Integrand size = 165, antiderivative size = 26 \[ \int \frac {\left (-1920 x-576 x^2+e^{4/x} \left (640 x+192 x^2\right )\right ) \log \left (-3+e^{4/x}\right ) \log \left (\log \left (-3+e^{4/x}\right )\right )+512 e^{4/x} \log \left (\frac {5}{\log \left (\log \left (-3+e^{4/x}\right )\right )}\right )+\left (-384 x+128 e^{4/x} x\right ) \log \left (-3+e^{4/x}\right ) \log \left (\log \left (-3+e^{4/x}\right )\right ) \log ^2\left (\frac {5}{\log \left (\log \left (-3+e^{4/x}\right )\right )}\right )}{\left (-3+e^{4/x}\right ) \log \left (-3+e^{4/x}\right ) \log \left (\log \left (-3+e^{4/x}\right )\right )} \, dx=64 x^2 \left (5+x+\log ^2\left (\frac {5}{\log \left (\log \left (-3+e^{4/x}\right )\right )}\right )\right ) \]
Time = 0.33 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.27 \[ \int \frac {\left (-1920 x-576 x^2+e^{4/x} \left (640 x+192 x^2\right )\right ) \log \left (-3+e^{4/x}\right ) \log \left (\log \left (-3+e^{4/x}\right )\right )+512 e^{4/x} \log \left (\frac {5}{\log \left (\log \left (-3+e^{4/x}\right )\right )}\right )+\left (-384 x+128 e^{4/x} x\right ) \log \left (-3+e^{4/x}\right ) \log \left (\log \left (-3+e^{4/x}\right )\right ) \log ^2\left (\frac {5}{\log \left (\log \left (-3+e^{4/x}\right )\right )}\right )}{\left (-3+e^{4/x}\right ) \log \left (-3+e^{4/x}\right ) \log \left (\log \left (-3+e^{4/x}\right )\right )} \, dx=64 \left (5 x^2+x^3+x^2 \log ^2\left (\frac {5}{\log \left (\log \left (-3+e^{4/x}\right )\right )}\right )\right ) \]
Integrate[((-1920*x - 576*x^2 + E^(4/x)*(640*x + 192*x^2))*Log[-3 + E^(4/x )]*Log[Log[-3 + E^(4/x)]] + 512*E^(4/x)*Log[5/Log[Log[-3 + E^(4/x)]]] + (- 384*x + 128*E^(4/x)*x)*Log[-3 + E^(4/x)]*Log[Log[-3 + E^(4/x)]]*Log[5/Log[ Log[-3 + E^(4/x)]]]^2)/((-3 + E^(4/x))*Log[-3 + E^(4/x)]*Log[Log[-3 + E^(4 /x)]]),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.
| \(\displaystyle \int \frac {\left (-576 x^2+e^{4/x} \left (192 x^2+640 x\right )-1920 x\right ) \log \left (e^{4/x}-3\right ) \log \left (\log \left (e^{4/x}-3\right )\right )+\left (128 e^{4/x} x-384 x\right ) \log \left (e^{4/x}-3\right ) \log \left (\log \left (e^{4/x}-3\right )\right ) \log ^2\left (\frac {5}{\log \left (\log \left (e^{4/x}-3\right )\right )}\right )+512 e^{4/x} \log \left (\frac {5}{\log \left (\log \left (e^{4/x}-3\right )\right )}\right )}{\left (e^{4/x}-3\right ) \log \left (e^{4/x}-3\right ) \log \left (\log \left (e^{4/x}-3\right )\right )} \, dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int 64 \left (x \left (3 x+2 \log ^2\left (\frac {5}{\log \left (\log \left (e^{4/x}-3\right )\right )}\right )+10\right )+\frac {8 e^{4/x} \log \left (\frac {5}{\log \left (\log \left (e^{4/x}-3\right )\right )}\right )}{\left (e^{4/x}-3\right ) \log \left (e^{4/x}-3\right ) \log \left (\log \left (e^{4/x}-3\right )\right )}\right )dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 64 \int \left (x \left (2 \log ^2\left (\frac {5}{\log \left (\log \left (-3+e^{4/x}\right )\right )}\right )+3 x+10\right )-\frac {8 e^{4/x} \log \left (\frac {5}{\log \left (\log \left (-3+e^{4/x}\right )\right )}\right )}{\left (3-e^{4/x}\right ) \log \left (-3+e^{4/x}\right ) \log \left (\log \left (-3+e^{4/x}\right )\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 64 \left (2 \int x \log ^2\left (\frac {5}{\log \left (\log \left (-3+e^{4/x}\right )\right )}\right )dx-8 \int \frac {e^{4/x} \log \left (\frac {5}{\log \left (\log \left (-3+e^{4/x}\right )\right )}\right )}{\left (3-e^{4/x}\right ) \log \left (-3+e^{4/x}\right ) \log \left (\log \left (-3+e^{4/x}\right )\right )}dx+x^3+5 x^2\right )\) |
Int[((-1920*x - 576*x^2 + E^(4/x)*(640*x + 192*x^2))*Log[-3 + E^(4/x)]*Log [Log[-3 + E^(4/x)]] + 512*E^(4/x)*Log[5/Log[Log[-3 + E^(4/x)]]] + (-384*x + 128*E^(4/x)*x)*Log[-3 + E^(4/x)]*Log[Log[-3 + E^(4/x)]]*Log[5/Log[Log[-3 + E^(4/x)]]]^2)/((-3 + E^(4/x))*Log[-3 + E^(4/x)]*Log[Log[-3 + E^(4/x)]]) ,x]
3.30.79.3.1 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]]
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Time = 32.24 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.31
| method | result | size |
| parallelrisch | \(64 {\ln \left (\frac {5}{\ln \left (\ln \left ({\mathrm e}^{\frac {4}{x}}-3\right )\right )}\right )}^{2} x^{2}+64 x^{3}+320 x^{2}\) | \(34\) |
| risch | \(64 x^{2} {\ln \left (\ln \left (\ln \left ({\mathrm e}^{\frac {4}{x}}-3\right )\right )\right )}^{2}-128 \ln \left (5\right ) x^{2} \ln \left (\ln \left (\ln \left ({\mathrm e}^{\frac {4}{x}}-3\right )\right )\right )+64 x^{2} \ln \left (5\right )^{2}+64 x^{3}+320 x^{2}\) | \(57\) |
int(((128*x*exp(4/x)-384*x)*ln(exp(4/x)-3)*ln(ln(exp(4/x)-3))*ln(5/ln(ln(e xp(4/x)-3)))^2+512*exp(4/x)*ln(5/ln(ln(exp(4/x)-3)))+((192*x^2+640*x)*exp( 4/x)-576*x^2-1920*x)*ln(exp(4/x)-3)*ln(ln(exp(4/x)-3)))/(exp(4/x)-3)/ln(ex p(4/x)-3)/ln(ln(exp(4/x)-3)),x,method=_RETURNVERBOSE)
Time = 0.27 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.27 \[ \int \frac {\left (-1920 x-576 x^2+e^{4/x} \left (640 x+192 x^2\right )\right ) \log \left (-3+e^{4/x}\right ) \log \left (\log \left (-3+e^{4/x}\right )\right )+512 e^{4/x} \log \left (\frac {5}{\log \left (\log \left (-3+e^{4/x}\right )\right )}\right )+\left (-384 x+128 e^{4/x} x\right ) \log \left (-3+e^{4/x}\right ) \log \left (\log \left (-3+e^{4/x}\right )\right ) \log ^2\left (\frac {5}{\log \left (\log \left (-3+e^{4/x}\right )\right )}\right )}{\left (-3+e^{4/x}\right ) \log \left (-3+e^{4/x}\right ) \log \left (\log \left (-3+e^{4/x}\right )\right )} \, dx=64 \, x^{2} \log \left (\frac {5}{\log \left (\log \left (e^{\frac {4}{x}} - 3\right )\right )}\right )^{2} + 64 \, x^{3} + 320 \, x^{2} \]
integrate(((128*x*exp(4/x)-384*x)*log(exp(4/x)-3)*log(log(exp(4/x)-3))*log (5/log(log(exp(4/x)-3)))^2+512*exp(4/x)*log(5/log(log(exp(4/x)-3)))+((192* x^2+640*x)*exp(4/x)-576*x^2-1920*x)*log(exp(4/x)-3)*log(log(exp(4/x)-3)))/ (exp(4/x)-3)/log(exp(4/x)-3)/log(log(exp(4/x)-3)),x, algorithm=\
Timed out. \[ \int \frac {\left (-1920 x-576 x^2+e^{4/x} \left (640 x+192 x^2\right )\right ) \log \left (-3+e^{4/x}\right ) \log \left (\log \left (-3+e^{4/x}\right )\right )+512 e^{4/x} \log \left (\frac {5}{\log \left (\log \left (-3+e^{4/x}\right )\right )}\right )+\left (-384 x+128 e^{4/x} x\right ) \log \left (-3+e^{4/x}\right ) \log \left (\log \left (-3+e^{4/x}\right )\right ) \log ^2\left (\frac {5}{\log \left (\log \left (-3+e^{4/x}\right )\right )}\right )}{\left (-3+e^{4/x}\right ) \log \left (-3+e^{4/x}\right ) \log \left (\log \left (-3+e^{4/x}\right )\right )} \, dx=\text {Timed out} \]
integrate(((128*x*exp(4/x)-384*x)*ln(exp(4/x)-3)*ln(ln(exp(4/x)-3))*ln(5/l n(ln(exp(4/x)-3)))**2+512*exp(4/x)*ln(5/ln(ln(exp(4/x)-3)))+((192*x**2+640 *x)*exp(4/x)-576*x**2-1920*x)*ln(exp(4/x)-3)*ln(ln(exp(4/x)-3)))/(exp(4/x) -3)/ln(exp(4/x)-3)/ln(ln(exp(4/x)-3)),x)
Leaf count of result is larger than twice the leaf count of optimal. 53 vs. \(2 (25) = 50\).
Time = 0.37 (sec) , antiderivative size = 53, normalized size of antiderivative = 2.04 \[ \int \frac {\left (-1920 x-576 x^2+e^{4/x} \left (640 x+192 x^2\right )\right ) \log \left (-3+e^{4/x}\right ) \log \left (\log \left (-3+e^{4/x}\right )\right )+512 e^{4/x} \log \left (\frac {5}{\log \left (\log \left (-3+e^{4/x}\right )\right )}\right )+\left (-384 x+128 e^{4/x} x\right ) \log \left (-3+e^{4/x}\right ) \log \left (\log \left (-3+e^{4/x}\right )\right ) \log ^2\left (\frac {5}{\log \left (\log \left (-3+e^{4/x}\right )\right )}\right )}{\left (-3+e^{4/x}\right ) \log \left (-3+e^{4/x}\right ) \log \left (\log \left (-3+e^{4/x}\right )\right )} \, dx=-128 \, x^{2} \log \left (5\right ) \log \left (\log \left (\log \left (e^{\frac {4}{x}} - 3\right )\right )\right ) + 64 \, x^{2} \log \left (\log \left (\log \left (e^{\frac {4}{x}} - 3\right )\right )\right )^{2} + 64 \, {\left (\log \left (5\right )^{2} + 5\right )} x^{2} + 64 \, x^{3} \]
integrate(((128*x*exp(4/x)-384*x)*log(exp(4/x)-3)*log(log(exp(4/x)-3))*log (5/log(log(exp(4/x)-3)))^2+512*exp(4/x)*log(5/log(log(exp(4/x)-3)))+((192* x^2+640*x)*exp(4/x)-576*x^2-1920*x)*log(exp(4/x)-3)*log(log(exp(4/x)-3)))/ (exp(4/x)-3)/log(exp(4/x)-3)/log(log(exp(4/x)-3)),x, algorithm=\
-128*x^2*log(5)*log(log(log(e^(4/x) - 3))) + 64*x^2*log(log(log(e^(4/x) - 3)))^2 + 64*(log(5)^2 + 5)*x^2 + 64*x^3
\[ \int \frac {\left (-1920 x-576 x^2+e^{4/x} \left (640 x+192 x^2\right )\right ) \log \left (-3+e^{4/x}\right ) \log \left (\log \left (-3+e^{4/x}\right )\right )+512 e^{4/x} \log \left (\frac {5}{\log \left (\log \left (-3+e^{4/x}\right )\right )}\right )+\left (-384 x+128 e^{4/x} x\right ) \log \left (-3+e^{4/x}\right ) \log \left (\log \left (-3+e^{4/x}\right )\right ) \log ^2\left (\frac {5}{\log \left (\log \left (-3+e^{4/x}\right )\right )}\right )}{\left (-3+e^{4/x}\right ) \log \left (-3+e^{4/x}\right ) \log \left (\log \left (-3+e^{4/x}\right )\right )} \, dx=\int { \frac {64 \, {\left (2 \, {\left (x e^{\frac {4}{x}} - 3 \, x\right )} \log \left (\frac {5}{\log \left (\log \left (e^{\frac {4}{x}} - 3\right )\right )}\right )^{2} \log \left (e^{\frac {4}{x}} - 3\right ) \log \left (\log \left (e^{\frac {4}{x}} - 3\right )\right ) - {\left (9 \, x^{2} - {\left (3 \, x^{2} + 10 \, x\right )} e^{\frac {4}{x}} + 30 \, x\right )} \log \left (e^{\frac {4}{x}} - 3\right ) \log \left (\log \left (e^{\frac {4}{x}} - 3\right )\right ) + 8 \, e^{\frac {4}{x}} \log \left (\frac {5}{\log \left (\log \left (e^{\frac {4}{x}} - 3\right )\right )}\right )\right )}}{{\left (e^{\frac {4}{x}} - 3\right )} \log \left (e^{\frac {4}{x}} - 3\right ) \log \left (\log \left (e^{\frac {4}{x}} - 3\right )\right )} \,d x } \]
integrate(((128*x*exp(4/x)-384*x)*log(exp(4/x)-3)*log(log(exp(4/x)-3))*log (5/log(log(exp(4/x)-3)))^2+512*exp(4/x)*log(5/log(log(exp(4/x)-3)))+((192* x^2+640*x)*exp(4/x)-576*x^2-1920*x)*log(exp(4/x)-3)*log(log(exp(4/x)-3)))/ (exp(4/x)-3)/log(exp(4/x)-3)/log(log(exp(4/x)-3)),x, algorithm=\
integrate(64*(2*(x*e^(4/x) - 3*x)*log(5/log(log(e^(4/x) - 3)))^2*log(e^(4/ x) - 3)*log(log(e^(4/x) - 3)) - (9*x^2 - (3*x^2 + 10*x)*e^(4/x) + 30*x)*lo g(e^(4/x) - 3)*log(log(e^(4/x) - 3)) + 8*e^(4/x)*log(5/log(log(e^(4/x) - 3 ))))/((e^(4/x) - 3)*log(e^(4/x) - 3)*log(log(e^(4/x) - 3))), x)
Time = 12.85 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.96 \[ \int \frac {\left (-1920 x-576 x^2+e^{4/x} \left (640 x+192 x^2\right )\right ) \log \left (-3+e^{4/x}\right ) \log \left (\log \left (-3+e^{4/x}\right )\right )+512 e^{4/x} \log \left (\frac {5}{\log \left (\log \left (-3+e^{4/x}\right )\right )}\right )+\left (-384 x+128 e^{4/x} x\right ) \log \left (-3+e^{4/x}\right ) \log \left (\log \left (-3+e^{4/x}\right )\right ) \log ^2\left (\frac {5}{\log \left (\log \left (-3+e^{4/x}\right )\right )}\right )}{\left (-3+e^{4/x}\right ) \log \left (-3+e^{4/x}\right ) \log \left (\log \left (-3+e^{4/x}\right )\right )} \, dx=64\,x^2\,\left ({\ln \left (\frac {5}{\ln \left (\ln \left ({\mathrm {e}}^{4/x}-3\right )\right )}\right )}^2+x+5\right ) \]
int(-(log(exp(4/x) - 3)*log(log(exp(4/x) - 3))*(1920*x - exp(4/x)*(640*x + 192*x^2) + 576*x^2) - 512*exp(4/x)*log(5/log(log(exp(4/x) - 3))) + log(ex p(4/x) - 3)*log(log(exp(4/x) - 3))*log(5/log(log(exp(4/x) - 3)))^2*(384*x - 128*x*exp(4/x)))/(log(exp(4/x) - 3)*log(log(exp(4/x) - 3))*(exp(4/x) - 3 )),x)