Integrand size = 59, antiderivative size = 24 \[ \int \frac {-16 e^5-16 x+e^5 (-16+8 x) \log \left (-\frac {x}{-2+x}\right )}{e^{10} (-2+x)-2 x^2+x^3+e^5 \left (-4 x+2 x^2\right )} \, dx=\frac {1}{3}+\frac {8 x \log \left (\frac {x}{2-x}\right )}{e^5+x} \] Output:
8*x*ln(x/(2-x))/(exp(5)+x)+1/3
Time = 0.06 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.46 \[ \int \frac {-16 e^5-16 x+e^5 (-16+8 x) \log \left (-\frac {x}{-2+x}\right )}{e^{10} (-2+x)-2 x^2+x^3+e^5 \left (-4 x+2 x^2\right )} \, dx=8 \left (-\log (2-x)+\log (x)-\frac {e^5 \log \left (\frac {x}{2-x}\right )}{e^5+x}\right ) \] Input:
Integrate[(-16*E^5 - 16*x + E^5*(-16 + 8*x)*Log[-(x/(-2 + x))])/(E^10*(-2 + x) - 2*x^2 + x^3 + E^5*(-4*x + 2*x^2)),x]
Output:
8*(-Log[2 - x] + Log[x] - (E^5*Log[x/(2 - x)])/(E^5 + x))
Leaf count is larger than twice the leaf count of optimal. \(74\) vs. \(2(24)=48\).
Time = 0.84 (sec) , antiderivative size = 74, normalized size of antiderivative = 3.08, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.034, Rules used = {2463, 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.
\(\displaystyle \int \frac {-16 x+e^5 (8 x-16) \log \left (-\frac {x}{x-2}\right )-16 e^5}{x^3-2 x^2+e^5 \left (2 x^2-4 x\right )+e^{10} (x-2)} \, dx\) |
\(\Big \downarrow \) 2463 |
\(\displaystyle \int \left (\frac {-16 x+e^5 (8 x-16) \log \left (-\frac {x}{x-2}\right )-16 e^5}{\left (2+e^5\right )^2 (x-2)}-\frac {-16 x+e^5 (8 x-16) \log \left (-\frac {x}{x-2}\right )-16 e^5}{\left (2+e^5\right )^2 \left (x+e^5\right )}-\frac {-16 x+e^5 (8 x-16) \log \left (-\frac {x}{x-2}\right )-16 e^5}{\left (2+e^5\right ) \left (x+e^5\right )^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {16 \log (2-x)}{2+e^5}+\frac {8 x \log \left (\frac {x}{2-x}\right )}{x+e^5}+\frac {16 \log \left (x+e^5\right )}{2+e^5}-\frac {16 \log \left (\frac {x+e^5}{2-x}\right )}{2+e^5}\) |
Input:
Int[(-16*E^5 - 16*x + E^5*(-16 + 8*x)*Log[-(x/(-2 + x))])/(E^10*(-2 + x) - 2*x^2 + x^3 + E^5*(-4*x + 2*x^2)),x]
Output:
(-16*Log[2 - x])/(2 + E^5) + (8*x*Log[x/(2 - x)])/(E^5 + x) + (16*Log[E^5 + x])/(2 + E^5) - (16*Log[(E^5 + x)/(2 - x)])/(2 + E^5)
Int[(u_.)*(Px_)^(p_), x_Symbol] :> With[{Qx = Factor[Px]}, Int[ExpandIntegr and[u, Qx^p, x], x] /; !SumQ[NonfreeFactors[Qx, x]]] /; PolyQ[Px, x] && Gt Q[Expon[Px, x], 2] && !BinomialQ[Px, x] && !TrinomialQ[Px, x] && ILtQ[p, 0]
Time = 0.49 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.79
method | result | size |
norman | \(\frac {8 x \ln \left (-\frac {x}{-2+x}\right )}{{\mathrm e}^{5}+x}\) | \(19\) |
parallelrisch | \(\frac {8 x \ln \left (-\frac {x}{-2+x}\right )}{{\mathrm e}^{5}+x}\) | \(19\) |
risch | \(-\frac {8 \,{\mathrm e}^{5} \ln \left (-\frac {x}{-2+x}\right )}{{\mathrm e}^{5}+x}-8 \ln \left (-2+x \right )+8 \ln \left (x \right )\) | \(31\) |
parts | \(\frac {16 \ln \left ({\mathrm e}^{5}+x \right )}{{\mathrm e}^{5}+2}-\frac {16 \ln \left (-2+x \right )}{{\mathrm e}^{5}+2}+8 \,{\mathrm e}^{5} \left (\frac {\left (\ln \left (\frac {4 \,{\mathrm e}^{5} \left (-1-\frac {2}{-2+x}\right )+{\mathrm e}^{10} \left (-1-\frac {2}{-2+x}\right )+2 \,{\mathrm e}^{5}-2 \sqrt {\left ({\mathrm e}^{5}\right )^{2}-{\mathrm e}^{10}}+{\mathrm e}^{10}-4-\frac {8}{-2+x}}{{\mathrm e}^{10}+2 \,{\mathrm e}^{5}-2 \sqrt {\left ({\mathrm e}^{5}\right )^{2}-{\mathrm e}^{10}}}\right )-\ln \left (\frac {4 \,{\mathrm e}^{5} \left (-1-\frac {2}{-2+x}\right )+{\mathrm e}^{10} \left (-1-\frac {2}{-2+x}\right )+2 \,{\mathrm e}^{5}+2 \sqrt {\left ({\mathrm e}^{5}\right )^{2}-{\mathrm e}^{10}}+{\mathrm e}^{10}-4-\frac {8}{-2+x}}{{\mathrm e}^{10}+2 \,{\mathrm e}^{5}+2 \sqrt {\left ({\mathrm e}^{5}\right )^{2}-{\mathrm e}^{10}}}\right )\right ) \ln \left (-1-\frac {2}{-2+x}\right )}{2 \sqrt {\left ({\mathrm e}^{5}\right )^{2}-{\mathrm e}^{10}}}+\frac {\operatorname {dilog}\left (\frac {4 \,{\mathrm e}^{5} \left (-1-\frac {2}{-2+x}\right )+{\mathrm e}^{10} \left (-1-\frac {2}{-2+x}\right )+2 \,{\mathrm e}^{5}-2 \sqrt {\left ({\mathrm e}^{5}\right )^{2}-{\mathrm e}^{10}}+{\mathrm e}^{10}-4-\frac {8}{-2+x}}{{\mathrm e}^{10}+2 \,{\mathrm e}^{5}-2 \sqrt {\left ({\mathrm e}^{5}\right )^{2}-{\mathrm e}^{10}}}\right )-\operatorname {dilog}\left (\frac {4 \,{\mathrm e}^{5} \left (-1-\frac {2}{-2+x}\right )+{\mathrm e}^{10} \left (-1-\frac {2}{-2+x}\right )+2 \,{\mathrm e}^{5}+2 \sqrt {\left ({\mathrm e}^{5}\right )^{2}-{\mathrm e}^{10}}+{\mathrm e}^{10}-4-\frac {8}{-2+x}}{{\mathrm e}^{10}+2 \,{\mathrm e}^{5}+2 \sqrt {\left ({\mathrm e}^{5}\right )^{2}-{\mathrm e}^{10}}}\right )}{2 \sqrt {\left ({\mathrm e}^{5}\right )^{2}-{\mathrm e}^{10}}}\right )\) | \(382\) |
derivativedivides | \(\frac {16 \ln \left (\left ({\mathrm e}^{5}+2\right ) \left (-1-\frac {2}{-2+x}\right )+{\mathrm e}^{5}\right )}{{\mathrm e}^{5}+2}+16 \,{\mathrm e}^{5} \left (\frac {\left (\ln \left (\frac {4 \,{\mathrm e}^{5} \left (-1-\frac {2}{-2+x}\right )+{\mathrm e}^{10} \left (-1-\frac {2}{-2+x}\right )+2 \,{\mathrm e}^{5}-2 \sqrt {\left ({\mathrm e}^{5}\right )^{2}-{\mathrm e}^{10}}+{\mathrm e}^{10}-4-\frac {8}{-2+x}}{{\mathrm e}^{10}+2 \,{\mathrm e}^{5}-2 \sqrt {\left ({\mathrm e}^{5}\right )^{2}-{\mathrm e}^{10}}}\right )-\ln \left (\frac {4 \,{\mathrm e}^{5} \left (-1-\frac {2}{-2+x}\right )+{\mathrm e}^{10} \left (-1-\frac {2}{-2+x}\right )+2 \,{\mathrm e}^{5}+2 \sqrt {\left ({\mathrm e}^{5}\right )^{2}-{\mathrm e}^{10}}+{\mathrm e}^{10}-4-\frac {8}{-2+x}}{{\mathrm e}^{10}+2 \,{\mathrm e}^{5}+2 \sqrt {\left ({\mathrm e}^{5}\right )^{2}-{\mathrm e}^{10}}}\right )\right ) \ln \left (-1-\frac {2}{-2+x}\right )}{4 \sqrt {\left ({\mathrm e}^{5}\right )^{2}-{\mathrm e}^{10}}}+\frac {\operatorname {dilog}\left (\frac {4 \,{\mathrm e}^{5} \left (-1-\frac {2}{-2+x}\right )+{\mathrm e}^{10} \left (-1-\frac {2}{-2+x}\right )+2 \,{\mathrm e}^{5}-2 \sqrt {\left ({\mathrm e}^{5}\right )^{2}-{\mathrm e}^{10}}+{\mathrm e}^{10}-4-\frac {8}{-2+x}}{{\mathrm e}^{10}+2 \,{\mathrm e}^{5}-2 \sqrt {\left ({\mathrm e}^{5}\right )^{2}-{\mathrm e}^{10}}}\right )-\operatorname {dilog}\left (\frac {4 \,{\mathrm e}^{5} \left (-1-\frac {2}{-2+x}\right )+{\mathrm e}^{10} \left (-1-\frac {2}{-2+x}\right )+2 \,{\mathrm e}^{5}+2 \sqrt {\left ({\mathrm e}^{5}\right )^{2}-{\mathrm e}^{10}}+{\mathrm e}^{10}-4-\frac {8}{-2+x}}{{\mathrm e}^{10}+2 \,{\mathrm e}^{5}+2 \sqrt {\left ({\mathrm e}^{5}\right )^{2}-{\mathrm e}^{10}}}\right )}{4 \sqrt {\left ({\mathrm e}^{5}\right )^{2}-{\mathrm e}^{10}}}\right )\) | \(383\) |
default | \(\frac {16 \ln \left (\left ({\mathrm e}^{5}+2\right ) \left (-1-\frac {2}{-2+x}\right )+{\mathrm e}^{5}\right )}{{\mathrm e}^{5}+2}+16 \,{\mathrm e}^{5} \left (\frac {\left (\ln \left (\frac {4 \,{\mathrm e}^{5} \left (-1-\frac {2}{-2+x}\right )+{\mathrm e}^{10} \left (-1-\frac {2}{-2+x}\right )+2 \,{\mathrm e}^{5}-2 \sqrt {\left ({\mathrm e}^{5}\right )^{2}-{\mathrm e}^{10}}+{\mathrm e}^{10}-4-\frac {8}{-2+x}}{{\mathrm e}^{10}+2 \,{\mathrm e}^{5}-2 \sqrt {\left ({\mathrm e}^{5}\right )^{2}-{\mathrm e}^{10}}}\right )-\ln \left (\frac {4 \,{\mathrm e}^{5} \left (-1-\frac {2}{-2+x}\right )+{\mathrm e}^{10} \left (-1-\frac {2}{-2+x}\right )+2 \,{\mathrm e}^{5}+2 \sqrt {\left ({\mathrm e}^{5}\right )^{2}-{\mathrm e}^{10}}+{\mathrm e}^{10}-4-\frac {8}{-2+x}}{{\mathrm e}^{10}+2 \,{\mathrm e}^{5}+2 \sqrt {\left ({\mathrm e}^{5}\right )^{2}-{\mathrm e}^{10}}}\right )\right ) \ln \left (-1-\frac {2}{-2+x}\right )}{4 \sqrt {\left ({\mathrm e}^{5}\right )^{2}-{\mathrm e}^{10}}}+\frac {\operatorname {dilog}\left (\frac {4 \,{\mathrm e}^{5} \left (-1-\frac {2}{-2+x}\right )+{\mathrm e}^{10} \left (-1-\frac {2}{-2+x}\right )+2 \,{\mathrm e}^{5}-2 \sqrt {\left ({\mathrm e}^{5}\right )^{2}-{\mathrm e}^{10}}+{\mathrm e}^{10}-4-\frac {8}{-2+x}}{{\mathrm e}^{10}+2 \,{\mathrm e}^{5}-2 \sqrt {\left ({\mathrm e}^{5}\right )^{2}-{\mathrm e}^{10}}}\right )-\operatorname {dilog}\left (\frac {4 \,{\mathrm e}^{5} \left (-1-\frac {2}{-2+x}\right )+{\mathrm e}^{10} \left (-1-\frac {2}{-2+x}\right )+2 \,{\mathrm e}^{5}+2 \sqrt {\left ({\mathrm e}^{5}\right )^{2}-{\mathrm e}^{10}}+{\mathrm e}^{10}-4-\frac {8}{-2+x}}{{\mathrm e}^{10}+2 \,{\mathrm e}^{5}+2 \sqrt {\left ({\mathrm e}^{5}\right )^{2}-{\mathrm e}^{10}}}\right )}{4 \sqrt {\left ({\mathrm e}^{5}\right )^{2}-{\mathrm e}^{10}}}\right )\) | \(383\) |
Input:
int(((8*x-16)*exp(5)*ln(-x/(-2+x))-16*exp(5)-16*x)/((-2+x)*exp(5)^2+(2*x^2 -4*x)*exp(5)+x^3-2*x^2),x,method=_RETURNVERBOSE)
Output:
8*x*ln(-x/(-2+x))/(exp(5)+x)
Time = 0.07 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.75 \[ \int \frac {-16 e^5-16 x+e^5 (-16+8 x) \log \left (-\frac {x}{-2+x}\right )}{e^{10} (-2+x)-2 x^2+x^3+e^5 \left (-4 x+2 x^2\right )} \, dx=\frac {8 \, x \log \left (-\frac {x}{x - 2}\right )}{x + e^{5}} \] Input:
integrate(((8*x-16)*exp(5)*log(-x/(-2+x))-16*exp(5)-16*x)/((-2+x)*exp(5)^2 +(2*x^2-4*x)*exp(5)+x^3-2*x^2),x, algorithm="fricas")
Output:
8*x*log(-x/(x - 2))/(x + e^5)
Time = 0.11 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.21 \[ \int \frac {-16 e^5-16 x+e^5 (-16+8 x) \log \left (-\frac {x}{-2+x}\right )}{e^{10} (-2+x)-2 x^2+x^3+e^5 \left (-4 x+2 x^2\right )} \, dx=8 \log {\left (x \right )} - 8 \log {\left (x - 2 \right )} - \frac {8 e^{5} \log {\left (- \frac {x}{x - 2} \right )}}{x + e^{5}} \] Input:
integrate(((8*x-16)*exp(5)*ln(-x/(-2+x))-16*exp(5)-16*x)/((-2+x)*exp(5)**2 +(2*x**2-4*x)*exp(5)+x**3-2*x**2),x)
Output:
8*log(x) - 8*log(x - 2) - 8*exp(5)*log(-x/(x - 2))/(x + exp(5))
Leaf count of result is larger than twice the leaf count of optimal. 168 vs. \(2 (20) = 40\).
Time = 0.17 (sec) , antiderivative size = 168, normalized size of antiderivative = 7.00 \[ \int \frac {-16 e^5-16 x+e^5 (-16+8 x) \log \left (-\frac {x}{-2+x}\right )}{e^{10} (-2+x)-2 x^2+x^3+e^5 \left (-4 x+2 x^2\right )} \, dx=16 \, {\left (\frac {\log \left (x + e^{5}\right )}{e^{10} + 4 \, e^{5} + 4} - \frac {\log \left (x - 2\right )}{e^{10} + 4 \, e^{5} + 4} - \frac {1}{x {\left (e^{5} + 2\right )} + e^{10} + 2 \, e^{5}}\right )} e^{5} - \frac {8 \, {\left ({\left (e^{10} + 2 \, e^{5}\right )} \log \left (x\right ) + {\left (x e^{5} - 2 \, e^{5}\right )} \log \left (-x + 2\right )\right )}}{x {\left (e^{5} + 2\right )} + e^{10} + 2 \, e^{5}} + \frac {16 \, e^{5}}{x {\left (e^{5} + 2\right )} + e^{10} + 2 \, e^{5}} + \frac {32 \, \log \left (x + e^{5}\right )}{e^{10} + 4 \, e^{5} + 4} - \frac {16 \, \log \left (x + e^{5}\right )}{e^{5} + 2} - \frac {32 \, \log \left (x - 2\right )}{e^{10} + 4 \, e^{5} + 4} + 8 \, \log \left (x\right ) \] Input:
integrate(((8*x-16)*exp(5)*log(-x/(-2+x))-16*exp(5)-16*x)/((-2+x)*exp(5)^2 +(2*x^2-4*x)*exp(5)+x^3-2*x^2),x, algorithm="maxima")
Output:
16*(log(x + e^5)/(e^10 + 4*e^5 + 4) - log(x - 2)/(e^10 + 4*e^5 + 4) - 1/(x *(e^5 + 2) + e^10 + 2*e^5))*e^5 - 8*((e^10 + 2*e^5)*log(x) + (x*e^5 - 2*e^ 5)*log(-x + 2))/(x*(e^5 + 2) + e^10 + 2*e^5) + 16*e^5/(x*(e^5 + 2) + e^10 + 2*e^5) + 32*log(x + e^5)/(e^10 + 4*e^5 + 4) - 16*log(x + e^5)/(e^5 + 2) - 32*log(x - 2)/(e^10 + 4*e^5 + 4) + 8*log(x)
Leaf count of result is larger than twice the leaf count of optimal. 41 vs. \(2 (20) = 40\).
Time = 0.12 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.71 \[ \int \frac {-16 e^5-16 x+e^5 (-16+8 x) \log \left (-\frac {x}{-2+x}\right )}{e^{10} (-2+x)-2 x^2+x^3+e^5 \left (-4 x+2 x^2\right )} \, dx=\frac {16 \, x \log \left (-\frac {x}{x - 2}\right )}{{\left (x - 2\right )} {\left (\frac {x e^{5}}{x - 2} + \frac {2 \, x}{x - 2} - e^{5}\right )}} \] Input:
integrate(((8*x-16)*exp(5)*log(-x/(-2+x))-16*exp(5)-16*x)/((-2+x)*exp(5)^2 +(2*x^2-4*x)*exp(5)+x^3-2*x^2),x, algorithm="giac")
Output:
16*x*log(-x/(x - 2))/((x - 2)*(x*e^5/(x - 2) + 2*x/(x - 2) - e^5))
Time = 0.87 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {-16 e^5-16 x+e^5 (-16+8 x) \log \left (-\frac {x}{-2+x}\right )}{e^{10} (-2+x)-2 x^2+x^3+e^5 \left (-4 x+2 x^2\right )} \, dx=16\,\mathrm {atanh}\left (x-1\right )-\frac {8\,{\mathrm {e}}^5\,\ln \left (-\frac {x}{x-2}\right )}{x+{\mathrm {e}}^5} \] Input:
int((16*x + 16*exp(5) - exp(5)*log(-x/(x - 2))*(8*x - 16))/(exp(5)*(4*x - 2*x^2) - exp(10)*(x - 2) + 2*x^2 - x^3),x)
Output:
16*atanh(x - 1) - (8*exp(5)*log(-x/(x - 2)))/(x + exp(5))
Time = 0.24 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.79 \[ \int \frac {-16 e^5-16 x+e^5 (-16+8 x) \log \left (-\frac {x}{-2+x}\right )}{e^{10} (-2+x)-2 x^2+x^3+e^5 \left (-4 x+2 x^2\right )} \, dx=\frac {8 \,\mathrm {log}\left (-\frac {x}{x -2}\right ) x}{e^{5}+x} \] Input:
int(((8*x-16)*exp(5)*log(-x/(-2+x))-16*exp(5)-16*x)/((-2+x)*exp(5)^2+(2*x^ 2-4*x)*exp(5)+x^3-2*x^2),x)
Output:
(8*log(( - x)/(x - 2))*x)/(e**5 + x)