Integrand size = 120, antiderivative size = 26 \[ \int \frac {40 x-4 x^2-4 x^3+e^{e^x} \left (4 x^2-4 e^x x^3\right )+4 x^2 \log \left (-e^x\right )}{25-10 x+x^2+e^{2 e^x} x^2+e^{e^x} \left (10 x-2 x^2\right )+\left (10 x-2 x^2+2 e^{e^x} x^2\right ) \log \left (-e^x\right )+x^2 \log ^2\left (-e^x\right )} \, dx=\frac {4 x}{e^{e^x}+\frac {5-x}{x}+\log \left (-e^x\right )} \]
Time = 0.45 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \frac {40 x-4 x^2-4 x^3+e^{e^x} \left (4 x^2-4 e^x x^3\right )+4 x^2 \log \left (-e^x\right )}{25-10 x+x^2+e^{2 e^x} x^2+e^{e^x} \left (10 x-2 x^2\right )+\left (10 x-2 x^2+2 e^{e^x} x^2\right ) \log \left (-e^x\right )+x^2 \log ^2\left (-e^x\right )} \, dx=\frac {4 x^2}{5+\left (-1+e^{e^x}\right ) x+x \log \left (-e^x\right )} \]
Integrate[(40*x - 4*x^2 - 4*x^3 + E^E^x*(4*x^2 - 4*E^x*x^3) + 4*x^2*Log[-E ^x])/(25 - 10*x + x^2 + E^(2*E^x)*x^2 + E^E^x*(10*x - 2*x^2) + (10*x - 2*x ^2 + 2*E^E^x*x^2)*Log[-E^x] + x^2*Log[-E^x]^2),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.
\(\displaystyle \int \frac {-4 x^3-4 x^2+4 x^2 \log \left (-e^x\right )+e^{e^x} \left (4 x^2-4 e^x x^3\right )+40 x}{e^{2 e^x} x^2+x^2+e^{e^x} \left (10 x-2 x^2\right )+x^2 \log ^2\left (-e^x\right )+\left (2 e^{e^x} x^2-2 x^2+10 x\right ) \log \left (-e^x\right )-10 x+25} \, dx\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle \int \frac {4 x \left (-\left (\left (e^{x+e^x}+1\right ) x^2\right )+e^{e^x} x-x+x \log \left (-e^x\right )+10\right )}{\left (\left (e^{e^x}-1\right ) x+x \log \left (-e^x\right )+5\right )^2}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle 4 \int \frac {x \left (-\left (\left (1+e^{x+e^x}\right ) x^2\right )+e^{e^x} x+\log \left (-e^x\right ) x-x+10\right )}{\left (-\left (\left (1-e^{e^x}\right ) x\right )+\log \left (-e^x\right ) x+5\right )^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle 4 \int \left (-\frac {e^{x+e^x} x^3}{\left (e^{e^x} x+\log \left (-e^x\right ) x-x+5\right )^2}-\frac {x^3}{\left (e^{e^x} x+\log \left (-e^x\right ) x-x+5\right )^2}+\frac {e^{e^x} x^2}{\left (e^{e^x} x+\log \left (-e^x\right ) x-x+5\right )^2}+\frac {\log \left (-e^x\right ) x^2}{\left (e^{e^x} x+\log \left (-e^x\right ) x-x+5\right )^2}-\frac {x^2}{\left (e^{e^x} x+\log \left (-e^x\right ) x-x+5\right )^2}+\frac {10 x}{\left (e^{e^x} x+\log \left (-e^x\right ) x-x+5\right )^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 4 \left (-\int \frac {x^3}{\left (e^{e^x} x+\log \left (-e^x\right ) x-x+5\right )^2}dx-\int \frac {e^{x+e^x} x^3}{\left (e^{e^x} x+\log \left (-e^x\right ) x-x+5\right )^2}dx-\int \frac {x^2}{\left (e^{e^x} x+\log \left (-e^x\right ) x-x+5\right )^2}dx+\int \frac {e^{e^x} x^2}{\left (e^{e^x} x+\log \left (-e^x\right ) x-x+5\right )^2}dx+\int \frac {x^2 \log \left (-e^x\right )}{\left (e^{e^x} x+\log \left (-e^x\right ) x-x+5\right )^2}dx+10 \int \frac {x}{\left (e^{e^x} x+\log \left (-e^x\right ) x-x+5\right )^2}dx\right )\) |
Int[(40*x - 4*x^2 - 4*x^3 + E^E^x*(4*x^2 - 4*E^x*x^3) + 4*x^2*Log[-E^x])/( 25 - 10*x + x^2 + E^(2*E^x)*x^2 + E^E^x*(10*x - 2*x^2) + (10*x - 2*x^2 + 2 *E^E^x*x^2)*Log[-E^x] + x^2*Log[-E^x]^2),x]
3.11.94.3.1 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]]
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Time = 1.40 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.96
method | result | size |
parallelrisch | \(\frac {4 x^{2}}{x \ln \left (-{\mathrm e}^{x}\right )+x \,{\mathrm e}^{{\mathrm e}^{x}}-x +5}\) | \(25\) |
default | \(\frac {\left (-4 \ln \left (-{\mathrm e}^{x}\right )+4 x +4\right ) x -4 x \,{\mathrm e}^{{\mathrm e}^{x}}-20}{x^{2}+x \,{\mathrm e}^{{\mathrm e}^{x}}+x \left (\ln \left (-{\mathrm e}^{x}\right )-x \right )-x +5}\) | \(50\) |
risch | \(\frac {8 i x^{2}}{-2 \pi x \operatorname {csgn}\left (i {\mathrm e}^{x}\right )^{3}+2 \pi x \operatorname {csgn}\left (i {\mathrm e}^{x}\right )^{2}-2 \pi x +2 i x \,{\mathrm e}^{{\mathrm e}^{x}}+2 i x \ln \left ({\mathrm e}^{x}\right )-2 i x +10 i}\) | \(58\) |
int((4*x^2*ln(-exp(x))+(-4*exp(x)*x^3+4*x^2)*exp(exp(x))-4*x^3-4*x^2+40*x) /(x^2*ln(-exp(x))^2+(2*exp(exp(x))*x^2-2*x^2+10*x)*ln(-exp(x))+x^2*exp(exp (x))^2+(-2*x^2+10*x)*exp(exp(x))+x^2-10*x+25),x,method=_RETURNVERBOSE)
Result contains complex when optimal does not.
Time = 0.25 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.92 \[ \int \frac {40 x-4 x^2-4 x^3+e^{e^x} \left (4 x^2-4 e^x x^3\right )+4 x^2 \log \left (-e^x\right )}{25-10 x+x^2+e^{2 e^x} x^2+e^{e^x} \left (10 x-2 x^2\right )+\left (10 x-2 x^2+2 e^{e^x} x^2\right ) \log \left (-e^x\right )+x^2 \log ^2\left (-e^x\right )} \, dx=\frac {4 \, x^{2}}{i \, \pi x + x^{2} + x e^{\left (e^{x}\right )} - x + 5} \]
integrate((4*x^2*log(-exp(x))+(-4*exp(x)*x^3+4*x^2)*exp(exp(x))-4*x^3-4*x^ 2+40*x)/(x^2*log(-exp(x))^2+(2*exp(exp(x))*x^2-2*x^2+10*x)*log(-exp(x))+x^ 2*exp(exp(x))^2+(-2*x^2+10*x)*exp(exp(x))+x^2-10*x+25),x, algorithm=\
Result contains complex when optimal does not.
Time = 0.18 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.85 \[ \int \frac {40 x-4 x^2-4 x^3+e^{e^x} \left (4 x^2-4 e^x x^3\right )+4 x^2 \log \left (-e^x\right )}{25-10 x+x^2+e^{2 e^x} x^2+e^{e^x} \left (10 x-2 x^2\right )+\left (10 x-2 x^2+2 e^{e^x} x^2\right ) \log \left (-e^x\right )+x^2 \log ^2\left (-e^x\right )} \, dx=\frac {4 x^{2}}{x^{2} + x e^{e^{x}} - x + i \pi x + 5} \]
integrate((4*x**2*ln(-exp(x))+(-4*exp(x)*x**3+4*x**2)*exp(exp(x))-4*x**3-4 *x**2+40*x)/(x**2*ln(-exp(x))**2+(2*exp(exp(x))*x**2-2*x**2+10*x)*ln(-exp( x))+x**2*exp(exp(x))**2+(-2*x**2+10*x)*exp(exp(x))+x**2-10*x+25),x)
Time = 0.26 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.92 \[ \int \frac {40 x-4 x^2-4 x^3+e^{e^x} \left (4 x^2-4 e^x x^3\right )+4 x^2 \log \left (-e^x\right )}{25-10 x+x^2+e^{2 e^x} x^2+e^{e^x} \left (10 x-2 x^2\right )+\left (10 x-2 x^2+2 e^{e^x} x^2\right ) \log \left (-e^x\right )+x^2 \log ^2\left (-e^x\right )} \, dx=\frac {4 \, x^{2}}{x e^{\left (e^{x}\right )} + x \log \left (-e^{x}\right ) - x + 5} \]
integrate((4*x^2*log(-exp(x))+(-4*exp(x)*x^3+4*x^2)*exp(exp(x))-4*x^3-4*x^ 2+40*x)/(x^2*log(-exp(x))^2+(2*exp(exp(x))*x^2-2*x^2+10*x)*log(-exp(x))+x^ 2*exp(exp(x))^2+(-2*x^2+10*x)*exp(exp(x))+x^2-10*x+25),x, algorithm=\
Leaf count of result is larger than twice the leaf count of optimal. 81 vs. \(2 (25) = 50\).
Time = 0.30 (sec) , antiderivative size = 81, normalized size of antiderivative = 3.12 \[ \int \frac {40 x-4 x^2-4 x^3+e^{e^x} \left (4 x^2-4 e^x x^3\right )+4 x^2 \log \left (-e^x\right )}{25-10 x+x^2+e^{2 e^x} x^2+e^{e^x} \left (10 x-2 x^2\right )+\left (10 x-2 x^2+2 e^{e^x} x^2\right ) \log \left (-e^x\right )+x^2 \log ^2\left (-e^x\right )} \, dx=\frac {4 \, {\left (x^{4} + x^{3} e^{\left (e^{x}\right )} - x^{3} + 5 \, x^{2}\right )}}{\pi ^{2} x^{2} + x^{4} + 2 \, x^{3} e^{\left (e^{x}\right )} - 2 \, x^{3} + x^{2} e^{\left (2 \, e^{x}\right )} - 2 \, x^{2} e^{\left (e^{x}\right )} + 11 \, x^{2} + 10 \, x e^{\left (e^{x}\right )} - 10 \, x + 25} \]
integrate((4*x^2*log(-exp(x))+(-4*exp(x)*x^3+4*x^2)*exp(exp(x))-4*x^3-4*x^ 2+40*x)/(x^2*log(-exp(x))^2+(2*exp(exp(x))*x^2-2*x^2+10*x)*log(-exp(x))+x^ 2*exp(exp(x))^2+(-2*x^2+10*x)*exp(exp(x))+x^2-10*x+25),x, algorithm=\
4*(x^4 + x^3*e^(e^x) - x^3 + 5*x^2)/(pi^2*x^2 + x^4 + 2*x^3*e^(e^x) - 2*x^ 3 + x^2*e^(2*e^x) - 2*x^2*e^(e^x) + 11*x^2 + 10*x*e^(e^x) - 10*x + 25)
Time = 13.59 (sec) , antiderivative size = 98, normalized size of antiderivative = 3.77 \[ \int \frac {40 x-4 x^2-4 x^3+e^{e^x} \left (4 x^2-4 e^x x^3\right )+4 x^2 \log \left (-e^x\right )}{25-10 x+x^2+e^{2 e^x} x^2+e^{e^x} \left (10 x-2 x^2\right )+\left (10 x-2 x^2+2 e^{e^x} x^2\right ) \log \left (-e^x\right )+x^2 \log ^2\left (-e^x\right )} \, dx=\frac {4\,\left (5\,x^3\,{\mathrm {e}}^x-x^4\,{\mathrm {e}}^x+x^5\,{\mathrm {e}}^x+5\,x^2-x^4+\pi \,x^4\,{\mathrm {e}}^x\,1{}\mathrm {i}\right )}{\left (x\,{\mathrm {e}}^{{\mathrm {e}}^x}-x+x^2+5+\pi \,x\,1{}\mathrm {i}\right )\,\left (x^3\,{\mathrm {e}}^x-x^2\,{\mathrm {e}}^x+5\,x\,{\mathrm {e}}^x-x^2+5+\pi \,x^2\,{\mathrm {e}}^x\,1{}\mathrm {i}\right )} \]
int(-(exp(exp(x))*(4*x^3*exp(x) - 4*x^2) - 40*x + 4*x^2 + 4*x^3 - 4*x^2*lo g(-exp(x)))/(exp(exp(x))*(10*x - 2*x^2) - 10*x + x^2*log(-exp(x))^2 + x^2 + x^2*exp(2*exp(x)) + log(-exp(x))*(10*x + 2*x^2*exp(exp(x)) - 2*x^2) + 25 ),x)