\(\int \frac {e^{\frac {x^2+2 e^3 x^4+e^{10-4 x} x^4+e^6 x^6+e^{2-2 x} (2 e^3 x^3+2 e^6 x^5)}{e^{4-4 x}+2 e^{2-2 x} x+x^2}} (4 e^{12-6 x} x^3+4 e^3 x^4+4 e^6 x^6+e^{4-4 x} (12 e^6 x^4+e^3 (6 x^2+4 x^3))+e^{2-2 x} (2 x+4 x^2+12 e^6 x^5+e^3 (10 x^3+4 x^4)))}{-4 e^{6-6 x}-12 e^{4-4 x} x-12 e^{2-2 x} x^2-4 x^3+e^{\frac {x^2+2 e^3 x^4+e^{10-4 x} x^4+e^6 x^6+e^{2-2 x} (2 e^3 x^3+2 e^6 x^5)}{e^{4-4 x}+2 e^{2-2 x} x+x^2}} (e^{6-6 x}+3 e^{4-4 x} x+3 e^{2-2 x} x^2+x^3)} \, dx\) [1461]

Optimal result

Integrand size = 337, antiderivative size = 34 \[ \int \frac {e^{\frac {x^2+2 e^3 x^4+e^{10-4 x} x^4+e^6 x^6+e^{2-2 x} \left (2 e^3 x^3+2 e^6 x^5\right )}{e^{4-4 x}+2 e^{2-2 x} x+x^2}} \left (4 e^{12-6 x} x^3+4 e^3 x^4+4 e^6 x^6+e^{4-4 x} \left (12 e^6 x^4+e^3 \left (6 x^2+4 x^3\right )\right )+e^{2-2 x} \left (2 x+4 x^2+12 e^6 x^5+e^3 \left (10 x^3+4 x^4\right )\right )\right )}{-4 e^{6-6 x}-12 e^{4-4 x} x-12 e^{2-2 x} x^2-4 x^3+e^{\frac {x^2+2 e^3 x^4+e^{10-4 x} x^4+e^6 x^6+e^{2-2 x} \left (2 e^3 x^3+2 e^6 x^5\right )}{e^{4-4 x}+2 e^{2-2 x} x+x^2}} \left (e^{6-6 x}+3 e^{4-4 x} x+3 e^{2-2 x} x^2+x^3\right )} \, dx=\log \left (\frac {3}{2} \left (-4+e^{\left (e^3 x^2+\frac {x}{e^{2 (1-x)}+x}\right )^2}\right )\right ) \] Output:

ln(3/2*exp((x^2*exp(3)+x/(x+exp(2-2*x)))^2)-6)
 

Mathematica [A] (warning: unable to verify)

Time = 0.45 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.97 \[ \int \frac {e^{\frac {x^2+2 e^3 x^4+e^{10-4 x} x^4+e^6 x^6+e^{2-2 x} \left (2 e^3 x^3+2 e^6 x^5\right )}{e^{4-4 x}+2 e^{2-2 x} x+x^2}} \left (4 e^{12-6 x} x^3+4 e^3 x^4+4 e^6 x^6+e^{4-4 x} \left (12 e^6 x^4+e^3 \left (6 x^2+4 x^3\right )\right )+e^{2-2 x} \left (2 x+4 x^2+12 e^6 x^5+e^3 \left (10 x^3+4 x^4\right )\right )\right )}{-4 e^{6-6 x}-12 e^{4-4 x} x-12 e^{2-2 x} x^2-4 x^3+e^{\frac {x^2+2 e^3 x^4+e^{10-4 x} x^4+e^6 x^6+e^{2-2 x} \left (2 e^3 x^3+2 e^6 x^5\right )}{e^{4-4 x}+2 e^{2-2 x} x+x^2}} \left (e^{6-6 x}+3 e^{4-4 x} x+3 e^{2-2 x} x^2+x^3\right )} \, dx=\log \left (4-e^{1+2 e^3 x^2+e^6 x^4+\frac {e^4}{\left (e^2+e^{2 x} x\right )^2}-\frac {2 \left (e^2+e^5 x^2\right )}{e^2+e^{2 x} x}}\right ) \] Input:

Integrate[(E^((x^2 + 2*E^3*x^4 + E^(10 - 4*x)*x^4 + E^6*x^6 + E^(2 - 2*x)* 
(2*E^3*x^3 + 2*E^6*x^5))/(E^(4 - 4*x) + 2*E^(2 - 2*x)*x + x^2))*(4*E^(12 - 
 6*x)*x^3 + 4*E^3*x^4 + 4*E^6*x^6 + E^(4 - 4*x)*(12*E^6*x^4 + E^3*(6*x^2 + 
 4*x^3)) + E^(2 - 2*x)*(2*x + 4*x^2 + 12*E^6*x^5 + E^3*(10*x^3 + 4*x^4)))) 
/(-4*E^(6 - 6*x) - 12*E^(4 - 4*x)*x - 12*E^(2 - 2*x)*x^2 - 4*x^3 + E^((x^2 
 + 2*E^3*x^4 + E^(10 - 4*x)*x^4 + E^6*x^6 + E^(2 - 2*x)*(2*E^3*x^3 + 2*E^6 
*x^5))/(E^(4 - 4*x) + 2*E^(2 - 2*x)*x + x^2))*(E^(6 - 6*x) + 3*E^(4 - 4*x) 
*x + 3*E^(2 - 2*x)*x^2 + x^3)),x]
 

Output:

Log[4 - E^(1 + 2*E^3*x^2 + E^6*x^4 + E^4/(E^2 + E^(2*x)*x)^2 - (2*(E^2 + E 
^5*x^2))/(E^2 + E^(2*x)*x))]
 

Rubi [A] (verified)

Time = 34.86 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.41, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.009, Rules used = {7292, 7259, 16}

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.

\(\displaystyle \int \frac {\left (4 e^6 x^6+4 e^3 x^4+4 e^{12-6 x} x^3+e^{4-4 x} \left (12 e^6 x^4+e^3 \left (4 x^3+6 x^2\right )\right )+e^{2-2 x} \left (12 e^6 x^5+4 x^2+e^3 \left (4 x^4+10 x^3\right )+2 x\right )\right ) \exp \left (\frac {e^6 x^6+e^{10-4 x} x^4+2 e^3 x^4+x^2+e^{2-2 x} \left (2 e^6 x^5+2 e^3 x^3\right )}{x^2+2 e^{2-2 x} x+e^{4-4 x}}\right )}{\left (x^3+3 e^{2-2 x} x^2+3 e^{4-4 x} x+e^{6-6 x}\right ) \exp \left (\frac {e^6 x^6+e^{10-4 x} x^4+2 e^3 x^4+x^2+e^{2-2 x} \left (2 e^6 x^5+2 e^3 x^3\right )}{x^2+2 e^{2-2 x} x+e^{4-4 x}}\right )-4 x^3-12 e^{2-2 x} x^2-12 e^{4-4 x} x-4 e^{6-6 x}} \, dx\)

\(\Big \downarrow \) 7292

\(\displaystyle \int \frac {\left (-4 e^6 x^6-4 e^3 x^4-4 e^{12-6 x} x^3-e^{4-4 x} \left (12 e^6 x^4+e^3 \left (4 x^3+6 x^2\right )\right )-e^{2-2 x} \left (12 e^6 x^5+4 x^2+e^3 \left (4 x^4+10 x^3\right )+2 x\right )\right ) \exp \left (\frac {x^2 \left (e^{2 x+3} x^2+e^5 x+e^{2 x}\right )^2}{\left (e^{2 x} x+e^2\right )^2}+6 x\right )}{\left (e^{2 x} x+e^2\right )^3 \left (4-\exp \left (\frac {x^2 \left (e^{2 x+3} x^2+e^5 x+e^{2 x}\right )^2}{\left (e^{2 x} x+e^2\right )^2}\right )\right )}dx\)

\(\Big \downarrow \) 7259

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

\(\Big \downarrow \) 16

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

Input:

Int[(E^((x^2 + 2*E^3*x^4 + E^(10 - 4*x)*x^4 + E^6*x^6 + E^(2 - 2*x)*(2*E^3 
*x^3 + 2*E^6*x^5))/(E^(4 - 4*x) + 2*E^(2 - 2*x)*x + x^2))*(4*E^(12 - 6*x)* 
x^3 + 4*E^3*x^4 + 4*E^6*x^6 + E^(4 - 4*x)*(12*E^6*x^4 + E^3*(6*x^2 + 4*x^3 
)) + E^(2 - 2*x)*(2*x + 4*x^2 + 12*E^6*x^5 + E^3*(10*x^3 + 4*x^4))))/(-4*E 
^(6 - 6*x) - 12*E^(4 - 4*x)*x - 12*E^(2 - 2*x)*x^2 - 4*x^3 + E^((x^2 + 2*E 
^3*x^4 + E^(10 - 4*x)*x^4 + E^6*x^6 + E^(2 - 2*x)*(2*E^3*x^3 + 2*E^6*x^5)) 
/(E^(4 - 4*x) + 2*E^(2 - 2*x)*x + x^2))*(E^(6 - 6*x) + 3*E^(4 - 4*x)*x + 3 
*E^(2 - 2*x)*x^2 + x^3)),x]
 

Output:

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

Defintions of rubi rules used

rule 16

Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + 
b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
 

rule 7259

Int[(u_)*((a_) + (b_.)*(v_)^(p_.)*(w_)^(p_.))^(m_.), x_Symbol] :> With[{c = 
 Simplify[u/(w*D[v, x] + v*D[w, x])]}, Simp[c   Subst[Int[(a + b*x^p)^m, x] 
, x, v*w], x] /; FreeQ[c, x]] /; FreeQ[{a, b, m, p}, x] && IntegerQ[p]
 

rule 7292

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(87\) vs. \(2(27)=54\).

Time = 10.11 (sec) , antiderivative size = 88, normalized size of antiderivative = 2.59

method	result	size
parallelrisch	\(\ln \left ({\mathrm e}^{\frac {x^{4} {\mathrm e}^{6} {\mathrm e}^{4-4 x}+\left (2 x^{5} {\mathrm e}^{6}+2 x^{3} {\mathrm e}^{3}\right ) {\mathrm e}^{2-2 x}+{\mathrm e}^{6} x^{6}+2 x^{4} {\mathrm e}^{3}+x^{2}}{{\mathrm e}^{4-4 x}+2 x \,{\mathrm e}^{2-2 x}+x^{2}}}-4\right )\)	\(88\)
risch	\(x^{4} {\mathrm e}^{6}+\frac {x^{2} \left (2 x^{2} {\mathrm e}^{3}+2 x \,{\mathrm e}^{5-2 x}+1\right )}{\left (x +{\mathrm e}^{2-2 x}\right )^{2}}-\frac {2 x^{5} {\mathrm e}^{-2 x +8}+{\mathrm e}^{6} x^{6}+2 x^{3} {\mathrm e}^{5-2 x}+2 x^{4} {\mathrm e}^{3}+x^{4} {\mathrm e}^{10-4 x}+x^{2}}{{\mathrm e}^{4-4 x}+2 x \,{\mathrm e}^{2-2 x}+x^{2}}+\ln \left ({\mathrm e}^{\frac {x^{2} \left (x^{4} {\mathrm e}^{6}+2 \,{\mathrm e}^{-2 x +8} x^{3}+2 x^{2} {\mathrm e}^{3}+{\mathrm e}^{10-4 x} x^{2}+2 x \,{\mathrm e}^{5-2 x}+1\right )}{{\mathrm e}^{4-4 x}+2 x \,{\mathrm e}^{2-2 x}+x^{2}}}-4\right )\)	\(186\)

Input:

int((4*x^3*exp(3)^2*exp(2-2*x)^3+(12*x^4*exp(3)^2+(4*x^3+6*x^2)*exp(3))*ex 
p(2-2*x)^2+(12*x^5*exp(3)^2+(4*x^4+10*x^3)*exp(3)+4*x^2+2*x)*exp(2-2*x)+4* 
x^6*exp(3)^2+4*x^4*exp(3))*exp((x^4*exp(3)^2*exp(2-2*x)^2+(2*x^5*exp(3)^2+ 
2*x^3*exp(3))*exp(2-2*x)+x^6*exp(3)^2+2*x^4*exp(3)+x^2)/(exp(2-2*x)^2+2*x* 
exp(2-2*x)+x^2))/((exp(2-2*x)^3+3*x*exp(2-2*x)^2+3*x^2*exp(2-2*x)+x^3)*exp 
((x^4*exp(3)^2*exp(2-2*x)^2+(2*x^5*exp(3)^2+2*x^3*exp(3))*exp(2-2*x)+x^6*e 
xp(3)^2+2*x^4*exp(3)+x^2)/(exp(2-2*x)^2+2*x*exp(2-2*x)+x^2))-4*exp(2-2*x)^ 
3-12*x*exp(2-2*x)^2-12*x^2*exp(2-2*x)-4*x^3),x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 80 vs. \(2 (27) = 54\).

Time = 0.10 (sec) , antiderivative size = 80, normalized size of antiderivative = 2.35 \[ \int \frac {e^{\frac {x^2+2 e^3 x^4+e^{10-4 x} x^4+e^6 x^6+e^{2-2 x} \left (2 e^3 x^3+2 e^6 x^5\right )}{e^{4-4 x}+2 e^{2-2 x} x+x^2}} \left (4 e^{12-6 x} x^3+4 e^3 x^4+4 e^6 x^6+e^{4-4 x} \left (12 e^6 x^4+e^3 \left (6 x^2+4 x^3\right )\right )+e^{2-2 x} \left (2 x+4 x^2+12 e^6 x^5+e^3 \left (10 x^3+4 x^4\right )\right )\right )}{-4 e^{6-6 x}-12 e^{4-4 x} x-12 e^{2-2 x} x^2-4 x^3+e^{\frac {x^2+2 e^3 x^4+e^{10-4 x} x^4+e^6 x^6+e^{2-2 x} \left (2 e^3 x^3+2 e^6 x^5\right )}{e^{4-4 x}+2 e^{2-2 x} x+x^2}} \left (e^{6-6 x}+3 e^{4-4 x} x+3 e^{2-2 x} x^2+x^3\right )} \, dx=\log \left (e^{\left (\frac {x^{6} e^{10} + 2 \, x^{4} e^{7} + x^{4} e^{\left (-4 \, x + 14\right )} + x^{2} e^{4} + 2 \, {\left (x^{5} e^{8} + x^{3} e^{5}\right )} e^{\left (-2 \, x + 4\right )}}{x^{2} e^{4} + 2 \, x e^{\left (-2 \, x + 6\right )} + e^{\left (-4 \, x + 8\right )}}\right )} - 4\right ) \] Input:

integrate((4*x^3*exp(3)^2*exp(2-2*x)^3+(12*x^4*exp(3)^2+(4*x^3+6*x^2)*exp( 
3))*exp(2-2*x)^2+(12*x^5*exp(3)^2+(4*x^4+10*x^3)*exp(3)+4*x^2+2*x)*exp(2-2 
*x)+4*x^6*exp(3)^2+4*x^4*exp(3))*exp((x^4*exp(3)^2*exp(2-2*x)^2+(2*x^5*exp 
(3)^2+2*x^3*exp(3))*exp(2-2*x)+x^6*exp(3)^2+2*x^4*exp(3)+x^2)/(exp(2-2*x)^ 
2+2*x*exp(2-2*x)+x^2))/((exp(2-2*x)^3+3*x*exp(2-2*x)^2+3*x^2*exp(2-2*x)+x^ 
3)*exp((x^4*exp(3)^2*exp(2-2*x)^2+(2*x^5*exp(3)^2+2*x^3*exp(3))*exp(2-2*x) 
+x^6*exp(3)^2+2*x^4*exp(3)+x^2)/(exp(2-2*x)^2+2*x*exp(2-2*x)+x^2))-4*exp(2 
-2*x)^3-12*x*exp(2-2*x)^2-12*x^2*exp(2-2*x)-4*x^3),x, algorithm="fricas")
 

Output:

log(e^((x^6*e^10 + 2*x^4*e^7 + x^4*e^(-4*x + 14) + x^2*e^4 + 2*(x^5*e^8 + 
x^3*e^5)*e^(-2*x + 4))/(x^2*e^4 + 2*x*e^(-2*x + 6) + e^(-4*x + 8))) - 4)
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 80 vs. \(2 (26) = 52\).

Time = 0.79 (sec) , antiderivative size = 80, normalized size of antiderivative = 2.35 \[ \int \frac {e^{\frac {x^2+2 e^3 x^4+e^{10-4 x} x^4+e^6 x^6+e^{2-2 x} \left (2 e^3 x^3+2 e^6 x^5\right )}{e^{4-4 x}+2 e^{2-2 x} x+x^2}} \left (4 e^{12-6 x} x^3+4 e^3 x^4+4 e^6 x^6+e^{4-4 x} \left (12 e^6 x^4+e^3 \left (6 x^2+4 x^3\right )\right )+e^{2-2 x} \left (2 x+4 x^2+12 e^6 x^5+e^3 \left (10 x^3+4 x^4\right )\right )\right )}{-4 e^{6-6 x}-12 e^{4-4 x} x-12 e^{2-2 x} x^2-4 x^3+e^{\frac {x^2+2 e^3 x^4+e^{10-4 x} x^4+e^6 x^6+e^{2-2 x} \left (2 e^3 x^3+2 e^6 x^5\right )}{e^{4-4 x}+2 e^{2-2 x} x+x^2}} \left (e^{6-6 x}+3 e^{4-4 x} x+3 e^{2-2 x} x^2+x^3\right )} \, dx=\log {\left (e^{\frac {x^{6} e^{6} + x^{4} e^{6} e^{4 - 4 x} + 2 x^{4} e^{3} + x^{2} + \left (2 x^{5} e^{6} + 2 x^{3} e^{3}\right ) e^{2 - 2 x}}{x^{2} + 2 x e^{2 - 2 x} + e^{4 - 4 x}}} - 4 \right )} \] Input:

integrate((4*x**3*exp(3)**2*exp(2-2*x)**3+(12*x**4*exp(3)**2+(4*x**3+6*x** 
2)*exp(3))*exp(2-2*x)**2+(12*x**5*exp(3)**2+(4*x**4+10*x**3)*exp(3)+4*x**2 
+2*x)*exp(2-2*x)+4*x**6*exp(3)**2+4*x**4*exp(3))*exp((x**4*exp(3)**2*exp(2 
-2*x)**2+(2*x**5*exp(3)**2+2*x**3*exp(3))*exp(2-2*x)+x**6*exp(3)**2+2*x**4 
*exp(3)+x**2)/(exp(2-2*x)**2+2*x*exp(2-2*x)+x**2))/((exp(2-2*x)**3+3*x*exp 
(2-2*x)**2+3*x**2*exp(2-2*x)+x**3)*exp((x**4*exp(3)**2*exp(2-2*x)**2+(2*x* 
*5*exp(3)**2+2*x**3*exp(3))*exp(2-2*x)+x**6*exp(3)**2+2*x**4*exp(3)+x**2)/ 
(exp(2-2*x)**2+2*x*exp(2-2*x)+x**2))-4*exp(2-2*x)**3-12*x*exp(2-2*x)**2-12 
*x**2*exp(2-2*x)-4*x**3),x)
 

Output:

log(exp((x**6*exp(6) + x**4*exp(6)*exp(4 - 4*x) + 2*x**4*exp(3) + x**2 + ( 
2*x**5*exp(6) + 2*x**3*exp(3))*exp(2 - 2*x))/(x**2 + 2*x*exp(2 - 2*x) + ex 
p(4 - 4*x))) - 4)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 167 vs. \(2 (27) = 54\).

Time = 1.16 (sec) , antiderivative size = 167, normalized size of antiderivative = 4.91 \[ \int \frac {e^{\frac {x^2+2 e^3 x^4+e^{10-4 x} x^4+e^6 x^6+e^{2-2 x} \left (2 e^3 x^3+2 e^6 x^5\right )}{e^{4-4 x}+2 e^{2-2 x} x+x^2}} \left (4 e^{12-6 x} x^3+4 e^3 x^4+4 e^6 x^6+e^{4-4 x} \left (12 e^6 x^4+e^3 \left (6 x^2+4 x^3\right )\right )+e^{2-2 x} \left (2 x+4 x^2+12 e^6 x^5+e^3 \left (10 x^3+4 x^4\right )\right )\right )}{-4 e^{6-6 x}-12 e^{4-4 x} x-12 e^{2-2 x} x^2-4 x^3+e^{\frac {x^2+2 e^3 x^4+e^{10-4 x} x^4+e^6 x^6+e^{2-2 x} \left (2 e^3 x^3+2 e^6 x^5\right )}{e^{4-4 x}+2 e^{2-2 x} x+x^2}} \left (e^{6-6 x}+3 e^{4-4 x} x+3 e^{2-2 x} x^2+x^3\right )} \, dx=\frac {x^{4} e^{8} + {\left (x^{5} e^{6} + 2 \, x^{3} e^{3}\right )} e^{\left (2 \, x\right )} - 2 \, e^{2}}{x e^{\left (2 \, x\right )} + e^{2}} + \log \left ({\left (e^{\left (x^{4} e^{6} + 2 \, x^{2} e^{3} + \frac {e^{4}}{x^{2} e^{\left (4 \, x\right )} + 2 \, x e^{\left (2 \, x + 2\right )} + e^{4}} + 2 \, e^{\left (-4 \, x + 7\right )} + 1\right )} - 4 \, e^{\left (2 \, x e^{\left (-2 \, x + 5\right )} + \frac {2 \, e^{9}}{x e^{\left (6 \, x\right )} + e^{\left (4 \, x + 2\right )}} + \frac {2 \, e^{2}}{x e^{\left (2 \, x\right )} + e^{2}}\right )}\right )} e^{\left (-x^{4} e^{6} - 2 \, x^{2} e^{3} - 2 \, e^{\left (-4 \, x + 7\right )} - 1\right )}\right ) \] Input:

integrate((4*x^3*exp(3)^2*exp(2-2*x)^3+(12*x^4*exp(3)^2+(4*x^3+6*x^2)*exp( 
3))*exp(2-2*x)^2+(12*x^5*exp(3)^2+(4*x^4+10*x^3)*exp(3)+4*x^2+2*x)*exp(2-2 
*x)+4*x^6*exp(3)^2+4*x^4*exp(3))*exp((x^4*exp(3)^2*exp(2-2*x)^2+(2*x^5*exp 
(3)^2+2*x^3*exp(3))*exp(2-2*x)+x^6*exp(3)^2+2*x^4*exp(3)+x^2)/(exp(2-2*x)^ 
2+2*x*exp(2-2*x)+x^2))/((exp(2-2*x)^3+3*x*exp(2-2*x)^2+3*x^2*exp(2-2*x)+x^ 
3)*exp((x^4*exp(3)^2*exp(2-2*x)^2+(2*x^5*exp(3)^2+2*x^3*exp(3))*exp(2-2*x) 
+x^6*exp(3)^2+2*x^4*exp(3)+x^2)/(exp(2-2*x)^2+2*x*exp(2-2*x)+x^2))-4*exp(2 
-2*x)^3-12*x*exp(2-2*x)^2-12*x^2*exp(2-2*x)-4*x^3),x, algorithm="maxima")
 

Output:

(x^4*e^8 + (x^5*e^6 + 2*x^3*e^3)*e^(2*x) - 2*e^2)/(x*e^(2*x) + e^2) + log( 
(e^(x^4*e^6 + 2*x^2*e^3 + e^4/(x^2*e^(4*x) + 2*x*e^(2*x + 2) + e^4) + 2*e^ 
(-4*x + 7) + 1) - 4*e^(2*x*e^(-2*x + 5) + 2*e^9/(x*e^(6*x) + e^(4*x + 2)) 
+ 2*e^2/(x*e^(2*x) + e^2)))*e^(-x^4*e^6 - 2*x^2*e^3 - 2*e^(-4*x + 7) - 1))
 

Giac [F(-1)]

Timed out. \[ \int \frac {e^{\frac {x^2+2 e^3 x^4+e^{10-4 x} x^4+e^6 x^6+e^{2-2 x} \left (2 e^3 x^3+2 e^6 x^5\right )}{e^{4-4 x}+2 e^{2-2 x} x+x^2}} \left (4 e^{12-6 x} x^3+4 e^3 x^4+4 e^6 x^6+e^{4-4 x} \left (12 e^6 x^4+e^3 \left (6 x^2+4 x^3\right )\right )+e^{2-2 x} \left (2 x+4 x^2+12 e^6 x^5+e^3 \left (10 x^3+4 x^4\right )\right )\right )}{-4 e^{6-6 x}-12 e^{4-4 x} x-12 e^{2-2 x} x^2-4 x^3+e^{\frac {x^2+2 e^3 x^4+e^{10-4 x} x^4+e^6 x^6+e^{2-2 x} \left (2 e^3 x^3+2 e^6 x^5\right )}{e^{4-4 x}+2 e^{2-2 x} x+x^2}} \left (e^{6-6 x}+3 e^{4-4 x} x+3 e^{2-2 x} x^2+x^3\right )} \, dx=\text {Timed out} \] Input:

integrate((4*x^3*exp(3)^2*exp(2-2*x)^3+(12*x^4*exp(3)^2+(4*x^3+6*x^2)*exp( 
3))*exp(2-2*x)^2+(12*x^5*exp(3)^2+(4*x^4+10*x^3)*exp(3)+4*x^2+2*x)*exp(2-2 
*x)+4*x^6*exp(3)^2+4*x^4*exp(3))*exp((x^4*exp(3)^2*exp(2-2*x)^2+(2*x^5*exp 
(3)^2+2*x^3*exp(3))*exp(2-2*x)+x^6*exp(3)^2+2*x^4*exp(3)+x^2)/(exp(2-2*x)^ 
2+2*x*exp(2-2*x)+x^2))/((exp(2-2*x)^3+3*x*exp(2-2*x)^2+3*x^2*exp(2-2*x)+x^ 
3)*exp((x^4*exp(3)^2*exp(2-2*x)^2+(2*x^5*exp(3)^2+2*x^3*exp(3))*exp(2-2*x) 
+x^6*exp(3)^2+2*x^4*exp(3)+x^2)/(exp(2-2*x)^2+2*x*exp(2-2*x)+x^2))-4*exp(2 
-2*x)^3-12*x*exp(2-2*x)^2-12*x^2*exp(2-2*x)-4*x^3),x, algorithm="giac")
 

Output:

Timed out
 

Mupad [B] (verification not implemented)

Time = 2.46 (sec) , antiderivative size = 191, normalized size of antiderivative = 5.62 \[ \int \frac {e^{\frac {x^2+2 e^3 x^4+e^{10-4 x} x^4+e^6 x^6+e^{2-2 x} \left (2 e^3 x^3+2 e^6 x^5\right )}{e^{4-4 x}+2 e^{2-2 x} x+x^2}} \left (4 e^{12-6 x} x^3+4 e^3 x^4+4 e^6 x^6+e^{4-4 x} \left (12 e^6 x^4+e^3 \left (6 x^2+4 x^3\right )\right )+e^{2-2 x} \left (2 x+4 x^2+12 e^6 x^5+e^3 \left (10 x^3+4 x^4\right )\right )\right )}{-4 e^{6-6 x}-12 e^{4-4 x} x-12 e^{2-2 x} x^2-4 x^3+e^{\frac {x^2+2 e^3 x^4+e^{10-4 x} x^4+e^6 x^6+e^{2-2 x} \left (2 e^3 x^3+2 e^6 x^5\right )}{e^{4-4 x}+2 e^{2-2 x} x+x^2}} \left (e^{6-6 x}+3 e^{4-4 x} x+3 e^{2-2 x} x^2+x^3\right )} \, dx=\ln \left ({\mathrm {e}}^{\frac {2\,x^4\,{\mathrm {e}}^3}{{\mathrm {e}}^{-4\,x}\,{\mathrm {e}}^4+x^2+2\,x\,{\mathrm {e}}^{-2\,x}\,{\mathrm {e}}^2}}\,{\mathrm {e}}^{\frac {x^6\,{\mathrm {e}}^6}{{\mathrm {e}}^{-4\,x}\,{\mathrm {e}}^4+x^2+2\,x\,{\mathrm {e}}^{-2\,x}\,{\mathrm {e}}^2}}\,{\mathrm {e}}^{\frac {2\,x^3\,{\mathrm {e}}^{-2\,x}\,{\mathrm {e}}^5}{{\mathrm {e}}^{-4\,x}\,{\mathrm {e}}^4+x^2+2\,x\,{\mathrm {e}}^{-2\,x}\,{\mathrm {e}}^2}}\,{\mathrm {e}}^{\frac {2\,x^5\,{\mathrm {e}}^{-2\,x}\,{\mathrm {e}}^8}{{\mathrm {e}}^{-4\,x}\,{\mathrm {e}}^4+x^2+2\,x\,{\mathrm {e}}^{-2\,x}\,{\mathrm {e}}^2}}\,{\mathrm {e}}^{\frac {x^4\,{\mathrm {e}}^{-4\,x}\,{\mathrm {e}}^{10}}{{\mathrm {e}}^{-4\,x}\,{\mathrm {e}}^4+x^2+2\,x\,{\mathrm {e}}^{-2\,x}\,{\mathrm {e}}^2}}\,{\mathrm {e}}^{\frac {x^2}{{\mathrm {e}}^{-4\,x}\,{\mathrm {e}}^4+x^2+2\,x\,{\mathrm {e}}^{-2\,x}\,{\mathrm {e}}^2}}-4\right ) \] Input:

int(-(exp((exp(2 - 2*x)*(2*x^3*exp(3) + 2*x^5*exp(6)) + 2*x^4*exp(3) + x^6 
*exp(6) + x^2 + x^4*exp(6)*exp(4 - 4*x))/(exp(4 - 4*x) + 2*x*exp(2 - 2*x) 
+ x^2))*(exp(4 - 4*x)*(exp(3)*(6*x^2 + 4*x^3) + 12*x^4*exp(6)) + exp(2 - 2 
*x)*(2*x + exp(3)*(10*x^3 + 4*x^4) + 12*x^5*exp(6) + 4*x^2) + 4*x^4*exp(3) 
 + 4*x^6*exp(6) + 4*x^3*exp(6)*exp(6 - 6*x)))/(4*exp(6 - 6*x) - exp((exp(2 
 - 2*x)*(2*x^3*exp(3) + 2*x^5*exp(6)) + 2*x^4*exp(3) + x^6*exp(6) + x^2 + 
x^4*exp(6)*exp(4 - 4*x))/(exp(4 - 4*x) + 2*x*exp(2 - 2*x) + x^2))*(exp(6 - 
 6*x) + 3*x*exp(4 - 4*x) + 3*x^2*exp(2 - 2*x) + x^3) + 12*x*exp(4 - 4*x) + 
 12*x^2*exp(2 - 2*x) + 4*x^3),x)
 

Output:

log(exp((2*x^4*exp(3))/(exp(-4*x)*exp(4) + x^2 + 2*x*exp(-2*x)*exp(2)))*ex 
p((x^6*exp(6))/(exp(-4*x)*exp(4) + x^2 + 2*x*exp(-2*x)*exp(2)))*exp((2*x^3 
*exp(-2*x)*exp(5))/(exp(-4*x)*exp(4) + x^2 + 2*x*exp(-2*x)*exp(2)))*exp((2 
*x^5*exp(-2*x)*exp(8))/(exp(-4*x)*exp(4) + x^2 + 2*x*exp(-2*x)*exp(2)))*ex 
p((x^4*exp(-4*x)*exp(10))/(exp(-4*x)*exp(4) + x^2 + 2*x*exp(-2*x)*exp(2))) 
*exp(x^2/(exp(-4*x)*exp(4) + x^2 + 2*x*exp(-2*x)*exp(2))) - 4)
 

Reduce [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 352, normalized size of antiderivative = 10.35 \[ \int \frac {e^{\frac {x^2+2 e^3 x^4+e^{10-4 x} x^4+e^6 x^6+e^{2-2 x} \left (2 e^3 x^3+2 e^6 x^5\right )}{e^{4-4 x}+2 e^{2-2 x} x+x^2}} \left (4 e^{12-6 x} x^3+4 e^3 x^4+4 e^6 x^6+e^{4-4 x} \left (12 e^6 x^4+e^3 \left (6 x^2+4 x^3\right )\right )+e^{2-2 x} \left (2 x+4 x^2+12 e^6 x^5+e^3 \left (10 x^3+4 x^4\right )\right )\right )}{-4 e^{6-6 x}-12 e^{4-4 x} x-12 e^{2-2 x} x^2-4 x^3+e^{\frac {x^2+2 e^3 x^4+e^{10-4 x} x^4+e^6 x^6+e^{2-2 x} \left (2 e^3 x^3+2 e^6 x^5\right )}{e^{4-4 x}+2 e^{2-2 x} x+x^2}} \left (e^{6-6 x}+3 e^{4-4 x} x+3 e^{2-2 x} x^2+x^3\right )} \, dx=\frac {e^{4 x} \mathrm {log}\left (4 e^{\frac {2 e^{2 x} e^{5} x^{3}+2 e^{2 x} e^{2} x +2 e^{7} x^{2}+e^{4}}{e^{4 x} x^{2}+2 e^{2 x} e^{2} x +e^{4}}}-e^{e^{6} x^{4}+2 e^{3} x^{2}} e \right ) x^{2}+e^{4 x} x^{2}+2 e^{2 x} \mathrm {log}\left (4 e^{\frac {2 e^{2 x} e^{5} x^{3}+2 e^{2 x} e^{2} x +2 e^{7} x^{2}+e^{4}}{e^{4 x} x^{2}+2 e^{2 x} e^{2} x +e^{4}}}-e^{e^{6} x^{4}+2 e^{3} x^{2}} e \right ) e^{2} x -2 e^{2 x} e^{5} x^{3}+\mathrm {log}\left (4 e^{\frac {2 e^{2 x} e^{5} x^{3}+2 e^{2 x} e^{2} x +2 e^{7} x^{2}+e^{4}}{e^{4 x} x^{2}+2 e^{2 x} e^{2} x +e^{4}}}-e^{e^{6} x^{4}+2 e^{3} x^{2}} e \right ) e^{4}-2 e^{7} x^{2}}{e^{4 x} x^{2}+2 e^{2 x} e^{2} x +e^{4}} \] Input:

int((4*x^3*exp(3)^2*exp(2-2*x)^3+(12*x^4*exp(3)^2+(4*x^3+6*x^2)*exp(3))*ex 
p(2-2*x)^2+(12*x^5*exp(3)^2+(4*x^4+10*x^3)*exp(3)+4*x^2+2*x)*exp(2-2*x)+4* 
x^6*exp(3)^2+4*x^4*exp(3))*exp((x^4*exp(3)^2*exp(2-2*x)^2+(2*x^5*exp(3)^2+ 
2*x^3*exp(3))*exp(2-2*x)+x^6*exp(3)^2+2*x^4*exp(3)+x^2)/(exp(2-2*x)^2+2*x* 
exp(2-2*x)+x^2))/((exp(2-2*x)^3+3*x*exp(2-2*x)^2+3*x^2*exp(2-2*x)+x^3)*exp 
((x^4*exp(3)^2*exp(2-2*x)^2+(2*x^5*exp(3)^2+2*x^3*exp(3))*exp(2-2*x)+x^6*e 
xp(3)^2+2*x^4*exp(3)+x^2)/(exp(2-2*x)^2+2*x*exp(2-2*x)+x^2))-4*exp(2-2*x)^ 
3-12*x*exp(2-2*x)^2-12*x^2*exp(2-2*x)-4*x^3),x)
 

Output:

(e**(4*x)*log(4*e**((2*e**(2*x)*e**5*x**3 + 2*e**(2*x)*e**2*x + 2*e**7*x** 
2 + e**4)/(e**(4*x)*x**2 + 2*e**(2*x)*e**2*x + e**4)) - e**(e**6*x**4 + 2* 
e**3*x**2)*e)*x**2 + e**(4*x)*x**2 + 2*e**(2*x)*log(4*e**((2*e**(2*x)*e**5 
*x**3 + 2*e**(2*x)*e**2*x + 2*e**7*x**2 + e**4)/(e**(4*x)*x**2 + 2*e**(2*x 
)*e**2*x + e**4)) - e**(e**6*x**4 + 2*e**3*x**2)*e)*e**2*x - 2*e**(2*x)*e* 
*5*x**3 + log(4*e**((2*e**(2*x)*e**5*x**3 + 2*e**(2*x)*e**2*x + 2*e**7*x** 
2 + e**4)/(e**(4*x)*x**2 + 2*e**(2*x)*e**2*x + e**4)) - e**(e**6*x**4 + 2* 
e**3*x**2)*e)*e**4 - 2*e**7*x**2)/(e**(4*x)*x**2 + 2*e**(2*x)*e**2*x + e** 
4)