Integrand size = 170, antiderivative size = 27 \[ \int \frac {-2 x+2 e^{2 x} x+e^x \left (2-2 x^2\right )+\left (e^{2 x} \left (4 x+2 x^2\right )+e^x \left (-4 x^2-2 x^3\right )\right ) \log (x)+\left (-2 x+e^x (1+x)+\left (-2 e^x x^2+e^{2 x} \left (x+x^2\right )\right ) \log (x)\right ) \log \left (\frac {2}{x^2+2 e^x x^3 \log (x)+e^{2 x} x^4 \log ^2(x)}\right )}{\left (1+e^x x \log (x)\right ) \log ^2\left (\frac {2}{x^2+2 e^x x^3 \log (x)+e^{2 x} x^4 \log ^2(x)}\right )} \, dx=\frac {\left (e^x-x\right ) x}{\log \left (\frac {2}{\left (x+e^x x^2 \log (x)\right )^2}\right )} \]
Time = 0.14 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.04 \[ \int \frac {-2 x+2 e^{2 x} x+e^x \left (2-2 x^2\right )+\left (e^{2 x} \left (4 x+2 x^2\right )+e^x \left (-4 x^2-2 x^3\right )\right ) \log (x)+\left (-2 x+e^x (1+x)+\left (-2 e^x x^2+e^{2 x} \left (x+x^2\right )\right ) \log (x)\right ) \log \left (\frac {2}{x^2+2 e^x x^3 \log (x)+e^{2 x} x^4 \log ^2(x)}\right )}{\left (1+e^x x \log (x)\right ) \log ^2\left (\frac {2}{x^2+2 e^x x^3 \log (x)+e^{2 x} x^4 \log ^2(x)}\right )} \, dx=\frac {\left (e^x-x\right ) x}{\log \left (\frac {2}{x^2 \left (1+e^x x \log (x)\right )^2}\right )} \]
Integrate[(-2*x + 2*E^(2*x)*x + E^x*(2 - 2*x^2) + (E^(2*x)*(4*x + 2*x^2) + E^x*(-4*x^2 - 2*x^3))*Log[x] + (-2*x + E^x*(1 + x) + (-2*E^x*x^2 + E^(2*x )*(x + x^2))*Log[x])*Log[2/(x^2 + 2*E^x*x^3*Log[x] + E^(2*x)*x^4*Log[x]^2) ])/((1 + E^x*x*Log[x])*Log[2/(x^2 + 2*E^x*x^3*Log[x] + E^(2*x)*x^4*Log[x]^ 2)]^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.
| \(\displaystyle \int \frac {e^x \left (2-2 x^2\right )+\left (e^{2 x} \left (2 x^2+4 x\right )+e^x \left (-2 x^3-4 x^2\right )\right ) \log (x)+\left (\left (e^{2 x} \left (x^2+x\right )-2 e^x x^2\right ) \log (x)-2 x+e^x (x+1)\right ) \log \left (\frac {2}{e^{2 x} x^4 \log ^2(x)+2 e^x x^3 \log (x)+x^2}\right )+2 e^{2 x} x-2 x}{\left (e^x x \log (x)+1\right ) \log ^2\left (\frac {2}{e^{2 x} x^4 \log ^2(x)+2 e^x x^3 \log (x)+x^2}\right )} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {e^x \left (2-2 x^2\right )+\left (e^{2 x} \left (2 x^2+4 x\right )+e^x \left (-2 x^3-4 x^2\right )\right ) \log (x)+\left (\left (e^{2 x} \left (x^2+x\right )-2 e^x x^2\right ) \log (x)-2 x+e^x (x+1)\right ) \log \left (\frac {2}{e^{2 x} x^4 \log ^2(x)+2 e^x x^3 \log (x)+x^2}\right )+2 e^{2 x} x-2 x}{\left (e^x x \log (x)+1\right ) \log ^2\left (\frac {2}{x^2 \left (e^x x \log (x)+1\right )^2}\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {e^x \left (x \log \left (\frac {2}{x^2 \left (e^x x \log (x)+1\right )^2}\right ) \log (x)+\log \left (\frac {2}{x^2 \left (e^x x \log (x)+1\right )^2}\right ) \log (x)+2 x \log (x)+4 \log (x)+2\right )}{\log (x) \log ^2\left (\frac {2}{x^2 \left (e^x x \log (x)+1\right )^2}\right )}+\frac {2 \left (x^3 \log ^2(x)+x^2 \log ^2(x)+x^2 \log (x)+x \log (x)+\log (x)+1\right )}{x \log ^2(x) \left (e^x x \log (x)+1\right ) \log ^2\left (\frac {2}{x^2 \left (e^x x \log (x)+1\right )^2}\right )}-\frac {2 \left (x^3 \log ^2(x)+2 x^2 \log ^2(x)+x^2 \log ^2(x) \log \left (\frac {2}{x^2 \left (e^x x \log (x)+1\right )^2}\right )+x^2 \log (x)+x \log (x)+\log (x)+1\right )}{x \log ^2(x) \log ^2\left (\frac {2}{x^2 \left (e^x x \log (x)+1\right )^2}\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 4 \int \frac {e^x}{\log ^2\left (\frac {2}{x^2 \left (e^x x \log (x)+1\right )^2}\right )}dx-4 \int \frac {x}{\log ^2\left (\frac {2}{x^2 \left (e^x x \log (x)+1\right )^2}\right )}dx+2 \int \frac {e^x x}{\log ^2\left (\frac {2}{x^2 \left (e^x x \log (x)+1\right )^2}\right )}dx-2 \int \frac {x^2}{\log ^2\left (\frac {2}{x^2 \left (e^x x \log (x)+1\right )^2}\right )}dx-2 \int \frac {1}{x \log ^2(x) \log ^2\left (\frac {2}{x^2 \left (e^x x \log (x)+1\right )^2}\right )}dx-2 \int \frac {1}{\log (x) \log ^2\left (\frac {2}{x^2 \left (e^x x \log (x)+1\right )^2}\right )}dx+2 \int \frac {e^x}{\log (x) \log ^2\left (\frac {2}{x^2 \left (e^x x \log (x)+1\right )^2}\right )}dx-2 \int \frac {1}{x \log (x) \log ^2\left (\frac {2}{x^2 \left (e^x x \log (x)+1\right )^2}\right )}dx-2 \int \frac {x}{\log (x) \log ^2\left (\frac {2}{x^2 \left (e^x x \log (x)+1\right )^2}\right )}dx+2 \int \frac {x}{\left (e^x x \log (x)+1\right ) \log ^2\left (\frac {2}{x^2 \left (e^x x \log (x)+1\right )^2}\right )}dx+2 \int \frac {x^2}{\left (e^x x \log (x)+1\right ) \log ^2\left (\frac {2}{x^2 \left (e^x x \log (x)+1\right )^2}\right )}dx+2 \int \frac {1}{x \log ^2(x) \left (e^x x \log (x)+1\right ) \log ^2\left (\frac {2}{x^2 \left (e^x x \log (x)+1\right )^2}\right )}dx+2 \int \frac {1}{\log (x) \left (e^x x \log (x)+1\right ) \log ^2\left (\frac {2}{x^2 \left (e^x x \log (x)+1\right )^2}\right )}dx+2 \int \frac {1}{x \log (x) \left (e^x x \log (x)+1\right ) \log ^2\left (\frac {2}{x^2 \left (e^x x \log (x)+1\right )^2}\right )}dx+2 \int \frac {x}{\log (x) \left (e^x x \log (x)+1\right ) \log ^2\left (\frac {2}{x^2 \left (e^x x \log (x)+1\right )^2}\right )}dx+\int \frac {e^x}{\log \left (\frac {2}{x^2 \left (e^x x \log (x)+1\right )^2}\right )}dx-2 \int \frac {x}{\log \left (\frac {2}{x^2 \left (e^x x \log (x)+1\right )^2}\right )}dx+\int \frac {e^x x}{\log \left (\frac {2}{x^2 \left (e^x x \log (x)+1\right )^2}\right )}dx\) |
Int[(-2*x + 2*E^(2*x)*x + E^x*(2 - 2*x^2) + (E^(2*x)*(4*x + 2*x^2) + E^x*( -4*x^2 - 2*x^3))*Log[x] + (-2*x + E^x*(1 + x) + (-2*E^x*x^2 + E^(2*x)*(x + x^2))*Log[x])*Log[2/(x^2 + 2*E^x*x^3*Log[x] + E^(2*x)*x^4*Log[x]^2)])/((1 + E^x*x*Log[x])*Log[2/(x^2 + 2*E^x*x^3*Log[x] + E^(2*x)*x^4*Log[x]^2)]^2) ,x]
3.20.83.3.1 Defintions of rubi rules used
Time = 2.92 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.67
| method | result | size |
| parallelrisch | \(-\frac {-2 \,{\mathrm e}^{x} x +2 x^{2}}{2 \ln \left (\frac {2}{x^{2} \left ({\mathrm e}^{2 x} \ln \left (x \right )^{2} x^{2}+2 x \,{\mathrm e}^{x} \ln \left (x \right )+1\right )}\right )}\) | \(45\) |
| risch | \(\frac {2 i \left (x -{\mathrm e}^{x}\right ) x}{\pi \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right )-2 \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{2}+\pi \operatorname {csgn}\left (i x^{2}\right )^{3}-\pi \,\operatorname {csgn}\left (\frac {i}{x^{2}}\right ) \operatorname {csgn}\left (\frac {i}{\left (x \,{\mathrm e}^{x} \ln \left (x \right )+1\right )^{2}}\right ) \operatorname {csgn}\left (\frac {i}{x^{2} \left (x \,{\mathrm e}^{x} \ln \left (x \right )+1\right )^{2}}\right )+\pi \,\operatorname {csgn}\left (\frac {i}{x^{2}}\right ) \operatorname {csgn}\left (\frac {i}{x^{2} \left (x \,{\mathrm e}^{x} \ln \left (x \right )+1\right )^{2}}\right )^{2}+\pi \,\operatorname {csgn}\left (\frac {i}{\left (x \,{\mathrm e}^{x} \ln \left (x \right )+1\right )^{2}}\right ) \operatorname {csgn}\left (\frac {i}{x^{2} \left (x \,{\mathrm e}^{x} \ln \left (x \right )+1\right )^{2}}\right )^{2}+\pi {\operatorname {csgn}\left (i \left (x \,{\mathrm e}^{x} \ln \left (x \right )+1\right )\right )}^{2} \operatorname {csgn}\left (i \left (x \,{\mathrm e}^{x} \ln \left (x \right )+1\right )^{2}\right )-2 \pi \,\operatorname {csgn}\left (i \left (x \,{\mathrm e}^{x} \ln \left (x \right )+1\right )\right ) {\operatorname {csgn}\left (i \left (x \,{\mathrm e}^{x} \ln \left (x \right )+1\right )^{2}\right )}^{2}+\pi {\operatorname {csgn}\left (i \left (x \,{\mathrm e}^{x} \ln \left (x \right )+1\right )^{2}\right )}^{3}-\pi \operatorname {csgn}\left (\frac {i}{x^{2} \left (x \,{\mathrm e}^{x} \ln \left (x \right )+1\right )^{2}}\right )^{3}-2 i \ln \left (2\right )+4 i \ln \left (x \right )+4 i \ln \left (x \,{\mathrm e}^{x} \ln \left (x \right )+1\right )}\) | \(285\) |
int(((((x^2+x)*exp(x)^2-2*exp(x)*x^2)*ln(x)+(1+x)*exp(x)-2*x)*ln(2/(x^4*ex p(x)^2*ln(x)^2+2*x^3*exp(x)*ln(x)+x^2))+((2*x^2+4*x)*exp(x)^2+(-2*x^3-4*x^ 2)*exp(x))*ln(x)+2*x*exp(x)^2+(-2*x^2+2)*exp(x)-2*x)/(x*exp(x)*ln(x)+1)/ln (2/(x^4*exp(x)^2*ln(x)^2+2*x^3*exp(x)*ln(x)+x^2))^2,x,method=_RETURNVERBOS E)
Time = 0.24 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.59 \[ \int \frac {-2 x+2 e^{2 x} x+e^x \left (2-2 x^2\right )+\left (e^{2 x} \left (4 x+2 x^2\right )+e^x \left (-4 x^2-2 x^3\right )\right ) \log (x)+\left (-2 x+e^x (1+x)+\left (-2 e^x x^2+e^{2 x} \left (x+x^2\right )\right ) \log (x)\right ) \log \left (\frac {2}{x^2+2 e^x x^3 \log (x)+e^{2 x} x^4 \log ^2(x)}\right )}{\left (1+e^x x \log (x)\right ) \log ^2\left (\frac {2}{x^2+2 e^x x^3 \log (x)+e^{2 x} x^4 \log ^2(x)}\right )} \, dx=-\frac {x^{2} - x e^{x}}{\log \left (\frac {2}{x^{4} e^{\left (2 \, x\right )} \log \left (x\right )^{2} + 2 \, x^{3} e^{x} \log \left (x\right ) + x^{2}}\right )} \]
integrate(((((x^2+x)*exp(x)^2-2*exp(x)*x^2)*log(x)+(1+x)*exp(x)-2*x)*log(2 /(x^4*exp(x)^2*log(x)^2+2*x^3*exp(x)*log(x)+x^2))+((2*x^2+4*x)*exp(x)^2+(- 2*x^3-4*x^2)*exp(x))*log(x)+2*x*exp(x)^2+(-2*x^2+2)*exp(x)-2*x)/(x*exp(x)* log(x)+1)/log(2/(x^4*exp(x)^2*log(x)^2+2*x^3*exp(x)*log(x)+x^2))^2,x, algo rithm=\
Time = 0.31 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.44 \[ \int \frac {-2 x+2 e^{2 x} x+e^x \left (2-2 x^2\right )+\left (e^{2 x} \left (4 x+2 x^2\right )+e^x \left (-4 x^2-2 x^3\right )\right ) \log (x)+\left (-2 x+e^x (1+x)+\left (-2 e^x x^2+e^{2 x} \left (x+x^2\right )\right ) \log (x)\right ) \log \left (\frac {2}{x^2+2 e^x x^3 \log (x)+e^{2 x} x^4 \log ^2(x)}\right )}{\left (1+e^x x \log (x)\right ) \log ^2\left (\frac {2}{x^2+2 e^x x^3 \log (x)+e^{2 x} x^4 \log ^2(x)}\right )} \, dx=\frac {- x^{2} + x e^{x}}{\log {\left (\frac {2}{x^{4} e^{2 x} \log {\left (x \right )}^{2} + 2 x^{3} e^{x} \log {\left (x \right )} + x^{2}} \right )}} \]
integrate(((((x**2+x)*exp(x)**2-2*exp(x)*x**2)*ln(x)+(1+x)*exp(x)-2*x)*ln( 2/(x**4*exp(x)**2*ln(x)**2+2*x**3*exp(x)*ln(x)+x**2))+((2*x**2+4*x)*exp(x) **2+(-2*x**3-4*x**2)*exp(x))*ln(x)+2*x*exp(x)**2+(-2*x**2+2)*exp(x)-2*x)/( x*exp(x)*ln(x)+1)/ln(2/(x**4*exp(x)**2*ln(x)**2+2*x**3*exp(x)*ln(x)+x**2)) **2,x)
Time = 0.33 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.15 \[ \int \frac {-2 x+2 e^{2 x} x+e^x \left (2-2 x^2\right )+\left (e^{2 x} \left (4 x+2 x^2\right )+e^x \left (-4 x^2-2 x^3\right )\right ) \log (x)+\left (-2 x+e^x (1+x)+\left (-2 e^x x^2+e^{2 x} \left (x+x^2\right )\right ) \log (x)\right ) \log \left (\frac {2}{x^2+2 e^x x^3 \log (x)+e^{2 x} x^4 \log ^2(x)}\right )}{\left (1+e^x x \log (x)\right ) \log ^2\left (\frac {2}{x^2+2 e^x x^3 \log (x)+e^{2 x} x^4 \log ^2(x)}\right )} \, dx=-\frac {x^{2} - x e^{x}}{\log \left (2\right ) - 2 \, \log \left (x e^{x} \log \left (x\right ) + 1\right ) - 2 \, \log \left (x\right )} \]
integrate(((((x^2+x)*exp(x)^2-2*exp(x)*x^2)*log(x)+(1+x)*exp(x)-2*x)*log(2 /(x^4*exp(x)^2*log(x)^2+2*x^3*exp(x)*log(x)+x^2))+((2*x^2+4*x)*exp(x)^2+(- 2*x^3-4*x^2)*exp(x))*log(x)+2*x*exp(x)^2+(-2*x^2+2)*exp(x)-2*x)/(x*exp(x)* log(x)+1)/log(2/(x^4*exp(x)^2*log(x)^2+2*x^3*exp(x)*log(x)+x^2))^2,x, algo rithm=\
Time = 1.28 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.63 \[ \int \frac {-2 x+2 e^{2 x} x+e^x \left (2-2 x^2\right )+\left (e^{2 x} \left (4 x+2 x^2\right )+e^x \left (-4 x^2-2 x^3\right )\right ) \log (x)+\left (-2 x+e^x (1+x)+\left (-2 e^x x^2+e^{2 x} \left (x+x^2\right )\right ) \log (x)\right ) \log \left (\frac {2}{x^2+2 e^x x^3 \log (x)+e^{2 x} x^4 \log ^2(x)}\right )}{\left (1+e^x x \log (x)\right ) \log ^2\left (\frac {2}{x^2+2 e^x x^3 \log (x)+e^{2 x} x^4 \log ^2(x)}\right )} \, dx=-\frac {x^{2} - x e^{x}}{\log \left (2\right ) - \log \left (x^{2} e^{\left (2 \, x\right )} \log \left (x\right )^{2} + 2 \, x e^{x} \log \left (x\right ) + 1\right ) - 2 \, \log \left (x\right )} \]
integrate(((((x^2+x)*exp(x)^2-2*exp(x)*x^2)*log(x)+(1+x)*exp(x)-2*x)*log(2 /(x^4*exp(x)^2*log(x)^2+2*x^3*exp(x)*log(x)+x^2))+((2*x^2+4*x)*exp(x)^2+(- 2*x^3-4*x^2)*exp(x))*log(x)+2*x*exp(x)^2+(-2*x^2+2)*exp(x)-2*x)/(x*exp(x)* log(x)+1)/log(2/(x^4*exp(x)^2*log(x)^2+2*x^3*exp(x)*log(x)+x^2))^2,x, algo rithm=\
Timed out. \[ \int \frac {-2 x+2 e^{2 x} x+e^x \left (2-2 x^2\right )+\left (e^{2 x} \left (4 x+2 x^2\right )+e^x \left (-4 x^2-2 x^3\right )\right ) \log (x)+\left (-2 x+e^x (1+x)+\left (-2 e^x x^2+e^{2 x} \left (x+x^2\right )\right ) \log (x)\right ) \log \left (\frac {2}{x^2+2 e^x x^3 \log (x)+e^{2 x} x^4 \log ^2(x)}\right )}{\left (1+e^x x \log (x)\right ) \log ^2\left (\frac {2}{x^2+2 e^x x^3 \log (x)+e^{2 x} x^4 \log ^2(x)}\right )} \, dx=\int -\frac {2\,x-2\,x\,{\mathrm {e}}^{2\,x}-\ln \left (x\right )\,\left ({\mathrm {e}}^{2\,x}\,\left (2\,x^2+4\,x\right )-{\mathrm {e}}^x\,\left (2\,x^3+4\,x^2\right )\right )+\ln \left (\frac {2}{x^2+x^4\,{\mathrm {e}}^{2\,x}\,{\ln \left (x\right )}^2+2\,x^3\,{\mathrm {e}}^x\,\ln \left (x\right )}\right )\,\left (2\,x+\ln \left (x\right )\,\left (2\,x^2\,{\mathrm {e}}^x-{\mathrm {e}}^{2\,x}\,\left (x^2+x\right )\right )-{\mathrm {e}}^x\,\left (x+1\right )\right )+{\mathrm {e}}^x\,\left (2\,x^2-2\right )}{{\ln \left (\frac {2}{x^2+x^4\,{\mathrm {e}}^{2\,x}\,{\ln \left (x\right )}^2+2\,x^3\,{\mathrm {e}}^x\,\ln \left (x\right )}\right )}^2\,\left (x\,{\mathrm {e}}^x\,\ln \left (x\right )+1\right )} \,d x \]
int(-(2*x - 2*x*exp(2*x) - log(x)*(exp(2*x)*(4*x + 2*x^2) - exp(x)*(4*x^2 + 2*x^3)) + log(2/(x^2 + x^4*exp(2*x)*log(x)^2 + 2*x^3*exp(x)*log(x)))*(2* x + log(x)*(2*x^2*exp(x) - exp(2*x)*(x + x^2)) - exp(x)*(x + 1)) + exp(x)* (2*x^2 - 2))/(log(2/(x^2 + x^4*exp(2*x)*log(x)^2 + 2*x^3*exp(x)*log(x)))^2 *(x*exp(x)*log(x) + 1)),x)
int(-(2*x - 2*x*exp(2*x) - log(x)*(exp(2*x)*(4*x + 2*x^2) - exp(x)*(4*x^2 + 2*x^3)) + log(2/(x^2 + x^4*exp(2*x)*log(x)^2 + 2*x^3*exp(x)*log(x)))*(2* x + log(x)*(2*x^2*exp(x) - exp(2*x)*(x + x^2)) - exp(x)*(x + 1)) + exp(x)* (2*x^2 - 2))/(log(2/(x^2 + x^4*exp(2*x)*log(x)^2 + 2*x^3*exp(x)*log(x)))^2 *(x*exp(x)*log(x) + 1)), x)