Integrand size = 65, antiderivative size = 27 \[ \int \frac {2 e^3 x-x \log (x)+\left (-1+2 x-x^2+e^3 (-2+2 x)\right ) \log \left (-\frac {\log (3)}{1+2 e^3-x}\right )}{x+2 e^3 x-x^2} \, dx=x+\log (2)-(1-x+\log (x)) \log \left (\frac {\log (3)}{-1-2 e^3+x}\right ) \] Output:
ln(2)+x-(1+ln(x)-x)*ln(ln(3)/(x-1-2*exp(3)))
Leaf count is larger than twice the leaf count of optimal. \(72\) vs. \(2(27)=54\).
Time = 0.06 (sec) , antiderivative size = 72, normalized size of antiderivative = 2.67 \[ \int \frac {2 e^3 x-x \log (x)+\left (-1+2 x-x^2+e^3 (-2+2 x)\right ) \log \left (-\frac {\log (3)}{1+2 e^3-x}\right )}{x+2 e^3 x-x^2} \, dx=x-\left (2 e^3-\log \left (1+2 e^3\right )\right ) \log \left (1+2 e^3-x\right )-\left (1+2 e^3-x+\log \left (\frac {x}{1+2 e^3}\right )\right ) \log \left (-\frac {\log (3)}{1+2 e^3-x}\right ) \] Input:
Integrate[(2*E^3*x - x*Log[x] + (-1 + 2*x - x^2 + E^3*(-2 + 2*x))*Log[-(Lo g[3]/(1 + 2*E^3 - x))])/(x + 2*E^3*x - x^2),x]
Output:
x - (2*E^3 - Log[1 + 2*E^3])*Log[1 + 2*E^3 - x] - (1 + 2*E^3 - x + Log[x/( 1 + 2*E^3)])*Log[-(Log[3]/(1 + 2*E^3 - x))]
Leaf count is larger than twice the leaf count of optimal. \(91\) vs. \(2(27)=54\).
Time = 0.60 (sec) , antiderivative size = 91, normalized size of antiderivative = 3.37, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {6, 2026, 7293, 2009}
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.
| \(\displaystyle \int \frac {\left (-x^2+2 x+e^3 (2 x-2)-1\right ) \log \left (-\frac {\log (3)}{-x+2 e^3+1}\right )+2 e^3 x+x (-\log (x))}{-x^2+2 e^3 x+x} \, dx\) |
| \(\Big \downarrow \) 6 |
| \(\displaystyle \int \frac {\left (-x^2+2 x+e^3 (2 x-2)-1\right ) \log \left (-\frac {\log (3)}{-x+2 e^3+1}\right )+2 e^3 x+x (-\log (x))}{\left (1+2 e^3\right ) x-x^2}dx\) |
| \(\Big \downarrow \) 2026 |
| \(\displaystyle \int \frac {\left (-x^2+2 x+e^3 (2 x-2)-1\right ) \log \left (-\frac {\log (3)}{-x+2 e^3+1}\right )+2 e^3 x+x (-\log (x))}{\left (-x+2 e^3+1\right ) x}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {2 e^3-\log (x)}{-x+2 e^3+1}+\frac {(x-1) \log \left (-\frac {\log (3)}{-x+2 e^3+1}\right )}{x}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle x-\left (2 e^3-\log \left (1+2 e^3\right )\right ) \log \left (-x+2 e^3+1\right )-\left (-x+2 e^3+1\right ) \log \left (-\frac {\log (3)}{-x+2 e^3+1}\right )-\log \left (\frac {x}{1+2 e^3}\right ) \log \left (-\frac {\log (3)}{-x+2 e^3+1}\right )\) |
Input:
Int[(2*E^3*x - x*Log[x] + (-1 + 2*x - x^2 + E^3*(-2 + 2*x))*Log[-(Log[3]/( 1 + 2*E^3 - x))])/(x + 2*E^3*x - x^2),x]
Output:
x - (2*E^3 - Log[1 + 2*E^3])*Log[1 + 2*E^3 - x] - (1 + 2*E^3 - x)*Log[-(Lo g[3]/(1 + 2*E^3 - x))] - Log[x/(1 + 2*E^3)]*Log[-(Log[3]/(1 + 2*E^3 - x))]
Int[(u_.)*((v_.) + (a_.)*(Fx_) + (b_.)*(Fx_))^(p_.), x_Symbol] :> Int[u*(v + (a + b)*Fx)^p, x] /; FreeQ[{a, b}, x] && !FreeQ[Fx, x]
Int[(Fx_.)*(Px_)^(p_.), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Int[x^(p *r)*ExpandToSum[Px/x^r, x]^p*Fx, x] /; IGtQ[r, 0]] /; PolyQ[Px, x] && Integ erQ[p] && !MonomialQ[Px, x] && (ILtQ[p, 0] || !PolyQ[u, x])
Leaf count of result is larger than twice the leaf count of optimal. \(58\) vs. \(2(26)=52\).
Time = 1.41 (sec) , antiderivative size = 59, normalized size of antiderivative = 2.19
| method | result | size |
| norman | \(x -\ln \left (-\frac {\ln \left (3\right )}{2 \,{\mathrm e}^{3}-x +1}\right )+\ln \left (-\frac {\ln \left (3\right )}{2 \,{\mathrm e}^{3}-x +1}\right ) x -\ln \left (x \right ) \ln \left (-\frac {\ln \left (3\right )}{2 \,{\mathrm e}^{3}-x +1}\right )\) | \(59\) |
| parallelrisch | \(4 \,{\mathrm e}^{3} \ln \left (x -1-2 \,{\mathrm e}^{3}\right )+4 \ln \left (-\frac {\ln \left (3\right )}{2 \,{\mathrm e}^{3}-x +1}\right ) {\mathrm e}^{3}+2+\ln \left (-\frac {\ln \left (3\right )}{2 \,{\mathrm e}^{3}-x +1}\right ) x -\ln \left (x \right ) \ln \left (-\frac {\ln \left (3\right )}{2 \,{\mathrm e}^{3}-x +1}\right )+4 \,{\mathrm e}^{3}+3 \ln \left (x -1-2 \,{\mathrm e}^{3}\right )+x +2 \ln \left (-\frac {\ln \left (3\right )}{2 \,{\mathrm e}^{3}-x +1}\right )\) | \(106\) |
| default | \(-\left (\frac {\operatorname {dilog}\left (1+\frac {2 \,{\mathrm e}^{3}+1}{x -1-2 \,{\mathrm e}^{3}}\right )}{2 \,{\mathrm e}^{3}+1}+\frac {\ln \left (\frac {1}{x -1-2 \,{\mathrm e}^{3}}\right ) \ln \left (1+\frac {2 \,{\mathrm e}^{3}+1}{x -1-2 \,{\mathrm e}^{3}}\right )}{2 \,{\mathrm e}^{3}+1}\right ) \left (2 \,{\mathrm e}^{3}+1\right )+\frac {\ln \left (\frac {1}{x -1-2 \,{\mathrm e}^{3}}\right )^{2}}{2}+\left (x -1-2 \,{\mathrm e}^{3}\right ) \ln \left (\frac {1}{x -1-2 \,{\mathrm e}^{3}}\right )+x -1-2 \,{\mathrm e}^{3}+\ln \left (\ln \left (3\right )\right ) \left (x -\ln \left (x \right )\right )-2 \,{\mathrm e}^{3} \ln \left (2 \,{\mathrm e}^{3}-x +1\right )+\left (\ln \left (x \right )-\ln \left (\frac {x}{2 \,{\mathrm e}^{3}+1}\right )\right ) \ln \left (\frac {2 \,{\mathrm e}^{3}-x +1}{2 \,{\mathrm e}^{3}+1}\right )-\operatorname {dilog}\left (\frac {x}{2 \,{\mathrm e}^{3}+1}\right )\) | \(188\) |
| parts | \(-\ln \left (3\right ) \left (\frac {\left (\frac {\operatorname {dilog}\left (\frac {\frac {\left (2 \,{\mathrm e}^{3}+1\right ) \ln \left (3\right )}{x -1-2 \,{\mathrm e}^{3}}+\ln \left (3\right )}{\ln \left (3\right )}\right )}{2 \,{\mathrm e}^{3}+1}+\frac {\ln \left (\frac {\ln \left (3\right )}{x -1-2 \,{\mathrm e}^{3}}\right ) \ln \left (\frac {\frac {\left (2 \,{\mathrm e}^{3}+1\right ) \ln \left (3\right )}{x -1-2 \,{\mathrm e}^{3}}+\ln \left (3\right )}{\ln \left (3\right )}\right )}{2 \,{\mathrm e}^{3}+1}\right ) \left (2 \,{\mathrm e}^{3}+1\right )}{\ln \left (3\right )}-\frac {\ln \left (\frac {\ln \left (3\right )}{x -1-2 \,{\mathrm e}^{3}}\right )^{2}}{2 \ln \left (3\right )}-\frac {\left (x -1-2 \,{\mathrm e}^{3}\right ) \ln \left (\frac {\ln \left (3\right )}{x -1-2 \,{\mathrm e}^{3}}\right )}{\ln \left (3\right )}-\frac {x -1-2 \,{\mathrm e}^{3}}{\ln \left (3\right )}\right )-2 \,{\mathrm e}^{3} \ln \left (2 \,{\mathrm e}^{3}-x +1\right )+\left (\ln \left (x \right )-\ln \left (\frac {x}{2 \,{\mathrm e}^{3}+1}\right )\right ) \ln \left (\frac {2 \,{\mathrm e}^{3}-x +1}{2 \,{\mathrm e}^{3}+1}\right )-\operatorname {dilog}\left (\frac {x}{2 \,{\mathrm e}^{3}+1}\right )\) | \(227\) |
| risch | \(\left (\ln \left (x \right )-x \right ) \ln \left ({\mathrm e}^{3}-\frac {x}{2}+\frac {1}{2}\right )+i \pi \ln \left (\left (-2 \pi \operatorname {csgn}\left (\frac {i}{{\mathrm e}^{3}-\frac {x}{2}+\frac {1}{2}}\right )^{2}+2 \pi \operatorname {csgn}\left (\frac {i}{{\mathrm e}^{3}-\frac {x}{2}+\frac {1}{2}}\right )^{3}+2 \pi -2 i \ln \left (\ln \left (3\right )\right )+2 i \ln \left (2\right )-2 i\right ) x \right ) \operatorname {csgn}\left (\frac {i}{{\mathrm e}^{3}-\frac {x}{2}+\frac {1}{2}}\right )^{2}+i \pi x \operatorname {csgn}\left (\frac {i}{{\mathrm e}^{3}-\frac {x}{2}+\frac {1}{2}}\right )^{3}-i \pi \ln \left (\left (-2 \pi \operatorname {csgn}\left (\frac {i}{{\mathrm e}^{3}-\frac {x}{2}+\frac {1}{2}}\right )^{2}+2 \pi \operatorname {csgn}\left (\frac {i}{{\mathrm e}^{3}-\frac {x}{2}+\frac {1}{2}}\right )^{3}+2 \pi -2 i \ln \left (\ln \left (3\right )\right )+2 i \ln \left (2\right )-2 i\right ) x \right )-i \pi \ln \left (\left (-2 \pi \operatorname {csgn}\left (\frac {i}{{\mathrm e}^{3}-\frac {x}{2}+\frac {1}{2}}\right )^{2}+2 \pi \operatorname {csgn}\left (\frac {i}{{\mathrm e}^{3}-\frac {x}{2}+\frac {1}{2}}\right )^{3}+2 \pi -2 i \ln \left (\ln \left (3\right )\right )+2 i \ln \left (2\right )-2 i\right ) x \right ) \operatorname {csgn}\left (\frac {i}{{\mathrm e}^{3}-\frac {x}{2}+\frac {1}{2}}\right )^{3}-i \pi x \operatorname {csgn}\left (\frac {i}{{\mathrm e}^{3}-\frac {x}{2}+\frac {1}{2}}\right )^{2}+\ln \left (\ln \left (3\right )\right ) x -x \ln \left (2\right )+x +i \pi x -\ln \left (\ln \left (3\right )\right ) \ln \left (\left (-2 \pi \operatorname {csgn}\left (\frac {i}{{\mathrm e}^{3}-\frac {x}{2}+\frac {1}{2}}\right )^{2}+2 \pi \operatorname {csgn}\left (\frac {i}{{\mathrm e}^{3}-\frac {x}{2}+\frac {1}{2}}\right )^{3}+2 \pi -2 i \ln \left (\ln \left (3\right )\right )+2 i \ln \left (2\right )-2 i\right ) x \right )+\ln \left (2\right ) \ln \left (\left (-2 \pi \operatorname {csgn}\left (\frac {i}{{\mathrm e}^{3}-\frac {x}{2}+\frac {1}{2}}\right )^{2}+2 \pi \operatorname {csgn}\left (\frac {i}{{\mathrm e}^{3}-\frac {x}{2}+\frac {1}{2}}\right )^{3}+2 \pi -2 i \ln \left (\ln \left (3\right )\right )+2 i \ln \left (2\right )-2 i\right ) x \right )+\ln \left (2 \,{\mathrm e}^{3}-x +1\right )\) | \(413\) |
Input:
int((-x*ln(x)+((-2+2*x)*exp(3)-x^2+2*x-1)*ln(-ln(3)/(2*exp(3)-x+1))+2*x*ex p(3))/(2*x*exp(3)-x^2+x),x,method=_RETURNVERBOSE)
Output:
x-ln(-ln(3)/(2*exp(3)-x+1))+ln(-ln(3)/(2*exp(3)-x+1))*x-ln(x)*ln(-ln(3)/(2 *exp(3)-x+1))
Time = 0.07 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.33 \[ \int \frac {2 e^3 x-x \log (x)+\left (-1+2 x-x^2+e^3 (-2+2 x)\right ) \log \left (-\frac {\log (3)}{1+2 e^3-x}\right )}{x+2 e^3 x-x^2} \, dx={\left (x - 1\right )} \log \left (\frac {\log \left (3\right )}{x - 2 \, e^{3} - 1}\right ) - \log \left (x\right ) \log \left (\frac {\log \left (3\right )}{x - 2 \, e^{3} - 1}\right ) + x \] Input:
integrate((-x*log(x)+((2*x-2)*exp(3)-x^2+2*x-1)*log(-log(3)/(2*exp(3)-x+1) )+2*x*exp(3))/(2*x*exp(3)-x^2+x),x, algorithm="fricas")
Output:
(x - 1)*log(log(3)/(x - 2*e^3 - 1)) - log(x)*log(log(3)/(x - 2*e^3 - 1)) + x
Time = 0.17 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.15 \[ \int \frac {2 e^3 x-x \log (x)+\left (-1+2 x-x^2+e^3 (-2+2 x)\right ) \log \left (-\frac {\log (3)}{1+2 e^3-x}\right )}{x+2 e^3 x-x^2} \, dx=x + \left (x - \log {\left (x \right )}\right ) \log {\left (- \frac {\log {\left (3 \right )}}{- x + 1 + 2 e^{3}} \right )} + \log {\left (x - 2 e^{3} - 1 \right )} \] Input:
integrate((-x*ln(x)+((2*x-2)*exp(3)-x**2+2*x-1)*ln(-ln(3)/(2*exp(3)-x+1))+ 2*x*exp(3))/(2*x*exp(3)-x**2+x),x)
Output:
x + (x - log(x))*log(-log(3)/(-x + 1 + 2*exp(3))) + log(x - 2*exp(3) - 1)
Time = 0.15 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.78 \[ \int \frac {2 e^3 x-x \log (x)+\left (-1+2 x-x^2+e^3 (-2+2 x)\right ) \log \left (-\frac {\log (3)}{1+2 e^3-x}\right )}{x+2 e^3 x-x^2} \, dx=x {\left (\log \left (\log \left (3\right )\right ) + 1\right )} - {\left (x - 2 \, e^{3} - \log \left (x\right ) - 1\right )} \log \left (x - 2 \, e^{3} - 1\right ) - 2 \, e^{3} \log \left (x - 2 \, e^{3} - 1\right ) - \log \left (x\right ) \log \left (\log \left (3\right )\right ) \] Input:
integrate((-x*log(x)+((2*x-2)*exp(3)-x^2+2*x-1)*log(-log(3)/(2*exp(3)-x+1) )+2*x*exp(3))/(2*x*exp(3)-x^2+x),x, algorithm="maxima")
Output:
x*(log(log(3)) + 1) - (x - 2*e^3 - log(x) - 1)*log(x - 2*e^3 - 1) - 2*e^3* log(x - 2*e^3 - 1) - log(x)*log(log(3))
Leaf count of result is larger than twice the leaf count of optimal. 279 vs. \(2 (25) = 50\).
Time = 0.15 (sec) , antiderivative size = 279, normalized size of antiderivative = 10.33 \[ \int \frac {2 e^3 x-x \log (x)+\left (-1+2 x-x^2+e^3 (-2+2 x)\right ) \log \left (-\frac {\log (3)}{1+2 e^3-x}\right )}{x+2 e^3 x-x^2} \, dx=-\frac {2 \, \pi ^{2} e^{3} \mathrm {sgn}\left (x - 2 \, e^{3} - 1\right ) \mathrm {sgn}\left (x\right ) - 8 \, \pi ^{2} e^{6} \mathrm {sgn}\left (x - 2 \, e^{3} - 1\right ) - 2 \, \pi ^{2} e^{3} \mathrm {sgn}\left (x - 2 \, e^{3} - 1\right ) + 6 \, \pi ^{2} e^{3} \mathrm {sgn}\left (x\right ) + \pi ^{2} \mathrm {sgn}\left (x - 2 \, e^{3} - 1\right ) \mathrm {sgn}\left (x\right ) + 8 \, \pi ^{2} e^{6} - 6 \, \pi ^{2} e^{3} + 8 \, x e^{3} \log \left ({\left | x - 2 \, e^{3} - 1 \right |}\right ) - 16 \, e^{6} \log \left ({\left | x - 2 \, e^{3} - 1 \right |}\right )^{2} - 8 \, e^{3} \log \left ({\left | x - 2 \, e^{3} - 1 \right |}\right ) \log \left ({\left | x \right |}\right ) - 8 \, x e^{3} \log \left (\log \left (3\right )\right ) + 8 \, e^{3} \log \left ({\left | x \right |}\right ) \log \left (\log \left (3\right )\right ) - \pi ^{2} \mathrm {sgn}\left (x - 2 \, e^{3} - 1\right ) + 3 \, \pi ^{2} \mathrm {sgn}\left (x\right ) - 3 \, \pi ^{2} - 8 \, x e^{3} + 4 \, x \log \left ({\left | x - 2 \, e^{3} - 1 \right |}\right ) - 4 \, \log \left ({\left | x - 2 \, e^{3} - 1 \right |}\right ) \log \left ({\left | x \right |}\right ) - 4 \, x \log \left (\log \left (3\right )\right ) + 4 \, \log \left ({\left | x \right |}\right ) \log \left (\log \left (3\right )\right ) - 4 \, x}{4 \, {\left (2 \, e^{3} + 1\right )}} - \frac {4 \, e^{6} \log \left (x - 2 \, e^{3} - 1\right )^{2} - 2 \, e^{3} \log \left (x - 2 \, e^{3} - 1\right ) - \log \left (x - 2 \, e^{3} - 1\right )}{2 \, e^{3} + 1} \] Input:
integrate((-x*log(x)+((2*x-2)*exp(3)-x^2+2*x-1)*log(-log(3)/(2*exp(3)-x+1) )+2*x*exp(3))/(2*x*exp(3)-x^2+x),x, algorithm="giac")
Output:
-1/4*(2*pi^2*e^3*sgn(x - 2*e^3 - 1)*sgn(x) - 8*pi^2*e^6*sgn(x - 2*e^3 - 1) - 2*pi^2*e^3*sgn(x - 2*e^3 - 1) + 6*pi^2*e^3*sgn(x) + pi^2*sgn(x - 2*e^3 - 1)*sgn(x) + 8*pi^2*e^6 - 6*pi^2*e^3 + 8*x*e^3*log(abs(x - 2*e^3 - 1)) - 16*e^6*log(abs(x - 2*e^3 - 1))^2 - 8*e^3*log(abs(x - 2*e^3 - 1))*log(abs(x )) - 8*x*e^3*log(log(3)) + 8*e^3*log(abs(x))*log(log(3)) - pi^2*sgn(x - 2* e^3 - 1) + 3*pi^2*sgn(x) - 3*pi^2 - 8*x*e^3 + 4*x*log(abs(x - 2*e^3 - 1)) - 4*log(abs(x - 2*e^3 - 1))*log(abs(x)) - 4*x*log(log(3)) + 4*log(abs(x))* log(log(3)) - 4*x)/(2*e^3 + 1) - (4*e^6*log(x - 2*e^3 - 1)^2 - 2*e^3*log(x - 2*e^3 - 1) - log(x - 2*e^3 - 1))/(2*e^3 + 1)
Time = 2.09 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.22 \[ \int \frac {2 e^3 x-x \log (x)+\left (-1+2 x-x^2+e^3 (-2+2 x)\right ) \log \left (-\frac {\log (3)}{1+2 e^3-x}\right )}{x+2 e^3 x-x^2} \, dx=x+\ln \left (x-2\,{\mathrm {e}}^3-1\right )+\ln \left (-\frac {\ln \left (3\right )}{2\,{\mathrm {e}}^3-x+1}\right )\,\left (x-\ln \left (x\right )\right ) \] Input:
int((2*x*exp(3) + log(-log(3)/(2*exp(3) - x + 1))*(2*x - x^2 + exp(3)*(2*x - 2) - 1) - x*log(x))/(x + 2*x*exp(3) - x^2),x)
Output:
x + log(x - 2*exp(3) - 1) + log(-log(3)/(2*exp(3) - x + 1))*(x - log(x))
Time = 0.20 (sec) , antiderivative size = 99, normalized size of antiderivative = 3.67 \[ \int \frac {2 e^3 x-x \log (x)+\left (-1+2 x-x^2+e^3 (-2+2 x)\right ) \log \left (-\frac {\log (3)}{1+2 e^3-x}\right )}{x+2 e^3 x-x^2} \, dx=-2 \,\mathrm {log}\left (2 e^{3}-x +1\right ) e^{3}-\mathrm {log}\left (-\frac {\mathrm {log}\left (3\right )}{2 e^{3}-x +1}\right ) \mathrm {log}\left (x \right )-2 \,\mathrm {log}\left (-\frac {\mathrm {log}\left (3\right )}{2 e^{3}-x +1}\right ) e^{3}+\mathrm {log}\left (-\frac {\mathrm {log}\left (3\right )}{2 e^{3}-x +1}\right ) x -\mathrm {log}\left (-\frac {\mathrm {log}\left (3\right )}{2 e^{3}-x +1}\right )+x \] Input:
int((-x*log(x)+((2*x-2)*exp(3)-x^2+2*x-1)*log(-log(3)/(2*exp(3)-x+1))+2*x* exp(3))/(2*x*exp(3)-x^2+x),x)
Output:
- 2*log(2*e**3 - x + 1)*e**3 - log(( - log(3))/(2*e**3 - x + 1))*log(x) - 2*log(( - log(3))/(2*e**3 - x + 1))*e**3 + log(( - log(3))/(2*e**3 - x + 1))*x - log(( - log(3))/(2*e**3 - x + 1)) + x