Integrand size = 204, antiderivative size = 35 \[ \int \frac {e^{1+e^5} x+e^{-4+2 e^5} \left (e^3 (-1+x)-2 e^6 x+e^{3+x} \left (2 x-x^2\right )\right )-e^{-1+2 e^5} x \log (x)}{x+e^{-2+e^5} \left (2 e^x x+e^3 \left (-6 x+2 x^2\right )\right )+e^{-4+2 e^5} \left (e^{2 x} x+e^{3+x} \left (-6 x+2 x^2\right )+e^6 \left (9 x-6 x^2+x^3\right )\right )+\left (-2 e^{-2+e^5} x+e^{-4+2 e^5} \left (-2 e^x x+e^3 \left (6 x-2 x^2\right )\right )\right ) \log (x)+e^{-4+2 e^5} x \log ^2(x)} \, dx=\frac {1-x}{3-x-\frac {e^{2-e^5}+e^x-\log (x)}{e^3}} \]
Time = 0.12 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.26 \[ \int \frac {e^{1+e^5} x+e^{-4+2 e^5} \left (e^3 (-1+x)-2 e^6 x+e^{3+x} \left (2 x-x^2\right )\right )-e^{-1+2 e^5} x \log (x)}{x+e^{-2+e^5} \left (2 e^x x+e^3 \left (-6 x+2 x^2\right )\right )+e^{-4+2 e^5} \left (e^{2 x} x+e^{3+x} \left (-6 x+2 x^2\right )+e^6 \left (9 x-6 x^2+x^3\right )\right )+\left (-2 e^{-2+e^5} x+e^{-4+2 e^5} \left (-2 e^x x+e^3 \left (6 x-2 x^2\right )\right )\right ) \log (x)+e^{-4+2 e^5} x \log ^2(x)} \, dx=\frac {e^{3+e^5} (-1+x)}{e^2+e^{e^5+x}+e^{3+e^5} (-3+x)-e^{e^5} \log (x)} \]
Integrate[(E^(1 + E^5)*x + E^(-4 + 2*E^5)*(E^3*(-1 + x) - 2*E^6*x + E^(3 + x)*(2*x - x^2)) - E^(-1 + 2*E^5)*x*Log[x])/(x + E^(-2 + E^5)*(2*E^x*x + E ^3*(-6*x + 2*x^2)) + E^(-4 + 2*E^5)*(E^(2*x)*x + E^(3 + x)*(-6*x + 2*x^2) + E^6*(9*x - 6*x^2 + x^3)) + (-2*E^(-2 + E^5)*x + E^(-4 + 2*E^5)*(-2*E^x*x + E^3*(6*x - 2*x^2)))*Log[x] + E^(-4 + 2*E^5)*x*Log[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.
| \(\displaystyle \int \frac {e^{2 e^5-4} \left (e^{x+3} \left (2 x-x^2\right )+e^3 (x-1)-2 e^6 x\right )+e^{1+e^5} x-e^{2 e^5-1} x \log (x)}{e^{e^5-2} \left (e^3 \left (2 x^2-6 x\right )+2 e^x x\right )+\left (e^{2 e^5-4} \left (e^3 \left (6 x-2 x^2\right )-2 e^x x\right )-2 e^{e^5-2} x\right ) \log (x)+e^{2 e^5-4} \left (e^{x+3} \left (2 x^2-6 x\right )+e^6 \left (x^3-6 x^2+9 x\right )+e^{2 x} x\right )+x+e^{2 e^5-4} x \log ^2(x)} \, dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {e^{3+e^5} \left (e^{e^5} (x-1)-e^{x+e^5} (x-2) x+e^2 \left (1-2 e^{1+e^5}\right ) x-e^{e^5} x \log (x)\right )}{x \left (e^{3+e^5} (x-3)+e^{x+e^5}-e^{e^5} \log (x)+e^2\right )^2}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle e^{3+e^5} \int -\frac {e^{e^5} (1-x)-e^{x+e^5} (2-x) x-e^2 \left (1-2 e^{1+e^5}\right ) x+e^{e^5} x \log (x)}{x \left (-e^{3+e^5} (3-x)+e^{x+e^5}-e^{e^5} \log (x)+e^2\right )^2}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -e^{3+e^5} \int \frac {e^{e^5} (1-x)-e^{x+e^5} (2-x) x-e^2 \left (1-2 e^{1+e^5}\right ) x+e^{e^5} x \log (x)}{x \left (-e^{3+e^5} (3-x)+e^{x+e^5}-e^{e^5} \log (x)+e^2\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -e^{3+e^5} \int \left (\frac {x-2}{e^{3+e^5} x+e^{x+e^5}-e^{e^5} \log (x)+e^2 \left (1-3 e^{1+e^5}\right )}+\frac {(1-x) \left (e^{3+e^5} x^2-e^{e^5} \log (x) x+e^2 \left (1-4 e^{1+e^5}\right ) x+e^{e^5}\right )}{x \left (e^{3+e^5} x+e^{x+e^5}-e^{e^5} \log (x)+e^2 \left (1-3 e^{1+e^5}\right )\right )^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -e^{3+e^5} \left (-e^{3+e^5} \int \frac {x^2}{\left (e^{3+e^5} x+e^{x+e^5}-e^{e^5} \log (x)+e^2 \left (1-3 e^{1+e^5}\right )\right )^2}dx+e^2 \left (1-4 e^{1+e^5}\right ) \int \frac {1}{\left (e^{3+e^5} x+e^{x+e^5}-e^{e^5} \log (x)+e^2 \left (1-3 e^{1+e^5}\right )\right )^2}dx-e^{e^5} \int \frac {1}{\left (e^{3+e^5} x+e^{x+e^5}-e^{e^5} \log (x)+e^2 \left (1-3 e^{1+e^5}\right )\right )^2}dx+e^{e^5} \int \frac {1}{x \left (e^{3+e^5} x+e^{x+e^5}-e^{e^5} \log (x)+e^2 \left (1-3 e^{1+e^5}\right )\right )^2}dx-e^2 \left (1-4 e^{1+e^5}\right ) \int \frac {x}{\left (e^{3+e^5} x+e^{x+e^5}-e^{e^5} \log (x)+e^2 \left (1-3 e^{1+e^5}\right )\right )^2}dx+e^{3+e^5} \int \frac {x}{\left (e^{3+e^5} x+e^{x+e^5}-e^{e^5} \log (x)+e^2 \left (1-3 e^{1+e^5}\right )\right )^2}dx-e^{e^5} \int \frac {\log (x)}{\left (e^{3+e^5} x+e^{x+e^5}-e^{e^5} \log (x)+e^2 \left (1-3 e^{1+e^5}\right )\right )^2}dx+e^{e^5} \int \frac {x \log (x)}{\left (e^{3+e^5} x+e^{x+e^5}-e^{e^5} \log (x)+e^2 \left (1-3 e^{1+e^5}\right )\right )^2}dx+\int \frac {x}{e^{3+e^5} x+e^{x+e^5}-e^{e^5} \log (x)+e^2 \left (1-3 e^{1+e^5}\right )}dx+2 \int \frac {1}{-e^{3+e^5} x-e^{x+e^5}+e^{e^5} \log (x)-e^2 \left (1-3 e^{1+e^5}\right )}dx\right )\) |
Int[(E^(1 + E^5)*x + E^(-4 + 2*E^5)*(E^3*(-1 + x) - 2*E^6*x + E^(3 + x)*(2 *x - x^2)) - E^(-1 + 2*E^5)*x*Log[x])/(x + E^(-2 + E^5)*(2*E^x*x + E^3*(-6 *x + 2*x^2)) + E^(-4 + 2*E^5)*(E^(2*x)*x + E^(3 + x)*(-6*x + 2*x^2) + E^6* (9*x - 6*x^2 + x^3)) + (-2*E^(-2 + E^5)*x + E^(-4 + 2*E^5)*(-2*E^x*x + E^3 *(6*x - 2*x^2)))*Log[x] + E^(-4 + 2*E^5)*x*Log[x]^2),x]
3.14.36.3.1 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 = 0.20 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.09
| method | result | size |
| risch | \(\frac {\left (-1+x \right ) {\mathrm e}^{{\mathrm e}^{5}}}{{\mathrm e}^{{\mathrm e}^{5}-3+x}-\ln \left (x \right ) {\mathrm e}^{{\mathrm e}^{5}-3}+x \,{\mathrm e}^{{\mathrm e}^{5}}+{\mathrm e}^{-1}-3 \,{\mathrm e}^{{\mathrm e}^{5}}}\) | \(38\) |
| parallelrisch | \(\frac {\left ({\mathrm e}^{2 \,{\mathrm e}^{5}-4} {\mathrm e}^{3} x -{\mathrm e}^{2 \,{\mathrm e}^{5}-4} {\mathrm e}^{3}\right ) {\mathrm e}^{2-{\mathrm e}^{5}}}{x \,{\mathrm e}^{3} {\mathrm e}^{{\mathrm e}^{5}-2}-3 \,{\mathrm e}^{3} {\mathrm e}^{{\mathrm e}^{5}-2}+{\mathrm e}^{{\mathrm e}^{5}-2} {\mathrm e}^{x}-{\mathrm e}^{{\mathrm e}^{5}-2} \ln \left (x \right )+1}\) | \(71\) |
int((-x*exp(3)*exp(exp(5)-2)^2*ln(x)+((-x^2+2*x)*exp(3)*exp(x)-2*x*exp(3)^ 2+(-1+x)*exp(3))*exp(exp(5)-2)^2+x*exp(3)*exp(exp(5)-2))/(x*exp(exp(5)-2)^ 2*ln(x)^2+((-2*exp(x)*x+(-2*x^2+6*x)*exp(3))*exp(exp(5)-2)^2-2*x*exp(exp(5 )-2))*ln(x)+(x*exp(x)^2+(2*x^2-6*x)*exp(3)*exp(x)+(x^3-6*x^2+9*x)*exp(3)^2 )*exp(exp(5)-2)^2+(2*exp(x)*x+(2*x^2-6*x)*exp(3))*exp(exp(5)-2)+x),x,metho d=_RETURNVERBOSE)
(-1+x)*exp(exp(5))/(exp(exp(5)-3+x)-ln(x)*exp(exp(5)-3)+x*exp(exp(5))+exp( -1)-3*exp(exp(5)))
Time = 0.24 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.14 \[ \int \frac {e^{1+e^5} x+e^{-4+2 e^5} \left (e^3 (-1+x)-2 e^6 x+e^{3+x} \left (2 x-x^2\right )\right )-e^{-1+2 e^5} x \log (x)}{x+e^{-2+e^5} \left (2 e^x x+e^3 \left (-6 x+2 x^2\right )\right )+e^{-4+2 e^5} \left (e^{2 x} x+e^{3+x} \left (-6 x+2 x^2\right )+e^6 \left (9 x-6 x^2+x^3\right )\right )+\left (-2 e^{-2+e^5} x+e^{-4+2 e^5} \left (-2 e^x x+e^3 \left (6 x-2 x^2\right )\right )\right ) \log (x)+e^{-4+2 e^5} x \log ^2(x)} \, dx=\frac {{\left (x - 1\right )} e^{\left (e^{5} + 7\right )}}{{\left ({\left (x - 3\right )} e^{6} + e^{\left (x + 3\right )}\right )} e^{\left (e^{5} + 1\right )} - e^{\left (e^{5} + 4\right )} \log \left (x\right ) + e^{6}} \]
integrate((-x*exp(3)*exp(exp(5)-2)^2*log(x)+((-x^2+2*x)*exp(3)*exp(x)-2*x* exp(3)^2+(-1+x)*exp(3))*exp(exp(5)-2)^2+x*exp(3)*exp(exp(5)-2))/(x*exp(exp (5)-2)^2*log(x)^2+((-2*exp(x)*x+(-2*x^2+6*x)*exp(3))*exp(exp(5)-2)^2-2*x*e xp(exp(5)-2))*log(x)+(x*exp(x)^2+(2*x^2-6*x)*exp(3)*exp(x)+(x^3-6*x^2+9*x) *exp(3)^2)*exp(exp(5)-2)^2+(2*exp(x)*x+(2*x^2-6*x)*exp(3))*exp(exp(5)-2)+x ),x, algorithm=\
Leaf count of result is larger than twice the leaf count of optimal. 58 vs. \(2 (22) = 44\).
Time = 0.17 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.66 \[ \int \frac {e^{1+e^5} x+e^{-4+2 e^5} \left (e^3 (-1+x)-2 e^6 x+e^{3+x} \left (2 x-x^2\right )\right )-e^{-1+2 e^5} x \log (x)}{x+e^{-2+e^5} \left (2 e^x x+e^3 \left (-6 x+2 x^2\right )\right )+e^{-4+2 e^5} \left (e^{2 x} x+e^{3+x} \left (-6 x+2 x^2\right )+e^6 \left (9 x-6 x^2+x^3\right )\right )+\left (-2 e^{-2+e^5} x+e^{-4+2 e^5} \left (-2 e^x x+e^3 \left (6 x-2 x^2\right )\right )\right ) \log (x)+e^{-4+2 e^5} x \log ^2(x)} \, dx=\frac {x e^{3} e^{e^{5}} - e^{3} e^{e^{5}}}{x e^{3} e^{e^{5}} + e^{x} e^{e^{5}} - e^{e^{5}} \log {\left (x \right )} - 3 e^{3} e^{e^{5}} + e^{2}} \]
integrate((-x*exp(3)*exp(exp(5)-2)**2*ln(x)+((-x**2+2*x)*exp(3)*exp(x)-2*x *exp(3)**2+(-1+x)*exp(3))*exp(exp(5)-2)**2+x*exp(3)*exp(exp(5)-2))/(x*exp( exp(5)-2)**2*ln(x)**2+((-2*exp(x)*x+(-2*x**2+6*x)*exp(3))*exp(exp(5)-2)**2 -2*x*exp(exp(5)-2))*ln(x)+(x*exp(x)**2+(2*x**2-6*x)*exp(3)*exp(x)+(x**3-6* x**2+9*x)*exp(3)**2)*exp(exp(5)-2)**2+(2*exp(x)*x+(2*x**2-6*x)*exp(3))*exp (exp(5)-2)+x),x)
(x*exp(3)*exp(exp(5)) - exp(3)*exp(exp(5)))/(x*exp(3)*exp(exp(5)) + exp(x) *exp(exp(5)) - exp(exp(5))*log(x) - 3*exp(3)*exp(exp(5)) + exp(2))
Time = 0.27 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.34 \[ \int \frac {e^{1+e^5} x+e^{-4+2 e^5} \left (e^3 (-1+x)-2 e^6 x+e^{3+x} \left (2 x-x^2\right )\right )-e^{-1+2 e^5} x \log (x)}{x+e^{-2+e^5} \left (2 e^x x+e^3 \left (-6 x+2 x^2\right )\right )+e^{-4+2 e^5} \left (e^{2 x} x+e^{3+x} \left (-6 x+2 x^2\right )+e^6 \left (9 x-6 x^2+x^3\right )\right )+\left (-2 e^{-2+e^5} x+e^{-4+2 e^5} \left (-2 e^x x+e^3 \left (6 x-2 x^2\right )\right )\right ) \log (x)+e^{-4+2 e^5} x \log ^2(x)} \, dx=\frac {x e^{\left (e^{5} + 3\right )} - e^{\left (e^{5} + 3\right )}}{x e^{\left (e^{5} + 3\right )} - e^{\left (e^{5}\right )} \log \left (x\right ) + e^{2} + e^{\left (x + e^{5}\right )} - 3 \, e^{\left (e^{5} + 3\right )}} \]
integrate((-x*exp(3)*exp(exp(5)-2)^2*log(x)+((-x^2+2*x)*exp(3)*exp(x)-2*x* exp(3)^2+(-1+x)*exp(3))*exp(exp(5)-2)^2+x*exp(3)*exp(exp(5)-2))/(x*exp(exp (5)-2)^2*log(x)^2+((-2*exp(x)*x+(-2*x^2+6*x)*exp(3))*exp(exp(5)-2)^2-2*x*e xp(exp(5)-2))*log(x)+(x*exp(x)^2+(2*x^2-6*x)*exp(3)*exp(x)+(x^3-6*x^2+9*x) *exp(3)^2)*exp(exp(5)-2)^2+(2*exp(x)*x+(2*x^2-6*x)*exp(3))*exp(exp(5)-2)+x ),x, algorithm=\
(x*e^(e^5 + 3) - e^(e^5 + 3))/(x*e^(e^5 + 3) - e^(e^5)*log(x) + e^2 + e^(x + e^5) - 3*e^(e^5 + 3))
Timed out. \[ \int \frac {e^{1+e^5} x+e^{-4+2 e^5} \left (e^3 (-1+x)-2 e^6 x+e^{3+x} \left (2 x-x^2\right )\right )-e^{-1+2 e^5} x \log (x)}{x+e^{-2+e^5} \left (2 e^x x+e^3 \left (-6 x+2 x^2\right )\right )+e^{-4+2 e^5} \left (e^{2 x} x+e^{3+x} \left (-6 x+2 x^2\right )+e^6 \left (9 x-6 x^2+x^3\right )\right )+\left (-2 e^{-2+e^5} x+e^{-4+2 e^5} \left (-2 e^x x+e^3 \left (6 x-2 x^2\right )\right )\right ) \log (x)+e^{-4+2 e^5} x \log ^2(x)} \, dx=\text {Timed out} \]
integrate((-x*exp(3)*exp(exp(5)-2)^2*log(x)+((-x^2+2*x)*exp(3)*exp(x)-2*x* exp(3)^2+(-1+x)*exp(3))*exp(exp(5)-2)^2+x*exp(3)*exp(exp(5)-2))/(x*exp(exp (5)-2)^2*log(x)^2+((-2*exp(x)*x+(-2*x^2+6*x)*exp(3))*exp(exp(5)-2)^2-2*x*e xp(exp(5)-2))*log(x)+(x*exp(x)^2+(2*x^2-6*x)*exp(3)*exp(x)+(x^3-6*x^2+9*x) *exp(3)^2)*exp(exp(5)-2)^2+(2*exp(x)*x+(2*x^2-6*x)*exp(3))*exp(exp(5)-2)+x ),x, algorithm=\
Timed out. \[ \int \frac {e^{1+e^5} x+e^{-4+2 e^5} \left (e^3 (-1+x)-2 e^6 x+e^{3+x} \left (2 x-x^2\right )\right )-e^{-1+2 e^5} x \log (x)}{x+e^{-2+e^5} \left (2 e^x x+e^3 \left (-6 x+2 x^2\right )\right )+e^{-4+2 e^5} \left (e^{2 x} x+e^{3+x} \left (-6 x+2 x^2\right )+e^6 \left (9 x-6 x^2+x^3\right )\right )+\left (-2 e^{-2+e^5} x+e^{-4+2 e^5} \left (-2 e^x x+e^3 \left (6 x-2 x^2\right )\right )\right ) \log (x)+e^{-4+2 e^5} x \log ^2(x)} \, dx=\int \frac {{\mathrm {e}}^{2\,{\mathrm {e}}^5-4}\,\left ({\mathrm {e}}^3\,\left (x-1\right )-2\,x\,{\mathrm {e}}^6+{\mathrm {e}}^3\,{\mathrm {e}}^x\,\left (2\,x-x^2\right )\right )+x\,{\mathrm {e}}^{{\mathrm {e}}^5-2}\,{\mathrm {e}}^3-x\,{\mathrm {e}}^3\,{\mathrm {e}}^{2\,{\mathrm {e}}^5-4}\,\ln \left (x\right )}{x\,{\mathrm {e}}^{2\,{\mathrm {e}}^5-4}\,{\ln \left (x\right )}^2+\left ({\mathrm {e}}^{2\,{\mathrm {e}}^5-4}\,\left ({\mathrm {e}}^3\,\left (6\,x-2\,x^2\right )-2\,x\,{\mathrm {e}}^x\right )-2\,x\,{\mathrm {e}}^{{\mathrm {e}}^5-2}\right )\,\ln \left (x\right )+x+{\mathrm {e}}^{2\,{\mathrm {e}}^5-4}\,\left ({\mathrm {e}}^6\,\left (x^3-6\,x^2+9\,x\right )+x\,{\mathrm {e}}^{2\,x}-{\mathrm {e}}^3\,{\mathrm {e}}^x\,\left (6\,x-2\,x^2\right )\right )-{\mathrm {e}}^{{\mathrm {e}}^5-2}\,\left ({\mathrm {e}}^3\,\left (6\,x-2\,x^2\right )-2\,x\,{\mathrm {e}}^x\right )} \,d x \]
int((exp(2*exp(5) - 4)*(exp(3)*(x - 1) - 2*x*exp(6) + exp(3)*exp(x)*(2*x - x^2)) + x*exp(exp(5) - 2)*exp(3) - x*exp(3)*exp(2*exp(5) - 4)*log(x))/(x + exp(2*exp(5) - 4)*(exp(6)*(9*x - 6*x^2 + x^3) + x*exp(2*x) - exp(3)*exp( x)*(6*x - 2*x^2)) - exp(exp(5) - 2)*(exp(3)*(6*x - 2*x^2) - 2*x*exp(x)) - log(x)*(2*x*exp(exp(5) - 2) - exp(2*exp(5) - 4)*(exp(3)*(6*x - 2*x^2) - 2* x*exp(x))) + x*exp(2*exp(5) - 4)*log(x)^2),x)
int((exp(2*exp(5) - 4)*(exp(3)*(x - 1) - 2*x*exp(6) + exp(3)*exp(x)*(2*x - x^2)) + x*exp(exp(5) - 2)*exp(3) - x*exp(3)*exp(2*exp(5) - 4)*log(x))/(x + exp(2*exp(5) - 4)*(exp(6)*(9*x - 6*x^2 + x^3) + x*exp(2*x) - exp(3)*exp( x)*(6*x - 2*x^2)) - exp(exp(5) - 2)*(exp(3)*(6*x - 2*x^2) - 2*x*exp(x)) - log(x)*(2*x*exp(exp(5) - 2) - exp(2*exp(5) - 4)*(exp(3)*(6*x - 2*x^2) - 2* x*exp(x))) + x*exp(2*exp(5) - 4)*log(x)^2), x)