3.19.33 \(\int \frac {2 x-3 x^2+x^3-x^4+5 e^{-e^x-x} (1-x-e^x x+x^2)+(-5 e^{-e^x-x}-x+x^2) \log (-5 e^{-e^x-x} x-x^2+x^3)}{5 e^{-e^x-x} x^2+x^3-x^4} \, dx\)

Optimal. Leaf size=28 \[ x+\frac {\log \left (x \left (-5 e^{-e^x-x}+(-1+x) x\right )\right )}{x} \]

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Rubi [A]  time = 31.45, antiderivative size = 32, normalized size of antiderivative = 1.14, number of steps used = 76, number of rules used = 5, integrand size = 117, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {6742, 6688, 1620, 2551, 77} \begin {gather*} \frac {\log \left (-\left ((1-x) x^2\right )-5 e^{-x-e^x} x\right )}{x}+x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(2*x - 3*x^2 + x^3 - x^4 + 5*E^(-E^x - x)*(1 - x - E^x*x + x^2) + (-5*E^(-E^x - x) - x + x^2)*Log[-5*E^(-E
^x - x)*x - x^2 + x^3])/(5*E^(-E^x - x)*x^2 + x^3 - x^4),x]

[Out]

x + Log[-5*E^(-E^x - x)*x - (1 - x)*x^2]/x

Rule 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rule 1620

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[Px*(a + b*x)
^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && PolyQ[Px, x] && (IntegersQ[m, n] || IGtQ[m, -2]) &&
GtQ[Expon[Px, x], 2]

Rule 2551

Int[Log[u_]*((a_.) + (b_.)*(x_))^(m_.), x_Symbol] :> Simp[((a + b*x)^(m + 1)*Log[u])/(b*(m + 1)), x] - Dist[1/
(b*(m + 1)), Int[SimplifyIntegrand[((a + b*x)^(m + 1)*D[u, x])/u, x], x], x] /; FreeQ[{a, b, m}, x] && Inverse
FunctionFreeQ[u, x] && NeQ[m, -1]

Rule 6688

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {5 e^{-e^x} \left (5-e^{e^x}+e^{e^x} x+e^{e^x} x^2\right )}{(-1+x) x^2 \left (-5-e^{e^x+x} x+e^{e^x+x} x^2\right )}+\frac {e^{-e^x} \left (5-2 e^{e^x}+3 e^{e^x} x-e^{e^x} x^2+e^{e^x} x^3+e^{e^x} \log \left (x \left (-5 e^{-e^x-x}+(-1+x) x\right )\right )-e^{e^x} x \log \left (x \left (-5 e^{-e^x-x}+(-1+x) x\right )\right )\right )}{(-1+x) x^2}\right ) \, dx\\ &=5 \int \frac {e^{-e^x} \left (5-e^{e^x}+e^{e^x} x+e^{e^x} x^2\right )}{(-1+x) x^2 \left (-5-e^{e^x+x} x+e^{e^x+x} x^2\right )} \, dx+\int \frac {e^{-e^x} \left (5-2 e^{e^x}+3 e^{e^x} x-e^{e^x} x^2+e^{e^x} x^3+e^{e^x} \log \left (x \left (-5 e^{-e^x-x}+(-1+x) x\right )\right )-e^{e^x} x \log \left (x \left (-5 e^{-e^x-x}+(-1+x) x\right )\right )\right )}{(-1+x) x^2} \, dx\\ &=5 \int \frac {e^{-e^x} \left (5+e^{e^x} \left (-1+x+x^2\right )\right )}{(1-x) x^2 \left (5-e^{e^x+x} (-1+x) x\right )} \, dx+\int \frac {e^{-e^x} \left (-5-e^{e^x} \left (-2+3 x-x^2+x^3\right )+e^{e^x} (-1+x) \log \left (x \left (-5 e^{-e^x-x}+(-1+x) x\right )\right )\right )}{(1-x) x^2} \, dx\\ &=5 \int \left (\frac {e^{-e^x} \left (5-e^{e^x}+e^{e^x} x+e^{e^x} x^2\right )}{(-1+x) \left (-5-e^{e^x+x} x+e^{e^x+x} x^2\right )}-\frac {e^{-e^x} \left (5-e^{e^x}+e^{e^x} x+e^{e^x} x^2\right )}{x^2 \left (-5-e^{e^x+x} x+e^{e^x+x} x^2\right )}-\frac {e^{-e^x} \left (5-e^{e^x}+e^{e^x} x+e^{e^x} x^2\right )}{x \left (-5-e^{e^x+x} x+e^{e^x+x} x^2\right )}\right ) \, dx+\int \left (\frac {5 e^{-e^x}}{(-1+x) x^2}+\frac {-2+3 x-x^2+x^3+\log \left (x \left (-5 e^{-e^x-x}+(-1+x) x\right )\right )-x \log \left (x \left (-5 e^{-e^x-x}+(-1+x) x\right )\right )}{(-1+x) x^2}\right ) \, dx\\ &=5 \int \frac {e^{-e^x}}{(-1+x) x^2} \, dx+5 \int \frac {e^{-e^x} \left (5-e^{e^x}+e^{e^x} x+e^{e^x} x^2\right )}{(-1+x) \left (-5-e^{e^x+x} x+e^{e^x+x} x^2\right )} \, dx-5 \int \frac {e^{-e^x} \left (5-e^{e^x}+e^{e^x} x+e^{e^x} x^2\right )}{x^2 \left (-5-e^{e^x+x} x+e^{e^x+x} x^2\right )} \, dx-5 \int \frac {e^{-e^x} \left (5-e^{e^x}+e^{e^x} x+e^{e^x} x^2\right )}{x \left (-5-e^{e^x+x} x+e^{e^x+x} x^2\right )} \, dx+\int \frac {-2+3 x-x^2+x^3+\log \left (x \left (-5 e^{-e^x-x}+(-1+x) x\right )\right )-x \log \left (x \left (-5 e^{-e^x-x}+(-1+x) x\right )\right )}{(-1+x) x^2} \, dx\\ &=5 \int \left (\frac {e^{-e^x}}{-1+x}-\frac {e^{-e^x}}{x^2}-\frac {e^{-e^x}}{x}\right ) \, dx-5 \int \frac {e^{-e^x} \left (-5-e^{e^x} \left (-1+x+x^2\right )\right )}{x^2 \left (5-e^{e^x+x} (-1+x) x\right )} \, dx-5 \int \frac {e^{-e^x} \left (-5-e^{e^x} \left (-1+x+x^2\right )\right )}{x \left (5-e^{e^x+x} (-1+x) x\right )} \, dx+5 \int \frac {e^{-e^x} \left (5+e^{e^x} \left (-1+x+x^2\right )\right )}{(1-x) \left (5-e^{e^x+x} (-1+x) x\right )} \, dx+\int \frac {2-3 x+x^2-x^3+(-1+x) \log \left (x \left (-5 e^{-e^x-x}+(-1+x) x\right )\right )}{(1-x) x^2} \, dx\\ &=5 \int \frac {e^{-e^x}}{-1+x} \, dx-5 \int \frac {e^{-e^x}}{x^2} \, dx-5 \int \frac {e^{-e^x}}{x} \, dx-5 \int \left (\frac {1}{-5-e^{e^x+x} x+e^{e^x+x} x^2}-\frac {1}{x^2 \left (-5-e^{e^x+x} x+e^{e^x+x} x^2\right )}+\frac {5 e^{-e^x}}{x^2 \left (-5-e^{e^x+x} x+e^{e^x+x} x^2\right )}+\frac {1}{x \left (-5-e^{e^x+x} x+e^{e^x+x} x^2\right )}\right ) \, dx-5 \int \left (\frac {1}{-5-e^{e^x+x} x+e^{e^x+x} x^2}-\frac {1}{x \left (-5-e^{e^x+x} x+e^{e^x+x} x^2\right )}+\frac {5 e^{-e^x}}{x \left (-5-e^{e^x+x} x+e^{e^x+x} x^2\right )}+\frac {x}{-5-e^{e^x+x} x+e^{e^x+x} x^2}\right ) \, dx+5 \int \left (-\frac {1}{(-1+x) \left (-5-e^{e^x+x} x+e^{e^x+x} x^2\right )}+\frac {5 e^{-e^x}}{(-1+x) \left (-5-e^{e^x+x} x+e^{e^x+x} x^2\right )}+\frac {x}{(-1+x) \left (-5-e^{e^x+x} x+e^{e^x+x} x^2\right )}+\frac {x^2}{(-1+x) \left (-5-e^{e^x+x} x+e^{e^x+x} x^2\right )}\right ) \, dx+\int \left (\frac {-2+3 x-x^2+x^3}{(-1+x) x^2}-\frac {\log \left (-5 e^{-e^x-x} x+(-1+x) x^2\right )}{x^2}\right ) \, dx\\ &=5 \int \frac {e^{-e^x}}{-1+x} \, dx-5 \int \frac {e^{-e^x}}{x^2} \, dx-5 \int \frac {e^{-e^x}}{x} \, dx-2 \left (5 \int \frac {1}{-5-e^{e^x+x} x+e^{e^x+x} x^2} \, dx\right )-5 \int \frac {1}{(-1+x) \left (-5-e^{e^x+x} x+e^{e^x+x} x^2\right )} \, dx+5 \int \frac {1}{x^2 \left (-5-e^{e^x+x} x+e^{e^x+x} x^2\right )} \, dx-5 \int \frac {x}{-5-e^{e^x+x} x+e^{e^x+x} x^2} \, dx+5 \int \frac {x}{(-1+x) \left (-5-e^{e^x+x} x+e^{e^x+x} x^2\right )} \, dx+5 \int \frac {x^2}{(-1+x) \left (-5-e^{e^x+x} x+e^{e^x+x} x^2\right )} \, dx+25 \int \frac {e^{-e^x}}{(-1+x) \left (-5-e^{e^x+x} x+e^{e^x+x} x^2\right )} \, dx-25 \int \frac {e^{-e^x}}{x^2 \left (-5-e^{e^x+x} x+e^{e^x+x} x^2\right )} \, dx-25 \int \frac {e^{-e^x}}{x \left (-5-e^{e^x+x} x+e^{e^x+x} x^2\right )} \, dx+\int \frac {-2+3 x-x^2+x^3}{(-1+x) x^2} \, dx-\int \frac {\log \left (-5 e^{-e^x-x} x+(-1+x) x^2\right )}{x^2} \, dx\\ &=\frac {\log \left (-5 e^{-e^x-x} x-(1-x) x^2\right )}{x}+5 \int \frac {e^{-e^x}}{-1+x} \, dx-5 \int \frac {e^{-e^x}}{x^2} \, dx-5 \int \frac {e^{-e^x}}{x} \, dx-2 \left (5 \int \frac {1}{-5+e^{e^x+x} (-1+x) x} \, dx\right )+5 \int \frac {1}{x^2 \left (-5+e^{e^x+x} (-1+x) x\right )} \, dx-5 \int \frac {x}{-5+e^{e^x+x} (-1+x) x} \, dx-5 \int \frac {1}{(-1+x) \left (-5-e^{e^x+x} x+e^{e^x+x} x^2\right )} \, dx+5 \int \left (\frac {1}{-5-e^{e^x+x} x+e^{e^x+x} x^2}+\frac {1}{(-1+x) \left (-5-e^{e^x+x} x+e^{e^x+x} x^2\right )}\right ) \, dx+5 \int \left (\frac {1}{-5-e^{e^x+x} x+e^{e^x+x} x^2}+\frac {1}{(-1+x) \left (-5-e^{e^x+x} x+e^{e^x+x} x^2\right )}+\frac {x}{-5-e^{e^x+x} x+e^{e^x+x} x^2}\right ) \, dx-25 \int \frac {e^{-e^x}}{x^2 \left (-5+e^{e^x+x} (-1+x) x\right )} \, dx-25 \int \frac {e^{-e^x}}{x \left (-5+e^{e^x+x} (-1+x) x\right )} \, dx+25 \int \frac {e^{-e^x}}{(-1+x) \left (-5-e^{e^x+x} x+e^{e^x+x} x^2\right )} \, dx+\int \left (1+\frac {1}{-1+x}+\frac {2}{x^2}-\frac {1}{x}\right ) \, dx-\int \frac {5-\left (5+5 e^x-2 e^{e^x+x}\right ) x-3 e^{e^x+x} x^2}{x^2 \left (5-e^{e^x+x} (-1+x) x\right )} \, dx\\ &=-\frac {2}{x}+x+\log (1-x)-\log (x)+\frac {\log \left (-5 e^{-e^x-x} x-(1-x) x^2\right )}{x}+5 \int \frac {e^{-e^x}}{-1+x} \, dx-5 \int \frac {e^{-e^x}}{x^2} \, dx-5 \int \frac {e^{-e^x}}{x} \, dx-2 \left (5 \int \frac {1}{-5+e^{e^x+x} (-1+x) x} \, dx\right )+5 \int \frac {1}{x^2 \left (-5+e^{e^x+x} (-1+x) x\right )} \, dx-5 \int \frac {x}{-5+e^{e^x+x} (-1+x) x} \, dx+2 \left (5 \int \frac {1}{-5-e^{e^x+x} x+e^{e^x+x} x^2} \, dx\right )+5 \int \frac {1}{(-1+x) \left (-5-e^{e^x+x} x+e^{e^x+x} x^2\right )} \, dx+5 \int \frac {x}{-5-e^{e^x+x} x+e^{e^x+x} x^2} \, dx-25 \int \frac {e^{-e^x}}{x^2 \left (-5+e^{e^x+x} (-1+x) x\right )} \, dx-25 \int \frac {e^{-e^x}}{x \left (-5+e^{e^x+x} (-1+x) x\right )} \, dx+25 \int \frac {e^{-e^x}}{(-1+x) \left (-5-e^{e^x+x} x+e^{e^x+x} x^2\right )} \, dx-\int \left (\frac {e^{-e^x} \left (5-2 e^{e^x}+3 e^{e^x} x\right )}{(-1+x) x^2}+\frac {5 e^{-e^x} \left (5-e^{e^x}+e^{e^x} x+e^{e^x} x^2\right )}{(-1+x) x^2 \left (-5-e^{e^x+x} x+e^{e^x+x} x^2\right )}\right ) \, dx\\ &=-\frac {2}{x}+x+\log (1-x)-\log (x)+\frac {\log \left (-5 e^{-e^x-x} x-(1-x) x^2\right )}{x}+5 \int \frac {e^{-e^x}}{-1+x} \, dx-5 \int \frac {e^{-e^x}}{x^2} \, dx-5 \int \frac {e^{-e^x}}{x} \, dx+5 \int \frac {1}{x^2 \left (-5+e^{e^x+x} (-1+x) x\right )} \, dx+5 \int \frac {1}{(-1+x) \left (-5-e^{e^x+x} x+e^{e^x+x} x^2\right )} \, dx-5 \int \frac {e^{-e^x} \left (5-e^{e^x}+e^{e^x} x+e^{e^x} x^2\right )}{(-1+x) x^2 \left (-5-e^{e^x+x} x+e^{e^x+x} x^2\right )} \, dx-25 \int \frac {e^{-e^x}}{x^2 \left (-5+e^{e^x+x} (-1+x) x\right )} \, dx-25 \int \frac {e^{-e^x}}{x \left (-5+e^{e^x+x} (-1+x) x\right )} \, dx+25 \int \frac {e^{-e^x}}{(-1+x) \left (-5-e^{e^x+x} x+e^{e^x+x} x^2\right )} \, dx-\int \frac {e^{-e^x} \left (5-2 e^{e^x}+3 e^{e^x} x\right )}{(-1+x) x^2} \, dx\\ &=-\frac {2}{x}+x+\log (1-x)-\log (x)+\frac {\log \left (-5 e^{-e^x-x} x-(1-x) x^2\right )}{x}+5 \int \frac {e^{-e^x}}{-1+x} \, dx-5 \int \frac {e^{-e^x}}{x^2} \, dx-5 \int \frac {e^{-e^x}}{x} \, dx+5 \int \frac {1}{x^2 \left (-5+e^{e^x+x} (-1+x) x\right )} \, dx+5 \int \frac {1}{(-1+x) \left (-5-e^{e^x+x} x+e^{e^x+x} x^2\right )} \, dx-5 \int \frac {e^{-e^x} \left (5+e^{e^x} \left (-1+x+x^2\right )\right )}{(1-x) x^2 \left (5-e^{e^x+x} (-1+x) x\right )} \, dx-25 \int \frac {e^{-e^x}}{x^2 \left (-5+e^{e^x+x} (-1+x) x\right )} \, dx-25 \int \frac {e^{-e^x}}{x \left (-5+e^{e^x+x} (-1+x) x\right )} \, dx+25 \int \frac {e^{-e^x}}{(-1+x) \left (-5-e^{e^x+x} x+e^{e^x+x} x^2\right )} \, dx-\int \frac {2-5 e^{-e^x}-3 x}{(1-x) x^2} \, dx\\ &=-\frac {2}{x}+x+\log (1-x)-\log (x)+\frac {\log \left (-5 e^{-e^x-x} x-(1-x) x^2\right )}{x}+5 \int \frac {e^{-e^x}}{-1+x} \, dx-5 \int \frac {e^{-e^x}}{x^2} \, dx-5 \int \frac {e^{-e^x}}{x} \, dx+5 \int \frac {1}{x^2 \left (-5+e^{e^x+x} (-1+x) x\right )} \, dx+5 \int \frac {1}{(-1+x) \left (-5-e^{e^x+x} x+e^{e^x+x} x^2\right )} \, dx-5 \int \left (\frac {e^{-e^x} \left (5-e^{e^x}+e^{e^x} x+e^{e^x} x^2\right )}{(-1+x) \left (-5-e^{e^x+x} x+e^{e^x+x} x^2\right )}-\frac {e^{-e^x} \left (5-e^{e^x}+e^{e^x} x+e^{e^x} x^2\right )}{x^2 \left (-5-e^{e^x+x} x+e^{e^x+x} x^2\right )}-\frac {e^{-e^x} \left (5-e^{e^x}+e^{e^x} x+e^{e^x} x^2\right )}{x \left (-5-e^{e^x+x} x+e^{e^x+x} x^2\right )}\right ) \, dx-25 \int \frac {e^{-e^x}}{x^2 \left (-5+e^{e^x+x} (-1+x) x\right )} \, dx-25 \int \frac {e^{-e^x}}{x \left (-5+e^{e^x+x} (-1+x) x\right )} \, dx+25 \int \frac {e^{-e^x}}{(-1+x) \left (-5-e^{e^x+x} x+e^{e^x+x} x^2\right )} \, dx-\int \left (\frac {5 e^{-e^x}}{(-1+x) x^2}+\frac {-2+3 x}{(-1+x) x^2}\right ) \, dx\\ &=-\frac {2}{x}+x+\log (1-x)-\log (x)+\frac {\log \left (-5 e^{-e^x-x} x-(1-x) x^2\right )}{x}+5 \int \frac {e^{-e^x}}{-1+x} \, dx-5 \int \frac {e^{-e^x}}{x^2} \, dx-5 \int \frac {e^{-e^x}}{(-1+x) x^2} \, dx-5 \int \frac {e^{-e^x}}{x} \, dx+5 \int \frac {1}{x^2 \left (-5+e^{e^x+x} (-1+x) x\right )} \, dx+5 \int \frac {1}{(-1+x) \left (-5-e^{e^x+x} x+e^{e^x+x} x^2\right )} \, dx-5 \int \frac {e^{-e^x} \left (5-e^{e^x}+e^{e^x} x+e^{e^x} x^2\right )}{(-1+x) \left (-5-e^{e^x+x} x+e^{e^x+x} x^2\right )} \, dx+5 \int \frac {e^{-e^x} \left (5-e^{e^x}+e^{e^x} x+e^{e^x} x^2\right )}{x^2 \left (-5-e^{e^x+x} x+e^{e^x+x} x^2\right )} \, dx+5 \int \frac {e^{-e^x} \left (5-e^{e^x}+e^{e^x} x+e^{e^x} x^2\right )}{x \left (-5-e^{e^x+x} x+e^{e^x+x} x^2\right )} \, dx-25 \int \frac {e^{-e^x}}{x^2 \left (-5+e^{e^x+x} (-1+x) x\right )} \, dx-25 \int \frac {e^{-e^x}}{x \left (-5+e^{e^x+x} (-1+x) x\right )} \, dx+25 \int \frac {e^{-e^x}}{(-1+x) \left (-5-e^{e^x+x} x+e^{e^x+x} x^2\right )} \, dx-\int \frac {-2+3 x}{(-1+x) x^2} \, dx\\ &=-\frac {2}{x}+x+\log (1-x)-\log (x)+\frac {\log \left (-5 e^{-e^x-x} x-(1-x) x^2\right )}{x}-5 \int \left (\frac {e^{-e^x}}{-1+x}-\frac {e^{-e^x}}{x^2}-\frac {e^{-e^x}}{x}\right ) \, dx+5 \int \frac {e^{-e^x}}{-1+x} \, dx-5 \int \frac {e^{-e^x}}{x^2} \, dx-5 \int \frac {e^{-e^x}}{x} \, dx+5 \int \frac {1}{x^2 \left (-5+e^{e^x+x} (-1+x) x\right )} \, dx+5 \int \frac {1}{(-1+x) \left (-5-e^{e^x+x} x+e^{e^x+x} x^2\right )} \, dx+5 \int \frac {e^{-e^x} \left (-5-e^{e^x} \left (-1+x+x^2\right )\right )}{x^2 \left (5-e^{e^x+x} (-1+x) x\right )} \, dx+5 \int \frac {e^{-e^x} \left (-5-e^{e^x} \left (-1+x+x^2\right )\right )}{x \left (5-e^{e^x+x} (-1+x) x\right )} \, dx-5 \int \frac {e^{-e^x} \left (5+e^{e^x} \left (-1+x+x^2\right )\right )}{(1-x) \left (5-e^{e^x+x} (-1+x) x\right )} \, dx-25 \int \frac {e^{-e^x}}{x^2 \left (-5+e^{e^x+x} (-1+x) x\right )} \, dx-25 \int \frac {e^{-e^x}}{x \left (-5+e^{e^x+x} (-1+x) x\right )} \, dx+25 \int \frac {e^{-e^x}}{(-1+x) \left (-5-e^{e^x+x} x+e^{e^x+x} x^2\right )} \, dx-\int \left (\frac {1}{-1+x}+\frac {2}{x^2}-\frac {1}{x}\right ) \, dx\\ &=x+\frac {\log \left (-5 e^{-e^x-x} x-(1-x) x^2\right )}{x}+5 \int \frac {1}{x^2 \left (-5+e^{e^x+x} (-1+x) x\right )} \, dx+5 \int \frac {1}{(-1+x) \left (-5-e^{e^x+x} x+e^{e^x+x} x^2\right )} \, dx+5 \int \left (\frac {1}{-5-e^{e^x+x} x+e^{e^x+x} x^2}-\frac {1}{x^2 \left (-5-e^{e^x+x} x+e^{e^x+x} x^2\right )}+\frac {5 e^{-e^x}}{x^2 \left (-5-e^{e^x+x} x+e^{e^x+x} x^2\right )}+\frac {1}{x \left (-5-e^{e^x+x} x+e^{e^x+x} x^2\right )}\right ) \, dx+5 \int \left (\frac {1}{-5-e^{e^x+x} x+e^{e^x+x} x^2}-\frac {1}{x \left (-5-e^{e^x+x} x+e^{e^x+x} x^2\right )}+\frac {5 e^{-e^x}}{x \left (-5-e^{e^x+x} x+e^{e^x+x} x^2\right )}+\frac {x}{-5-e^{e^x+x} x+e^{e^x+x} x^2}\right ) \, dx-5 \int \left (-\frac {1}{(-1+x) \left (-5-e^{e^x+x} x+e^{e^x+x} x^2\right )}+\frac {5 e^{-e^x}}{(-1+x) \left (-5-e^{e^x+x} x+e^{e^x+x} x^2\right )}+\frac {x}{(-1+x) \left (-5-e^{e^x+x} x+e^{e^x+x} x^2\right )}+\frac {x^2}{(-1+x) \left (-5-e^{e^x+x} x+e^{e^x+x} x^2\right )}\right ) \, dx-25 \int \frac {e^{-e^x}}{x^2 \left (-5+e^{e^x+x} (-1+x) x\right )} \, dx-25 \int \frac {e^{-e^x}}{x \left (-5+e^{e^x+x} (-1+x) x\right )} \, dx+25 \int \frac {e^{-e^x}}{(-1+x) \left (-5-e^{e^x+x} x+e^{e^x+x} x^2\right )} \, dx\\ &=x+\frac {\log \left (-5 e^{-e^x-x} x-(1-x) x^2\right )}{x}+5 \int \frac {1}{x^2 \left (-5+e^{e^x+x} (-1+x) x\right )} \, dx+2 \left (5 \int \frac {1}{-5-e^{e^x+x} x+e^{e^x+x} x^2} \, dx\right )+2 \left (5 \int \frac {1}{(-1+x) \left (-5-e^{e^x+x} x+e^{e^x+x} x^2\right )} \, dx\right )-5 \int \frac {1}{x^2 \left (-5-e^{e^x+x} x+e^{e^x+x} x^2\right )} \, dx+5 \int \frac {x}{-5-e^{e^x+x} x+e^{e^x+x} x^2} \, dx-5 \int \frac {x}{(-1+x) \left (-5-e^{e^x+x} x+e^{e^x+x} x^2\right )} \, dx-5 \int \frac {x^2}{(-1+x) \left (-5-e^{e^x+x} x+e^{e^x+x} x^2\right )} \, dx-25 \int \frac {e^{-e^x}}{x^2 \left (-5+e^{e^x+x} (-1+x) x\right )} \, dx-25 \int \frac {e^{-e^x}}{x \left (-5+e^{e^x+x} (-1+x) x\right )} \, dx+25 \int \frac {e^{-e^x}}{x^2 \left (-5-e^{e^x+x} x+e^{e^x+x} x^2\right )} \, dx+25 \int \frac {e^{-e^x}}{x \left (-5-e^{e^x+x} x+e^{e^x+x} x^2\right )} \, dx\\ &=x+\frac {\log \left (-5 e^{-e^x-x} x-(1-x) x^2\right )}{x}+2 \left (5 \int \frac {1}{-5+e^{e^x+x} (-1+x) x} \, dx\right )+5 \int \frac {x}{-5+e^{e^x+x} (-1+x) x} \, dx+2 \left (5 \int \frac {1}{(-1+x) \left (-5-e^{e^x+x} x+e^{e^x+x} x^2\right )} \, dx\right )-5 \int \left (\frac {1}{-5-e^{e^x+x} x+e^{e^x+x} x^2}+\frac {1}{(-1+x) \left (-5-e^{e^x+x} x+e^{e^x+x} x^2\right )}\right ) \, dx-5 \int \left (\frac {1}{-5-e^{e^x+x} x+e^{e^x+x} x^2}+\frac {1}{(-1+x) \left (-5-e^{e^x+x} x+e^{e^x+x} x^2\right )}+\frac {x}{-5-e^{e^x+x} x+e^{e^x+x} x^2}\right ) \, dx\\ &=x+\frac {\log \left (-5 e^{-e^x-x} x-(1-x) x^2\right )}{x}+2 \left (5 \int \frac {1}{-5+e^{e^x+x} (-1+x) x} \, dx\right )+5 \int \frac {x}{-5+e^{e^x+x} (-1+x) x} \, dx-2 \left (5 \int \frac {1}{-5-e^{e^x+x} x+e^{e^x+x} x^2} \, dx\right )-5 \int \frac {x}{-5-e^{e^x+x} x+e^{e^x+x} x^2} \, dx\\ &=x+\frac {\log \left (-5 e^{-e^x-x} x-(1-x) x^2\right )}{x}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.17, size = 29, normalized size = 1.04 \begin {gather*} 1+x+\frac {\log \left (x \left (-5 e^{-e^x-x}+(-1+x) x\right )\right )}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(2*x - 3*x^2 + x^3 - x^4 + 5*E^(-E^x - x)*(1 - x - E^x*x + x^2) + (-5*E^(-E^x - x) - x + x^2)*Log[-5
*E^(-E^x - x)*x - x^2 + x^3])/(5*E^(-E^x - x)*x^2 + x^3 - x^4),x]

[Out]

1 + x + Log[x*(-5*E^(-E^x - x) + (-1 + x)*x)]/x

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fricas [A]  time = 0.64, size = 32, normalized size = 1.14 \begin {gather*} \frac {x^{2} + \log \left (x^{3} - x^{2} - x e^{\left (-x - e^{x} + \log \relax (5)\right )}\right )}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-exp(log(5/exp(exp(x)))-x)+x^2-x)*log(-x*exp(log(5/exp(exp(x)))-x)+x^3-x^2)+(-exp(x)*x+x^2-x+1)*ex
p(log(5/exp(exp(x)))-x)-x^4+x^3-3*x^2+2*x)/(x^2*exp(log(5/exp(exp(x)))-x)-x^4+x^3),x, algorithm="fricas")

[Out]

(x^2 + log(x^3 - x^2 - x*e^(-x - e^x + log(5))))/x

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giac [A]  time = 0.24, size = 34, normalized size = 1.21 \begin {gather*} \frac {x^{2} - e^{x} + \log \left (x^{2} e^{\left (x + e^{x}\right )} - x e^{\left (x + e^{x}\right )} - 5\right ) + \log \relax (x)}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-exp(log(5/exp(exp(x)))-x)+x^2-x)*log(-x*exp(log(5/exp(exp(x)))-x)+x^3-x^2)+(-exp(x)*x+x^2-x+1)*ex
p(log(5/exp(exp(x)))-x)-x^4+x^3-3*x^2+2*x)/(x^2*exp(log(5/exp(exp(x)))-x)-x^4+x^3),x, algorithm="giac")

[Out]

(x^2 - e^x + log(x^2*e^(x + e^x) - x*e^(x + e^x) - 5) + log(x))/x

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maple [F]  time = 0.24, size = 0, normalized size = 0.00 \[\int \frac {\left (-{\mathrm e}^{\ln \left (5 \,{\mathrm e}^{-{\mathrm e}^{x}}\right )-x}+x^{2}-x \right ) \ln \left (-x \,{\mathrm e}^{\ln \left (5 \,{\mathrm e}^{-{\mathrm e}^{x}}\right )-x}+x^{3}-x^{2}\right )+\left (-{\mathrm e}^{x} x +x^{2}-x +1\right ) {\mathrm e}^{\ln \left (5 \,{\mathrm e}^{-{\mathrm e}^{x}}\right )-x}-x^{4}+x^{3}-3 x^{2}+2 x}{x^{2} {\mathrm e}^{\ln \left (5 \,{\mathrm e}^{-{\mathrm e}^{x}}\right )-x}-x^{4}+x^{3}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-exp(ln(5/exp(exp(x)))-x)+x^2-x)*ln(-x*exp(ln(5/exp(exp(x)))-x)+x^3-x^2)+(-exp(x)*x+x^2-x+1)*exp(ln(5/ex
p(exp(x)))-x)-x^4+x^3-3*x^2+2*x)/(x^2*exp(ln(5/exp(exp(x)))-x)-x^4+x^3),x)

[Out]

int(((-exp(ln(5/exp(exp(x)))-x)+x^2-x)*ln(-x*exp(ln(5/exp(exp(x)))-x)+x^3-x^2)+(-exp(x)*x+x^2-x+1)*exp(ln(5/ex
p(exp(x)))-x)-x^4+x^3-3*x^2+2*x)/(x^2*exp(ln(5/exp(exp(x)))-x)-x^4+x^3),x)

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maxima [A]  time = 0.56, size = 30, normalized size = 1.07 \begin {gather*} \frac {x^{2} - e^{x} + \log \left ({\left (x^{2} - x\right )} e^{\left (x + e^{x}\right )} - 5\right ) + \log \relax (x)}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-exp(log(5/exp(exp(x)))-x)+x^2-x)*log(-x*exp(log(5/exp(exp(x)))-x)+x^3-x^2)+(-exp(x)*x+x^2-x+1)*ex
p(log(5/exp(exp(x)))-x)-x^4+x^3-3*x^2+2*x)/(x^2*exp(log(5/exp(exp(x)))-x)-x^4+x^3),x, algorithm="maxima")

[Out]

(x^2 - e^x + log((x^2 - x)*e^(x + e^x) - 5) + log(x))/x

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mupad [B]  time = 1.48, size = 28, normalized size = 1.00 \begin {gather*} x+\frac {\ln \left (x^3-x^2-5\,x\,{\mathrm {e}}^{-x}\,{\mathrm {e}}^{-{\mathrm {e}}^x}\right )}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(log(5*exp(-exp(x))) - x)*(x + x*exp(x) - x^2 - 1) - 2*x + log(x^3 - x^2 - x*exp(log(5*exp(-exp(x)))
- x))*(x + exp(log(5*exp(-exp(x))) - x) - x^2) + 3*x^2 - x^3 + x^4)/(x^2*exp(log(5*exp(-exp(x))) - x) + x^3 -
x^4),x)

[Out]

x + log(x^3 - x^2 - 5*x*exp(-x)*exp(-exp(x)))/x

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sympy [A]  time = 1.28, size = 22, normalized size = 0.79 \begin {gather*} x + \frac {\log {\left (x^{3} - x^{2} - 5 x e^{- x} e^{- e^{x}} \right )}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-exp(ln(5/exp(exp(x)))-x)+x**2-x)*ln(-x*exp(ln(5/exp(exp(x)))-x)+x**3-x**2)+(-exp(x)*x+x**2-x+1)*e
xp(ln(5/exp(exp(x)))-x)-x**4+x**3-3*x**2+2*x)/(x**2*exp(ln(5/exp(exp(x)))-x)-x**4+x**3),x)

[Out]

x + log(x**3 - x**2 - 5*x*exp(-x)*exp(-exp(x)))/x

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