Integrand size = 106, antiderivative size = 33 \[ \int \frac {25 x^2+10 x^5+x^8+e^5 \left (25 x+5 x^4\right )+\left (-25 x^2+15 e^5 x^4-10 x^5-x^8\right ) \log (x)+\left (-25-10 x^3-x^6+\left (-25-10 x^3-x^6\right ) \log (x)\right ) \log (\log (3))}{\left (25 x^2+10 x^5+x^8\right ) \log ^2(x)} \, dx=\frac {-x+\frac {-\frac {5 e^5}{\frac {5}{x}+x^2}+\log (\log (3))}{x}}{\log (x)} \] Output:
((ln(ln(3))-5*exp(5)/(x^2+5/x))/x-x)/ln(x)
Time = 0.03 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.85 \[ \int \frac {25 x^2+10 x^5+x^8+e^5 \left (25 x+5 x^4\right )+\left (-25 x^2+15 e^5 x^4-10 x^5-x^8\right ) \log (x)+\left (-25-10 x^3-x^6+\left (-25-10 x^3-x^6\right ) \log (x)\right ) \log (\log (3))}{\left (25 x^2+10 x^5+x^8\right ) \log ^2(x)} \, dx=\frac {-x-\frac {5 e^5}{5+x^3}+\frac {\log (\log (3))}{x}}{\log (x)} \] Input:
Integrate[(25*x^2 + 10*x^5 + x^8 + E^5*(25*x + 5*x^4) + (-25*x^2 + 15*E^5* x^4 - 10*x^5 - x^8)*Log[x] + (-25 - 10*x^3 - x^6 + (-25 - 10*x^3 - x^6)*Lo g[x])*Log[Log[3]])/((25*x^2 + 10*x^5 + x^8)*Log[x]^2),x]
Output:
(-x - (5*E^5)/(5 + x^3) + Log[Log[3]]/x)/Log[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.
\(\displaystyle \int \frac {x^8+10 x^5+e^5 \left (5 x^4+25 x\right )+25 x^2+\log (\log (3)) \left (-x^6-10 x^3+\left (-x^6-10 x^3-25\right ) \log (x)-25\right )+\left (-x^8-10 x^5+15 e^5 x^4-25 x^2\right ) \log (x)}{\left (x^8+10 x^5+25 x^2\right ) \log ^2(x)} \, dx\) |
\(\Big \downarrow \) 2026 |
\(\displaystyle \int \frac {x^8+10 x^5+e^5 \left (5 x^4+25 x\right )+25 x^2+\log (\log (3)) \left (-x^6-10 x^3+\left (-x^6-10 x^3-25\right ) \log (x)-25\right )+\left (-x^8-10 x^5+15 e^5 x^4-25 x^2\right ) \log (x)}{x^2 \left (x^6+10 x^3+25\right ) \log ^2(x)}dx\) |
\(\Big \downarrow \) 1380 |
\(\displaystyle \int \frac {x^8+10 x^5+5 e^5 \left (x^4+5 x\right )+25 x^2-\log (\log (3)) \left (x^6+10 x^3+\left (x^6+10 x^3+25\right ) \log (x)+25\right )-\left (x^8+10 x^5-15 e^5 x^4+25 x^2\right ) \log (x)}{x^2 \left (x^3+5\right )^2 \log ^2(x)}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {x^5-x^3 \log (\log (3))+5 x^2+5 e^5 x-5 \log (\log (3))}{x^2 \left (x^3+5\right ) \log ^2(x)}+\frac {-x^8-x^6 \log (\log (3))-10 x^5+15 e^5 x^4-10 x^3 \log (\log (3))-25 x^2-25 \log (\log (3))}{x^2 \left (x^3+5\right )^2 \log (x)}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \int \frac {x^5-\log (\log (3)) x^3+5 x^2+5 e^5 x-5 \log (\log (3))}{x^2 \left (x^3+5\right ) \log ^2(x)}dx+\int \frac {-x^8-\log (\log (3)) x^6-10 x^5+15 e^5 x^4-10 \log (\log (3)) x^3-25 x^2-25 \log (\log (3))}{x^2 \left (x^3+5\right )^2 \log (x)}dx\) |
Input:
Int[(25*x^2 + 10*x^5 + x^8 + E^5*(25*x + 5*x^4) + (-25*x^2 + 15*E^5*x^4 - 10*x^5 - x^8)*Log[x] + (-25 - 10*x^3 - x^6 + (-25 - 10*x^3 - x^6)*Log[x])* Log[Log[3]])/((25*x^2 + 10*x^5 + x^8)*Log[x]^2),x]
Output:
$Aborted
Time = 14.13 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.33
method | result | size |
default | \(\frac {\ln \left (\ln \left (3\right )\right ) x^{3}-5 x^{2}-x^{5}-5 x \,{\mathrm e}^{5}+5 \ln \left (\ln \left (3\right )\right )}{x \left (x^{3}+5\right ) \ln \left (x \right )}\) | \(44\) |
risch | \(\frac {\ln \left (\ln \left (3\right )\right ) x^{3}-5 x^{2}-x^{5}-5 x \,{\mathrm e}^{5}+5 \ln \left (\ln \left (3\right )\right )}{x \left (x^{3}+5\right ) \ln \left (x \right )}\) | \(44\) |
parallelrisch | \(\frac {\ln \left (\ln \left (3\right )\right ) x^{3}-5 x^{2}-x^{5}-5 x \,{\mathrm e}^{5}+5 \ln \left (\ln \left (3\right )\right )}{x \left (x^{3}+5\right ) \ln \left (x \right )}\) | \(44\) |
Input:
int((((-x^6-10*x^3-25)*ln(x)-x^6-10*x^3-25)*ln(ln(3))+(15*x^4*exp(5)-x^8-1 0*x^5-25*x^2)*ln(x)+(5*x^4+25*x)*exp(5)+x^8+10*x^5+25*x^2)/(x^8+10*x^5+25* x^2)/ln(x)^2,x,method=_RETURNVERBOSE)
Output:
(ln(ln(3))*x^3-5*x^2-x^5-5*x*exp(5)+5*ln(ln(3)))/x/(x^3+5)/ln(x)
Time = 0.06 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.18 \[ \int \frac {25 x^2+10 x^5+x^8+e^5 \left (25 x+5 x^4\right )+\left (-25 x^2+15 e^5 x^4-10 x^5-x^8\right ) \log (x)+\left (-25-10 x^3-x^6+\left (-25-10 x^3-x^6\right ) \log (x)\right ) \log (\log (3))}{\left (25 x^2+10 x^5+x^8\right ) \log ^2(x)} \, dx=-\frac {x^{5} + 5 \, x^{2} + 5 \, x e^{5} - {\left (x^{3} + 5\right )} \log \left (\log \left (3\right )\right )}{{\left (x^{4} + 5 \, x\right )} \log \left (x\right )} \] Input:
integrate((((-x^6-10*x^3-25)*log(x)-x^6-10*x^3-25)*log(log(3))+(15*x^4*exp (5)-x^8-10*x^5-25*x^2)*log(x)+(5*x^4+25*x)*exp(5)+x^8+10*x^5+25*x^2)/(x^8+ 10*x^5+25*x^2)/log(x)^2,x, algorithm="fricas")
Output:
-(x^5 + 5*x^2 + 5*x*e^5 - (x^3 + 5)*log(log(3)))/((x^4 + 5*x)*log(x))
Time = 0.11 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.18 \[ \int \frac {25 x^2+10 x^5+x^8+e^5 \left (25 x+5 x^4\right )+\left (-25 x^2+15 e^5 x^4-10 x^5-x^8\right ) \log (x)+\left (-25-10 x^3-x^6+\left (-25-10 x^3-x^6\right ) \log (x)\right ) \log (\log (3))}{\left (25 x^2+10 x^5+x^8\right ) \log ^2(x)} \, dx=\frac {- x^{5} + x^{3} \log {\left (\log {\left (3 \right )} \right )} - 5 x^{2} - 5 x e^{5} + 5 \log {\left (\log {\left (3 \right )} \right )}}{\left (x^{4} + 5 x\right ) \log {\left (x \right )}} \] Input:
integrate((((-x**6-10*x**3-25)*ln(x)-x**6-10*x**3-25)*ln(ln(3))+(15*x**4*e xp(5)-x**8-10*x**5-25*x**2)*ln(x)+(5*x**4+25*x)*exp(5)+x**8+10*x**5+25*x** 2)/(x**8+10*x**5+25*x**2)/ln(x)**2,x)
Output:
(-x**5 + x**3*log(log(3)) - 5*x**2 - 5*x*exp(5) + 5*log(log(3)))/((x**4 + 5*x)*log(x))
Time = 0.15 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.27 \[ \int \frac {25 x^2+10 x^5+x^8+e^5 \left (25 x+5 x^4\right )+\left (-25 x^2+15 e^5 x^4-10 x^5-x^8\right ) \log (x)+\left (-25-10 x^3-x^6+\left (-25-10 x^3-x^6\right ) \log (x)\right ) \log (\log (3))}{\left (25 x^2+10 x^5+x^8\right ) \log ^2(x)} \, dx=-\frac {x^{5} - x^{3} \log \left (\log \left (3\right )\right ) + 5 \, x^{2} + 5 \, x e^{5} - 5 \, \log \left (\log \left (3\right )\right )}{{\left (x^{4} + 5 \, x\right )} \log \left (x\right )} \] Input:
integrate((((-x^6-10*x^3-25)*log(x)-x^6-10*x^3-25)*log(log(3))+(15*x^4*exp (5)-x^8-10*x^5-25*x^2)*log(x)+(5*x^4+25*x)*exp(5)+x^8+10*x^5+25*x^2)/(x^8+ 10*x^5+25*x^2)/log(x)^2,x, algorithm="maxima")
Output:
-(x^5 - x^3*log(log(3)) + 5*x^2 + 5*x*e^5 - 5*log(log(3)))/((x^4 + 5*x)*lo g(x))
Time = 0.13 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.30 \[ \int \frac {25 x^2+10 x^5+x^8+e^5 \left (25 x+5 x^4\right )+\left (-25 x^2+15 e^5 x^4-10 x^5-x^8\right ) \log (x)+\left (-25-10 x^3-x^6+\left (-25-10 x^3-x^6\right ) \log (x)\right ) \log (\log (3))}{\left (25 x^2+10 x^5+x^8\right ) \log ^2(x)} \, dx=-\frac {x^{5} - x^{3} \log \left (\log \left (3\right )\right ) + 5 \, x^{2} + 5 \, x e^{5} - 5 \, \log \left (\log \left (3\right )\right )}{x^{4} \log \left (x\right ) + 5 \, x \log \left (x\right )} \] Input:
integrate((((-x^6-10*x^3-25)*log(x)-x^6-10*x^3-25)*log(log(3))+(15*x^4*exp (5)-x^8-10*x^5-25*x^2)*log(x)+(5*x^4+25*x)*exp(5)+x^8+10*x^5+25*x^2)/(x^8+ 10*x^5+25*x^2)/log(x)^2,x, algorithm="giac")
Output:
-(x^5 - x^3*log(log(3)) + 5*x^2 + 5*x*e^5 - 5*log(log(3)))/(x^4*log(x) + 5 *x*log(x))
Time = 1.99 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.30 \[ \int \frac {25 x^2+10 x^5+x^8+e^5 \left (25 x+5 x^4\right )+\left (-25 x^2+15 e^5 x^4-10 x^5-x^8\right ) \log (x)+\left (-25-10 x^3-x^6+\left (-25-10 x^3-x^6\right ) \log (x)\right ) \log (\log (3))}{\left (25 x^2+10 x^5+x^8\right ) \log ^2(x)} \, dx=-\frac {x^5-\ln \left (\ln \left (3\right )\right )\,x^3+5\,x^2+5\,{\mathrm {e}}^5\,x-5\,\ln \left (\ln \left (3\right )\right )}{x\,\ln \left (x\right )\,\left (x^3+5\right )} \] Input:
int((exp(5)*(25*x + 5*x^4) - log(log(3))*(log(x)*(10*x^3 + x^6 + 25) + 10* x^3 + x^6 + 25) - log(x)*(25*x^2 - 15*x^4*exp(5) + 10*x^5 + x^8) + 25*x^2 + 10*x^5 + x^8)/(log(x)^2*(25*x^2 + 10*x^5 + x^8)),x)
Output:
-(5*x*exp(5) - x^3*log(log(3)) - 5*log(log(3)) + 5*x^2 + x^5)/(x*log(x)*(x ^3 + 5))
Time = 0.18 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.33 \[ \int \frac {25 x^2+10 x^5+x^8+e^5 \left (25 x+5 x^4\right )+\left (-25 x^2+15 e^5 x^4-10 x^5-x^8\right ) \log (x)+\left (-25-10 x^3-x^6+\left (-25-10 x^3-x^6\right ) \log (x)\right ) \log (\log (3))}{\left (25 x^2+10 x^5+x^8\right ) \log ^2(x)} \, dx=\frac {\mathrm {log}\left (\mathrm {log}\left (3\right )\right ) x^{3}+5 \,\mathrm {log}\left (\mathrm {log}\left (3\right )\right )-5 e^{5} x -x^{5}-5 x^{2}}{\mathrm {log}\left (x \right ) x \left (x^{3}+5\right )} \] Input:
int((((-x^6-10*x^3-25)*log(x)-x^6-10*x^3-25)*log(log(3))+(15*x^4*exp(5)-x^ 8-10*x^5-25*x^2)*log(x)+(5*x^4+25*x)*exp(5)+x^8+10*x^5+25*x^2)/(x^8+10*x^5 +25*x^2)/log(x)^2,x)
Output:
(log(log(3))*x**3 + 5*log(log(3)) - 5*e**5*x - x**5 - 5*x**2)/(log(x)*x*(x **3 + 5))