Integrand size = 180, antiderivative size = 34 \[ \int \frac {\left (-5400+2160 x^3-216 x^6+e^{e^{\frac {x}{-20+4 x^3}}} \left (200-80 x^3+8 x^6\right )\right ) \log \left (\frac {1}{5} \left (27-e^{e^{\frac {x}{-20+4 x^3}}}\right )\right )+e^{e^{\frac {x}{-20+4 x^3}}+\frac {x}{-20+4 x^3}} \left (5 x+2 x^4\right ) \log \left (x^2\right )}{\left (-2700 x+1080 x^4-108 x^7+e^{e^{\frac {x}{-20+4 x^3}}} \left (100 x-40 x^4+4 x^7\right )\right ) \log ^2\left (\frac {1}{5} \left (27-e^{e^{\frac {x}{-20+4 x^3}}}\right )\right )} \, dx=\frac {\log \left (x^2\right )}{\log \left (5+\frac {1}{5} \left (2-e^{e^{\frac {x}{4 \left (-5+x^3\right )}}}\right )\right )} \]
Time = 0.36 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.94 \[ \int \frac {\left (-5400+2160 x^3-216 x^6+e^{e^{\frac {x}{-20+4 x^3}}} \left (200-80 x^3+8 x^6\right )\right ) \log \left (\frac {1}{5} \left (27-e^{e^{\frac {x}{-20+4 x^3}}}\right )\right )+e^{e^{\frac {x}{-20+4 x^3}}+\frac {x}{-20+4 x^3}} \left (5 x+2 x^4\right ) \log \left (x^2\right )}{\left (-2700 x+1080 x^4-108 x^7+e^{e^{\frac {x}{-20+4 x^3}}} \left (100 x-40 x^4+4 x^7\right )\right ) \log ^2\left (\frac {1}{5} \left (27-e^{e^{\frac {x}{-20+4 x^3}}}\right )\right )} \, dx=\frac {\log \left (x^2\right )}{\log \left (\frac {1}{5} \left (27-e^{e^{\frac {x}{4 \left (-5+x^3\right )}}}\right )\right )} \]
Integrate[((-5400 + 2160*x^3 - 216*x^6 + E^E^(x/(-20 + 4*x^3))*(200 - 80*x ^3 + 8*x^6))*Log[(27 - E^E^(x/(-20 + 4*x^3)))/5] + E^(E^(x/(-20 + 4*x^3)) + x/(-20 + 4*x^3))*(5*x + 2*x^4)*Log[x^2])/((-2700*x + 1080*x^4 - 108*x^7 + E^E^(x/(-20 + 4*x^3))*(100*x - 40*x^4 + 4*x^7))*Log[(27 - E^E^(x/(-20 + 4*x^3)))/5]^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.
| \(\displaystyle \int \frac {\left (-216 x^6+2160 x^3+e^{e^{\frac {x}{4 x^3-20}}} \left (8 x^6-80 x^3+200\right )-5400\right ) \log \left (\frac {1}{5} \left (27-e^{e^{\frac {x}{4 x^3-20}}}\right )\right )+e^{\frac {x}{4 x^3-20}+e^{\frac {x}{4 x^3-20}}} \left (2 x^4+5 x\right ) \log \left (x^2\right )}{\left (-108 x^7+1080 x^4+e^{e^{\frac {x}{4 x^3-20}}} \left (4 x^7-40 x^4+100 x\right )-2700 x\right ) \log ^2\left (\frac {1}{5} \left (27-e^{e^{\frac {x}{4 x^3-20}}}\right )\right )} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {-\left (\left (-216 x^6+2160 x^3+e^{e^{\frac {x}{4 x^3-20}}} \left (8 x^6-80 x^3+200\right )-5400\right ) \log \left (\frac {1}{5} \left (27-e^{e^{\frac {x}{4 x^3-20}}}\right )\right )\right )-e^{\frac {x}{4 x^3-20}+e^{\frac {x}{4 x^3-20}}} \left (2 x^4+5 x\right ) \log \left (x^2\right )}{4 \left (27-e^{e^{\frac {x}{4 \left (x^3-5\right )}}}\right ) x \left (5-x^3\right )^2 \log ^2\left (\frac {1}{5} \left (27-e^{e^{\frac {x}{4 x^3-20}}}\right )\right )}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{4} \int \frac {8 \left (27 x^6-270 x^3-e^{e^{-\frac {x}{4 \left (5-x^3\right )}}} \left (x^6-10 x^3+25\right )+675\right ) \log \left (\frac {1}{5} \left (27-e^{e^{-\frac {x}{4 \left (5-x^3\right )}}}\right )\right )-e^{e^{-\frac {x}{4 \left (5-x^3\right )}}-\frac {x}{4 \left (5-x^3\right )}} \left (2 x^4+5 x\right ) \log \left (x^2\right )}{\left (27-e^{e^{-\frac {x}{4 \left (5-x^3\right )}}}\right ) x \left (5-x^3\right )^2 \log ^2\left (\frac {1}{5} \left (27-e^{e^{-\frac {x}{4 \left (5-x^3\right )}}}\right )\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {1}{4} \int \left (\frac {e^{\frac {x}{4 \left (x^3-5\right )}+e^{\frac {x}{4 \left (x^3-5\right )}}} \left (2 x^3+5\right ) \log \left (x^2\right )}{\left (-27+e^{e^{\frac {x}{4 \left (x^3-5\right )}}}\right ) \left (x^3-5\right )^2 \log ^2\left (\frac {1}{5} \left (27-e^{e^{\frac {x}{4 \left (x^3-5\right )}}}\right )\right )}+\frac {8}{x \log \left (\frac {1}{5} \left (27-e^{e^{\frac {x}{4 \left (x^3-5\right )}}}\right )\right )}\right )dx\) |
| \(\Big \downarrow \) 7299 |
| \(\displaystyle \frac {1}{4} \int \left (\frac {e^{\frac {x}{4 \left (x^3-5\right )}+e^{\frac {x}{4 \left (x^3-5\right )}}} \left (2 x^3+5\right ) \log \left (x^2\right )}{\left (-27+e^{e^{\frac {x}{4 \left (x^3-5\right )}}}\right ) \left (x^3-5\right )^2 \log ^2\left (\frac {1}{5} \left (27-e^{e^{\frac {x}{4 \left (x^3-5\right )}}}\right )\right )}+\frac {8}{x \log \left (\frac {1}{5} \left (27-e^{e^{\frac {x}{4 \left (x^3-5\right )}}}\right )\right )}\right )dx\) |
Int[((-5400 + 2160*x^3 - 216*x^6 + E^E^(x/(-20 + 4*x^3))*(200 - 80*x^3 + 8 *x^6))*Log[(27 - E^E^(x/(-20 + 4*x^3)))/5] + E^(E^(x/(-20 + 4*x^3)) + x/(- 20 + 4*x^3))*(5*x + 2*x^4)*Log[x^2])/((-2700*x + 1080*x^4 - 108*x^7 + E^E^ (x/(-20 + 4*x^3))*(100*x - 40*x^4 + 4*x^7))*Log[(27 - E^E^(x/(-20 + 4*x^3) ))/5]^2),x]
3.5.32.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.51 (sec) , antiderivative size = 76, normalized size of antiderivative = 2.24
\[\frac {4 \ln \left (x \right )-i \pi \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right )+2 i \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{2}-i \pi \operatorname {csgn}\left (i x^{2}\right )^{3}}{2 \ln \left (-\frac {{\mathrm e}^{{\mathrm e}^{\frac {x}{4 x^{3}-20}}}}{5}+\frac {27}{5}\right )}\]
int((((8*x^6-80*x^3+200)*exp(exp(x/(4*x^3-20)))-216*x^6+2160*x^3-5400)*ln( -1/5*exp(exp(x/(4*x^3-20)))+27/5)+(2*x^4+5*x)*exp(x/(4*x^3-20))*ln(x^2)*ex p(exp(x/(4*x^3-20))))/((4*x^7-40*x^4+100*x)*exp(exp(x/(4*x^3-20)))-108*x^7 +1080*x^4-2700*x)/ln(-1/5*exp(exp(x/(4*x^3-20)))+27/5)^2,x)
1/2*(4*ln(x)-I*Pi*csgn(I*x)^2*csgn(I*x^2)+2*I*Pi*csgn(I*x)*csgn(I*x^2)^2-I *Pi*csgn(I*x^2)^3)/ln(-1/5*exp(exp(1/4*x/(x^3-5)))+27/5)
Leaf count of result is larger than twice the leaf count of optimal. 65 vs. \(2 (24) = 48\).
Time = 0.26 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.91 \[ \int \frac {\left (-5400+2160 x^3-216 x^6+e^{e^{\frac {x}{-20+4 x^3}}} \left (200-80 x^3+8 x^6\right )\right ) \log \left (\frac {1}{5} \left (27-e^{e^{\frac {x}{-20+4 x^3}}}\right )\right )+e^{e^{\frac {x}{-20+4 x^3}}+\frac {x}{-20+4 x^3}} \left (5 x+2 x^4\right ) \log \left (x^2\right )}{\left (-2700 x+1080 x^4-108 x^7+e^{e^{\frac {x}{-20+4 x^3}}} \left (100 x-40 x^4+4 x^7\right )\right ) \log ^2\left (\frac {1}{5} \left (27-e^{e^{\frac {x}{-20+4 x^3}}}\right )\right )} \, dx=\frac {\log \left (x^{2}\right )}{\log \left (-\frac {1}{5} \, {\left (e^{\left (\frac {4 \, {\left (x^{3} - 5\right )} e^{\left (\frac {x}{4 \, {\left (x^{3} - 5\right )}}\right )} + x}{4 \, {\left (x^{3} - 5\right )}}\right )} - 27 \, e^{\left (\frac {x}{4 \, {\left (x^{3} - 5\right )}}\right )}\right )} e^{\left (-\frac {x}{4 \, {\left (x^{3} - 5\right )}}\right )}\right )} \]
integrate((((8*x^6-80*x^3+200)*exp(exp(x/(4*x^3-20)))-216*x^6+2160*x^3-540 0)*log(-1/5*exp(exp(x/(4*x^3-20)))+27/5)+(2*x^4+5*x)*exp(x/(4*x^3-20))*log (x^2)*exp(exp(x/(4*x^3-20))))/((4*x^7-40*x^4+100*x)*exp(exp(x/(4*x^3-20))) -108*x^7+1080*x^4-2700*x)/log(-1/5*exp(exp(x/(4*x^3-20)))+27/5)^2,x, algor ithm=\
log(x^2)/log(-1/5*(e^(1/4*(4*(x^3 - 5)*e^(1/4*x/(x^3 - 5)) + x)/(x^3 - 5)) - 27*e^(1/4*x/(x^3 - 5)))*e^(-1/4*x/(x^3 - 5)))
Time = 10.87 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.65 \[ \int \frac {\left (-5400+2160 x^3-216 x^6+e^{e^{\frac {x}{-20+4 x^3}}} \left (200-80 x^3+8 x^6\right )\right ) \log \left (\frac {1}{5} \left (27-e^{e^{\frac {x}{-20+4 x^3}}}\right )\right )+e^{e^{\frac {x}{-20+4 x^3}}+\frac {x}{-20+4 x^3}} \left (5 x+2 x^4\right ) \log \left (x^2\right )}{\left (-2700 x+1080 x^4-108 x^7+e^{e^{\frac {x}{-20+4 x^3}}} \left (100 x-40 x^4+4 x^7\right )\right ) \log ^2\left (\frac {1}{5} \left (27-e^{e^{\frac {x}{-20+4 x^3}}}\right )\right )} \, dx=\frac {\log {\left (x^{2} \right )}}{\log {\left (\frac {27}{5} - \frac {e^{e^{\frac {x}{4 x^{3} - 20}}}}{5} \right )}} \]
integrate((((8*x**6-80*x**3+200)*exp(exp(x/(4*x**3-20)))-216*x**6+2160*x** 3-5400)*ln(-1/5*exp(exp(x/(4*x**3-20)))+27/5)+(2*x**4+5*x)*exp(x/(4*x**3-2 0))*ln(x**2)*exp(exp(x/(4*x**3-20))))/((4*x**7-40*x**4+100*x)*exp(exp(x/(4 *x**3-20)))-108*x**7+1080*x**4-2700*x)/ln(-1/5*exp(exp(x/(4*x**3-20)))+27/ 5)**2,x)
Time = 0.51 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.82 \[ \int \frac {\left (-5400+2160 x^3-216 x^6+e^{e^{\frac {x}{-20+4 x^3}}} \left (200-80 x^3+8 x^6\right )\right ) \log \left (\frac {1}{5} \left (27-e^{e^{\frac {x}{-20+4 x^3}}}\right )\right )+e^{e^{\frac {x}{-20+4 x^3}}+\frac {x}{-20+4 x^3}} \left (5 x+2 x^4\right ) \log \left (x^2\right )}{\left (-2700 x+1080 x^4-108 x^7+e^{e^{\frac {x}{-20+4 x^3}}} \left (100 x-40 x^4+4 x^7\right )\right ) \log ^2\left (\frac {1}{5} \left (27-e^{e^{\frac {x}{-20+4 x^3}}}\right )\right )} \, dx=-\frac {2 \, \log \left (x\right )}{\log \left (5\right ) - \log \left (-e^{\left (e^{\left (\frac {x}{4 \, {\left (x^{3} - 5\right )}}\right )}\right )} + 27\right )} \]
integrate((((8*x^6-80*x^3+200)*exp(exp(x/(4*x^3-20)))-216*x^6+2160*x^3-540 0)*log(-1/5*exp(exp(x/(4*x^3-20)))+27/5)+(2*x^4+5*x)*exp(x/(4*x^3-20))*log (x^2)*exp(exp(x/(4*x^3-20))))/((4*x^7-40*x^4+100*x)*exp(exp(x/(4*x^3-20))) -108*x^7+1080*x^4-2700*x)/log(-1/5*exp(exp(x/(4*x^3-20)))+27/5)^2,x, algor ithm=\
\[ \int \frac {\left (-5400+2160 x^3-216 x^6+e^{e^{\frac {x}{-20+4 x^3}}} \left (200-80 x^3+8 x^6\right )\right ) \log \left (\frac {1}{5} \left (27-e^{e^{\frac {x}{-20+4 x^3}}}\right )\right )+e^{e^{\frac {x}{-20+4 x^3}}+\frac {x}{-20+4 x^3}} \left (5 x+2 x^4\right ) \log \left (x^2\right )}{\left (-2700 x+1080 x^4-108 x^7+e^{e^{\frac {x}{-20+4 x^3}}} \left (100 x-40 x^4+4 x^7\right )\right ) \log ^2\left (\frac {1}{5} \left (27-e^{e^{\frac {x}{-20+4 x^3}}}\right )\right )} \, dx=\int { -\frac {{\left (2 \, x^{4} + 5 \, x\right )} e^{\left (\frac {x}{4 \, {\left (x^{3} - 5\right )}} + e^{\left (\frac {x}{4 \, {\left (x^{3} - 5\right )}}\right )}\right )} \log \left (x^{2}\right ) - 8 \, {\left (27 \, x^{6} - 270 \, x^{3} - {\left (x^{6} - 10 \, x^{3} + 25\right )} e^{\left (e^{\left (\frac {x}{4 \, {\left (x^{3} - 5\right )}}\right )}\right )} + 675\right )} \log \left (-\frac {1}{5} \, e^{\left (e^{\left (\frac {x}{4 \, {\left (x^{3} - 5\right )}}\right )}\right )} + \frac {27}{5}\right )}{4 \, {\left (27 \, x^{7} - 270 \, x^{4} - {\left (x^{7} - 10 \, x^{4} + 25 \, x\right )} e^{\left (e^{\left (\frac {x}{4 \, {\left (x^{3} - 5\right )}}\right )}\right )} + 675 \, x\right )} \log \left (-\frac {1}{5} \, e^{\left (e^{\left (\frac {x}{4 \, {\left (x^{3} - 5\right )}}\right )}\right )} + \frac {27}{5}\right )^{2}} \,d x } \]
integrate((((8*x^6-80*x^3+200)*exp(exp(x/(4*x^3-20)))-216*x^6+2160*x^3-540 0)*log(-1/5*exp(exp(x/(4*x^3-20)))+27/5)+(2*x^4+5*x)*exp(x/(4*x^3-20))*log (x^2)*exp(exp(x/(4*x^3-20))))/((4*x^7-40*x^4+100*x)*exp(exp(x/(4*x^3-20))) -108*x^7+1080*x^4-2700*x)/log(-1/5*exp(exp(x/(4*x^3-20)))+27/5)^2,x, algor ithm=\
integrate(-1/4*((2*x^4 + 5*x)*e^(1/4*x/(x^3 - 5) + e^(1/4*x/(x^3 - 5)))*lo g(x^2) - 8*(27*x^6 - 270*x^3 - (x^6 - 10*x^3 + 25)*e^(e^(1/4*x/(x^3 - 5))) + 675)*log(-1/5*e^(e^(1/4*x/(x^3 - 5))) + 27/5))/((27*x^7 - 270*x^4 - (x^ 7 - 10*x^4 + 25*x)*e^(e^(1/4*x/(x^3 - 5))) + 675*x)*log(-1/5*e^(e^(1/4*x/( x^3 - 5))) + 27/5)^2), x)
Timed out. \[ \int \frac {\left (-5400+2160 x^3-216 x^6+e^{e^{\frac {x}{-20+4 x^3}}} \left (200-80 x^3+8 x^6\right )\right ) \log \left (\frac {1}{5} \left (27-e^{e^{\frac {x}{-20+4 x^3}}}\right )\right )+e^{e^{\frac {x}{-20+4 x^3}}+\frac {x}{-20+4 x^3}} \left (5 x+2 x^4\right ) \log \left (x^2\right )}{\left (-2700 x+1080 x^4-108 x^7+e^{e^{\frac {x}{-20+4 x^3}}} \left (100 x-40 x^4+4 x^7\right )\right ) \log ^2\left (\frac {1}{5} \left (27-e^{e^{\frac {x}{-20+4 x^3}}}\right )\right )} \, dx=\int -\frac {\ln \left (\frac {27}{5}-\frac {{\mathrm {e}}^{{\mathrm {e}}^{\frac {x}{4\,x^3-20}}}}{5}\right )\,\left ({\mathrm {e}}^{{\mathrm {e}}^{\frac {x}{4\,x^3-20}}}\,\left (8\,x^6-80\,x^3+200\right )+2160\,x^3-216\,x^6-5400\right )+\ln \left (x^2\right )\,{\mathrm {e}}^{\frac {x}{4\,x^3-20}}\,{\mathrm {e}}^{{\mathrm {e}}^{\frac {x}{4\,x^3-20}}}\,\left (2\,x^4+5\,x\right )}{{\ln \left (\frac {27}{5}-\frac {{\mathrm {e}}^{{\mathrm {e}}^{\frac {x}{4\,x^3-20}}}}{5}\right )}^2\,\left (2700\,x-{\mathrm {e}}^{{\mathrm {e}}^{\frac {x}{4\,x^3-20}}}\,\left (4\,x^7-40\,x^4+100\,x\right )-1080\,x^4+108\,x^7\right )} \,d x \]
int(-(log(27/5 - exp(exp(x/(4*x^3 - 20)))/5)*(exp(exp(x/(4*x^3 - 20)))*(8* x^6 - 80*x^3 + 200) + 2160*x^3 - 216*x^6 - 5400) + log(x^2)*exp(x/(4*x^3 - 20))*exp(exp(x/(4*x^3 - 20)))*(5*x + 2*x^4))/(log(27/5 - exp(exp(x/(4*x^3 - 20)))/5)^2*(2700*x - exp(exp(x/(4*x^3 - 20)))*(100*x - 40*x^4 + 4*x^7) - 1080*x^4 + 108*x^7)),x)
int(-(log(27/5 - exp(exp(x/(4*x^3 - 20)))/5)*(exp(exp(x/(4*x^3 - 20)))*(8* x^6 - 80*x^3 + 200) + 2160*x^3 - 216*x^6 - 5400) + log(x^2)*exp(x/(4*x^3 - 20))*exp(exp(x/(4*x^3 - 20)))*(5*x + 2*x^4))/(log(27/5 - exp(exp(x/(4*x^3 - 20)))/5)^2*(2700*x - exp(exp(x/(4*x^3 - 20)))*(100*x - 40*x^4 + 4*x^7) - 1080*x^4 + 108*x^7)), x)