\(\int \frac {-e^{1+x} x^2+2 e^{10+e^2+e^{8-x^2}-x^2} x^3+(2 e^{2+e^2+e^{8-x^2}} x+2 e^{1+x} x) \log (e^{2+e^2+e^{8-x^2}}+e^{1+x})}{(e^{2+e^2+e^{8-x^2}}+e^{1+x}) \log ^2(e^{2+e^2+e^{8-x^2}}+e^{1+x})} \, dx\) [700]

Optimal result

Integrand size = 139, antiderivative size = 29 \[ \int \frac {-e^{1+x} x^2+2 e^{10+e^2+e^{8-x^2}-x^2} x^3+\left (2 e^{2+e^2+e^{8-x^2}} x+2 e^{1+x} x\right ) \log \left (e^{2+e^2+e^{8-x^2}}+e^{1+x}\right )}{\left (e^{2+e^2+e^{8-x^2}}+e^{1+x}\right ) \log ^2\left (e^{2+e^2+e^{8-x^2}}+e^{1+x}\right )} \, dx=\frac {x^2}{\log \left (e^{2+e^2+e^{8-x^2}}+e^{1+x}\right )} \] Output:

x^2/ln(exp(exp(-x^2+8)+exp(2)+2)+exp(1+x))
 

Mathematica [A] (verified)

Time = 0.22 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00 \[ \int \frac {-e^{1+x} x^2+2 e^{10+e^2+e^{8-x^2}-x^2} x^3+\left (2 e^{2+e^2+e^{8-x^2}} x+2 e^{1+x} x\right ) \log \left (e^{2+e^2+e^{8-x^2}}+e^{1+x}\right )}{\left (e^{2+e^2+e^{8-x^2}}+e^{1+x}\right ) \log ^2\left (e^{2+e^2+e^{8-x^2}}+e^{1+x}\right )} \, dx=\frac {x^2}{1+\log \left (e^{1+e^2+e^{8-x^2}}+e^x\right )} \] Input:

Integrate[(-(E^(1 + x)*x^2) + 2*E^(10 + E^2 + E^(8 - x^2) - x^2)*x^3 + (2* 
E^(2 + E^2 + E^(8 - x^2))*x + 2*E^(1 + x)*x)*Log[E^(2 + E^2 + E^(8 - x^2)) 
 + E^(1 + x)])/((E^(2 + E^2 + E^(8 - x^2)) + E^(1 + x))*Log[E^(2 + E^2 + E 
^(8 - x^2)) + E^(1 + x)]^2),x]
 

Output:

x^2/(1 + Log[E^(1 + E^2 + E^(8 - x^2)) + E^x])
 

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.

\(\displaystyle \int \frac {-e^{x+1} x^2+\left (2 e^{e^{8-x^2}+2+e^2} x+2 e^{x+1} x\right ) \log \left (e^{e^{8-x^2}+2+e^2}+e^{x+1}\right )+2 e^{-x^2+e^{8-x^2}+e^2+10} x^3}{\left (e^{e^{8-x^2}+2+e^2}+e^{x+1}\right ) \log ^2\left (e^{e^{8-x^2}+2+e^2}+e^{x+1}\right )} \, dx\)

\(\Big \downarrow \) 7292

\(\displaystyle \int \frac {-e^{x+1} x^2+\left (2 e^{e^{8-x^2}+2+e^2} x+2 e^{x+1} x\right ) \log \left (e^{e^{8-x^2}+2+e^2}+e^{x+1}\right )+2 e^{-x^2+e^{8-x^2}+e^2+10} x^3}{e \left (e^{e^{8-x^2}+1+e^2}+e^x\right ) \log ^2\left (e \left (e^{e^{8-x^2}+1+e^2}+e^x\right )\right )}dx\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int -\frac {-2 e^{-x^2+e^{8-x^2}+e^2+10} x^3+e^{x+1} x^2-2 \left (e^{2+e^2+e^{8-x^2}} x+e^{x+1} x\right ) \log \left (e^{2+e^2+e^{8-x^2}}+e^{x+1}\right )}{\left (e^{1+e^2+e^{8-x^2}}+e^x\right ) \log ^2\left (e \left (e^{1+e^2+e^{8-x^2}}+e^x\right )\right )}dx}{e}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {\int \frac {-2 e^{-x^2+e^{8-x^2}+e^2+10} x^3+e^{x+1} x^2-2 \left (e^{2+e^2+e^{8-x^2}} x+e^{x+1} x\right ) \log \left (e^{2+e^2+e^{8-x^2}}+e^{x+1}\right )}{\left (e^{1+e^2+e^{8-x^2}}+e^x\right ) \log ^2\left (e \left (e^{1+e^2+e^{8-x^2}}+e^x\right )\right )}dx}{e}\)

\(\Big \downarrow \) 7293

\(\displaystyle -\frac {\int \left (\frac {2 e^{-x^2+e^{8-x^2}+10 \left (1+\frac {e^2}{10}\right )} x^3}{\left (-e^{1+e^2+e^{8-x^2}}-e^x\right ) \left (\log \left (e^{1+e^2+e^{8-x^2}}+e^x\right )+1\right )^2}-\frac {e x \left (-e^x x+2 e^{1+e^2+e^{8-x^2}}+2 e^x+2 e^{1+e^2+e^{8-x^2}} \log \left (e^{1+e^2+e^{8-x^2}}+e^x\right )+2 e^x \log \left (e^{1+e^2+e^{8-x^2}}+e^x\right )\right )}{\left (e^{1+e^2+e^{8-x^2}}+e^x\right ) \left (\log \left (e^{1+e^2+e^{8-x^2}}+e^x\right )+1\right )^2}\right )dx}{e}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {e \int \frac {x^2}{\left (\log \left (e^{1+e^2+e^{8-x^2}}+e^x\right )+1\right )^2}dx-e \int \frac {e^{1+e^2+e^{8-x^2}} x^2}{\left (e^{1+e^2+e^{8-x^2}}+e^x\right ) \left (\log \left (e^{1+e^2+e^{8-x^2}}+e^x\right )+1\right )^2}dx-2 e \int \frac {x}{\log \left (e^{1+e^2+e^{8-x^2}}+e^x\right )+1}dx+2 \int \frac {e^{-x^2+e^{8-x^2}+10 \left (1+\frac {e^2}{10}\right )} x^3}{\left (-e^{1+e^2+e^{8-x^2}}-e^x\right ) \left (\log \left (e^{1+e^2+e^{8-x^2}}+e^x\right )+1\right )^2}dx}{e}\)

Input:

Int[(-(E^(1 + x)*x^2) + 2*E^(10 + E^2 + E^(8 - x^2) - x^2)*x^3 + (2*E^(2 + 
 E^2 + E^(8 - x^2))*x + 2*E^(1 + x)*x)*Log[E^(2 + E^2 + E^(8 - x^2)) + E^( 
1 + x)])/((E^(2 + E^2 + E^(8 - x^2)) + E^(1 + x))*Log[E^(2 + E^2 + E^(8 - 
x^2)) + E^(1 + x)]^2),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 1.60 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.90

method	result	size
risch	\(\frac {x^{2}}{\ln \left ({\mathrm e}^{{\mathrm e}^{-x^{2}+8}+{\mathrm e}^{2}+2}+{\mathrm e}^{1+x}\right )}\)	\(26\)
parallelrisch	\(\frac {x^{2}}{\ln \left ({\mathrm e}^{{\mathrm e}^{-x^{2}+8}+{\mathrm e}^{2}+2}+{\mathrm e}^{1+x}\right )}\)	\(26\)

Input:

int(((2*x*exp(exp(-x^2+8)+exp(2)+2)+2*x*exp(1+x))*ln(exp(exp(-x^2+8)+exp(2 
)+2)+exp(1+x))+2*x^3*exp(-x^2+8)*exp(exp(-x^2+8)+exp(2)+2)-x^2*exp(1+x))/( 
exp(exp(-x^2+8)+exp(2)+2)+exp(1+x))/ln(exp(exp(-x^2+8)+exp(2)+2)+exp(1+x)) 
^2,x,method=_RETURNVERBOSE)
 

Output:

x^2/ln(exp(exp(-x^2+8)+exp(2)+2)+exp(1+x))
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.86 \[ \int \frac {-e^{1+x} x^2+2 e^{10+e^2+e^{8-x^2}-x^2} x^3+\left (2 e^{2+e^2+e^{8-x^2}} x+2 e^{1+x} x\right ) \log \left (e^{2+e^2+e^{8-x^2}}+e^{1+x}\right )}{\left (e^{2+e^2+e^{8-x^2}}+e^{1+x}\right ) \log ^2\left (e^{2+e^2+e^{8-x^2}}+e^{1+x}\right )} \, dx=\frac {x^{2}}{\log \left (e^{\left (x + 1\right )} + e^{\left (e^{2} + e^{\left (-x^{2} + 8\right )} + 2\right )}\right )} \] Input:

integrate(((2*x*exp(exp(-x^2+8)+exp(2)+2)+2*x*exp(1+x))*log(exp(exp(-x^2+8 
)+exp(2)+2)+exp(1+x))+2*x^3*exp(-x^2+8)*exp(exp(-x^2+8)+exp(2)+2)-x^2*exp( 
1+x))/(exp(exp(-x^2+8)+exp(2)+2)+exp(1+x))/log(exp(exp(-x^2+8)+exp(2)+2)+e 
xp(1+x))^2,x, algorithm="fricas")
 

Output:

x^2/log(e^(x + 1) + e^(e^2 + e^(-x^2 + 8) + 2))
 

Sympy [A] (verification not implemented)

Time = 1.44 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.76 \[ \int \frac {-e^{1+x} x^2+2 e^{10+e^2+e^{8-x^2}-x^2} x^3+\left (2 e^{2+e^2+e^{8-x^2}} x+2 e^{1+x} x\right ) \log \left (e^{2+e^2+e^{8-x^2}}+e^{1+x}\right )}{\left (e^{2+e^2+e^{8-x^2}}+e^{1+x}\right ) \log ^2\left (e^{2+e^2+e^{8-x^2}}+e^{1+x}\right )} \, dx=\frac {x^{2}}{\log {\left (e^{x + 1} + e^{e^{8 - x^{2}} + 2 + e^{2}} \right )}} \] Input:

integrate(((2*x*exp(exp(-x**2+8)+exp(2)+2)+2*x*exp(1+x))*ln(exp(exp(-x**2+ 
8)+exp(2)+2)+exp(1+x))+2*x**3*exp(-x**2+8)*exp(exp(-x**2+8)+exp(2)+2)-x**2 
*exp(1+x))/(exp(exp(-x**2+8)+exp(2)+2)+exp(1+x))/ln(exp(exp(-x**2+8)+exp(2 
)+2)+exp(1+x))**2,x)
 

Output:

x**2/log(exp(x + 1) + exp(exp(8 - x**2) + 2 + exp(2)))
 

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.86 \[ \int \frac {-e^{1+x} x^2+2 e^{10+e^2+e^{8-x^2}-x^2} x^3+\left (2 e^{2+e^2+e^{8-x^2}} x+2 e^{1+x} x\right ) \log \left (e^{2+e^2+e^{8-x^2}}+e^{1+x}\right )}{\left (e^{2+e^2+e^{8-x^2}}+e^{1+x}\right ) \log ^2\left (e^{2+e^2+e^{8-x^2}}+e^{1+x}\right )} \, dx=\frac {x^{2}}{\log \left (e^{x} + e^{\left (e^{2} + e^{\left (-x^{2} + 8\right )} + 1\right )}\right ) + 1} \] Input:

integrate(((2*x*exp(exp(-x^2+8)+exp(2)+2)+2*x*exp(1+x))*log(exp(exp(-x^2+8 
)+exp(2)+2)+exp(1+x))+2*x^3*exp(-x^2+8)*exp(exp(-x^2+8)+exp(2)+2)-x^2*exp( 
1+x))/(exp(exp(-x^2+8)+exp(2)+2)+exp(1+x))/log(exp(exp(-x^2+8)+exp(2)+2)+e 
xp(1+x))^2,x, algorithm="maxima")
 

Output:

x^2/(log(e^x + e^(e^2 + e^(-x^2 + 8) + 1)) + 1)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 786 vs. \(2 (25) = 50\).

Time = 13.04 (sec) , antiderivative size = 786, normalized size of antiderivative = 27.10 \[ \int \frac {-e^{1+x} x^2+2 e^{10+e^2+e^{8-x^2}-x^2} x^3+\left (2 e^{2+e^2+e^{8-x^2}} x+2 e^{1+x} x\right ) \log \left (e^{2+e^2+e^{8-x^2}}+e^{1+x}\right )}{\left (e^{2+e^2+e^{8-x^2}}+e^{1+x}\right ) \log ^2\left (e^{2+e^2+e^{8-x^2}}+e^{1+x}\right )} \, dx=\text {Too large to display} \] Input:

integrate(((2*x*exp(exp(-x^2+8)+exp(2)+2)+2*x*exp(1+x))*log(exp(exp(-x^2+8 
)+exp(2)+2)+exp(1+x))+2*x^3*exp(-x^2+8)*exp(exp(-x^2+8)+exp(2)+2)-x^2*exp( 
1+x))/(exp(exp(-x^2+8)+exp(2)+2)+exp(1+x))/log(exp(exp(-x^2+8)+exp(2)+2)+e 
xp(1+x))^2,x, algorithm="giac")
 

Output:

(4*x^4*e^(-2*x^2 + 2*e^2 + 2*e^(-x^2 + 8) + 18)*log(e^x + e^(e^2 + e^(-x^2 
 + 8) + 1)) + 4*x^4*e^(-2*x^2 + 2*e^2 + 2*e^(-x^2 + 8) + 18) - 2*x^3*e^(-x 
^2 + x + e^2 + e^(-x^2 + 8) + 9)*log((e^(-x^2 + x + 8) + e^(-x^2 + e^2 + e 
^(-x^2 + 8) + 9))*e^(x^2 - 8)) - 2*x^3*e^(-x^2 + x + e^2 + e^(-x^2 + 8) + 
9)*log(e^x + e^(e^2 + e^(-x^2 + 8) + 1)) - 4*x^3*e^(-x^2 + x + e^2 + e^(-x 
^2 + 8) + 9) + x^2*e^(2*x)*log((e^(-x^2 + x + 8) + e^(-x^2 + e^2 + e^(-x^2 
 + 8) + 9))*e^(x^2 - 8)) + x^2*e^(2*x))/(4*x^2*e^(-2*x^2 + 2*e^2 + 2*e^(-x 
^2 + 8) + 18)*log((e^(-x^2 + x + 8) + e^(-x^2 + e^2 + e^(-x^2 + 8) + 9))*e 
^(x^2 - 8))*log(e^x + e^(e^2 + e^(-x^2 + 8) + 1)) + 4*x^2*e^(-2*x^2 + 2*e^ 
2 + 2*e^(-x^2 + 8) + 18)*log((e^(-x^2 + x + 8) + e^(-x^2 + e^2 + e^(-x^2 + 
 8) + 9))*e^(x^2 - 8)) + 4*x^2*e^(-2*x^2 + 2*e^2 + 2*e^(-x^2 + 8) + 18)*lo 
g(e^x + e^(e^2 + e^(-x^2 + 8) + 1)) - 4*x*e^(-x^2 + x + e^2 + e^(-x^2 + 8) 
 + 9)*log((e^(-x^2 + x + 8) + e^(-x^2 + e^2 + e^(-x^2 + 8) + 9))*e^(x^2 - 
8))*log(e^x + e^(e^2 + e^(-x^2 + 8) + 1)) + 4*x^2*e^(-2*x^2 + 2*e^2 + 2*e^ 
(-x^2 + 8) + 18) - 4*x*e^(-x^2 + x + e^2 + e^(-x^2 + 8) + 9)*log((e^(-x^2 
+ x + 8) + e^(-x^2 + e^2 + e^(-x^2 + 8) + 9))*e^(x^2 - 8)) - 4*x*e^(-x^2 + 
 x + e^2 + e^(-x^2 + 8) + 9)*log(e^x + e^(e^2 + e^(-x^2 + 8) + 1)) + e^(2* 
x)*log((e^(-x^2 + x + 8) + e^(-x^2 + e^2 + e^(-x^2 + 8) + 9))*e^(x^2 - 8)) 
*log(e^x + e^(e^2 + e^(-x^2 + 8) + 1)) - 4*x*e^(-x^2 + x + e^2 + e^(-x^2 + 
 8) + 9) + e^(2*x)*log((e^(-x^2 + x + 8) + e^(-x^2 + e^2 + e^(-x^2 + 8)...
 

Mupad [B] (verification not implemented)

Time = 2.92 (sec) , antiderivative size = 286, normalized size of antiderivative = 9.86 \[ \int \frac {-e^{1+x} x^2+2 e^{10+e^2+e^{8-x^2}-x^2} x^3+\left (2 e^{2+e^2+e^{8-x^2}} x+2 e^{1+x} x\right ) \log \left (e^{2+e^2+e^{8-x^2}}+e^{1+x}\right )}{\left (e^{2+e^2+e^{8-x^2}}+e^{1+x}\right ) \log ^2\left (e^{2+e^2+e^{8-x^2}}+e^{1+x}\right )} \, dx=\frac {x^2-\frac {2\,x\,\ln \left (\mathrm {e}\,{\mathrm {e}}^x+{\mathrm {e}}^2\,{\mathrm {e}}^{{\mathrm {e}}^8\,{\mathrm {e}}^{-x^2}}\,{\mathrm {e}}^{{\mathrm {e}}^2}\right )\,\left ({\mathrm {e}}^{x+1}+{\mathrm {e}}^{{\mathrm {e}}^2+{\mathrm {e}}^8\,{\mathrm {e}}^{-x^2}+2}\right )}{{\mathrm {e}}^{x+1}-2\,x\,{\mathrm {e}}^{{\mathrm {e}}^2+{\mathrm {e}}^8\,{\mathrm {e}}^{-x^2}-x^2+10}}}{\ln \left (\mathrm {e}\,{\mathrm {e}}^x+{\mathrm {e}}^2\,{\mathrm {e}}^{{\mathrm {e}}^8\,{\mathrm {e}}^{-x^2}}\,{\mathrm {e}}^{{\mathrm {e}}^2}\right )}-{\mathrm {e}}^{x^2-8}+\frac {4\,x^2\,{\mathrm {e}}^{-x^2+2\,x+10}-{\mathrm {e}}^{2\,x+2}+4\,x^3\,{\mathrm {e}}^{-x^2+2\,x+10}+4\,x^3\,{\mathrm {e}}^{-2\,x^2+2\,x+18}+x\,{\mathrm {e}}^{2\,x+2}-2\,x\,{\mathrm {e}}^{-x^2+2\,x+10}+2\,x^2\,{\mathrm {e}}^{2\,x+2}}{\left ({\mathrm {e}}^{x+1}-2\,x\,{\mathrm {e}}^{{\mathrm {e}}^2+{\mathrm {e}}^8\,{\mathrm {e}}^{-x^2}-x^2+10}\right )\,\left (x\,{\mathrm {e}}^{-x^2+x+9}-{\mathrm {e}}^{-x^2+x+9}+2\,x^2\,{\mathrm {e}}^{-x^2+x+9}+2\,x^2\,{\mathrm {e}}^{-2\,x^2+x+17}\right )} \] Input:

int((log(exp(exp(2) + exp(8 - x^2) + 2) + exp(x + 1))*(2*x*exp(x + 1) + 2* 
x*exp(exp(2) + exp(8 - x^2) + 2)) - x^2*exp(x + 1) + 2*x^3*exp(exp(2) + ex 
p(8 - x^2) + 2)*exp(8 - x^2))/(log(exp(exp(2) + exp(8 - x^2) + 2) + exp(x 
+ 1))^2*(exp(exp(2) + exp(8 - x^2) + 2) + exp(x + 1))),x)
 

Output:

(x^2 - (2*x*log(exp(1)*exp(x) + exp(2)*exp(exp(8)*exp(-x^2))*exp(exp(2)))* 
(exp(x + 1) + exp(exp(2) + exp(8)*exp(-x^2) + 2)))/(exp(x + 1) - 2*x*exp(e 
xp(2) + exp(8)*exp(-x^2) - x^2 + 10)))/log(exp(1)*exp(x) + exp(2)*exp(exp( 
8)*exp(-x^2))*exp(exp(2))) - exp(x^2 - 8) + (4*x^2*exp(2*x - x^2 + 10) - e 
xp(2*x + 2) + 4*x^3*exp(2*x - x^2 + 10) + 4*x^3*exp(2*x - 2*x^2 + 18) + x* 
exp(2*x + 2) - 2*x*exp(2*x - x^2 + 10) + 2*x^2*exp(2*x + 2))/((exp(x + 1) 
- 2*x*exp(exp(2) + exp(8)*exp(-x^2) - x^2 + 10))*(x*exp(x - x^2 + 9) - exp 
(x - x^2 + 9) + 2*x^2*exp(x - x^2 + 9) + 2*x^2*exp(x - 2*x^2 + 17)))
 

Reduce [B] (verification not implemented)

Time = 0.29 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.38 \[ \int \frac {-e^{1+x} x^2+2 e^{10+e^2+e^{8-x^2}-x^2} x^3+\left (2 e^{2+e^2+e^{8-x^2}} x+2 e^{1+x} x\right ) \log \left (e^{2+e^2+e^{8-x^2}}+e^{1+x}\right )}{\left (e^{2+e^2+e^{8-x^2}}+e^{1+x}\right ) \log ^2\left (e^{2+e^2+e^{8-x^2}}+e^{1+x}\right )} \, dx=\frac {x^{2}}{\mathrm {log}\left (e^{\frac {e^{x^{2}} e^{2}+e^{8}}{e^{x^{2}}}} e^{2}+e^{x} e \right )} \] Input:

int(((2*x*exp(exp(-x^2+8)+exp(2)+2)+2*x*exp(1+x))*log(exp(exp(-x^2+8)+exp( 
2)+2)+exp(1+x))+2*x^3*exp(-x^2+8)*exp(exp(-x^2+8)+exp(2)+2)-x^2*exp(1+x))/ 
(exp(exp(-x^2+8)+exp(2)+2)+exp(1+x))/log(exp(exp(-x^2+8)+exp(2)+2)+exp(1+x 
))^2,x)
 

Output:

x**2/log(e**((e**(x**2)*e**2 + e**8)/e**(x**2))*e**2 + e**x*e)