Integrand size = 122, antiderivative size = 32 \[ \int \frac {-2 x^2+3 x^3+3 x^5+e^x \left (2 x-3 x^3-4 x^4+x^5\right )+\left (3 x^2+e^x \left (-2 x-x^2\right )\right ) \log (x)}{108 x^3-108 x^5+36 x^7-4 x^9+\left (108 x^2-72 x^4+12 x^6\right ) \log (x)+\left (36 x-12 x^3\right ) \log ^2(x)+4 \log ^3(x)} \, dx=\frac {x+\frac {x \left (-e^x+x\right )}{4 \left (-3+x^2-\frac {\log (x)}{x}\right )^2}}{x} \]
Time = 0.41 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.84 \[ \int \frac {-2 x^2+3 x^3+3 x^5+e^x \left (2 x-3 x^3-4 x^4+x^5\right )+\left (3 x^2+e^x \left (-2 x-x^2\right )\right ) \log (x)}{108 x^3-108 x^5+36 x^7-4 x^9+\left (108 x^2-72 x^4+12 x^6\right ) \log (x)+\left (36 x-12 x^3\right ) \log ^2(x)+4 \log ^3(x)} \, dx=-\frac {\left (e^x-x\right ) x^2}{4 \left (3 x-x^3+\log (x)\right )^2} \]
Integrate[(-2*x^2 + 3*x^3 + 3*x^5 + E^x*(2*x - 3*x^3 - 4*x^4 + x^5) + (3*x ^2 + E^x*(-2*x - x^2))*Log[x])/(108*x^3 - 108*x^5 + 36*x^7 - 4*x^9 + (108* x^2 - 72*x^4 + 12*x^6)*Log[x] + (36*x - 12*x^3)*Log[x]^2 + 4*Log[x]^3),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.
\(\displaystyle \int \frac {3 x^5+3 x^3-2 x^2+\left (3 x^2+e^x \left (-x^2-2 x\right )\right ) \log (x)+e^x \left (x^5-4 x^4-3 x^3+2 x\right )}{-4 x^9+36 x^7-108 x^5+108 x^3+\left (36 x-12 x^3\right ) \log ^2(x)+\left (12 x^6-72 x^4+108 x^2\right ) \log (x)+4 \log ^3(x)} \, dx\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle \int \frac {x \left (x \left (3 x^3+3 x-2\right )+e^x \left (x^4-4 x^3-3 x^2+2\right )+3 x \log (x)-e^x (x+2) \log (x)\right )}{4 \left (-x^3+3 x+\log (x)\right )^3}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{4} \int -\frac {x \left (x \left (-3 x^3-3 x+2\right )-e^x \left (x^4-4 x^3-3 x^2+2\right )-3 x \log (x)+e^x (x+2) \log (x)\right )}{\left (-x^3+3 x+\log (x)\right )^3}dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {1}{4} \int \frac {x \left (x \left (-3 x^3-3 x+2\right )-e^x \left (x^4-4 x^3-3 x^2+2\right )-3 x \log (x)+e^x (x+2) \log (x)\right )}{\left (-x^3+3 x+\log (x)\right )^3}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle -\frac {1}{4} \int \left (\frac {\left (3 x^3+3 x+3 \log (x)-2\right ) x^2}{\left (x^3-3 x-\log (x)\right )^3}+\frac {e^x \left (x^4-4 x^3-3 x^2-\log (x) x-2 \log (x)+2\right ) x}{\left (x^3-3 x-\log (x)\right )^3}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{4} \left (6 \int \frac {x^3}{\left (x^3-3 x-\log (x)\right )^3}dx-6 \int \frac {x^5}{\left (x^3-3 x-\log (x)\right )^3}dx+2 \int \frac {x^2}{\left (x^3-3 x-\log (x)\right )^3}dx+3 \int \frac {x^2}{\left (x^3-3 x-\log (x)\right )^2}dx-\frac {e^x x \left (-x^4+3 x^2+x \log (x)\right )}{\left (-x^3+3 x+\log (x)\right )^3}\right )\) |
Int[(-2*x^2 + 3*x^3 + 3*x^5 + E^x*(2*x - 3*x^3 - 4*x^4 + x^5) + (3*x^2 + E ^x*(-2*x - x^2))*Log[x])/(108*x^3 - 108*x^5 + 36*x^7 - 4*x^9 + (108*x^2 - 72*x^4 + 12*x^6)*Log[x] + (36*x - 12*x^3)*Log[x]^2 + 4*Log[x]^3),x]
3.15.82.3.1 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 = 1.67 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.78
method | result | size |
risch | \(\frac {\left (x -{\mathrm e}^{x}\right ) x^{2}}{4 \left (x^{3}-3 x -\ln \left (x \right )\right )^{2}}\) | \(25\) |
parallelrisch | \(\frac {6 x^{3}-6 \,{\mathrm e}^{x} x^{2}}{24 x^{6}-144 x^{4}-48 x^{3} \ln \left (x \right )+216 x^{2}+144 x \ln \left (x \right )+24 \ln \left (x \right )^{2}}\) | \(48\) |
int((((-x^2-2*x)*exp(x)+3*x^2)*ln(x)+(x^5-4*x^4-3*x^3+2*x)*exp(x)+3*x^5+3* x^3-2*x^2)/(4*ln(x)^3+(-12*x^3+36*x)*ln(x)^2+(12*x^6-72*x^4+108*x^2)*ln(x) -4*x^9+36*x^7-108*x^5+108*x^3),x,method=_RETURNVERBOSE)
Time = 0.24 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.38 \[ \int \frac {-2 x^2+3 x^3+3 x^5+e^x \left (2 x-3 x^3-4 x^4+x^5\right )+\left (3 x^2+e^x \left (-2 x-x^2\right )\right ) \log (x)}{108 x^3-108 x^5+36 x^7-4 x^9+\left (108 x^2-72 x^4+12 x^6\right ) \log (x)+\left (36 x-12 x^3\right ) \log ^2(x)+4 \log ^3(x)} \, dx=\frac {x^{3} - x^{2} e^{x}}{4 \, {\left (x^{6} - 6 \, x^{4} + 9 \, x^{2} - 2 \, {\left (x^{3} - 3 \, x\right )} \log \left (x\right ) + \log \left (x\right )^{2}\right )}} \]
integrate((((-x^2-2*x)*exp(x)+3*x^2)*log(x)+(x^5-4*x^4-3*x^3+2*x)*exp(x)+3 *x^5+3*x^3-2*x^2)/(4*log(x)^3+(-12*x^3+36*x)*log(x)^2+(12*x^6-72*x^4+108*x ^2)*log(x)-4*x^9+36*x^7-108*x^5+108*x^3),x, algorithm=\
Leaf count of result is larger than twice the leaf count of optimal. 80 vs. \(2 (24) = 48\).
Time = 0.17 (sec) , antiderivative size = 80, normalized size of antiderivative = 2.50 \[ \int \frac {-2 x^2+3 x^3+3 x^5+e^x \left (2 x-3 x^3-4 x^4+x^5\right )+\left (3 x^2+e^x \left (-2 x-x^2\right )\right ) \log (x)}{108 x^3-108 x^5+36 x^7-4 x^9+\left (108 x^2-72 x^4+12 x^6\right ) \log (x)+\left (36 x-12 x^3\right ) \log ^2(x)+4 \log ^3(x)} \, dx=\frac {x^{3}}{4 x^{6} - 24 x^{4} + 36 x^{2} + \left (- 8 x^{3} + 24 x\right ) \log {\left (x \right )} + 4 \log {\left (x \right )}^{2}} - \frac {x^{2} e^{x}}{4 x^{6} - 24 x^{4} - 8 x^{3} \log {\left (x \right )} + 36 x^{2} + 24 x \log {\left (x \right )} + 4 \log {\left (x \right )}^{2}} \]
integrate((((-x**2-2*x)*exp(x)+3*x**2)*ln(x)+(x**5-4*x**4-3*x**3+2*x)*exp( x)+3*x**5+3*x**3-2*x**2)/(4*ln(x)**3+(-12*x**3+36*x)*ln(x)**2+(12*x**6-72* x**4+108*x**2)*ln(x)-4*x**9+36*x**7-108*x**5+108*x**3),x)
x**3/(4*x**6 - 24*x**4 + 36*x**2 + (-8*x**3 + 24*x)*log(x) + 4*log(x)**2) - x**2*exp(x)/(4*x**6 - 24*x**4 - 8*x**3*log(x) + 36*x**2 + 24*x*log(x) + 4*log(x)**2)
Time = 0.23 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.38 \[ \int \frac {-2 x^2+3 x^3+3 x^5+e^x \left (2 x-3 x^3-4 x^4+x^5\right )+\left (3 x^2+e^x \left (-2 x-x^2\right )\right ) \log (x)}{108 x^3-108 x^5+36 x^7-4 x^9+\left (108 x^2-72 x^4+12 x^6\right ) \log (x)+\left (36 x-12 x^3\right ) \log ^2(x)+4 \log ^3(x)} \, dx=\frac {x^{3} - x^{2} e^{x}}{4 \, {\left (x^{6} - 6 \, x^{4} + 9 \, x^{2} - 2 \, {\left (x^{3} - 3 \, x\right )} \log \left (x\right ) + \log \left (x\right )^{2}\right )}} \]
integrate((((-x^2-2*x)*exp(x)+3*x^2)*log(x)+(x^5-4*x^4-3*x^3+2*x)*exp(x)+3 *x^5+3*x^3-2*x^2)/(4*log(x)^3+(-12*x^3+36*x)*log(x)^2+(12*x^6-72*x^4+108*x ^2)*log(x)-4*x^9+36*x^7-108*x^5+108*x^3),x, algorithm=\
Time = 0.28 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.41 \[ \int \frac {-2 x^2+3 x^3+3 x^5+e^x \left (2 x-3 x^3-4 x^4+x^5\right )+\left (3 x^2+e^x \left (-2 x-x^2\right )\right ) \log (x)}{108 x^3-108 x^5+36 x^7-4 x^9+\left (108 x^2-72 x^4+12 x^6\right ) \log (x)+\left (36 x-12 x^3\right ) \log ^2(x)+4 \log ^3(x)} \, dx=\frac {x^{3} - x^{2} e^{x}}{4 \, {\left (x^{6} - 6 \, x^{4} - 2 \, x^{3} \log \left (x\right ) + 9 \, x^{2} + 6 \, x \log \left (x\right ) + \log \left (x\right )^{2}\right )}} \]
integrate((((-x^2-2*x)*exp(x)+3*x^2)*log(x)+(x^5-4*x^4-3*x^3+2*x)*exp(x)+3 *x^5+3*x^3-2*x^2)/(4*log(x)^3+(-12*x^3+36*x)*log(x)^2+(12*x^6-72*x^4+108*x ^2)*log(x)-4*x^9+36*x^7-108*x^5+108*x^3),x, algorithm=\
Timed out. \[ \int \frac {-2 x^2+3 x^3+3 x^5+e^x \left (2 x-3 x^3-4 x^4+x^5\right )+\left (3 x^2+e^x \left (-2 x-x^2\right )\right ) \log (x)}{108 x^3-108 x^5+36 x^7-4 x^9+\left (108 x^2-72 x^4+12 x^6\right ) \log (x)+\left (36 x-12 x^3\right ) \log ^2(x)+4 \log ^3(x)} \, dx=\int \frac {{\mathrm {e}}^x\,\left (x^5-4\,x^4-3\,x^3+2\,x\right )-\ln \left (x\right )\,\left ({\mathrm {e}}^x\,\left (x^2+2\,x\right )-3\,x^2\right )-2\,x^2+3\,x^3+3\,x^5}{{\ln \left (x\right )}^2\,\left (36\,x-12\,x^3\right )+4\,{\ln \left (x\right )}^3+\ln \left (x\right )\,\left (12\,x^6-72\,x^4+108\,x^2\right )+108\,x^3-108\,x^5+36\,x^7-4\,x^9} \,d x \]
int((exp(x)*(2*x - 3*x^3 - 4*x^4 + x^5) - log(x)*(exp(x)*(2*x + x^2) - 3*x ^2) - 2*x^2 + 3*x^3 + 3*x^5)/(log(x)^2*(36*x - 12*x^3) + 4*log(x)^3 + log( x)*(108*x^2 - 72*x^4 + 12*x^6) + 108*x^3 - 108*x^5 + 36*x^7 - 4*x^9),x)