\(\int \frac {e^{\frac {-1-e^5}{-3 e^5 x+e^{5+e^4} x}} (-1+e^5 (-1-3 x)+e^{5+e^4} x)}{-3 e^5 x^3+e^{5+e^4} x^3+e^{\frac {2 (-1-e^5)}{-3 e^5 x+e^{5+e^4} x}} (-3 e^5 x+e^{5+e^4} x)+e^{\frac {-1-e^5}{-3 e^5 x+e^{5+e^4} x}} (-6 e^5 x^2+2 e^{5+e^4} x^2)} \, dx\) [1378]

Optimal result

Integrand size = 167, antiderivative size = 28 \[ \int \frac {e^{\frac {-1-e^5}{-3 e^5 x+e^{5+e^4} x}} \left (-1+e^5 (-1-3 x)+e^{5+e^4} x\right )}{-3 e^5 x^3+e^{5+e^4} x^3+e^{\frac {2 \left (-1-e^5\right )}{-3 e^5 x+e^{5+e^4} x}} \left (-3 e^5 x+e^{5+e^4} x\right )+e^{\frac {-1-e^5}{-3 e^5 x+e^{5+e^4} x}} \left (-6 e^5 x^2+2 e^{5+e^4} x^2\right )} \, dx=\frac {x}{e^{\frac {1+\frac {1}{e^5}}{\left (3-e^{e^4}\right ) x}}+x} \] Output:

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

Mathematica [A] (verified)

Time = 0.18 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.18 \[ \int \frac {e^{\frac {-1-e^5}{-3 e^5 x+e^{5+e^4} x}} \left (-1+e^5 (-1-3 x)+e^{5+e^4} x\right )}{-3 e^5 x^3+e^{5+e^4} x^3+e^{\frac {2 \left (-1-e^5\right )}{-3 e^5 x+e^{5+e^4} x}} \left (-3 e^5 x+e^{5+e^4} x\right )+e^{\frac {-1-e^5}{-3 e^5 x+e^{5+e^4} x}} \left (-6 e^5 x^2+2 e^{5+e^4} x^2\right )} \, dx=\frac {x}{e^{\frac {1+e^5}{3 e^5 x-e^{5+e^4} x}}+x} \] Input:

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*x 
 + E^(5 + E^4)*x))*(-6*E^5*x^2 + 2*E^(5 + E^4)*x^2)),x]
 

Output:

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

Rubi [A] (verified)

Time = 7.16 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.64, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.024, Rules used = {6, 7292, 27, 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.

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

\(\Big \downarrow \) 6

\(\displaystyle \int \frac {e^{\frac {-1-e^5}{e^{5+e^4} x-3 e^5 x}} \left (e^5 (-3 x-1)+e^{5+e^4} x-1\right )}{\left (e^{5+e^4}-3 e^5\right ) x^3+e^{\frac {-1-e^5}{e^{5+e^4} x-3 e^5 x}} \left (2 e^{5+e^4} x^2-6 e^5 x^2\right )+e^{\frac {2 \left (-1-e^5\right )}{e^{5+e^4} x-3 e^5 x}} \left (e^{5+e^4} x-3 e^5 x\right )}dx\)

\(\Big \downarrow \) 7292

\(\displaystyle \int \frac {\left (e^5 \left (3-e^{e^4}\right ) x+e^5+1\right ) \exp \left (-\frac {-\frac {2}{e^{e^4}-3}-\frac {2}{e^5 \left (e^{e^4}-3\right )}}{x}-\frac {-1-e^5}{e^5 \left (3-e^{e^4}\right ) x}-\frac {5 e^{e^4}}{e^{e^4}-3}+\frac {15}{e^{e^4}-3}\right )}{\left (3-e^{e^4}\right ) x \left (e^{\frac {1}{e^5 \left (e^{e^4}-3\right ) x}+\frac {1}{\left (e^{e^4}-3\right ) x}} x+1\right )^2}dx\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {\exp \left (-5+\frac {1+e^5}{e^5 \left (3-e^{e^4}\right ) x}-\frac {2 \left (1+\frac {1}{e^5}\right )}{\left (3-e^{e^4}\right ) x}\right ) \left (e^5 \left (3-e^{e^4}\right ) x+e^5+1\right )}{x \left (e^{-\frac {1}{e^5 \left (3-e^{e^4}\right ) x}-\frac {1}{\left (3-e^{e^4}\right ) x}} x+1\right )^2}dx}{3-e^{e^4}}\)

\(\Big \downarrow \) 7237

\(\displaystyle -\frac {1}{e^{-\frac {1}{e^5 \left (3-e^{e^4}\right ) x}-\frac {1}{\left (3-e^{e^4}\right ) x}} x+1}\)

Input:

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^( 
5 + E^4)*x))*(-6*E^5*x^2 + 2*E^(5 + E^4)*x^2)),x]
 

Output:

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

Defintions of rubi rules used

rule 6

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

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 7237

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]
 

rule 7292

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

Maple [A] (verified)

Time = 1.64 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07

method	result	size
risch	\(\frac {x}{x +{\mathrm e}^{\frac {{\mathrm e}^{5}+1}{x \left (-{\mathrm e}^{5+{\mathrm e}^{4}}+3 \,{\mathrm e}^{5}\right )}}}\)	\(30\)
parallelrisch	\(\frac {\left (x \,{\mathrm e}^{5} {\mathrm e}^{{\mathrm e}^{4}}-3 x \,{\mathrm e}^{5}\right ) {\mathrm e}^{-5}}{\left ({\mathrm e}^{{\mathrm e}^{4}}-3\right ) \left ({\mathrm e}^{-\frac {\left ({\mathrm e}^{5}+1\right ) {\mathrm e}^{-5}}{x \left ({\mathrm e}^{{\mathrm e}^{4}}-3\right )}}+x \right )}\)	\(51\)
norman	\(-\frac {{\mathrm e}^{\frac {-{\mathrm e}^{5}-1}{x \,{\mathrm e}^{5} {\mathrm e}^{{\mathrm e}^{4}}-3 x \,{\mathrm e}^{5}}}}{x +{\mathrm e}^{\frac {-{\mathrm e}^{5}-1}{x \,{\mathrm e}^{5} {\mathrm e}^{{\mathrm e}^{4}}-3 x \,{\mathrm e}^{5}}}}\)	\(53\)

Input:

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=_RETURNVERBOSE)
 

Output:

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

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int \frac {e^{\frac {-1-e^5}{-3 e^5 x+e^{5+e^4} x}} \left (-1+e^5 (-1-3 x)+e^{5+e^4} x\right )}{-3 e^5 x^3+e^{5+e^4} x^3+e^{\frac {2 \left (-1-e^5\right )}{-3 e^5 x+e^{5+e^4} x}} \left (-3 e^5 x+e^{5+e^4} x\right )+e^{\frac {-1-e^5}{-3 e^5 x+e^{5+e^4} x}} \left (-6 e^5 x^2+2 e^{5+e^4} x^2\right )} \, dx=\frac {x}{x + e^{\left (\frac {e^{5} + 1}{3 \, x e^{5} - x e^{\left (e^{4} + 5\right )}}\right )}} \] Input:

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="fricas")
 

Output:

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

Sympy [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.96 \[ \int \frac {e^{\frac {-1-e^5}{-3 e^5 x+e^{5+e^4} x}} \left (-1+e^5 (-1-3 x)+e^{5+e^4} x\right )}{-3 e^5 x^3+e^{5+e^4} x^3+e^{\frac {2 \left (-1-e^5\right )}{-3 e^5 x+e^{5+e^4} x}} \left (-3 e^5 x+e^{5+e^4} x\right )+e^{\frac {-1-e^5}{-3 e^5 x+e^{5+e^4} x}} \left (-6 e^5 x^2+2 e^{5+e^4} x^2\right )} \, dx=\frac {x}{x + e^{\frac {- e^{5} - 1}{- 3 x e^{5} + x e^{5} e^{e^{4}}}}} \] Input:

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)
 

Output:

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

Maxima [A] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.43 \[ \int \frac {e^{\frac {-1-e^5}{-3 e^5 x+e^{5+e^4} x}} \left (-1+e^5 (-1-3 x)+e^{5+e^4} x\right )}{-3 e^5 x^3+e^{5+e^4} x^3+e^{\frac {2 \left (-1-e^5\right )}{-3 e^5 x+e^{5+e^4} x}} \left (-3 e^5 x+e^{5+e^4} x\right )+e^{\frac {-1-e^5}{-3 e^5 x+e^{5+e^4} x}} \left (-6 e^5 x^2+2 e^{5+e^4} x^2\right )} \, dx=-\frac {1}{x e^{\left (-\frac {1}{x {\left (3 \, e^{5} - e^{\left (e^{4} + 5\right )}\right )}} + \frac {1}{x {\left (e^{\left (e^{4}\right )} - 3\right )}}\right )} + 1} \] Input:

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="maxima")
 

Output:

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

Giac [F]

\[ \int \frac {e^{\frac {-1-e^5}{-3 e^5 x+e^{5+e^4} x}} \left (-1+e^5 (-1-3 x)+e^{5+e^4} x\right )}{-3 e^5 x^3+e^{5+e^4} x^3+e^{\frac {2 \left (-1-e^5\right )}{-3 e^5 x+e^{5+e^4} x}} \left (-3 e^5 x+e^{5+e^4} x\right )+e^{\frac {-1-e^5}{-3 e^5 x+e^{5+e^4} x}} \left (-6 e^5 x^2+2 e^{5+e^4} x^2\right )} \, dx=\int { \frac {{\left ({\left (3 \, x + 1\right )} e^{5} - x e^{\left (e^{4} + 5\right )} + 1\right )} e^{\left (\frac {e^{5} + 1}{3 \, x e^{5} - x e^{\left (e^{4} + 5\right )}}\right )}}{3 \, x^{3} e^{5} - x^{3} e^{\left (e^{4} + 5\right )} + {\left (3 \, x e^{5} - x e^{\left (e^{4} + 5\right )}\right )} e^{\left (\frac {2 \, {\left (e^{5} + 1\right )}}{3 \, x e^{5} - x e^{\left (e^{4} + 5\right )}}\right )} + 2 \, {\left (3 \, x^{2} e^{5} - x^{2} e^{\left (e^{4} + 5\right )}\right )} e^{\left (\frac {e^{5} + 1}{3 \, x e^{5} - x e^{\left (e^{4} + 5\right )}}\right )}} \,d x } \] Input:

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="giac")
 

Output:

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

Mupad [B] (verification not implemented)

Time = 3.32 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.57 \[ \int \frac {e^{\frac {-1-e^5}{-3 e^5 x+e^{5+e^4} x}} \left (-1+e^5 (-1-3 x)+e^{5+e^4} x\right )}{-3 e^5 x^3+e^{5+e^4} x^3+e^{\frac {2 \left (-1-e^5\right )}{-3 e^5 x+e^{5+e^4} x}} \left (-3 e^5 x+e^{5+e^4} x\right )+e^{\frac {-1-e^5}{-3 e^5 x+e^{5+e^4} x}} \left (-6 e^5 x^2+2 e^{5+e^4} x^2\right )} \, dx=\frac {x}{x+{\mathrm {e}}^{\frac {{\mathrm {e}}^5}{3\,x\,{\mathrm {e}}^5-x\,{\mathrm {e}}^5\,{\mathrm {e}}^{{\mathrm {e}}^4}}}\,{\mathrm {e}}^{\frac {1}{3\,x\,{\mathrm {e}}^5-x\,{\mathrm {e}}^5\,{\mathrm {e}}^{{\mathrm {e}}^4}}}} \] Input:

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)*e 
xp(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(exp(4))),x)
 

Output:

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)))))
 

Reduce [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 62, normalized size of antiderivative = 2.21 \[ \int \frac {e^{\frac {-1-e^5}{-3 e^5 x+e^{5+e^4} x}} \left (-1+e^5 (-1-3 x)+e^{5+e^4} x\right )}{-3 e^5 x^3+e^{5+e^4} x^3+e^{\frac {2 \left (-1-e^5\right )}{-3 e^5 x+e^{5+e^4} x}} \left (-3 e^5 x+e^{5+e^4} x\right )+e^{\frac {-1-e^5}{-3 e^5 x+e^{5+e^4} x}} \left (-6 e^5 x^2+2 e^{5+e^4} x^2\right )} \, dx=\frac {e^{\frac {e^{5}+1}{e^{e^{4}} e^{5} x -3 e^{5} x}} x}{e^{\frac {e^{5}+1}{e^{e^{4}} e^{5} x -3 e^{5} x}} x +1} \] Input:

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)
 

Output:

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