Integrand size = 112, antiderivative size = 32 \[ \int \frac {-x^2+e^6 \left (9+16 x-x^2\right )+\left (x^2+e^6 \left (-3-8 x+x^2\right )\right ) \log \left (-\frac {2 e^6 x^3}{x^2+e^6 \left (-3-8 x+x^2\right )}\right )}{\left (x^2+e^6 \left (-3-8 x+x^2\right )\right ) \log ^2\left (-\frac {2 e^6 x^3}{x^2+e^6 \left (-3-8 x+x^2\right )}\right )} \, dx=\frac {x}{\log \left (\frac {x^2}{4+\frac {3-x \left (x+\frac {x}{e^6}\right )}{2 x}}\right )} \] Output:
x/ln(x^2/(1/2*(3-x*(x/exp(3)^2+x))/x+4))
Time = 0.05 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.97 \[ \int \frac {-x^2+e^6 \left (9+16 x-x^2\right )+\left (x^2+e^6 \left (-3-8 x+x^2\right )\right ) \log \left (-\frac {2 e^6 x^3}{x^2+e^6 \left (-3-8 x+x^2\right )}\right )}{\left (x^2+e^6 \left (-3-8 x+x^2\right )\right ) \log ^2\left (-\frac {2 e^6 x^3}{x^2+e^6 \left (-3-8 x+x^2\right )}\right )} \, dx=\frac {x}{\log \left (-\frac {2 e^6 x^3}{x^2+e^6 \left (-3-8 x+x^2\right )}\right )} \] Input:
Integrate[(-x^2 + E^6*(9 + 16*x - x^2) + (x^2 + E^6*(-3 - 8*x + x^2))*Log[ (-2*E^6*x^3)/(x^2 + E^6*(-3 - 8*x + x^2))])/((x^2 + E^6*(-3 - 8*x + x^2))* Log[(-2*E^6*x^3)/(x^2 + E^6*(-3 - 8*x + x^2))]^2),x]
Output:
x/Log[(-2*E^6*x^3)/(x^2 + E^6*(-3 - 8*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.
\(\displaystyle \int \frac {-x^2+e^6 \left (-x^2+16 x+9\right )+\left (x^2+e^6 \left (x^2-8 x-3\right )\right ) \log \left (-\frac {2 e^6 x^3}{x^2+e^6 \left (x^2-8 x-3\right )}\right )}{\left (x^2+e^6 \left (x^2-8 x-3\right )\right ) \log ^2\left (-\frac {2 e^6 x^3}{x^2+e^6 \left (x^2-8 x-3\right )}\right )} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {x^2-e^6 \left (-x^2+16 x+9\right )-\left (x^2+e^6 \left (x^2-8 x-3\right )\right ) \log \left (-\frac {2 e^6 x^3}{x^2+e^6 \left (x^2-8 x-3\right )}\right )}{\left (-\left (\left (1+e^6\right ) x^2\right )+8 e^6 x+3 e^6\right ) \log ^2\left (-\frac {2 e^6 x^3}{\left (1+e^6\right ) x^2-8 e^6 x-3 e^6}\right )}dx\) |
\(\Big \downarrow \) 7279 |
\(\displaystyle \int \left (\frac {\left (1+e^6\right ) x^2-16 e^6 x-9 e^6}{\left (-\left (\left (1+e^6\right ) x^2\right )+8 e^6 x+3 e^6\right ) \log ^2\left (-\frac {2 e^6 x^3}{\left (1+e^6\right ) x^2-8 e^6 x-3 e^6}\right )}+\frac {1}{\log \left (-\frac {2 e^6 x^3}{\left (1+e^6\right ) x^2-8 e^6 x-3 e^6}\right )}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\int \frac {1}{\log ^2\left (-\frac {2 e^6 x^3}{\left (1+e^6\right ) x^2-8 e^6 x-3 e^6}\right )}dx+\frac {6 e^3 \left (1+e^6\right ) \int \frac {1}{\left (-2 \left (1+e^6\right ) x-2 e^3 \sqrt {3+19 e^6}+8 e^6\right ) \log ^2\left (-\frac {2 e^6 x^3}{\left (1+e^6\right ) x^2-8 e^6 x-3 e^6}\right )}dx}{\sqrt {3+19 e^6}}+8 e^6 \left (1+\frac {4 e^3}{\sqrt {3+19 e^6}}\right ) \int \frac {1}{\left (2 \left (1+e^6\right ) x-2 e^3 \sqrt {3+19 e^6}-8 e^6\right ) \log ^2\left (-\frac {2 e^6 x^3}{\left (1+e^6\right ) x^2-8 e^6 x-3 e^6}\right )}dx+\frac {6 e^3 \left (1+e^6\right ) \int \frac {1}{\left (2 \left (1+e^6\right ) x-2 e^3 \sqrt {3+19 e^6}-8 e^6\right ) \log ^2\left (-\frac {2 e^6 x^3}{\left (1+e^6\right ) x^2-8 e^6 x-3 e^6}\right )}dx}{\sqrt {3+19 e^6}}+8 e^6 \left (1-\frac {4 e^3}{\sqrt {3+19 e^6}}\right ) \int \frac {1}{\left (2 \left (1+e^6\right ) x+2 e^3 \sqrt {3+19 e^6}-8 e^6\right ) \log ^2\left (-\frac {2 e^6 x^3}{\left (1+e^6\right ) x^2-8 e^6 x-3 e^6}\right )}dx+\int \frac {1}{\log \left (-\frac {2 e^6 x^3}{\left (1+e^6\right ) x^2-8 e^6 x-3 e^6}\right )}dx\) |
Input:
Int[(-x^2 + E^6*(9 + 16*x - x^2) + (x^2 + E^6*(-3 - 8*x + x^2))*Log[(-2*E^ 6*x^3)/(x^2 + E^6*(-3 - 8*x + x^2))])/((x^2 + E^6*(-3 - 8*x + x^2))*Log[(- 2*E^6*x^3)/(x^2 + E^6*(-3 - 8*x + x^2))]^2),x]
Output:
$Aborted
Time = 6.38 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.94
method | result | size |
risch | \(\frac {x}{\ln \left (-\frac {2 x^{3} {\mathrm e}^{6}}{\left (x^{2}-8 x -3\right ) {\mathrm e}^{6}+x^{2}}\right )}\) | \(30\) |
norman | \(\frac {x}{\ln \left (-\frac {2 x^{3} {\mathrm e}^{6}}{\left (x^{2}-8 x -3\right ) {\mathrm e}^{6}+x^{2}}\right )}\) | \(34\) |
parallelrisch | \(\frac {x}{\ln \left (-\frac {2 x^{3} {\mathrm e}^{6}}{x^{2} {\mathrm e}^{6}-8 x \,{\mathrm e}^{6}+x^{2}-3 \,{\mathrm e}^{6}}\right )}\) | \(42\) |
Input:
int((((x^2-8*x-3)*exp(3)^2+x^2)*ln(-2*x^3*exp(3)^2/((x^2-8*x-3)*exp(3)^2+x ^2))+(-x^2+16*x+9)*exp(3)^2-x^2)/((x^2-8*x-3)*exp(3)^2+x^2)/ln(-2*x^3*exp( 3)^2/((x^2-8*x-3)*exp(3)^2+x^2))^2,x,method=_RETURNVERBOSE)
Output:
x/ln(-2*x^3*exp(6)/((x^2-8*x-3)*exp(6)+x^2))
Time = 0.07 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.91 \[ \int \frac {-x^2+e^6 \left (9+16 x-x^2\right )+\left (x^2+e^6 \left (-3-8 x+x^2\right )\right ) \log \left (-\frac {2 e^6 x^3}{x^2+e^6 \left (-3-8 x+x^2\right )}\right )}{\left (x^2+e^6 \left (-3-8 x+x^2\right )\right ) \log ^2\left (-\frac {2 e^6 x^3}{x^2+e^6 \left (-3-8 x+x^2\right )}\right )} \, dx=\frac {x}{\log \left (-\frac {2 \, x^{3} e^{6}}{x^{2} + {\left (x^{2} - 8 \, x - 3\right )} e^{6}}\right )} \] Input:
integrate((((x^2-8*x-3)*exp(3)^2+x^2)*log(-2*x^3*exp(3)^2/((x^2-8*x-3)*exp (3)^2+x^2))+(-x^2+16*x+9)*exp(3)^2-x^2)/((x^2-8*x-3)*exp(3)^2+x^2)/log(-2* x^3*exp(3)^2/((x^2-8*x-3)*exp(3)^2+x^2))^2,x, algorithm="fricas")
Output:
x/log(-2*x^3*e^6/(x^2 + (x^2 - 8*x - 3)*e^6))
Time = 0.14 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.84 \[ \int \frac {-x^2+e^6 \left (9+16 x-x^2\right )+\left (x^2+e^6 \left (-3-8 x+x^2\right )\right ) \log \left (-\frac {2 e^6 x^3}{x^2+e^6 \left (-3-8 x+x^2\right )}\right )}{\left (x^2+e^6 \left (-3-8 x+x^2\right )\right ) \log ^2\left (-\frac {2 e^6 x^3}{x^2+e^6 \left (-3-8 x+x^2\right )}\right )} \, dx=\frac {x}{\log {\left (- \frac {2 x^{3} e^{6}}{x^{2} + \left (x^{2} - 8 x - 3\right ) e^{6}} \right )}} \] Input:
integrate((((x**2-8*x-3)*exp(3)**2+x**2)*ln(-2*x**3*exp(3)**2/((x**2-8*x-3 )*exp(3)**2+x**2))+(-x**2+16*x+9)*exp(3)**2-x**2)/((x**2-8*x-3)*exp(3)**2+ x**2)/ln(-2*x**3*exp(3)**2/((x**2-8*x-3)*exp(3)**2+x**2))**2,x)
Output:
x/log(-2*x**3*exp(6)/(x**2 + (x**2 - 8*x - 3)*exp(6)))
Time = 0.17 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.06 \[ \int \frac {-x^2+e^6 \left (9+16 x-x^2\right )+\left (x^2+e^6 \left (-3-8 x+x^2\right )\right ) \log \left (-\frac {2 e^6 x^3}{x^2+e^6 \left (-3-8 x+x^2\right )}\right )}{\left (x^2+e^6 \left (-3-8 x+x^2\right )\right ) \log ^2\left (-\frac {2 e^6 x^3}{x^2+e^6 \left (-3-8 x+x^2\right )}\right )} \, dx=\frac {x}{\log \left (2\right ) - \log \left (-x^{2} {\left (e^{6} + 1\right )} + 8 \, x e^{6} + 3 \, e^{6}\right ) + 3 \, \log \left (x\right ) + 6} \] Input:
integrate((((x^2-8*x-3)*exp(3)^2+x^2)*log(-2*x^3*exp(3)^2/((x^2-8*x-3)*exp (3)^2+x^2))+(-x^2+16*x+9)*exp(3)^2-x^2)/((x^2-8*x-3)*exp(3)^2+x^2)/log(-2* x^3*exp(3)^2/((x^2-8*x-3)*exp(3)^2+x^2))^2,x, algorithm="maxima")
Output:
x/(log(2) - log(-x^2*(e^6 + 1) + 8*x*e^6 + 3*e^6) + 3*log(x) + 6)
Time = 0.22 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.03 \[ \int \frac {-x^2+e^6 \left (9+16 x-x^2\right )+\left (x^2+e^6 \left (-3-8 x+x^2\right )\right ) \log \left (-\frac {2 e^6 x^3}{x^2+e^6 \left (-3-8 x+x^2\right )}\right )}{\left (x^2+e^6 \left (-3-8 x+x^2\right )\right ) \log ^2\left (-\frac {2 e^6 x^3}{x^2+e^6 \left (-3-8 x+x^2\right )}\right )} \, dx=\frac {x}{\log \left (-\frac {2 \, x^{3}}{x^{2} e^{6} + x^{2} - 8 \, x e^{6} - 3 \, e^{6}}\right ) + 6} \] Input:
integrate((((x^2-8*x-3)*exp(3)^2+x^2)*log(-2*x^3*exp(3)^2/((x^2-8*x-3)*exp (3)^2+x^2))+(-x^2+16*x+9)*exp(3)^2-x^2)/((x^2-8*x-3)*exp(3)^2+x^2)/log(-2* x^3*exp(3)^2/((x^2-8*x-3)*exp(3)^2+x^2))^2,x, algorithm="giac")
Output:
x/(log(-2*x^3/(x^2*e^6 + x^2 - 8*x*e^6 - 3*e^6)) + 6)
Time = 4.50 (sec) , antiderivative size = 77, normalized size of antiderivative = 2.41 \[ \int \frac {-x^2+e^6 \left (9+16 x-x^2\right )+\left (x^2+e^6 \left (-3-8 x+x^2\right )\right ) \log \left (-\frac {2 e^6 x^3}{x^2+e^6 \left (-3-8 x+x^2\right )}\right )}{\left (x^2+e^6 \left (-3-8 x+x^2\right )\right ) \log ^2\left (-\frac {2 e^6 x^3}{x^2+e^6 \left (-3-8 x+x^2\right )}\right )} \, dx=\frac {x+x\,{\mathrm {e}}^6-8\,\ln \left (\frac {2\,x^3\,{\mathrm {e}}^6}{{\mathrm {e}}^6\,\left (-x^2+8\,x+3\right )-x^2}\right )\,{\mathrm {e}}^6}{\ln \left (\frac {2\,x^3\,{\mathrm {e}}^6}{{\mathrm {e}}^6\,\left (-x^2+8\,x+3\right )-x^2}\right )\,\left ({\mathrm {e}}^6+1\right )} \] Input:
int((log((2*x^3*exp(6))/(exp(6)*(8*x - x^2 + 3) - x^2))*(exp(6)*(8*x - x^2 + 3) - x^2) - exp(6)*(16*x - x^2 + 9) + x^2)/(log((2*x^3*exp(6))/(exp(6)* (8*x - x^2 + 3) - x^2))^2*(exp(6)*(8*x - x^2 + 3) - x^2)),x)
Output:
(x + x*exp(6) - 8*log((2*x^3*exp(6))/(exp(6)*(8*x - x^2 + 3) - x^2))*exp(6 ))/(log((2*x^3*exp(6))/(exp(6)*(8*x - x^2 + 3) - x^2))*(exp(6) + 1))
Time = 0.20 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.16 \[ \int \frac {-x^2+e^6 \left (9+16 x-x^2\right )+\left (x^2+e^6 \left (-3-8 x+x^2\right )\right ) \log \left (-\frac {2 e^6 x^3}{x^2+e^6 \left (-3-8 x+x^2\right )}\right )}{\left (x^2+e^6 \left (-3-8 x+x^2\right )\right ) \log ^2\left (-\frac {2 e^6 x^3}{x^2+e^6 \left (-3-8 x+x^2\right )}\right )} \, dx=\frac {x}{\mathrm {log}\left (-\frac {2 e^{6} x^{3}}{e^{6} x^{2}-8 e^{6} x -3 e^{6}+x^{2}}\right )} \] Input:
int((((x^2-8*x-3)*exp(3)^2+x^2)*log(-2*x^3*exp(3)^2/((x^2-8*x-3)*exp(3)^2+ x^2))+(-x^2+16*x+9)*exp(3)^2-x^2)/((x^2-8*x-3)*exp(3)^2+x^2)/log(-2*x^3*ex p(3)^2/((x^2-8*x-3)*exp(3)^2+x^2))^2,x)
Output:
x/log(( - 2*e**6*x**3)/(e**6*x**2 - 8*e**6*x - 3*e**6 + x**2))