\(\int \frac {-10 x^2-2 e^3 x^2+e^{e^{\frac {5 x+25 x^4+5 e^3 x^4}{5+e^3}}+\frac {5 x+25 x^4+5 e^3 x^4}{5+e^3}} (5 x+100 x^4+20 e^3 x^4)+(e^{e^{\frac {5 x+25 x^4+5 e^3 x^4}{5+e^3}}} (-5-e^3)+5 x^2+e^3 x^2) \log (-e^{e^{\frac {5 x+25 x^4+5 e^3 x^4}{5+e^3}}}+x^2)}{-5 x^4-e^3 x^4+e^{e^{\frac {5 x+25 x^4+5 e^3 x^4}{5+e^3}}} (5 x^2+e^3 x^2)} \, dx\) [426]

Optimal result

Integrand size = 234, antiderivative size = 30 \[ \int \frac {-10 x^2-2 e^3 x^2+e^{e^{\frac {5 x+25 x^4+5 e^3 x^4}{5+e^3}}+\frac {5 x+25 x^4+5 e^3 x^4}{5+e^3}} \left (5 x+100 x^4+20 e^3 x^4\right )+\left (e^{e^{\frac {5 x+25 x^4+5 e^3 x^4}{5+e^3}}} \left (-5-e^3\right )+5 x^2+e^3 x^2\right ) \log \left (-e^{e^{\frac {5 x+25 x^4+5 e^3 x^4}{5+e^3}}}+x^2\right )}{-5 x^4-e^3 x^4+e^{e^{\frac {5 x+25 x^4+5 e^3 x^4}{5+e^3}}} \left (5 x^2+e^3 x^2\right )} \, dx=\frac {\log \left (-e^{e^{5 \left (\frac {x}{5+e^3}+x^4\right )}}+x^2\right )}{x} \] Output:

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

Mathematica [A] (verified)

Time = 0.50 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.03 \[ \int \frac {-10 x^2-2 e^3 x^2+e^{e^{\frac {5 x+25 x^4+5 e^3 x^4}{5+e^3}}+\frac {5 x+25 x^4+5 e^3 x^4}{5+e^3}} \left (5 x+100 x^4+20 e^3 x^4\right )+\left (e^{e^{\frac {5 x+25 x^4+5 e^3 x^4}{5+e^3}}} \left (-5-e^3\right )+5 x^2+e^3 x^2\right ) \log \left (-e^{e^{\frac {5 x+25 x^4+5 e^3 x^4}{5+e^3}}}+x^2\right )}{-5 x^4-e^3 x^4+e^{e^{\frac {5 x+25 x^4+5 e^3 x^4}{5+e^3}}} \left (5 x^2+e^3 x^2\right )} \, dx=\frac {\log \left (-e^{e^{\frac {5 x}{5+e^3}+5 x^4}}+x^2\right )}{x} \] Input:

Integrate[(-10*x^2 - 2*E^3*x^2 + E^(E^((5*x + 25*x^4 + 5*E^3*x^4)/(5 + E^3 
)) + (5*x + 25*x^4 + 5*E^3*x^4)/(5 + E^3))*(5*x + 100*x^4 + 20*E^3*x^4) + 
(E^E^((5*x + 25*x^4 + 5*E^3*x^4)/(5 + E^3))*(-5 - E^3) + 5*x^2 + E^3*x^2)* 
Log[-E^E^((5*x + 25*x^4 + 5*E^3*x^4)/(5 + E^3)) + x^2])/(-5*x^4 - E^3*x^4 
+ E^E^((5*x + 25*x^4 + 5*E^3*x^4)/(5 + E^3))*(5*x^2 + E^3*x^2)),x]
 

Output:

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

\(\displaystyle \int \frac {\left (20 e^3 x^4+100 x^4+5 x\right ) \exp \left (\frac {5 e^3 x^4+25 x^4+5 x}{5+e^3}+e^{\frac {5 e^3 x^4+25 x^4+5 x}{5+e^3}}\right )-2 e^3 x^2-10 x^2+\left (\left (-5-e^3\right ) e^{e^{\frac {5 e^3 x^4+25 x^4+5 x}{5+e^3}}}+e^3 x^2+5 x^2\right ) \log \left (x^2-e^{e^{\frac {5 e^3 x^4+25 x^4+5 x}{5+e^3}}}\right )}{-e^3 x^4-5 x^4+e^{e^{\frac {5 e^3 x^4+25 x^4+5 x}{5+e^3}}} \left (e^3 x^2+5 x^2\right )} \, dx\)

\(\Big \downarrow \) 6

\(\displaystyle \int \frac {\left (20 e^3 x^4+100 x^4+5 x\right ) \exp \left (\frac {5 e^3 x^4+25 x^4+5 x}{5+e^3}+e^{\frac {5 e^3 x^4+25 x^4+5 x}{5+e^3}}\right )-2 e^3 x^2-10 x^2+\left (\left (-5-e^3\right ) e^{e^{\frac {5 e^3 x^4+25 x^4+5 x}{5+e^3}}}+e^3 x^2+5 x^2\right ) \log \left (x^2-e^{e^{\frac {5 e^3 x^4+25 x^4+5 x}{5+e^3}}}\right )}{\left (-5-e^3\right ) x^4+e^{e^{\frac {5 e^3 x^4+25 x^4+5 x}{5+e^3}}} \left (e^3 x^2+5 x^2\right )}dx\)

\(\Big \downarrow \) 6

\(\displaystyle \int \frac {\left (20 e^3 x^4+100 x^4+5 x\right ) \exp \left (\frac {5 e^3 x^4+25 x^4+5 x}{5+e^3}+e^{\frac {5 e^3 x^4+25 x^4+5 x}{5+e^3}}\right )+\left (-10-2 e^3\right ) x^2+\left (\left (-5-e^3\right ) e^{e^{\frac {5 e^3 x^4+25 x^4+5 x}{5+e^3}}}+e^3 x^2+5 x^2\right ) \log \left (x^2-e^{e^{\frac {5 e^3 x^4+25 x^4+5 x}{5+e^3}}}\right )}{\left (-5-e^3\right ) x^4+e^{e^{\frac {5 e^3 x^4+25 x^4+5 x}{5+e^3}}} \left (e^3 x^2+5 x^2\right )}dx\)

\(\Big \downarrow \) 7292

\(\displaystyle \int \frac {\left (20 e^3 x^4+100 x^4+5 x\right ) \exp \left (\frac {5 e^3 x^4+25 x^4+5 x}{5+e^3}+e^{\frac {5 e^3 x^4+25 x^4+5 x}{5+e^3}}\right )+\left (-10-2 e^3\right ) x^2+\left (\left (-5-e^3\right ) e^{e^{\frac {5 e^3 x^4+25 x^4+5 x}{5+e^3}}}+e^3 x^2+5 x^2\right ) \log \left (x^2-e^{e^{\frac {5 e^3 x^4+25 x^4+5 x}{5+e^3}}}\right )}{\left (5+e^3\right ) x^2 \left (e^{e^{5 x \left (x^3+\frac {1}{5+e^3}\right )}}-x^2\right )}dx\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int -\frac {2 \left (5+e^3\right ) x^2-5 \exp \left (\frac {5 \left (e^3 x^4+5 x^4+x\right )}{5+e^3}+e^{\frac {5 \left (e^3 x^4+5 x^4+x\right )}{5+e^3}}\right ) \left (4 e^3 x^4+20 x^4+x\right )+\left (-e^3 x^2-5 x^2+e^{e^{\frac {5 \left (e^3 x^4+5 x^4+x\right )}{5+e^3}}} \left (5+e^3\right )\right ) \log \left (x^2-e^{e^{\frac {5 \left (e^3 x^4+5 x^4+x\right )}{5+e^3}}}\right )}{x^2 \left (e^{e^{5 x \left (x^3+\frac {1}{5+e^3}\right )}}-x^2\right )}dx}{5+e^3}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {\int \frac {2 \left (5+e^3\right ) x^2-5 \exp \left (\frac {5 \left (e^3 x^4+5 x^4+x\right )}{5+e^3}+e^{\frac {5 \left (e^3 x^4+5 x^4+x\right )}{5+e^3}}\right ) \left (4 e^3 x^4+20 x^4+x\right )+\left (-e^3 x^2-5 x^2+e^{e^{\frac {5 \left (e^3 x^4+5 x^4+x\right )}{5+e^3}}} \left (5+e^3\right )\right ) \log \left (x^2-e^{e^{\frac {5 \left (e^3 x^4+5 x^4+x\right )}{5+e^3}}}\right )}{x^2 \left (e^{e^{5 x \left (x^3+\frac {1}{5+e^3}\right )}}-x^2\right )}dx}{5+e^3}\)

\(\Big \downarrow \) 7293

\(\displaystyle -\frac {\int \left (\frac {5 \exp \left (\frac {25 \left (1+\frac {e^3}{5}\right ) x^4+5 x+5 e^{5 x \left (x^3+\frac {1}{5+e^3}\right )} \left (1+\frac {e^3}{5}\right )}{5+e^3}\right ) \left (-4 \left (5+e^3\right ) x^3-1\right )}{x \left (e^{e^{5 x \left (x^3+\frac {1}{5+e^3}\right )}}-x^2\right )}+\frac {\left (5+e^3\right ) \left (\log \left (x^2-e^{e^{5 x \left (x^3+\frac {1}{5+e^3}\right )}}\right ) x^2-2 x^2-e^{e^{5 x \left (x^3+\frac {1}{5+e^3}\right )}} \log \left (x^2-e^{e^{5 x \left (x^3+\frac {1}{5+e^3}\right )}}\right )\right )}{x^2 \left (x^2-e^{e^{5 x \left (x^3+\frac {1}{5+e^3}\right )}}\right )}\right )dx}{5+e^3}\)

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 7299

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

Input:

Int[(-10*x^2 - 2*E^3*x^2 + E^(E^((5*x + 25*x^4 + 5*E^3*x^4)/(5 + E^3)) + ( 
5*x + 25*x^4 + 5*E^3*x^4)/(5 + E^3))*(5*x + 100*x^4 + 20*E^3*x^4) + (E^E^( 
(5*x + 25*x^4 + 5*E^3*x^4)/(5 + E^3))*(-5 - E^3) + 5*x^2 + E^3*x^2)*Log[-E 
^E^((5*x + 25*x^4 + 5*E^3*x^4)/(5 + E^3)) + x^2])/(-5*x^4 - E^3*x^4 + E^E^ 
((5*x + 25*x^4 + 5*E^3*x^4)/(5 + E^3))*(5*x^2 + E^3*x^2)),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 17.42 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.20

method	result	size
risch	\(\frac {\ln \left (-{\mathrm e}^{{\mathrm e}^{\frac {5 x \left (x^{3} {\mathrm e}^{3}+5 x^{3}+1\right )}{{\mathrm e}^{3}+5}}}+x^{2}\right )}{x}\)	\(36\)
parallelrisch	\(\frac {{\mathrm e}^{3} \ln \left (-{\mathrm e}^{{\mathrm e}^{\frac {5 x^{4} {\mathrm e}^{3}+25 x^{4}+5 x}{{\mathrm e}^{3}+5}}}+x^{2}\right )+5 \ln \left (-{\mathrm e}^{{\mathrm e}^{\frac {5 x^{4} {\mathrm e}^{3}+25 x^{4}+5 x}{{\mathrm e}^{3}+5}}}+x^{2}\right )}{x \left ({\mathrm e}^{3}+5\right )}\)	\(81\)

Input:

int((((-exp(3)-5)*exp(exp((5*x^4*exp(3)+25*x^4+5*x)/(exp(3)+5)))+x^2*exp(3 
)+5*x^2)*ln(-exp(exp((5*x^4*exp(3)+25*x^4+5*x)/(exp(3)+5)))+x^2)+(20*x^4*e 
xp(3)+100*x^4+5*x)*exp((5*x^4*exp(3)+25*x^4+5*x)/(exp(3)+5))*exp(exp((5*x^ 
4*exp(3)+25*x^4+5*x)/(exp(3)+5)))-2*x^2*exp(3)-10*x^2)/((x^2*exp(3)+5*x^2) 
*exp(exp((5*x^4*exp(3)+25*x^4+5*x)/(exp(3)+5)))-x^4*exp(3)-5*x^4),x,method 
=_RETURNVERBOSE)
 

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 108 vs. \(2 (28) = 56\).

Time = 0.08 (sec) , antiderivative size = 108, normalized size of antiderivative = 3.60 \[ \int \frac {-10 x^2-2 e^3 x^2+e^{e^{\frac {5 x+25 x^4+5 e^3 x^4}{5+e^3}}+\frac {5 x+25 x^4+5 e^3 x^4}{5+e^3}} \left (5 x+100 x^4+20 e^3 x^4\right )+\left (e^{e^{\frac {5 x+25 x^4+5 e^3 x^4}{5+e^3}}} \left (-5-e^3\right )+5 x^2+e^3 x^2\right ) \log \left (-e^{e^{\frac {5 x+25 x^4+5 e^3 x^4}{5+e^3}}}+x^2\right )}{-5 x^4-e^3 x^4+e^{e^{\frac {5 x+25 x^4+5 e^3 x^4}{5+e^3}}} \left (5 x^2+e^3 x^2\right )} \, dx=\frac {\log \left ({\left (x^{2} e^{\left (\frac {5 \, {\left (x^{4} e^{3} + 5 \, x^{4} + x\right )}}{e^{3} + 5}\right )} - e^{\left (\frac {5 \, x^{4} e^{3} + 25 \, x^{4} + {\left (e^{3} + 5\right )} e^{\left (\frac {5 \, {\left (x^{4} e^{3} + 5 \, x^{4} + x\right )}}{e^{3} + 5}\right )} + 5 \, x}{e^{3} + 5}\right )}\right )} e^{\left (-\frac {5 \, {\left (x^{4} e^{3} + 5 \, x^{4} + x\right )}}{e^{3} + 5}\right )}\right )}{x} \] Input:

integrate((((-exp(3)-5)*exp(exp((5*x^4*exp(3)+25*x^4+5*x)/(exp(3)+5)))+x^2 
*exp(3)+5*x^2)*log(-exp(exp((5*x^4*exp(3)+25*x^4+5*x)/(exp(3)+5)))+x^2)+(2 
0*x^4*exp(3)+100*x^4+5*x)*exp((5*x^4*exp(3)+25*x^4+5*x)/(exp(3)+5))*exp(ex 
p((5*x^4*exp(3)+25*x^4+5*x)/(exp(3)+5)))-2*x^2*exp(3)-10*x^2)/((x^2*exp(3) 
+5*x^2)*exp(exp((5*x^4*exp(3)+25*x^4+5*x)/(exp(3)+5)))-x^4*exp(3)-5*x^4),x 
, algorithm="fricas")
 

Output:

log((x^2*e^(5*(x^4*e^3 + 5*x^4 + x)/(e^3 + 5)) - e^((5*x^4*e^3 + 25*x^4 + 
(e^3 + 5)*e^(5*(x^4*e^3 + 5*x^4 + x)/(e^3 + 5)) + 5*x)/(e^3 + 5)))*e^(-5*( 
x^4*e^3 + 5*x^4 + x)/(e^3 + 5)))/x
 

Sympy [A] (verification not implemented)

Time = 5.57 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.03 \[ \int \frac {-10 x^2-2 e^3 x^2+e^{e^{\frac {5 x+25 x^4+5 e^3 x^4}{5+e^3}}+\frac {5 x+25 x^4+5 e^3 x^4}{5+e^3}} \left (5 x+100 x^4+20 e^3 x^4\right )+\left (e^{e^{\frac {5 x+25 x^4+5 e^3 x^4}{5+e^3}}} \left (-5-e^3\right )+5 x^2+e^3 x^2\right ) \log \left (-e^{e^{\frac {5 x+25 x^4+5 e^3 x^4}{5+e^3}}}+x^2\right )}{-5 x^4-e^3 x^4+e^{e^{\frac {5 x+25 x^4+5 e^3 x^4}{5+e^3}}} \left (5 x^2+e^3 x^2\right )} \, dx=\frac {\log {\left (x^{2} - e^{e^{\frac {25 x^{4} + 5 x^{4} e^{3} + 5 x}{5 + e^{3}}}} \right )}}{x} \] Input:

integrate((((-exp(3)-5)*exp(exp((5*x**4*exp(3)+25*x**4+5*x)/(exp(3)+5)))+x 
**2*exp(3)+5*x**2)*ln(-exp(exp((5*x**4*exp(3)+25*x**4+5*x)/(exp(3)+5)))+x* 
*2)+(20*x**4*exp(3)+100*x**4+5*x)*exp((5*x**4*exp(3)+25*x**4+5*x)/(exp(3)+ 
5))*exp(exp((5*x**4*exp(3)+25*x**4+5*x)/(exp(3)+5)))-2*x**2*exp(3)-10*x**2 
)/((x**2*exp(3)+5*x**2)*exp(exp((5*x**4*exp(3)+25*x**4+5*x)/(exp(3)+5)))-x 
**4*exp(3)-5*x**4),x)
 

Output:

log(x**2 - exp(exp((25*x**4 + 5*x**4*exp(3) + 5*x)/(5 + exp(3)))))/x
 

Maxima [A] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.57 \[ \int \frac {-10 x^2-2 e^3 x^2+e^{e^{\frac {5 x+25 x^4+5 e^3 x^4}{5+e^3}}+\frac {5 x+25 x^4+5 e^3 x^4}{5+e^3}} \left (5 x+100 x^4+20 e^3 x^4\right )+\left (e^{e^{\frac {5 x+25 x^4+5 e^3 x^4}{5+e^3}}} \left (-5-e^3\right )+5 x^2+e^3 x^2\right ) \log \left (-e^{e^{\frac {5 x+25 x^4+5 e^3 x^4}{5+e^3}}}+x^2\right )}{-5 x^4-e^3 x^4+e^{e^{\frac {5 x+25 x^4+5 e^3 x^4}{5+e^3}}} \left (5 x^2+e^3 x^2\right )} \, dx=\frac {\log \left (x^{2} - e^{\left (e^{\left (\frac {5 \, x^{4} e^{3}}{e^{3} + 5} + \frac {25 \, x^{4}}{e^{3} + 5} + \frac {5 \, x}{e^{3} + 5}\right )}\right )}\right )}{x} \] Input:

integrate((((-exp(3)-5)*exp(exp((5*x^4*exp(3)+25*x^4+5*x)/(exp(3)+5)))+x^2 
*exp(3)+5*x^2)*log(-exp(exp((5*x^4*exp(3)+25*x^4+5*x)/(exp(3)+5)))+x^2)+(2 
0*x^4*exp(3)+100*x^4+5*x)*exp((5*x^4*exp(3)+25*x^4+5*x)/(exp(3)+5))*exp(ex 
p((5*x^4*exp(3)+25*x^4+5*x)/(exp(3)+5)))-2*x^2*exp(3)-10*x^2)/((x^2*exp(3) 
+5*x^2)*exp(exp((5*x^4*exp(3)+25*x^4+5*x)/(exp(3)+5)))-x^4*exp(3)-5*x^4),x 
, algorithm="maxima")
 

Output:

log(x^2 - e^(e^(5*x^4*e^3/(e^3 + 5) + 25*x^4/(e^3 + 5) + 5*x/(e^3 + 5))))/ 
x
 

Giac [F(-2)]

Exception generated. \[ \int \frac {-10 x^2-2 e^3 x^2+e^{e^{\frac {5 x+25 x^4+5 e^3 x^4}{5+e^3}}+\frac {5 x+25 x^4+5 e^3 x^4}{5+e^3}} \left (5 x+100 x^4+20 e^3 x^4\right )+\left (e^{e^{\frac {5 x+25 x^4+5 e^3 x^4}{5+e^3}}} \left (-5-e^3\right )+5 x^2+e^3 x^2\right ) \log \left (-e^{e^{\frac {5 x+25 x^4+5 e^3 x^4}{5+e^3}}}+x^2\right )}{-5 x^4-e^3 x^4+e^{e^{\frac {5 x+25 x^4+5 e^3 x^4}{5+e^3}}} \left (5 x^2+e^3 x^2\right )} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((((-exp(3)-5)*exp(exp((5*x^4*exp(3)+25*x^4+5*x)/(exp(3)+5)))+x^2 
*exp(3)+5*x^2)*log(-exp(exp((5*x^4*exp(3)+25*x^4+5*x)/(exp(3)+5)))+x^2)+(2 
0*x^4*exp(3)+100*x^4+5*x)*exp((5*x^4*exp(3)+25*x^4+5*x)/(exp(3)+5))*exp(ex 
p((5*x^4*exp(3)+25*x^4+5*x)/(exp(3)+5)))-2*x^2*exp(3)-10*x^2)/((x^2*exp(3) 
+5*x^2)*exp(exp((5*x^4*exp(3)+25*x^4+5*x)/(exp(3)+5)))-x^4*exp(3)-5*x^4),x 
, algorithm="giac")
 

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Unable to divide, perhaps due to ro 
unding error%%%{8000,[0,2,18,3,15]%%%}+%%%{200000,[0,2,18,3,12]%%%}+%%%{20 
00000,[0,
 

Mupad [B] (verification not implemented)

Time = 0.83 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.63 \[ \int \frac {-10 x^2-2 e^3 x^2+e^{e^{\frac {5 x+25 x^4+5 e^3 x^4}{5+e^3}}+\frac {5 x+25 x^4+5 e^3 x^4}{5+e^3}} \left (5 x+100 x^4+20 e^3 x^4\right )+\left (e^{e^{\frac {5 x+25 x^4+5 e^3 x^4}{5+e^3}}} \left (-5-e^3\right )+5 x^2+e^3 x^2\right ) \log \left (-e^{e^{\frac {5 x+25 x^4+5 e^3 x^4}{5+e^3}}}+x^2\right )}{-5 x^4-e^3 x^4+e^{e^{\frac {5 x+25 x^4+5 e^3 x^4}{5+e^3}}} \left (5 x^2+e^3 x^2\right )} \, dx=\frac {\ln \left (x^2-{\mathrm {e}}^{{\mathrm {e}}^{\frac {5\,x^4\,{\mathrm {e}}^3}{{\mathrm {e}}^3+5}}\,{\mathrm {e}}^{\frac {5\,x}{{\mathrm {e}}^3+5}}\,{\mathrm {e}}^{\frac {25\,x^4}{{\mathrm {e}}^3+5}}}\right )}{x} \] Input:

int(-(log(x^2 - exp(exp((5*x + 5*x^4*exp(3) + 25*x^4)/(exp(3) + 5))))*(x^2 
*exp(3) - exp(exp((5*x + 5*x^4*exp(3) + 25*x^4)/(exp(3) + 5)))*(exp(3) + 5 
) + 5*x^2) - 2*x^2*exp(3) - 10*x^2 + exp(exp((5*x + 5*x^4*exp(3) + 25*x^4) 
/(exp(3) + 5)))*exp((5*x + 5*x^4*exp(3) + 25*x^4)/(exp(3) + 5))*(5*x + 20* 
x^4*exp(3) + 100*x^4))/(x^4*exp(3) + 5*x^4 - exp(exp((5*x + 5*x^4*exp(3) + 
 25*x^4)/(exp(3) + 5)))*(x^2*exp(3) + 5*x^2)),x)
 

Output:

log(x^2 - exp(exp((5*x^4*exp(3))/(exp(3) + 5))*exp((5*x)/(exp(3) + 5))*exp 
((25*x^4)/(exp(3) + 5))))/x
 

Reduce [B] (verification not implemented)

Time = 0.35 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.33 \[ \int \frac {-10 x^2-2 e^3 x^2+e^{e^{\frac {5 x+25 x^4+5 e^3 x^4}{5+e^3}}+\frac {5 x+25 x^4+5 e^3 x^4}{5+e^3}} \left (5 x+100 x^4+20 e^3 x^4\right )+\left (e^{e^{\frac {5 x+25 x^4+5 e^3 x^4}{5+e^3}}} \left (-5-e^3\right )+5 x^2+e^3 x^2\right ) \log \left (-e^{e^{\frac {5 x+25 x^4+5 e^3 x^4}{5+e^3}}}+x^2\right )}{-5 x^4-e^3 x^4+e^{e^{\frac {5 x+25 x^4+5 e^3 x^4}{5+e^3}}} \left (5 x^2+e^3 x^2\right )} \, dx=\frac {\mathrm {log}\left (-e^{e^{\frac {5 e^{3} x^{4}+25 x^{4}+5 x}{e^{3}+5}}}+x^{2}\right )}{x} \] Input:

int((((-exp(3)-5)*exp(exp((5*x^4*exp(3)+25*x^4+5*x)/(exp(3)+5)))+x^2*exp(3 
)+5*x^2)*log(-exp(exp((5*x^4*exp(3)+25*x^4+5*x)/(exp(3)+5)))+x^2)+(20*x^4* 
exp(3)+100*x^4+5*x)*exp((5*x^4*exp(3)+25*x^4+5*x)/(exp(3)+5))*exp(exp((5*x 
^4*exp(3)+25*x^4+5*x)/(exp(3)+5)))-2*x^2*exp(3)-10*x^2)/((x^2*exp(3)+5*x^2 
)*exp(exp((5*x^4*exp(3)+25*x^4+5*x)/(exp(3)+5)))-x^4*exp(3)-5*x^4),x)
 

Output:

log( - e**(e**((5*e**3*x**4 + 25*x**4 + 5*x)/(e**3 + 5))) + x**2)/x