Integrand size = 172, antiderivative size = 28 \[ \int \frac {-2 x^2 \log (x) \log \left (\frac {5 e^x \log (x)}{x}\right )+e^{10} \log ^2\left (\frac {5 e^x \log (x)}{x}\right ) \left (-32-16 x-2 x^2+\left (32-16 x-14 x^2-2 x^3\right ) \log (x)+\left (-8 x-2 x^2\right ) \log (x) \log \left (\frac {5 e^x \log (x)}{x}\right )\right )+e^5 \log \left (\frac {5 e^x \log (x)}{x}\right ) \left (8 x+2 x^2+\left (-8 x+6 x^2+2 x^3\right ) \log (x)+\left (8 x+4 x^2\right ) \log (x) \log \left (\frac {5 e^x \log (x)}{x}\right )\right )}{x \log (x) \log \left (\frac {5 e^x \log (x)}{x}\right )} \, dx=3-\left (-x+e^5 (4+x) \log \left (\frac {5 e^x \log (x)}{x}\right )\right )^2 \]
Leaf count is larger than twice the leaf count of optimal. \(180\) vs. \(2(28)=56\).
Time = 0.69 (sec) , antiderivative size = 180, normalized size of antiderivative = 6.43 \[ \int \frac {-2 x^2 \log (x) \log \left (\frac {5 e^x \log (x)}{x}\right )+e^{10} \log ^2\left (\frac {5 e^x \log (x)}{x}\right ) \left (-32-16 x-2 x^2+\left (32-16 x-14 x^2-2 x^3\right ) \log (x)+\left (-8 x-2 x^2\right ) \log (x) \log \left (\frac {5 e^x \log (x)}{x}\right )\right )+e^5 \log \left (\frac {5 e^x \log (x)}{x}\right ) \left (8 x+2 x^2+\left (-8 x+6 x^2+2 x^3\right ) \log (x)+\left (8 x+4 x^2\right ) \log (x) \log \left (\frac {5 e^x \log (x)}{x}\right )\right )}{x \log (x) \log \left (\frac {5 e^x \log (x)}{x}\right )} \, dx=-x^2+16 e^{10} x^2-16 e^{10} \log ^2\left (\frac {\log (x)}{x}\right )-32 e^{10} \log (x) \left (x+\log \left (\frac {\log (x)}{x}\right )-\log \left (\frac {5 e^x \log (x)}{x}\right )\right )+32 e^{10} \log (\log (x)) \left (x+\log \left (\frac {\log (x)}{x}\right )-\log \left (\frac {5 e^x \log (x)}{x}\right )\right )+8 e^5 x \log \left (\frac {5 e^x \log (x)}{x}\right )-32 e^{10} x \log \left (\frac {5 e^x \log (x)}{x}\right )+2 e^5 x^2 \log \left (\frac {5 e^x \log (x)}{x}\right )-8 e^{10} x \log ^2\left (\frac {5 e^x \log (x)}{x}\right )-e^{10} x^2 \log ^2\left (\frac {5 e^x \log (x)}{x}\right ) \]
Integrate[(-2*x^2*Log[x]*Log[(5*E^x*Log[x])/x] + E^10*Log[(5*E^x*Log[x])/x ]^2*(-32 - 16*x - 2*x^2 + (32 - 16*x - 14*x^2 - 2*x^3)*Log[x] + (-8*x - 2* x^2)*Log[x]*Log[(5*E^x*Log[x])/x]) + E^5*Log[(5*E^x*Log[x])/x]*(8*x + 2*x^ 2 + (-8*x + 6*x^2 + 2*x^3)*Log[x] + (8*x + 4*x^2)*Log[x]*Log[(5*E^x*Log[x] )/x]))/(x*Log[x]*Log[(5*E^x*Log[x])/x]),x]
-x^2 + 16*E^10*x^2 - 16*E^10*Log[Log[x]/x]^2 - 32*E^10*Log[x]*(x + Log[Log [x]/x] - Log[(5*E^x*Log[x])/x]) + 32*E^10*Log[Log[x]]*(x + Log[Log[x]/x] - Log[(5*E^x*Log[x])/x]) + 8*E^5*x*Log[(5*E^x*Log[x])/x] - 32*E^10*x*Log[(5 *E^x*Log[x])/x] + 2*E^5*x^2*Log[(5*E^x*Log[x])/x] - 8*E^10*x*Log[(5*E^x*Lo g[x])/x]^2 - E^10*x^2*Log[(5*E^x*Log[x])/x]^2
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.
| \(\displaystyle \int \frac {-2 x^2 \log (x) \log \left (\frac {5 e^x \log (x)}{x}\right )+e^{10} \log ^2\left (\frac {5 e^x \log (x)}{x}\right ) \left (-2 x^2+\left (-2 x^2-8 x\right ) \log (x) \log \left (\frac {5 e^x \log (x)}{x}\right )+\left (-2 x^3-14 x^2-16 x+32\right ) \log (x)-16 x-32\right )+e^5 \log \left (\frac {5 e^x \log (x)}{x}\right ) \left (2 x^2+\left (4 x^2+8 x\right ) \log (x) \log \left (\frac {5 e^x \log (x)}{x}\right )+\left (2 x^3+6 x^2-8 x\right ) \log (x)+8 x\right )}{x \log (x) \log \left (\frac {5 e^x \log (x)}{x}\right )} \, dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {2 \left (x-e^5 (x+4) \log \left (\frac {5 e^x \log (x)}{x}\right )\right ) \left (\log (x) \left (e^5 \left (x^2+3 x-4\right )-x+e^5 x \log \left (\frac {5 e^x \log (x)}{x}\right )\right )+e^5 (x+4)\right )}{x \log (x)}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 \int \frac {\left (x-e^5 (x+4) \log \left (\frac {5 e^x \log (x)}{x}\right )\right ) \left (e^5 (x+4)-\log (x) \left (-e^5 \log \left (\frac {5 e^x \log (x)}{x}\right ) x+x+e^5 \left (-x^2-3 x+4\right )\right )\right )}{x \log (x)}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 2 \int \left (-e^{10} (x+4) \log ^2\left (\frac {5 e^x \log (x)}{x}\right )+\frac {e^5 \left (-e^5 \log (x) x^3+2 \left (1-\frac {7 e^5}{2}\right ) \log (x) x^2-e^5 x^2+4 \left (1-2 e^5\right ) \log (x) x-8 e^5 x+16 e^5 \log (x)-16 e^5\right ) \log \left (\frac {5 e^x \log (x)}{x}\right )}{x \log (x)}+\frac {e^5 \log (x) x^2-\left (1-3 e^5\right ) \log (x) x+e^5 x-4 e^5 \log (x)+4 e^5}{\log (x)}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 2 \left (8 e^{10} \int \frac {\operatorname {LogIntegral}(x)}{x \log (x)}dx-4 e^{10} \int \log ^2\left (\frac {5 e^x \log (x)}{x}\right )dx-e^{10} \int x \log ^2\left (\frac {5 e^x \log (x)}{x}\right )dx+16 e^{10} \int \frac {\log \left (\frac {5 e^x \log (x)}{x}\right )}{x}dx-16 e^{10} \int \frac {\log \left (\frac {5 e^x \log (x)}{x}\right )}{x \log (x)}dx-e^{10} \int \frac {x \log \left (\frac {5 e^x \log (x)}{x}\right )}{\log (x)}dx-\frac {1}{2} e^5 \left (2-7 e^5\right ) \operatorname {ExpIntegralEi}(2 \log (x))-8 e^{10} \operatorname {ExpIntegralEi}(2 \log (x))+e^5 \operatorname {ExpIntegralEi}(2 \log (x))+\frac {1}{3} e^{10} \operatorname {ExpIntegralEi}(3 \log (x))+8 e^{10} x \operatorname {LogIntegral}(x)-4 e^5 \left (1-2 e^5\right ) \operatorname {LogIntegral}(x)+4 e^5 \operatorname {LogIntegral}(x)-8 e^{10} \operatorname {LogIntegral}(x) \log (x)-8 e^{10} \operatorname {LogIntegral}(x) \log \left (\frac {5 e^x \log (x)}{x}\right )+\frac {e^{10} x^4}{12}-\frac {1}{6} e^5 \left (2-7 e^5\right ) x^3-\frac {e^{10} x^3}{9}+\frac {e^5 x^3}{3}-\frac {1}{3} e^{10} x^3 \log \left (\frac {5 e^x \log (x)}{x}\right )-2 e^5 \left (1-2 e^5\right ) x^2-\frac {1}{2} \left (1-3 e^5\right ) x^2+\frac {1}{4} e^5 \left (2-7 e^5\right ) x^2+\frac {1}{2} e^5 \left (2-7 e^5\right ) x^2 \log \left (\frac {5 e^x \log (x)}{x}\right )+4 e^5 \left (1-2 e^5\right ) x+8 e^{10} x-4 e^5 x+4 e^5 \left (1-2 e^5\right ) x \log \left (\frac {5 e^x \log (x)}{x}\right )\right )\) |
Int[(-2*x^2*Log[x]*Log[(5*E^x*Log[x])/x] + E^10*Log[(5*E^x*Log[x])/x]^2*(- 32 - 16*x - 2*x^2 + (32 - 16*x - 14*x^2 - 2*x^3)*Log[x] + (-8*x - 2*x^2)*L og[x]*Log[(5*E^x*Log[x])/x]) + E^5*Log[(5*E^x*Log[x])/x]*(8*x + 2*x^2 + (- 8*x + 6*x^2 + 2*x^3)*Log[x] + (8*x + 4*x^2)*Log[x]*Log[(5*E^x*Log[x])/x])) /(x*Log[x]*Log[(5*E^x*Log[x])/x]),x]
3.17.11.3.1 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]]
Leaf count of result is larger than twice the leaf count of optimal. \(185\) vs. \(2(28)=56\).
Time = 16.49 (sec) , antiderivative size = 186, normalized size of antiderivative = 6.64
| method | result | size |
| parallelrisch | \(\frac {-\ln \left (\frac {5 \,{\mathrm e}^{x} \ln \left (x \right )}{x}\right )^{4} {\mathrm e}^{10} x^{2}+2 \,{\mathrm e}^{\ln \left (\ln \left (\frac {5 \,{\mathrm e}^{x} \ln \left (x \right )}{x}\right )\right )+5} \ln \left (\frac {5 \,{\mathrm e}^{x} \ln \left (x \right )}{x}\right )^{2} x^{2}-8 \ln \left (\frac {5 \,{\mathrm e}^{x} \ln \left (x \right )}{x}\right )^{4} {\mathrm e}^{10} x -\ln \left (\frac {5 \,{\mathrm e}^{x} \ln \left (x \right )}{x}\right )^{2} x^{2}+8 \,{\mathrm e}^{\ln \left (\ln \left (\frac {5 \,{\mathrm e}^{x} \ln \left (x \right )}{x}\right )\right )+5} \ln \left (\frac {5 \,{\mathrm e}^{x} \ln \left (x \right )}{x}\right )^{2} x -16 \ln \left (\frac {5 \,{\mathrm e}^{x} \ln \left (x \right )}{x}\right )^{4} {\mathrm e}^{10}}{\ln \left (\frac {5 \,{\mathrm e}^{x} \ln \left (x \right )}{x}\right )^{2}}\) | \(186\) |
| risch | \(\text {Expression too large to display}\) | \(5910\) |
int((((-2*x^2-8*x)*ln(x)*ln(5*exp(x)*ln(x)/x)+(-2*x^3-14*x^2-16*x+32)*ln(x )-2*x^2-16*x-32)*exp(ln(ln(5*exp(x)*ln(x)/x))+5)^2+((4*x^2+8*x)*ln(x)*ln(5 *exp(x)*ln(x)/x)+(2*x^3+6*x^2-8*x)*ln(x)+2*x^2+8*x)*exp(ln(ln(5*exp(x)*ln( x)/x))+5)-2*x^2*ln(x)*ln(5*exp(x)*ln(x)/x))/x/ln(x)/ln(5*exp(x)*ln(x)/x),x ,method=_RETURNVERBOSE)
(-exp(ln(ln(5*exp(x)*ln(x)/x))+5)^2*ln(5*exp(x)*ln(x)/x)^2*x^2+2*exp(ln(ln (5*exp(x)*ln(x)/x))+5)*ln(5*exp(x)*ln(x)/x)^2*x^2-8*exp(ln(ln(5*exp(x)*ln( x)/x))+5)^2*ln(5*exp(x)*ln(x)/x)^2*x-ln(5*exp(x)*ln(x)/x)^2*x^2+8*exp(ln(l n(5*exp(x)*ln(x)/x))+5)*ln(5*exp(x)*ln(x)/x)^2*x-16*exp(ln(ln(5*exp(x)*ln( x)/x))+5)^2*ln(5*exp(x)*ln(x)/x)^2)/ln(5*exp(x)*ln(x)/x)^2
Time = 0.27 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.82 \[ \int \frac {-2 x^2 \log (x) \log \left (\frac {5 e^x \log (x)}{x}\right )+e^{10} \log ^2\left (\frac {5 e^x \log (x)}{x}\right ) \left (-32-16 x-2 x^2+\left (32-16 x-14 x^2-2 x^3\right ) \log (x)+\left (-8 x-2 x^2\right ) \log (x) \log \left (\frac {5 e^x \log (x)}{x}\right )\right )+e^5 \log \left (\frac {5 e^x \log (x)}{x}\right ) \left (8 x+2 x^2+\left (-8 x+6 x^2+2 x^3\right ) \log (x)+\left (8 x+4 x^2\right ) \log (x) \log \left (\frac {5 e^x \log (x)}{x}\right )\right )}{x \log (x) \log \left (\frac {5 e^x \log (x)}{x}\right )} \, dx=-{\left (x^{2} + 8 \, x + 16\right )} e^{10} \log \left (\frac {5 \, e^{x} \log \left (x\right )}{x}\right )^{2} + 2 \, {\left (x^{2} + 4 \, x\right )} e^{5} \log \left (\frac {5 \, e^{x} \log \left (x\right )}{x}\right ) - x^{2} \]
integrate((((-2*x^2-8*x)*log(x)*log(5*exp(x)*log(x)/x)+(-2*x^3-14*x^2-16*x +32)*log(x)-2*x^2-16*x-32)*exp(log(log(5*exp(x)*log(x)/x))+5)^2+((4*x^2+8* x)*log(x)*log(5*exp(x)*log(x)/x)+(2*x^3+6*x^2-8*x)*log(x)+2*x^2+8*x)*exp(l og(log(5*exp(x)*log(x)/x))+5)-2*x^2*log(x)*log(5*exp(x)*log(x)/x))/x/log(x )/log(5*exp(x)*log(x)/x),x, algorithm=\
Leaf count of result is larger than twice the leaf count of optimal. 63 vs. \(2 (22) = 44\).
Time = 0.34 (sec) , antiderivative size = 63, normalized size of antiderivative = 2.25 \[ \int \frac {-2 x^2 \log (x) \log \left (\frac {5 e^x \log (x)}{x}\right )+e^{10} \log ^2\left (\frac {5 e^x \log (x)}{x}\right ) \left (-32-16 x-2 x^2+\left (32-16 x-14 x^2-2 x^3\right ) \log (x)+\left (-8 x-2 x^2\right ) \log (x) \log \left (\frac {5 e^x \log (x)}{x}\right )\right )+e^5 \log \left (\frac {5 e^x \log (x)}{x}\right ) \left (8 x+2 x^2+\left (-8 x+6 x^2+2 x^3\right ) \log (x)+\left (8 x+4 x^2\right ) \log (x) \log \left (\frac {5 e^x \log (x)}{x}\right )\right )}{x \log (x) \log \left (\frac {5 e^x \log (x)}{x}\right )} \, dx=- x^{2} + \left (2 x^{2} e^{5} + 8 x e^{5}\right ) \log {\left (\frac {5 e^{x} \log {\left (x \right )}}{x} \right )} + \left (- x^{2} e^{10} - 8 x e^{10} - 16 e^{10}\right ) \log {\left (\frac {5 e^{x} \log {\left (x \right )}}{x} \right )}^{2} \]
integrate((((-2*x**2-8*x)*ln(x)*ln(5*exp(x)*ln(x)/x)+(-2*x**3-14*x**2-16*x +32)*ln(x)-2*x**2-16*x-32)*exp(ln(ln(5*exp(x)*ln(x)/x))+5)**2+((4*x**2+8*x )*ln(x)*ln(5*exp(x)*ln(x)/x)+(2*x**3+6*x**2-8*x)*ln(x)+2*x**2+8*x)*exp(ln( ln(5*exp(x)*ln(x)/x))+5)-2*x**2*ln(x)*ln(5*exp(x)*ln(x)/x))/x/ln(x)/ln(5*e xp(x)*ln(x)/x),x)
-x**2 + (2*x**2*exp(5) + 8*x*exp(5))*log(5*exp(x)*log(x)/x) + (-x**2*exp(1 0) - 8*x*exp(10) - 16*exp(10))*log(5*exp(x)*log(x)/x)**2
\[ \int \frac {-2 x^2 \log (x) \log \left (\frac {5 e^x \log (x)}{x}\right )+e^{10} \log ^2\left (\frac {5 e^x \log (x)}{x}\right ) \left (-32-16 x-2 x^2+\left (32-16 x-14 x^2-2 x^3\right ) \log (x)+\left (-8 x-2 x^2\right ) \log (x) \log \left (\frac {5 e^x \log (x)}{x}\right )\right )+e^5 \log \left (\frac {5 e^x \log (x)}{x}\right ) \left (8 x+2 x^2+\left (-8 x+6 x^2+2 x^3\right ) \log (x)+\left (8 x+4 x^2\right ) \log (x) \log \left (\frac {5 e^x \log (x)}{x}\right )\right )}{x \log (x) \log \left (\frac {5 e^x \log (x)}{x}\right )} \, dx=\int { -\frac {2 \, {\left (x^{2} \log \left (x\right ) \log \left (\frac {5 \, e^{x} \log \left (x\right )}{x}\right ) + {\left ({\left (x^{2} + 4 \, x\right )} \log \left (x\right ) \log \left (\frac {5 \, e^{x} \log \left (x\right )}{x}\right ) + x^{2} + {\left (x^{3} + 7 \, x^{2} + 8 \, x - 16\right )} \log \left (x\right ) + 8 \, x + 16\right )} e^{\left (2 \, \log \left (\log \left (\frac {5 \, e^{x} \log \left (x\right )}{x}\right )\right ) + 10\right )} - {\left (2 \, {\left (x^{2} + 2 \, x\right )} \log \left (x\right ) \log \left (\frac {5 \, e^{x} \log \left (x\right )}{x}\right ) + x^{2} + {\left (x^{3} + 3 \, x^{2} - 4 \, x\right )} \log \left (x\right ) + 4 \, x\right )} e^{\left (\log \left (\log \left (\frac {5 \, e^{x} \log \left (x\right )}{x}\right )\right ) + 5\right )}\right )}}{x \log \left (x\right ) \log \left (\frac {5 \, e^{x} \log \left (x\right )}{x}\right )} \,d x } \]
integrate((((-2*x^2-8*x)*log(x)*log(5*exp(x)*log(x)/x)+(-2*x^3-14*x^2-16*x +32)*log(x)-2*x^2-16*x-32)*exp(log(log(5*exp(x)*log(x)/x))+5)^2+((4*x^2+8* x)*log(x)*log(5*exp(x)*log(x)/x)+(2*x^3+6*x^2-8*x)*log(x)+2*x^2+8*x)*exp(l og(log(5*exp(x)*log(x)/x))+5)-2*x^2*log(x)*log(5*exp(x)*log(x)/x))/x/log(x )/log(5*exp(x)*log(x)/x),x, algorithm=\
-x^4*e^10 - 2/3*(3*(log(5) + 4)*e^10 - 2*e^5)*x^3 + 2/3*x^3*e^5 - ((log(5) ^2 + 16*log(5) + 16)*e^10 - (2*log(5) + 5)*e^5)*x^2 + 3*x^2*e^5 - (x^2*e^1 0 + 8*x*e^10 + 16*e^10)*log(x)^2 - (x^2*e^10 + 8*x*e^10 + 16*e^10)*log(log (x))^2 - 8*((log(5)^2 + 4*log(5))*e^10 - (log(5) + 1)*e^5)*x - x^2 - 8*x*e ^5 + 2*Ei(2*log(x))*e^5 + 8*Ei(log(x))*e^5 + 2*(x^3*e^10 + ((log(5) + 8)*e ^10 - e^5)*x^2 + 4*(2*(log(5) + 2)*e^10 - e^5)*x + 16*e^10*log(5))*log(x) - 2*(x^3*e^10 + ((log(5) + 8)*e^10 - e^5)*x^2 + 4*(2*(log(5) + 2)*e^10 - e ^5)*x - (x^2*e^10 + 8*x*e^10 + 16*e^10)*log(x))*log(log(x)) - 2*integrate( (x^2*e^5 + 4*x*e^5 + 16*e^10*log(5))/(x*log(x)), x)
Leaf count of result is larger than twice the leaf count of optimal. 338 vs. \(2 (28) = 56\).
Time = 0.35 (sec) , antiderivative size = 338, normalized size of antiderivative = 12.07 \[ \int \frac {-2 x^2 \log (x) \log \left (\frac {5 e^x \log (x)}{x}\right )+e^{10} \log ^2\left (\frac {5 e^x \log (x)}{x}\right ) \left (-32-16 x-2 x^2+\left (32-16 x-14 x^2-2 x^3\right ) \log (x)+\left (-8 x-2 x^2\right ) \log (x) \log \left (\frac {5 e^x \log (x)}{x}\right )\right )+e^5 \log \left (\frac {5 e^x \log (x)}{x}\right ) \left (8 x+2 x^2+\left (-8 x+6 x^2+2 x^3\right ) \log (x)+\left (8 x+4 x^2\right ) \log (x) \log \left (\frac {5 e^x \log (x)}{x}\right )\right )}{x \log (x) \log \left (\frac {5 e^x \log (x)}{x}\right )} \, dx=-x^{4} e^{10} - 2 \, x^{3} e^{10} \log \left (5\right ) - x^{2} e^{10} \log \left (5\right )^{2} + 2 \, x^{3} e^{10} \log \left (x\right ) + 2 \, x^{2} e^{10} \log \left (5\right ) \log \left (x\right ) - x^{2} e^{10} \log \left (x\right )^{2} - 2 \, x^{3} e^{10} \log \left (\log \left (x\right )\right ) - 2 \, x^{2} e^{10} \log \left (5\right ) \log \left (\log \left (x\right )\right ) + 2 \, x^{2} e^{10} \log \left (x\right ) \log \left (\log \left (x\right )\right ) - x^{2} e^{10} \log \left (\log \left (x\right )\right )^{2} - 8 \, x^{3} e^{10} + 2 \, x^{3} e^{5} - 16 \, x^{2} e^{10} \log \left (5\right ) + 2 \, x^{2} e^{5} \log \left (5\right ) - 8 \, x e^{10} \log \left (5\right )^{2} + 16 \, x^{2} e^{10} \log \left (x\right ) - 2 \, x^{2} e^{5} \log \left (x\right ) + 16 \, x e^{10} \log \left (5\right ) \log \left (x\right ) - 8 \, x e^{10} \log \left (x\right )^{2} - 16 \, x^{2} e^{10} \log \left (\log \left (x\right )\right ) + 2 \, x^{2} e^{5} \log \left (\log \left (x\right )\right ) - 16 \, x e^{10} \log \left (5\right ) \log \left (\log \left (x\right )\right ) + 16 \, x e^{10} \log \left (x\right ) \log \left (\log \left (x\right )\right ) - 8 \, x e^{10} \log \left (\log \left (x\right )\right )^{2} - 16 \, x^{2} e^{10} + 8 \, x^{2} e^{5} - 32 \, x e^{10} \log \left (5\right ) + 8 \, x e^{5} \log \left (5\right ) + 32 \, x e^{10} \log \left (x\right ) - 8 \, x e^{5} \log \left (x\right ) + 32 \, e^{10} \log \left (5\right ) \log \left (x\right ) - 16 \, e^{10} \log \left (x\right )^{2} - 32 \, x e^{10} \log \left (\log \left (x\right )\right ) + 8 \, x e^{5} \log \left (\log \left (x\right )\right ) - 32 \, e^{10} \log \left (5\right ) \log \left (\log \left (x\right )\right ) + 32 \, e^{10} \log \left (x\right ) \log \left (\log \left (x\right )\right ) - 16 \, e^{10} \log \left (\log \left (x\right )\right )^{2} - x^{2} \]
integrate((((-2*x^2-8*x)*log(x)*log(5*exp(x)*log(x)/x)+(-2*x^3-14*x^2-16*x +32)*log(x)-2*x^2-16*x-32)*exp(log(log(5*exp(x)*log(x)/x))+5)^2+((4*x^2+8* x)*log(x)*log(5*exp(x)*log(x)/x)+(2*x^3+6*x^2-8*x)*log(x)+2*x^2+8*x)*exp(l og(log(5*exp(x)*log(x)/x))+5)-2*x^2*log(x)*log(5*exp(x)*log(x)/x))/x/log(x )/log(5*exp(x)*log(x)/x),x, algorithm=\
-x^4*e^10 - 2*x^3*e^10*log(5) - x^2*e^10*log(5)^2 + 2*x^3*e^10*log(x) + 2* x^2*e^10*log(5)*log(x) - x^2*e^10*log(x)^2 - 2*x^3*e^10*log(log(x)) - 2*x^ 2*e^10*log(5)*log(log(x)) + 2*x^2*e^10*log(x)*log(log(x)) - x^2*e^10*log(l og(x))^2 - 8*x^3*e^10 + 2*x^3*e^5 - 16*x^2*e^10*log(5) + 2*x^2*e^5*log(5) - 8*x*e^10*log(5)^2 + 16*x^2*e^10*log(x) - 2*x^2*e^5*log(x) + 16*x*e^10*lo g(5)*log(x) - 8*x*e^10*log(x)^2 - 16*x^2*e^10*log(log(x)) + 2*x^2*e^5*log( log(x)) - 16*x*e^10*log(5)*log(log(x)) + 16*x*e^10*log(x)*log(log(x)) - 8* x*e^10*log(log(x))^2 - 16*x^2*e^10 + 8*x^2*e^5 - 32*x*e^10*log(5) + 8*x*e^ 5*log(5) + 32*x*e^10*log(x) - 8*x*e^5*log(x) + 32*e^10*log(5)*log(x) - 16* e^10*log(x)^2 - 32*x*e^10*log(log(x)) + 8*x*e^5*log(log(x)) - 32*e^10*log( 5)*log(log(x)) + 32*e^10*log(x)*log(log(x)) - 16*e^10*log(log(x))^2 - x^2
Time = 9.40 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.29 \[ \int \frac {-2 x^2 \log (x) \log \left (\frac {5 e^x \log (x)}{x}\right )+e^{10} \log ^2\left (\frac {5 e^x \log (x)}{x}\right ) \left (-32-16 x-2 x^2+\left (32-16 x-14 x^2-2 x^3\right ) \log (x)+\left (-8 x-2 x^2\right ) \log (x) \log \left (\frac {5 e^x \log (x)}{x}\right )\right )+e^5 \log \left (\frac {5 e^x \log (x)}{x}\right ) \left (8 x+2 x^2+\left (-8 x+6 x^2+2 x^3\right ) \log (x)+\left (8 x+4 x^2\right ) \log (x) \log \left (\frac {5 e^x \log (x)}{x}\right )\right )}{x \log (x) \log \left (\frac {5 e^x \log (x)}{x}\right )} \, dx=-{\left (4\,\ln \left (\frac {5\,{\mathrm {e}}^x\,\ln \left (x\right )}{x}\right )\,{\mathrm {e}}^5-x+x\,\ln \left (\frac {5\,{\mathrm {e}}^x\,\ln \left (x\right )}{x}\right )\,{\mathrm {e}}^5\right )}^2 \]
int(-(exp(2*log(log((5*exp(x)*log(x))/x)) + 10)*(16*x + 2*x^2 + log(x)*(16 *x + 14*x^2 + 2*x^3 - 32) + log((5*exp(x)*log(x))/x)*log(x)*(8*x + 2*x^2) + 32) - exp(log(log((5*exp(x)*log(x))/x)) + 5)*(8*x + 2*x^2 + log(x)*(6*x^ 2 - 8*x + 2*x^3) + log((5*exp(x)*log(x))/x)*log(x)*(8*x + 4*x^2)) + 2*x^2* log((5*exp(x)*log(x))/x)*log(x))/(x*log((5*exp(x)*log(x))/x)*log(x)),x)