Integrand size = 117, antiderivative size = 23 \[ \int \frac {-10 \log (5 x)+\left (-5 x-10 x^2 \log (5 x)\right ) \log \left (x^2\right )-5 \log \left (x^2\right ) \log \left (e^{x^2} \log \left (x^2\right )\right )}{\left (4 x+4 x^2+x^3\right ) \log \left (x^2\right )+\left (4 x+2 x^2\right ) \log (5 x) \log \left (x^2\right ) \log \left (e^{x^2} \log \left (x^2\right )\right )+x \log ^2(5 x) \log \left (x^2\right ) \log ^2\left (e^{x^2} \log \left (x^2\right )\right )} \, dx=\frac {5}{2+x+\log (5 x) \log \left (e^{x^2} \log \left (x^2\right )\right )} \]
Time = 0.34 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int \frac {-10 \log (5 x)+\left (-5 x-10 x^2 \log (5 x)\right ) \log \left (x^2\right )-5 \log \left (x^2\right ) \log \left (e^{x^2} \log \left (x^2\right )\right )}{\left (4 x+4 x^2+x^3\right ) \log \left (x^2\right )+\left (4 x+2 x^2\right ) \log (5 x) \log \left (x^2\right ) \log \left (e^{x^2} \log \left (x^2\right )\right )+x \log ^2(5 x) \log \left (x^2\right ) \log ^2\left (e^{x^2} \log \left (x^2\right )\right )} \, dx=\frac {5}{2+x+\log (5 x) \log \left (e^{x^2} \log \left (x^2\right )\right )} \]
Integrate[(-10*Log[5*x] + (-5*x - 10*x^2*Log[5*x])*Log[x^2] - 5*Log[x^2]*L og[E^x^2*Log[x^2]])/((4*x + 4*x^2 + x^3)*Log[x^2] + (4*x + 2*x^2)*Log[5*x] *Log[x^2]*Log[E^x^2*Log[x^2]] + x*Log[5*x]^2*Log[x^2]*Log[E^x^2*Log[x^2]]^ 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.
| \(\displaystyle \int \frac {\left (-10 x^2 \log (5 x)-5 x\right ) \log \left (x^2\right )-5 \log \left (x^2\right ) \log \left (e^{x^2} \log \left (x^2\right )\right )-10 \log (5 x)}{x \log ^2(5 x) \log \left (x^2\right ) \log ^2\left (e^{x^2} \log \left (x^2\right )\right )+\left (2 x^2+4 x\right ) \log (5 x) \log \left (x^2\right ) \log \left (e^{x^2} \log \left (x^2\right )\right )+\left (x^3+4 x^2+4 x\right ) \log \left (x^2\right )} \, dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {5 \left (-2 \log (5 x) \left (x^2 \log \left (x^2\right )+1\right )-\log \left (x^2\right ) \left (\log \left (e^{x^2} \log \left (x^2\right )\right )+x\right )\right )}{x \log \left (x^2\right ) \left (\log (5 x) \log \left (e^{x^2} \log \left (x^2\right )\right )+x+2\right )^2}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 5 \int -\frac {2 \log (5 x) \left (\log \left (x^2\right ) x^2+1\right )+\log \left (x^2\right ) \left (x+\log \left (e^{x^2} \log \left (x^2\right )\right )\right )}{x \log \left (x^2\right ) \left (x+\log (5 x) \log \left (e^{x^2} \log \left (x^2\right )\right )+2\right )^2}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -5 \int \frac {2 \log (5 x) \left (\log \left (x^2\right ) x^2+1\right )+\log \left (x^2\right ) \left (x+\log \left (e^{x^2} \log \left (x^2\right )\right )\right )}{x \log \left (x^2\right ) \left (x+\log (5 x) \log \left (e^{x^2} \log \left (x^2\right )\right )+2\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -5 \int \left (\frac {2 x^2 \log \left (x^2\right ) \log ^2(5 x)+2 \log ^2(5 x)+x \log \left (x^2\right ) \log (5 x)-x \log \left (x^2\right )-2 \log \left (x^2\right )}{x \log (5 x) \log \left (x^2\right ) \left (x+\log (5 x) \log \left (e^{x^2} \log \left (x^2\right )\right )+2\right )^2}+\frac {1}{x \log (5 x) \left (x+\log (5 x) \log \left (e^{x^2} \log \left (x^2\right )\right )+2\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -5 \left (\int \frac {1}{\left (x+\log (5 x) \log \left (e^{x^2} \log \left (x^2\right )\right )+2\right )^2}dx-\int \frac {1}{\log (5 x) \left (x+\log (5 x) \log \left (e^{x^2} \log \left (x^2\right )\right )+2\right )^2}dx-2 \int \frac {1}{x \log (5 x) \left (x+\log (5 x) \log \left (e^{x^2} \log \left (x^2\right )\right )+2\right )^2}dx+2 \int \frac {x \log (5 x)}{\left (x+\log (5 x) \log \left (e^{x^2} \log \left (x^2\right )\right )+2\right )^2}dx+2 \int \frac {\log (5 x)}{x \log \left (x^2\right ) \left (x+\log (5 x) \log \left (e^{x^2} \log \left (x^2\right )\right )+2\right )^2}dx+\int \frac {1}{x \log (5 x) \left (x+\log (5 x) \log \left (e^{x^2} \log \left (x^2\right )\right )+2\right )}dx\right )\) |
Int[(-10*Log[5*x] + (-5*x - 10*x^2*Log[5*x])*Log[x^2] - 5*Log[x^2]*Log[E^x ^2*Log[x^2]])/((4*x + 4*x^2 + x^3)*Log[x^2] + (4*x + 2*x^2)*Log[5*x]*Log[x ^2]*Log[E^x^2*Log[x^2]] + x*Log[5*x]^2*Log[x^2]*Log[E^x^2*Log[x^2]]^2),x]
3.30.15.3.1 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]]
Timed out.
\[\int \frac {-5 \ln \left (x^{2}\right ) \ln \left ({\mathrm e}^{x^{2}} \ln \left (x^{2}\right )\right )+\left (-10 x^{2} \ln \left (5 x \right )-5 x \right ) \ln \left (x^{2}\right )-10 \ln \left (5 x \right )}{x \ln \left (5 x \right )^{2} \ln \left (x^{2}\right ) \ln \left ({\mathrm e}^{x^{2}} \ln \left (x^{2}\right )\right )^{2}+\left (2 x^{2}+4 x \right ) \ln \left (5 x \right ) \ln \left (x^{2}\right ) \ln \left ({\mathrm e}^{x^{2}} \ln \left (x^{2}\right )\right )+\left (x^{3}+4 x^{2}+4 x \right ) \ln \left (x^{2}\right )}d x\]
int((-5*ln(x^2)*ln(exp(x^2)*ln(x^2))+(-10*x^2*ln(5*x)-5*x)*ln(x^2)-10*ln(5 *x))/(x*ln(5*x)^2*ln(x^2)*ln(exp(x^2)*ln(x^2))^2+(2*x^2+4*x)*ln(5*x)*ln(x^ 2)*ln(exp(x^2)*ln(x^2))+(x^3+4*x^2+4*x)*ln(x^2)),x)
int((-5*ln(x^2)*ln(exp(x^2)*ln(x^2))+(-10*x^2*ln(5*x)-5*x)*ln(x^2)-10*ln(5 *x))/(x*ln(5*x)^2*ln(x^2)*ln(exp(x^2)*ln(x^2))^2+(2*x^2+4*x)*ln(5*x)*ln(x^ 2)*ln(exp(x^2)*ln(x^2))+(x^3+4*x^2+4*x)*ln(x^2)),x)
Time = 0.26 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.39 \[ \int \frac {-10 \log (5 x)+\left (-5 x-10 x^2 \log (5 x)\right ) \log \left (x^2\right )-5 \log \left (x^2\right ) \log \left (e^{x^2} \log \left (x^2\right )\right )}{\left (4 x+4 x^2+x^3\right ) \log \left (x^2\right )+\left (4 x+2 x^2\right ) \log (5 x) \log \left (x^2\right ) \log \left (e^{x^2} \log \left (x^2\right )\right )+x \log ^2(5 x) \log \left (x^2\right ) \log ^2\left (e^{x^2} \log \left (x^2\right )\right )} \, dx=\frac {5}{\log \left (-2 \, e^{\left (x^{2}\right )} \log \left (5\right ) + 2 \, e^{\left (x^{2}\right )} \log \left (5 \, x\right )\right ) \log \left (5 \, x\right ) + x + 2} \]
integrate((-5*log(x^2)*log(exp(x^2)*log(x^2))+(-10*x^2*log(5*x)-5*x)*log(x ^2)-10*log(5*x))/(x*log(5*x)^2*log(x^2)*log(exp(x^2)*log(x^2))^2+(2*x^2+4* x)*log(5*x)*log(x^2)*log(exp(x^2)*log(x^2))+(x^3+4*x^2+4*x)*log(x^2)),x, a lgorithm=\
Time = 0.27 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.13 \[ \int \frac {-10 \log (5 x)+\left (-5 x-10 x^2 \log (5 x)\right ) \log \left (x^2\right )-5 \log \left (x^2\right ) \log \left (e^{x^2} \log \left (x^2\right )\right )}{\left (4 x+4 x^2+x^3\right ) \log \left (x^2\right )+\left (4 x+2 x^2\right ) \log (5 x) \log \left (x^2\right ) \log \left (e^{x^2} \log \left (x^2\right )\right )+x \log ^2(5 x) \log \left (x^2\right ) \log ^2\left (e^{x^2} \log \left (x^2\right )\right )} \, dx=\frac {5}{x + \log {\left (5 x \right )} \log {\left (\left (2 \log {\left (5 x \right )} - \log {\left (25 \right )}\right ) e^{x^{2}} \right )} + 2} \]
integrate((-5*ln(x**2)*ln(exp(x**2)*ln(x**2))+(-10*x**2*ln(5*x)-5*x)*ln(x* *2)-10*ln(5*x))/(x*ln(5*x)**2*ln(x**2)*ln(exp(x**2)*ln(x**2))**2+(2*x**2+4 *x)*ln(5*x)*ln(x**2)*ln(exp(x**2)*ln(x**2))+(x**3+4*x**2+4*x)*ln(x**2)),x)
Time = 0.38 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.57 \[ \int \frac {-10 \log (5 x)+\left (-5 x-10 x^2 \log (5 x)\right ) \log \left (x^2\right )-5 \log \left (x^2\right ) \log \left (e^{x^2} \log \left (x^2\right )\right )}{\left (4 x+4 x^2+x^3\right ) \log \left (x^2\right )+\left (4 x+2 x^2\right ) \log (5 x) \log \left (x^2\right ) \log \left (e^{x^2} \log \left (x^2\right )\right )+x \log ^2(5 x) \log \left (x^2\right ) \log ^2\left (e^{x^2} \log \left (x^2\right )\right )} \, dx=\frac {5}{x^{2} \log \left (5\right ) + \log \left (5\right ) \log \left (2\right ) + {\left (x^{2} + \log \left (2\right )\right )} \log \left (x\right ) + {\left (\log \left (5\right ) + \log \left (x\right )\right )} \log \left (\log \left (x\right )\right ) + x + 2} \]
integrate((-5*log(x^2)*log(exp(x^2)*log(x^2))+(-10*x^2*log(5*x)-5*x)*log(x ^2)-10*log(5*x))/(x*log(5*x)^2*log(x^2)*log(exp(x^2)*log(x^2))^2+(2*x^2+4* x)*log(5*x)*log(x^2)*log(exp(x^2)*log(x^2))+(x^3+4*x^2+4*x)*log(x^2)),x, a lgorithm=\
5/(x^2*log(5) + log(5)*log(2) + (x^2 + log(2))*log(x) + (log(5) + log(x))* log(log(x)) + x + 2)
Leaf count of result is larger than twice the leaf count of optimal. 697 vs. \(2 (22) = 44\).
Time = 0.67 (sec) , antiderivative size = 697, normalized size of antiderivative = 30.30 \[ \int \frac {-10 \log (5 x)+\left (-5 x-10 x^2 \log (5 x)\right ) \log \left (x^2\right )-5 \log \left (x^2\right ) \log \left (e^{x^2} \log \left (x^2\right )\right )}{\left (4 x+4 x^2+x^3\right ) \log \left (x^2\right )+\left (4 x+2 x^2\right ) \log (5 x) \log \left (x^2\right ) \log \left (e^{x^2} \log \left (x^2\right )\right )+x \log ^2(5 x) \log \left (x^2\right ) \log ^2\left (e^{x^2} \log \left (x^2\right )\right )} \, dx=\text {Too large to display} \]
integrate((-5*log(x^2)*log(exp(x^2)*log(x^2))+(-10*x^2*log(5*x)-5*x)*log(x ^2)-10*log(5*x))/(x*log(5*x)^2*log(x^2)*log(exp(x^2)*log(x^2))^2+(2*x^2+4* x)*log(5*x)*log(x^2)*log(exp(x^2)*log(x^2))+(x^3+4*x^2+4*x)*log(x^2)),x, a lgorithm=\
5*(2*x^2*log(5)^2*log(x^2)*log(x) + 4*x^2*log(5)*log(x^2)*log(x)^2 + 2*x^2 *log(x^2)*log(x)^3 + x*log(5)*log(x^2)*log(x) + x*log(x^2)*log(x)^2 + log( 5)^2*log(x^2) - x*log(x^2)*log(x) + 2*log(5)*log(x^2)*log(x) + log(x^2)*lo g(x)^2 - 2*log(x^2)*log(x))/(2*x^4*log(5)^3*log(x^2)*log(x) + 6*x^4*log(5) ^2*log(x^2)*log(x)^2 + 6*x^4*log(5)*log(x^2)*log(x)^3 + 2*x^4*log(x^2)*log (x)^4 + 2*x^2*log(5)^3*log(x^2)*log(x)*log(log(x^2)) + 6*x^2*log(5)^2*log( x^2)*log(x)^2*log(log(x^2)) + 6*x^2*log(5)*log(x^2)*log(x)^3*log(log(x^2)) + 2*x^2*log(x^2)*log(x)^4*log(log(x^2)) + 3*x^3*log(5)^2*log(x^2)*log(x) + 6*x^3*log(5)*log(x^2)*log(x)^2 + 3*x^3*log(x^2)*log(x)^3 + 2*x^2*log(5)^ 3*log(x) - x^3*log(5)*log(x^2)*log(x) + 4*x^2*log(5)^2*log(x^2)*log(x) + 6 *x^2*log(5)^2*log(x)^2 - x^3*log(x^2)*log(x)^2 + 8*x^2*log(5)*log(x^2)*log (x)^2 + 6*x^2*log(5)*log(x)^3 + 4*x^2*log(x^2)*log(x)^3 + 2*x^2*log(x)^4 + x*log(5)^2*log(x^2)*log(x)*log(log(x^2)) + 2*x*log(5)*log(x^2)*log(x)^2*l og(log(x^2)) + x*log(x^2)*log(x)^3*log(log(x^2)) - x^2*log(5)*log(x^2)*log (x) - x^2*log(x^2)*log(x)^2 + 2*log(5)^3*log(x)*log(log(x^2)) - x*log(5)*l og(x^2)*log(x)*log(log(x^2)) + 6*log(5)^2*log(x)^2*log(log(x^2)) - x*log(x ^2)*log(x)^2*log(log(x^2)) + 6*log(5)*log(x)^3*log(log(x^2)) + 2*log(x)^4* log(log(x^2)) + 2*x*log(5)^2*log(x) - x^2*log(x^2)*log(x) + 2*x*log(5)*log (x^2)*log(x) + 4*x*log(5)*log(x)^2 + 2*x*log(x^2)*log(x)^2 + 2*x*log(x)^3 - 2*log(5)*log(x^2)*log(x)*log(log(x^2)) - 2*log(x^2)*log(x)^2*log(log(...
Time = 9.18 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.96 \[ \int \frac {-10 \log (5 x)+\left (-5 x-10 x^2 \log (5 x)\right ) \log \left (x^2\right )-5 \log \left (x^2\right ) \log \left (e^{x^2} \log \left (x^2\right )\right )}{\left (4 x+4 x^2+x^3\right ) \log \left (x^2\right )+\left (4 x+2 x^2\right ) \log (5 x) \log \left (x^2\right ) \log \left (e^{x^2} \log \left (x^2\right )\right )+x \log ^2(5 x) \log \left (x^2\right ) \log ^2\left (e^{x^2} \log \left (x^2\right )\right )} \, dx=\frac {5}{x+\ln \left (5\,x\right )\,\ln \left (\ln \left (x^2\right )\,{\mathrm {e}}^{x^2}\right )+2} \]