Integrand size = 132, antiderivative size = 33 \[ \int \frac {-6 x+e^x (12+3 \log (3))+e^{2 e^{-x}+x} \left (24-2 x^2+\left (6-x^2\right ) \log (3)\right )+\left (-3 e^x+e^{2 e^{-x}+x} \left (-6+x^2\right )\right ) \log \left (-\frac {e^{2 e^{-x}} x}{-3+e^{2 e^{-x}} \left (-6+x^2\right )}\right )}{-3 e^x x^2+e^{2 e^{-x}+x} \left (-6 x^2+x^4\right )} \, dx=\frac {3+\log (3)-\log \left (\frac {x}{6+3 e^{-2 e^{-x}}-x^2}\right )}{x} \] Output:
(3-ln(x/(3/exp(2/exp(x))+6-x^2))+ln(3))/x
Time = 0.11 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.36 \[ \int \frac {-6 x+e^x (12+3 \log (3))+e^{2 e^{-x}+x} \left (24-2 x^2+\left (6-x^2\right ) \log (3)\right )+\left (-3 e^x+e^{2 e^{-x}+x} \left (-6+x^2\right )\right ) \log \left (-\frac {e^{2 e^{-x}} x}{-3+e^{2 e^{-x}} \left (-6+x^2\right )}\right )}{-3 e^x x^2+e^{2 e^{-x}+x} \left (-6 x^2+x^4\right )} \, dx=\frac {18+\log (729)-6 \log \left (-\frac {e^{2 e^{-x}} x}{-3+e^{2 e^{-x}} \left (-6+x^2\right )}\right )}{6 x} \] Input:
Integrate[(-6*x + E^x*(12 + 3*Log[3]) + E^(2/E^x + x)*(24 - 2*x^2 + (6 - x ^2)*Log[3]) + (-3*E^x + E^(2/E^x + x)*(-6 + x^2))*Log[-((E^(2/E^x)*x)/(-3 + E^(2/E^x)*(-6 + x^2)))])/(-3*E^x*x^2 + E^(2/E^x + x)*(-6*x^2 + x^4)),x]
Output:
(18 + Log[729] - 6*Log[-((E^(2/E^x)*x)/(-3 + E^(2/E^x)*(-6 + x^2)))])/(6*x )
Time = 4.37 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.58, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.015, Rules used = {7293, 2009}
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.
| \(\displaystyle \int \frac {e^{x+2 e^{-x}} \left (-2 x^2+\left (6-x^2\right ) \log (3)+24\right )+\left (e^{x+2 e^{-x}} \left (x^2-6\right )-3 e^x\right ) \log \left (-\frac {e^{2 e^{-x}} x}{e^{2 e^{-x}} \left (x^2-6\right )-3}\right )-6 x+e^x (12+3 \log (3))}{e^{x+2 e^{-x}} \left (x^4-6 x^2\right )-3 e^x x^2} \, dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {-e^{2 e^{-x}} x^2 \log \left (-\frac {e^{2 e^{-x}} x}{e^{2 e^{-x}} \left (x^2-6\right )-3}\right )+2 e^{2 e^{-x}} x^2 \left (1+\frac {\log (3)}{2}\right )+6 e^{2 e^{-x}} \log \left (-\frac {e^{2 e^{-x}} x}{e^{2 e^{-x}} \left (x^2-6\right )-3}\right )+3 \log \left (-\frac {e^{2 e^{-x}} x}{e^{2 e^{-x}} \left (x^2-6\right )-3}\right )-24 e^{2 e^{-x}} \left (1+\frac {\log (3)}{4}\right )-12 \left (1+\frac {\log (3)}{4}\right )}{x^2 \left (-e^{2 e^{-x}} x^2+6 e^{2 e^{-x}}+3\right )}-\frac {6 e^{-x}}{x \left (e^{2 e^{-x}} x^2-6 e^{2 e^{-x}}-3\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\log \left (\frac {e^{2 e^{-x}} x}{e^{2 e^{-x}} \left (6-x^2\right )+3}\right )}{x}-\frac {1}{x}+\frac {4+\log (3)}{x}\) |
Input:
Int[(-6*x + E^x*(12 + 3*Log[3]) + E^(2/E^x + x)*(24 - 2*x^2 + (6 - x^2)*Lo g[3]) + (-3*E^x + E^(2/E^x + x)*(-6 + x^2))*Log[-((E^(2/E^x)*x)/(-3 + E^(2 /E^x)*(-6 + x^2)))])/(-3*E^x*x^2 + E^(2/E^x + x)*(-6*x^2 + x^4)),x]
Output:
-x^(-1) + (4 + Log[3])/x - Log[(E^(2/E^x)*x)/(3 + E^(2/E^x)*(6 - x^2))]/x
Time = 69.62 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.76
| method | result | size |
| parallelrisch | \(\frac {\left (\ln \left (3\right ) {\mathrm e}^{x}-{\mathrm e}^{x} \ln \left (-\frac {x \,{\mathrm e}^{2 \,{\mathrm e}^{-x}}}{{\mathrm e}^{2 \,{\mathrm e}^{-x}} x^{2}-6 \,{\mathrm e}^{2 \,{\mathrm e}^{-x}}-3}\right )+3 \,{\mathrm e}^{x}\right ) {\mathrm e}^{-x}}{x}\) | \(58\) |
| risch | \(-\frac {\ln \left ({\mathrm e}^{2 \,{\mathrm e}^{-x}}\right )}{x}+\frac {-2 i \pi -i \pi \operatorname {csgn}\left (\frac {i x \,{\mathrm e}^{2 \,{\mathrm e}^{-x}}}{{\mathrm e}^{2 \,{\mathrm e}^{-x}} x^{2}-6 \,{\mathrm e}^{2 \,{\mathrm e}^{-x}}-3}\right )^{3}-i \pi \,\operatorname {csgn}\left (\frac {i}{{\mathrm e}^{2 \,{\mathrm e}^{-x}} x^{2}-6 \,{\mathrm e}^{2 \,{\mathrm e}^{-x}}-3}\right ) \operatorname {csgn}\left (\frac {i {\mathrm e}^{2 \,{\mathrm e}^{-x}}}{{\mathrm e}^{2 \,{\mathrm e}^{-x}} x^{2}-6 \,{\mathrm e}^{2 \,{\mathrm e}^{-x}}-3}\right )^{2}+2 i \pi \operatorname {csgn}\left (\frac {i x \,{\mathrm e}^{2 \,{\mathrm e}^{-x}}}{{\mathrm e}^{2 \,{\mathrm e}^{-x}} x^{2}-6 \,{\mathrm e}^{2 \,{\mathrm e}^{-x}}-3}\right )^{2}-i \pi \,\operatorname {csgn}\left (\frac {i {\mathrm e}^{2 \,{\mathrm e}^{-x}}}{{\mathrm e}^{2 \,{\mathrm e}^{-x}} x^{2}-6 \,{\mathrm e}^{2 \,{\mathrm e}^{-x}}-3}\right ) \operatorname {csgn}\left (\frac {i x \,{\mathrm e}^{2 \,{\mathrm e}^{-x}}}{{\mathrm e}^{2 \,{\mathrm e}^{-x}} x^{2}-6 \,{\mathrm e}^{2 \,{\mathrm e}^{-x}}-3}\right )^{2}-i \pi \operatorname {csgn}\left (\frac {i x \,{\mathrm e}^{2 \,{\mathrm e}^{-x}}}{{\mathrm e}^{2 \,{\mathrm e}^{-x}} x^{2}-6 \,{\mathrm e}^{2 \,{\mathrm e}^{-x}}-3}\right )^{2} \operatorname {csgn}\left (i x \right )-i \pi \,\operatorname {csgn}\left (i {\mathrm e}^{2 \,{\mathrm e}^{-x}}\right ) \operatorname {csgn}\left (\frac {i {\mathrm e}^{2 \,{\mathrm e}^{-x}}}{{\mathrm e}^{2 \,{\mathrm e}^{-x}} x^{2}-6 \,{\mathrm e}^{2 \,{\mathrm e}^{-x}}-3}\right )^{2}+i \pi \operatorname {csgn}\left (\frac {i {\mathrm e}^{2 \,{\mathrm e}^{-x}}}{{\mathrm e}^{2 \,{\mathrm e}^{-x}} x^{2}-6 \,{\mathrm e}^{2 \,{\mathrm e}^{-x}}-3}\right )^{3}+i \pi \,\operatorname {csgn}\left (\frac {i {\mathrm e}^{2 \,{\mathrm e}^{-x}}}{{\mathrm e}^{2 \,{\mathrm e}^{-x}} x^{2}-6 \,{\mathrm e}^{2 \,{\mathrm e}^{-x}}-3}\right ) \operatorname {csgn}\left (\frac {i x \,{\mathrm e}^{2 \,{\mathrm e}^{-x}}}{{\mathrm e}^{2 \,{\mathrm e}^{-x}} x^{2}-6 \,{\mathrm e}^{2 \,{\mathrm e}^{-x}}-3}\right ) \operatorname {csgn}\left (i x \right )+i \pi \,\operatorname {csgn}\left (\frac {i}{{\mathrm e}^{2 \,{\mathrm e}^{-x}} x^{2}-6 \,{\mathrm e}^{2 \,{\mathrm e}^{-x}}-3}\right ) \operatorname {csgn}\left (i {\mathrm e}^{2 \,{\mathrm e}^{-x}}\right ) \operatorname {csgn}\left (\frac {i {\mathrm e}^{2 \,{\mathrm e}^{-x}}}{{\mathrm e}^{2 \,{\mathrm e}^{-x}} x^{2}-6 \,{\mathrm e}^{2 \,{\mathrm e}^{-x}}-3}\right )+6+2 \ln \left (3\right )-2 \ln \left (x \right )+2 \ln \left ({\mathrm e}^{2 \,{\mathrm e}^{-x}} x^{2}-6 \,{\mathrm e}^{2 \,{\mathrm e}^{-x}}-3\right )}{2 x}\) | \(587\) |
Input:
int((((x^2-6)*exp(x)*exp(2/exp(x))-3*exp(x))*ln(-x*exp(2/exp(x))/((x^2-6)* exp(2/exp(x))-3))+((-x^2+6)*ln(3)-2*x^2+24)*exp(x)*exp(2/exp(x))+(3*ln(3)+ 12)*exp(x)-6*x)/((x^4-6*x^2)*exp(x)*exp(2/exp(x))-3*exp(x)*x^2),x,method=_ RETURNVERBOSE)
Output:
(ln(3)*exp(x)-exp(x)*ln(-x*exp(2/exp(x))/(x^2*exp(2/exp(x))-6*exp(2/exp(x) )-3))+3*exp(x))/exp(x)/x
Time = 0.09 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.55 \[ \int \frac {-6 x+e^x (12+3 \log (3))+e^{2 e^{-x}+x} \left (24-2 x^2+\left (6-x^2\right ) \log (3)\right )+\left (-3 e^x+e^{2 e^{-x}+x} \left (-6+x^2\right )\right ) \log \left (-\frac {e^{2 e^{-x}} x}{-3+e^{2 e^{-x}} \left (-6+x^2\right )}\right )}{-3 e^x x^2+e^{2 e^{-x}+x} \left (-6 x^2+x^4\right )} \, dx=\frac {\log \left (3\right ) - \log \left (-\frac {x e^{\left ({\left (x e^{x} + 2\right )} e^{\left (-x\right )}\right )}}{{\left (x^{2} - 6\right )} e^{\left ({\left (x e^{x} + 2\right )} e^{\left (-x\right )}\right )} - 3 \, e^{x}}\right ) + 3}{x} \] Input:
integrate((((x^2-6)*exp(x)*exp(2/exp(x))-3*exp(x))*log(-x*exp(2/exp(x))/(( x^2-6)*exp(2/exp(x))-3))+((-x^2+6)*log(3)-2*x^2+24)*exp(x)*exp(2/exp(x))+( 3*log(3)+12)*exp(x)-6*x)/((x^4-6*x^2)*exp(x)*exp(2/exp(x))-3*exp(x)*x^2),x , algorithm="fricas")
Output:
(log(3) - log(-x*e^((x*e^x + 2)*e^(-x))/((x^2 - 6)*e^((x*e^x + 2)*e^(-x)) - 3*e^x)) + 3)/x
Time = 0.53 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.09 \[ \int \frac {-6 x+e^x (12+3 \log (3))+e^{2 e^{-x}+x} \left (24-2 x^2+\left (6-x^2\right ) \log (3)\right )+\left (-3 e^x+e^{2 e^{-x}+x} \left (-6+x^2\right )\right ) \log \left (-\frac {e^{2 e^{-x}} x}{-3+e^{2 e^{-x}} \left (-6+x^2\right )}\right )}{-3 e^x x^2+e^{2 e^{-x}+x} \left (-6 x^2+x^4\right )} \, dx=- \frac {\log {\left (- \frac {x e^{2 e^{- x}}}{\left (x^{2} - 6\right ) e^{2 e^{- x}} - 3} \right )}}{x} - \frac {-3 - \log {\left (3 \right )}}{x} \] Input:
integrate((((x**2-6)*exp(x)*exp(2/exp(x))-3*exp(x))*ln(-x*exp(2/exp(x))/(( x**2-6)*exp(2/exp(x))-3))+((-x**2+6)*ln(3)-2*x**2+24)*exp(x)*exp(2/exp(x)) +(3*ln(3)+12)*exp(x)-6*x)/((x**4-6*x**2)*exp(x)*exp(2/exp(x))-3*exp(x)*x** 2),x)
Output:
-log(-x*exp(2*exp(-x))/((x**2 - 6)*exp(2*exp(-x)) - 3))/x - (-3 - log(3))/ x
Time = 0.18 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.15 \[ \int \frac {-6 x+e^x (12+3 \log (3))+e^{2 e^{-x}+x} \left (24-2 x^2+\left (6-x^2\right ) \log (3)\right )+\left (-3 e^x+e^{2 e^{-x}+x} \left (-6+x^2\right )\right ) \log \left (-\frac {e^{2 e^{-x}} x}{-3+e^{2 e^{-x}} \left (-6+x^2\right )}\right )}{-3 e^x x^2+e^{2 e^{-x}+x} \left (-6 x^2+x^4\right )} \, dx=-\frac {2 \, e^{\left (-x\right )} - \log \left (3\right ) - \log \left (-{\left (x^{2} - 6\right )} e^{\left (2 \, e^{\left (-x\right )}\right )} + 3\right ) + \log \left (x\right ) - 3}{x} \] Input:
integrate((((x^2-6)*exp(x)*exp(2/exp(x))-3*exp(x))*log(-x*exp(2/exp(x))/(( x^2-6)*exp(2/exp(x))-3))+((-x^2+6)*log(3)-2*x^2+24)*exp(x)*exp(2/exp(x))+( 3*log(3)+12)*exp(x)-6*x)/((x^4-6*x^2)*exp(x)*exp(2/exp(x))-3*exp(x)*x^2),x , algorithm="maxima")
Output:
-(2*e^(-x) - log(3) - log(-(x^2 - 6)*e^(2*e^(-x)) + 3) + log(x) - 3)/x
Result contains higher order function than in optimal. Order 4 vs. order 3.
Time = 0.17 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.82 \[ \int \frac {-6 x+e^x (12+3 \log (3))+e^{2 e^{-x}+x} \left (24-2 x^2+\left (6-x^2\right ) \log (3)\right )+\left (-3 e^x+e^{2 e^{-x}+x} \left (-6+x^2\right )\right ) \log \left (-\frac {e^{2 e^{-x}} x}{-3+e^{2 e^{-x}} \left (-6+x^2\right )}\right )}{-3 e^x x^2+e^{2 e^{-x}+x} \left (-6 x^2+x^4\right )} \, dx=-\frac {2 \, x {\rm Ei}\left (-x\right ) + 2 \, e^{\left (-x\right )} - \log \left (3\right ) - \log \left (x^{2} e^{\left (2 \, e^{\left (-x\right )}\right )} - 6 \, e^{\left (2 \, e^{\left (-x\right )}\right )} - 3\right ) + \log \left (-x\right ) - 3}{x} + 2 \, {\rm Ei}\left (-x\right ) \] Input:
integrate((((x^2-6)*exp(x)*exp(2/exp(x))-3*exp(x))*log(-x*exp(2/exp(x))/(( x^2-6)*exp(2/exp(x))-3))+((-x^2+6)*log(3)-2*x^2+24)*exp(x)*exp(2/exp(x))+( 3*log(3)+12)*exp(x)-6*x)/((x^4-6*x^2)*exp(x)*exp(2/exp(x))-3*exp(x)*x^2),x , algorithm="giac")
Output:
-(2*x*Ei(-x) + 2*e^(-x) - log(3) - log(x^2*e^(2*e^(-x)) - 6*e^(2*e^(-x)) - 3) + log(-x) - 3)/x + 2*Ei(-x)
Time = 2.64 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.09 \[ \int \frac {-6 x+e^x (12+3 \log (3))+e^{2 e^{-x}+x} \left (24-2 x^2+\left (6-x^2\right ) \log (3)\right )+\left (-3 e^x+e^{2 e^{-x}+x} \left (-6+x^2\right )\right ) \log \left (-\frac {e^{2 e^{-x}} x}{-3+e^{2 e^{-x}} \left (-6+x^2\right )}\right )}{-3 e^x x^2+e^{2 e^{-x}+x} \left (-6 x^2+x^4\right )} \, dx=-\frac {\ln \left (-\frac {x\,{\mathrm {e}}^{2\,{\mathrm {e}}^{-x}}}{3\,\left ({\mathrm {e}}^{2\,{\mathrm {e}}^{-x}}\,\left (x^2-6\right )-3\right )}\right )-3}{x} \] Input:
int((6*x + log(-(x*exp(2*exp(-x)))/(exp(2*exp(-x))*(x^2 - 6) - 3))*(3*exp( x) - exp(2*exp(-x))*exp(x)*(x^2 - 6)) - exp(x)*(3*log(3) + 12) + exp(2*exp (-x))*exp(x)*(log(3)*(x^2 - 6) + 2*x^2 - 24))/(3*x^2*exp(x) + exp(2*exp(-x ))*exp(x)*(6*x^2 - x^4)),x)
Output:
-(log(-(x*exp(2*exp(-x)))/(3*(exp(2*exp(-x))*(x^2 - 6) - 3))) - 3)/x
\[ \int \frac {-6 x+e^x (12+3 \log (3))+e^{2 e^{-x}+x} \left (24-2 x^2+\left (6-x^2\right ) \log (3)\right )+\left (-3 e^x+e^{2 e^{-x}+x} \left (-6+x^2\right )\right ) \log \left (-\frac {e^{2 e^{-x}} x}{-3+e^{2 e^{-x}} \left (-6+x^2\right )}\right )}{-3 e^x x^2+e^{2 e^{-x}+x} \left (-6 x^2+x^4\right )} \, dx=6 \left (\int \frac {e^{\frac {2}{e^{x}}}}{e^{\frac {2}{e^{x}}} x^{4}-6 e^{\frac {2}{e^{x}}} x^{2}-3 x^{2}}d x \right ) \mathrm {log}\left (3\right )+24 \left (\int \frac {e^{\frac {2}{e^{x}}}}{e^{\frac {2}{e^{x}}} x^{4}-6 e^{\frac {2}{e^{x}}} x^{2}-3 x^{2}}d x \right )-\left (\int \frac {e^{\frac {2}{e^{x}}}}{e^{\frac {2}{e^{x}}} x^{2}-6 e^{\frac {2}{e^{x}}}-3}d x \right ) \mathrm {log}\left (3\right )-2 \left (\int \frac {e^{\frac {2}{e^{x}}}}{e^{\frac {2}{e^{x}}} x^{2}-6 e^{\frac {2}{e^{x}}}-3}d x \right )-3 \left (\int \frac {\mathrm {log}\left (-\frac {e^{\frac {2}{e^{x}}} x}{e^{\frac {2}{e^{x}}} x^{2}-6 e^{\frac {2}{e^{x}}}-3}\right )}{e^{\frac {2}{e^{x}}} x^{4}-6 e^{\frac {2}{e^{x}}} x^{2}-3 x^{2}}d x \right )-6 \left (\int \frac {e^{\frac {2}{e^{x}}} \mathrm {log}\left (-\frac {e^{\frac {2}{e^{x}}} x}{e^{\frac {2}{e^{x}}} x^{2}-6 e^{\frac {2}{e^{x}}}-3}\right )}{e^{\frac {2}{e^{x}}} x^{4}-6 e^{\frac {2}{e^{x}}} x^{2}-3 x^{2}}d x \right )+\int \frac {e^{\frac {2}{e^{x}}} \mathrm {log}\left (-\frac {e^{\frac {2}{e^{x}}} x}{e^{\frac {2}{e^{x}}} x^{2}-6 e^{\frac {2}{e^{x}}}-3}\right )}{e^{\frac {2}{e^{x}}} x^{2}-6 e^{\frac {2}{e^{x}}}-3}d x -6 \left (\int \frac {1}{e^{\frac {e^{x} x +2}{e^{x}}} x^{3}-6 e^{\frac {e^{x} x +2}{e^{x}}} x -3 e^{x} x}d x \right )+3 \left (\int \frac {1}{e^{\frac {2}{e^{x}}} x^{4}-6 e^{\frac {2}{e^{x}}} x^{2}-3 x^{2}}d x \right ) \mathrm {log}\left (3\right )+12 \left (\int \frac {1}{e^{\frac {2}{e^{x}}} x^{4}-6 e^{\frac {2}{e^{x}}} x^{2}-3 x^{2}}d x \right ) \] Input:
int((((x^2-6)*exp(x)*exp(2/exp(x))-3*exp(x))*log(-x*exp(2/exp(x))/((x^2-6) *exp(2/exp(x))-3))+((-x^2+6)*log(3)-2*x^2+24)*exp(x)*exp(2/exp(x))+(3*log( 3)+12)*exp(x)-6*x)/((x^4-6*x^2)*exp(x)*exp(2/exp(x))-3*exp(x)*x^2),x)
Output:
6*int(e**(2/e**x)/(e**(2/e**x)*x**4 - 6*e**(2/e**x)*x**2 - 3*x**2),x)*log( 3) + 24*int(e**(2/e**x)/(e**(2/e**x)*x**4 - 6*e**(2/e**x)*x**2 - 3*x**2),x ) - int(e**(2/e**x)/(e**(2/e**x)*x**2 - 6*e**(2/e**x) - 3),x)*log(3) - 2*i nt(e**(2/e**x)/(e**(2/e**x)*x**2 - 6*e**(2/e**x) - 3),x) - 3*int(log(( - e **(2/e**x)*x)/(e**(2/e**x)*x**2 - 6*e**(2/e**x) - 3))/(e**(2/e**x)*x**4 - 6*e**(2/e**x)*x**2 - 3*x**2),x) - 6*int((e**(2/e**x)*log(( - e**(2/e**x)*x )/(e**(2/e**x)*x**2 - 6*e**(2/e**x) - 3)))/(e**(2/e**x)*x**4 - 6*e**(2/e** x)*x**2 - 3*x**2),x) + int((e**(2/e**x)*log(( - e**(2/e**x)*x)/(e**(2/e**x )*x**2 - 6*e**(2/e**x) - 3)))/(e**(2/e**x)*x**2 - 6*e**(2/e**x) - 3),x) - 6*int(1/(e**((e**x*x + 2)/e**x)*x**3 - 6*e**((e**x*x + 2)/e**x)*x - 3*e**x *x),x) + 3*int(1/(e**(2/e**x)*x**4 - 6*e**(2/e**x)*x**2 - 3*x**2),x)*log(3 ) + 12*int(1/(e**(2/e**x)*x**4 - 6*e**(2/e**x)*x**2 - 3*x**2),x)