Integrand size = 185, antiderivative size = 27 \[ \int \frac {13 e^3 x+52 x^2+e^x \left (24 x+12 e^3 x+24 x^2\right )+\left (e^x \left (-12 e^3-24 x\right )-13 e^3 x-26 x^2\right ) \log \left (\frac {1}{3} \left (13 e^3 x+26 x^2+e^x \left (12 e^3+24 x\right )\right )\right ) \log \left (\log \left (\frac {1}{3} \left (13 e^3 x+26 x^2+e^x \left (12 e^3+24 x\right )\right )\right )\right )}{\left (13 e^3 x^3+26 x^4+e^x \left (12 e^3 x^2+24 x^3\right )\right ) \log \left (\frac {1}{3} \left (13 e^3 x+26 x^2+e^x \left (12 e^3+24 x\right )\right )\right )} \, dx=\frac {\log \left (\log \left (\left (e^3+2 x\right ) \left (\frac {x}{3}+4 \left (e^x+x\right )\right )\right )\right )}{x} \]
Time = 0.12 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.96 \[ \int \frac {13 e^3 x+52 x^2+e^x \left (24 x+12 e^3 x+24 x^2\right )+\left (e^x \left (-12 e^3-24 x\right )-13 e^3 x-26 x^2\right ) \log \left (\frac {1}{3} \left (13 e^3 x+26 x^2+e^x \left (12 e^3+24 x\right )\right )\right ) \log \left (\log \left (\frac {1}{3} \left (13 e^3 x+26 x^2+e^x \left (12 e^3+24 x\right )\right )\right )\right )}{\left (13 e^3 x^3+26 x^4+e^x \left (12 e^3 x^2+24 x^3\right )\right ) \log \left (\frac {1}{3} \left (13 e^3 x+26 x^2+e^x \left (12 e^3+24 x\right )\right )\right )} \, dx=\frac {\log \left (\log \left (\frac {1}{3} \left (e^3+2 x\right ) \left (12 e^x+13 x\right )\right )\right )}{x} \]
Integrate[(13*E^3*x + 52*x^2 + E^x*(24*x + 12*E^3*x + 24*x^2) + (E^x*(-12* E^3 - 24*x) - 13*E^3*x - 26*x^2)*Log[(13*E^3*x + 26*x^2 + E^x*(12*E^3 + 24 *x))/3]*Log[Log[(13*E^3*x + 26*x^2 + E^x*(12*E^3 + 24*x))/3]])/((13*E^3*x^ 3 + 26*x^4 + E^x*(12*E^3*x^2 + 24*x^3))*Log[(13*E^3*x + 26*x^2 + E^x*(12*E ^3 + 24*x))/3]),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.
| \(\displaystyle \int \frac {52 x^2+e^x \left (24 x^2+12 e^3 x+24 x\right )+\left (-26 x^2-13 e^3 x+e^x \left (-24 x-12 e^3\right )\right ) \log \left (\frac {1}{3} \left (26 x^2+13 e^3 x+e^x \left (24 x+12 e^3\right )\right )\right ) \log \left (\log \left (\frac {1}{3} \left (26 x^2+13 e^3 x+e^x \left (24 x+12 e^3\right )\right )\right )\right )+13 e^3 x}{\left (26 x^4+13 e^3 x^3+e^x \left (24 x^3+12 e^3 x^2\right )\right ) \log \left (\frac {1}{3} \left (26 x^2+13 e^3 x+e^x \left (24 x+12 e^3\right )\right )\right )} \, dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\frac {x \left (52 x+12 e^{x+3}+24 e^x (x+1)+13 e^3\right )}{\left (2 x+e^3\right ) \left (13 x+12 e^x\right ) \log \left (\frac {1}{3} \left (2 x+e^3\right ) \left (13 x+12 e^x\right )\right )}-\log \left (\log \left (\frac {1}{3} \left (2 x+e^3\right ) \left (13 x+12 e^x\right )\right )\right )}{x^2}dx\) |
| \(\Big \downarrow \) 2010 |
| \(\displaystyle \int \left (\frac {2 x^2+2 \left (1+\frac {e^3}{2}\right ) x-2 x \log \left (\frac {1}{3} \left (2 x+e^3\right ) \left (13 x+12 e^x\right )\right ) \log \left (\log \left (\frac {1}{3} \left (2 x+e^3\right ) \left (13 x+12 e^x\right )\right )\right )-e^3 \log \left (\frac {1}{3} \left (2 x+e^3\right ) \left (13 x+12 e^x\right )\right ) \log \left (\log \left (\frac {1}{3} \left (2 x+e^3\right ) \left (13 x+12 e^x\right )\right )\right )}{x^2 \left (2 x+e^3\right ) \log \left (\frac {1}{3} \left (2 x+e^3\right ) \left (13 x+12 e^x\right )\right )}-\frac {13 (x-1)}{x \left (13 x+12 e^x\right ) \log \left (\frac {1}{3} \left (2 x+e^3\right ) \left (13 x+12 e^x\right )\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\int \frac {\log \left (\log \left (\frac {1}{3} \left (2 x+e^3\right ) \left (13 x+12 e^x\right )\right )\right )}{x^2}dx+\frac {\left (2+e^3\right ) \int \frac {1}{x \log \left (\frac {1}{3} \left (2 x+e^3\right ) \left (13 x+12 e^x\right )\right )}dx}{e^3}-\frac {4 \int \frac {1}{\left (2 x+e^3\right ) \log \left (\frac {1}{3} \left (2 x+e^3\right ) \left (13 x+12 e^x\right )\right )}dx}{e^3}-13 \int \frac {1}{\left (13 x+12 e^x\right ) \log \left (\frac {1}{3} \left (2 x+e^3\right ) \left (13 x+12 e^x\right )\right )}dx+13 \int \frac {1}{x \left (13 x+12 e^x\right ) \log \left (\frac {1}{3} \left (2 x+e^3\right ) \left (13 x+12 e^x\right )\right )}dx\) |
Int[(13*E^3*x + 52*x^2 + E^x*(24*x + 12*E^3*x + 24*x^2) + (E^x*(-12*E^3 - 24*x) - 13*E^3*x - 26*x^2)*Log[(13*E^3*x + 26*x^2 + E^x*(12*E^3 + 24*x))/3 ]*Log[Log[(13*E^3*x + 26*x^2 + E^x*(12*E^3 + 24*x))/3]])/((13*E^3*x^3 + 26 *x^4 + E^x*(12*E^3*x^2 + 24*x^3))*Log[(13*E^3*x + 26*x^2 + E^x*(12*E^3 + 2 4*x))/3]),x]
3.5.36.3.1 Defintions of rubi rules used
Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x] , x] /; FreeQ[{c, m}, x] && SumQ[u] && !LinearQ[u, x] && !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Time = 15.95 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.11
| method | result | size |
| parallelrisch | \(\frac {\ln \left (\ln \left (\frac {\left (12 \,{\mathrm e}^{3}+24 x \right ) {\mathrm e}^{x}}{3}+\frac {13 x \,{\mathrm e}^{3}}{3}+\frac {26 x^{2}}{3}\right )\right )}{x}\) | \(30\) |
| risch | \(\frac {\ln \left (-\ln \left (3\right )+\ln \left (x +\frac {12 \,{\mathrm e}^{x}}{13}\right )+\ln \left (2 x +{\mathrm e}^{3}\right )-\frac {i \pi \,\operatorname {csgn}\left (i \left (x +\frac {12 \,{\mathrm e}^{x}}{13}\right ) \left (2 x +{\mathrm e}^{3}\right )\right ) \left (-\operatorname {csgn}\left (i \left (x +\frac {12 \,{\mathrm e}^{x}}{13}\right ) \left (2 x +{\mathrm e}^{3}\right )\right )+\operatorname {csgn}\left (i \left (x +\frac {12 \,{\mathrm e}^{x}}{13}\right )\right )\right ) \left (-\operatorname {csgn}\left (i \left (x +\frac {12 \,{\mathrm e}^{x}}{13}\right ) \left (2 x +{\mathrm e}^{3}\right )\right )+\operatorname {csgn}\left (i \left (2 x +{\mathrm e}^{3}\right )\right )\right )}{2}\right )}{x}\) | \(103\) |
int((((-12*exp(3)-24*x)*exp(x)-13*x*exp(3)-26*x^2)*ln(1/3*(12*exp(3)+24*x) *exp(x)+13/3*x*exp(3)+26/3*x^2)*ln(ln(1/3*(12*exp(3)+24*x)*exp(x)+13/3*x*e xp(3)+26/3*x^2))+(12*x*exp(3)+24*x^2+24*x)*exp(x)+13*x*exp(3)+52*x^2)/((12 *x^2*exp(3)+24*x^3)*exp(x)+13*x^3*exp(3)+26*x^4)/ln(1/3*(12*exp(3)+24*x)*e xp(x)+13/3*x*exp(3)+26/3*x^2),x,method=_RETURNVERBOSE)
Time = 0.24 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00 \[ \int \frac {13 e^3 x+52 x^2+e^x \left (24 x+12 e^3 x+24 x^2\right )+\left (e^x \left (-12 e^3-24 x\right )-13 e^3 x-26 x^2\right ) \log \left (\frac {1}{3} \left (13 e^3 x+26 x^2+e^x \left (12 e^3+24 x\right )\right )\right ) \log \left (\log \left (\frac {1}{3} \left (13 e^3 x+26 x^2+e^x \left (12 e^3+24 x\right )\right )\right )\right )}{\left (13 e^3 x^3+26 x^4+e^x \left (12 e^3 x^2+24 x^3\right )\right ) \log \left (\frac {1}{3} \left (13 e^3 x+26 x^2+e^x \left (12 e^3+24 x\right )\right )\right )} \, dx=\frac {\log \left (\log \left (\frac {26}{3} \, x^{2} + \frac {13}{3} \, x e^{3} + 4 \, {\left (2 \, x + e^{3}\right )} e^{x}\right )\right )}{x} \]
integrate((((-12*exp(3)-24*x)*exp(x)-13*x*exp(3)-26*x^2)*log(1/3*(12*exp(3 )+24*x)*exp(x)+13/3*x*exp(3)+26/3*x^2)*log(log(1/3*(12*exp(3)+24*x)*exp(x) +13/3*x*exp(3)+26/3*x^2))+(12*x*exp(3)+24*x^2+24*x)*exp(x)+13*x*exp(3)+52* x^2)/((12*x^2*exp(3)+24*x^3)*exp(x)+13*x^3*exp(3)+26*x^4)/log(1/3*(12*exp( 3)+24*x)*exp(x)+13/3*x*exp(3)+26/3*x^2),x, algorithm=\
Time = 2.06 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.15 \[ \int \frac {13 e^3 x+52 x^2+e^x \left (24 x+12 e^3 x+24 x^2\right )+\left (e^x \left (-12 e^3-24 x\right )-13 e^3 x-26 x^2\right ) \log \left (\frac {1}{3} \left (13 e^3 x+26 x^2+e^x \left (12 e^3+24 x\right )\right )\right ) \log \left (\log \left (\frac {1}{3} \left (13 e^3 x+26 x^2+e^x \left (12 e^3+24 x\right )\right )\right )\right )}{\left (13 e^3 x^3+26 x^4+e^x \left (12 e^3 x^2+24 x^3\right )\right ) \log \left (\frac {1}{3} \left (13 e^3 x+26 x^2+e^x \left (12 e^3+24 x\right )\right )\right )} \, dx=\frac {\log {\left (\log {\left (\frac {26 x^{2}}{3} + \frac {13 x e^{3}}{3} + \left (8 x + 4 e^{3}\right ) e^{x} \right )} \right )}}{x} \]
integrate((((-12*exp(3)-24*x)*exp(x)-13*x*exp(3)-26*x**2)*ln(1/3*(12*exp(3 )+24*x)*exp(x)+13/3*x*exp(3)+26/3*x**2)*ln(ln(1/3*(12*exp(3)+24*x)*exp(x)+ 13/3*x*exp(3)+26/3*x**2))+(12*x*exp(3)+24*x**2+24*x)*exp(x)+13*x*exp(3)+52 *x**2)/((12*x**2*exp(3)+24*x**3)*exp(x)+13*x**3*exp(3)+26*x**4)/ln(1/3*(12 *exp(3)+24*x)*exp(x)+13/3*x*exp(3)+26/3*x**2),x)
Time = 0.40 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.96 \[ \int \frac {13 e^3 x+52 x^2+e^x \left (24 x+12 e^3 x+24 x^2\right )+\left (e^x \left (-12 e^3-24 x\right )-13 e^3 x-26 x^2\right ) \log \left (\frac {1}{3} \left (13 e^3 x+26 x^2+e^x \left (12 e^3+24 x\right )\right )\right ) \log \left (\log \left (\frac {1}{3} \left (13 e^3 x+26 x^2+e^x \left (12 e^3+24 x\right )\right )\right )\right )}{\left (13 e^3 x^3+26 x^4+e^x \left (12 e^3 x^2+24 x^3\right )\right ) \log \left (\frac {1}{3} \left (13 e^3 x+26 x^2+e^x \left (12 e^3+24 x\right )\right )\right )} \, dx=\frac {\log \left (-\log \left (3\right ) + \log \left (13 \, x + 12 \, e^{x}\right ) + \log \left (2 \, x + e^{3}\right )\right )}{x} \]
integrate((((-12*exp(3)-24*x)*exp(x)-13*x*exp(3)-26*x^2)*log(1/3*(12*exp(3 )+24*x)*exp(x)+13/3*x*exp(3)+26/3*x^2)*log(log(1/3*(12*exp(3)+24*x)*exp(x) +13/3*x*exp(3)+26/3*x^2))+(12*x*exp(3)+24*x^2+24*x)*exp(x)+13*x*exp(3)+52* x^2)/((12*x^2*exp(3)+24*x^3)*exp(x)+13*x^3*exp(3)+26*x^4)/log(1/3*(12*exp( 3)+24*x)*exp(x)+13/3*x*exp(3)+26/3*x^2),x, algorithm=\
\[ \int \frac {13 e^3 x+52 x^2+e^x \left (24 x+12 e^3 x+24 x^2\right )+\left (e^x \left (-12 e^3-24 x\right )-13 e^3 x-26 x^2\right ) \log \left (\frac {1}{3} \left (13 e^3 x+26 x^2+e^x \left (12 e^3+24 x\right )\right )\right ) \log \left (\log \left (\frac {1}{3} \left (13 e^3 x+26 x^2+e^x \left (12 e^3+24 x\right )\right )\right )\right )}{\left (13 e^3 x^3+26 x^4+e^x \left (12 e^3 x^2+24 x^3\right )\right ) \log \left (\frac {1}{3} \left (13 e^3 x+26 x^2+e^x \left (12 e^3+24 x\right )\right )\right )} \, dx=\int { -\frac {{\left (26 \, x^{2} + 13 \, x e^{3} + 12 \, {\left (2 \, x + e^{3}\right )} e^{x}\right )} \log \left (\frac {26}{3} \, x^{2} + \frac {13}{3} \, x e^{3} + 4 \, {\left (2 \, x + e^{3}\right )} e^{x}\right ) \log \left (\log \left (\frac {26}{3} \, x^{2} + \frac {13}{3} \, x e^{3} + 4 \, {\left (2 \, x + e^{3}\right )} e^{x}\right )\right ) - 52 \, x^{2} - 13 \, x e^{3} - 12 \, {\left (2 \, x^{2} + x e^{3} + 2 \, x\right )} e^{x}}{{\left (26 \, x^{4} + 13 \, x^{3} e^{3} + 12 \, {\left (2 \, x^{3} + x^{2} e^{3}\right )} e^{x}\right )} \log \left (\frac {26}{3} \, x^{2} + \frac {13}{3} \, x e^{3} + 4 \, {\left (2 \, x + e^{3}\right )} e^{x}\right )} \,d x } \]
integrate((((-12*exp(3)-24*x)*exp(x)-13*x*exp(3)-26*x^2)*log(1/3*(12*exp(3 )+24*x)*exp(x)+13/3*x*exp(3)+26/3*x^2)*log(log(1/3*(12*exp(3)+24*x)*exp(x) +13/3*x*exp(3)+26/3*x^2))+(12*x*exp(3)+24*x^2+24*x)*exp(x)+13*x*exp(3)+52* x^2)/((12*x^2*exp(3)+24*x^3)*exp(x)+13*x^3*exp(3)+26*x^4)/log(1/3*(12*exp( 3)+24*x)*exp(x)+13/3*x*exp(3)+26/3*x^2),x, algorithm=\
integrate(-((26*x^2 + 13*x*e^3 + 12*(2*x + e^3)*e^x)*log(26/3*x^2 + 13/3*x *e^3 + 4*(2*x + e^3)*e^x)*log(log(26/3*x^2 + 13/3*x*e^3 + 4*(2*x + e^3)*e^ x)) - 52*x^2 - 13*x*e^3 - 12*(2*x^2 + x*e^3 + 2*x)*e^x)/((26*x^4 + 13*x^3* e^3 + 12*(2*x^3 + x^2*e^3)*e^x)*log(26/3*x^2 + 13/3*x*e^3 + 4*(2*x + e^3)* e^x)), x)
Time = 9.63 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.74 \[ \int \frac {13 e^3 x+52 x^2+e^x \left (24 x+12 e^3 x+24 x^2\right )+\left (e^x \left (-12 e^3-24 x\right )-13 e^3 x-26 x^2\right ) \log \left (\frac {1}{3} \left (13 e^3 x+26 x^2+e^x \left (12 e^3+24 x\right )\right )\right ) \log \left (\log \left (\frac {1}{3} \left (13 e^3 x+26 x^2+e^x \left (12 e^3+24 x\right )\right )\right )\right )}{\left (13 e^3 x^3+26 x^4+e^x \left (12 e^3 x^2+24 x^3\right )\right ) \log \left (\frac {1}{3} \left (13 e^3 x+26 x^2+e^x \left (12 e^3+24 x\right )\right )\right )} \, dx=\frac {\ln \left (\ln \left (\frac {{\mathrm {e}}^x\,\left (24\,x+12\,{\mathrm {e}}^3\right )}{3}+\frac {13\,x\,{\mathrm {e}}^3}{3}+\frac {26\,x^2}{3}\right )\right )\,\left (2\,x^2+{\mathrm {e}}^3\,x\right )}{x^2\,\left (2\,x+{\mathrm {e}}^3\right )} \]
int((13*x*exp(3) + 52*x^2 + exp(x)*(24*x + 12*x*exp(3) + 24*x^2) - log((ex p(x)*(24*x + 12*exp(3)))/3 + (13*x*exp(3))/3 + (26*x^2)/3)*log(log((exp(x) *(24*x + 12*exp(3)))/3 + (13*x*exp(3))/3 + (26*x^2)/3))*(exp(x)*(24*x + 12 *exp(3)) + 13*x*exp(3) + 26*x^2))/(log((exp(x)*(24*x + 12*exp(3)))/3 + (13 *x*exp(3))/3 + (26*x^2)/3)*(exp(x)*(12*x^2*exp(3) + 24*x^3) + 13*x^3*exp(3 ) + 26*x^4)),x)