\(\int \frac {e^3+e^{3 e^x}+3 e^{2+e^x}+3 e^{1+2 e^x}+e^{\frac {75-30 e+3 e^2+3 e^{2 e^x}+(-30+6 e) \log (5)+3 \log ^2(5)+e^{e^x} (-30+6 e+6 \log (5))}{e^2+e^{2 e^x}+2 e^{1+e^x}}} (e^{2 e^x+x} (30-6 \log (5))+e^{e^x+x} (-150+30 e+(60-6 e) \log (5)-6 \log ^2(5)))}{e^3+e^{3 e^x}+3 e^{2+e^x}+3 e^{1+2 e^x}} \, dx\) [2550]

Optimal result

Integrand size = 183, antiderivative size = 24 \[ \int \frac {e^3+e^{3 e^x}+3 e^{2+e^x}+3 e^{1+2 e^x}+e^{\frac {75-30 e+3 e^2+3 e^{2 e^x}+(-30+6 e) \log (5)+3 \log ^2(5)+e^{e^x} (-30+6 e+6 \log (5))}{e^2+e^{2 e^x}+2 e^{1+e^x}}} \left (e^{2 e^x+x} (30-6 \log (5))+e^{e^x+x} \left (-150+30 e+(60-6 e) \log (5)-6 \log ^2(5)\right )\right )}{e^3+e^{3 e^x}+3 e^{2+e^x}+3 e^{1+2 e^x}} \, dx=e^{3 \left (1+\frac {-5+\log (5)}{e+e^{e^x}}\right )^2}+x \] Output:

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

Mathematica [A] (verified)

Time = 0.81 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.12 \[ \int \frac {e^3+e^{3 e^x}+3 e^{2+e^x}+3 e^{1+2 e^x}+e^{\frac {75-30 e+3 e^2+3 e^{2 e^x}+(-30+6 e) \log (5)+3 \log ^2(5)+e^{e^x} (-30+6 e+6 \log (5))}{e^2+e^{2 e^x}+2 e^{1+e^x}}} \left (e^{2 e^x+x} (30-6 \log (5))+e^{e^x+x} \left (-150+30 e+(60-6 e) \log (5)-6 \log ^2(5)\right )\right )}{e^3+e^{3 e^x}+3 e^{2+e^x}+3 e^{1+2 e^x}} \, dx=e^{\frac {3 \left (-5+e+e^{e^x}+\log (5)\right )^2}{\left (e+e^{e^x}\right )^2}}+x \] Input:

Integrate[(E^3 + E^(3*E^x) + 3*E^(2 + E^x) + 3*E^(1 + 2*E^x) + E^((75 - 30 
*E + 3*E^2 + 3*E^(2*E^x) + (-30 + 6*E)*Log[5] + 3*Log[5]^2 + E^E^x*(-30 + 
6*E + 6*Log[5]))/(E^2 + E^(2*E^x) + 2*E^(1 + E^x)))*(E^(2*E^x + x)*(30 - 6 
*Log[5]) + E^(E^x + x)*(-150 + 30*E + (60 - 6*E)*Log[5] - 6*Log[5]^2)))/(E 
^3 + E^(3*E^x) + 3*E^(2 + E^x) + 3*E^(1 + 2*E^x)),x]
 

Output:

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

\(\displaystyle \int \frac {\left (e^{x+e^x} \left (-150+30 e-6 \log ^2(5)+(60-6 e) \log (5)\right )+e^{x+2 e^x} (30-6 \log (5))\right ) \exp \left (\frac {3 e^{2 e^x}+e^{e^x} (-30+6 e+6 \log (5))+75-30 e+3 e^2+3 \log ^2(5)+(6 e-30) \log (5)}{e^{2 e^x}+2 e^{e^x+1}+e^2}\right )+e^{3 e^x}+3 e^{e^x+2}+3 e^{2 e^x+1}+e^3}{e^{3 e^x}+3 e^{e^x+2}+3 e^{2 e^x+1}+e^3} \, dx\)

\(\Big \downarrow \) 2720

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 2009

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

Input:

Int[(E^3 + E^(3*E^x) + 3*E^(2 + E^x) + 3*E^(1 + 2*E^x) + E^((75 - 30*E + 3 
*E^2 + 3*E^(2*E^x) + (-30 + 6*E)*Log[5] + 3*Log[5]^2 + E^E^x*(-30 + 6*E + 
6*Log[5]))/(E^2 + E^(2*E^x) + 2*E^(1 + E^x)))*(E^(2*E^x + x)*(30 - 6*Log[5 
]) + E^(E^x + x)*(-150 + 30*E + (60 - 6*E)*Log[5] - 6*Log[5]^2)))/(E^3 + E 
^(3*E^x) + 3*E^(2 + E^x) + 3*E^(1 + 2*E^x)),x]
 

Output:

$Aborted
 

Maple [F(-1)]

Timed out.

\[\int \frac {\left (\left (-6 \ln \left (5\right )+30\right ) {\mathrm e}^{x} {\mathrm e}^{2 \,{\mathrm e}^{x}}+\left (-6 \ln \left (5\right )^{2}+\left (-6 \,{\mathrm e}+60\right ) \ln \left (5\right )+30 \,{\mathrm e}-150\right ) {\mathrm e}^{x} {\mathrm e}^{{\mathrm e}^{x}}\right ) {\mathrm e}^{\frac {3 \,{\mathrm e}^{2 \,{\mathrm e}^{x}}+\left (6 \ln \left (5\right )+6 \,{\mathrm e}-30\right ) {\mathrm e}^{{\mathrm e}^{x}}+3 \ln \left (5\right )^{2}+\left (6 \,{\mathrm e}-30\right ) \ln \left (5\right )+3 \,{\mathrm e}^{2}-30 \,{\mathrm e}+75}{{\mathrm e}^{2 \,{\mathrm e}^{x}}+2 \,{\mathrm e} \,{\mathrm e}^{{\mathrm e}^{x}}+{\mathrm e}^{2}}}+{\mathrm e}^{3 \,{\mathrm e}^{x}}+3 \,{\mathrm e} \,{\mathrm e}^{2 \,{\mathrm e}^{x}}+3 \,{\mathrm e}^{2} {\mathrm e}^{{\mathrm e}^{x}}+{\mathrm e}^{3}}{{\mathrm e}^{3 \,{\mathrm e}^{x}}+3 \,{\mathrm e} \,{\mathrm e}^{2 \,{\mathrm e}^{x}}+3 \,{\mathrm e}^{2} {\mathrm e}^{{\mathrm e}^{x}}+{\mathrm e}^{3}}d x\]

Input:

int((((-6*ln(5)+30)*exp(x)*exp(exp(x))^2+(-6*ln(5)^2+(-6*exp(1)+60)*ln(5)+ 
30*exp(1)-150)*exp(x)*exp(exp(x)))*exp((3*exp(exp(x))^2+(6*ln(5)+6*exp(1)- 
30)*exp(exp(x))+3*ln(5)^2+(6*exp(1)-30)*ln(5)+3*exp(1)^2-30*exp(1)+75)/(ex 
p(exp(x))^2+2*exp(1)*exp(exp(x))+exp(1)^2))+exp(exp(x))^3+3*exp(1)*exp(exp 
(x))^2+3*exp(1)^2*exp(exp(x))+exp(1)^3)/(exp(exp(x))^3+3*exp(1)*exp(exp(x) 
)^2+3*exp(1)^2*exp(exp(x))+exp(1)^3),x)
 

Output:

int((((-6*ln(5)+30)*exp(x)*exp(exp(x))^2+(-6*ln(5)^2+(-6*exp(1)+60)*ln(5)+ 
30*exp(1)-150)*exp(x)*exp(exp(x)))*exp((3*exp(exp(x))^2+(6*ln(5)+6*exp(1)- 
30)*exp(exp(x))+3*ln(5)^2+(6*exp(1)-30)*ln(5)+3*exp(1)^2-30*exp(1)+75)/(ex 
p(exp(x))^2+2*exp(1)*exp(exp(x))+exp(1)^2))+exp(exp(x))^3+3*exp(1)*exp(exp 
(x))^2+3*exp(1)^2*exp(exp(x))+exp(1)^3)/(exp(exp(x))^3+3*exp(1)*exp(exp(x) 
)^2+3*exp(1)^2*exp(exp(x))+exp(1)^3),x)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 83 vs. \(2 (22) = 44\).

Time = 0.11 (sec) , antiderivative size = 83, normalized size of antiderivative = 3.46 \[ \int \frac {e^3+e^{3 e^x}+3 e^{2+e^x}+3 e^{1+2 e^x}+e^{\frac {75-30 e+3 e^2+3 e^{2 e^x}+(-30+6 e) \log (5)+3 \log ^2(5)+e^{e^x} (-30+6 e+6 \log (5))}{e^2+e^{2 e^x}+2 e^{1+e^x}}} \left (e^{2 e^x+x} (30-6 \log (5))+e^{e^x+x} \left (-150+30 e+(60-6 e) \log (5)-6 \log ^2(5)\right )\right )}{e^3+e^{3 e^x}+3 e^{2+e^x}+3 e^{1+2 e^x}} \, dx=x + e^{\left (\frac {3 \, {\left ({\left (2 \, {\left (e - 5\right )} \log \left (5\right ) + \log \left (5\right )^{2} + e^{2} - 10 \, e + 25\right )} e^{\left (2 \, x\right )} + 2 \, {\left (e + \log \left (5\right ) - 5\right )} e^{\left (2 \, x + e^{x}\right )} + e^{\left (2 \, x + 2 \, e^{x}\right )}\right )}}{e^{\left (2 \, x + 2 \, e^{x}\right )} + 2 \, e^{\left (2 \, x + e^{x} + 1\right )} + e^{\left (2 \, x + 2\right )}}\right )} \] Input:

integrate((((-6*log(5)+30)*exp(x)*exp(exp(x))^2+(-6*log(5)^2+(-6*exp(1)+60 
)*log(5)+30*exp(1)-150)*exp(x)*exp(exp(x)))*exp((3*exp(exp(x))^2+(6*log(5) 
+6*exp(1)-30)*exp(exp(x))+3*log(5)^2+(6*exp(1)-30)*log(5)+3*exp(1)^2-30*ex 
p(1)+75)/(exp(exp(x))^2+2*exp(1)*exp(exp(x))+exp(1)^2))+exp(exp(x))^3+3*ex 
p(1)*exp(exp(x))^2+3*exp(1)^2*exp(exp(x))+exp(1)^3)/(exp(exp(x))^3+3*exp(1 
)*exp(exp(x))^2+3*exp(1)^2*exp(exp(x))+exp(1)^3),x, algorithm="fricas")
 

Output:

x + e^(3*((2*(e - 5)*log(5) + log(5)^2 + e^2 - 10*e + 25)*e^(2*x) + 2*(e + 
 log(5) - 5)*e^(2*x + e^x) + e^(2*x + 2*e^x))/(e^(2*x + 2*e^x) + 2*e^(2*x 
+ e^x + 1) + e^(2*x + 2)))
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 76 vs. \(2 (20) = 40\).

Time = 0.51 (sec) , antiderivative size = 76, normalized size of antiderivative = 3.17 \[ \int \frac {e^3+e^{3 e^x}+3 e^{2+e^x}+3 e^{1+2 e^x}+e^{\frac {75-30 e+3 e^2+3 e^{2 e^x}+(-30+6 e) \log (5)+3 \log ^2(5)+e^{e^x} (-30+6 e+6 \log (5))}{e^2+e^{2 e^x}+2 e^{1+e^x}}} \left (e^{2 e^x+x} (30-6 \log (5))+e^{e^x+x} \left (-150+30 e+(60-6 e) \log (5)-6 \log ^2(5)\right )\right )}{e^3+e^{3 e^x}+3 e^{2+e^x}+3 e^{1+2 e^x}} \, dx=x + e^{\frac {3 e^{2 e^{x}} + \left (-30 + 6 \log {\left (5 \right )} + 6 e\right ) e^{e^{x}} - 30 e + \left (-30 + 6 e\right ) \log {\left (5 \right )} + 3 \log {\left (5 \right )}^{2} + 3 e^{2} + 75}{e^{2 e^{x}} + 2 e e^{e^{x}} + e^{2}}} \] Input:

integrate((((-6*ln(5)+30)*exp(x)*exp(exp(x))**2+(-6*ln(5)**2+(-6*exp(1)+60 
)*ln(5)+30*exp(1)-150)*exp(x)*exp(exp(x)))*exp((3*exp(exp(x))**2+(6*ln(5)+ 
6*exp(1)-30)*exp(exp(x))+3*ln(5)**2+(6*exp(1)-30)*ln(5)+3*exp(1)**2-30*exp 
(1)+75)/(exp(exp(x))**2+2*exp(1)*exp(exp(x))+exp(1)**2))+exp(exp(x))**3+3* 
exp(1)*exp(exp(x))**2+3*exp(1)**2*exp(exp(x))+exp(1)**3)/(exp(exp(x))**3+3 
*exp(1)*exp(exp(x))**2+3*exp(1)**2*exp(exp(x))+exp(1)**3),x)
 

Output:

x + exp((3*exp(2*exp(x)) + (-30 + 6*log(5) + 6*E)*exp(exp(x)) - 30*E + (-3 
0 + 6*E)*log(5) + 3*log(5)**2 + 3*exp(2) + 75)/(exp(2*exp(x)) + 2*E*exp(ex 
p(x)) + exp(2)))
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 296 vs. \(2 (22) = 44\).

Time = 0.28 (sec) , antiderivative size = 296, normalized size of antiderivative = 12.33 \[ \int \frac {e^3+e^{3 e^x}+3 e^{2+e^x}+3 e^{1+2 e^x}+e^{\frac {75-30 e+3 e^2+3 e^{2 e^x}+(-30+6 e) \log (5)+3 \log ^2(5)+e^{e^x} (-30+6 e+6 \log (5))}{e^2+e^{2 e^x}+2 e^{1+e^x}}} \left (e^{2 e^x+x} (30-6 \log (5))+e^{e^x+x} \left (-150+30 e+(60-6 e) \log (5)-6 \log ^2(5)\right )\right )}{e^3+e^{3 e^x}+3 e^{2+e^x}+3 e^{1+2 e^x}} \, dx={\left (x e^{\left (\frac {30 \, e}{e^{2} + e^{\left (2 \, e^{x}\right )} + 2 \, e^{\left (e^{x} + 1\right )}} + \frac {30 \, e^{\left (e^{x}\right )}}{e^{2} + e^{\left (2 \, e^{x}\right )} + 2 \, e^{\left (e^{x} + 1\right )}} + \frac {30 \, \log \left (5\right )}{e^{2} + e^{\left (2 \, e^{x}\right )} + 2 \, e^{\left (e^{x} + 1\right )}}\right )} + e^{\left (\frac {6 \, e \log \left (5\right )}{e^{2} + e^{\left (2 \, e^{x}\right )} + 2 \, e^{\left (e^{x} + 1\right )}} + \frac {6 \, e^{\left (e^{x}\right )} \log \left (5\right )}{e^{2} + e^{\left (2 \, e^{x}\right )} + 2 \, e^{\left (e^{x} + 1\right )}} + \frac {3 \, \log \left (5\right )^{2}}{e^{2} + e^{\left (2 \, e^{x}\right )} + 2 \, e^{\left (e^{x} + 1\right )}} + \frac {3 \, e^{2}}{e^{2} + e^{\left (2 \, e^{x}\right )} + 2 \, e^{\left (e^{x} + 1\right )}} + \frac {3 \, e^{\left (2 \, e^{x}\right )}}{e^{2} + e^{\left (2 \, e^{x}\right )} + 2 \, e^{\left (e^{x} + 1\right )}} + \frac {6 \, e^{\left (e^{x} + 1\right )}}{e^{2} + e^{\left (2 \, e^{x}\right )} + 2 \, e^{\left (e^{x} + 1\right )}} + \frac {75}{e^{2} + e^{\left (2 \, e^{x}\right )} + 2 \, e^{\left (e^{x} + 1\right )}}\right )}\right )} e^{\left (-\frac {30 \, e}{e^{2} + e^{\left (2 \, e^{x}\right )} + 2 \, e^{\left (e^{x} + 1\right )}} - \frac {30 \, e^{\left (e^{x}\right )}}{e^{2} + e^{\left (2 \, e^{x}\right )} + 2 \, e^{\left (e^{x} + 1\right )}} - \frac {30 \, \log \left (5\right )}{e^{2} + e^{\left (2 \, e^{x}\right )} + 2 \, e^{\left (e^{x} + 1\right )}}\right )} \] Input:

integrate((((-6*log(5)+30)*exp(x)*exp(exp(x))^2+(-6*log(5)^2+(-6*exp(1)+60 
)*log(5)+30*exp(1)-150)*exp(x)*exp(exp(x)))*exp((3*exp(exp(x))^2+(6*log(5) 
+6*exp(1)-30)*exp(exp(x))+3*log(5)^2+(6*exp(1)-30)*log(5)+3*exp(1)^2-30*ex 
p(1)+75)/(exp(exp(x))^2+2*exp(1)*exp(exp(x))+exp(1)^2))+exp(exp(x))^3+3*ex 
p(1)*exp(exp(x))^2+3*exp(1)^2*exp(exp(x))+exp(1)^3)/(exp(exp(x))^3+3*exp(1 
)*exp(exp(x))^2+3*exp(1)^2*exp(exp(x))+exp(1)^3),x, algorithm="maxima")
 

Output:

(x*e^(30*e/(e^2 + e^(2*e^x) + 2*e^(e^x + 1)) + 30*e^(e^x)/(e^2 + e^(2*e^x) 
 + 2*e^(e^x + 1)) + 30*log(5)/(e^2 + e^(2*e^x) + 2*e^(e^x + 1))) + e^(6*e* 
log(5)/(e^2 + e^(2*e^x) + 2*e^(e^x + 1)) + 6*e^(e^x)*log(5)/(e^2 + e^(2*e^ 
x) + 2*e^(e^x + 1)) + 3*log(5)^2/(e^2 + e^(2*e^x) + 2*e^(e^x + 1)) + 3*e^2 
/(e^2 + e^(2*e^x) + 2*e^(e^x + 1)) + 3*e^(2*e^x)/(e^2 + e^(2*e^x) + 2*e^(e 
^x + 1)) + 6*e^(e^x + 1)/(e^2 + e^(2*e^x) + 2*e^(e^x + 1)) + 75/(e^2 + e^( 
2*e^x) + 2*e^(e^x + 1))))*e^(-30*e/(e^2 + e^(2*e^x) + 2*e^(e^x + 1)) - 30* 
e^(e^x)/(e^2 + e^(2*e^x) + 2*e^(e^x + 1)) - 30*log(5)/(e^2 + e^(2*e^x) + 2 
*e^(e^x + 1)))
 

Giac [F]

\[ \int \frac {e^3+e^{3 e^x}+3 e^{2+e^x}+3 e^{1+2 e^x}+e^{\frac {75-30 e+3 e^2+3 e^{2 e^x}+(-30+6 e) \log (5)+3 \log ^2(5)+e^{e^x} (-30+6 e+6 \log (5))}{e^2+e^{2 e^x}+2 e^{1+e^x}}} \left (e^{2 e^x+x} (30-6 \log (5))+e^{e^x+x} \left (-150+30 e+(60-6 e) \log (5)-6 \log ^2(5)\right )\right )}{e^3+e^{3 e^x}+3 e^{2+e^x}+3 e^{1+2 e^x}} \, dx=\int { -\frac {6 \, {\left ({\left (\log \left (5\right ) - 5\right )} e^{\left (x + 2 \, e^{x}\right )} + {\left ({\left (e - 10\right )} \log \left (5\right ) + \log \left (5\right )^{2} - 5 \, e + 25\right )} e^{\left (x + e^{x}\right )}\right )} e^{\left (\frac {3 \, {\left (2 \, {\left (e + \log \left (5\right ) - 5\right )} e^{\left (e^{x}\right )} + 2 \, {\left (e - 5\right )} \log \left (5\right ) + \log \left (5\right )^{2} + e^{2} - 10 \, e + e^{\left (2 \, e^{x}\right )} + 25\right )}}{e^{2} + e^{\left (2 \, e^{x}\right )} + 2 \, e^{\left (e^{x} + 1\right )}}\right )} - e^{3} - e^{\left (3 \, e^{x}\right )} - 3 \, e^{\left (2 \, e^{x} + 1\right )} - 3 \, e^{\left (e^{x} + 2\right )}}{e^{3} + e^{\left (3 \, e^{x}\right )} + 3 \, e^{\left (2 \, e^{x} + 1\right )} + 3 \, e^{\left (e^{x} + 2\right )}} \,d x } \] Input:

integrate((((-6*log(5)+30)*exp(x)*exp(exp(x))^2+(-6*log(5)^2+(-6*exp(1)+60 
)*log(5)+30*exp(1)-150)*exp(x)*exp(exp(x)))*exp((3*exp(exp(x))^2+(6*log(5) 
+6*exp(1)-30)*exp(exp(x))+3*log(5)^2+(6*exp(1)-30)*log(5)+3*exp(1)^2-30*ex 
p(1)+75)/(exp(exp(x))^2+2*exp(1)*exp(exp(x))+exp(1)^2))+exp(exp(x))^3+3*ex 
p(1)*exp(exp(x))^2+3*exp(1)^2*exp(exp(x))+exp(1)^3)/(exp(exp(x))^3+3*exp(1 
)*exp(exp(x))^2+3*exp(1)^2*exp(exp(x))+exp(1)^3),x, algorithm="giac")
 

Output:

integrate(-(6*((log(5) - 5)*e^(x + 2*e^x) + ((e - 10)*log(5) + log(5)^2 - 
5*e + 25)*e^(x + e^x))*e^(3*(2*(e + log(5) - 5)*e^(e^x) + 2*(e - 5)*log(5) 
 + log(5)^2 + e^2 - 10*e + e^(2*e^x) + 25)/(e^2 + e^(2*e^x) + 2*e^(e^x + 1 
))) - e^3 - e^(3*e^x) - 3*e^(2*e^x + 1) - 3*e^(e^x + 2))/(e^3 + e^(3*e^x) 
+ 3*e^(2*e^x + 1) + 3*e^(e^x + 2)), x)
 

Mupad [B] (verification not implemented)

Time = 0.47 (sec) , antiderivative size = 234, normalized size of antiderivative = 9.75 \[ \int \frac {e^3+e^{3 e^x}+3 e^{2+e^x}+3 e^{1+2 e^x}+e^{\frac {75-30 e+3 e^2+3 e^{2 e^x}+(-30+6 e) \log (5)+3 \log ^2(5)+e^{e^x} (-30+6 e+6 \log (5))}{e^2+e^{2 e^x}+2 e^{1+e^x}}} \left (e^{2 e^x+x} (30-6 \log (5))+e^{e^x+x} \left (-150+30 e+(60-6 e) \log (5)-6 \log ^2(5)\right )\right )}{e^3+e^{3 e^x}+3 e^{2+e^x}+3 e^{1+2 e^x}} \, dx=x+\frac {5^{\frac {6\,{\mathrm {e}}^{{\mathrm {e}}^x}}{{\mathrm {e}}^2+{\mathrm {e}}^{2\,{\mathrm {e}}^x}+2\,{\mathrm {e}}^{{\mathrm {e}}^x}\,\mathrm {e}}}\,5^{\frac {6\,\mathrm {e}}{{\mathrm {e}}^2+{\mathrm {e}}^{2\,{\mathrm {e}}^x}+2\,{\mathrm {e}}^{{\mathrm {e}}^x}\,\mathrm {e}}}\,{\mathrm {e}}^{\frac {3\,{\mathrm {e}}^{2\,{\mathrm {e}}^x}}{{\mathrm {e}}^2+{\mathrm {e}}^{2\,{\mathrm {e}}^x}+2\,{\mathrm {e}}^{{\mathrm {e}}^x}\,\mathrm {e}}}\,{\mathrm {e}}^{\frac {6\,{\mathrm {e}}^{{\mathrm {e}}^x}\,\mathrm {e}}{{\mathrm {e}}^2+{\mathrm {e}}^{2\,{\mathrm {e}}^x}+2\,{\mathrm {e}}^{{\mathrm {e}}^x}\,\mathrm {e}}}\,{\mathrm {e}}^{\frac {3\,{\ln \left (5\right )}^2}{{\mathrm {e}}^2+{\mathrm {e}}^{2\,{\mathrm {e}}^x}+2\,{\mathrm {e}}^{{\mathrm {e}}^x}\,\mathrm {e}}}\,{\mathrm {e}}^{\frac {75}{{\mathrm {e}}^2+{\mathrm {e}}^{2\,{\mathrm {e}}^x}+2\,{\mathrm {e}}^{{\mathrm {e}}^x}\,\mathrm {e}}}\,{\mathrm {e}}^{-\frac {30\,{\mathrm {e}}^{{\mathrm {e}}^x}}{{\mathrm {e}}^2+{\mathrm {e}}^{2\,{\mathrm {e}}^x}+2\,{\mathrm {e}}^{{\mathrm {e}}^x}\,\mathrm {e}}}\,{\mathrm {e}}^{\frac {3\,{\mathrm {e}}^2}{{\mathrm {e}}^2+{\mathrm {e}}^{2\,{\mathrm {e}}^x}+2\,{\mathrm {e}}^{{\mathrm {e}}^x}\,\mathrm {e}}}\,{\mathrm {e}}^{-\frac {30\,\mathrm {e}}{{\mathrm {e}}^2+{\mathrm {e}}^{2\,{\mathrm {e}}^x}+2\,{\mathrm {e}}^{{\mathrm {e}}^x}\,\mathrm {e}}}}{5^{\frac {30}{{\mathrm {e}}^2+{\mathrm {e}}^{2\,{\mathrm {e}}^x}+2\,{\mathrm {e}}^{{\mathrm {e}}^x}\,\mathrm {e}}}} \] Input:

int((exp(3) + exp(3*exp(x)) + 3*exp(exp(x))*exp(2) + 3*exp(1)*exp(2*exp(x) 
) - exp((3*exp(2) - 30*exp(1) + 3*exp(2*exp(x)) + exp(exp(x))*(6*exp(1) + 
6*log(5) - 30) + 3*log(5)^2 + log(5)*(6*exp(1) - 30) + 75)/(exp(2) + exp(2 
*exp(x)) + 2*exp(exp(x))*exp(1)))*(exp(2*exp(x))*exp(x)*(6*log(5) - 30) + 
exp(exp(x))*exp(x)*(6*log(5)^2 - 30*exp(1) + log(5)*(6*exp(1) - 60) + 150) 
))/(exp(3) + exp(3*exp(x)) + 3*exp(exp(x))*exp(2) + 3*exp(1)*exp(2*exp(x)) 
),x)
 

Output:

x + (5^((6*exp(exp(x)))/(exp(2) + exp(2*exp(x)) + 2*exp(exp(x))*exp(1)))*5 
^((6*exp(1))/(exp(2) + exp(2*exp(x)) + 2*exp(exp(x))*exp(1)))*exp((3*exp(2 
*exp(x)))/(exp(2) + exp(2*exp(x)) + 2*exp(exp(x))*exp(1)))*exp((6*exp(exp( 
x))*exp(1))/(exp(2) + exp(2*exp(x)) + 2*exp(exp(x))*exp(1)))*exp((3*log(5) 
^2)/(exp(2) + exp(2*exp(x)) + 2*exp(exp(x))*exp(1)))*exp(75/(exp(2) + exp( 
2*exp(x)) + 2*exp(exp(x))*exp(1)))*exp(-(30*exp(exp(x)))/(exp(2) + exp(2*e 
xp(x)) + 2*exp(exp(x))*exp(1)))*exp((3*exp(2))/(exp(2) + exp(2*exp(x)) + 2 
*exp(exp(x))*exp(1)))*exp(-(30*exp(1))/(exp(2) + exp(2*exp(x)) + 2*exp(exp 
(x))*exp(1))))/5^(30/(exp(2) + exp(2*exp(x)) + 2*exp(exp(x))*exp(1)))
 

Reduce [B] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 134, normalized size of antiderivative = 5.58 \[ \int \frac {e^3+e^{3 e^x}+3 e^{2+e^x}+3 e^{1+2 e^x}+e^{\frac {75-30 e+3 e^2+3 e^{2 e^x}+(-30+6 e) \log (5)+3 \log ^2(5)+e^{e^x} (-30+6 e+6 \log (5))}{e^2+e^{2 e^x}+2 e^{1+e^x}}} \left (e^{2 e^x+x} (30-6 \log (5))+e^{e^x+x} \left (-150+30 e+(60-6 e) \log (5)-6 \log ^2(5)\right )\right )}{e^3+e^{3 e^x}+3 e^{2+e^x}+3 e^{1+2 e^x}} \, dx=\frac {e^{\frac {6 e^{e^{x}} \mathrm {log}\left (5\right )+3 \mathrm {log}\left (5\right )^{2}+6 \,\mathrm {log}\left (5\right ) e +75}{e^{2 e^{x}}+2 e^{e^{x}} e +e^{2}}} e^{3}+e^{\frac {30 e^{e^{x}}+30 \,\mathrm {log}\left (5\right )+30 e}{e^{2 e^{x}}+2 e^{e^{x}} e +e^{2}}} x}{e^{\frac {30 e^{e^{x}}+30 \,\mathrm {log}\left (5\right )+30 e}{e^{2 e^{x}}+2 e^{e^{x}} e +e^{2}}}} \] Input:

int((((-6*log(5)+30)*exp(x)*exp(exp(x))^2+(-6*log(5)^2+(-6*exp(1)+60)*log( 
5)+30*exp(1)-150)*exp(x)*exp(exp(x)))*exp((3*exp(exp(x))^2+(6*log(5)+6*exp 
(1)-30)*exp(exp(x))+3*log(5)^2+(6*exp(1)-30)*log(5)+3*exp(1)^2-30*exp(1)+7 
5)/(exp(exp(x))^2+2*exp(1)*exp(exp(x))+exp(1)^2))+exp(exp(x))^3+3*exp(1)*e 
xp(exp(x))^2+3*exp(1)^2*exp(exp(x))+exp(1)^3)/(exp(exp(x))^3+3*exp(1)*exp( 
exp(x))^2+3*exp(1)^2*exp(exp(x))+exp(1)^3),x)
 

Output:

(e**((6*e**(e**x)*log(5) + 3*log(5)**2 + 6*log(5)*e + 75)/(e**(2*e**x) + 2 
*e**(e**x)*e + e**2))*e**3 + e**((30*e**(e**x) + 30*log(5) + 30*e)/(e**(2* 
e**x) + 2*e**(e**x)*e + e**2))*x)/e**((30*e**(e**x) + 30*log(5) + 30*e)/(e 
**(2*e**x) + 2*e**(e**x)*e + e**2))