∫ e −1−e5 _____________________ −3e5x+e5+e4x (−1+e5(−1−3x)+e5+e4 x)_________________ −3e5x3+e5+e4x3+e 2(−1−e5) _____________________ −3e5x+e5+e4x (−3e5x+e5+e4x)+e −1−e5 ____________________

Optimal. Leaf size=28 \[ \frac {x}{e^{\frac {1+\frac {1}{e^5}}{\left (3-e^{e^4}\right ) x}}+x} \]

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Rubi [A] time = 7.12, antiderivative size = 31, normalized size of antiderivative = 1.11, number of steps used = 4, number of rules used = 4, integrand size = 167, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.024, Rules used = {6, 6688, 12, 6687} \begin {gather*} \frac {x}{x+e^{\frac {1+e^5}{e^5 \left (3-e^{e^4}\right ) x}}} \end {gather*}

Int[(E^((-1 - E^5)/(-3*E^5*x + E^(5 + E^4)*x))*(-1 + E^5*(-1 - 3*x) + E^(5 + E^4)*x))/(-3*E^5*x^3 + E^(5 + E^4

)*x^3 + E^((2*(-1 - E^5))/(-3*E^5*x + E^(5 + E^4)*x))*(-3*E^5*x + E^(5 + E^4)*x) + E^((-1 - E^5)/(-3*E^5*x + E

Int[(u_.)*((w_.) + (a_.)*(v_) + (b_.)*(v_))^(p_.), x_Symbol] :> Int[u*((a + b)*v + w)^p, x] /; FreeQ[{a, b}, x

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] && !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Int[(u_)*(y_)^(m_.)*(z_)^(n_.), x_Symbol] :> With[{q = DerivativeDivides[y*z, u*z^(n - m), x]}, Simp[(q*y^(m +

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

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{\frac {-1-e^5}{-3 e^5 x+e^{5+e^4} x}} \left (-1+e^5 (-1-3 x)+e^{5+e^4} x\right )}{\left (-3 e^5+e^{5+e^4}\right ) x^3+e^{\frac {2 \left (-1-e^5\right )}{-3 e^5 x+e^{5+e^4} x}} \left (-3 e^5 x+e^{5+e^4} x\right )+e^{\frac {-1-e^5}{-3 e^5 x+e^{5+e^4} x}} \left (-6 e^5 x^2+2 e^{5+e^4} x^2\right )} \, dx\\ &=\int \frac {\exp \left (\frac {1+e^5-5 e^5 \left (3-e^{e^4}\right ) x}{e^5 \left (3-e^{e^4}\right ) x}\right ) \left (1+e^5+e^5 \left (3-e^{e^4}\right ) x\right )}{\left (3-e^{e^4}\right ) x \left (e^{\frac {1+e^5}{3 e^5 x-e^{5+e^4} x}}+x\right )^2} \, dx\\ &=\frac {\int \frac {\exp \left (\frac {1+e^5-5 e^5 \left (3-e^{e^4}\right ) x}{e^5 \left (3-e^{e^4}\right ) x}\right ) \left (1+e^5+e^5 \left (3-e^{e^4}\right ) x\right )}{x \left (e^{\frac {1+e^5}{3 e^5 x-e^{5+e^4} x}}+x\right )^2} \, dx}{3-e^{e^4}}\\ &=\frac {x}{e^{\frac {1+e^5}{e^5 \left (3-e^{e^4}\right ) x}}+x}\\ \end {aligned} \end {gather*}

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Mathematica [A] time = 0.27, size = 33, normalized size = 1.18 \begin {gather*} \frac {x}{e^{\frac {1+e^5}{3 e^5 x-e^{5+e^4} x}}+x} \end {gather*}

Integrate[(E^((-1 - E^5)/(-3*E^5*x + E^(5 + E^4)*x))*(-1 + E^5*(-1 - 3*x) + E^(5 + E^4)*x))/(-3*E^5*x^3 + E^(5

+ E^4)*x^3 + E^((2*(-1 - E^5))/(-3*E^5*x + E^(5 + E^4)*x))*(-3*E^5*x + E^(5 + E^4)*x) + E^((-1 - E^5)/(-3*E^5

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fricas [A] time = 0.61, size = 28, normalized size = 1.00 \begin {gather*} \frac {x}{x + e^{\left (\frac {e^{5} + 1}{3 \, x e^{5} - x e^{\left (e^{4} + 5\right )}}\right )}} \end {gather*}

integrate((x*exp(5)*exp(exp(4))+(-3*x-1)*exp(5)-1)*exp((-exp(5)-1)/(x*exp(5)*exp(exp(4))-3*x*exp(5)))/((x*exp(

5)*exp(exp(4))-3*x*exp(5))*exp((-exp(5)-1)/(x*exp(5)*exp(exp(4))-3*x*exp(5)))^2+(2*x^2*exp(5)*exp(exp(4))-6*x^

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

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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left ({\left (3 \, x + 1\right )} e^{5} - x e^{\left (e^{4} + 5\right )} + 1\right )} e^{\left (\frac {e^{5} + 1}{3 \, x e^{5} - x e^{\left (e^{4} + 5\right )}}\right )}}{3 \, x^{3} e^{5} - x^{3} e^{\left (e^{4} + 5\right )} + {\left (3 \, x e^{5} - x e^{\left (e^{4} + 5\right )}\right )} e^{\left (\frac {2 \, {\left (e^{5} + 1\right )}}{3 \, x e^{5} - x e^{\left (e^{4} + 5\right )}}\right )} + 2 \, {\left (3 \, x^{2} e^{5} - x^{2} e^{\left (e^{4} + 5\right )}\right )} e^{\left (\frac {e^{5} + 1}{3 \, x e^{5} - x e^{\left (e^{4} + 5\right )}}\right )}}\,{d x} \end {gather*}

integrate((x*exp(5)*exp(exp(4))+(-3*x-1)*exp(5)-1)*exp((-exp(5)-1)/(x*exp(5)*exp(exp(4))-3*x*exp(5)))/((x*exp(

5)*exp(exp(4))-3*x*exp(5))*exp((-exp(5)-1)/(x*exp(5)*exp(exp(4))-3*x*exp(5)))^2+(2*x^2*exp(5)*exp(exp(4))-6*x^

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

integrate(((3*x + 1)*e^5 - x*e^(e^4 + 5) + 1)*e^((e^5 + 1)/(3*x*e^5 - x*e^(e^4 + 5)))/(3*x^3*e^5 - x^3*e^(e^4

+ 5) + (3*x*e^5 - x*e^(e^4 + 5))*e^(2*(e^5 + 1)/(3*x*e^5 - x*e^(e^4 + 5))) + 2*(3*x^2*e^5 - x^2*e^(e^4 + 5))*e

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method	result	size

risch	\(\frac {x}{x +{\mathrm e}^{\frac {{\mathrm e}^{5}+1}{x \left (3 \,{\mathrm e}^{5}-{\mathrm e}^{5+{\mathrm e}^{4}}\right )}}}\)	\(30\)
norman	\(-\frac {{\mathrm e}^{\frac {-{\mathrm e}^{5}-1}{x \,{\mathrm e}^{5} {\mathrm e}^{{\mathrm e}^{4}}-3 x \,{\mathrm e}^{5}}}}{x +{\mathrm e}^{\frac {-{\mathrm e}^{5}-1}{x \,{\mathrm e}^{5} {\mathrm e}^{{\mathrm e}^{4}}-3 x \,{\mathrm e}^{5}}}}\)	\(53\)

int((x*exp(5)*exp(exp(4))+(-3*x-1)*exp(5)-1)*exp((-exp(5)-1)/(x*exp(5)*exp(exp(4))-3*x*exp(5)))/((x*exp(5)*exp

(exp(4))-3*x*exp(5))*exp((-exp(5)-1)/(x*exp(5)*exp(exp(4))-3*x*exp(5)))^2+(2*x^2*exp(5)*exp(exp(4))-6*x^2*exp(

5))*exp((-exp(5)-1)/(x*exp(5)*exp(exp(4))-3*x*exp(5)))+x^3*exp(5)*exp(exp(4))-3*x^3*exp(5)),x,method=_RETURNVE

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maxima [A] time = 0.66, size = 40, normalized size = 1.43 \begin {gather*} -\frac {1}{x e^{\left (-\frac {1}{x {\left (3 \, e^{5} - e^{\left (e^{4} + 5\right )}\right )}} + \frac {1}{x {\left (e^{\left (e^{4}\right )} - 3\right )}}\right )} + 1} \end {gather*}

integrate((x*exp(5)*exp(exp(4))+(-3*x-1)*exp(5)-1)*exp((-exp(5)-1)/(x*exp(5)*exp(exp(4))-3*x*exp(5)))/((x*exp(

5)*exp(exp(4))-3*x*exp(5))*exp((-exp(5)-1)/(x*exp(5)*exp(exp(4))-3*x*exp(5)))^2+(2*x^2*exp(5)*exp(exp(4))-6*x^

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

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mupad [B] time = 6.12, size = 44, normalized size = 1.57 \begin {gather*} \frac {x}{x+{\mathrm {e}}^{\frac {{\mathrm {e}}^5}{3\,x\,{\mathrm {e}}^5-x\,{\mathrm {e}}^5\,{\mathrm {e}}^{{\mathrm {e}}^4}}}\,{\mathrm {e}}^{\frac {1}{3\,x\,{\mathrm {e}}^5-x\,{\mathrm {e}}^5\,{\mathrm {e}}^{{\mathrm {e}}^4}}}} \end {gather*}

int((exp((exp(5) + 1)/(3*x*exp(5) - x*exp(5)*exp(exp(4))))*(exp(5)*(3*x + 1) - x*exp(5)*exp(exp(4)) + 1))/(exp

((exp(5) + 1)/(3*x*exp(5) - x*exp(5)*exp(exp(4))))*(6*x^2*exp(5) - 2*x^2*exp(5)*exp(exp(4))) + exp((2*(exp(5)

+ 1))/(3*x*exp(5) - x*exp(5)*exp(exp(4))))*(3*x*exp(5) - x*exp(5)*exp(exp(4))) + 3*x^3*exp(5) - x^3*exp(5)*exp

x/(x + exp(exp(5)/(3*x*exp(5) - x*exp(5)*exp(exp(4))))*exp(1/(3*x*exp(5) - x*exp(5)*exp(exp(4)))))

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sympy [A] time = 0.24, size = 27, normalized size = 0.96 \begin {gather*} \frac {x}{x + e^{\frac {- e^{5} - 1}{- 3 x e^{5} + x e^{5} e^{e^{4}}}}} \end {gather*}

integrate((x*exp(5)*exp(exp(4))+(-3*x-1)*exp(5)-1)*exp((-exp(5)-1)/(x*exp(5)*exp(exp(4))-3*x*exp(5)))/((x*exp(

5)*exp(exp(4))-3*x*exp(5))*exp((-exp(5)-1)/(x*exp(5)*exp(exp(4))-3*x*exp(5)))**2+(2*x**2*exp(5)*exp(exp(4))-6*

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

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3.74.78 \(\int \frac {e^{\frac {-1-e^5}{-3 e^5 x+e^{5+e^4} x}} (-1+e^5 (-1-3 x)+e^{5+e^4} x)}{-3 e^5 x^3+e^{5+e^4} x^3+e^{\frac {2 (-1-e^5)}{-3 e^5 x+e^{5+e^4} x}} (-3 e^5 x+e^{5+e^4} x)+e^{\frac {-1-e^5}{-3 e^5 x+e^{5+e^4} x}} (-6 e^5 x^2+2 e^{5+e^4} x^2)} \, dx\)