Integrand size = 119, antiderivative size = 25 \[ \int \frac {e^{4/3} \left (-x+3 x^2-2 x^3+x^4\right )+e^{4/3} \left (-2 x+4 x^2-2 x^3\right ) \log (x)+e^{4/3} \left (-2 x+x^2\right ) \log ^2(x)}{5-10 x+15 x^2-10 x^3+5 x^4+\left (10-20 x+20 x^2-10 x^3\right ) \log (x)+\left (5-10 x+5 x^2\right ) \log ^2(x)} \, dx=\frac {e^{4/3} x^2}{5 \left (-1+x+\frac {1}{x-\log (x)}\right )} \]
Time = 0.27 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.72 \[ \int \frac {e^{4/3} \left (-x+3 x^2-2 x^3+x^4\right )+e^{4/3} \left (-2 x+4 x^2-2 x^3\right ) \log (x)+e^{4/3} \left (-2 x+x^2\right ) \log ^2(x)}{5-10 x+15 x^2-10 x^3+5 x^4+\left (10-20 x+20 x^2-10 x^3\right ) \log (x)+\left (5-10 x+5 x^2\right ) \log ^2(x)} \, dx=\frac {1}{5} e^{4/3} \left (\frac {1}{-1+x}+x+\frac {x^2}{(-1+x) \left (-1+x-x^2-\log (x)+x \log (x)\right )}\right ) \]
Integrate[(E^(4/3)*(-x + 3*x^2 - 2*x^3 + x^4) + E^(4/3)*(-2*x + 4*x^2 - 2* x^3)*Log[x] + E^(4/3)*(-2*x + x^2)*Log[x]^2)/(5 - 10*x + 15*x^2 - 10*x^3 + 5*x^4 + (10 - 20*x + 20*x^2 - 10*x^3)*Log[x] + (5 - 10*x + 5*x^2)*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.
\(\displaystyle \int \frac {e^{4/3} \left (x^2-2 x\right ) \log ^2(x)+e^{4/3} \left (-2 x^3+4 x^2-2 x\right ) \log (x)+e^{4/3} \left (x^4-2 x^3+3 x^2-x\right )}{5 x^4-10 x^3+15 x^2+\left (5 x^2-10 x+5\right ) \log ^2(x)+\left (-10 x^3+20 x^2-20 x+10\right ) \log (x)-10 x+5} \, dx\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle \int \frac {e^{4/3} x \left (x^3-2 x^2+3 x+(x-2) \log ^2(x)-2 (x-1)^2 \log (x)-1\right )}{5 \left (x^2-x-x \log (x)+\log (x)+1\right )^2}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{5} e^{4/3} \int -\frac {x \left (-x^3+2 x^2-3 x+(2-x) \log ^2(x)+2 (1-x)^2 \log (x)+1\right )}{\left (x^2-\log (x) x-x+\log (x)+1\right )^2}dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {1}{5} e^{4/3} \int \frac {x \left (-x^3+2 x^2-3 x+(2-x) \log ^2(x)+2 (1-x)^2 \log (x)+1\right )}{\left (x^2-\log (x) x-x+\log (x)+1\right )^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle -\frac {1}{5} e^{4/3} \int \left (-\frac {2 x}{(x-1)^2 \left (x^2-\log (x) x-x+\log (x)+1\right )}-\frac {(x-2) x}{(x-1)^2}-\frac {\left (x^3-3 x^2+2 x-1\right ) x}{(x-1)^2 \left (x^2-\log (x) x-x+\log (x)+1\right )^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {1}{5} e^{4/3} \left (\int \frac {1}{\left (x^2-\log (x) x-x+\log (x)+1\right )^2}dx+\int \frac {1}{(x-1)^2 \left (x^2-\log (x) x-x+\log (x)+1\right )^2}dx+2 \int \frac {1}{(x-1) \left (x^2-\log (x) x-x+\log (x)+1\right )^2}dx+\int \frac {x}{\left (x^2-\log (x) x-x+\log (x)+1\right )^2}dx-\int \frac {x^2}{\left (x^2-\log (x) x-x+\log (x)+1\right )^2}dx-2 \int \frac {1}{(x-1)^2 \left (x^2-\log (x) x-x+\log (x)+1\right )}dx-2 \int \frac {1}{(x-1) \left (x^2-\log (x) x-x+\log (x)+1\right )}dx+\frac {x^2}{1-x}\right )\) |
Int[(E^(4/3)*(-x + 3*x^2 - 2*x^3 + x^4) + E^(4/3)*(-2*x + 4*x^2 - 2*x^3)*L og[x] + E^(4/3)*(-2*x + x^2)*Log[x]^2)/(5 - 10*x + 15*x^2 - 10*x^3 + 5*x^4 + (10 - 20*x + 20*x^2 - 10*x^3)*Log[x] + (5 - 10*x + 5*x^2)*Log[x]^2),x]
3.13.4.3.1 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 = 0.33 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.48
method | result | size |
default | \(\frac {{\mathrm e}^{\frac {4}{3}} \left (x^{2} \ln \left (x \right )-x^{3}\right )}{5 x \ln \left (x \right )-5 x^{2}-5 \ln \left (x \right )+5 x -5}\) | \(37\) |
norman | \(\frac {\frac {{\mathrm e}^{\frac {4}{3}} x^{3}}{5}-\frac {{\mathrm e}^{\frac {4}{3}} x^{2} \ln \left (x \right )}{5}}{x^{2}-x \ln \left (x \right )-x +\ln \left (x \right )+1}\) | \(40\) |
parallelrisch | \(\frac {{\mathrm e}^{\frac {4}{3}} x^{3}-{\mathrm e}^{\frac {4}{3}} x^{2} \ln \left (x \right )}{5 x^{2}-5 x \ln \left (x \right )-5 x +5 \ln \left (x \right )+5}\) | \(40\) |
risch | \(\frac {{\mathrm e}^{\frac {4}{3}} \left (x^{2}-x +1\right )}{5 x -5}-\frac {{\mathrm e}^{\frac {4}{3}} x^{2}}{5 \left (-1+x \right ) \left (x^{2}-x \ln \left (x \right )-x +\ln \left (x \right )+1\right )}\) | \(48\) |
int(((x^2-2*x)*exp(2/3)^2*ln(x)^2+(-2*x^3+4*x^2-2*x)*exp(2/3)^2*ln(x)+(x^4 -2*x^3+3*x^2-x)*exp(2/3)^2)/((5*x^2-10*x+5)*ln(x)^2+(-10*x^3+20*x^2-20*x+1 0)*ln(x)+5*x^4-10*x^3+15*x^2-10*x+5),x,method=_RETURNVERBOSE)
Leaf count of result is larger than twice the leaf count of optimal. 48 vs. \(2 (20) = 40\).
Time = 0.26 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.92 \[ \int \frac {e^{4/3} \left (-x+3 x^2-2 x^3+x^4\right )+e^{4/3} \left (-2 x+4 x^2-2 x^3\right ) \log (x)+e^{4/3} \left (-2 x+x^2\right ) \log ^2(x)}{5-10 x+15 x^2-10 x^3+5 x^4+\left (10-20 x+20 x^2-10 x^3\right ) \log (x)+\left (5-10 x+5 x^2\right ) \log ^2(x)} \, dx=-\frac {{\left (x^{2} - x + 1\right )} e^{\frac {4}{3}} \log \left (x\right ) - {\left (x^{3} - x^{2} + x - 1\right )} e^{\frac {4}{3}}}{5 \, {\left (x^{2} - {\left (x - 1\right )} \log \left (x\right ) - x + 1\right )}} \]
integrate(((x^2-2*x)*exp(2/3)^2*log(x)^2+(-2*x^3+4*x^2-2*x)*exp(2/3)^2*log (x)+(x^4-2*x^3+3*x^2-x)*exp(2/3)^2)/((5*x^2-10*x+5)*log(x)^2+(-10*x^3+20*x ^2-20*x+10)*log(x)+5*x^4-10*x^3+15*x^2-10*x+5),x, algorithm=\
-1/5*((x^2 - x + 1)*e^(4/3)*log(x) - (x^3 - x^2 + x - 1)*e^(4/3))/(x^2 - ( x - 1)*log(x) - x + 1)
Leaf count of result is larger than twice the leaf count of optimal. 54 vs. \(2 (19) = 38\).
Time = 0.16 (sec) , antiderivative size = 54, normalized size of antiderivative = 2.16 \[ \int \frac {e^{4/3} \left (-x+3 x^2-2 x^3+x^4\right )+e^{4/3} \left (-2 x+4 x^2-2 x^3\right ) \log (x)+e^{4/3} \left (-2 x+x^2\right ) \log ^2(x)}{5-10 x+15 x^2-10 x^3+5 x^4+\left (10-20 x+20 x^2-10 x^3\right ) \log (x)+\left (5-10 x+5 x^2\right ) \log ^2(x)} \, dx=\frac {x^{2} e^{\frac {4}{3}}}{- 5 x^{3} + 10 x^{2} - 10 x + \left (5 x^{2} - 10 x + 5\right ) \log {\left (x \right )} + 5} + \frac {x e^{\frac {4}{3}}}{5} + \frac {e^{\frac {4}{3}}}{5 x - 5} \]
integrate(((x**2-2*x)*exp(2/3)**2*ln(x)**2+(-2*x**3+4*x**2-2*x)*exp(2/3)** 2*ln(x)+(x**4-2*x**3+3*x**2-x)*exp(2/3)**2)/((5*x**2-10*x+5)*ln(x)**2+(-10 *x**3+20*x**2-20*x+10)*ln(x)+5*x**4-10*x**3+15*x**2-10*x+5),x)
x**2*exp(4/3)/(-5*x**3 + 10*x**2 - 10*x + (5*x**2 - 10*x + 5)*log(x) + 5) + x*exp(4/3)/5 + exp(4/3)/(5*x - 5)
Leaf count of result is larger than twice the leaf count of optimal. 59 vs. \(2 (20) = 40\).
Time = 0.23 (sec) , antiderivative size = 59, normalized size of antiderivative = 2.36 \[ \int \frac {e^{4/3} \left (-x+3 x^2-2 x^3+x^4\right )+e^{4/3} \left (-2 x+4 x^2-2 x^3\right ) \log (x)+e^{4/3} \left (-2 x+x^2\right ) \log ^2(x)}{5-10 x+15 x^2-10 x^3+5 x^4+\left (10-20 x+20 x^2-10 x^3\right ) \log (x)+\left (5-10 x+5 x^2\right ) \log ^2(x)} \, dx=\frac {x^{3} e^{\frac {4}{3}} - x^{2} e^{\frac {4}{3}} + x e^{\frac {4}{3}} - {\left (x^{2} e^{\frac {4}{3}} - x e^{\frac {4}{3}} + e^{\frac {4}{3}}\right )} \log \left (x\right ) - e^{\frac {4}{3}}}{5 \, {\left (x^{2} - {\left (x - 1\right )} \log \left (x\right ) - x + 1\right )}} \]
integrate(((x^2-2*x)*exp(2/3)^2*log(x)^2+(-2*x^3+4*x^2-2*x)*exp(2/3)^2*log (x)+(x^4-2*x^3+3*x^2-x)*exp(2/3)^2)/((5*x^2-10*x+5)*log(x)^2+(-10*x^3+20*x ^2-20*x+10)*log(x)+5*x^4-10*x^3+15*x^2-10*x+5),x, algorithm=\
1/5*(x^3*e^(4/3) - x^2*e^(4/3) + x*e^(4/3) - (x^2*e^(4/3) - x*e^(4/3) + e^ (4/3))*log(x) - e^(4/3))/(x^2 - (x - 1)*log(x) - x + 1)
Leaf count of result is larger than twice the leaf count of optimal. 62 vs. \(2 (20) = 40\).
Time = 0.26 (sec) , antiderivative size = 62, normalized size of antiderivative = 2.48 \[ \int \frac {e^{4/3} \left (-x+3 x^2-2 x^3+x^4\right )+e^{4/3} \left (-2 x+4 x^2-2 x^3\right ) \log (x)+e^{4/3} \left (-2 x+x^2\right ) \log ^2(x)}{5-10 x+15 x^2-10 x^3+5 x^4+\left (10-20 x+20 x^2-10 x^3\right ) \log (x)+\left (5-10 x+5 x^2\right ) \log ^2(x)} \, dx=\frac {x^{3} e^{\frac {4}{3}} - x^{2} e^{\frac {4}{3}} \log \left (x\right ) - x^{2} e^{\frac {4}{3}} + x e^{\frac {4}{3}} \log \left (x\right ) + x e^{\frac {4}{3}} - e^{\frac {4}{3}} \log \left (x\right ) - e^{\frac {4}{3}}}{5 \, {\left (x^{2} - x \log \left (x\right ) - x + \log \left (x\right ) + 1\right )}} \]
integrate(((x^2-2*x)*exp(2/3)^2*log(x)^2+(-2*x^3+4*x^2-2*x)*exp(2/3)^2*log (x)+(x^4-2*x^3+3*x^2-x)*exp(2/3)^2)/((5*x^2-10*x+5)*log(x)^2+(-10*x^3+20*x ^2-20*x+10)*log(x)+5*x^4-10*x^3+15*x^2-10*x+5),x, algorithm=\
1/5*(x^3*e^(4/3) - x^2*e^(4/3)*log(x) - x^2*e^(4/3) + x*e^(4/3)*log(x) + x *e^(4/3) - e^(4/3)*log(x) - e^(4/3))/(x^2 - x*log(x) - x + log(x) + 1)
Time = 8.63 (sec) , antiderivative size = 51, normalized size of antiderivative = 2.04 \[ \int \frac {e^{4/3} \left (-x+3 x^2-2 x^3+x^4\right )+e^{4/3} \left (-2 x+4 x^2-2 x^3\right ) \log (x)+e^{4/3} \left (-2 x+x^2\right ) \log ^2(x)}{5-10 x+15 x^2-10 x^3+5 x^4+\left (10-20 x+20 x^2-10 x^3\right ) \log (x)+\left (5-10 x+5 x^2\right ) \log ^2(x)} \, dx=-\frac {{\mathrm {e}}^{4/3}\,\left (\ln \left (x\right )-x+x^2\,\ln \left (x\right )-x\,\ln \left (x\right )+x^2-x^3+1\right )}{5\,\left (\ln \left (x\right )-x-x\,\ln \left (x\right )+x^2+1\right )} \]
int(-(exp(4/3)*(x - 3*x^2 + 2*x^3 - x^4) + exp(4/3)*log(x)*(2*x - 4*x^2 + 2*x^3) + exp(4/3)*log(x)^2*(2*x - x^2))/(log(x)^2*(5*x^2 - 10*x + 5) - 10* x + 15*x^2 - 10*x^3 + 5*x^4 - log(x)*(20*x - 20*x^2 + 10*x^3 - 10) + 5),x)