\(\int \frac {-2 x-e^5 x-x^2+e^x (-6 e^5 x^3-6 x^4)+e^{2 x} (-6 e^5 x^3-6 x^4)+(e^5+x+e^x (-18 e^5 x^2-18 x^3)+e^{2 x} (-18 e^5 x^2-18 x^3)) \log (e^5+x)+(e^x (-18 e^5 x-18 x^2)+e^{2 x} (-18 e^5 x-18 x^2)) \log ^2(e^5+x)+(e^x (-6 e^5-6 x)+e^{2 x} (-6 e^5-6 x)) \log ^3(e^5+x)}{e^5 x^3+x^4+(3 e^5 x^2+3 x^3) \log (e^5+x)+(3 e^5 x+3 x^2) \log ^2(e^5+x)+(e^5+x) \log ^3(e^5+x)} \, dx\) [228]

Optimal result

Integrand size = 255, antiderivative size = 22 \[ \int \frac {-2 x-e^5 x-x^2+e^x \left (-6 e^5 x^3-6 x^4\right )+e^{2 x} \left (-6 e^5 x^3-6 x^4\right )+\left (e^5+x+e^x \left (-18 e^5 x^2-18 x^3\right )+e^{2 x} \left (-18 e^5 x^2-18 x^3\right )\right ) \log \left (e^5+x\right )+\left (e^x \left (-18 e^5 x-18 x^2\right )+e^{2 x} \left (-18 e^5 x-18 x^2\right )\right ) \log ^2\left (e^5+x\right )+\left (e^x \left (-6 e^5-6 x\right )+e^{2 x} \left (-6 e^5-6 x\right )\right ) \log ^3\left (e^5+x\right )}{e^5 x^3+x^4+\left (3 e^5 x^2+3 x^3\right ) \log \left (e^5+x\right )+\left (3 e^5 x+3 x^2\right ) \log ^2\left (e^5+x\right )+\left (e^5+x\right ) \log ^3\left (e^5+x\right )} \, dx=-3 \left (1+e^x\right )^2+\frac {x}{\left (x+\log \left (e^5+x\right )\right )^2} \] Output:

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

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.14 \[ \int \frac {-2 x-e^5 x-x^2+e^x \left (-6 e^5 x^3-6 x^4\right )+e^{2 x} \left (-6 e^5 x^3-6 x^4\right )+\left (e^5+x+e^x \left (-18 e^5 x^2-18 x^3\right )+e^{2 x} \left (-18 e^5 x^2-18 x^3\right )\right ) \log \left (e^5+x\right )+\left (e^x \left (-18 e^5 x-18 x^2\right )+e^{2 x} \left (-18 e^5 x-18 x^2\right )\right ) \log ^2\left (e^5+x\right )+\left (e^x \left (-6 e^5-6 x\right )+e^{2 x} \left (-6 e^5-6 x\right )\right ) \log ^3\left (e^5+x\right )}{e^5 x^3+x^4+\left (3 e^5 x^2+3 x^3\right ) \log \left (e^5+x\right )+\left (3 e^5 x+3 x^2\right ) \log ^2\left (e^5+x\right )+\left (e^5+x\right ) \log ^3\left (e^5+x\right )} \, dx=-6 e^x-3 e^{2 x}+\frac {x}{\left (x+\log \left (e^5+x\right )\right )^2} \] Input:

Integrate[(-2*x - E^5*x - x^2 + E^x*(-6*E^5*x^3 - 6*x^4) + E^(2*x)*(-6*E^5 
*x^3 - 6*x^4) + (E^5 + x + E^x*(-18*E^5*x^2 - 18*x^3) + E^(2*x)*(-18*E^5*x 
^2 - 18*x^3))*Log[E^5 + x] + (E^x*(-18*E^5*x - 18*x^2) + E^(2*x)*(-18*E^5* 
x - 18*x^2))*Log[E^5 + x]^2 + (E^x*(-6*E^5 - 6*x) + E^(2*x)*(-6*E^5 - 6*x) 
)*Log[E^5 + x]^3)/(E^5*x^3 + x^4 + (3*E^5*x^2 + 3*x^3)*Log[E^5 + x] + (3*E 
^5*x + 3*x^2)*Log[E^5 + x]^2 + (E^5 + x)*Log[E^5 + x]^3),x]
 

Output:

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

Rubi [F]

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.

\(\displaystyle \int \frac {-x^2+\left (e^x \left (-18 x^2-18 e^5 x\right )+e^{2 x} \left (-18 x^2-18 e^5 x\right )\right ) \log ^2\left (x+e^5\right )+e^x \left (-6 x^4-6 e^5 x^3\right )+e^{2 x} \left (-6 x^4-6 e^5 x^3\right )+\left (e^x \left (-18 x^3-18 e^5 x^2\right )+e^{2 x} \left (-18 x^3-18 e^5 x^2\right )+x+e^5\right ) \log \left (x+e^5\right )-e^5 x-2 x+\left (e^x \left (-6 x-6 e^5\right )+e^{2 x} \left (-6 x-6 e^5\right )\right ) \log ^3\left (x+e^5\right )}{x^4+e^5 x^3+\left (3 x^2+3 e^5 x\right ) \log ^2\left (x+e^5\right )+\left (3 x^3+3 e^5 x^2\right ) \log \left (x+e^5\right )+\left (x+e^5\right ) \log ^3\left (x+e^5\right )} \, dx\)

\(\Big \downarrow \) 6

\(\displaystyle \int \frac {-x^2+\left (e^x \left (-18 x^2-18 e^5 x\right )+e^{2 x} \left (-18 x^2-18 e^5 x\right )\right ) \log ^2\left (x+e^5\right )+e^x \left (-6 x^4-6 e^5 x^3\right )+e^{2 x} \left (-6 x^4-6 e^5 x^3\right )+\left (e^x \left (-18 x^3-18 e^5 x^2\right )+e^{2 x} \left (-18 x^3-18 e^5 x^2\right )+x+e^5\right ) \log \left (x+e^5\right )+\left (-2-e^5\right ) x+\left (e^x \left (-6 x-6 e^5\right )+e^{2 x} \left (-6 x-6 e^5\right )\right ) \log ^3\left (x+e^5\right )}{x^4+e^5 x^3+\left (3 x^2+3 e^5 x\right ) \log ^2\left (x+e^5\right )+\left (3 x^3+3 e^5 x^2\right ) \log \left (x+e^5\right )+\left (x+e^5\right ) \log ^3\left (x+e^5\right )}dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {-\left (x+e^5\right ) \left (18 e^x x^2+18 e^{2 x} x^2-1\right ) \log \left (x+e^5\right )-x \left (6 e^x x^3+6 e^{2 x} x^3+6 e^{x+5} x^2+6 e^{2 x+5} x^2+x+e^5+2\right )-6 e^x \left (e^x+1\right ) \left (x+e^5\right ) \log ^3\left (x+e^5\right )-18 e^x \left (e^x+1\right ) x \left (x+e^5\right ) \log ^2\left (x+e^5\right )}{\left (x+e^5\right ) \left (x+\log \left (x+e^5\right )\right )^3}dx\)

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 2009

\(\displaystyle -\left (\left (2+e^5\right ) \int \frac {1}{\left (x+\log \left (x+e^5\right )\right )^3}dx\right )+e^5 \int \frac {1}{\left (x+\log \left (x+e^5\right )\right )^3}dx-2 \int \frac {x}{\left (x+\log \left (x+e^5\right )\right )^3}dx+e^5 \left (2+e^5\right ) \int \frac {1}{\left (x+e^5\right ) \left (x+\log \left (x+e^5\right )\right )^3}dx-e^{10} \int \frac {1}{\left (x+e^5\right ) \left (x+\log \left (x+e^5\right )\right )^3}dx+\int \frac {1}{\left (x+\log \left (x+e^5\right )\right )^2}dx-6 e^x-3 e^{2 x}\)

Input:

Int[(-2*x - E^5*x - x^2 + E^x*(-6*E^5*x^3 - 6*x^4) + E^(2*x)*(-6*E^5*x^3 - 
 6*x^4) + (E^5 + x + E^x*(-18*E^5*x^2 - 18*x^3) + E^(2*x)*(-18*E^5*x^2 - 1 
8*x^3))*Log[E^5 + x] + (E^x*(-18*E^5*x - 18*x^2) + E^(2*x)*(-18*E^5*x - 18 
*x^2))*Log[E^5 + x]^2 + (E^x*(-6*E^5 - 6*x) + E^(2*x)*(-6*E^5 - 6*x))*Log[ 
E^5 + x]^3)/(E^5*x^3 + x^4 + (3*E^5*x^2 + 3*x^3)*Log[E^5 + x] + (3*E^5*x + 
 3*x^2)*Log[E^5 + x]^2 + (E^5 + x)*Log[E^5 + x]^3),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 1.54 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.05

\[-3 \,{\mathrm e}^{2 x}-6 \,{\mathrm e}^{x}+\frac {x}{{\left (\ln \left ({\mathrm e}^{5}+x \right )+x \right )}^{2}}\]

Input:

int((((-6*exp(5)-6*x)*exp(x)^2+(-6*exp(5)-6*x)*exp(x))*ln(exp(5)+x)^3+((-1 
8*x*exp(5)-18*x^2)*exp(x)^2+(-18*x*exp(5)-18*x^2)*exp(x))*ln(exp(5)+x)^2+( 
(-18*x^2*exp(5)-18*x^3)*exp(x)^2+(-18*x^2*exp(5)-18*x^3)*exp(x)+exp(5)+x)* 
ln(exp(5)+x)+(-6*x^3*exp(5)-6*x^4)*exp(x)^2+(-6*x^3*exp(5)-6*x^4)*exp(x)-x 
*exp(5)-x^2-2*x)/((exp(5)+x)*ln(exp(5)+x)^3+(3*x*exp(5)+3*x^2)*ln(exp(5)+x 
)^2+(3*x^2*exp(5)+3*x^3)*ln(exp(5)+x)+x^3*exp(5)+x^4),x)
 

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 80 vs. \(2 (20) = 40\).

Time = 0.09 (sec) , antiderivative size = 80, normalized size of antiderivative = 3.64 \[ \int \frac {-2 x-e^5 x-x^2+e^x \left (-6 e^5 x^3-6 x^4\right )+e^{2 x} \left (-6 e^5 x^3-6 x^4\right )+\left (e^5+x+e^x \left (-18 e^5 x^2-18 x^3\right )+e^{2 x} \left (-18 e^5 x^2-18 x^3\right )\right ) \log \left (e^5+x\right )+\left (e^x \left (-18 e^5 x-18 x^2\right )+e^{2 x} \left (-18 e^5 x-18 x^2\right )\right ) \log ^2\left (e^5+x\right )+\left (e^x \left (-6 e^5-6 x\right )+e^{2 x} \left (-6 e^5-6 x\right )\right ) \log ^3\left (e^5+x\right )}{e^5 x^3+x^4+\left (3 e^5 x^2+3 x^3\right ) \log \left (e^5+x\right )+\left (3 e^5 x+3 x^2\right ) \log ^2\left (e^5+x\right )+\left (e^5+x\right ) \log ^3\left (e^5+x\right )} \, dx=-\frac {3 \, x^{2} e^{\left (2 \, x\right )} + 6 \, x^{2} e^{x} + 3 \, {\left (e^{\left (2 \, x\right )} + 2 \, e^{x}\right )} \log \left (x + e^{5}\right )^{2} + 6 \, {\left (x e^{\left (2 \, x\right )} + 2 \, x e^{x}\right )} \log \left (x + e^{5}\right ) - x}{x^{2} + 2 \, x \log \left (x + e^{5}\right ) + \log \left (x + e^{5}\right )^{2}} \] Input:

integrate((((-6*exp(5)-6*x)*exp(x)^2+(-6*exp(5)-6*x)*exp(x))*log(exp(5)+x) 
^3+((-18*x*exp(5)-18*x^2)*exp(x)^2+(-18*x*exp(5)-18*x^2)*exp(x))*log(exp(5 
)+x)^2+((-18*x^2*exp(5)-18*x^3)*exp(x)^2+(-18*x^2*exp(5)-18*x^3)*exp(x)+ex 
p(5)+x)*log(exp(5)+x)+(-6*x^3*exp(5)-6*x^4)*exp(x)^2+(-6*x^3*exp(5)-6*x^4) 
*exp(x)-x*exp(5)-x^2-2*x)/((exp(5)+x)*log(exp(5)+x)^3+(3*x*exp(5)+3*x^2)*l 
og(exp(5)+x)^2+(3*x^2*exp(5)+3*x^3)*log(exp(5)+x)+x^3*exp(5)+x^4),x, algor 
ithm="fricas")
 

Output:

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

Sympy [A] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.55 \[ \int \frac {-2 x-e^5 x-x^2+e^x \left (-6 e^5 x^3-6 x^4\right )+e^{2 x} \left (-6 e^5 x^3-6 x^4\right )+\left (e^5+x+e^x \left (-18 e^5 x^2-18 x^3\right )+e^{2 x} \left (-18 e^5 x^2-18 x^3\right )\right ) \log \left (e^5+x\right )+\left (e^x \left (-18 e^5 x-18 x^2\right )+e^{2 x} \left (-18 e^5 x-18 x^2\right )\right ) \log ^2\left (e^5+x\right )+\left (e^x \left (-6 e^5-6 x\right )+e^{2 x} \left (-6 e^5-6 x\right )\right ) \log ^3\left (e^5+x\right )}{e^5 x^3+x^4+\left (3 e^5 x^2+3 x^3\right ) \log \left (e^5+x\right )+\left (3 e^5 x+3 x^2\right ) \log ^2\left (e^5+x\right )+\left (e^5+x\right ) \log ^3\left (e^5+x\right )} \, dx=\frac {x}{x^{2} + 2 x \log {\left (x + e^{5} \right )} + \log {\left (x + e^{5} \right )}^{2}} - 3 e^{2 x} - 6 e^{x} \] Input:

integrate((((-6*exp(5)-6*x)*exp(x)**2+(-6*exp(5)-6*x)*exp(x))*ln(exp(5)+x) 
**3+((-18*x*exp(5)-18*x**2)*exp(x)**2+(-18*x*exp(5)-18*x**2)*exp(x))*ln(ex 
p(5)+x)**2+((-18*x**2*exp(5)-18*x**3)*exp(x)**2+(-18*x**2*exp(5)-18*x**3)* 
exp(x)+exp(5)+x)*ln(exp(5)+x)+(-6*x**3*exp(5)-6*x**4)*exp(x)**2+(-6*x**3*e 
xp(5)-6*x**4)*exp(x)-x*exp(5)-x**2-2*x)/((exp(5)+x)*ln(exp(5)+x)**3+(3*x*e 
xp(5)+3*x**2)*ln(exp(5)+x)**2+(3*x**2*exp(5)+3*x**3)*ln(exp(5)+x)+x**3*exp 
(5)+x**4),x)
 

Output:

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 80 vs. \(2 (20) = 40\).

Time = 0.08 (sec) , antiderivative size = 80, normalized size of antiderivative = 3.64 \[ \int \frac {-2 x-e^5 x-x^2+e^x \left (-6 e^5 x^3-6 x^4\right )+e^{2 x} \left (-6 e^5 x^3-6 x^4\right )+\left (e^5+x+e^x \left (-18 e^5 x^2-18 x^3\right )+e^{2 x} \left (-18 e^5 x^2-18 x^3\right )\right ) \log \left (e^5+x\right )+\left (e^x \left (-18 e^5 x-18 x^2\right )+e^{2 x} \left (-18 e^5 x-18 x^2\right )\right ) \log ^2\left (e^5+x\right )+\left (e^x \left (-6 e^5-6 x\right )+e^{2 x} \left (-6 e^5-6 x\right )\right ) \log ^3\left (e^5+x\right )}{e^5 x^3+x^4+\left (3 e^5 x^2+3 x^3\right ) \log \left (e^5+x\right )+\left (3 e^5 x+3 x^2\right ) \log ^2\left (e^5+x\right )+\left (e^5+x\right ) \log ^3\left (e^5+x\right )} \, dx=-\frac {3 \, x^{2} e^{\left (2 \, x\right )} + 6 \, x^{2} e^{x} + 3 \, {\left (e^{\left (2 \, x\right )} + 2 \, e^{x}\right )} \log \left (x + e^{5}\right )^{2} + 6 \, {\left (x e^{\left (2 \, x\right )} + 2 \, x e^{x}\right )} \log \left (x + e^{5}\right ) - x}{x^{2} + 2 \, x \log \left (x + e^{5}\right ) + \log \left (x + e^{5}\right )^{2}} \] Input:

integrate((((-6*exp(5)-6*x)*exp(x)^2+(-6*exp(5)-6*x)*exp(x))*log(exp(5)+x) 
^3+((-18*x*exp(5)-18*x^2)*exp(x)^2+(-18*x*exp(5)-18*x^2)*exp(x))*log(exp(5 
)+x)^2+((-18*x^2*exp(5)-18*x^3)*exp(x)^2+(-18*x^2*exp(5)-18*x^3)*exp(x)+ex 
p(5)+x)*log(exp(5)+x)+(-6*x^3*exp(5)-6*x^4)*exp(x)^2+(-6*x^3*exp(5)-6*x^4) 
*exp(x)-x*exp(5)-x^2-2*x)/((exp(5)+x)*log(exp(5)+x)^3+(3*x*exp(5)+3*x^2)*l 
og(exp(5)+x)^2+(3*x^2*exp(5)+3*x^3)*log(exp(5)+x)+x^3*exp(5)+x^4),x, algor 
ithm="maxima")
 

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 301 vs. \(2 (20) = 40\).

Time = 0.23 (sec) , antiderivative size = 301, normalized size of antiderivative = 13.68 \[ \int \frac {-2 x-e^5 x-x^2+e^x \left (-6 e^5 x^3-6 x^4\right )+e^{2 x} \left (-6 e^5 x^3-6 x^4\right )+\left (e^5+x+e^x \left (-18 e^5 x^2-18 x^3\right )+e^{2 x} \left (-18 e^5 x^2-18 x^3\right )\right ) \log \left (e^5+x\right )+\left (e^x \left (-18 e^5 x-18 x^2\right )+e^{2 x} \left (-18 e^5 x-18 x^2\right )\right ) \log ^2\left (e^5+x\right )+\left (e^x \left (-6 e^5-6 x\right )+e^{2 x} \left (-6 e^5-6 x\right )\right ) \log ^3\left (e^5+x\right )}{e^5 x^3+x^4+\left (3 e^5 x^2+3 x^3\right ) \log \left (e^5+x\right )+\left (3 e^5 x+3 x^2\right ) \log ^2\left (e^5+x\right )+\left (e^5+x\right ) \log ^3\left (e^5+x\right )} \, dx=-\frac {3 \, {\left (x + e^{5}\right )}^{2} e^{\left (2 \, x + 3 \, e^{5} + 5\right )} + 6 \, {\left (x + e^{5}\right )}^{2} e^{\left (x + 3 \, e^{5} + 5\right )} + 6 \, {\left (x + e^{5}\right )} e^{\left (2 \, x + 3 \, e^{5} + 5\right )} \log \left (x + e^{5}\right ) + 12 \, {\left (x + e^{5}\right )} e^{\left (x + 3 \, e^{5} + 5\right )} \log \left (x + e^{5}\right ) + 3 \, e^{\left (2 \, x + 3 \, e^{5} + 5\right )} \log \left (x + e^{5}\right )^{2} + 6 \, e^{\left (x + 3 \, e^{5} + 5\right )} \log \left (x + e^{5}\right )^{2} - 6 \, {\left (x + e^{5}\right )} e^{\left (2 \, x + 3 \, e^{5} + 10\right )} - 12 \, {\left (x + e^{5}\right )} e^{\left (x + 3 \, e^{5} + 10\right )} - {\left (x + e^{5}\right )} e^{\left (3 \, e^{5} + 5\right )} - 6 \, e^{\left (2 \, x + 3 \, e^{5} + 10\right )} \log \left (x + e^{5}\right ) - 12 \, e^{\left (x + 3 \, e^{5} + 10\right )} \log \left (x + e^{5}\right ) + 3 \, e^{\left (2 \, x + 3 \, e^{5} + 15\right )} + 6 \, e^{\left (x + 3 \, e^{5} + 15\right )} + e^{\left (3 \, e^{5} + 10\right )}}{{\left (x + e^{5}\right )}^{2} e^{\left (3 \, e^{5} + 5\right )} + 2 \, {\left (x + e^{5}\right )} e^{\left (3 \, e^{5} + 5\right )} \log \left (x + e^{5}\right ) + e^{\left (3 \, e^{5} + 5\right )} \log \left (x + e^{5}\right )^{2} - 2 \, {\left (x + e^{5}\right )} e^{\left (3 \, e^{5} + 10\right )} - 2 \, e^{\left (3 \, e^{5} + 10\right )} \log \left (x + e^{5}\right ) + e^{\left (3 \, e^{5} + 15\right )}} \] Input:

integrate((((-6*exp(5)-6*x)*exp(x)^2+(-6*exp(5)-6*x)*exp(x))*log(exp(5)+x) 
^3+((-18*x*exp(5)-18*x^2)*exp(x)^2+(-18*x*exp(5)-18*x^2)*exp(x))*log(exp(5 
)+x)^2+((-18*x^2*exp(5)-18*x^3)*exp(x)^2+(-18*x^2*exp(5)-18*x^3)*exp(x)+ex 
p(5)+x)*log(exp(5)+x)+(-6*x^3*exp(5)-6*x^4)*exp(x)^2+(-6*x^3*exp(5)-6*x^4) 
*exp(x)-x*exp(5)-x^2-2*x)/((exp(5)+x)*log(exp(5)+x)^3+(3*x*exp(5)+3*x^2)*l 
og(exp(5)+x)^2+(3*x^2*exp(5)+3*x^3)*log(exp(5)+x)+x^3*exp(5)+x^4),x, algor 
ithm="giac")
 

Output:

-(3*(x + e^5)^2*e^(2*x + 3*e^5 + 5) + 6*(x + e^5)^2*e^(x + 3*e^5 + 5) + 6* 
(x + e^5)*e^(2*x + 3*e^5 + 5)*log(x + e^5) + 12*(x + e^5)*e^(x + 3*e^5 + 5 
)*log(x + e^5) + 3*e^(2*x + 3*e^5 + 5)*log(x + e^5)^2 + 6*e^(x + 3*e^5 + 5 
)*log(x + e^5)^2 - 6*(x + e^5)*e^(2*x + 3*e^5 + 10) - 12*(x + e^5)*e^(x + 
3*e^5 + 10) - (x + e^5)*e^(3*e^5 + 5) - 6*e^(2*x + 3*e^5 + 10)*log(x + e^5 
) - 12*e^(x + 3*e^5 + 10)*log(x + e^5) + 3*e^(2*x + 3*e^5 + 15) + 6*e^(x + 
 3*e^5 + 15) + e^(3*e^5 + 10))/((x + e^5)^2*e^(3*e^5 + 5) + 2*(x + e^5)*e^ 
(3*e^5 + 5)*log(x + e^5) + e^(3*e^5 + 5)*log(x + e^5)^2 - 2*(x + e^5)*e^(3 
*e^5 + 10) - 2*e^(3*e^5 + 10)*log(x + e^5) + e^(3*e^5 + 15))
 

Mupad [B] (verification not implemented)

Time = 3.01 (sec) , antiderivative size = 181, normalized size of antiderivative = 8.23 \[ \int \frac {-2 x-e^5 x-x^2+e^x \left (-6 e^5 x^3-6 x^4\right )+e^{2 x} \left (-6 e^5 x^3-6 x^4\right )+\left (e^5+x+e^x \left (-18 e^5 x^2-18 x^3\right )+e^{2 x} \left (-18 e^5 x^2-18 x^3\right )\right ) \log \left (e^5+x\right )+\left (e^x \left (-18 e^5 x-18 x^2\right )+e^{2 x} \left (-18 e^5 x-18 x^2\right )\right ) \log ^2\left (e^5+x\right )+\left (e^x \left (-6 e^5-6 x\right )+e^{2 x} \left (-6 e^5-6 x\right )\right ) \log ^3\left (e^5+x\right )}{e^5 x^3+x^4+\left (3 e^5 x^2+3 x^3\right ) \log \left (e^5+x\right )+\left (3 e^5 x+3 x^2\right ) \log ^2\left (e^5+x\right )+\left (e^5+x\right ) \log ^3\left (e^5+x\right )} \, dx=\frac {\frac {x\,\left (x+{\mathrm {e}}^5+2\right )}{2\,\left (x+{\mathrm {e}}^5+1\right )}-\frac {\ln \left (x+{\mathrm {e}}^5\right )\,\left (x+{\mathrm {e}}^5\right )}{2\,\left (x+{\mathrm {e}}^5+1\right )}}{x^2+2\,x\,\ln \left (x+{\mathrm {e}}^5\right )+{\ln \left (x+{\mathrm {e}}^5\right )}^2}-6\,{\mathrm {e}}^x-3\,{\mathrm {e}}^{2\,x}+\frac {\frac {\left (x+{\mathrm {e}}^5\right )\,\left (x+2\,{\mathrm {e}}^5+{\mathrm {e}}^{10}+2\,x\,{\mathrm {e}}^5+x^2+1\right )}{2\,{\left (x+{\mathrm {e}}^5+1\right )}^3}-\frac {\ln \left (x+{\mathrm {e}}^5\right )\,\left (x+{\mathrm {e}}^5\right )}{2\,{\left (x+{\mathrm {e}}^5+1\right )}^3}}{x+\ln \left (x+{\mathrm {e}}^5\right )}+\frac {x+{\mathrm {e}}^5}{2\,x^3+\left (6\,{\mathrm {e}}^5+6\right )\,x^2+\left (12\,{\mathrm {e}}^5+6\,{\mathrm {e}}^{10}+6\right )\,x+6\,{\mathrm {e}}^5+6\,{\mathrm {e}}^{10}+2\,{\mathrm {e}}^{15}+2} \] Input:

int(-(2*x + x*exp(5) + log(x + exp(5))^2*(exp(x)*(18*x*exp(5) + 18*x^2) + 
exp(2*x)*(18*x*exp(5) + 18*x^2)) + exp(x)*(6*x^3*exp(5) + 6*x^4) - log(x + 
 exp(5))*(x + exp(5) - exp(x)*(18*x^2*exp(5) + 18*x^3) - exp(2*x)*(18*x^2* 
exp(5) + 18*x^3)) + log(x + exp(5))^3*(exp(x)*(6*x + 6*exp(5)) + exp(2*x)* 
(6*x + 6*exp(5))) + exp(2*x)*(6*x^3*exp(5) + 6*x^4) + x^2)/(log(x + exp(5) 
)^2*(3*x*exp(5) + 3*x^2) + log(x + exp(5))*(3*x^2*exp(5) + 3*x^3) + x^3*ex 
p(5) + log(x + exp(5))^3*(x + exp(5)) + x^4),x)
 

Output:

((x*(x + exp(5) + 2))/(2*(x + exp(5) + 1)) - (log(x + exp(5))*(x + exp(5)) 
)/(2*(x + exp(5) + 1)))/(log(x + exp(5))^2 + 2*x*log(x + exp(5)) + x^2) - 
6*exp(x) - 3*exp(2*x) + (((x + exp(5))*(x + 2*exp(5) + exp(10) + 2*x*exp(5 
) + x^2 + 1))/(2*(x + exp(5) + 1)^3) - (log(x + exp(5))*(x + exp(5)))/(2*( 
x + exp(5) + 1)^3))/(x + log(x + exp(5))) + (x + exp(5))/(6*exp(5) + 6*exp 
(10) + 2*exp(15) + x^2*(6*exp(5) + 6) + x*(12*exp(5) + 6*exp(10) + 6) + 2* 
x^3 + 2)
 

Reduce [B] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 98, normalized size of antiderivative = 4.45 \[ \int \frac {-2 x-e^5 x-x^2+e^x \left (-6 e^5 x^3-6 x^4\right )+e^{2 x} \left (-6 e^5 x^3-6 x^4\right )+\left (e^5+x+e^x \left (-18 e^5 x^2-18 x^3\right )+e^{2 x} \left (-18 e^5 x^2-18 x^3\right )\right ) \log \left (e^5+x\right )+\left (e^x \left (-18 e^5 x-18 x^2\right )+e^{2 x} \left (-18 e^5 x-18 x^2\right )\right ) \log ^2\left (e^5+x\right )+\left (e^x \left (-6 e^5-6 x\right )+e^{2 x} \left (-6 e^5-6 x\right )\right ) \log ^3\left (e^5+x\right )}{e^5 x^3+x^4+\left (3 e^5 x^2+3 x^3\right ) \log \left (e^5+x\right )+\left (3 e^5 x+3 x^2\right ) \log ^2\left (e^5+x\right )+\left (e^5+x\right ) \log ^3\left (e^5+x\right )} \, dx=\frac {-3 e^{2 x} \mathrm {log}\left (e^{5}+x \right )^{2}-6 e^{2 x} \mathrm {log}\left (e^{5}+x \right ) x -3 e^{2 x} x^{2}-6 e^{x} \mathrm {log}\left (e^{5}+x \right )^{2}-12 e^{x} \mathrm {log}\left (e^{5}+x \right ) x -6 e^{x} x^{2}+x}{\mathrm {log}\left (e^{5}+x \right )^{2}+2 \,\mathrm {log}\left (e^{5}+x \right ) x +x^{2}} \] Input:

int((((-6*exp(5)-6*x)*exp(x)^2+(-6*exp(5)-6*x)*exp(x))*log(exp(5)+x)^3+((- 
18*x*exp(5)-18*x^2)*exp(x)^2+(-18*x*exp(5)-18*x^2)*exp(x))*log(exp(5)+x)^2 
+((-18*x^2*exp(5)-18*x^3)*exp(x)^2+(-18*x^2*exp(5)-18*x^3)*exp(x)+exp(5)+x 
)*log(exp(5)+x)+(-6*x^3*exp(5)-6*x^4)*exp(x)^2+(-6*x^3*exp(5)-6*x^4)*exp(x 
)-x*exp(5)-x^2-2*x)/((exp(5)+x)*log(exp(5)+x)^3+(3*x*exp(5)+3*x^2)*log(exp 
(5)+x)^2+(3*x^2*exp(5)+3*x^3)*log(exp(5)+x)+x^3*exp(5)+x^4),x)
 

Output:

( - 3*e**(2*x)*log(e**5 + x)**2 - 6*e**(2*x)*log(e**5 + x)*x - 3*e**(2*x)* 
x**2 - 6*e**x*log(e**5 + x)**2 - 12*e**x*log(e**5 + x)*x - 6*e**x*x**2 + x 
)/(log(e**5 + x)**2 + 2*log(e**5 + x)*x + x**2)