Integrand size = 191, antiderivative size = 32 \[ \int \frac {-4 e^x x \log (x)+e^x \left (8 x+2 x^2\right ) \log ^2(x)+e^x \left (-8 x-4 x^2\right ) \log ^3(x)+e^x \left (-2+2 x+2 x^2\right ) \log ^4(x)}{5 e^{2 x} x^4-20 e^{2 x} x^4 \log (x)+\left (10 e^x x^2+e^{2 x} \left (-10 x^3+30 x^4\right )\right ) \log ^2(x)+\left (-20 e^x x^2+e^{2 x} \left (20 x^3-20 x^4\right )\right ) \log ^3(x)+\left (5+e^x \left (-10 x+10 x^2\right )+e^{2 x} \left (5 x^2-10 x^3+5 x^4\right )\right ) \log ^4(x)} \, dx=\frac {2 x}{5 \left (-x+e^x x \left (x-\left (-x+\frac {x}{\log (x)}\right )^2\right )\right )} \]
Time = 5.17 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.34 \[ \int \frac {-4 e^x x \log (x)+e^x \left (8 x+2 x^2\right ) \log ^2(x)+e^x \left (-8 x-4 x^2\right ) \log ^3(x)+e^x \left (-2+2 x+2 x^2\right ) \log ^4(x)}{5 e^{2 x} x^4-20 e^{2 x} x^4 \log (x)+\left (10 e^x x^2+e^{2 x} \left (-10 x^3+30 x^4\right )\right ) \log ^2(x)+\left (-20 e^x x^2+e^{2 x} \left (20 x^3-20 x^4\right )\right ) \log ^3(x)+\left (5+e^x \left (-10 x+10 x^2\right )+e^{2 x} \left (5 x^2-10 x^3+5 x^4\right )\right ) \log ^4(x)} \, dx=-\frac {2 \log ^2(x)}{5 \left (e^x x^2-2 e^x x^2 \log (x)+\left (1+e^x (-1+x) x\right ) \log ^2(x)\right )} \]
Integrate[(-4*E^x*x*Log[x] + E^x*(8*x + 2*x^2)*Log[x]^2 + E^x*(-8*x - 4*x^ 2)*Log[x]^3 + E^x*(-2 + 2*x + 2*x^2)*Log[x]^4)/(5*E^(2*x)*x^4 - 20*E^(2*x) *x^4*Log[x] + (10*E^x*x^2 + E^(2*x)*(-10*x^3 + 30*x^4))*Log[x]^2 + (-20*E^ x*x^2 + E^(2*x)*(20*x^3 - 20*x^4))*Log[x]^3 + (5 + E^x*(-10*x + 10*x^2) + E^(2*x)*(5*x^2 - 10*x^3 + 5*x^4))*Log[x]^4),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.
| \(\displaystyle \int \frac {e^x \left (2 x^2+2 x-2\right ) \log ^4(x)+e^x \left (-4 x^2-8 x\right ) \log ^3(x)+e^x \left (2 x^2+8 x\right ) \log ^2(x)-4 e^x x \log (x)}{5 e^{2 x} x^4-20 e^{2 x} x^4 \log (x)+\left (e^x \left (10 x^2-10 x\right )+e^{2 x} \left (5 x^4-10 x^3+5 x^2\right )+5\right ) \log ^4(x)+\left (e^{2 x} \left (20 x^3-20 x^4\right )-20 e^x x^2\right ) \log ^3(x)+\left (10 e^x x^2+e^{2 x} \left (30 x^4-10 x^3\right )\right ) \log ^2(x)} \, dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {2 e^x \log (x) \left (\left (x^2+x-1\right ) \log ^3(x)-2 x-2 x (x+2) \log ^2(x)+x (x+4) \log (x)\right )}{5 \left (e^x x^2-2 e^x x^2 \log (x)+\left (e^x (x-1) x+1\right ) \log ^2(x)\right )^2}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2}{5} \int -\frac {e^x \log (x) \left (\left (-x^2-x+1\right ) \log ^3(x)+2 x (x+2) \log ^2(x)-x (x+4) \log (x)+2 x\right )}{\left (e^x x^2-2 e^x \log (x) x^2+\left (1-e^x (1-x) x\right ) \log ^2(x)\right )^2}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {2}{5} \int \frac {e^x \log (x) \left (\left (-x^2-x+1\right ) \log ^3(x)+2 x (x+2) \log ^2(x)-x (x+4) \log (x)+2 x\right )}{\left (e^x x^2-2 e^x \log (x) x^2+\left (1-e^x (1-x) x\right ) \log ^2(x)\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {2}{5} \int \left (\frac {e^x \log ^4(x)}{\left (e^x x^2+e^x \log ^2(x) x^2-2 e^x \log (x) x^2-e^x \log ^2(x) x+\log ^2(x)\right )^2}-\frac {e^x x^2 \log ^4(x)}{\left (e^x x^2+e^x \log ^2(x) x^2-2 e^x \log (x) x^2-e^x \log ^2(x) x+\log ^2(x)\right )^2}-\frac {e^x x \log ^4(x)}{\left (e^x x^2+e^x \log ^2(x) x^2-2 e^x \log (x) x^2-e^x \log ^2(x) x+\log ^2(x)\right )^2}+\frac {2 e^x x^2 \log ^3(x)}{\left (e^x x^2+e^x \log ^2(x) x^2-2 e^x \log (x) x^2-e^x \log ^2(x) x+\log ^2(x)\right )^2}+\frac {4 e^x x \log ^3(x)}{\left (e^x x^2+e^x \log ^2(x) x^2-2 e^x \log (x) x^2-e^x \log ^2(x) x+\log ^2(x)\right )^2}-\frac {e^x x^2 \log ^2(x)}{\left (e^x x^2+e^x \log ^2(x) x^2-2 e^x \log (x) x^2-e^x \log ^2(x) x+\log ^2(x)\right )^2}-\frac {4 e^x x \log ^2(x)}{\left (e^x x^2+e^x \log ^2(x) x^2-2 e^x \log (x) x^2-e^x \log ^2(x) x+\log ^2(x)\right )^2}+\frac {2 e^x x \log (x)}{\left (e^x x^2+e^x \log ^2(x) x^2-2 e^x \log (x) x^2-e^x \log ^2(x) x+\log ^2(x)\right )^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {2}{5} \left (2 \int \frac {e^x x \log (x)}{\left (e^x x^2+e^x \log ^2(x) x^2-2 e^x \log (x) x^2-e^x \log ^2(x) x+\log ^2(x)\right )^2}dx-4 \int \frac {e^x x \log ^2(x)}{\left (e^x x^2+e^x \log ^2(x) x^2-2 e^x \log (x) x^2-e^x \log ^2(x) x+\log ^2(x)\right )^2}dx-\int \frac {e^x x^2 \log ^2(x)}{\left (e^x x^2+e^x \log ^2(x) x^2-2 e^x \log (x) x^2-e^x \log ^2(x) x+\log ^2(x)\right )^2}dx+\int \frac {e^x \log ^4(x)}{\left (e^x x^2+e^x \log ^2(x) x^2-2 e^x \log (x) x^2-e^x \log ^2(x) x+\log ^2(x)\right )^2}dx-\int \frac {e^x x \log ^4(x)}{\left (e^x x^2+e^x \log ^2(x) x^2-2 e^x \log (x) x^2-e^x \log ^2(x) x+\log ^2(x)\right )^2}dx-\int \frac {e^x x^2 \log ^4(x)}{\left (e^x x^2+e^x \log ^2(x) x^2-2 e^x \log (x) x^2-e^x \log ^2(x) x+\log ^2(x)\right )^2}dx+4 \int \frac {e^x x \log ^3(x)}{\left (e^x x^2+e^x \log ^2(x) x^2-2 e^x \log (x) x^2-e^x \log ^2(x) x+\log ^2(x)\right )^2}dx+2 \int \frac {e^x x^2 \log ^3(x)}{\left (e^x x^2+e^x \log ^2(x) x^2-2 e^x \log (x) x^2-e^x \log ^2(x) x+\log ^2(x)\right )^2}dx\right )\) |
Int[(-4*E^x*x*Log[x] + E^x*(8*x + 2*x^2)*Log[x]^2 + E^x*(-8*x - 4*x^2)*Log [x]^3 + E^x*(-2 + 2*x + 2*x^2)*Log[x]^4)/(5*E^(2*x)*x^4 - 20*E^(2*x)*x^4*L og[x] + (10*E^x*x^2 + E^(2*x)*(-10*x^3 + 30*x^4))*Log[x]^2 + (-20*E^x*x^2 + E^(2*x)*(20*x^3 - 20*x^4))*Log[x]^3 + (5 + E^x*(-10*x + 10*x^2) + E^(2*x )*(5*x^2 - 10*x^3 + 5*x^4))*Log[x]^4),x]
3.3.99.3.1 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]]
Leaf count of result is larger than twice the leaf count of optimal. \(80\) vs. \(2(29)=58\).
Time = 1.35 (sec) , antiderivative size = 81, normalized size of antiderivative = 2.53
| method | result | size |
| parallelrisch | \(\frac {2 x^{2} {\mathrm e}^{x} \ln \left (x \right )^{2}-4 x^{2} {\mathrm e}^{x} \ln \left (x \right )-2 x \,{\mathrm e}^{x} \ln \left (x \right )^{2}+2 \,{\mathrm e}^{x} x^{2}}{5 x^{2} {\mathrm e}^{x} \ln \left (x \right )^{2}-5 x \,{\mathrm e}^{x} \ln \left (x \right )^{2}-10 x^{2} {\mathrm e}^{x} \ln \left (x \right )+5 \,{\mathrm e}^{x} x^{2}+5 \ln \left (x \right )^{2}}\) | \(81\) |
| risch | \(-\frac {2}{5 \left ({\mathrm e}^{x} x^{2}-{\mathrm e}^{x} x +1\right )}-\frac {2 \,{\mathrm e}^{x} x^{2} \left (2 \ln \left (x \right )-1\right )}{5 \left ({\mathrm e}^{x} x^{2}-{\mathrm e}^{x} x +1\right ) \left (x^{2} {\mathrm e}^{x} \ln \left (x \right )^{2}-x \,{\mathrm e}^{x} \ln \left (x \right )^{2}-2 x^{2} {\mathrm e}^{x} \ln \left (x \right )+{\mathrm e}^{x} x^{2}+\ln \left (x \right )^{2}\right )}\) | \(88\) |
int(((2*x^2+2*x-2)*exp(x)*ln(x)^4+(-4*x^2-8*x)*exp(x)*ln(x)^3+(2*x^2+8*x)* exp(x)*ln(x)^2-4*x*exp(x)*ln(x))/(((5*x^4-10*x^3+5*x^2)*exp(x)^2+(10*x^2-1 0*x)*exp(x)+5)*ln(x)^4+((-20*x^4+20*x^3)*exp(x)^2-20*exp(x)*x^2)*ln(x)^3+( (30*x^4-10*x^3)*exp(x)^2+10*exp(x)*x^2)*ln(x)^2-20*x^4*exp(x)^2*ln(x)+5*ex p(x)^2*x^4),x,method=_RETURNVERBOSE)
1/5*(2*x^2*exp(x)*ln(x)^2-4*x^2*exp(x)*ln(x)-2*x*exp(x)*ln(x)^2+2*exp(x)*x ^2)/(x^2*exp(x)*ln(x)^2-x*exp(x)*ln(x)^2-2*x^2*exp(x)*ln(x)+exp(x)*x^2+ln( x)^2)
Time = 0.25 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.34 \[ \int \frac {-4 e^x x \log (x)+e^x \left (8 x+2 x^2\right ) \log ^2(x)+e^x \left (-8 x-4 x^2\right ) \log ^3(x)+e^x \left (-2+2 x+2 x^2\right ) \log ^4(x)}{5 e^{2 x} x^4-20 e^{2 x} x^4 \log (x)+\left (10 e^x x^2+e^{2 x} \left (-10 x^3+30 x^4\right )\right ) \log ^2(x)+\left (-20 e^x x^2+e^{2 x} \left (20 x^3-20 x^4\right )\right ) \log ^3(x)+\left (5+e^x \left (-10 x+10 x^2\right )+e^{2 x} \left (5 x^2-10 x^3+5 x^4\right )\right ) \log ^4(x)} \, dx=\frac {2 \, \log \left (x\right )^{2}}{5 \, {\left (2 \, x^{2} e^{x} \log \left (x\right ) - x^{2} e^{x} - {\left ({\left (x^{2} - x\right )} e^{x} + 1\right )} \log \left (x\right )^{2}\right )}} \]
integrate(((2*x^2+2*x-2)*exp(x)*log(x)^4+(-4*x^2-8*x)*exp(x)*log(x)^3+(2*x ^2+8*x)*exp(x)*log(x)^2-4*x*exp(x)*log(x))/(((5*x^4-10*x^3+5*x^2)*exp(x)^2 +(10*x^2-10*x)*exp(x)+5)*log(x)^4+((-20*x^4+20*x^3)*exp(x)^2-20*exp(x)*x^2 )*log(x)^3+((30*x^4-10*x^3)*exp(x)^2+10*exp(x)*x^2)*log(x)^2-20*x^4*exp(x) ^2*log(x)+5*exp(x)^2*x^4),x, algorithm=\
Leaf count of result is larger than twice the leaf count of optimal. 49 vs. \(2 (22) = 44\).
Time = 0.27 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.53 \[ \int \frac {-4 e^x x \log (x)+e^x \left (8 x+2 x^2\right ) \log ^2(x)+e^x \left (-8 x-4 x^2\right ) \log ^3(x)+e^x \left (-2+2 x+2 x^2\right ) \log ^4(x)}{5 e^{2 x} x^4-20 e^{2 x} x^4 \log (x)+\left (10 e^x x^2+e^{2 x} \left (-10 x^3+30 x^4\right )\right ) \log ^2(x)+\left (-20 e^x x^2+e^{2 x} \left (20 x^3-20 x^4\right )\right ) \log ^3(x)+\left (5+e^x \left (-10 x+10 x^2\right )+e^{2 x} \left (5 x^2-10 x^3+5 x^4\right )\right ) \log ^4(x)} \, dx=- \frac {2 \log {\left (x \right )}^{2}}{\left (5 x^{2} \log {\left (x \right )}^{2} - 10 x^{2} \log {\left (x \right )} + 5 x^{2} - 5 x \log {\left (x \right )}^{2}\right ) e^{x} + 5 \log {\left (x \right )}^{2}} \]
integrate(((2*x**2+2*x-2)*exp(x)*ln(x)**4+(-4*x**2-8*x)*exp(x)*ln(x)**3+(2 *x**2+8*x)*exp(x)*ln(x)**2-4*x*exp(x)*ln(x))/(((5*x**4-10*x**3+5*x**2)*exp (x)**2+(10*x**2-10*x)*exp(x)+5)*ln(x)**4+((-20*x**4+20*x**3)*exp(x)**2-20* exp(x)*x**2)*ln(x)**3+((30*x**4-10*x**3)*exp(x)**2+10*exp(x)*x**2)*ln(x)** 2-20*x**4*exp(x)**2*ln(x)+5*exp(x)**2*x**4),x)
-2*log(x)**2/((5*x**2*log(x)**2 - 10*x**2*log(x) + 5*x**2 - 5*x*log(x)**2) *exp(x) + 5*log(x)**2)
Time = 0.31 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.38 \[ \int \frac {-4 e^x x \log (x)+e^x \left (8 x+2 x^2\right ) \log ^2(x)+e^x \left (-8 x-4 x^2\right ) \log ^3(x)+e^x \left (-2+2 x+2 x^2\right ) \log ^4(x)}{5 e^{2 x} x^4-20 e^{2 x} x^4 \log (x)+\left (10 e^x x^2+e^{2 x} \left (-10 x^3+30 x^4\right )\right ) \log ^2(x)+\left (-20 e^x x^2+e^{2 x} \left (20 x^3-20 x^4\right )\right ) \log ^3(x)+\left (5+e^x \left (-10 x+10 x^2\right )+e^{2 x} \left (5 x^2-10 x^3+5 x^4\right )\right ) \log ^4(x)} \, dx=\frac {2 \, \log \left (x\right )^{2}}{5 \, {\left ({\left (2 \, x^{2} \log \left (x\right ) - {\left (x^{2} - x\right )} \log \left (x\right )^{2} - x^{2}\right )} e^{x} - \log \left (x\right )^{2}\right )}} \]
integrate(((2*x^2+2*x-2)*exp(x)*log(x)^4+(-4*x^2-8*x)*exp(x)*log(x)^3+(2*x ^2+8*x)*exp(x)*log(x)^2-4*x*exp(x)*log(x))/(((5*x^4-10*x^3+5*x^2)*exp(x)^2 +(10*x^2-10*x)*exp(x)+5)*log(x)^4+((-20*x^4+20*x^3)*exp(x)^2-20*exp(x)*x^2 )*log(x)^3+((30*x^4-10*x^3)*exp(x)^2+10*exp(x)*x^2)*log(x)^2-20*x^4*exp(x) ^2*log(x)+5*exp(x)^2*x^4),x, algorithm=\
Time = 0.85 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.47 \[ \int \frac {-4 e^x x \log (x)+e^x \left (8 x+2 x^2\right ) \log ^2(x)+e^x \left (-8 x-4 x^2\right ) \log ^3(x)+e^x \left (-2+2 x+2 x^2\right ) \log ^4(x)}{5 e^{2 x} x^4-20 e^{2 x} x^4 \log (x)+\left (10 e^x x^2+e^{2 x} \left (-10 x^3+30 x^4\right )\right ) \log ^2(x)+\left (-20 e^x x^2+e^{2 x} \left (20 x^3-20 x^4\right )\right ) \log ^3(x)+\left (5+e^x \left (-10 x+10 x^2\right )+e^{2 x} \left (5 x^2-10 x^3+5 x^4\right )\right ) \log ^4(x)} \, dx=-\frac {2 \, \log \left (x\right )^{2}}{5 \, {\left (x^{2} e^{x} \log \left (x\right )^{2} - 2 \, x^{2} e^{x} \log \left (x\right ) - x e^{x} \log \left (x\right )^{2} + x^{2} e^{x} + \log \left (x\right )^{2}\right )}} \]
integrate(((2*x^2+2*x-2)*exp(x)*log(x)^4+(-4*x^2-8*x)*exp(x)*log(x)^3+(2*x ^2+8*x)*exp(x)*log(x)^2-4*x*exp(x)*log(x))/(((5*x^4-10*x^3+5*x^2)*exp(x)^2 +(10*x^2-10*x)*exp(x)+5)*log(x)^4+((-20*x^4+20*x^3)*exp(x)^2-20*exp(x)*x^2 )*log(x)^3+((30*x^4-10*x^3)*exp(x)^2+10*exp(x)*x^2)*log(x)^2-20*x^4*exp(x) ^2*log(x)+5*exp(x)^2*x^4),x, algorithm=\
Timed out. \[ \int \frac {-4 e^x x \log (x)+e^x \left (8 x+2 x^2\right ) \log ^2(x)+e^x \left (-8 x-4 x^2\right ) \log ^3(x)+e^x \left (-2+2 x+2 x^2\right ) \log ^4(x)}{5 e^{2 x} x^4-20 e^{2 x} x^4 \log (x)+\left (10 e^x x^2+e^{2 x} \left (-10 x^3+30 x^4\right )\right ) \log ^2(x)+\left (-20 e^x x^2+e^{2 x} \left (20 x^3-20 x^4\right )\right ) \log ^3(x)+\left (5+e^x \left (-10 x+10 x^2\right )+e^{2 x} \left (5 x^2-10 x^3+5 x^4\right )\right ) \log ^4(x)} \, dx=\int -\frac {-{\mathrm {e}}^x\,\left (2\,x^2+2\,x-2\right )\,{\ln \left (x\right )}^4+{\mathrm {e}}^x\,\left (4\,x^2+8\,x\right )\,{\ln \left (x\right )}^3-{\mathrm {e}}^x\,\left (2\,x^2+8\,x\right )\,{\ln \left (x\right )}^2+4\,x\,{\mathrm {e}}^x\,\ln \left (x\right )}{{\ln \left (x\right )}^4\,\left ({\mathrm {e}}^{2\,x}\,\left (5\,x^4-10\,x^3+5\,x^2\right )-{\mathrm {e}}^x\,\left (10\,x-10\,x^2\right )+5\right )+5\,x^4\,{\mathrm {e}}^{2\,x}+{\ln \left (x\right )}^2\,\left (10\,x^2\,{\mathrm {e}}^x-{\mathrm {e}}^{2\,x}\,\left (10\,x^3-30\,x^4\right )\right )-{\ln \left (x\right )}^3\,\left (20\,x^2\,{\mathrm {e}}^x-{\mathrm {e}}^{2\,x}\,\left (20\,x^3-20\,x^4\right )\right )-20\,x^4\,{\mathrm {e}}^{2\,x}\,\ln \left (x\right )} \,d x \]
int(-(4*x*exp(x)*log(x) - exp(x)*log(x)^4*(2*x + 2*x^2 - 2) - exp(x)*log(x )^2*(8*x + 2*x^2) + exp(x)*log(x)^3*(8*x + 4*x^2))/(log(x)^4*(exp(2*x)*(5* x^2 - 10*x^3 + 5*x^4) - exp(x)*(10*x - 10*x^2) + 5) + 5*x^4*exp(2*x) + log (x)^2*(10*x^2*exp(x) - exp(2*x)*(10*x^3 - 30*x^4)) - log(x)^3*(20*x^2*exp( x) - exp(2*x)*(20*x^3 - 20*x^4)) - 20*x^4*exp(2*x)*log(x)),x)
int(-(4*x*exp(x)*log(x) - exp(x)*log(x)^4*(2*x + 2*x^2 - 2) - exp(x)*log(x )^2*(8*x + 2*x^2) + exp(x)*log(x)^3*(8*x + 4*x^2))/(log(x)^4*(exp(2*x)*(5* x^2 - 10*x^3 + 5*x^4) - exp(x)*(10*x - 10*x^2) + 5) + 5*x^4*exp(2*x) + log (x)^2*(10*x^2*exp(x) - exp(2*x)*(10*x^3 - 30*x^4)) - log(x)^3*(20*x^2*exp( x) - exp(2*x)*(20*x^3 - 20*x^4)) - 20*x^4*exp(2*x)*log(x)), x)