Integrand size = 144, antiderivative size = 24 \[ \int \frac {-2 x^2+\left (-2-4 x+e^x \left (-1-x+2 x^2\right )\right ) \log (-3-6 x)}{\left (4-24 x+128 x^3+e^{2 x} (1+2 x)+e^x \left (4-8 x-32 x^2\right )\right ) \log (-3-6 x)+\left (-4 x+8 x^2+32 x^3+e^x \left (-2 x-4 x^2\right )\right ) \log (-3-6 x) \log (\log (-3-6 x))+\left (x^2+2 x^3\right ) \log (-3-6 x) \log ^2(\log (-3-6 x))} \, dx=\frac {x}{-2-e^x+x (8+\log (\log (-3 (1+2 x))))} \]
Time = 0.73 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.96 \[ \int \frac {-2 x^2+\left (-2-4 x+e^x \left (-1-x+2 x^2\right )\right ) \log (-3-6 x)}{\left (4-24 x+128 x^3+e^{2 x} (1+2 x)+e^x \left (4-8 x-32 x^2\right )\right ) \log (-3-6 x)+\left (-4 x+8 x^2+32 x^3+e^x \left (-2 x-4 x^2\right )\right ) \log (-3-6 x) \log (\log (-3-6 x))+\left (x^2+2 x^3\right ) \log (-3-6 x) \log ^2(\log (-3-6 x))} \, dx=\frac {x}{-2-e^x+8 x+x \log (\log (-3-6 x))} \]
Integrate[(-2*x^2 + (-2 - 4*x + E^x*(-1 - x + 2*x^2))*Log[-3 - 6*x])/((4 - 24*x + 128*x^3 + E^(2*x)*(1 + 2*x) + E^x*(4 - 8*x - 32*x^2))*Log[-3 - 6*x ] + (-4*x + 8*x^2 + 32*x^3 + E^x*(-2*x - 4*x^2))*Log[-3 - 6*x]*Log[Log[-3 - 6*x]] + (x^2 + 2*x^3)*Log[-3 - 6*x]*Log[Log[-3 - 6*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.
\(\displaystyle \int \frac {\left (e^x \left (2 x^2-x-1\right )-4 x-2\right ) \log (-6 x-3)-2 x^2}{\left (2 x^3+x^2\right ) \log (-6 x-3) \log ^2(\log (-6 x-3))+\left (32 x^3+8 x^2+e^x \left (-4 x^2-2 x\right )-4 x\right ) \log (-6 x-3) \log (\log (-6 x-3))+\left (128 x^3+e^x \left (-32 x^2-8 x+4\right )-24 x+e^{2 x} (2 x+1)+4\right ) \log (-6 x-3)} \, dx\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle \int \frac {\left (e^x (x-1)-2\right ) (2 x+1) \log (-6 x-3)-2 x^2}{(2 x+1) \log (-6 x-3) \left (-8 x+e^x+x (-\log (\log (-6 x-3)))+2\right )^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {x \left (16 x^2 \log (-6 x-3)+2 x^2 \log (-6 x-3) \log (\log (-6 x-3))-2 x-12 x \log (-6 x-3)-x \log (-6 x-3) \log (\log (-6 x-3))-10 \log (-6 x-3)-\log (-6 x-3) \log (\log (-6 x-3))\right )}{(2 x+1) \log (-6 x-3) \left (8 x-e^x+x \log (\log (-6 x-3))-2\right )^2}-\frac {x-1}{8 x-e^x+x \log (\log (-6 x-3))-2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 8 \int \frac {x^2}{\left (\log (\log (-6 x-3)) x+8 x-e^x-2\right )^2}dx+\int \frac {x^2 \log (\log (-6 x-3))}{\left (\log (\log (-6 x-3)) x+8 x-e^x-2\right )^2}dx+\frac {1}{2} \int \frac {1}{\log (-6 x-3) \left (-\log (\log (-6 x-3)) x-8 x+e^x+2\right )^2}dx-\int \frac {1}{-\log (\log (-6 x-3)) x-8 x+e^x+2}dx-10 \int \frac {x}{\left (\log (\log (-6 x-3)) x+8 x-e^x-2\right )^2}dx-\int \frac {x}{\log (-6 x-3) \left (\log (\log (-6 x-3)) x+8 x-e^x-2\right )^2}dx-\frac {1}{2} \int \frac {1}{(2 x+1) \log (-6 x-3) \left (\log (\log (-6 x-3)) x+8 x-e^x-2\right )^2}dx-\int \frac {x \log (\log (-6 x-3))}{\left (\log (\log (-6 x-3)) x+8 x-e^x-2\right )^2}dx-\int \frac {x}{\log (\log (-6 x-3)) x+8 x-e^x-2}dx\) |
Int[(-2*x^2 + (-2 - 4*x + E^x*(-1 - x + 2*x^2))*Log[-3 - 6*x])/((4 - 24*x + 128*x^3 + E^(2*x)*(1 + 2*x) + E^x*(4 - 8*x - 32*x^2))*Log[-3 - 6*x] + (- 4*x + 8*x^2 + 32*x^3 + E^x*(-2*x - 4*x^2))*Log[-3 - 6*x]*Log[Log[-3 - 6*x] ] + (x^2 + 2*x^3)*Log[-3 - 6*x]*Log[Log[-3 - 6*x]]^2),x]
3.8.63.3.1 Defintions of rubi rules used
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Time = 5.01 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.96
method | result | size |
risch | \(\frac {x}{x \ln \left (\ln \left (-6 x -3\right )\right )+8 x -{\mathrm e}^{x}-2}\) | \(23\) |
parallelrisch | \(\frac {x}{x \ln \left (\ln \left (-6 x -3\right )\right )+8 x -{\mathrm e}^{x}-2}\) | \(23\) |
int((((2*x^2-x-1)*exp(x)-4*x-2)*ln(-6*x-3)-2*x^2)/((2*x^3+x^2)*ln(-6*x-3)* ln(ln(-6*x-3))^2+((-4*x^2-2*x)*exp(x)+32*x^3+8*x^2-4*x)*ln(-6*x-3)*ln(ln(- 6*x-3))+((1+2*x)*exp(x)^2+(-32*x^2-8*x+4)*exp(x)+128*x^3-24*x+4)*ln(-6*x-3 )),x,method=_RETURNVERBOSE)
Time = 0.25 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92 \[ \int \frac {-2 x^2+\left (-2-4 x+e^x \left (-1-x+2 x^2\right )\right ) \log (-3-6 x)}{\left (4-24 x+128 x^3+e^{2 x} (1+2 x)+e^x \left (4-8 x-32 x^2\right )\right ) \log (-3-6 x)+\left (-4 x+8 x^2+32 x^3+e^x \left (-2 x-4 x^2\right )\right ) \log (-3-6 x) \log (\log (-3-6 x))+\left (x^2+2 x^3\right ) \log (-3-6 x) \log ^2(\log (-3-6 x))} \, dx=\frac {x}{x \log \left (\log \left (-6 \, x - 3\right )\right ) + 8 \, x - e^{x} - 2} \]
integrate((((2*x^2-x-1)*exp(x)-4*x-2)*log(-6*x-3)-2*x^2)/((2*x^3+x^2)*log( -6*x-3)*log(log(-6*x-3))^2+((-4*x^2-2*x)*exp(x)+32*x^3+8*x^2-4*x)*log(-6*x -3)*log(log(-6*x-3))+((1+2*x)*exp(x)^2+(-32*x^2-8*x+4)*exp(x)+128*x^3-24*x +4)*log(-6*x-3)),x, algorithm=\
Time = 0.14 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92 \[ \int \frac {-2 x^2+\left (-2-4 x+e^x \left (-1-x+2 x^2\right )\right ) \log (-3-6 x)}{\left (4-24 x+128 x^3+e^{2 x} (1+2 x)+e^x \left (4-8 x-32 x^2\right )\right ) \log (-3-6 x)+\left (-4 x+8 x^2+32 x^3+e^x \left (-2 x-4 x^2\right )\right ) \log (-3-6 x) \log (\log (-3-6 x))+\left (x^2+2 x^3\right ) \log (-3-6 x) \log ^2(\log (-3-6 x))} \, dx=- \frac {x}{- x \log {\left (\log {\left (- 6 x - 3 \right )} \right )} - 8 x + e^{x} + 2} \]
integrate((((2*x**2-x-1)*exp(x)-4*x-2)*ln(-6*x-3)-2*x**2)/((2*x**3+x**2)*l n(-6*x-3)*ln(ln(-6*x-3))**2+((-4*x**2-2*x)*exp(x)+32*x**3+8*x**2-4*x)*ln(- 6*x-3)*ln(ln(-6*x-3))+((1+2*x)*exp(x)**2+(-32*x**2-8*x+4)*exp(x)+128*x**3- 24*x+4)*ln(-6*x-3)),x)
Result contains complex when optimal does not.
Time = 0.43 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.17 \[ \int \frac {-2 x^2+\left (-2-4 x+e^x \left (-1-x+2 x^2\right )\right ) \log (-3-6 x)}{\left (4-24 x+128 x^3+e^{2 x} (1+2 x)+e^x \left (4-8 x-32 x^2\right )\right ) \log (-3-6 x)+\left (-4 x+8 x^2+32 x^3+e^x \left (-2 x-4 x^2\right )\right ) \log (-3-6 x) \log (\log (-3-6 x))+\left (x^2+2 x^3\right ) \log (-3-6 x) \log ^2(\log (-3-6 x))} \, dx=\frac {x}{x \log \left (i \, \pi + \log \left (3\right ) + \log \left (2 \, x + 1\right )\right ) + 8 \, x - e^{x} - 2} \]
integrate((((2*x^2-x-1)*exp(x)-4*x-2)*log(-6*x-3)-2*x^2)/((2*x^3+x^2)*log( -6*x-3)*log(log(-6*x-3))^2+((-4*x^2-2*x)*exp(x)+32*x^3+8*x^2-4*x)*log(-6*x -3)*log(log(-6*x-3))+((1+2*x)*exp(x)^2+(-32*x^2-8*x+4)*exp(x)+128*x^3-24*x +4)*log(-6*x-3)),x, algorithm=\
Time = 0.39 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92 \[ \int \frac {-2 x^2+\left (-2-4 x+e^x \left (-1-x+2 x^2\right )\right ) \log (-3-6 x)}{\left (4-24 x+128 x^3+e^{2 x} (1+2 x)+e^x \left (4-8 x-32 x^2\right )\right ) \log (-3-6 x)+\left (-4 x+8 x^2+32 x^3+e^x \left (-2 x-4 x^2\right )\right ) \log (-3-6 x) \log (\log (-3-6 x))+\left (x^2+2 x^3\right ) \log (-3-6 x) \log ^2(\log (-3-6 x))} \, dx=\frac {x}{x \log \left (\log \left (-6 \, x - 3\right )\right ) + 8 \, x - e^{x} - 2} \]
integrate((((2*x^2-x-1)*exp(x)-4*x-2)*log(-6*x-3)-2*x^2)/((2*x^3+x^2)*log( -6*x-3)*log(log(-6*x-3))^2+((-4*x^2-2*x)*exp(x)+32*x^3+8*x^2-4*x)*log(-6*x -3)*log(log(-6*x-3))+((1+2*x)*exp(x)^2+(-32*x^2-8*x+4)*exp(x)+128*x^3-24*x +4)*log(-6*x-3)),x, algorithm=\
Time = 8.51 (sec) , antiderivative size = 226, normalized size of antiderivative = 9.42 \[ \int \frac {-2 x^2+\left (-2-4 x+e^x \left (-1-x+2 x^2\right )\right ) \log (-3-6 x)}{\left (4-24 x+128 x^3+e^{2 x} (1+2 x)+e^x \left (4-8 x-32 x^2\right )\right ) \log (-3-6 x)+\left (-4 x+8 x^2+32 x^3+e^x \left (-2 x-4 x^2\right )\right ) \log (-3-6 x) \log (\log (-3-6 x))+\left (x^2+2 x^3\right ) \log (-3-6 x) \log ^2(\log (-3-6 x))} \, dx=\frac {x\,{\left (\ln \left (-6\,x-3\right )+2\,x\,\ln \left (-6\,x-3\right )\right )}^2\,\left (2\,\ln \left (-6\,x-3\right )+4\,x\,\ln \left (-6\,x-3\right )+2\,x^2\right )+x\,{\mathrm {e}}^x\,{\left (\ln \left (-6\,x-3\right )+2\,x\,\ln \left (-6\,x-3\right )\right )}^2\,\left (\ln \left (-6\,x-3\right )+x\,\ln \left (-6\,x-3\right )-2\,x^2\,\ln \left (-6\,x-3\right )\right )}{\ln \left (-6\,x-3\right )\,\left (2\,x+1\right )\,\left (8\,x-{\mathrm {e}}^x+x\,\ln \left (\ln \left (-6\,x-3\right )\right )-2\right )\,\left (2\,{\ln \left (-6\,x-3\right )}^2+8\,x^2\,{\ln \left (-6\,x-3\right )}^2+{\mathrm {e}}^x\,{\ln \left (-6\,x-3\right )}^2+8\,x\,{\ln \left (-6\,x-3\right )}^2+2\,x^2\,\ln \left (-6\,x-3\right )+4\,x^3\,\ln \left (-6\,x-3\right )-4\,x^3\,{\mathrm {e}}^x\,{\ln \left (-6\,x-3\right )}^2+3\,x\,{\mathrm {e}}^x\,{\ln \left (-6\,x-3\right )}^2\right )} \]
int(-(log(- 6*x - 3)*(4*x + exp(x)*(x - 2*x^2 + 1) + 2) + 2*x^2)/(log(- 6* x - 3)*(exp(2*x)*(2*x + 1) - exp(x)*(8*x + 32*x^2 - 4) - 24*x + 128*x^3 + 4) - log(log(- 6*x - 3))*log(- 6*x - 3)*(4*x + exp(x)*(2*x + 4*x^2) - 8*x^ 2 - 32*x^3) + log(log(- 6*x - 3))^2*log(- 6*x - 3)*(x^2 + 2*x^3)),x)
(x*(log(- 6*x - 3) + 2*x*log(- 6*x - 3))^2*(2*log(- 6*x - 3) + 4*x*log(- 6 *x - 3) + 2*x^2) + x*exp(x)*(log(- 6*x - 3) + 2*x*log(- 6*x - 3))^2*(log(- 6*x - 3) + x*log(- 6*x - 3) - 2*x^2*log(- 6*x - 3)))/(log(- 6*x - 3)*(2*x + 1)*(8*x - exp(x) + x*log(log(- 6*x - 3)) - 2)*(2*log(- 6*x - 3)^2 + 8*x ^2*log(- 6*x - 3)^2 + exp(x)*log(- 6*x - 3)^2 + 8*x*log(- 6*x - 3)^2 + 2*x ^2*log(- 6*x - 3) + 4*x^3*log(- 6*x - 3) - 4*x^3*exp(x)*log(- 6*x - 3)^2 + 3*x*exp(x)*log(- 6*x - 3)^2))