Integrand size = 241, antiderivative size = 26 \[ \int \frac {2 x^4-6 x^5-2 x^6+e^{2+x} \left (-16 x-16 x^2-8 x^3\right )+\left (8 e^{2+x} x^2+2 x^5\right ) \log \left (\frac {e^{-2-x} \left (4 e^{2+x}+x^3\right )}{x^2}\right )+\left (2 x^3-6 x^4-2 x^5+e^{2+x} \left (-16-16 x-8 x^2\right )+\left (8 e^{2+x} x+2 x^4\right ) \log \left (\frac {e^{-2-x} \left (4 e^{2+x}+x^3\right )}{x^2}\right )\right ) \log \left (-1-x+\log \left (\frac {e^{-2-x} \left (4 e^{2+x}+x^3\right )}{x^2}\right )\right )}{-x^4-x^5+e^{2+x} \left (-4 x-4 x^2\right )+\left (4 e^{2+x} x+x^4\right ) \log \left (\frac {e^{-2-x} \left (4 e^{2+x}+x^3\right )}{x^2}\right )} \, dx=\left (x+\log \left (-1-x+\log \left (\frac {4}{x^2}+e^{-2-x} x\right )\right )\right )^2 \] Output:
(x+ln(ln(4/x^2+x/exp(2+x))-1-x))^2
Leaf count is larger than twice the leaf count of optimal. \(62\) vs. \(2(26)=52\).
Time = 0.10 (sec) , antiderivative size = 62, normalized size of antiderivative = 2.38 \[ \int \frac {2 x^4-6 x^5-2 x^6+e^{2+x} \left (-16 x-16 x^2-8 x^3\right )+\left (8 e^{2+x} x^2+2 x^5\right ) \log \left (\frac {e^{-2-x} \left (4 e^{2+x}+x^3\right )}{x^2}\right )+\left (2 x^3-6 x^4-2 x^5+e^{2+x} \left (-16-16 x-8 x^2\right )+\left (8 e^{2+x} x+2 x^4\right ) \log \left (\frac {e^{-2-x} \left (4 e^{2+x}+x^3\right )}{x^2}\right )\right ) \log \left (-1-x+\log \left (\frac {e^{-2-x} \left (4 e^{2+x}+x^3\right )}{x^2}\right )\right )}{-x^4-x^5+e^{2+x} \left (-4 x-4 x^2\right )+\left (4 e^{2+x} x+x^4\right ) \log \left (\frac {e^{-2-x} \left (4 e^{2+x}+x^3\right )}{x^2}\right )} \, dx=2 \left (\frac {x^2}{2}+x \log \left (-1-x+\log \left (\frac {4}{x^2}+e^{-2-x} x\right )\right )+\frac {1}{2} \log ^2\left (-1-x+\log \left (\frac {4}{x^2}+e^{-2-x} x\right )\right )\right ) \] Input:
Integrate[(2*x^4 - 6*x^5 - 2*x^6 + E^(2 + x)*(-16*x - 16*x^2 - 8*x^3) + (8 *E^(2 + x)*x^2 + 2*x^5)*Log[(E^(-2 - x)*(4*E^(2 + x) + x^3))/x^2] + (2*x^3 - 6*x^4 - 2*x^5 + E^(2 + x)*(-16 - 16*x - 8*x^2) + (8*E^(2 + x)*x + 2*x^4 )*Log[(E^(-2 - x)*(4*E^(2 + x) + x^3))/x^2])*Log[-1 - x + Log[(E^(-2 - x)* (4*E^(2 + x) + x^3))/x^2]])/(-x^4 - x^5 + E^(2 + x)*(-4*x - 4*x^2) + (4*E^ (2 + x)*x + x^4)*Log[(E^(-2 - x)*(4*E^(2 + x) + x^3))/x^2]),x]
Output:
2*(x^2/2 + x*Log[-1 - x + Log[4/x^2 + E^(-2 - x)*x]] + Log[-1 - x + Log[4/ x^2 + E^(-2 - x)*x]]^2/2)
Time = 1.24 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.017, Rules used = {7239, 27, 25, 7237}
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.
| \(\displaystyle \int \frac {-2 x^6-6 x^5+2 x^4+e^{x+2} \left (-8 x^3-16 x^2-16 x\right )+\left (2 x^5+8 e^{x+2} x^2\right ) \log \left (\frac {e^{-x-2} \left (x^3+4 e^{x+2}\right )}{x^2}\right )+\left (-2 x^5-6 x^4+2 x^3+e^{x+2} \left (-8 x^2-16 x-16\right )+\left (2 x^4+8 e^{x+2} x\right ) \log \left (\frac {e^{-x-2} \left (x^3+4 e^{x+2}\right )}{x^2}\right )\right ) \log \left (\log \left (\frac {e^{-x-2} \left (x^3+4 e^{x+2}\right )}{x^2}\right )-x-1\right )}{-x^5-x^4+e^{x+2} \left (-4 x^2-4 x\right )+\left (x^4+4 e^{x+2} x\right ) \log \left (\frac {e^{-x-2} \left (x^3+4 e^{x+2}\right )}{x^2}\right )} \, dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {2 \left (4 e^{x+2} \left (x^2+2 x+2\right )+\left (x^2+3 x-1\right ) x^3-\left (x^3+4 e^{x+2}\right ) x \log \left (\frac {4}{x^2}+e^{-x-2} x\right )\right ) \left (\log \left (\log \left (\frac {4}{x^2}+e^{-x-2} x\right )-x-1\right )+x\right )}{x \left (x^3+4 e^{x+2}\right ) \left (-\log \left (\frac {4}{x^2}+e^{-x-2} x\right )+x+1\right )}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 \int -\frac {\left (\left (-x^2-3 x+1\right ) x^3+\left (x^3+4 e^{x+2}\right ) \log \left (e^{-x-2} x+\frac {4}{x^2}\right ) x-4 e^{x+2} \left (x^2+2 x+2\right )\right ) \left (x+\log \left (-x+\log \left (e^{-x-2} x+\frac {4}{x^2}\right )-1\right )\right )}{x \left (x^3+4 e^{x+2}\right ) \left (x-\log \left (e^{-x-2} x+\frac {4}{x^2}\right )+1\right )}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -2 \int \frac {\left (\left (-x^2-3 x+1\right ) x^3+\left (x^3+4 e^{x+2}\right ) \log \left (e^{-x-2} x+\frac {4}{x^2}\right ) x-4 e^{x+2} \left (x^2+2 x+2\right )\right ) \left (x+\log \left (-x+\log \left (e^{-x-2} x+\frac {4}{x^2}\right )-1\right )\right )}{x \left (x^3+4 e^{x+2}\right ) \left (x-\log \left (e^{-x-2} x+\frac {4}{x^2}\right )+1\right )}dx\) |
| \(\Big \downarrow \) 7237 |
| \(\displaystyle \left (\log \left (\log \left (\frac {4}{x^2}+e^{-x-2} x\right )-x-1\right )+x\right )^2\) |
Input:
Int[(2*x^4 - 6*x^5 - 2*x^6 + E^(2 + x)*(-16*x - 16*x^2 - 8*x^3) + (8*E^(2 + x)*x^2 + 2*x^5)*Log[(E^(-2 - x)*(4*E^(2 + x) + x^3))/x^2] + (2*x^3 - 6*x ^4 - 2*x^5 + E^(2 + x)*(-16 - 16*x - 8*x^2) + (8*E^(2 + x)*x + 2*x^4)*Log[ (E^(-2 - x)*(4*E^(2 + x) + x^3))/x^2])*Log[-1 - x + Log[(E^(-2 - x)*(4*E^( 2 + x) + x^3))/x^2]])/(-x^4 - x^5 + E^(2 + x)*(-4*x - 4*x^2) + (4*E^(2 + x )*x + x^4)*Log[(E^(-2 - x)*(4*E^(2 + x) + x^3))/x^2]),x]
Output:
(x + Log[-1 - x + Log[4/x^2 + E^(-2 - x)*x]])^2
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(u_)*(y_)^(m_.), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Si mp[q*(y^(m + 1)/(m + 1)), x] /; !FalseQ[q]] /; FreeQ[m, x] && NeQ[m, -1]
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Leaf count of result is larger than twice the leaf count of optimal. \(121\) vs. \(2(25)=50\).
Time = 42.66 (sec) , antiderivative size = 122, normalized size of antiderivative = 4.69
| method | result | size |
| parallelrisch | \(x^{2}+2 x \ln \left (\ln \left (\frac {\left (4 \,{\mathrm e}^{2+x}+x^{3}\right ) {\mathrm e}^{-2-x}}{x^{2}}\right )-x -1\right )+{\ln \left (\ln \left (\frac {\left (4 \,{\mathrm e}^{2+x}+x^{3}\right ) {\mathrm e}^{-2-x}}{x^{2}}\right )-x -1\right )}^{2}+2 \ln \left (-\ln \left (\frac {\left (4 \,{\mathrm e}^{2+x}+x^{3}\right ) {\mathrm e}^{-2-x}}{x^{2}}\right )+x +1\right )-2 \ln \left (\ln \left (\frac {\left (4 \,{\mathrm e}^{2+x}+x^{3}\right ) {\mathrm e}^{-2-x}}{x^{2}}\right )-x -1\right )\) | \(122\) |
| risch | \(x^{2}+2 x \ln \left (-2 \ln \left (x \right )-\ln \left ({\mathrm e}^{2+x}\right )+\ln \left (4 \,{\mathrm e}^{2+x}+x^{3}\right )-\frac {i \pi \,\operatorname {csgn}\left (i {\mathrm e}^{-2-x} \left (4 \,{\mathrm e}^{2+x}+x^{3}\right )\right ) \left (-\operatorname {csgn}\left (i {\mathrm e}^{-2-x} \left (4 \,{\mathrm e}^{2+x}+x^{3}\right )\right )+\operatorname {csgn}\left (i {\mathrm e}^{-2-x}\right )\right ) \left (-\operatorname {csgn}\left (i {\mathrm e}^{-2-x} \left (4 \,{\mathrm e}^{2+x}+x^{3}\right )\right )+\operatorname {csgn}\left (i \left (4 \,{\mathrm e}^{2+x}+x^{3}\right )\right )\right )}{2}+\frac {i \pi \,\operatorname {csgn}\left (i x^{2}\right ) {\left (-\operatorname {csgn}\left (i x^{2}\right )+\operatorname {csgn}\left (i x \right )\right )}^{2}}{2}-\frac {i \pi \,\operatorname {csgn}\left (\frac {i \left (4 \,{\mathrm e}^{2+x}+x^{3}\right ) {\mathrm e}^{-2-x}}{x^{2}}\right ) \left (-\operatorname {csgn}\left (\frac {i \left (4 \,{\mathrm e}^{2+x}+x^{3}\right ) {\mathrm e}^{-2-x}}{x^{2}}\right )+\operatorname {csgn}\left (\frac {i}{x^{2}}\right )\right ) \left (-\operatorname {csgn}\left (\frac {i \left (4 \,{\mathrm e}^{2+x}+x^{3}\right ) {\mathrm e}^{-2-x}}{x^{2}}\right )+\operatorname {csgn}\left (i {\mathrm e}^{-2-x} \left (4 \,{\mathrm e}^{2+x}+x^{3}\right )\right )\right )}{2}-x -1\right )+{\ln \left (-2 \ln \left (x \right )-\ln \left ({\mathrm e}^{2+x}\right )+\ln \left (4 \,{\mathrm e}^{2+x}+x^{3}\right )-\frac {i \pi \,\operatorname {csgn}\left (i {\mathrm e}^{-2-x} \left (4 \,{\mathrm e}^{2+x}+x^{3}\right )\right ) \left (-\operatorname {csgn}\left (i {\mathrm e}^{-2-x} \left (4 \,{\mathrm e}^{2+x}+x^{3}\right )\right )+\operatorname {csgn}\left (i {\mathrm e}^{-2-x}\right )\right ) \left (-\operatorname {csgn}\left (i {\mathrm e}^{-2-x} \left (4 \,{\mathrm e}^{2+x}+x^{3}\right )\right )+\operatorname {csgn}\left (i \left (4 \,{\mathrm e}^{2+x}+x^{3}\right )\right )\right )}{2}+\frac {i \pi \,\operatorname {csgn}\left (i x^{2}\right ) {\left (-\operatorname {csgn}\left (i x^{2}\right )+\operatorname {csgn}\left (i x \right )\right )}^{2}}{2}-\frac {i \pi \,\operatorname {csgn}\left (\frac {i \left (4 \,{\mathrm e}^{2+x}+x^{3}\right ) {\mathrm e}^{-2-x}}{x^{2}}\right ) \left (-\operatorname {csgn}\left (\frac {i \left (4 \,{\mathrm e}^{2+x}+x^{3}\right ) {\mathrm e}^{-2-x}}{x^{2}}\right )+\operatorname {csgn}\left (\frac {i}{x^{2}}\right )\right ) \left (-\operatorname {csgn}\left (\frac {i \left (4 \,{\mathrm e}^{2+x}+x^{3}\right ) {\mathrm e}^{-2-x}}{x^{2}}\right )+\operatorname {csgn}\left (i {\mathrm e}^{-2-x} \left (4 \,{\mathrm e}^{2+x}+x^{3}\right )\right )\right )}{2}-x -1\right )}^{2}\) | \(522\) |
Input:
int((((8*x*exp(2+x)+2*x^4)*ln((4*exp(2+x)+x^3)/x^2/exp(2+x))+(-8*x^2-16*x- 16)*exp(2+x)-2*x^5-6*x^4+2*x^3)*ln(ln((4*exp(2+x)+x^3)/x^2/exp(2+x))-x-1)+ (8*x^2*exp(2+x)+2*x^5)*ln((4*exp(2+x)+x^3)/x^2/exp(2+x))+(-8*x^3-16*x^2-16 *x)*exp(2+x)-2*x^6-6*x^5+2*x^4)/((4*x*exp(2+x)+x^4)*ln((4*exp(2+x)+x^3)/x^ 2/exp(2+x))+(-4*x^2-4*x)*exp(2+x)-x^5-x^4),x,method=_RETURNVERBOSE)
Output:
x^2+2*x*ln(ln((4*exp(2+x)+x^3)/x^2/exp(2+x))-x-1)+ln(ln((4*exp(2+x)+x^3)/x ^2/exp(2+x))-x-1)^2+2*ln(-ln((4*exp(2+x)+x^3)/x^2/exp(2+x))+x+1)-2*ln(ln(( 4*exp(2+x)+x^3)/x^2/exp(2+x))-x-1)
Leaf count of result is larger than twice the leaf count of optimal. 63 vs. \(2 (25) = 50\).
Time = 0.10 (sec) , antiderivative size = 63, normalized size of antiderivative = 2.42 \[ \int \frac {2 x^4-6 x^5-2 x^6+e^{2+x} \left (-16 x-16 x^2-8 x^3\right )+\left (8 e^{2+x} x^2+2 x^5\right ) \log \left (\frac {e^{-2-x} \left (4 e^{2+x}+x^3\right )}{x^2}\right )+\left (2 x^3-6 x^4-2 x^5+e^{2+x} \left (-16-16 x-8 x^2\right )+\left (8 e^{2+x} x+2 x^4\right ) \log \left (\frac {e^{-2-x} \left (4 e^{2+x}+x^3\right )}{x^2}\right )\right ) \log \left (-1-x+\log \left (\frac {e^{-2-x} \left (4 e^{2+x}+x^3\right )}{x^2}\right )\right )}{-x^4-x^5+e^{2+x} \left (-4 x-4 x^2\right )+\left (4 e^{2+x} x+x^4\right ) \log \left (\frac {e^{-2-x} \left (4 e^{2+x}+x^3\right )}{x^2}\right )} \, dx=x^{2} + 2 \, x \log \left (-x + \log \left (\frac {{\left (x^{3} + 4 \, e^{\left (x + 2\right )}\right )} e^{\left (-x - 2\right )}}{x^{2}}\right ) - 1\right ) + \log \left (-x + \log \left (\frac {{\left (x^{3} + 4 \, e^{\left (x + 2\right )}\right )} e^{\left (-x - 2\right )}}{x^{2}}\right ) - 1\right )^{2} \] Input:
integrate((((8*x*exp(2+x)+2*x^4)*log((4*exp(2+x)+x^3)/x^2/exp(2+x))+(-8*x^ 2-16*x-16)*exp(2+x)-2*x^5-6*x^4+2*x^3)*log(log((4*exp(2+x)+x^3)/x^2/exp(2+ x))-x-1)+(8*x^2*exp(2+x)+2*x^5)*log((4*exp(2+x)+x^3)/x^2/exp(2+x))+(-8*x^3 -16*x^2-16*x)*exp(2+x)-2*x^6-6*x^5+2*x^4)/((4*x*exp(2+x)+x^4)*log((4*exp(2 +x)+x^3)/x^2/exp(2+x))+(-4*x^2-4*x)*exp(2+x)-x^5-x^4),x, algorithm="fricas ")
Output:
x^2 + 2*x*log(-x + log((x^3 + 4*e^(x + 2))*e^(-x - 2)/x^2) - 1) + log(-x + log((x^3 + 4*e^(x + 2))*e^(-x - 2)/x^2) - 1)^2
Leaf count of result is larger than twice the leaf count of optimal. 61 vs. \(2 (22) = 44\).
Time = 12.27 (sec) , antiderivative size = 61, normalized size of antiderivative = 2.35 \[ \int \frac {2 x^4-6 x^5-2 x^6+e^{2+x} \left (-16 x-16 x^2-8 x^3\right )+\left (8 e^{2+x} x^2+2 x^5\right ) \log \left (\frac {e^{-2-x} \left (4 e^{2+x}+x^3\right )}{x^2}\right )+\left (2 x^3-6 x^4-2 x^5+e^{2+x} \left (-16-16 x-8 x^2\right )+\left (8 e^{2+x} x+2 x^4\right ) \log \left (\frac {e^{-2-x} \left (4 e^{2+x}+x^3\right )}{x^2}\right )\right ) \log \left (-1-x+\log \left (\frac {e^{-2-x} \left (4 e^{2+x}+x^3\right )}{x^2}\right )\right )}{-x^4-x^5+e^{2+x} \left (-4 x-4 x^2\right )+\left (4 e^{2+x} x+x^4\right ) \log \left (\frac {e^{-2-x} \left (4 e^{2+x}+x^3\right )}{x^2}\right )} \, dx=x^{2} + 2 x \log {\left (- x + \log {\left (\frac {\left (x^{3} + 4 e^{x + 2}\right ) e^{- x - 2}}{x^{2}} \right )} - 1 \right )} + \log {\left (- x + \log {\left (\frac {\left (x^{3} + 4 e^{x + 2}\right ) e^{- x - 2}}{x^{2}} \right )} - 1 \right )}^{2} \] Input:
integrate((((8*x*exp(2+x)+2*x**4)*ln((4*exp(2+x)+x**3)/x**2/exp(2+x))+(-8* x**2-16*x-16)*exp(2+x)-2*x**5-6*x**4+2*x**3)*ln(ln((4*exp(2+x)+x**3)/x**2/ exp(2+x))-x-1)+(8*x**2*exp(2+x)+2*x**5)*ln((4*exp(2+x)+x**3)/x**2/exp(2+x) )+(-8*x**3-16*x**2-16*x)*exp(2+x)-2*x**6-6*x**5+2*x**4)/((4*x*exp(2+x)+x** 4)*ln((4*exp(2+x)+x**3)/x**2/exp(2+x))+(-4*x**2-4*x)*exp(2+x)-x**5-x**4),x )
Output:
x**2 + 2*x*log(-x + log((x**3 + 4*exp(x + 2))*exp(-x - 2)/x**2) - 1) + log (-x + log((x**3 + 4*exp(x + 2))*exp(-x - 2)/x**2) - 1)**2
Leaf count of result is larger than twice the leaf count of optimal. 51 vs. \(2 (25) = 50\).
Time = 0.13 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.96 \[ \int \frac {2 x^4-6 x^5-2 x^6+e^{2+x} \left (-16 x-16 x^2-8 x^3\right )+\left (8 e^{2+x} x^2+2 x^5\right ) \log \left (\frac {e^{-2-x} \left (4 e^{2+x}+x^3\right )}{x^2}\right )+\left (2 x^3-6 x^4-2 x^5+e^{2+x} \left (-16-16 x-8 x^2\right )+\left (8 e^{2+x} x+2 x^4\right ) \log \left (\frac {e^{-2-x} \left (4 e^{2+x}+x^3\right )}{x^2}\right )\right ) \log \left (-1-x+\log \left (\frac {e^{-2-x} \left (4 e^{2+x}+x^3\right )}{x^2}\right )\right )}{-x^4-x^5+e^{2+x} \left (-4 x-4 x^2\right )+\left (4 e^{2+x} x+x^4\right ) \log \left (\frac {e^{-2-x} \left (4 e^{2+x}+x^3\right )}{x^2}\right )} \, dx=x^{2} + 2 \, x \log \left (-2 \, x + \log \left (x^{3} + 4 \, e^{\left (x + 2\right )}\right ) - 2 \, \log \left (x\right ) - 3\right ) + \log \left (-2 \, x + \log \left (x^{3} + 4 \, e^{\left (x + 2\right )}\right ) - 2 \, \log \left (x\right ) - 3\right )^{2} \] Input:
integrate((((8*x*exp(2+x)+2*x^4)*log((4*exp(2+x)+x^3)/x^2/exp(2+x))+(-8*x^ 2-16*x-16)*exp(2+x)-2*x^5-6*x^4+2*x^3)*log(log((4*exp(2+x)+x^3)/x^2/exp(2+ x))-x-1)+(8*x^2*exp(2+x)+2*x^5)*log((4*exp(2+x)+x^3)/x^2/exp(2+x))+(-8*x^3 -16*x^2-16*x)*exp(2+x)-2*x^6-6*x^5+2*x^4)/((4*x*exp(2+x)+x^4)*log((4*exp(2 +x)+x^3)/x^2/exp(2+x))+(-4*x^2-4*x)*exp(2+x)-x^5-x^4),x, algorithm="maxima ")
Output:
x^2 + 2*x*log(-2*x + log(x^3 + 4*e^(x + 2)) - 2*log(x) - 3) + log(-2*x + l og(x^3 + 4*e^(x + 2)) - 2*log(x) - 3)^2
\[ \int \frac {2 x^4-6 x^5-2 x^6+e^{2+x} \left (-16 x-16 x^2-8 x^3\right )+\left (8 e^{2+x} x^2+2 x^5\right ) \log \left (\frac {e^{-2-x} \left (4 e^{2+x}+x^3\right )}{x^2}\right )+\left (2 x^3-6 x^4-2 x^5+e^{2+x} \left (-16-16 x-8 x^2\right )+\left (8 e^{2+x} x+2 x^4\right ) \log \left (\frac {e^{-2-x} \left (4 e^{2+x}+x^3\right )}{x^2}\right )\right ) \log \left (-1-x+\log \left (\frac {e^{-2-x} \left (4 e^{2+x}+x^3\right )}{x^2}\right )\right )}{-x^4-x^5+e^{2+x} \left (-4 x-4 x^2\right )+\left (4 e^{2+x} x+x^4\right ) \log \left (\frac {e^{-2-x} \left (4 e^{2+x}+x^3\right )}{x^2}\right )} \, dx=\int { \frac {2 \, {\left (x^{6} + 3 \, x^{5} - x^{4} + 4 \, {\left (x^{3} + 2 \, x^{2} + 2 \, x\right )} e^{\left (x + 2\right )} + {\left (x^{5} + 3 \, x^{4} - x^{3} + 4 \, {\left (x^{2} + 2 \, x + 2\right )} e^{\left (x + 2\right )} - {\left (x^{4} + 4 \, x e^{\left (x + 2\right )}\right )} \log \left (\frac {{\left (x^{3} + 4 \, e^{\left (x + 2\right )}\right )} e^{\left (-x - 2\right )}}{x^{2}}\right )\right )} \log \left (-x + \log \left (\frac {{\left (x^{3} + 4 \, e^{\left (x + 2\right )}\right )} e^{\left (-x - 2\right )}}{x^{2}}\right ) - 1\right ) - {\left (x^{5} + 4 \, x^{2} e^{\left (x + 2\right )}\right )} \log \left (\frac {{\left (x^{3} + 4 \, e^{\left (x + 2\right )}\right )} e^{\left (-x - 2\right )}}{x^{2}}\right )\right )}}{x^{5} + x^{4} + 4 \, {\left (x^{2} + x\right )} e^{\left (x + 2\right )} - {\left (x^{4} + 4 \, x e^{\left (x + 2\right )}\right )} \log \left (\frac {{\left (x^{3} + 4 \, e^{\left (x + 2\right )}\right )} e^{\left (-x - 2\right )}}{x^{2}}\right )} \,d x } \] Input:
integrate((((8*x*exp(2+x)+2*x^4)*log((4*exp(2+x)+x^3)/x^2/exp(2+x))+(-8*x^ 2-16*x-16)*exp(2+x)-2*x^5-6*x^4+2*x^3)*log(log((4*exp(2+x)+x^3)/x^2/exp(2+ x))-x-1)+(8*x^2*exp(2+x)+2*x^5)*log((4*exp(2+x)+x^3)/x^2/exp(2+x))+(-8*x^3 -16*x^2-16*x)*exp(2+x)-2*x^6-6*x^5+2*x^4)/((4*x*exp(2+x)+x^4)*log((4*exp(2 +x)+x^3)/x^2/exp(2+x))+(-4*x^2-4*x)*exp(2+x)-x^5-x^4),x, algorithm="giac")
Output:
integrate(2*(x^6 + 3*x^5 - x^4 + 4*(x^3 + 2*x^2 + 2*x)*e^(x + 2) + (x^5 + 3*x^4 - x^3 + 4*(x^2 + 2*x + 2)*e^(x + 2) - (x^4 + 4*x*e^(x + 2))*log((x^3 + 4*e^(x + 2))*e^(-x - 2)/x^2))*log(-x + log((x^3 + 4*e^(x + 2))*e^(-x - 2)/x^2) - 1) - (x^5 + 4*x^2*e^(x + 2))*log((x^3 + 4*e^(x + 2))*e^(-x - 2)/ x^2))/(x^5 + x^4 + 4*(x^2 + x)*e^(x + 2) - (x^4 + 4*x*e^(x + 2))*log((x^3 + 4*e^(x + 2))*e^(-x - 2)/x^2)), x)
Time = 3.03 (sec) , antiderivative size = 55, normalized size of antiderivative = 2.12 \[ \int \frac {2 x^4-6 x^5-2 x^6+e^{2+x} \left (-16 x-16 x^2-8 x^3\right )+\left (8 e^{2+x} x^2+2 x^5\right ) \log \left (\frac {e^{-2-x} \left (4 e^{2+x}+x^3\right )}{x^2}\right )+\left (2 x^3-6 x^4-2 x^5+e^{2+x} \left (-16-16 x-8 x^2\right )+\left (8 e^{2+x} x+2 x^4\right ) \log \left (\frac {e^{-2-x} \left (4 e^{2+x}+x^3\right )}{x^2}\right )\right ) \log \left (-1-x+\log \left (\frac {e^{-2-x} \left (4 e^{2+x}+x^3\right )}{x^2}\right )\right )}{-x^4-x^5+e^{2+x} \left (-4 x-4 x^2\right )+\left (4 e^{2+x} x+x^4\right ) \log \left (\frac {e^{-2-x} \left (4 e^{2+x}+x^3\right )}{x^2}\right )} \, dx=x^2+2\,x\,\ln \left (\ln \left (\frac {x^3\,{\mathrm {e}}^{-x}\,{\mathrm {e}}^{-2}+4}{x^2}\right )-x-1\right )+{\ln \left (\ln \left (\frac {x^3\,{\mathrm {e}}^{-x}\,{\mathrm {e}}^{-2}+4}{x^2}\right )-x-1\right )}^2 \] Input:
int((log(log((exp(- x - 2)*(4*exp(x + 2) + x^3))/x^2) - x - 1)*(exp(x + 2) *(16*x + 8*x^2 + 16) - log((exp(- x - 2)*(4*exp(x + 2) + x^3))/x^2)*(8*x*e xp(x + 2) + 2*x^4) - 2*x^3 + 6*x^4 + 2*x^5) + exp(x + 2)*(16*x + 16*x^2 + 8*x^3) - log((exp(- x - 2)*(4*exp(x + 2) + x^3))/x^2)*(8*x^2*exp(x + 2) + 2*x^5) - 2*x^4 + 6*x^5 + 2*x^6)/(exp(x + 2)*(4*x + 4*x^2) - log((exp(- x - 2)*(4*exp(x + 2) + x^3))/x^2)*(4*x*exp(x + 2) + x^4) + x^4 + x^5),x)
Output:
2*x*log(log((x^3*exp(-x)*exp(-2) + 4)/x^2) - x - 1) + x^2 + log(log((x^3*e xp(-x)*exp(-2) + 4)/x^2) - x - 1)^2
\[ \int \frac {2 x^4-6 x^5-2 x^6+e^{2+x} \left (-16 x-16 x^2-8 x^3\right )+\left (8 e^{2+x} x^2+2 x^5\right ) \log \left (\frac {e^{-2-x} \left (4 e^{2+x}+x^3\right )}{x^2}\right )+\left (2 x^3-6 x^4-2 x^5+e^{2+x} \left (-16-16 x-8 x^2\right )+\left (8 e^{2+x} x+2 x^4\right ) \log \left (\frac {e^{-2-x} \left (4 e^{2+x}+x^3\right )}{x^2}\right )\right ) \log \left (-1-x+\log \left (\frac {e^{-2-x} \left (4 e^{2+x}+x^3\right )}{x^2}\right )\right )}{-x^4-x^5+e^{2+x} \left (-4 x-4 x^2\right )+\left (4 e^{2+x} x+x^4\right ) \log \left (\frac {e^{-2-x} \left (4 e^{2+x}+x^3\right )}{x^2}\right )} \, dx=\text {too large to display} \] Input:
int((((8*x*exp(2+x)+2*x^4)*log((4*exp(2+x)+x^3)/x^2/exp(2+x))+(-8*x^2-16*x -16)*exp(2+x)-2*x^5-6*x^4+2*x^3)*log(log((4*exp(2+x)+x^3)/x^2/exp(2+x))-x- 1)+(8*x^2*exp(2+x)+2*x^5)*log((4*exp(2+x)+x^3)/x^2/exp(2+x))+(-8*x^3-16*x^ 2-16*x)*exp(2+x)-2*x^6-6*x^5+2*x^4)/((4*x*exp(2+x)+x^4)*log((4*exp(2+x)+x^ 3)/x^2/exp(2+x))+(-4*x^2-4*x)*exp(2+x)-x^5-x^4),x)
Output:
2*( - 8*int(e**x/(4*e**x*log((4*e**x*e**2 + x**3)/(e**x*e**2*x**2))*e**2 - 4*e**x*e**2*x - 4*e**x*e**2 + log((4*e**x*e**2 + x**3)/(e**x*e**2*x**2))* x**3 - x**4 - x**3),x)*e**2 - int(x**5/(4*e**x*log((4*e**x*e**2 + x**3)/(e **x*e**2*x**2))*e**2 - 4*e**x*e**2*x - 4*e**x*e**2 + log((4*e**x*e**2 + x* *3)/(e**x*e**2*x**2))*x**3 - x**4 - x**3),x) - 3*int(x**4/(4*e**x*log((4*e **x*e**2 + x**3)/(e**x*e**2*x**2))*e**2 - 4*e**x*e**2*x - 4*e**x*e**2 + lo g((4*e**x*e**2 + x**3)/(e**x*e**2*x**2))*x**3 - x**4 - x**3),x) + int(x**3 /(4*e**x*log((4*e**x*e**2 + x**3)/(e**x*e**2*x**2))*e**2 - 4*e**x*e**2*x - 4*e**x*e**2 + log((4*e**x*e**2 + x**3)/(e**x*e**2*x**2))*x**3 - x**4 - x* *3),x) - 4*int((e**x*x**2)/(4*e**x*log((4*e**x*e**2 + x**3)/(e**x*e**2*x** 2))*e**2 - 4*e**x*e**2*x - 4*e**x*e**2 + log((4*e**x*e**2 + x**3)/(e**x*e* *2*x**2))*x**3 - x**4 - x**3),x)*e**2 + 4*int((e**x*log(log((4*e**x*e**2 + x**3)/(e**x*e**2*x**2)) - x - 1)*log((4*e**x*e**2 + x**3)/(e**x*e**2*x**2 )))/(4*e**x*log((4*e**x*e**2 + x**3)/(e**x*e**2*x**2))*e**2 - 4*e**x*e**2* x - 4*e**x*e**2 + log((4*e**x*e**2 + x**3)/(e**x*e**2*x**2))*x**3 - x**4 - x**3),x)*e**2 - 4*int((e**x*log(log((4*e**x*e**2 + x**3)/(e**x*e**2*x**2) ) - x - 1)*x)/(4*e**x*log((4*e**x*e**2 + x**3)/(e**x*e**2*x**2))*e**2 - 4* e**x*e**2*x - 4*e**x*e**2 + log((4*e**x*e**2 + x**3)/(e**x*e**2*x**2))*x** 3 - x**4 - x**3),x)*e**2 - 8*int((e**x*log(log((4*e**x*e**2 + x**3)/(e**x* e**2*x**2)) - x - 1))/(4*e**x*log((4*e**x*e**2 + x**3)/(e**x*e**2*x**2)...