Integrand size = 141, antiderivative size = 33 \[ \int e^{-\frac {2 \left (2 x-2 e^x \log (2) \log (4)+2 \log ^2(2) \log (4)\right )}{x}} \left (2 x+e^{\frac {2 \left (2 x-2 e^x \log (2) \log (4)+2 \log ^2(2) \log (4)\right )}{x}} (-1+2 x)+e^x (-4+4 x) \log (2) \log (4)+4 \log ^2(2) \log (4)+e^{\frac {2 x-2 e^x \log (2) \log (4)+2 \log ^2(2) \log (4)}{x}} \left (-4 x+e^x (4-4 x) \log (2) \log (4)-4 \log ^2(2) \log (4)\right )\right ) \, dx=-x+\left (x-e^{-2 \left (1+\frac {\log (2) \left (-e^x+\log (2)\right ) \log (4)}{x}\right )} x\right )^2 \]
Time = 0.60 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.76 \[ \int e^{-\frac {2 \left (2 x-2 e^x \log (2) \log (4)+2 \log ^2(2) \log (4)\right )}{x}} \left (2 x+e^{\frac {2 \left (2 x-2 e^x \log (2) \log (4)+2 \log ^2(2) \log (4)\right )}{x}} (-1+2 x)+e^x (-4+4 x) \log (2) \log (4)+4 \log ^2(2) \log (4)+e^{\frac {2 x-2 e^x \log (2) \log (4)+2 \log ^2(2) \log (4)}{x}} \left (-4 x+e^x (4-4 x) \log (2) \log (4)-4 \log ^2(2) \log (4)\right )\right ) \, dx=x \left (-1+\left (1+e^{-\frac {4 \left (x+2 \log ^3(2)-e^x \log (2) \log (4)\right )}{x}}-2 e^{\frac {-2 x-4 \log ^3(2)+e^x \log ^2(4)}{x}}\right ) x\right ) \]
Integrate[(2*x + E^((2*(2*x - 2*E^x*Log[2]*Log[4] + 2*Log[2]^2*Log[4]))/x) *(-1 + 2*x) + E^x*(-4 + 4*x)*Log[2]*Log[4] + 4*Log[2]^2*Log[4] + E^((2*x - 2*E^x*Log[2]*Log[4] + 2*Log[2]^2*Log[4])/x)*(-4*x + E^x*(4 - 4*x)*Log[2]* Log[4] - 4*Log[2]^2*Log[4]))/E^((2*(2*x - 2*E^x*Log[2]*Log[4] + 2*Log[2]^2 *Log[4]))/x),x]
x*(-1 + (1 + E^((-4*(x + 2*Log[2]^3 - E^x*Log[2]*Log[4]))/x) - 2*E^((-2*x - 4*Log[2]^3 + E^x*Log[4]^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.
| \(\displaystyle \int \exp \left (-\frac {2 \left (2 x-2 e^x \log (2) \log (4)+2 \log ^2(2) \log (4)\right )}{x}\right ) \left ((2 x-1) \exp \left (\frac {2 \left (2 x-2 e^x \log (2) \log (4)+2 \log ^2(2) \log (4)\right )}{x}\right )+\left (-4 x+e^x (4-4 x) \log (2) \log (4)-4 \log ^2(2) \log (4)\right ) \exp \left (\frac {2 x-2 e^x \log (2) \log (4)+2 \log ^2(2) \log (4)}{x}\right )+2 x+e^x (4 x-4) \log (2) \log (4)+4 \log ^2(2) \log (4)\right ) \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \exp \left (-\frac {4 \left (x-e^x \log (2) \log (4)+\log ^2(2) \log (4)\right )}{x}\right ) \left ((2 x-1) \exp \left (\frac {2 \left (2 x-2 e^x \log (2) \log (4)+2 \log ^2(2) \log (4)\right )}{x}\right )+\left (-4 x+e^x (4-4 x) \log (2) \log (4)-4 \log ^2(2) \log (4)\right ) \exp \left (\frac {2 x-2 e^x \log (2) \log (4)+2 \log ^2(2) \log (4)}{x}\right )+2 x+e^x (4 x-4) \log (2) \log (4)+4 \log ^2(2) \log (4)\right )dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (4 (x-1) \log (2) \log (4) \exp \left (x-\frac {4 \left (x-e^x \log (2) \log (4)+\log ^2(2) \log (4)\right )}{x}\right )+2 x \exp \left (-\frac {4 \left (x-e^x \log (2) \log (4)+\log ^2(2) \log (4)\right )}{x}\right )-4 \left (x+e^x x \log (2) \log (4)-e^x \log (2) \log (4)+\log ^2(2) \log (4)\right ) \exp \left (-\frac {4 \left (x-e^x \log (2) \log (4)+\log ^2(2) \log (4)\right )}{x}-\frac {2 \log (2) \log (4) \left (e^x-\log (2)\right )}{x}+2\right )+4 \log ^2(2) \log (4) \exp \left (-\frac {4 \left (x-e^x \log (2) \log (4)+\log ^2(2) \log (4)\right )}{x}\right )+2 x-1\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 4 \log ^2(2) \log (4) \int \exp \left (-\frac {4 \left (x+\log ^2(2) \log (4)-e^x \log (2) \log (4)\right )}{x}\right )dx-4 \log (2) \log (4) \int \exp \left (x-\frac {4 \left (x+\log ^2(2) \log (4)-e^x \log (2) \log (4)\right )}{x}\right )dx+2 \int \exp \left (-\frac {4 \left (x+\log ^2(2) \log (4)-e^x \log (2) \log (4)\right )}{x}\right ) xdx+4 \log (2) \log (4) \int \exp \left (x-\frac {4 \left (x+\log ^2(2) \log (4)-e^x \log (2) \log (4)\right )}{x}\right ) xdx-\frac {2^{1-\frac {2 \log (4) \left (e^x-\log (2)\right )}{x}} \left (-e^x x \log (2) \log (4)+e^x \log (2) \log (4)-\log ^2(2) \log (4)\right ) \exp \left (2-\frac {4 \left (x-e^x \log (2) \log (4)+\log ^2(2) \log (4)\right )}{x}\right )}{-\frac {2 \left (x-e^x \log (2) \log (4)+\log ^2(2) \log (4)\right )}{x^2}-\frac {\log (2) \log (4) \left (e^x-\log (2)\right )}{x^2}+\frac {2 \left (1-e^x \log (2) \log (4)\right )}{x}+\frac {e^x \log (2) \log (4)}{x}}+x^2-x\) |
Int[(2*x + E^((2*(2*x - 2*E^x*Log[2]*Log[4] + 2*Log[2]^2*Log[4]))/x)*(-1 + 2*x) + E^x*(-4 + 4*x)*Log[2]*Log[4] + 4*Log[2]^2*Log[4] + E^((2*x - 2*E^x *Log[2]*Log[4] + 2*Log[2]^2*Log[4])/x)*(-4*x + E^x*(4 - 4*x)*Log[2]*Log[4] - 4*Log[2]^2*Log[4]))/E^((2*(2*x - 2*E^x*Log[2]*Log[4] + 2*Log[2]^2*Log[4 ]))/x),x]
3.30.100.3.1 Defintions of rubi rules used
Time = 0.44 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.97
| method | result | size |
| risch | \(x^{2}-x -2 x^{2} {\mathrm e}^{\frac {4 \ln \left (2\right )^{2} {\mathrm e}^{x}-4 \ln \left (2\right )^{3}-2 x}{x}}+x^{2} {\mathrm e}^{\frac {8 \ln \left (2\right )^{2} {\mathrm e}^{x}-8 \ln \left (2\right )^{3}-4 x}{x}}\) | \(65\) |
| norman | \(\left (x^{2}+x^{2} {\mathrm e}^{\frac {-8 \ln \left (2\right )^{2} {\mathrm e}^{x}+8 \ln \left (2\right )^{3}+4 x}{x}}-x \,{\mathrm e}^{\frac {-8 \ln \left (2\right )^{2} {\mathrm e}^{x}+8 \ln \left (2\right )^{3}+4 x}{x}}-2 x^{2} {\mathrm e}^{\frac {-4 \ln \left (2\right )^{2} {\mathrm e}^{x}+4 \ln \left (2\right )^{3}+2 x}{x}}\right ) {\mathrm e}^{-\frac {2 \left (-4 \ln \left (2\right )^{2} {\mathrm e}^{x}+4 \ln \left (2\right )^{3}+2 x \right )}{x}}\) | \(116\) |
| parallelrisch | \(-\frac {\left (-{\mathrm e}^{-\frac {4 \left (2 \ln \left (2\right )^{2} {\mathrm e}^{x}-2 \ln \left (2\right )^{3}-x \right )}{x}} x^{3}+2 \,{\mathrm e}^{-\frac {2 \left (2 \ln \left (2\right )^{2} {\mathrm e}^{x}-2 \ln \left (2\right )^{3}-x \right )}{x}} x^{3}+{\mathrm e}^{-\frac {4 \left (2 \ln \left (2\right )^{2} {\mathrm e}^{x}-2 \ln \left (2\right )^{3}-x \right )}{x}} x^{2}-x^{3}\right ) {\mathrm e}^{\frac {8 \ln \left (2\right )^{2} {\mathrm e}^{x}-8 \ln \left (2\right )^{3}-4 x}{x}}}{x}\) | \(128\) |
int(((-1+2*x)*exp((-4*ln(2)^2*exp(x)+4*ln(2)^3+2*x)/x)^2+(2*(4-4*x)*ln(2)^ 2*exp(x)-8*ln(2)^3-4*x)*exp((-4*ln(2)^2*exp(x)+4*ln(2)^3+2*x)/x)+2*(-4+4*x )*ln(2)^2*exp(x)+8*ln(2)^3+2*x)/exp((-4*ln(2)^2*exp(x)+4*ln(2)^3+2*x)/x)^2 ,x,method=_RETURNVERBOSE)
x^2-x-2*x^2*exp(2*(2*ln(2)^2*exp(x)-2*ln(2)^3-x)/x)+x^2*exp(4*(2*ln(2)^2*e xp(x)-2*ln(2)^3-x)/x)
Leaf count of result is larger than twice the leaf count of optimal. 94 vs. \(2 (31) = 62\).
Time = 0.26 (sec) , antiderivative size = 94, normalized size of antiderivative = 2.85 \[ \int e^{-\frac {2 \left (2 x-2 e^x \log (2) \log (4)+2 \log ^2(2) \log (4)\right )}{x}} \left (2 x+e^{\frac {2 \left (2 x-2 e^x \log (2) \log (4)+2 \log ^2(2) \log (4)\right )}{x}} (-1+2 x)+e^x (-4+4 x) \log (2) \log (4)+4 \log ^2(2) \log (4)+e^{\frac {2 x-2 e^x \log (2) \log (4)+2 \log ^2(2) \log (4)}{x}} \left (-4 x+e^x (4-4 x) \log (2) \log (4)-4 \log ^2(2) \log (4)\right )\right ) \, dx=-{\left (2 \, x^{2} e^{\left (-\frac {2 \, {\left (2 \, e^{x} \log \left (2\right )^{2} - 2 \, \log \left (2\right )^{3} - x\right )}}{x}\right )} - x^{2} - {\left (x^{2} - x\right )} e^{\left (-\frac {4 \, {\left (2 \, e^{x} \log \left (2\right )^{2} - 2 \, \log \left (2\right )^{3} - x\right )}}{x}\right )}\right )} e^{\left (\frac {4 \, {\left (2 \, e^{x} \log \left (2\right )^{2} - 2 \, \log \left (2\right )^{3} - x\right )}}{x}\right )} \]
integrate(((-1+2*x)*exp((-4*log(2)^2*exp(x)+4*log(2)^3+2*x)/x)^2+(2*(4-4*x )*log(2)^2*exp(x)-8*log(2)^3-4*x)*exp((-4*log(2)^2*exp(x)+4*log(2)^3+2*x)/ x)+2*(-4+4*x)*log(2)^2*exp(x)+8*log(2)^3+2*x)/exp((-4*log(2)^2*exp(x)+4*lo g(2)^3+2*x)/x)^2,x, algorithm=\
-(2*x^2*e^(-2*(2*e^x*log(2)^2 - 2*log(2)^3 - x)/x) - x^2 - (x^2 - x)*e^(-4 *(2*e^x*log(2)^2 - 2*log(2)^3 - x)/x))*e^(4*(2*e^x*log(2)^2 - 2*log(2)^3 - x)/x)
Leaf count of result is larger than twice the leaf count of optimal. 60 vs. \(2 (27) = 54\).
Time = 0.17 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.82 \[ \int e^{-\frac {2 \left (2 x-2 e^x \log (2) \log (4)+2 \log ^2(2) \log (4)\right )}{x}} \left (2 x+e^{\frac {2 \left (2 x-2 e^x \log (2) \log (4)+2 \log ^2(2) \log (4)\right )}{x}} (-1+2 x)+e^x (-4+4 x) \log (2) \log (4)+4 \log ^2(2) \log (4)+e^{\frac {2 x-2 e^x \log (2) \log (4)+2 \log ^2(2) \log (4)}{x}} \left (-4 x+e^x (4-4 x) \log (2) \log (4)-4 \log ^2(2) \log (4)\right )\right ) \, dx=x^{2} - 2 x^{2} e^{- \frac {2 x - 4 e^{x} \log {\left (2 \right )}^{2} + 4 \log {\left (2 \right )}^{3}}{x}} + x^{2} e^{- \frac {2 \cdot \left (2 x - 4 e^{x} \log {\left (2 \right )}^{2} + 4 \log {\left (2 \right )}^{3}\right )}{x}} - x \]
integrate(((-1+2*x)*exp((-4*ln(2)**2*exp(x)+4*ln(2)**3+2*x)/x)**2+(2*(4-4* x)*ln(2)**2*exp(x)-8*ln(2)**3-4*x)*exp((-4*ln(2)**2*exp(x)+4*ln(2)**3+2*x) /x)+2*(-4+4*x)*ln(2)**2*exp(x)+8*ln(2)**3+2*x)/exp((-4*ln(2)**2*exp(x)+4*l n(2)**3+2*x)/x)**2,x)
x**2 - 2*x**2*exp(-(2*x - 4*exp(x)*log(2)**2 + 4*log(2)**3)/x) + x**2*exp( -2*(2*x - 4*exp(x)*log(2)**2 + 4*log(2)**3)/x) - x
Leaf count of result is larger than twice the leaf count of optimal. 65 vs. \(2 (31) = 62\).
Time = 0.40 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.97 \[ \int e^{-\frac {2 \left (2 x-2 e^x \log (2) \log (4)+2 \log ^2(2) \log (4)\right )}{x}} \left (2 x+e^{\frac {2 \left (2 x-2 e^x \log (2) \log (4)+2 \log ^2(2) \log (4)\right )}{x}} (-1+2 x)+e^x (-4+4 x) \log (2) \log (4)+4 \log ^2(2) \log (4)+e^{\frac {2 x-2 e^x \log (2) \log (4)+2 \log ^2(2) \log (4)}{x}} \left (-4 x+e^x (4-4 x) \log (2) \log (4)-4 \log ^2(2) \log (4)\right )\right ) \, dx=x^{2} + {\left (x^{2} e^{\left (\frac {8 \, e^{x} \log \left (2\right )^{2}}{x}\right )} - 2 \, x^{2} e^{\left (\frac {4 \, e^{x} \log \left (2\right )^{2}}{x} + \frac {4 \, \log \left (2\right )^{3}}{x} + 2\right )}\right )} e^{\left (-\frac {8 \, \log \left (2\right )^{3}}{x} - 4\right )} - x \]
integrate(((-1+2*x)*exp((-4*log(2)^2*exp(x)+4*log(2)^3+2*x)/x)^2+(2*(4-4*x )*log(2)^2*exp(x)-8*log(2)^3-4*x)*exp((-4*log(2)^2*exp(x)+4*log(2)^3+2*x)/ x)+2*(-4+4*x)*log(2)^2*exp(x)+8*log(2)^3+2*x)/exp((-4*log(2)^2*exp(x)+4*lo g(2)^3+2*x)/x)^2,x, algorithm=\
x^2 + (x^2*e^(8*e^x*log(2)^2/x) - 2*x^2*e^(4*e^x*log(2)^2/x + 4*log(2)^3/x + 2))*e^(-8*log(2)^3/x - 4) - x
\[ \int e^{-\frac {2 \left (2 x-2 e^x \log (2) \log (4)+2 \log ^2(2) \log (4)\right )}{x}} \left (2 x+e^{\frac {2 \left (2 x-2 e^x \log (2) \log (4)+2 \log ^2(2) \log (4)\right )}{x}} (-1+2 x)+e^x (-4+4 x) \log (2) \log (4)+4 \log ^2(2) \log (4)+e^{\frac {2 x-2 e^x \log (2) \log (4)+2 \log ^2(2) \log (4)}{x}} \left (-4 x+e^x (4-4 x) \log (2) \log (4)-4 \log ^2(2) \log (4)\right )\right ) \, dx=\int { {\left (8 \, {\left (x - 1\right )} e^{x} \log \left (2\right )^{2} + 8 \, \log \left (2\right )^{3} - 4 \, {\left (2 \, {\left (x - 1\right )} e^{x} \log \left (2\right )^{2} + 2 \, \log \left (2\right )^{3} + x\right )} e^{\left (-\frac {2 \, {\left (2 \, e^{x} \log \left (2\right )^{2} - 2 \, \log \left (2\right )^{3} - x\right )}}{x}\right )} + {\left (2 \, x - 1\right )} e^{\left (-\frac {4 \, {\left (2 \, e^{x} \log \left (2\right )^{2} - 2 \, \log \left (2\right )^{3} - x\right )}}{x}\right )} + 2 \, x\right )} e^{\left (\frac {4 \, {\left (2 \, e^{x} \log \left (2\right )^{2} - 2 \, \log \left (2\right )^{3} - x\right )}}{x}\right )} \,d x } \]
integrate(((-1+2*x)*exp((-4*log(2)^2*exp(x)+4*log(2)^3+2*x)/x)^2+(2*(4-4*x )*log(2)^2*exp(x)-8*log(2)^3-4*x)*exp((-4*log(2)^2*exp(x)+4*log(2)^3+2*x)/ x)+2*(-4+4*x)*log(2)^2*exp(x)+8*log(2)^3+2*x)/exp((-4*log(2)^2*exp(x)+4*lo g(2)^3+2*x)/x)^2,x, algorithm=\
integrate((8*(x - 1)*e^x*log(2)^2 + 8*log(2)^3 - 4*(2*(x - 1)*e^x*log(2)^2 + 2*log(2)^3 + x)*e^(-2*(2*e^x*log(2)^2 - 2*log(2)^3 - x)/x) + (2*x - 1)* e^(-4*(2*e^x*log(2)^2 - 2*log(2)^3 - x)/x) + 2*x)*e^(4*(2*e^x*log(2)^2 - 2 *log(2)^3 - x)/x), x)
Time = 9.73 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.88 \[ \int e^{-\frac {2 \left (2 x-2 e^x \log (2) \log (4)+2 \log ^2(2) \log (4)\right )}{x}} \left (2 x+e^{\frac {2 \left (2 x-2 e^x \log (2) \log (4)+2 \log ^2(2) \log (4)\right )}{x}} (-1+2 x)+e^x (-4+4 x) \log (2) \log (4)+4 \log ^2(2) \log (4)+e^{\frac {2 x-2 e^x \log (2) \log (4)+2 \log ^2(2) \log (4)}{x}} \left (-4 x+e^x (4-4 x) \log (2) \log (4)-4 \log ^2(2) \log (4)\right )\right ) \, dx=x^2-x-2\,x^2\,{\mathrm {e}}^{\frac {4\,{\mathrm {e}}^x\,{\ln \left (2\right )}^2}{x}-\frac {4\,{\ln \left (2\right )}^3}{x}-2}+x^2\,{\mathrm {e}}^{\frac {8\,{\mathrm {e}}^x\,{\ln \left (2\right )}^2}{x}-\frac {8\,{\ln \left (2\right )}^3}{x}-4} \]
int(exp(-(2*(2*x - 4*exp(x)*log(2)^2 + 4*log(2)^3))/x)*(2*x - exp((2*x - 4 *exp(x)*log(2)^2 + 4*log(2)^3)/x)*(4*x + 8*log(2)^3 + 2*exp(x)*log(2)^2*(4 *x - 4)) + exp((2*(2*x - 4*exp(x)*log(2)^2 + 4*log(2)^3))/x)*(2*x - 1) + 8 *log(2)^3 + 2*exp(x)*log(2)^2*(4*x - 4)),x)