Integrand size = 170, antiderivative size = 35 \[ \int \frac {\frac {e^{4-2 e^x} \left (2 x+\left (2-2 x^2\right ) \log (4)+e^x \left (2 x^2+\left (2 x-2 x^3\right ) \log (4)\right )\right )}{x^2}+\left (-2 x+4 x^2 \log (4)\right ) \log \left (-x+\left (-1+x^2\right ) \log (4)\right )+\frac {e^{2-e^x} \left (2 x-4 x^2 \log (4)+\left (-2 x+\left (-2+2 x^2\right ) \log (4)+e^x \left (-2 x^2+\left (-2 x+2 x^3\right ) \log (4)\right )\right ) \log \left (-x+\left (-1+x^2\right ) \log (4)\right )\right )}{x}}{-9 x^2+\left (-9 x+9 x^3\right ) \log (4)} \, dx=\frac {1}{9} \left (\frac {e^{2-e^x}}{x}-\log \left (-x+\left (-1+x^2\right ) \log (4)\right )\right )^2 \] Output:
1/3*(exp(-ln(x)-exp(x)+2)-ln(2*(x^2-1)*ln(2)-x))*(1/3*exp(-ln(x)-exp(x)+2) -1/3*ln(2*(x^2-1)*ln(2)-x))
\[ \int \frac {\frac {e^{4-2 e^x} \left (2 x+\left (2-2 x^2\right ) \log (4)+e^x \left (2 x^2+\left (2 x-2 x^3\right ) \log (4)\right )\right )}{x^2}+\left (-2 x+4 x^2 \log (4)\right ) \log \left (-x+\left (-1+x^2\right ) \log (4)\right )+\frac {e^{2-e^x} \left (2 x-4 x^2 \log (4)+\left (-2 x+\left (-2+2 x^2\right ) \log (4)+e^x \left (-2 x^2+\left (-2 x+2 x^3\right ) \log (4)\right )\right ) \log \left (-x+\left (-1+x^2\right ) \log (4)\right )\right )}{x}}{-9 x^2+\left (-9 x+9 x^3\right ) \log (4)} \, dx=\int \frac {\frac {e^{4-2 e^x} \left (2 x+\left (2-2 x^2\right ) \log (4)+e^x \left (2 x^2+\left (2 x-2 x^3\right ) \log (4)\right )\right )}{x^2}+\left (-2 x+4 x^2 \log (4)\right ) \log \left (-x+\left (-1+x^2\right ) \log (4)\right )+\frac {e^{2-e^x} \left (2 x-4 x^2 \log (4)+\left (-2 x+\left (-2+2 x^2\right ) \log (4)+e^x \left (-2 x^2+\left (-2 x+2 x^3\right ) \log (4)\right )\right ) \log \left (-x+\left (-1+x^2\right ) \log (4)\right )\right )}{x}}{-9 x^2+\left (-9 x+9 x^3\right ) \log (4)} \, dx \] Input:
Integrate[((E^(4 - 2*E^x)*(2*x + (2 - 2*x^2)*Log[4] + E^x*(2*x^2 + (2*x - 2*x^3)*Log[4])))/x^2 + (-2*x + 4*x^2*Log[4])*Log[-x + (-1 + x^2)*Log[4]] + (E^(2 - E^x)*(2*x - 4*x^2*Log[4] + (-2*x + (-2 + 2*x^2)*Log[4] + E^x*(-2* x^2 + (-2*x + 2*x^3)*Log[4]))*Log[-x + (-1 + x^2)*Log[4]]))/x)/(-9*x^2 + ( -9*x + 9*x^3)*Log[4]),x]
Output:
Integrate[((E^(4 - 2*E^x)*(2*x + (2 - 2*x^2)*Log[4] + E^x*(2*x^2 + (2*x - 2*x^3)*Log[4])))/x^2 + (-2*x + 4*x^2*Log[4])*Log[-x + (-1 + x^2)*Log[4]] + (E^(2 - E^x)*(2*x - 4*x^2*Log[4] + (-2*x + (-2 + 2*x^2)*Log[4] + E^x*(-2* x^2 + (-2*x + 2*x^3)*Log[4]))*Log[-x + (-1 + x^2)*Log[4]]))/x)/(-9*x^2 + ( -9*x + 9*x^3)*Log[4]), 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.
| \(\displaystyle \int \frac {\left (4 x^2 \log (4)-2 x\right ) \log \left (\left (x^2-1\right ) \log (4)-x\right )+\frac {e^{4-2 e^x} \left (\left (2-2 x^2\right ) \log (4)+e^x \left (\left (2 x-2 x^3\right ) \log (4)+2 x^2\right )+2 x\right )}{x^2}+\frac {e^{2-e^x} \left (-4 x^2 \log (4)+\left (\left (2 x^2-2\right ) \log (4)+e^x \left (\left (2 x^3-2 x\right ) \log (4)-2 x^2\right )-2 x\right ) \log \left (\left (x^2-1\right ) \log (4)-x\right )+2 x\right )}{x}}{\left (9 x^3-9 x\right ) \log (4)-9 x^2} \, dx\) |
| \(\Big \downarrow \) 2026 |
| \(\displaystyle \int \frac {\left (4 x^2 \log (4)-2 x\right ) \log \left (\left (x^2-1\right ) \log (4)-x\right )+\frac {e^{4-2 e^x} \left (\left (2-2 x^2\right ) \log (4)+e^x \left (\left (2 x-2 x^3\right ) \log (4)+2 x^2\right )+2 x\right )}{x^2}+\frac {e^{2-e^x} \left (-4 x^2 \log (4)+\left (\left (2 x^2-2\right ) \log (4)+e^x \left (\left (2 x^3-2 x\right ) \log (4)-2 x^2\right )-2 x\right ) \log \left (\left (x^2-1\right ) \log (4)-x\right )+2 x\right )}{x}}{x \left (9 x^2 \log (4)-9 x-9 \log (4)\right )}dx\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle \int \left (\frac {2 e^{2-e^x} (x+1) (1-x) \log (4) \log \left (x^2 \log (4)-x-\log (4)\right )}{9 x^2 \left (x^2 (-\log (4))+x+\log (4)\right )}+\frac {2 (1-x \log (16)) \log \left (x^2 \log (4)-x-\log (4)\right )}{9 \left (x^2 (-\log (4))+x+\log (4)\right )}+\frac {2 e^{2-e^x} \log \left (x^2 \log (4)-x-\log (4)\right )}{9 x \left (x^2 (-\log (4))+x+\log (4)\right )}+\frac {2 e^{x-2 e^x+2} \left (e^{e^x} x \log \left (\left (x^2-1\right ) \log (4)-x\right )-e^2\right )}{9 x^2}+\frac {2 e^{2-e^x}}{9 x \left (x^2 \log (4)-x-\log (4)\right )}+\frac {2 e^{-2 \left (e^x-2\right )}}{9 x^2 \left (x^2 \log (4)-x-\log (4)\right )}-\frac {4 e^{2-e^x} \log (4)}{9 \left (x^2 \log (4)-x-\log (4)\right )}+\frac {2 e^{-2 \left (e^x-2\right )} (-x-1) (1-x) \log (4)}{9 x^3 \left (x^2 (-\log (4))+x+\log (4)\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {2 e^{-2 e^x} \left (e^{e^x} x^2 (x \log (16)-1)+e^{x+2} x \left (x^2 \log (4)-x-\log (4)\right )+e^2 \left (x^2 \log (4)-x-\log (4)\right )\right ) \left (e^2-e^{e^x} x \log \left (x^2 \log (4)-x-\log (4)\right )\right )}{9 x^3 \left (x^2 (-\log (4))+x+\log (4)\right )}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2}{9} \int -\frac {e^{-2 e^x} \left (e^{e^x} (1-x \log (16)) x^2+e^{x+2} \left (-\log (4) x^2+x+\log (4)\right ) x+e^2 \left (-\log (4) x^2+x+\log (4)\right )\right ) \left (e^2-e^{e^x} x \log \left (\log (4) x^2-x-\log (4)\right )\right )}{x^3 \left (-\log (4) x^2+x+\log (4)\right )}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {2}{9} \int \frac {e^{-2 e^x} \left (e^{e^x} (1-x \log (16)) x^2+e^{x+2} \left (-\log (4) x^2+x+\log (4)\right ) x+e^2 \left (-\log (4) x^2+x+\log (4)\right )\right ) \left (e^2-e^{e^x} x \log \left (\log (4) x^2-x-\log (4)\right )\right )}{x^3 \left (-\log (4) x^2+x+\log (4)\right )}dx\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle -\frac {2}{9} \int \left (\frac {e^{x-2 e^x+2} \left (e^2-e^{e^x} x \log \left (\log (4) x^2-x-\log (4)\right )\right )}{x^2}-\frac {e^{-2 e^x} \left (e^{e^x} \log (16) x^3-e^{e^x} x^2+e^2 \log (4) x^2-e^2 x-e^2 \log (4)\right ) \left (e^{e^x} x \log \left (\log (4) x^2-x-\log (4)\right )-e^2\right )}{x^3 \left (\log (4) x^2-x-\log (4)\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {2}{9} \left (\frac {\log (16) \log ^2\left (\log (16) x-\sqrt {1+4 \log ^2(4)}-1\right )}{4 \log (4)}+\frac {\log (16) \log ^2\left (\log (16) x+\sqrt {1+4 \log ^2(4)}-1\right )}{4 \log (4)}-\frac {\log (16) \log \left (\log (4) x^2-x-\log (4)\right ) \log \left (2 \log (4) x-\sqrt {1+4 \log ^2(4)}-1\right )}{2 \log (4)}+\frac {\log (16) \log \left (-\frac {-2 \log (4) x-\sqrt {1+4 \log ^2(4)}+1}{2 \sqrt {1+4 \log ^2(4)}}\right ) \log \left (2 \log (4) x-\sqrt {1+4 \log ^2(4)}-1\right )}{2 \log (4)}-\frac {\log (16) \log \left (\log (4) x^2-x-\log (4)\right ) \log \left (2 \log (4) x+\sqrt {1+4 \log ^2(4)}-1\right )}{2 \log (4)}+\frac {\log (16) \log \left (\frac {-2 \log (4) x+\sqrt {1+4 \log ^2(4)}+1}{2 \sqrt {1+4 \log ^2(4)}}\right ) \log \left (2 \log (4) x+\sqrt {1+4 \log ^2(4)}-1\right )}{2 \log (4)}+\frac {\log (16) \operatorname {PolyLog}\left (2,\frac {-2 \log (4) x+\sqrt {1+4 \log ^2(4)}+1}{2 \sqrt {1+4 \log ^2(4)}}\right )}{2 \log (4)}+\frac {\log (16) \operatorname {PolyLog}\left (2,-\frac {-\log (16) x-\sqrt {1+4 \log ^2(4)}+1}{2 \sqrt {1+4 \log ^2(4)}}\right )}{2 \log (4)}+\int \frac {e^{-2 \left (-2+e^x\right )}}{x^3}dx-\log \left (\log (4) x^2-x-\log (4)\right ) \int \frac {e^{2-e^x}}{x^2}dx+\int \frac {e^{x-2 e^x+4}}{x^2}dx+\frac {\int \frac {e^{2-e^x}}{x}dx}{\log (4)}-\log \left (\log (4) x^2-x-\log (4)\right ) \int \frac {e^{x-e^x+2}}{x}dx+\frac {\left (\sqrt {1+4 \log ^2(4)} \log ^2(16)-\log (4) \log (256)\right ) \int \frac {e^{2-e^x}}{2 \log (4) x-\sqrt {1+4 \log ^2(4)}-1}dx}{4 \log ^2(4)}-\frac {\left (\sqrt {1+4 \log ^2(4)} \log ^2(16)+\log (4) \log (256)\right ) \int \frac {e^{2-e^x}}{2 \log (4) x+\sqrt {1+4 \log ^2(4)}-1}dx}{4 \log ^2(4)}+\left (1+\frac {1}{\sqrt {1+4 \log ^2(4)}}\right ) \log (16) \int \frac {\int \frac {e^{2-e^x}}{x^2}dx}{2 \log (4) x-\sqrt {1+4 \log ^2(4)}-1}dx+\frac {2 \log (4) \int \frac {\int \frac {e^{2-e^x}}{x^2}dx}{-2 \log (4) x+\sqrt {1+4 \log ^2(4)}+1}dx}{\sqrt {1+4 \log ^2(4)}}+\left (1-\frac {1}{\sqrt {1+4 \log ^2(4)}}\right ) \log (16) \int \frac {\int \frac {e^{2-e^x}}{x^2}dx}{2 \log (4) x+\sqrt {1+4 \log ^2(4)}-1}dx+\frac {2 \log (4) \int \frac {\int \frac {e^{2-e^x}}{x^2}dx}{\log (16) x+\sqrt {1+4 \log ^2(4)}-1}dx}{\sqrt {1+4 \log ^2(4)}}+\left (1+\frac {1}{\sqrt {1+4 \log ^2(4)}}\right ) \log (16) \int \frac {\int \frac {e^{x-e^x+2}}{x}dx}{2 \log (4) x-\sqrt {1+4 \log ^2(4)}-1}dx+\frac {2 \log (4) \int \frac {\int \frac {e^{x-e^x+2}}{x}dx}{-2 \log (4) x+\sqrt {1+4 \log ^2(4)}+1}dx}{\sqrt {1+4 \log ^2(4)}}+\left (1-\frac {1}{\sqrt {1+4 \log ^2(4)}}\right ) \log (16) \int \frac {\int \frac {e^{x-e^x+2}}{x}dx}{2 \log (4) x+\sqrt {1+4 \log ^2(4)}-1}dx+\frac {2 \log (4) \int \frac {\int \frac {e^{x-e^x+2}}{x}dx}{\log (16) x+\sqrt {1+4 \log ^2(4)}-1}dx}{\sqrt {1+4 \log ^2(4)}}\right )\) |
Input:
Int[((E^(4 - 2*E^x)*(2*x + (2 - 2*x^2)*Log[4] + E^x*(2*x^2 + (2*x - 2*x^3) *Log[4])))/x^2 + (-2*x + 4*x^2*Log[4])*Log[-x + (-1 + x^2)*Log[4]] + (E^(2 - E^x)*(2*x - 4*x^2*Log[4] + (-2*x + (-2 + 2*x^2)*Log[4] + E^x*(-2*x^2 + (-2*x + 2*x^3)*Log[4]))*Log[-x + (-1 + x^2)*Log[4]]))/x)/(-9*x^2 + (-9*x + 9*x^3)*Log[4]),x]
Output:
$Aborted
Time = 199.82 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.66
| method | result | size |
| risch | \(\frac {{\ln \left (2 \left (x^{2}-1\right ) \ln \left (2\right )-x \right )}^{2}}{9}+\frac {{\mathrm e}^{-2 \,{\mathrm e}^{x}+4}}{9 x^{2}}-\frac {2 \ln \left (2 \left (x^{2}-1\right ) \ln \left (2\right )-x \right ) {\mathrm e}^{-{\mathrm e}^{x}+2}}{9 x}\) | \(58\) |
| parallelrisch | \(\frac {\frac {4 \,{\mathrm e}^{-2 \,{\mathrm e}^{x}+4} \ln \left (2\right )}{x^{2}}-8 \,{\mathrm e}^{-\ln \left (x \right )-{\mathrm e}^{x}+2} \ln \left (2 x^{2} \ln \left (2\right )-2 \ln \left (2\right )-x \right ) \ln \left (2\right )+4 \ln \left (2\right ) \ln \left (2 x^{2} \ln \left (2\right )-2 \ln \left (2\right )-x \right )^{2}}{36 \ln \left (2\right )}\) | \(78\) |
Input:
int((((2*(-2*x^3+2*x)*ln(2)+2*x^2)*exp(x)+2*(-2*x^2+2)*ln(2)+2*x)*exp(-ln( x)-exp(x)+2)^2+(((2*(2*x^3-2*x)*ln(2)-2*x^2)*exp(x)+2*(2*x^2-2)*ln(2)-2*x) *ln(2*(x^2-1)*ln(2)-x)-8*x^2*ln(2)+2*x)*exp(-ln(x)-exp(x)+2)+(8*x^2*ln(2)- 2*x)*ln(2*(x^2-1)*ln(2)-x))/(2*(9*x^3-9*x)*ln(2)-9*x^2),x,method=_RETURNVE RBOSE)
Output:
1/9*ln(2*(x^2-1)*ln(2)-x)^2+1/9/x^2*exp(-2*exp(x)+4)-2/9/x*ln(2*(x^2-1)*ln (2)-x)*exp(-exp(x)+2)
Time = 0.11 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.69 \[ \int \frac {\frac {e^{4-2 e^x} \left (2 x+\left (2-2 x^2\right ) \log (4)+e^x \left (2 x^2+\left (2 x-2 x^3\right ) \log (4)\right )\right )}{x^2}+\left (-2 x+4 x^2 \log (4)\right ) \log \left (-x+\left (-1+x^2\right ) \log (4)\right )+\frac {e^{2-e^x} \left (2 x-4 x^2 \log (4)+\left (-2 x+\left (-2+2 x^2\right ) \log (4)+e^x \left (-2 x^2+\left (-2 x+2 x^3\right ) \log (4)\right )\right ) \log \left (-x+\left (-1+x^2\right ) \log (4)\right )\right )}{x}}{-9 x^2+\left (-9 x+9 x^3\right ) \log (4)} \, dx=-\frac {2}{9} \, e^{\left (-e^{x} - \log \left (x\right ) + 2\right )} \log \left (2 \, {\left (x^{2} - 1\right )} \log \left (2\right ) - x\right ) + \frac {1}{9} \, \log \left (2 \, {\left (x^{2} - 1\right )} \log \left (2\right ) - x\right )^{2} + \frac {1}{9} \, e^{\left (-2 \, e^{x} - 2 \, \log \left (x\right ) + 4\right )} \] Input:
integrate((((2*(-2*x^3+2*x)*log(2)+2*x^2)*exp(x)+2*(-2*x^2+2)*log(2)+2*x)* exp(-log(x)-exp(x)+2)^2+(((2*(2*x^3-2*x)*log(2)-2*x^2)*exp(x)+2*(2*x^2-2)* log(2)-2*x)*log(2*(x^2-1)*log(2)-x)-8*x^2*log(2)+2*x)*exp(-log(x)-exp(x)+2 )+(8*x^2*log(2)-2*x)*log(2*(x^2-1)*log(2)-x))/(2*(9*x^3-9*x)*log(2)-9*x^2) ,x, algorithm="fricas")
Output:
-2/9*e^(-e^x - log(x) + 2)*log(2*(x^2 - 1)*log(2) - x) + 1/9*log(2*(x^2 - 1)*log(2) - x)^2 + 1/9*e^(-2*e^x - 2*log(x) + 4)
Leaf count of result is larger than twice the leaf count of optimal. 58 vs. \(2 (26) = 52\).
Time = 0.34 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.66 \[ \int \frac {\frac {e^{4-2 e^x} \left (2 x+\left (2-2 x^2\right ) \log (4)+e^x \left (2 x^2+\left (2 x-2 x^3\right ) \log (4)\right )\right )}{x^2}+\left (-2 x+4 x^2 \log (4)\right ) \log \left (-x+\left (-1+x^2\right ) \log (4)\right )+\frac {e^{2-e^x} \left (2 x-4 x^2 \log (4)+\left (-2 x+\left (-2+2 x^2\right ) \log (4)+e^x \left (-2 x^2+\left (-2 x+2 x^3\right ) \log (4)\right )\right ) \log \left (-x+\left (-1+x^2\right ) \log (4)\right )\right )}{x}}{-9 x^2+\left (-9 x+9 x^3\right ) \log (4)} \, dx=\frac {\log {\left (- x + \left (2 x^{2} - 2\right ) \log {\left (2 \right )} \right )}^{2}}{9} + \frac {- 18 x^{2} e^{2 - e^{x}} \log {\left (- x + \left (2 x^{2} - 2\right ) \log {\left (2 \right )} \right )} + 9 x e^{4 - 2 e^{x}}}{81 x^{3}} \] Input:
integrate((((2*(-2*x**3+2*x)*ln(2)+2*x**2)*exp(x)+2*(-2*x**2+2)*ln(2)+2*x) *exp(-ln(x)-exp(x)+2)**2+(((2*(2*x**3-2*x)*ln(2)-2*x**2)*exp(x)+2*(2*x**2- 2)*ln(2)-2*x)*ln(2*(x**2-1)*ln(2)-x)-8*x**2*ln(2)+2*x)*exp(-ln(x)-exp(x)+2 )+(8*x**2*ln(2)-2*x)*ln(2*(x**2-1)*ln(2)-x))/(2*(9*x**3-9*x)*ln(2)-9*x**2) ,x)
Output:
log(-x + (2*x**2 - 2)*log(2))**2/9 + (-18*x**2*exp(2 - exp(x))*log(-x + (2 *x**2 - 2)*log(2)) + 9*x*exp(4 - 2*exp(x)))/(81*x**3)
Time = 0.19 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.74 \[ \int \frac {\frac {e^{4-2 e^x} \left (2 x+\left (2-2 x^2\right ) \log (4)+e^x \left (2 x^2+\left (2 x-2 x^3\right ) \log (4)\right )\right )}{x^2}+\left (-2 x+4 x^2 \log (4)\right ) \log \left (-x+\left (-1+x^2\right ) \log (4)\right )+\frac {e^{2-e^x} \left (2 x-4 x^2 \log (4)+\left (-2 x+\left (-2+2 x^2\right ) \log (4)+e^x \left (-2 x^2+\left (-2 x+2 x^3\right ) \log (4)\right )\right ) \log \left (-x+\left (-1+x^2\right ) \log (4)\right )\right )}{x}}{-9 x^2+\left (-9 x+9 x^3\right ) \log (4)} \, dx=\frac {x^{2} \log \left (2 \, x^{2} \log \left (2\right ) - x - 2 \, \log \left (2\right )\right )^{2} - 2 \, x e^{\left (-e^{x} + 2\right )} \log \left (2 \, x^{2} \log \left (2\right ) - x - 2 \, \log \left (2\right )\right ) + e^{\left (-2 \, e^{x} + 4\right )}}{9 \, x^{2}} \] Input:
integrate((((2*(-2*x^3+2*x)*log(2)+2*x^2)*exp(x)+2*(-2*x^2+2)*log(2)+2*x)* exp(-log(x)-exp(x)+2)^2+(((2*(2*x^3-2*x)*log(2)-2*x^2)*exp(x)+2*(2*x^2-2)* log(2)-2*x)*log(2*(x^2-1)*log(2)-x)-8*x^2*log(2)+2*x)*exp(-log(x)-exp(x)+2 )+(8*x^2*log(2)-2*x)*log(2*(x^2-1)*log(2)-x))/(2*(9*x^3-9*x)*log(2)-9*x^2) ,x, algorithm="maxima")
Output:
1/9*(x^2*log(2*x^2*log(2) - x - 2*log(2))^2 - 2*x*e^(-e^x + 2)*log(2*x^2*l og(2) - x - 2*log(2)) + e^(-2*e^x + 4))/x^2
\[ \int \frac {\frac {e^{4-2 e^x} \left (2 x+\left (2-2 x^2\right ) \log (4)+e^x \left (2 x^2+\left (2 x-2 x^3\right ) \log (4)\right )\right )}{x^2}+\left (-2 x+4 x^2 \log (4)\right ) \log \left (-x+\left (-1+x^2\right ) \log (4)\right )+\frac {e^{2-e^x} \left (2 x-4 x^2 \log (4)+\left (-2 x+\left (-2+2 x^2\right ) \log (4)+e^x \left (-2 x^2+\left (-2 x+2 x^3\right ) \log (4)\right )\right ) \log \left (-x+\left (-1+x^2\right ) \log (4)\right )\right )}{x}}{-9 x^2+\left (-9 x+9 x^3\right ) \log (4)} \, dx=\int { \frac {2 \, {\left ({\left (4 \, x^{2} \log \left (2\right ) + {\left ({\left (x^{2} - 2 \, {\left (x^{3} - x\right )} \log \left (2\right )\right )} e^{x} - 2 \, {\left (x^{2} - 1\right )} \log \left (2\right ) + x\right )} \log \left (2 \, {\left (x^{2} - 1\right )} \log \left (2\right ) - x\right ) - x\right )} e^{\left (-e^{x} - \log \left (x\right ) + 2\right )} - {\left ({\left (x^{2} - 2 \, {\left (x^{3} - x\right )} \log \left (2\right )\right )} e^{x} - 2 \, {\left (x^{2} - 1\right )} \log \left (2\right ) + x\right )} e^{\left (-2 \, e^{x} - 2 \, \log \left (x\right ) + 4\right )} - {\left (4 \, x^{2} \log \left (2\right ) - x\right )} \log \left (2 \, {\left (x^{2} - 1\right )} \log \left (2\right ) - x\right )\right )}}{9 \, {\left (x^{2} - 2 \, {\left (x^{3} - x\right )} \log \left (2\right )\right )}} \,d x } \] Input:
integrate((((2*(-2*x^3+2*x)*log(2)+2*x^2)*exp(x)+2*(-2*x^2+2)*log(2)+2*x)* exp(-log(x)-exp(x)+2)^2+(((2*(2*x^3-2*x)*log(2)-2*x^2)*exp(x)+2*(2*x^2-2)* log(2)-2*x)*log(2*(x^2-1)*log(2)-x)-8*x^2*log(2)+2*x)*exp(-log(x)-exp(x)+2 )+(8*x^2*log(2)-2*x)*log(2*(x^2-1)*log(2)-x))/(2*(9*x^3-9*x)*log(2)-9*x^2) ,x, algorithm="giac")
Output:
integrate(2/9*((4*x^2*log(2) + ((x^2 - 2*(x^3 - x)*log(2))*e^x - 2*(x^2 - 1)*log(2) + x)*log(2*(x^2 - 1)*log(2) - x) - x)*e^(-e^x - log(x) + 2) - (( x^2 - 2*(x^3 - x)*log(2))*e^x - 2*(x^2 - 1)*log(2) + x)*e^(-2*e^x - 2*log( x) + 4) - (4*x^2*log(2) - x)*log(2*(x^2 - 1)*log(2) - x))/(x^2 - 2*(x^3 - x)*log(2)), x)
Timed out. \[ \int \frac {\frac {e^{4-2 e^x} \left (2 x+\left (2-2 x^2\right ) \log (4)+e^x \left (2 x^2+\left (2 x-2 x^3\right ) \log (4)\right )\right )}{x^2}+\left (-2 x+4 x^2 \log (4)\right ) \log \left (-x+\left (-1+x^2\right ) \log (4)\right )+\frac {e^{2-e^x} \left (2 x-4 x^2 \log (4)+\left (-2 x+\left (-2+2 x^2\right ) \log (4)+e^x \left (-2 x^2+\left (-2 x+2 x^3\right ) \log (4)\right )\right ) \log \left (-x+\left (-1+x^2\right ) \log (4)\right )\right )}{x}}{-9 x^2+\left (-9 x+9 x^3\right ) \log (4)} \, dx=\int \frac {{\mathrm {e}}^{2-\ln \left (x\right )-{\mathrm {e}}^x}\,\left (8\,x^2\,\ln \left (2\right )-2\,x+\ln \left (2\,\ln \left (2\right )\,\left (x^2-1\right )-x\right )\,\left (2\,x+{\mathrm {e}}^x\,\left (2\,\ln \left (2\right )\,\left (2\,x-2\,x^3\right )+2\,x^2\right )-2\,\ln \left (2\right )\,\left (2\,x^2-2\right )\right )\right )-{\mathrm {e}}^{4-2\,\ln \left (x\right )-2\,{\mathrm {e}}^x}\,\left (2\,x+{\mathrm {e}}^x\,\left (2\,\ln \left (2\right )\,\left (2\,x-2\,x^3\right )+2\,x^2\right )-2\,\ln \left (2\right )\,\left (2\,x^2-2\right )\right )+\ln \left (2\,\ln \left (2\right )\,\left (x^2-1\right )-x\right )\,\left (2\,x-8\,x^2\,\ln \left (2\right )\right )}{2\,\ln \left (2\right )\,\left (9\,x-9\,x^3\right )+9\,x^2} \,d x \] Input:
int((exp(2 - log(x) - exp(x))*(8*x^2*log(2) - 2*x + log(2*log(2)*(x^2 - 1) - x)*(2*x + exp(x)*(2*log(2)*(2*x - 2*x^3) + 2*x^2) - 2*log(2)*(2*x^2 - 2 ))) - exp(4 - 2*log(x) - 2*exp(x))*(2*x + exp(x)*(2*log(2)*(2*x - 2*x^3) + 2*x^2) - 2*log(2)*(2*x^2 - 2)) + log(2*log(2)*(x^2 - 1) - x)*(2*x - 8*x^2 *log(2)))/(2*log(2)*(9*x - 9*x^3) + 9*x^2),x)
Output:
int((exp(2 - log(x) - exp(x))*(8*x^2*log(2) - 2*x + log(2*log(2)*(x^2 - 1) - x)*(2*x + exp(x)*(2*log(2)*(2*x - 2*x^3) + 2*x^2) - 2*log(2)*(2*x^2 - 2 ))) - exp(4 - 2*log(x) - 2*exp(x))*(2*x + exp(x)*(2*log(2)*(2*x - 2*x^3) + 2*x^2) - 2*log(2)*(2*x^2 - 2)) + log(2*log(2)*(x^2 - 1) - x)*(2*x - 8*x^2 *log(2)))/(2*log(2)*(9*x - 9*x^3) + 9*x^2), x)
Time = 0.19 (sec) , antiderivative size = 74, normalized size of antiderivative = 2.11 \[ \int \frac {\frac {e^{4-2 e^x} \left (2 x+\left (2-2 x^2\right ) \log (4)+e^x \left (2 x^2+\left (2 x-2 x^3\right ) \log (4)\right )\right )}{x^2}+\left (-2 x+4 x^2 \log (4)\right ) \log \left (-x+\left (-1+x^2\right ) \log (4)\right )+\frac {e^{2-e^x} \left (2 x-4 x^2 \log (4)+\left (-2 x+\left (-2+2 x^2\right ) \log (4)+e^x \left (-2 x^2+\left (-2 x+2 x^3\right ) \log (4)\right )\right ) \log \left (-x+\left (-1+x^2\right ) \log (4)\right )\right )}{x}}{-9 x^2+\left (-9 x+9 x^3\right ) \log (4)} \, dx=\frac {e^{2 e^{x}} \mathrm {log}\left (2 \,\mathrm {log}\left (2\right ) x^{2}-2 \,\mathrm {log}\left (2\right )-x \right )^{2} x^{2}-2 e^{e^{x}} \mathrm {log}\left (2 \,\mathrm {log}\left (2\right ) x^{2}-2 \,\mathrm {log}\left (2\right )-x \right ) e^{2} x +e^{4}}{9 e^{2 e^{x}} x^{2}} \] Input:
int((((2*(-2*x^3+2*x)*log(2)+2*x^2)*exp(x)+2*(-2*x^2+2)*log(2)+2*x)*exp(-l og(x)-exp(x)+2)^2+(((2*(2*x^3-2*x)*log(2)-2*x^2)*exp(x)+2*(2*x^2-2)*log(2) -2*x)*log(2*(x^2-1)*log(2)-x)-8*x^2*log(2)+2*x)*exp(-log(x)-exp(x)+2)+(8*x ^2*log(2)-2*x)*log(2*(x^2-1)*log(2)-x))/(2*(9*x^3-9*x)*log(2)-9*x^2),x)
Output:
(e**(2*e**x)*log(2*log(2)*x**2 - 2*log(2) - x)**2*x**2 - 2*e**(e**x)*log(2 *log(2)*x**2 - 2*log(2) - x)*e**2*x + e**4)/(9*e**(2*e**x)*x**2)