Integrand size = 234, antiderivative size = 30 \[ \int \frac {-10 x^2-2 e^3 x^2+e^{e^{\frac {5 x+25 x^4+5 e^3 x^4}{5+e^3}}+\frac {5 x+25 x^4+5 e^3 x^4}{5+e^3}} \left (5 x+100 x^4+20 e^3 x^4\right )+\left (e^{e^{\frac {5 x+25 x^4+5 e^3 x^4}{5+e^3}}} \left (-5-e^3\right )+5 x^2+e^3 x^2\right ) \log \left (-e^{e^{\frac {5 x+25 x^4+5 e^3 x^4}{5+e^3}}}+x^2\right )}{-5 x^4-e^3 x^4+e^{e^{\frac {5 x+25 x^4+5 e^3 x^4}{5+e^3}}} \left (5 x^2+e^3 x^2\right )} \, dx=\frac {\log \left (-e^{e^{5 \left (\frac {x}{5+e^3}+x^4\right )}}+x^2\right )}{x} \]
Time = 0.71 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.03 \[ \int \frac {-10 x^2-2 e^3 x^2+e^{e^{\frac {5 x+25 x^4+5 e^3 x^4}{5+e^3}}+\frac {5 x+25 x^4+5 e^3 x^4}{5+e^3}} \left (5 x+100 x^4+20 e^3 x^4\right )+\left (e^{e^{\frac {5 x+25 x^4+5 e^3 x^4}{5+e^3}}} \left (-5-e^3\right )+5 x^2+e^3 x^2\right ) \log \left (-e^{e^{\frac {5 x+25 x^4+5 e^3 x^4}{5+e^3}}}+x^2\right )}{-5 x^4-e^3 x^4+e^{e^{\frac {5 x+25 x^4+5 e^3 x^4}{5+e^3}}} \left (5 x^2+e^3 x^2\right )} \, dx=\frac {\log \left (-e^{e^{\frac {5 x}{5+e^3}+5 x^4}}+x^2\right )}{x} \]
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]
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 |
| \(\displaystyle -\frac {\int \left (\frac {5 \exp \left (5 x \left (x^3+\frac {1}{5+e^3}\right )+e^{5 x \left (x^3+\frac {1}{5+e^3}\right )}\right ) \left (-4 \left (5+e^3\right ) x^3-1\right )}{x \left (e^{e^{5 x \left (x^3+\frac {1}{5+e^3}\right )}}-x^2\right )}+\frac {\left (5+e^3\right ) \left (\log \left (x^2-e^{e^{5 x \left (x^3+\frac {1}{5+e^3}\right )}}\right ) x^2-2 x^2-e^{e^{5 x \left (x^3+\frac {1}{5+e^3}\right )}} \log \left (x^2-e^{e^{5 x \left (x^3+\frac {1}{5+e^3}\right )}}\right )\right )}{x^2 \left (x^2-e^{e^{5 x \left (x^3+\frac {1}{5+e^3}\right )}}\right )}\right )dx}{5+e^3}\) |
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]
3.5.26.3.1 Defintions of rubi rules used
Int[(u_.)*((v_.) + (a_.)*(Fx_) + (b_.)*(Fx_))^(p_.), x_Symbol] :> Int[u*(v + (a + b)*Fx)^p, x] /; FreeQ[{a, b}, x] && !FreeQ[Fx, x]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Time = 14.23 (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 | \(\frac {{\mathrm e}^{3} \ln \left (-{\mathrm e}^{{\mathrm e}^{\frac {5 x^{4} {\mathrm e}^{3}+25 x^{4}+5 x}{{\mathrm e}^{3}+5}}}+x^{2}\right )+5 \ln \left (-{\mathrm e}^{{\mathrm e}^{\frac {5 x^{4} {\mathrm e}^{3}+25 x^{4}+5 x}{{\mathrm e}^{3}+5}}}+x^{2}\right )}{x \left ({\mathrm e}^{3}+5\right )}\) | \(81\) |
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)
Leaf count of result is larger than twice the leaf count of optimal. 108 vs. \(2 (28) = 56\).
Time = 0.25 (sec) , antiderivative size = 108, normalized size of antiderivative = 3.60 \[ \int \frac {-10 x^2-2 e^3 x^2+e^{e^{\frac {5 x+25 x^4+5 e^3 x^4}{5+e^3}}+\frac {5 x+25 x^4+5 e^3 x^4}{5+e^3}} \left (5 x+100 x^4+20 e^3 x^4\right )+\left (e^{e^{\frac {5 x+25 x^4+5 e^3 x^4}{5+e^3}}} \left (-5-e^3\right )+5 x^2+e^3 x^2\right ) \log \left (-e^{e^{\frac {5 x+25 x^4+5 e^3 x^4}{5+e^3}}}+x^2\right )}{-5 x^4-e^3 x^4+e^{e^{\frac {5 x+25 x^4+5 e^3 x^4}{5+e^3}}} \left (5 x^2+e^3 x^2\right )} \, dx=\frac {\log \left ({\left (x^{2} e^{\left (\frac {5 \, {\left (x^{4} e^{3} + 5 \, x^{4} + x\right )}}{e^{3} + 5}\right )} - e^{\left (\frac {5 \, x^{4} e^{3} + 25 \, x^{4} + {\left (e^{3} + 5\right )} e^{\left (\frac {5 \, {\left (x^{4} e^{3} + 5 \, x^{4} + x\right )}}{e^{3} + 5}\right )} + 5 \, x}{e^{3} + 5}\right )}\right )} e^{\left (-\frac {5 \, {\left (x^{4} e^{3} + 5 \, x^{4} + x\right )}}{e^{3} + 5}\right )}\right )}{x} \]
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=\
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
Time = 5.16 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.03 \[ \int \frac {-10 x^2-2 e^3 x^2+e^{e^{\frac {5 x+25 x^4+5 e^3 x^4}{5+e^3}}+\frac {5 x+25 x^4+5 e^3 x^4}{5+e^3}} \left (5 x+100 x^4+20 e^3 x^4\right )+\left (e^{e^{\frac {5 x+25 x^4+5 e^3 x^4}{5+e^3}}} \left (-5-e^3\right )+5 x^2+e^3 x^2\right ) \log \left (-e^{e^{\frac {5 x+25 x^4+5 e^3 x^4}{5+e^3}}}+x^2\right )}{-5 x^4-e^3 x^4+e^{e^{\frac {5 x+25 x^4+5 e^3 x^4}{5+e^3}}} \left (5 x^2+e^3 x^2\right )} \, dx=\frac {\log {\left (x^{2} - e^{e^{\frac {25 x^{4} + 5 x^{4} e^{3} + 5 x}{5 + e^{3}}}} \right )}}{x} \]
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)
Time = 0.37 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.57 \[ \int \frac {-10 x^2-2 e^3 x^2+e^{e^{\frac {5 x+25 x^4+5 e^3 x^4}{5+e^3}}+\frac {5 x+25 x^4+5 e^3 x^4}{5+e^3}} \left (5 x+100 x^4+20 e^3 x^4\right )+\left (e^{e^{\frac {5 x+25 x^4+5 e^3 x^4}{5+e^3}}} \left (-5-e^3\right )+5 x^2+e^3 x^2\right ) \log \left (-e^{e^{\frac {5 x+25 x^4+5 e^3 x^4}{5+e^3}}}+x^2\right )}{-5 x^4-e^3 x^4+e^{e^{\frac {5 x+25 x^4+5 e^3 x^4}{5+e^3}}} \left (5 x^2+e^3 x^2\right )} \, dx=\frac {\log \left (x^{2} - e^{\left (e^{\left (\frac {5 \, x^{4} e^{3}}{e^{3} + 5} + \frac {25 \, x^{4}}{e^{3} + 5} + \frac {5 \, x}{e^{3} + 5}\right )}\right )}\right )}{x} \]
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=\
Leaf count of result is larger than twice the leaf count of optimal. 129 vs. \(2 (28) = 56\).
Time = 2.95 (sec) , antiderivative size = 129, normalized size of antiderivative = 4.30 \[ \int \frac {-10 x^2-2 e^3 x^2+e^{e^{\frac {5 x+25 x^4+5 e^3 x^4}{5+e^3}}+\frac {5 x+25 x^4+5 e^3 x^4}{5+e^3}} \left (5 x+100 x^4+20 e^3 x^4\right )+\left (e^{e^{\frac {5 x+25 x^4+5 e^3 x^4}{5+e^3}}} \left (-5-e^3\right )+5 x^2+e^3 x^2\right ) \log \left (-e^{e^{\frac {5 x+25 x^4+5 e^3 x^4}{5+e^3}}}+x^2\right )}{-5 x^4-e^3 x^4+e^{e^{\frac {5 x+25 x^4+5 e^3 x^4}{5+e^3}}} \left (5 x^2+e^3 x^2\right )} \, dx=\frac {\log \left ({\left (x^{2} e^{\left (\frac {5 \, {\left (x^{4} e^{3} + 5 \, x^{4} + x\right )}}{e^{3} + 5}\right )} - e^{\left (\frac {5 \, x^{4} e^{3} + 25 \, x^{4} + 5 \, x + 5 \, e^{\left (\frac {5 \, {\left (x^{4} e^{3} + 5 \, x^{4} + x\right )}}{e^{3} + 5}\right )} + e^{\left (\frac {5 \, {\left (x^{4} e^{3} + 5 \, x^{4} + x\right )}}{e^{3} + 5} + 3\right )}}{e^{3} + 5}\right )}\right )} e^{\left (-\frac {5 \, {\left (x^{4} e^{3} + 5 \, x^{4} + x\right )}}{e^{3} + 5}\right )}\right )}{x} \]
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=\
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 + 5*x + 5*e^(5*(x^4*e^3 + 5*x^4 + x)/(e^3 + 5)) + e^(5*(x^4*e^3 + 5*x^4 + x) /(e^3 + 5) + 3))/(e^3 + 5)))*e^(-5*(x^4*e^3 + 5*x^4 + x)/(e^3 + 5)))/x
Time = 8.84 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.63 \[ \int \frac {-10 x^2-2 e^3 x^2+e^{e^{\frac {5 x+25 x^4+5 e^3 x^4}{5+e^3}}+\frac {5 x+25 x^4+5 e^3 x^4}{5+e^3}} \left (5 x+100 x^4+20 e^3 x^4\right )+\left (e^{e^{\frac {5 x+25 x^4+5 e^3 x^4}{5+e^3}}} \left (-5-e^3\right )+5 x^2+e^3 x^2\right ) \log \left (-e^{e^{\frac {5 x+25 x^4+5 e^3 x^4}{5+e^3}}}+x^2\right )}{-5 x^4-e^3 x^4+e^{e^{\frac {5 x+25 x^4+5 e^3 x^4}{5+e^3}}} \left (5 x^2+e^3 x^2\right )} \, dx=\frac {\ln \left (x^2-{\mathrm {e}}^{{\mathrm {e}}^{\frac {5\,x^4\,{\mathrm {e}}^3}{{\mathrm {e}}^3+5}}\,{\mathrm {e}}^{\frac {5\,x}{{\mathrm {e}}^3+5}}\,{\mathrm {e}}^{\frac {25\,x^4}{{\mathrm {e}}^3+5}}}\right )}{x} \]
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)