Integrand size = 79, antiderivative size = 26 \[ \int \frac {6-2 x-6 x^2+2 e^3 x^2+4 x^3+\left (-6-x+e^3 x+2 x^2\right ) \log (x)+\left (4 x^2+2 x \log (x)\right ) \log \left (\frac {2 x}{2 x+\log (x)}\right )}{4 x^2+2 x \log (x)} \, dx=(-3+x) \left (\frac {1}{2} \left (e^3+x\right )+\log \left (\frac {2 x}{2 x+\log (x)}\right )\right ) \] Output:
(1/2*exp(3)+1/2*x+ln(2*x/(2*x+ln(x))))*(-3+x)
Time = 0.15 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.65 \[ \int \frac {6-2 x-6 x^2+2 e^3 x^2+4 x^3+\left (-6-x+e^3 x+2 x^2\right ) \log (x)+\left (4 x^2+2 x \log (x)\right ) \log \left (\frac {2 x}{2 x+\log (x)}\right )}{4 x^2+2 x \log (x)} \, dx=\frac {1}{2} \left (\left (-3+e^3\right ) x+x^2-6 \log (x)+2 x \log \left (\frac {2 x}{2 x+\log (x)}\right )+6 \log (2 x+\log (x))\right ) \] Input:
Integrate[(6 - 2*x - 6*x^2 + 2*E^3*x^2 + 4*x^3 + (-6 - x + E^3*x + 2*x^2)* Log[x] + (4*x^2 + 2*x*Log[x])*Log[(2*x)/(2*x + Log[x])])/(4*x^2 + 2*x*Log[ x]),x]
Output:
((-3 + E^3)*x + x^2 - 6*Log[x] + 2*x*Log[(2*x)/(2*x + Log[x])] + 6*Log[2*x + Log[x]])/2
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.
\(\displaystyle \int \frac {4 x^3+2 e^3 x^2-6 x^2+\left (2 x^2+e^3 x-x-6\right ) \log (x)+\left (4 x^2+2 x \log (x)\right ) \log \left (\frac {2 x}{2 x+\log (x)}\right )-2 x+6}{4 x^2+2 x \log (x)} \, dx\) |
\(\Big \downarrow \) 6 |
\(\displaystyle \int \frac {4 x^3+\left (2 e^3-6\right ) x^2+\left (2 x^2+e^3 x-x-6\right ) \log (x)+\left (4 x^2+2 x \log (x)\right ) \log \left (\frac {2 x}{2 x+\log (x)}\right )-2 x+6}{4 x^2+2 x \log (x)}dx\) |
\(\Big \downarrow \) 3041 |
\(\displaystyle \int \frac {4 x^3+\left (2 e^3-6\right ) x^2+\left (2 x^2+e^3 x-x-6\right ) \log (x)+\left (4 x^2+2 x \log (x)\right ) \log \left (\frac {2 x}{2 x+\log (x)}\right )-2 x+6}{x (4 x+2 \log (x))}dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {4 x^3+\left (2 e^3-6\right ) x^2+\left (2 x^2+e^3 x-x-6\right ) \log (x)+\left (4 x^2+2 x \log (x)\right ) \log \left (\frac {2 x}{2 x+\log (x)}\right )-2 x+6}{2 x (2 x+\log (x))}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{2} \int \frac {4 x^3-2 \left (3-e^3\right ) x^2-2 x-\left (-2 x^2-e^3 x+x+6\right ) \log (x)+2 \left (2 x^2+\log (x) x\right ) \log \left (\frac {2 x}{2 x+\log (x)}\right )+6}{x (2 x+\log (x))}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \frac {1}{2} \int \left (\frac {4 x^3+2 \log (x) x^2-6 \left (1-\frac {e^3}{3}\right ) x^2-\left (1-e^3\right ) \log (x) x-2 x-6 \log (x)+6}{x (2 x+\log (x))}+2 \log \left (\frac {2 x}{2 x+\log (x)}\right )\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{2} \left (-2 \int \frac {1}{-2 x-\log (x)}dx+10 \int \frac {1}{2 x+\log (x)}dx+6 \int \frac {1}{x (2 x+\log (x))}dx+x^2-\left (1-e^3\right ) x-2 x+2 x \log \left (\frac {2 x}{2 x+\log (x)}\right )-6 \log (x)\right )\) |
Input:
Int[(6 - 2*x - 6*x^2 + 2*E^3*x^2 + 4*x^3 + (-6 - x + E^3*x + 2*x^2)*Log[x] + (4*x^2 + 2*x*Log[x])*Log[(2*x)/(2*x + Log[x])])/(4*x^2 + 2*x*Log[x]),x]
Output:
$Aborted
Time = 0.51 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.65
method | result | size |
parallelrisch | \(\frac {x \,{\mathrm e}^{3}}{2}+\frac {x^{2}}{2}+\ln \left (\frac {2 x}{2 x +\ln \left (x \right )}\right ) x -\frac {3 x}{2}-3 \ln \left (\frac {2 x}{2 x +\ln \left (x \right )}\right )\) | \(43\) |
risch | \(-x \ln \left (x +\frac {\ln \left (x \right )}{2}\right )+x \ln \left (x \right )+\frac {i \pi x \,\operatorname {csgn}\left (\frac {i}{x +\frac {\ln \left (x \right )}{2}}\right ) \operatorname {csgn}\left (\frac {i x}{x +\frac {\ln \left (x \right )}{2}}\right )^{2}}{2}-\frac {i \pi x \,\operatorname {csgn}\left (\frac {i}{x +\frac {\ln \left (x \right )}{2}}\right ) \operatorname {csgn}\left (\frac {i x}{x +\frac {\ln \left (x \right )}{2}}\right ) \operatorname {csgn}\left (i x \right )}{2}-\frac {i \pi x \operatorname {csgn}\left (\frac {i x}{x +\frac {\ln \left (x \right )}{2}}\right )^{3}}{2}+\frac {i \pi x \operatorname {csgn}\left (\frac {i x}{x +\frac {\ln \left (x \right )}{2}}\right )^{2} \operatorname {csgn}\left (i x \right )}{2}+\frac {x \,{\mathrm e}^{3}}{2}+\frac {x^{2}}{2}-\frac {3 x}{2}-3 \ln \left (x \right )+3 \ln \left (2 x +\ln \left (x \right )\right )\) | \(154\) |
Input:
int(((2*x*ln(x)+4*x^2)*ln(2*x/(2*x+ln(x)))+(x*exp(3)+2*x^2-x-6)*ln(x)+2*x^ 2*exp(3)+4*x^3-6*x^2-2*x+6)/(2*x*ln(x)+4*x^2),x,method=_RETURNVERBOSE)
Output:
1/2*x*exp(3)+1/2*x^2+ln(2*x/(2*x+ln(x)))*x-3/2*x-3*ln(2*x/(2*x+ln(x)))
Time = 0.08 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.15 \[ \int \frac {6-2 x-6 x^2+2 e^3 x^2+4 x^3+\left (-6-x+e^3 x+2 x^2\right ) \log (x)+\left (4 x^2+2 x \log (x)\right ) \log \left (\frac {2 x}{2 x+\log (x)}\right )}{4 x^2+2 x \log (x)} \, dx=\frac {1}{2} \, x^{2} + \frac {1}{2} \, x e^{3} + {\left (x - 3\right )} \log \left (\frac {2 \, x}{2 \, x + \log \left (x\right )}\right ) - \frac {3}{2} \, x \] Input:
integrate(((2*x*log(x)+4*x^2)*log(2*x/(2*x+log(x)))+(x*exp(3)+2*x^2-x-6)*l og(x)+2*x^2*exp(3)+4*x^3-6*x^2-2*x+6)/(2*x*log(x)+4*x^2),x, algorithm="fri cas")
Output:
1/2*x^2 + 1/2*x*e^3 + (x - 3)*log(2*x/(2*x + log(x))) - 3/2*x
Time = 0.22 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.58 \[ \int \frac {6-2 x-6 x^2+2 e^3 x^2+4 x^3+\left (-6-x+e^3 x+2 x^2\right ) \log (x)+\left (4 x^2+2 x \log (x)\right ) \log \left (\frac {2 x}{2 x+\log (x)}\right )}{4 x^2+2 x \log (x)} \, dx=\frac {x^{2}}{2} + x \log {\left (\frac {2 x}{2 x + \log {\left (x \right )}} \right )} + \frac {x \left (-3 + e^{3}\right )}{2} - 3 \log {\left (x \right )} + 3 \log {\left (2 x + \log {\left (x \right )} \right )} \] Input:
integrate(((2*x*ln(x)+4*x**2)*ln(2*x/(2*x+ln(x)))+(x*exp(3)+2*x**2-x-6)*ln (x)+2*x**2*exp(3)+4*x**3-6*x**2-2*x+6)/(2*x*ln(x)+4*x**2),x)
Output:
x**2/2 + x*log(2*x/(2*x + log(x))) + x*(-3 + exp(3))/2 - 3*log(x) + 3*log( 2*x + log(x))
Time = 0.17 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.35 \[ \int \frac {6-2 x-6 x^2+2 e^3 x^2+4 x^3+\left (-6-x+e^3 x+2 x^2\right ) \log (x)+\left (4 x^2+2 x \log (x)\right ) \log \left (\frac {2 x}{2 x+\log (x)}\right )}{4 x^2+2 x \log (x)} \, dx=\frac {1}{2} \, x^{2} + \frac {1}{2} \, x {\left (e^{3} + 2 \, \log \left (2\right ) - 3\right )} - {\left (x - 3\right )} \log \left (2 \, x + \log \left (x\right )\right ) + {\left (x - 3\right )} \log \left (x\right ) \] Input:
integrate(((2*x*log(x)+4*x^2)*log(2*x/(2*x+log(x)))+(x*exp(3)+2*x^2-x-6)*l og(x)+2*x^2*exp(3)+4*x^3-6*x^2-2*x+6)/(2*x*log(x)+4*x^2),x, algorithm="max ima")
Output:
1/2*x^2 + 1/2*x*(e^3 + 2*log(2) - 3) - (x - 3)*log(2*x + log(x)) + (x - 3) *log(x)
Time = 0.13 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.73 \[ \int \frac {6-2 x-6 x^2+2 e^3 x^2+4 x^3+\left (-6-x+e^3 x+2 x^2\right ) \log (x)+\left (4 x^2+2 x \log (x)\right ) \log \left (\frac {2 x}{2 x+\log (x)}\right )}{4 x^2+2 x \log (x)} \, dx=\frac {1}{2} \, x^{2} + \frac {1}{2} \, x e^{3} + x \log \left (2\right ) - x \log \left (2 \, x + \log \left (x\right )\right ) + x \log \left (x\right ) - \frac {3}{2} \, x + 3 \, \log \left (2 \, x + \log \left (x\right )\right ) - 3 \, \log \left (x\right ) \] Input:
integrate(((2*x*log(x)+4*x^2)*log(2*x/(2*x+log(x)))+(x*exp(3)+2*x^2-x-6)*l og(x)+2*x^2*exp(3)+4*x^3-6*x^2-2*x+6)/(2*x*log(x)+4*x^2),x, algorithm="gia c")
Output:
1/2*x^2 + 1/2*x*e^3 + x*log(2) - x*log(2*x + log(x)) + x*log(x) - 3/2*x + 3*log(2*x + log(x)) - 3*log(x)
Time = 2.85 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.58 \[ \int \frac {6-2 x-6 x^2+2 e^3 x^2+4 x^3+\left (-6-x+e^3 x+2 x^2\right ) \log (x)+\left (4 x^2+2 x \log (x)\right ) \log \left (\frac {2 x}{2 x+\log (x)}\right )}{4 x^2+2 x \log (x)} \, dx=3\,\ln \left (2\,x+\ln \left (x\right )\right )-3\,\ln \left (x\right )+x\,\ln \left (\frac {2\,x}{2\,x+\ln \left (x\right )}\right )+\frac {x^2}{2}+x\,\left (\frac {{\mathrm {e}}^3}{2}-\frac {3}{2}\right ) \] Input:
int((log((2*x)/(2*x + log(x)))*(2*x*log(x) + 4*x^2) - 2*x - log(x)*(x - x* exp(3) - 2*x^2 + 6) + 2*x^2*exp(3) - 6*x^2 + 4*x^3 + 6)/(2*x*log(x) + 4*x^ 2),x)
Output:
3*log(2*x + log(x)) - 3*log(x) + x*log((2*x)/(2*x + log(x))) + x^2/2 + x*( exp(3)/2 - 3/2)
Time = 0.22 (sec) , antiderivative size = 56, normalized size of antiderivative = 2.15 \[ \int \frac {6-2 x-6 x^2+2 e^3 x^2+4 x^3+\left (-6-x+e^3 x+2 x^2\right ) \log (x)+\left (4 x^2+2 x \log (x)\right ) \log \left (\frac {2 x}{2 x+\log (x)}\right )}{4 x^2+2 x \log (x)} \, dx=\frac {5 \,\mathrm {log}\left (\mathrm {log}\left (x \right )+2 x \right )}{2}+\mathrm {log}\left (\frac {2 x}{\mathrm {log}\left (x \right )+2 x}\right ) x -\frac {\mathrm {log}\left (\frac {2 x}{\mathrm {log}\left (x \right )+2 x}\right )}{2}-\frac {5 \,\mathrm {log}\left (x \right )}{2}+\frac {e^{3} x}{2}+\frac {x^{2}}{2}-\frac {3 x}{2} \] Input:
int(((2*x*log(x)+4*x^2)*log(2*x/(2*x+log(x)))+(x*exp(3)+2*x^2-x-6)*log(x)+ 2*x^2*exp(3)+4*x^3-6*x^2-2*x+6)/(2*x*log(x)+4*x^2),x)
Output:
(5*log(log(x) + 2*x) + 2*log((2*x)/(log(x) + 2*x))*x - log((2*x)/(log(x) + 2*x)) - 5*log(x) + e**3*x + x**2 - 3*x)/2