Integrand size = 164, antiderivative size = 34 \[ \int \frac {-3 x^2+6 x^3+e^x \left (2 x^2+6 x^3-2 x^4\right )+\left (-6 x^2+e^x \left (-8 x-4 x^2+4 x^3\right )\right ) \log \left (\frac {1}{3} e^{-2 x} \left (-3 x+e^x \left (-4-2 x+2 x^2\right )\right )\right ) \log \left (\log \left (\frac {1}{3} e^{-2 x} \left (-3 x+e^x \left (-4-2 x+2 x^2\right )\right )\right )\right )}{\left (-3 x+e^x \left (-4-2 x+2 x^2\right )\right ) \log \left (\frac {1}{3} e^{-2 x} \left (-3 x+e^x \left (-4-2 x+2 x^2\right )\right )\right )} \, dx=x^2 \log \left (\log \left (\frac {1}{3} e^{-x} \left (-4-\left (2+3 e^{-x}\right ) x+2 x^2\right )\right )\right ) \]
Time = 0.12 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.94 \[ \int \frac {-3 x^2+6 x^3+e^x \left (2 x^2+6 x^3-2 x^4\right )+\left (-6 x^2+e^x \left (-8 x-4 x^2+4 x^3\right )\right ) \log \left (\frac {1}{3} e^{-2 x} \left (-3 x+e^x \left (-4-2 x+2 x^2\right )\right )\right ) \log \left (\log \left (\frac {1}{3} e^{-2 x} \left (-3 x+e^x \left (-4-2 x+2 x^2\right )\right )\right )\right )}{\left (-3 x+e^x \left (-4-2 x+2 x^2\right )\right ) \log \left (\frac {1}{3} e^{-2 x} \left (-3 x+e^x \left (-4-2 x+2 x^2\right )\right )\right )} \, dx=x^2 \log \left (\log \left (\frac {1}{3} e^{-2 x} \left (-3 x+2 e^x \left (-2-x+x^2\right )\right )\right )\right ) \]
Integrate[(-3*x^2 + 6*x^3 + E^x*(2*x^2 + 6*x^3 - 2*x^4) + (-6*x^2 + E^x*(- 8*x - 4*x^2 + 4*x^3))*Log[(-3*x + E^x*(-4 - 2*x + 2*x^2))/(3*E^(2*x))]*Log [Log[(-3*x + E^x*(-4 - 2*x + 2*x^2))/(3*E^(2*x))]])/((-3*x + E^x*(-4 - 2*x + 2*x^2))*Log[(-3*x + E^x*(-4 - 2*x + 2*x^2))/(3*E^(2*x))]),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.
| \(\displaystyle \int \frac {6 x^3-3 x^2+\left (e^x \left (4 x^3-4 x^2-8 x\right )-6 x^2\right ) \log \left (\frac {1}{3} e^{-2 x} \left (e^x \left (2 x^2-2 x-4\right )-3 x\right )\right ) \log \left (\log \left (\frac {1}{3} e^{-2 x} \left (e^x \left (2 x^2-2 x-4\right )-3 x\right )\right )\right )+e^x \left (-2 x^4+6 x^3+2 x^2\right )}{\left (e^x \left (2 x^2-2 x-4\right )-3 x\right ) \log \left (\frac {1}{3} e^{-2 x} \left (e^x \left (2 x^2-2 x-4\right )-3 x\right )\right )} \, dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {3 x^2 \left (x^3-2 x+2\right )}{\left (x^2-x-2\right ) \left (2 e^x x^2-2 e^x x-3 x-4 e^x\right ) \log \left (\frac {1}{3} e^{-2 x} \left (2 e^x \left (x^2-x-2\right )-3 x\right )\right )}-\frac {x \left (x^3-3 x^2-2 x^2 \log \left (\frac {1}{3} e^{-2 x} \left (2 e^x \left (x^2-x-2\right )-3 x\right )\right ) \log \left (\log \left (\frac {1}{3} e^{-2 x} \left (2 e^x \left (x^2-x-2\right )-3 x\right )\right )\right )+2 x \log \left (\frac {1}{3} e^{-2 x} \left (2 e^x \left (x^2-x-2\right )-3 x\right )\right ) \log \left (\log \left (\frac {1}{3} e^{-2 x} \left (2 e^x \left (x^2-x-2\right )-3 x\right )\right )\right )+4 \log \left (\frac {1}{3} e^{-2 x} \left (2 e^x \left (x^2-x-2\right )-3 x\right )\right ) \log \left (\log \left (\frac {1}{3} e^{-2 x} \left (2 e^x \left (x^2-x-2\right )-3 x\right )\right )\right )-x\right )}{\left (x^2-x-2\right ) \log \left (\frac {1}{3} e^{-2 x} \left (2 e^x \left (x^2-x-2\right )-3 x\right )\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \int \frac {1}{\log \left (\frac {1}{3} e^{-2 x} \left (2 e^x \left (x^2-x-2\right )-3 x\right )\right )}dx+4 \int \frac {1}{(x-2) \log \left (\frac {1}{3} e^{-2 x} \left (2 e^x \left (x^2-x-2\right )-3 x\right )\right )}dx+2 \int \frac {x}{\log \left (\frac {1}{3} e^{-2 x} \left (2 e^x \left (x^2-x-2\right )-3 x\right )\right )}dx-\int \frac {x^2}{\log \left (\frac {1}{3} e^{-2 x} \left (2 e^x \left (x^2-x-2\right )-3 x\right )\right )}dx+\int \frac {1}{(x+1) \log \left (\frac {1}{3} e^{-2 x} \left (2 e^x \left (x^2-x-2\right )-3 x\right )\right )}dx+15 \int \frac {1}{\left (2 e^x x^2-2 e^x x-3 x-4 e^x\right ) \log \left (\frac {1}{3} e^{-2 x} \left (2 e^x \left (x^2-x-2\right )-3 x\right )\right )}dx+24 \int \frac {1}{(x-2) \left (2 e^x x^2-2 e^x x-3 x-4 e^x\right ) \log \left (\frac {1}{3} e^{-2 x} \left (2 e^x \left (x^2-x-2\right )-3 x\right )\right )}dx+3 \int \frac {x}{\left (2 e^x x^2-2 e^x x-3 x-4 e^x\right ) \log \left (\frac {1}{3} e^{-2 x} \left (2 e^x \left (x^2-x-2\right )-3 x\right )\right )}dx+3 \int \frac {x^2}{\left (2 e^x x^2-2 e^x x-3 x-4 e^x\right ) \log \left (\frac {1}{3} e^{-2 x} \left (2 e^x \left (x^2-x-2\right )-3 x\right )\right )}dx-3 \int \frac {1}{(x+1) \left (2 e^x x^2-2 e^x x-3 x-4 e^x\right ) \log \left (\frac {1}{3} e^{-2 x} \left (2 e^x \left (x^2-x-2\right )-3 x\right )\right )}dx+2 \int x \log \left (\log \left (\frac {1}{3} e^{-2 x} \left (2 e^x \left (x^2-x-2\right )-3 x\right )\right )\right )dx+3 \int \frac {x^3}{\left (2 e^x x^2-2 e^x x-3 x-4 e^x\right ) \log \left (\frac {1}{3} e^{-2 x} \left (2 e^x \left (x^2-x-2\right )-3 x\right )\right )}dx\) |
Int[(-3*x^2 + 6*x^3 + E^x*(2*x^2 + 6*x^3 - 2*x^4) + (-6*x^2 + E^x*(-8*x - 4*x^2 + 4*x^3))*Log[(-3*x + E^x*(-4 - 2*x + 2*x^2))/(3*E^(2*x))]*Log[Log[( -3*x + E^x*(-4 - 2*x + 2*x^2))/(3*E^(2*x))]])/((-3*x + E^x*(-4 - 2*x + 2*x ^2))*Log[(-3*x + E^x*(-4 - 2*x + 2*x^2))/(3*E^(2*x))]),x]
3.17.83.3.1 Defintions of rubi rules used
Time = 5.70 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.88
| method | result | size |
| parallelrisch | \(\ln \left (\ln \left (\frac {\left (\left (2 x^{2}-2 x -4\right ) {\mathrm e}^{x}-3 x \right ) {\mathrm e}^{-2 x}}{3}\right )\right ) x^{2}\) | \(30\) |
| risch | \(x^{2} \ln \left (\ln \left (2\right )-\ln \left (3\right )-2 \ln \left ({\mathrm e}^{x}\right )+\ln \left ({\mathrm e}^{x} x^{2}+\left (-{\mathrm e}^{x}-\frac {3}{2}\right ) x -2 \,{\mathrm e}^{x}\right )+\frac {i \pi \,\operatorname {csgn}\left (i {\mathrm e}^{2 x}\right ) {\left (-\operatorname {csgn}\left (i {\mathrm e}^{2 x}\right )+\operatorname {csgn}\left (i {\mathrm e}^{x}\right )\right )}^{2}}{2}-\frac {i \pi \,\operatorname {csgn}\left (i {\mathrm e}^{-2 x} \left ({\mathrm e}^{x} x^{2}+\left (-{\mathrm e}^{x}-\frac {3}{2}\right ) x -2 \,{\mathrm e}^{x}\right )\right ) \left (-\operatorname {csgn}\left (i {\mathrm e}^{-2 x} \left ({\mathrm e}^{x} x^{2}+\left (-{\mathrm e}^{x}-\frac {3}{2}\right ) x -2 \,{\mathrm e}^{x}\right )\right )+\operatorname {csgn}\left (i {\mathrm e}^{-2 x}\right )\right ) \left (-\operatorname {csgn}\left (i {\mathrm e}^{-2 x} \left ({\mathrm e}^{x} x^{2}+\left (-{\mathrm e}^{x}-\frac {3}{2}\right ) x -2 \,{\mathrm e}^{x}\right )\right )+\operatorname {csgn}\left (i \left ({\mathrm e}^{x} x^{2}+\left (-{\mathrm e}^{x}-\frac {3}{2}\right ) x -2 \,{\mathrm e}^{x}\right )\right )\right )}{2}\right )\) | \(191\) |
int((((4*x^3-4*x^2-8*x)*exp(x)-6*x^2)*ln(1/3*((2*x^2-2*x-4)*exp(x)-3*x)/ex p(x)^2)*ln(ln(1/3*((2*x^2-2*x-4)*exp(x)-3*x)/exp(x)^2))+(-2*x^4+6*x^3+2*x^ 2)*exp(x)+6*x^3-3*x^2)/((2*x^2-2*x-4)*exp(x)-3*x)/ln(1/3*((2*x^2-2*x-4)*ex p(x)-3*x)/exp(x)^2),x,method=_RETURNVERBOSE)
Time = 0.24 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.82 \[ \int \frac {-3 x^2+6 x^3+e^x \left (2 x^2+6 x^3-2 x^4\right )+\left (-6 x^2+e^x \left (-8 x-4 x^2+4 x^3\right )\right ) \log \left (\frac {1}{3} e^{-2 x} \left (-3 x+e^x \left (-4-2 x+2 x^2\right )\right )\right ) \log \left (\log \left (\frac {1}{3} e^{-2 x} \left (-3 x+e^x \left (-4-2 x+2 x^2\right )\right )\right )\right )}{\left (-3 x+e^x \left (-4-2 x+2 x^2\right )\right ) \log \left (\frac {1}{3} e^{-2 x} \left (-3 x+e^x \left (-4-2 x+2 x^2\right )\right )\right )} \, dx=x^{2} \log \left (\log \left (\frac {1}{3} \, {\left (2 \, {\left (x^{2} - x - 2\right )} e^{x} - 3 \, x\right )} e^{\left (-2 \, x\right )}\right )\right ) \]
integrate((((4*x^3-4*x^2-8*x)*exp(x)-6*x^2)*log(1/3*((2*x^2-2*x-4)*exp(x)- 3*x)/exp(x)^2)*log(log(1/3*((2*x^2-2*x-4)*exp(x)-3*x)/exp(x)^2))+(-2*x^4+6 *x^3+2*x^2)*exp(x)+6*x^3-3*x^2)/((2*x^2-2*x-4)*exp(x)-3*x)/log(1/3*((2*x^2 -2*x-4)*exp(x)-3*x)/exp(x)^2),x, algorithm=\
Time = 2.22 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.65 \[ \int \frac {-3 x^2+6 x^3+e^x \left (2 x^2+6 x^3-2 x^4\right )+\left (-6 x^2+e^x \left (-8 x-4 x^2+4 x^3\right )\right ) \log \left (\frac {1}{3} e^{-2 x} \left (-3 x+e^x \left (-4-2 x+2 x^2\right )\right )\right ) \log \left (\log \left (\frac {1}{3} e^{-2 x} \left (-3 x+e^x \left (-4-2 x+2 x^2\right )\right )\right )\right )}{\left (-3 x+e^x \left (-4-2 x+2 x^2\right )\right ) \log \left (\frac {1}{3} e^{-2 x} \left (-3 x+e^x \left (-4-2 x+2 x^2\right )\right )\right )} \, dx=\left (x^{2} - 8\right ) \log {\left (\log {\left (\left (- x + \frac {\left (2 x^{2} - 2 x - 4\right ) e^{x}}{3}\right ) e^{- 2 x} \right )} \right )} + 8 \log {\left (\log {\left (\left (- x + \frac {\left (2 x^{2} - 2 x - 4\right ) e^{x}}{3}\right ) e^{- 2 x} \right )} \right )} \]
integrate((((4*x**3-4*x**2-8*x)*exp(x)-6*x**2)*ln(1/3*((2*x**2-2*x-4)*exp( x)-3*x)/exp(x)**2)*ln(ln(1/3*((2*x**2-2*x-4)*exp(x)-3*x)/exp(x)**2))+(-2*x **4+6*x**3+2*x**2)*exp(x)+6*x**3-3*x**2)/((2*x**2-2*x-4)*exp(x)-3*x)/ln(1/ 3*((2*x**2-2*x-4)*exp(x)-3*x)/exp(x)**2),x)
(x**2 - 8)*log(log((-x + (2*x**2 - 2*x - 4)*exp(x)/3)*exp(-2*x))) + 8*log( log((-x + (2*x**2 - 2*x - 4)*exp(x)/3)*exp(-2*x)))
Time = 0.38 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.88 \[ \int \frac {-3 x^2+6 x^3+e^x \left (2 x^2+6 x^3-2 x^4\right )+\left (-6 x^2+e^x \left (-8 x-4 x^2+4 x^3\right )\right ) \log \left (\frac {1}{3} e^{-2 x} \left (-3 x+e^x \left (-4-2 x+2 x^2\right )\right )\right ) \log \left (\log \left (\frac {1}{3} e^{-2 x} \left (-3 x+e^x \left (-4-2 x+2 x^2\right )\right )\right )\right )}{\left (-3 x+e^x \left (-4-2 x+2 x^2\right )\right ) \log \left (\frac {1}{3} e^{-2 x} \left (-3 x+e^x \left (-4-2 x+2 x^2\right )\right )\right )} \, dx=x^{2} \log \left (-2 \, x - \log \left (3\right ) + \log \left (2 \, {\left (x^{2} - x - 2\right )} e^{x} - 3 \, x\right )\right ) \]
integrate((((4*x^3-4*x^2-8*x)*exp(x)-6*x^2)*log(1/3*((2*x^2-2*x-4)*exp(x)- 3*x)/exp(x)^2)*log(log(1/3*((2*x^2-2*x-4)*exp(x)-3*x)/exp(x)^2))+(-2*x^4+6 *x^3+2*x^2)*exp(x)+6*x^3-3*x^2)/((2*x^2-2*x-4)*exp(x)-3*x)/log(1/3*((2*x^2 -2*x-4)*exp(x)-3*x)/exp(x)^2),x, algorithm=\
\[ \int \frac {-3 x^2+6 x^3+e^x \left (2 x^2+6 x^3-2 x^4\right )+\left (-6 x^2+e^x \left (-8 x-4 x^2+4 x^3\right )\right ) \log \left (\frac {1}{3} e^{-2 x} \left (-3 x+e^x \left (-4-2 x+2 x^2\right )\right )\right ) \log \left (\log \left (\frac {1}{3} e^{-2 x} \left (-3 x+e^x \left (-4-2 x+2 x^2\right )\right )\right )\right )}{\left (-3 x+e^x \left (-4-2 x+2 x^2\right )\right ) \log \left (\frac {1}{3} e^{-2 x} \left (-3 x+e^x \left (-4-2 x+2 x^2\right )\right )\right )} \, dx=\int { \frac {6 \, x^{3} - 2 \, {\left (3 \, x^{2} - 2 \, {\left (x^{3} - x^{2} - 2 \, x\right )} e^{x}\right )} \log \left (\frac {1}{3} \, {\left (2 \, {\left (x^{2} - x - 2\right )} e^{x} - 3 \, x\right )} e^{\left (-2 \, x\right )}\right ) \log \left (\log \left (\frac {1}{3} \, {\left (2 \, {\left (x^{2} - x - 2\right )} e^{x} - 3 \, x\right )} e^{\left (-2 \, x\right )}\right )\right ) - 3 \, x^{2} - 2 \, {\left (x^{4} - 3 \, x^{3} - x^{2}\right )} e^{x}}{{\left (2 \, {\left (x^{2} - x - 2\right )} e^{x} - 3 \, x\right )} \log \left (\frac {1}{3} \, {\left (2 \, {\left (x^{2} - x - 2\right )} e^{x} - 3 \, x\right )} e^{\left (-2 \, x\right )}\right )} \,d x } \]
integrate((((4*x^3-4*x^2-8*x)*exp(x)-6*x^2)*log(1/3*((2*x^2-2*x-4)*exp(x)- 3*x)/exp(x)^2)*log(log(1/3*((2*x^2-2*x-4)*exp(x)-3*x)/exp(x)^2))+(-2*x^4+6 *x^3+2*x^2)*exp(x)+6*x^3-3*x^2)/((2*x^2-2*x-4)*exp(x)-3*x)/log(1/3*((2*x^2 -2*x-4)*exp(x)-3*x)/exp(x)^2),x, algorithm=\
integrate((6*x^3 - 2*(3*x^2 - 2*(x^3 - x^2 - 2*x)*e^x)*log(1/3*(2*(x^2 - x - 2)*e^x - 3*x)*e^(-2*x))*log(log(1/3*(2*(x^2 - x - 2)*e^x - 3*x)*e^(-2*x ))) - 3*x^2 - 2*(x^4 - 3*x^3 - x^2)*e^x)/((2*(x^2 - x - 2)*e^x - 3*x)*log( 1/3*(2*(x^2 - x - 2)*e^x - 3*x)*e^(-2*x))), x)
Time = 10.86 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.82 \[ \int \frac {-3 x^2+6 x^3+e^x \left (2 x^2+6 x^3-2 x^4\right )+\left (-6 x^2+e^x \left (-8 x-4 x^2+4 x^3\right )\right ) \log \left (\frac {1}{3} e^{-2 x} \left (-3 x+e^x \left (-4-2 x+2 x^2\right )\right )\right ) \log \left (\log \left (\frac {1}{3} e^{-2 x} \left (-3 x+e^x \left (-4-2 x+2 x^2\right )\right )\right )\right )}{\left (-3 x+e^x \left (-4-2 x+2 x^2\right )\right ) \log \left (\frac {1}{3} e^{-2 x} \left (-3 x+e^x \left (-4-2 x+2 x^2\right )\right )\right )} \, dx=x^2\,\ln \left (\ln \left (-{\mathrm {e}}^{-2\,x}\,\left (x+\frac {{\mathrm {e}}^x\,\left (-2\,x^2+2\,x+4\right )}{3}\right )\right )\right ) \]