Integrand size = 103, antiderivative size = 33 \[ \int \frac {e^{\frac {e^x+x^2}{x^3 \log \left (-2+e^{3/2}\right ) \log (4+2 x)}} \left (-e^x x-x^3+\left (-2 x^2-x^3+e^x \left (-6-x+x^2\right )\right ) \log (4+2 x)\right )}{\left (2 x^4+x^5\right ) \log \left (-2+e^{3/2}\right ) \log ^2(4+2 x)} \, dx=e^{\frac {\frac {e^x}{x}+x}{x^2 \log \left (-2+e^{3/2}\right ) \log (4+2 x)}} \]
Time = 0.11 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.94 \[ \int \frac {e^{\frac {e^x+x^2}{x^3 \log \left (-2+e^{3/2}\right ) \log (4+2 x)}} \left (-e^x x-x^3+\left (-2 x^2-x^3+e^x \left (-6-x+x^2\right )\right ) \log (4+2 x)\right )}{\left (2 x^4+x^5\right ) \log \left (-2+e^{3/2}\right ) \log ^2(4+2 x)} \, dx=e^{\frac {e^x+x^2}{x^3 \log \left (-2+e^{3/2}\right ) \log (2 (2+x))}} \]
Integrate[(E^((E^x + x^2)/(x^3*Log[-2 + E^(3/2)]*Log[4 + 2*x]))*(-(E^x*x) - x^3 + (-2*x^2 - x^3 + E^x*(-6 - x + x^2))*Log[4 + 2*x]))/((2*x^4 + x^5)* Log[-2 + E^(3/2)]*Log[4 + 2*x]^2),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.
\(\displaystyle \int \frac {e^{\frac {x^2+e^x}{x^3 \log \left (e^{3/2}-2\right ) \log (2 x+4)}} \left (-x^3+\left (-x^3-2 x^2+e^x \left (x^2-x-6\right )\right ) \log (2 x+4)-e^x x\right )}{\left (x^5+2 x^4\right ) \log \left (e^{3/2}-2\right ) \log ^2(2 x+4)} \, dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\int -\frac {e^{\frac {x^2+e^x}{x^3 \log \left (-2+e^{3/2}\right ) \log (2 x+4)}} \left (x^3+e^x x+\left (x^3+2 x^2+e^x \left (-x^2+x+6\right )\right ) \log (2 x+4)\right )}{\left (x^5+2 x^4\right ) \log ^2(2 x+4)}dx}{\log \left (e^{3/2}-2\right )}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {\int \frac {e^{\frac {x^2+e^x}{x^3 \log \left (-2+e^{3/2}\right ) \log (2 x+4)}} \left (x^3+e^x x+\left (x^3+2 x^2+e^x \left (-x^2+x+6\right )\right ) \log (2 x+4)\right )}{\left (x^5+2 x^4\right ) \log ^2(2 x+4)}dx}{\log \left (e^{3/2}-2\right )}\) |
\(\Big \downarrow \) 2026 |
\(\displaystyle -\frac {\int \frac {e^{\frac {x^2+e^x}{x^3 \log \left (-2+e^{3/2}\right ) \log (2 x+4)}} \left (x^3+e^x x+\left (x^3+2 x^2+e^x \left (-x^2+x+6\right )\right ) \log (2 x+4)\right )}{x^4 (x+2) \log ^2(2 x+4)}dx}{\log \left (e^{3/2}-2\right )}\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle -\frac {\int \left (\frac {e^{\frac {x^2+e^x}{x^3 \log \left (-2+e^{3/2}\right ) \log (2 x+4)}} (\log (2 (x+2)) x+x+2 \log (2 (x+2)))}{x^2 (x+2) \log ^2(2 x+4)}+\frac {\exp \left (x+\frac {x^2+e^x}{\log \left (-2+e^{3/2}\right ) \log (2 x+4) x^3}\right ) \left (-\log (2 (x+2)) x^2+\log (2 (x+2)) x+x+6 \log (2 (x+2))\right )}{x^4 (x+2) \log ^2(2 x+4)}\right )dx}{\log \left (e^{3/2}-2\right )}\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle -\frac {\int \frac {e^{\frac {x^2+e^x}{x^3 \log \left (-2+e^{3/2}\right ) \log (2 x+4)}} \left (x \left (x^2+e^x\right )+(x+2) \left (x^2-e^x (x-3)\right ) \log (2 (x+2))\right )}{x^4 (x+2) \log ^2(2 x+4)}dx}{\log \left (e^{3/2}-2\right )}\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle -\frac {\int \left (\frac {e^{\frac {x^2+e^x}{x^3 \log \left (-2+e^{3/2}\right ) \log (2 x+4)}} (\log (2 (x+2)) x+x+2 \log (2 (x+2)))}{x^2 (x+2) \log ^2(2 x+4)}+\frac {\exp \left (x+\frac {x^2+e^x}{\log \left (-2+e^{3/2}\right ) \log (2 x+4) x^3}\right ) \left (-\log (2 (x+2)) x^2+\log (2 (x+2)) x+x+6 \log (2 (x+2))\right )}{x^4 (x+2) \log ^2(2 x+4)}\right )dx}{\log \left (e^{3/2}-2\right )}\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle -\frac {\int \frac {e^{\frac {x^2+e^x}{x^3 \log \left (-2+e^{3/2}\right ) \log (2 x+4)}} \left (x \left (x^2+e^x\right )+(x+2) \left (x^2-e^x (x-3)\right ) \log (2 (x+2))\right )}{x^4 (x+2) \log ^2(2 x+4)}dx}{\log \left (e^{3/2}-2\right )}\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle -\frac {\int \left (\frac {e^{\frac {x^2+e^x}{x^3 \log \left (-2+e^{3/2}\right ) \log (2 x+4)}} (\log (2 (x+2)) x+x+2 \log (2 (x+2)))}{x^2 (x+2) \log ^2(2 x+4)}+\frac {\exp \left (x+\frac {x^2+e^x}{\log \left (-2+e^{3/2}\right ) \log (2 x+4) x^3}\right ) \left (-\log (2 (x+2)) x^2+\log (2 (x+2)) x+x+6 \log (2 (x+2))\right )}{x^4 (x+2) \log ^2(2 x+4)}\right )dx}{\log \left (e^{3/2}-2\right )}\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle -\frac {\int \frac {e^{\frac {x^2+e^x}{x^3 \log \left (-2+e^{3/2}\right ) \log (2 x+4)}} \left (x \left (x^2+e^x\right )+(x+2) \left (x^2-e^x (x-3)\right ) \log (2 (x+2))\right )}{x^4 (x+2) \log ^2(2 x+4)}dx}{\log \left (e^{3/2}-2\right )}\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle -\frac {\int \left (\frac {e^{\frac {x^2+e^x}{x^3 \log \left (-2+e^{3/2}\right ) \log (2 x+4)}} (\log (2 (x+2)) x+x+2 \log (2 (x+2)))}{x^2 (x+2) \log ^2(2 x+4)}+\frac {\exp \left (x+\frac {x^2+e^x}{\log \left (-2+e^{3/2}\right ) \log (2 x+4) x^3}\right ) \left (-\log (2 (x+2)) x^2+\log (2 (x+2)) x+x+6 \log (2 (x+2))\right )}{x^4 (x+2) \log ^2(2 x+4)}\right )dx}{\log \left (e^{3/2}-2\right )}\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle -\frac {\int \frac {e^{\frac {x^2+e^x}{x^3 \log \left (-2+e^{3/2}\right ) \log (2 x+4)}} \left (x \left (x^2+e^x\right )+(x+2) \left (x^2-e^x (x-3)\right ) \log (2 (x+2))\right )}{x^4 (x+2) \log ^2(2 x+4)}dx}{\log \left (e^{3/2}-2\right )}\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle -\frac {\int \left (\frac {e^{\frac {x^2+e^x}{x^3 \log \left (-2+e^{3/2}\right ) \log (2 x+4)}} (\log (2 (x+2)) x+x+2 \log (2 (x+2)))}{x^2 (x+2) \log ^2(2 x+4)}+\frac {\exp \left (x+\frac {x^2+e^x}{\log \left (-2+e^{3/2}\right ) \log (2 x+4) x^3}\right ) \left (-\log (2 (x+2)) x^2+\log (2 (x+2)) x+x+6 \log (2 (x+2))\right )}{x^4 (x+2) \log ^2(2 x+4)}\right )dx}{\log \left (e^{3/2}-2\right )}\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle -\frac {\int \frac {e^{\frac {x^2+e^x}{x^3 \log \left (-2+e^{3/2}\right ) \log (2 x+4)}} \left (x \left (x^2+e^x\right )+(x+2) \left (x^2-e^x (x-3)\right ) \log (2 (x+2))\right )}{x^4 (x+2) \log ^2(2 x+4)}dx}{\log \left (e^{3/2}-2\right )}\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle -\frac {\int \left (\frac {e^{\frac {x^2+e^x}{x^3 \log \left (-2+e^{3/2}\right ) \log (2 x+4)}} (\log (2 (x+2)) x+x+2 \log (2 (x+2)))}{x^2 (x+2) \log ^2(2 x+4)}+\frac {\exp \left (x+\frac {x^2+e^x}{\log \left (-2+e^{3/2}\right ) \log (2 x+4) x^3}\right ) \left (-\log (2 (x+2)) x^2+\log (2 (x+2)) x+x+6 \log (2 (x+2))\right )}{x^4 (x+2) \log ^2(2 x+4)}\right )dx}{\log \left (e^{3/2}-2\right )}\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle -\frac {\int \frac {e^{\frac {x^2+e^x}{x^3 \log \left (-2+e^{3/2}\right ) \log (2 x+4)}} \left (x \left (x^2+e^x\right )+(x+2) \left (x^2-e^x (x-3)\right ) \log (2 (x+2))\right )}{x^4 (x+2) \log ^2(2 x+4)}dx}{\log \left (e^{3/2}-2\right )}\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle -\frac {\int \left (\frac {e^{\frac {x^2+e^x}{x^3 \log \left (-2+e^{3/2}\right ) \log (2 x+4)}} (\log (2 (x+2)) x+x+2 \log (2 (x+2)))}{x^2 (x+2) \log ^2(2 x+4)}+\frac {\exp \left (x+\frac {x^2+e^x}{\log \left (-2+e^{3/2}\right ) \log (2 x+4) x^3}\right ) \left (-\log (2 (x+2)) x^2+\log (2 (x+2)) x+x+6 \log (2 (x+2))\right )}{x^4 (x+2) \log ^2(2 x+4)}\right )dx}{\log \left (e^{3/2}-2\right )}\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle -\frac {\int \frac {e^{\frac {x^2+e^x}{x^3 \log \left (-2+e^{3/2}\right ) \log (2 x+4)}} \left (x \left (x^2+e^x\right )+(x+2) \left (x^2-e^x (x-3)\right ) \log (2 (x+2))\right )}{x^4 (x+2) \log ^2(2 x+4)}dx}{\log \left (e^{3/2}-2\right )}\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle -\frac {\int \left (\frac {e^{\frac {x^2+e^x}{x^3 \log \left (-2+e^{3/2}\right ) \log (2 x+4)}} (\log (2 (x+2)) x+x+2 \log (2 (x+2)))}{x^2 (x+2) \log ^2(2 x+4)}+\frac {\exp \left (x+\frac {x^2+e^x}{\log \left (-2+e^{3/2}\right ) \log (2 x+4) x^3}\right ) \left (-\log (2 (x+2)) x^2+\log (2 (x+2)) x+x+6 \log (2 (x+2))\right )}{x^4 (x+2) \log ^2(2 x+4)}\right )dx}{\log \left (e^{3/2}-2\right )}\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle -\frac {\int \frac {e^{\frac {x^2+e^x}{x^3 \log \left (-2+e^{3/2}\right ) \log (2 x+4)}} \left (x \left (x^2+e^x\right )+(x+2) \left (x^2-e^x (x-3)\right ) \log (2 (x+2))\right )}{x^4 (x+2) \log ^2(2 x+4)}dx}{\log \left (e^{3/2}-2\right )}\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle -\frac {\int \left (\frac {e^{\frac {x^2+e^x}{x^3 \log \left (-2+e^{3/2}\right ) \log (2 x+4)}} (\log (2 (x+2)) x+x+2 \log (2 (x+2)))}{x^2 (x+2) \log ^2(2 x+4)}+\frac {\exp \left (x+\frac {x^2+e^x}{\log \left (-2+e^{3/2}\right ) \log (2 x+4) x^3}\right ) \left (-\log (2 (x+2)) x^2+\log (2 (x+2)) x+x+6 \log (2 (x+2))\right )}{x^4 (x+2) \log ^2(2 x+4)}\right )dx}{\log \left (e^{3/2}-2\right )}\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle -\frac {\int \frac {e^{\frac {x^2+e^x}{x^3 \log \left (-2+e^{3/2}\right ) \log (2 x+4)}} \left (x \left (x^2+e^x\right )+(x+2) \left (x^2-e^x (x-3)\right ) \log (2 (x+2))\right )}{x^4 (x+2) \log ^2(2 x+4)}dx}{\log \left (e^{3/2}-2\right )}\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle -\frac {\int \left (\frac {e^{\frac {x^2+e^x}{x^3 \log \left (-2+e^{3/2}\right ) \log (2 x+4)}} (\log (2 (x+2)) x+x+2 \log (2 (x+2)))}{x^2 (x+2) \log ^2(2 x+4)}+\frac {\exp \left (x+\frac {x^2+e^x}{\log \left (-2+e^{3/2}\right ) \log (2 x+4) x^3}\right ) \left (-\log (2 (x+2)) x^2+\log (2 (x+2)) x+x+6 \log (2 (x+2))\right )}{x^4 (x+2) \log ^2(2 x+4)}\right )dx}{\log \left (e^{3/2}-2\right )}\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle -\frac {\int \frac {e^{\frac {x^2+e^x}{x^3 \log \left (-2+e^{3/2}\right ) \log (2 x+4)}} \left (x \left (x^2+e^x\right )+(x+2) \left (x^2-e^x (x-3)\right ) \log (2 (x+2))\right )}{x^4 (x+2) \log ^2(2 x+4)}dx}{\log \left (e^{3/2}-2\right )}\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle -\frac {\int \left (\frac {e^{\frac {x^2+e^x}{x^3 \log \left (-2+e^{3/2}\right ) \log (2 x+4)}} (\log (2 (x+2)) x+x+2 \log (2 (x+2)))}{x^2 (x+2) \log ^2(2 x+4)}+\frac {\exp \left (x+\frac {x^2+e^x}{\log \left (-2+e^{3/2}\right ) \log (2 x+4) x^3}\right ) \left (-\log (2 (x+2)) x^2+\log (2 (x+2)) x+x+6 \log (2 (x+2))\right )}{x^4 (x+2) \log ^2(2 x+4)}\right )dx}{\log \left (e^{3/2}-2\right )}\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle -\frac {\int \frac {e^{\frac {x^2+e^x}{x^3 \log \left (-2+e^{3/2}\right ) \log (2 x+4)}} \left (x \left (x^2+e^x\right )+(x+2) \left (x^2-e^x (x-3)\right ) \log (2 (x+2))\right )}{x^4 (x+2) \log ^2(2 x+4)}dx}{\log \left (e^{3/2}-2\right )}\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle -\frac {\int \left (\frac {e^{\frac {x^2+e^x}{x^3 \log \left (-2+e^{3/2}\right ) \log (2 x+4)}} (\log (2 (x+2)) x+x+2 \log (2 (x+2)))}{x^2 (x+2) \log ^2(2 x+4)}+\frac {\exp \left (x+\frac {x^2+e^x}{\log \left (-2+e^{3/2}\right ) \log (2 x+4) x^3}\right ) \left (-\log (2 (x+2)) x^2+\log (2 (x+2)) x+x+6 \log (2 (x+2))\right )}{x^4 (x+2) \log ^2(2 x+4)}\right )dx}{\log \left (e^{3/2}-2\right )}\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle -\frac {\int \frac {e^{\frac {x^2+e^x}{x^3 \log \left (-2+e^{3/2}\right ) \log (2 x+4)}} \left (x \left (x^2+e^x\right )+(x+2) \left (x^2-e^x (x-3)\right ) \log (2 (x+2))\right )}{x^4 (x+2) \log ^2(2 x+4)}dx}{\log \left (e^{3/2}-2\right )}\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle -\frac {\int \left (\frac {e^{\frac {x^2+e^x}{x^3 \log \left (-2+e^{3/2}\right ) \log (2 x+4)}} (\log (2 (x+2)) x+x+2 \log (2 (x+2)))}{x^2 (x+2) \log ^2(2 x+4)}+\frac {\exp \left (x+\frac {x^2+e^x}{\log \left (-2+e^{3/2}\right ) \log (2 x+4) x^3}\right ) \left (-\log (2 (x+2)) x^2+\log (2 (x+2)) x+x+6 \log (2 (x+2))\right )}{x^4 (x+2) \log ^2(2 x+4)}\right )dx}{\log \left (e^{3/2}-2\right )}\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle -\frac {\int \frac {e^{\frac {x^2+e^x}{x^3 \log \left (-2+e^{3/2}\right ) \log (2 x+4)}} \left (x \left (x^2+e^x\right )+(x+2) \left (x^2-e^x (x-3)\right ) \log (2 (x+2))\right )}{x^4 (x+2) \log ^2(2 x+4)}dx}{\log \left (e^{3/2}-2\right )}\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle -\frac {\int \left (\frac {e^{\frac {x^2+e^x}{x^3 \log \left (-2+e^{3/2}\right ) \log (2 x+4)}} (\log (2 (x+2)) x+x+2 \log (2 (x+2)))}{x^2 (x+2) \log ^2(2 x+4)}+\frac {\exp \left (x+\frac {x^2+e^x}{\log \left (-2+e^{3/2}\right ) \log (2 x+4) x^3}\right ) \left (-\log (2 (x+2)) x^2+\log (2 (x+2)) x+x+6 \log (2 (x+2))\right )}{x^4 (x+2) \log ^2(2 x+4)}\right )dx}{\log \left (e^{3/2}-2\right )}\) |
Int[(E^((E^x + x^2)/(x^3*Log[-2 + E^(3/2)]*Log[4 + 2*x]))*(-(E^x*x) - x^3 + (-2*x^2 - x^3 + E^x*(-6 - x + x^2))*Log[4 + 2*x]))/((2*x^4 + x^5)*Log[-2 + E^(3/2)]*Log[4 + 2*x]^2),x]
3.29.21.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(Fx_.)*(Px_)^(p_.), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Int[x^(p *r)*ExpandToSum[Px/x^r, x]^p*Fx, x] /; IGtQ[r, 0]] /; PolyQ[Px, x] && Integ erQ[p] && !MonomialQ[Px, x] && (ILtQ[p, 0] || !PolyQ[u, x])
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Time = 0.14 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.82
\[{\mathrm e}^{\frac {x^{2}+{\mathrm e}^{x}}{x^{3} \ln \left (4+2 x \right ) \ln \left ({\mathrm e}^{\frac {3}{2}}-2\right )}}\]
int((((x^2-x-6)*exp(x)-x^3-2*x^2)*ln(4+2*x)-exp(x)*x-x^3)*exp((x^2+exp(x)) /x^3/ln(4+2*x)/ln(exp(3/2)-2))/(x^5+2*x^4)/ln(4+2*x)^2/ln(exp(3/2)-2),x)
Time = 0.30 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.79 \[ \int \frac {e^{\frac {e^x+x^2}{x^3 \log \left (-2+e^{3/2}\right ) \log (4+2 x)}} \left (-e^x x-x^3+\left (-2 x^2-x^3+e^x \left (-6-x+x^2\right )\right ) \log (4+2 x)\right )}{\left (2 x^4+x^5\right ) \log \left (-2+e^{3/2}\right ) \log ^2(4+2 x)} \, dx=e^{\left (\frac {x^{2} + e^{x}}{x^{3} \log \left (2 \, x + 4\right ) \log \left (e^{\frac {3}{2}} - 2\right )}\right )} \]
integrate((((x^2-x-6)*exp(x)-x^3-2*x^2)*log(4+2*x)-exp(x)*x-x^3)*exp((x^2+ exp(x))/x^3/log(4+2*x)/log(exp(3/2)-2))/(x^5+2*x^4)/log(4+2*x)^2/log(exp(3 /2)-2),x, algorithm=\
Time = 0.56 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.79 \[ \int \frac {e^{\frac {e^x+x^2}{x^3 \log \left (-2+e^{3/2}\right ) \log (4+2 x)}} \left (-e^x x-x^3+\left (-2 x^2-x^3+e^x \left (-6-x+x^2\right )\right ) \log (4+2 x)\right )}{\left (2 x^4+x^5\right ) \log \left (-2+e^{3/2}\right ) \log ^2(4+2 x)} \, dx=e^{\frac {x^{2} + e^{x}}{x^{3} \log {\left (-2 + e^{\frac {3}{2}} \right )} \log {\left (2 x + 4 \right )}}} \]
integrate((((x**2-x-6)*exp(x)-x**3-2*x**2)*ln(4+2*x)-exp(x)*x-x**3)*exp((x **2+exp(x))/x**3/ln(4+2*x)/ln(exp(3/2)-2))/(x**5+2*x**4)/ln(4+2*x)**2/ln(e xp(3/2)-2),x)
Exception generated. \[ \int \frac {e^{\frac {e^x+x^2}{x^3 \log \left (-2+e^{3/2}\right ) \log (4+2 x)}} \left (-e^x x-x^3+\left (-2 x^2-x^3+e^x \left (-6-x+x^2\right )\right ) \log (4+2 x)\right )}{\left (2 x^4+x^5\right ) \log \left (-2+e^{3/2}\right ) \log ^2(4+2 x)} \, dx=\text {Exception raised: RuntimeError} \]
integrate((((x^2-x-6)*exp(x)-x^3-2*x^2)*log(4+2*x)-exp(x)*x-x^3)*exp((x^2+ exp(x))/x^3/log(4+2*x)/log(exp(3/2)-2))/(x^5+2*x^4)/log(4+2*x)^2/log(exp(3 /2)-2),x, algorithm=\
Exception raised: RuntimeError >> ECL says: In function CAR, the value of the first argument is 0which is not of the expected type LIST
Time = 0.53 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.27 \[ \int \frac {e^{\frac {e^x+x^2}{x^3 \log \left (-2+e^{3/2}\right ) \log (4+2 x)}} \left (-e^x x-x^3+\left (-2 x^2-x^3+e^x \left (-6-x+x^2\right )\right ) \log (4+2 x)\right )}{\left (2 x^4+x^5\right ) \log \left (-2+e^{3/2}\right ) \log ^2(4+2 x)} \, dx=e^{\left (\frac {1}{x \log \left (2 \, x + 4\right ) \log \left (e^{\frac {3}{2}} - 2\right )} + \frac {e^{x}}{x^{3} \log \left (2 \, x + 4\right ) \log \left (e^{\frac {3}{2}} - 2\right )}\right )} \]
integrate((((x^2-x-6)*exp(x)-x^3-2*x^2)*log(4+2*x)-exp(x)*x-x^3)*exp((x^2+ exp(x))/x^3/log(4+2*x)/log(exp(3/2)-2))/(x^5+2*x^4)/log(4+2*x)^2/log(exp(3 /2)-2),x, algorithm=\
Time = 14.64 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.79 \[ \int \frac {e^{\frac {e^x+x^2}{x^3 \log \left (-2+e^{3/2}\right ) \log (4+2 x)}} \left (-e^x x-x^3+\left (-2 x^2-x^3+e^x \left (-6-x+x^2\right )\right ) \log (4+2 x)\right )}{\left (2 x^4+x^5\right ) \log \left (-2+e^{3/2}\right ) \log ^2(4+2 x)} \, dx={\mathrm {e}}^{\frac {{\mathrm {e}}^x+x^2}{x^3\,\ln \left ({\mathrm {e}}^{3/2}-2\right )\,\ln \left (2\,x+4\right )}} \]