)*log(-1+x)+ (-1+x)*exp(-x+4)-5*x^2+5)/(((x^2-x)*log(-1+x)+(x^2-x)*exp(-x+4))*log(x)-5* x^2+5*x),x, algorithm="maxima")
Output:
x + log(x) + log((e^x*log(x - 1)*log(x) + e^4*log(x) - 5*e^x)*e^(-x)/log(x )) + log(log(x))
Time = 0.13 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.92 \[ \int \frac {5+e^{4-x} (-1+x)-5 x^2+(-1+x) \log (-1+x)+\left (e^{4-x} (-1+x)+x+\left (-1+x^2\right ) \log (-1+x)\right ) \log (x)}{5 x-5 x^2+\left (e^{4-x} \left (-x+x^2\right )+\left (-x+x^2\right ) \log (-1+x)\right ) \log (x)} \, dx=x + \log \left (e^{\left (-x + 4\right )} \log \left (x\right ) + \log \left (x - 1\right ) \log \left (x\right ) - 5\right ) + \log \left (x\right ) \] Input:
integrate((((x^2-1)*log(-1+x)+(-1+x)*exp(-x+4)+x)*log(x)+(-1+x)*log(-1+x)+ (-1+x)*exp(-x+4)-5*x^2+5)/(((x^2-x)*log(-1+x)+(x^2-x)*exp(-x+4))*log(x)-5* x^2+5*x),x, algorithm="giac")
Output:
x + log(e^(-x + 4)*log(x) + log(x - 1)*log(x) - 5) + log(x)
Time = 2.86 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.24 \[ \int \frac {5+e^{4-x} (-1+x)-5 x^2+(-1+x) \log (-1+x)+\left (e^{4-x} (-1+x)+x+\left (-1+x^2\right ) \log (-1+x)\right ) \log (x)}{5 x-5 x^2+\left (e^{4-x} \left (-x+x^2\right )+\left (-x+x^2\right ) \log (-1+x)\right ) \log (x)} \, dx=x+\ln \left (\ln \left (x\right )\right )+\ln \left (\frac {\ln \left (x-1\right )\,\ln \left (x\right )+{\mathrm {e}}^{4-x}\,\ln \left (x\right )-5}{\ln \left (x\right )}\right )+\ln \left (x\right ) \] Input:
int(-(log(x - 1)*(x - 1) + log(x)*(x + exp(4 - x)*(x - 1) + log(x - 1)*(x^ 2 - 1)) + exp(4 - x)*(x - 1) - 5*x^2 + 5)/(log(x)*(exp(4 - x)*(x - x^2) + log(x - 1)*(x - x^2)) - 5*x + 5*x^2),x)
Output:
x + log(log(x)) + log((log(x - 1)*log(x) + exp(4 - x)*log(x) - 5)/log(x)) + log(x)
Time = 0.18 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.04 \[ \int \frac {5+e^{4-x} (-1+x)-5 x^2+(-1+x) \log (-1+x)+\left (e^{4-x} (-1+x)+x+\left (-1+x^2\right ) \log (-1+x)\right ) \log (x)}{5 x-5 x^2+\left (e^{4-x} \left (-x+x^2\right )+\left (-x+x^2\right ) \log (-1+x)\right ) \log (x)} \, dx=\mathrm {log}\left (e^{x} \mathrm {log}\left (x -1\right ) \mathrm {log}\left (x \right )-5 e^{x}+\mathrm {log}\left (x \right ) e^{4}\right )+\mathrm {log}\left (x \right ) \] Input:
int((((x^2-1)*log(-1+x)+(-1+x)*exp(-x+4)+x)*log(x)+(-1+x)*log(-1+x)+(-1+x) *exp(-x+4)-5*x^2+5)/(((x^2-x)*log(-1+x)+(x^2-x)*exp(-x+4))*log(x)-5*x^2+5* x),x)
Output:
log(e**x*log(x - 1)*log(x) - 5*e**x + log(x)*e**4) + log(x)