Integrand size = 352, antiderivative size = 35 \[ \int \frac {-3 e^3 x^2+3 x^3+e^{-2-2 x} \left (-18 e^3+18 x\right )+e^{-1-x} \left (-15 e^3 x+12 x^2-3 x^3\right )+\left (e^{-2-2 x} \left (-18+18 e^3\right )-3 x^2+3 e^3 x^2+e^{-1-x} \left (-12 x+15 e^3 x+3 x^2\right )\right ) \log (x)+\left (18 e^{-2-2 x}+15 e^{-1-x} x+3 x^2+\left (-18 e^{-2-2 x}-15 e^{-1-x} x-3 x^2\right ) \log (x)\right ) \log \left (\frac {3 e^{-1-x} x+x^2}{2 e^{-1-x}+x}\right )}{6 e^{4-2 x} x^2+5 e^{5-x} x^3+e^6 x^4+\left (-12 e^{1-2 x} x^2-10 e^{2-x} x^3-2 e^3 x^4\right ) \log \left (\frac {3 e^{-1-x} x+x^2}{2 e^{-1-x}+x}\right )+\left (6 e^{-2-2 x} x^2+5 e^{-1-x} x^3+x^4\right ) \log ^2\left (\frac {3 e^{-1-x} x+x^2}{2 e^{-1-x}+x}\right )} \, dx=\frac {3 (-x+\log (x))}{x \left (-e^3+\log \left (x+\frac {x}{2+e^{1+x} x}\right )\right )} \] Output:
3*(ln(x)-x)/(ln(x/(2+x/exp(-1-x))+x)-exp(3))/x
Time = 0.20 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.20 \[ \int \frac {-3 e^3 x^2+3 x^3+e^{-2-2 x} \left (-18 e^3+18 x\right )+e^{-1-x} \left (-15 e^3 x+12 x^2-3 x^3\right )+\left (e^{-2-2 x} \left (-18+18 e^3\right )-3 x^2+3 e^3 x^2+e^{-1-x} \left (-12 x+15 e^3 x+3 x^2\right )\right ) \log (x)+\left (18 e^{-2-2 x}+15 e^{-1-x} x+3 x^2+\left (-18 e^{-2-2 x}-15 e^{-1-x} x-3 x^2\right ) \log (x)\right ) \log \left (\frac {3 e^{-1-x} x+x^2}{2 e^{-1-x}+x}\right )}{6 e^{4-2 x} x^2+5 e^{5-x} x^3+e^6 x^4+\left (-12 e^{1-2 x} x^2-10 e^{2-x} x^3-2 e^3 x^4\right ) \log \left (\frac {3 e^{-1-x} x+x^2}{2 e^{-1-x}+x}\right )+\left (6 e^{-2-2 x} x^2+5 e^{-1-x} x^3+x^4\right ) \log ^2\left (\frac {3 e^{-1-x} x+x^2}{2 e^{-1-x}+x}\right )} \, dx=\frac {3 (-x+\log (x))}{x \left (-e^3+\log \left (\frac {x \left (3+e^{1+x} x\right )}{2+e^{1+x} x}\right )\right )} \] Input:
Integrate[(-3*E^3*x^2 + 3*x^3 + E^(-2 - 2*x)*(-18*E^3 + 18*x) + E^(-1 - x) *(-15*E^3*x + 12*x^2 - 3*x^3) + (E^(-2 - 2*x)*(-18 + 18*E^3) - 3*x^2 + 3*E ^3*x^2 + E^(-1 - x)*(-12*x + 15*E^3*x + 3*x^2))*Log[x] + (18*E^(-2 - 2*x) + 15*E^(-1 - x)*x + 3*x^2 + (-18*E^(-2 - 2*x) - 15*E^(-1 - x)*x - 3*x^2)*L og[x])*Log[(3*E^(-1 - x)*x + x^2)/(2*E^(-1 - x) + x)])/(6*E^(4 - 2*x)*x^2 + 5*E^(5 - x)*x^3 + E^6*x^4 + (-12*E^(1 - 2*x)*x^2 - 10*E^(2 - x)*x^3 - 2* E^3*x^4)*Log[(3*E^(-1 - x)*x + x^2)/(2*E^(-1 - x) + x)] + (6*E^(-2 - 2*x)* x^2 + 5*E^(-1 - x)*x^3 + x^4)*Log[(3*E^(-1 - x)*x + x^2)/(2*E^(-1 - x) + x )]^2),x]
Output:
(3*(-x + Log[x]))/(x*(-E^3 + Log[(x*(3 + E^(1 + x)*x))/(2 + E^(1 + 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.
\(\displaystyle \int \frac {3 x^3-3 e^3 x^2+\left (3 e^3 x^2-3 x^2+e^{-x-1} \left (3 x^2+15 e^3 x-12 x\right )+\left (18 e^3-18\right ) e^{-2 x-2}\right ) \log (x)+\left (3 x^2+\left (-3 x^2-15 e^{-x-1} x-18 e^{-2 x-2}\right ) \log (x)+15 e^{-x-1} x+18 e^{-2 x-2}\right ) \log \left (\frac {x^2+3 e^{-x-1} x}{x+2 e^{-x-1}}\right )+e^{-x-1} \left (-3 x^3+12 x^2-15 e^3 x\right )+e^{-2 x-2} \left (18 x-18 e^3\right )}{e^6 x^4+5 e^{5-x} x^3+6 e^{4-2 x} x^2+\left (x^4+5 e^{-x-1} x^3+6 e^{-2 x-2} x^2\right ) \log ^2\left (\frac {x^2+3 e^{-x-1} x}{x+2 e^{-x-1}}\right )+\left (-2 e^3 x^4-10 e^{2-x} x^3-12 e^{1-2 x} x^2\right ) \log \left (\frac {x^2+3 e^{-x-1} x}{x+2 e^{-x-1}}\right )} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {e^{2 x+2} \left (3 x^3-3 e^3 x^2+\left (3 e^3 x^2-3 x^2+e^{-x-1} \left (3 x^2+15 e^3 x-12 x\right )+\left (18 e^3-18\right ) e^{-2 x-2}\right ) \log (x)+\left (3 x^2+\left (-3 x^2-15 e^{-x-1} x-18 e^{-2 x-2}\right ) \log (x)+15 e^{-x-1} x+18 e^{-2 x-2}\right ) \log \left (\frac {x^2+3 e^{-x-1} x}{x+2 e^{-x-1}}\right )+e^{-x-1} \left (-3 x^3+12 x^2-15 e^3 x\right )+e^{-2 x-2} \left (18 x-18 e^3\right )\right )}{x^2 \left (e^{2 x+2} x^2+5 e^{x+1} x+6\right ) \left (e^3-\log \left (\frac {x \left (e^{x+1} x+3\right )}{e^{x+1} x+2}\right )\right )^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {3 \left (x-\left (1-e^3\right ) \log (x)-\log (x) \log \left (\frac {x \left (e^{x+1} x+3\right )}{e^{x+1} x+2}\right )+\log \left (\frac {x \left (e^{x+1} x+3\right )}{e^{x+1} x+2}\right )-e^3\right )}{x^2 \left (e^3-\log \left (\frac {x \left (e^{x+1} x+3\right )}{e^{x+1} x+2}\right )\right )^2}-\frac {e^{x+1} (x+1) (x-\log (x))}{2 x \left (e^3-\log \left (\frac {x \left (e^{x+1} x+3\right )}{e^{x+1} x+2}\right )\right )^2}+\frac {3 e^{2 x+2} (x+1) (x-\log (x))}{2 \left (e^{x+1} x+2\right ) \left (e^3-\log \left (\frac {x \left (e^{x+1} x+3\right )}{e^{x+1} x+2}\right )\right )^2}-\frac {e^{2 x+2} (x+1) (x-\log (x))}{\left (e^{x+1} x+3\right ) \left (e^3-\log \left (\frac {x \left (e^{x+1} x+3\right )}{e^{x+1} x+2}\right )\right )^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {1}{2} \int \frac {e^{x+1}}{\left (e^3-\log \left (\frac {x \left (e^{x+1} x+3\right )}{e^{x+1} x+2}\right )\right )^2}dx+3 \int \frac {1}{x \left (e^3-\log \left (\frac {x \left (e^{x+1} x+3\right )}{e^{x+1} x+2}\right )\right )^2}dx-\frac {1}{2} \int \frac {e^{x+1} x}{\left (e^3-\log \left (\frac {x \left (e^{x+1} x+3\right )}{e^{x+1} x+2}\right )\right )^2}dx+\frac {3}{2} \int \frac {e^{2 x+2} x}{\left (e^{x+1} x+2\right ) \left (e^3-\log \left (\frac {x \left (e^{x+1} x+3\right )}{e^{x+1} x+2}\right )\right )^2}dx+\frac {3}{2} \int \frac {e^{2 x+2} x^2}{\left (e^{x+1} x+2\right ) \left (e^3-\log \left (\frac {x \left (e^{x+1} x+3\right )}{e^{x+1} x+2}\right )\right )^2}dx-\int \frac {e^{2 x+2} x}{\left (e^{x+1} x+3\right ) \left (e^3-\log \left (\frac {x \left (e^{x+1} x+3\right )}{e^{x+1} x+2}\right )\right )^2}dx-\int \frac {e^{2 x+2} x^2}{\left (e^{x+1} x+3\right ) \left (e^3-\log \left (\frac {x \left (e^{x+1} x+3\right )}{e^{x+1} x+2}\right )\right )^2}dx+\frac {1}{2} \int \frac {e^{x+1} \log (x)}{\left (e^3-\log \left (\frac {x \left (e^{x+1} x+3\right )}{e^{x+1} x+2}\right )\right )^2}dx-3 \int \frac {\log (x)}{x^2 \left (e^3-\log \left (\frac {x \left (e^{x+1} x+3\right )}{e^{x+1} x+2}\right )\right )^2}dx+\frac {1}{2} \int \frac {e^{x+1} \log (x)}{x \left (e^3-\log \left (\frac {x \left (e^{x+1} x+3\right )}{e^{x+1} x+2}\right )\right )^2}dx-\frac {3}{2} \int \frac {e^{2 x+2} \log (x)}{\left (e^{x+1} x+2\right ) \left (e^3-\log \left (\frac {x \left (e^{x+1} x+3\right )}{e^{x+1} x+2}\right )\right )^2}dx-\frac {3}{2} \int \frac {e^{2 x+2} x \log (x)}{\left (e^{x+1} x+2\right ) \left (e^3-\log \left (\frac {x \left (e^{x+1} x+3\right )}{e^{x+1} x+2}\right )\right )^2}dx+\int \frac {e^{2 x+2} \log (x)}{\left (e^{x+1} x+3\right ) \left (e^3-\log \left (\frac {x \left (e^{x+1} x+3\right )}{e^{x+1} x+2}\right )\right )^2}dx+\int \frac {e^{2 x+2} x \log (x)}{\left (e^{x+1} x+3\right ) \left (e^3-\log \left (\frac {x \left (e^{x+1} x+3\right )}{e^{x+1} x+2}\right )\right )^2}dx-3 \int \frac {1}{x^2 \left (e^3-\log \left (\frac {x \left (e^{x+1} x+3\right )}{e^{x+1} x+2}\right )\right )}dx+3 \int \frac {\log (x)}{x^2 \left (e^3-\log \left (\frac {x \left (e^{x+1} x+3\right )}{e^{x+1} x+2}\right )\right )}dx\) |
Input:
Int[(-3*E^3*x^2 + 3*x^3 + E^(-2 - 2*x)*(-18*E^3 + 18*x) + E^(-1 - x)*(-15* E^3*x + 12*x^2 - 3*x^3) + (E^(-2 - 2*x)*(-18 + 18*E^3) - 3*x^2 + 3*E^3*x^2 + E^(-1 - x)*(-12*x + 15*E^3*x + 3*x^2))*Log[x] + (18*E^(-2 - 2*x) + 15*E ^(-1 - x)*x + 3*x^2 + (-18*E^(-2 - 2*x) - 15*E^(-1 - x)*x - 3*x^2)*Log[x]) *Log[(3*E^(-1 - x)*x + x^2)/(2*E^(-1 - x) + x)])/(6*E^(4 - 2*x)*x^2 + 5*E^ (5 - x)*x^3 + E^6*x^4 + (-12*E^(1 - 2*x)*x^2 - 10*E^(2 - x)*x^3 - 2*E^3*x^ 4)*Log[(3*E^(-1 - x)*x + x^2)/(2*E^(-1 - x) + x)] + (6*E^(-2 - 2*x)*x^2 + 5*E^(-1 - x)*x^3 + x^4)*Log[(3*E^(-1 - x)*x + x^2)/(2*E^(-1 - x) + x)]^2), x]
Output:
$Aborted
Time = 58.84 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.29
method | result | size |
parallelrisch | \(\frac {3 x -3 \ln \left (x \right )}{x \left ({\mathrm e}^{3}-\ln \left (\frac {x \left (3 \,{\mathrm e}^{-1-x}+x \right )}{2 \,{\mathrm e}^{-1-x}+x}\right )\right )}\) | \(45\) |
risch | \(-\frac {6 \left (x -\ln \left (x \right )\right )}{x \left (-i \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (\frac {i \left (3 \,{\mathrm e}^{-1-x}+x \right )}{2 \,{\mathrm e}^{-1-x}+x}\right ) \operatorname {csgn}\left (\frac {i x \left (3 \,{\mathrm e}^{-1-x}+x \right )}{2 \,{\mathrm e}^{-1-x}+x}\right )+i \pi \,\operatorname {csgn}\left (i x \right ) {\operatorname {csgn}\left (\frac {i x \left (3 \,{\mathrm e}^{-1-x}+x \right )}{2 \,{\mathrm e}^{-1-x}+x}\right )}^{2}-i \pi \,\operatorname {csgn}\left (\frac {i}{2 \,{\mathrm e}^{-1-x}+x}\right ) \operatorname {csgn}\left (i \left (3 \,{\mathrm e}^{-1-x}+x \right )\right ) \operatorname {csgn}\left (\frac {i \left (3 \,{\mathrm e}^{-1-x}+x \right )}{2 \,{\mathrm e}^{-1-x}+x}\right )+i \pi \,\operatorname {csgn}\left (\frac {i}{2 \,{\mathrm e}^{-1-x}+x}\right ) {\operatorname {csgn}\left (\frac {i \left (3 \,{\mathrm e}^{-1-x}+x \right )}{2 \,{\mathrm e}^{-1-x}+x}\right )}^{2}+i \pi \,\operatorname {csgn}\left (i \left (3 \,{\mathrm e}^{-1-x}+x \right )\right ) {\operatorname {csgn}\left (\frac {i \left (3 \,{\mathrm e}^{-1-x}+x \right )}{2 \,{\mathrm e}^{-1-x}+x}\right )}^{2}-i \pi {\operatorname {csgn}\left (\frac {i \left (3 \,{\mathrm e}^{-1-x}+x \right )}{2 \,{\mathrm e}^{-1-x}+x}\right )}^{3}+i \pi \,\operatorname {csgn}\left (\frac {i \left (3 \,{\mathrm e}^{-1-x}+x \right )}{2 \,{\mathrm e}^{-1-x}+x}\right ) {\operatorname {csgn}\left (\frac {i x \left (3 \,{\mathrm e}^{-1-x}+x \right )}{2 \,{\mathrm e}^{-1-x}+x}\right )}^{2}-i \pi {\operatorname {csgn}\left (\frac {i x \left (3 \,{\mathrm e}^{-1-x}+x \right )}{2 \,{\mathrm e}^{-1-x}+x}\right )}^{3}-2 \,{\mathrm e}^{3}+2 \ln \left (x \right )-2 \ln \left (2 \,{\mathrm e}^{-1-x}+x \right )+2 \ln \left (3 \,{\mathrm e}^{-1-x}+x \right )\right )}\) | \(427\) |
Input:
int((((-18*exp(-1-x)^2-15*x*exp(-1-x)-3*x^2)*ln(x)+18*exp(-1-x)^2+15*x*exp (-1-x)+3*x^2)*ln((3*x*exp(-1-x)+x^2)/(2*exp(-1-x)+x))+((18*exp(3)-18)*exp( -1-x)^2+(15*x*exp(3)+3*x^2-12*x)*exp(-1-x)+3*x^2*exp(3)-3*x^2)*ln(x)+(-18* exp(3)+18*x)*exp(-1-x)^2+(-15*x*exp(3)-3*x^3+12*x^2)*exp(-1-x)-3*x^2*exp(3 )+3*x^3)/((6*x^2*exp(-1-x)^2+5*x^3*exp(-1-x)+x^4)*ln((3*x*exp(-1-x)+x^2)/( 2*exp(-1-x)+x))^2+(-12*x^2*exp(3)*exp(-1-x)^2-10*x^3*exp(3)*exp(-1-x)-2*x^ 4*exp(3))*ln((3*x*exp(-1-x)+x^2)/(2*exp(-1-x)+x))+6*x^2*exp(3)^2*exp(-1-x) ^2+5*x^3*exp(3)^2*exp(-1-x)+x^4*exp(3)^2),x,method=_RETURNVERBOSE)
Output:
1/x*(3*x-3*ln(x))/(exp(3)-ln(x*(3*exp(-1-x)+x)/(2*exp(-1-x)+x)))
Time = 0.11 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.46 \[ \int \frac {-3 e^3 x^2+3 x^3+e^{-2-2 x} \left (-18 e^3+18 x\right )+e^{-1-x} \left (-15 e^3 x+12 x^2-3 x^3\right )+\left (e^{-2-2 x} \left (-18+18 e^3\right )-3 x^2+3 e^3 x^2+e^{-1-x} \left (-12 x+15 e^3 x+3 x^2\right )\right ) \log (x)+\left (18 e^{-2-2 x}+15 e^{-1-x} x+3 x^2+\left (-18 e^{-2-2 x}-15 e^{-1-x} x-3 x^2\right ) \log (x)\right ) \log \left (\frac {3 e^{-1-x} x+x^2}{2 e^{-1-x}+x}\right )}{6 e^{4-2 x} x^2+5 e^{5-x} x^3+e^6 x^4+\left (-12 e^{1-2 x} x^2-10 e^{2-x} x^3-2 e^3 x^4\right ) \log \left (\frac {3 e^{-1-x} x+x^2}{2 e^{-1-x}+x}\right )+\left (6 e^{-2-2 x} x^2+5 e^{-1-x} x^3+x^4\right ) \log ^2\left (\frac {3 e^{-1-x} x+x^2}{2 e^{-1-x}+x}\right )} \, dx=\frac {3 \, {\left (x - \log \left (x\right )\right )}}{x e^{3} - x \log \left (\frac {x^{2} e^{6} + 3 \, x e^{\left (-x + 5\right )}}{x e^{6} + 2 \, e^{\left (-x + 5\right )}}\right )} \] Input:
integrate((((-18*exp(-1-x)^2-15*x*exp(-1-x)-3*x^2)*log(x)+18*exp(-1-x)^2+1 5*x*exp(-1-x)+3*x^2)*log((3*x*exp(-1-x)+x^2)/(2*exp(-1-x)+x))+((18*exp(3)- 18)*exp(-1-x)^2+(15*x*exp(3)+3*x^2-12*x)*exp(-1-x)+3*x^2*exp(3)-3*x^2)*log (x)+(-18*exp(3)+18*x)*exp(-1-x)^2+(-15*x*exp(3)-3*x^3+12*x^2)*exp(-1-x)-3* x^2*exp(3)+3*x^3)/((6*x^2*exp(-1-x)^2+5*x^3*exp(-1-x)+x^4)*log((3*x*exp(-1 -x)+x^2)/(2*exp(-1-x)+x))^2+(-12*x^2*exp(3)*exp(-1-x)^2-10*x^3*exp(3)*exp( -1-x)-2*x^4*exp(3))*log((3*x*exp(-1-x)+x^2)/(2*exp(-1-x)+x))+6*x^2*exp(3)^ 2*exp(-1-x)^2+5*x^3*exp(3)^2*exp(-1-x)+x^4*exp(3)^2),x, algorithm="fricas" )
Output:
3*(x - log(x))/(x*e^3 - x*log((x^2*e^6 + 3*x*e^(-x + 5))/(x*e^6 + 2*e^(-x + 5))))
Time = 0.38 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.11 \[ \int \frac {-3 e^3 x^2+3 x^3+e^{-2-2 x} \left (-18 e^3+18 x\right )+e^{-1-x} \left (-15 e^3 x+12 x^2-3 x^3\right )+\left (e^{-2-2 x} \left (-18+18 e^3\right )-3 x^2+3 e^3 x^2+e^{-1-x} \left (-12 x+15 e^3 x+3 x^2\right )\right ) \log (x)+\left (18 e^{-2-2 x}+15 e^{-1-x} x+3 x^2+\left (-18 e^{-2-2 x}-15 e^{-1-x} x-3 x^2\right ) \log (x)\right ) \log \left (\frac {3 e^{-1-x} x+x^2}{2 e^{-1-x}+x}\right )}{6 e^{4-2 x} x^2+5 e^{5-x} x^3+e^6 x^4+\left (-12 e^{1-2 x} x^2-10 e^{2-x} x^3-2 e^3 x^4\right ) \log \left (\frac {3 e^{-1-x} x+x^2}{2 e^{-1-x}+x}\right )+\left (6 e^{-2-2 x} x^2+5 e^{-1-x} x^3+x^4\right ) \log ^2\left (\frac {3 e^{-1-x} x+x^2}{2 e^{-1-x}+x}\right )} \, dx=\frac {- 3 x + 3 \log {\left (x \right )}}{x \log {\left (\frac {x^{2} + 3 x e^{- x - 1}}{x + 2 e^{- x - 1}} \right )} - x e^{3}} \] Input:
integrate((((-18*exp(-1-x)**2-15*x*exp(-1-x)-3*x**2)*ln(x)+18*exp(-1-x)**2 +15*x*exp(-1-x)+3*x**2)*ln((3*x*exp(-1-x)+x**2)/(2*exp(-1-x)+x))+((18*exp( 3)-18)*exp(-1-x)**2+(15*x*exp(3)+3*x**2-12*x)*exp(-1-x)+3*x**2*exp(3)-3*x* *2)*ln(x)+(-18*exp(3)+18*x)*exp(-1-x)**2+(-15*x*exp(3)-3*x**3+12*x**2)*exp (-1-x)-3*x**2*exp(3)+3*x**3)/((6*x**2*exp(-1-x)**2+5*x**3*exp(-1-x)+x**4)* ln((3*x*exp(-1-x)+x**2)/(2*exp(-1-x)+x))**2+(-12*x**2*exp(3)*exp(-1-x)**2- 10*x**3*exp(3)*exp(-1-x)-2*x**4*exp(3))*ln((3*x*exp(-1-x)+x**2)/(2*exp(-1- x)+x))+6*x**2*exp(3)**2*exp(-1-x)**2+5*x**3*exp(3)**2*exp(-1-x)+x**4*exp(3 )**2),x)
Output:
(-3*x + 3*log(x))/(x*log((x**2 + 3*x*exp(-x - 1))/(x + 2*exp(-x - 1))) - x *exp(3))
Time = 0.42 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.23 \[ \int \frac {-3 e^3 x^2+3 x^3+e^{-2-2 x} \left (-18 e^3+18 x\right )+e^{-1-x} \left (-15 e^3 x+12 x^2-3 x^3\right )+\left (e^{-2-2 x} \left (-18+18 e^3\right )-3 x^2+3 e^3 x^2+e^{-1-x} \left (-12 x+15 e^3 x+3 x^2\right )\right ) \log (x)+\left (18 e^{-2-2 x}+15 e^{-1-x} x+3 x^2+\left (-18 e^{-2-2 x}-15 e^{-1-x} x-3 x^2\right ) \log (x)\right ) \log \left (\frac {3 e^{-1-x} x+x^2}{2 e^{-1-x}+x}\right )}{6 e^{4-2 x} x^2+5 e^{5-x} x^3+e^6 x^4+\left (-12 e^{1-2 x} x^2-10 e^{2-x} x^3-2 e^3 x^4\right ) \log \left (\frac {3 e^{-1-x} x+x^2}{2 e^{-1-x}+x}\right )+\left (6 e^{-2-2 x} x^2+5 e^{-1-x} x^3+x^4\right ) \log ^2\left (\frac {3 e^{-1-x} x+x^2}{2 e^{-1-x}+x}\right )} \, dx=\frac {3 \, {\left (x - \log \left (x\right )\right )}}{x e^{3} - x \log \left (x e^{\left (x + 1\right )} + 3\right ) + x \log \left (x e^{\left (x + 1\right )} + 2\right ) - x \log \left (x\right )} \] Input:
integrate((((-18*exp(-1-x)^2-15*x*exp(-1-x)-3*x^2)*log(x)+18*exp(-1-x)^2+1 5*x*exp(-1-x)+3*x^2)*log((3*x*exp(-1-x)+x^2)/(2*exp(-1-x)+x))+((18*exp(3)- 18)*exp(-1-x)^2+(15*x*exp(3)+3*x^2-12*x)*exp(-1-x)+3*x^2*exp(3)-3*x^2)*log (x)+(-18*exp(3)+18*x)*exp(-1-x)^2+(-15*x*exp(3)-3*x^3+12*x^2)*exp(-1-x)-3* x^2*exp(3)+3*x^3)/((6*x^2*exp(-1-x)^2+5*x^3*exp(-1-x)+x^4)*log((3*x*exp(-1 -x)+x^2)/(2*exp(-1-x)+x))^2+(-12*x^2*exp(3)*exp(-1-x)^2-10*x^3*exp(3)*exp( -1-x)-2*x^4*exp(3))*log((3*x*exp(-1-x)+x^2)/(2*exp(-1-x)+x))+6*x^2*exp(3)^ 2*exp(-1-x)^2+5*x^3*exp(3)^2*exp(-1-x)+x^4*exp(3)^2),x, algorithm="maxima" )
Output:
3*(x - log(x))/(x*e^3 - x*log(x*e^(x + 1) + 3) + x*log(x*e^(x + 1) + 2) - x*log(x))
Leaf count of result is larger than twice the leaf count of optimal. 1129 vs. \(2 (33) = 66\).
Time = 2.96 (sec) , antiderivative size = 1129, normalized size of antiderivative = 32.26 \[ \int \frac {-3 e^3 x^2+3 x^3+e^{-2-2 x} \left (-18 e^3+18 x\right )+e^{-1-x} \left (-15 e^3 x+12 x^2-3 x^3\right )+\left (e^{-2-2 x} \left (-18+18 e^3\right )-3 x^2+3 e^3 x^2+e^{-1-x} \left (-12 x+15 e^3 x+3 x^2\right )\right ) \log (x)+\left (18 e^{-2-2 x}+15 e^{-1-x} x+3 x^2+\left (-18 e^{-2-2 x}-15 e^{-1-x} x-3 x^2\right ) \log (x)\right ) \log \left (\frac {3 e^{-1-x} x+x^2}{2 e^{-1-x}+x}\right )}{6 e^{4-2 x} x^2+5 e^{5-x} x^3+e^6 x^4+\left (-12 e^{1-2 x} x^2-10 e^{2-x} x^3-2 e^3 x^4\right ) \log \left (\frac {3 e^{-1-x} x+x^2}{2 e^{-1-x}+x}\right )+\left (6 e^{-2-2 x} x^2+5 e^{-1-x} x^3+x^4\right ) \log ^2\left (\frac {3 e^{-1-x} x+x^2}{2 e^{-1-x}+x}\right )} \, dx=\text {Too large to display} \] Input:
integrate((((-18*exp(-1-x)^2-15*x*exp(-1-x)-3*x^2)*log(x)+18*exp(-1-x)^2+1 5*x*exp(-1-x)+3*x^2)*log((3*x*exp(-1-x)+x^2)/(2*exp(-1-x)+x))+((18*exp(3)- 18)*exp(-1-x)^2+(15*x*exp(3)+3*x^2-12*x)*exp(-1-x)+3*x^2*exp(3)-3*x^2)*log (x)+(-18*exp(3)+18*x)*exp(-1-x)^2+(-15*x*exp(3)-3*x^3+12*x^2)*exp(-1-x)-3* x^2*exp(3)+3*x^3)/((6*x^2*exp(-1-x)^2+5*x^3*exp(-1-x)+x^4)*log((3*x*exp(-1 -x)+x^2)/(2*exp(-1-x)+x))^2+(-12*x^2*exp(3)*exp(-1-x)^2-10*x^3*exp(3)*exp( -1-x)-2*x^4*exp(3))*log((3*x*exp(-1-x)+x^2)/(2*exp(-1-x)+x))+6*x^2*exp(3)^ 2*exp(-1-x)^2+5*x^3*exp(3)^2*exp(-1-x)+x^4*exp(3)^2),x, algorithm="giac")
Output:
3/2*(pi^2*sgn(-(x - 5)*e^6 - 5*e^6 - 2*e^(-x + 5))*sgn(x) + pi^2*sgn(-(x - 5)*e^6 - 5*e^6 - 3*e^(-x + 5))*sgn(x) - pi^2*sgn(-(x - 5)*e^6 - 5*e^6 - 2 *e^(-x + 5)) - pi^2*sgn(-(x - 5)*e^6 - 5*e^6 - 3*e^(-x + 5)) + 4*(x - 5)*e ^3 + 4*(x - 5)*log(abs(-(x - 5)*e^6 - 5*e^6 - 2*e^(-x + 5))) - 4*(x - 5)*l og(abs(-(x - 5)*e^6 - 5*e^6 - 3*e^(-x + 5))) - 4*(x - 5)*log(abs(x)) - 4*e ^3*log(abs(x)) - 4*log(abs(-(x - 5)*e^6 - 5*e^6 - 2*e^(-x + 5)))*log(abs(x )) + 4*log(abs(-(x - 5)*e^6 - 5*e^6 - 3*e^(-x + 5)))*log(abs(x)) + 4*log(a bs(x))^2 + 20*e^3 + 20*log(abs(-(x - 5)*e^6 - 5*e^6 - 2*e^(-x + 5))) - 20* log(abs(-(x - 5)*e^6 - 5*e^6 - 3*e^(-x + 5))) - 20*log(abs(x)))/(pi^2*(x - 5)*sgn(-(x - 5)*e^6 - 5*e^6 - 2*e^(-x + 5))*sgn(-(x - 5)*e^6 - 5*e^6 - 3* e^(-x + 5)) + pi^2*(x - 5)*sgn(-(x - 5)*e^6 - 5*e^6 - 2*e^(-x + 5))*sgn(x) + pi^2*(x - 5)*sgn(-(x - 5)*e^6 - 5*e^6 - 3*e^(-x + 5))*sgn(x) + pi^2*(x - 5)*sgn(-(x - 5)*e^6 - 5*e^6 - 2*e^(-x + 5)) + pi^2*(x - 5)*sgn(-(x - 5)* e^6 - 5*e^6 - 3*e^(-x + 5)) + 5*pi^2*sgn(-(x - 5)*e^6 - 5*e^6 - 2*e^(-x + 5))*sgn(-(x - 5)*e^6 - 5*e^6 - 3*e^(-x + 5)) + pi^2*(x - 5)*sgn(x) + 5*pi^ 2*sgn(-(x - 5)*e^6 - 5*e^6 - 2*e^(-x + 5))*sgn(x) + 5*pi^2*sgn(-(x - 5)*e^ 6 - 5*e^6 - 3*e^(-x + 5))*sgn(x) + 2*pi^2*(x - 5) + 4*(x - 5)*e^3*log(abs( -(x - 5)*e^6 - 5*e^6 - 2*e^(-x + 5))) + 2*(x - 5)*log(abs(-(x - 5)*e^6 - 5 *e^6 - 2*e^(-x + 5)))^2 - 4*(x - 5)*e^3*log(abs(-(x - 5)*e^6 - 5*e^6 - 3*e ^(-x + 5))) - 4*(x - 5)*log(abs(-(x - 5)*e^6 - 5*e^6 - 2*e^(-x + 5)))*l...
Timed out. \[ \int \frac {-3 e^3 x^2+3 x^3+e^{-2-2 x} \left (-18 e^3+18 x\right )+e^{-1-x} \left (-15 e^3 x+12 x^2-3 x^3\right )+\left (e^{-2-2 x} \left (-18+18 e^3\right )-3 x^2+3 e^3 x^2+e^{-1-x} \left (-12 x+15 e^3 x+3 x^2\right )\right ) \log (x)+\left (18 e^{-2-2 x}+15 e^{-1-x} x+3 x^2+\left (-18 e^{-2-2 x}-15 e^{-1-x} x-3 x^2\right ) \log (x)\right ) \log \left (\frac {3 e^{-1-x} x+x^2}{2 e^{-1-x}+x}\right )}{6 e^{4-2 x} x^2+5 e^{5-x} x^3+e^6 x^4+\left (-12 e^{1-2 x} x^2-10 e^{2-x} x^3-2 e^3 x^4\right ) \log \left (\frac {3 e^{-1-x} x+x^2}{2 e^{-1-x}+x}\right )+\left (6 e^{-2-2 x} x^2+5 e^{-1-x} x^3+x^4\right ) \log ^2\left (\frac {3 e^{-1-x} x+x^2}{2 e^{-1-x}+x}\right )} \, dx=\int \frac {\ln \left (x\right )\,\left ({\mathrm {e}}^{-2\,x-2}\,\left (18\,{\mathrm {e}}^3-18\right )+3\,x^2\,{\mathrm {e}}^3+{\mathrm {e}}^{-x-1}\,\left (15\,x\,{\mathrm {e}}^3-12\,x+3\,x^2\right )-3\,x^2\right )-{\mathrm {e}}^{-x-1}\,\left (3\,x^3-12\,x^2+15\,{\mathrm {e}}^3\,x\right )+\ln \left (\frac {3\,x\,{\mathrm {e}}^{-x-1}+x^2}{x+2\,{\mathrm {e}}^{-x-1}}\right )\,\left (18\,{\mathrm {e}}^{-2\,x-2}-\ln \left (x\right )\,\left (18\,{\mathrm {e}}^{-2\,x-2}+15\,x\,{\mathrm {e}}^{-x-1}+3\,x^2\right )+15\,x\,{\mathrm {e}}^{-x-1}+3\,x^2\right )-3\,x^2\,{\mathrm {e}}^3+3\,x^3+{\mathrm {e}}^{-2\,x-2}\,\left (18\,x-18\,{\mathrm {e}}^3\right )}{{\ln \left (\frac {3\,x\,{\mathrm {e}}^{-x-1}+x^2}{x+2\,{\mathrm {e}}^{-x-1}}\right )}^2\,\left (5\,x^3\,{\mathrm {e}}^{-x-1}+6\,x^2\,{\mathrm {e}}^{-2\,x-2}+x^4\right )+x^4\,{\mathrm {e}}^6-\ln \left (\frac {3\,x\,{\mathrm {e}}^{-x-1}+x^2}{x+2\,{\mathrm {e}}^{-x-1}}\right )\,\left (2\,x^4\,{\mathrm {e}}^3+12\,x^2\,{\mathrm {e}}^{1-2\,x}+10\,x^3\,{\mathrm {e}}^{2-x}\right )+6\,x^2\,{\mathrm {e}}^{4-2\,x}+5\,x^3\,{\mathrm {e}}^{5-x}} \,d x \] Input:
int((log(x)*(exp(- 2*x - 2)*(18*exp(3) - 18) + 3*x^2*exp(3) + exp(- x - 1) *(15*x*exp(3) - 12*x + 3*x^2) - 3*x^2) - exp(- x - 1)*(15*x*exp(3) - 12*x^ 2 + 3*x^3) + log((3*x*exp(- x - 1) + x^2)/(x + 2*exp(- x - 1)))*(18*exp(- 2*x - 2) - log(x)*(18*exp(- 2*x - 2) + 15*x*exp(- x - 1) + 3*x^2) + 15*x*e xp(- x - 1) + 3*x^2) - 3*x^2*exp(3) + 3*x^3 + exp(- 2*x - 2)*(18*x - 18*ex p(3)))/(log((3*x*exp(- x - 1) + x^2)/(x + 2*exp(- x - 1)))^2*(5*x^3*exp(- x - 1) + 6*x^2*exp(- 2*x - 2) + x^4) - log((3*x*exp(- x - 1) + x^2)/(x + 2 *exp(- x - 1)))*(2*x^4*exp(3) + 10*x^3*exp(3)*exp(- x - 1) + 12*x^2*exp(3) *exp(- 2*x - 2)) + x^4*exp(6) + 5*x^3*exp(6)*exp(- x - 1) + 6*x^2*exp(6)*e xp(- 2*x - 2)),x)
Output:
int((log(x)*(exp(- 2*x - 2)*(18*exp(3) - 18) + 3*x^2*exp(3) + exp(- x - 1) *(15*x*exp(3) - 12*x + 3*x^2) - 3*x^2) - exp(- x - 1)*(15*x*exp(3) - 12*x^ 2 + 3*x^3) + log((3*x*exp(- x - 1) + x^2)/(x + 2*exp(- x - 1)))*(18*exp(- 2*x - 2) - log(x)*(18*exp(- 2*x - 2) + 15*x*exp(- x - 1) + 3*x^2) + 15*x*e xp(- x - 1) + 3*x^2) - 3*x^2*exp(3) + 3*x^3 + exp(- 2*x - 2)*(18*x - 18*ex p(3)))/(log((3*x*exp(- x - 1) + x^2)/(x + 2*exp(- x - 1)))^2*(5*x^3*exp(- x - 1) + 6*x^2*exp(- 2*x - 2) + x^4) + x^4*exp(6) - log((3*x*exp(- x - 1) + x^2)/(x + 2*exp(- x - 1)))*(2*x^4*exp(3) + 12*x^2*exp(1 - 2*x) + 10*x^3* exp(2 - x)) + 6*x^2*exp(4 - 2*x) + 5*x^3*exp(5 - x)), x)
Time = 0.25 (sec) , antiderivative size = 74, normalized size of antiderivative = 2.11 \[ \int \frac {-3 e^3 x^2+3 x^3+e^{-2-2 x} \left (-18 e^3+18 x\right )+e^{-1-x} \left (-15 e^3 x+12 x^2-3 x^3\right )+\left (e^{-2-2 x} \left (-18+18 e^3\right )-3 x^2+3 e^3 x^2+e^{-1-x} \left (-12 x+15 e^3 x+3 x^2\right )\right ) \log (x)+\left (18 e^{-2-2 x}+15 e^{-1-x} x+3 x^2+\left (-18 e^{-2-2 x}-15 e^{-1-x} x-3 x^2\right ) \log (x)\right ) \log \left (\frac {3 e^{-1-x} x+x^2}{2 e^{-1-x}+x}\right )}{6 e^{4-2 x} x^2+5 e^{5-x} x^3+e^6 x^4+\left (-12 e^{1-2 x} x^2-10 e^{2-x} x^3-2 e^3 x^4\right ) \log \left (\frac {3 e^{-1-x} x+x^2}{2 e^{-1-x}+x}\right )+\left (6 e^{-2-2 x} x^2+5 e^{-1-x} x^3+x^4\right ) \log ^2\left (\frac {3 e^{-1-x} x+x^2}{2 e^{-1-x}+x}\right )} \, dx=\frac {-3 \,\mathrm {log}\left (\frac {e^{x} e \,x^{2}+3 x}{e^{x} e x +2}\right ) x +3 \,\mathrm {log}\left (x \right ) e^{3}}{e^{3} x \left (\mathrm {log}\left (\frac {e^{x} e \,x^{2}+3 x}{e^{x} e x +2}\right )-e^{3}\right )} \] Input:
int((((-18*exp(-1-x)^2-15*x*exp(-1-x)-3*x^2)*log(x)+18*exp(-1-x)^2+15*x*ex p(-1-x)+3*x^2)*log((3*x*exp(-1-x)+x^2)/(2*exp(-1-x)+x))+((18*exp(3)-18)*ex p(-1-x)^2+(15*x*exp(3)+3*x^2-12*x)*exp(-1-x)+3*x^2*exp(3)-3*x^2)*log(x)+(- 18*exp(3)+18*x)*exp(-1-x)^2+(-15*x*exp(3)-3*x^3+12*x^2)*exp(-1-x)-3*x^2*ex p(3)+3*x^3)/((6*x^2*exp(-1-x)^2+5*x^3*exp(-1-x)+x^4)*log((3*x*exp(-1-x)+x^ 2)/(2*exp(-1-x)+x))^2+(-12*x^2*exp(3)*exp(-1-x)^2-10*x^3*exp(3)*exp(-1-x)- 2*x^4*exp(3))*log((3*x*exp(-1-x)+x^2)/(2*exp(-1-x)+x))+6*x^2*exp(3)^2*exp( -1-x)^2+5*x^3*exp(3)^2*exp(-1-x)+x^4*exp(3)^2),x)
Output:
(3*( - log((e**x*e*x**2 + 3*x)/(e**x*e*x + 2))*x + log(x)*e**3))/(e**3*x*( log((e**x*e*x**2 + 3*x)/(e**x*e*x + 2)) - e**3))