Integrand size = 102, antiderivative size = 19 \[ \int \frac {1+2 x+9 x^2+\left (-x-3 x^2+5 x^3+7 x^4\right ) \log ^2\left (\frac {-x-x^2}{-1-2 x+7 x^2}\right )}{\left (-x-3 x^2+5 x^3+7 x^4\right ) \log ^2\left (\frac {-x-x^2}{-1-2 x+7 x^2}\right )} \, dx=x+\frac {1}{\log \left (\frac {1+x}{2+\frac {1}{x}-7 x}\right )} \] Output:
x+1/ln((1+x)/(2-7*x+1/x))
Time = 0.03 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.16 \[ \int \frac {1+2 x+9 x^2+\left (-x-3 x^2+5 x^3+7 x^4\right ) \log ^2\left (\frac {-x-x^2}{-1-2 x+7 x^2}\right )}{\left (-x-3 x^2+5 x^3+7 x^4\right ) \log ^2\left (\frac {-x-x^2}{-1-2 x+7 x^2}\right )} \, dx=x+\frac {1}{\log \left (\frac {x (1+x)}{1+2 x-7 x^2}\right )} \] Input:
Integrate[(1 + 2*x + 9*x^2 + (-x - 3*x^2 + 5*x^3 + 7*x^4)*Log[(-x - x^2)/( -1 - 2*x + 7*x^2)]^2)/((-x - 3*x^2 + 5*x^3 + 7*x^4)*Log[(-x - x^2)/(-1 - 2 *x + 7*x^2)]^2),x]
Output:
x + Log[(x*(1 + x))/(1 + 2*x - 7*x^2)]^(-1)
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.
\(\displaystyle \int \frac {9 x^2+\left (7 x^4+5 x^3-3 x^2-x\right ) \log ^2\left (\frac {-x^2-x}{7 x^2-2 x-1}\right )+2 x+1}{\left (7 x^4+5 x^3-3 x^2-x\right ) \log ^2\left (\frac {-x^2-x}{7 x^2-2 x-1}\right )} \, dx\) |
\(\Big \downarrow \) 2026 |
\(\displaystyle \int \frac {9 x^2+\left (7 x^4+5 x^3-3 x^2-x\right ) \log ^2\left (\frac {-x^2-x}{7 x^2-2 x-1}\right )+2 x+1}{x \left (7 x^3+5 x^2-3 x-1\right ) \log ^2\left (\frac {-x^2-x}{7 x^2-2 x-1}\right )}dx\) |
\(\Big \downarrow \) 2463 |
\(\displaystyle \int \left (\frac {9 x^2+\left (7 x^4+5 x^3-3 x^2-x\right ) \log ^2\left (\frac {-x^2-x}{7 x^2-2 x-1}\right )+2 x+1}{8 x (x+1) \log ^2\left (\frac {-x^2-x}{7 x^2-2 x-1}\right )}+\frac {(9-7 x) \left (9 x^2+\left (7 x^4+5 x^3-3 x^2-x\right ) \log ^2\left (\frac {-x^2-x}{7 x^2-2 x-1}\right )+2 x+1\right )}{8 x \left (7 x^2-2 x-1\right ) \log ^2\left (\frac {-x^2-x}{7 x^2-2 x-1}\right )}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {7 \int \frac {1}{\left (-14 x+4 \sqrt {2}+2\right ) \log ^2\left (\frac {x (x+1)}{-7 x^2+2 x+1}\right )}dx}{\sqrt {2}}-\int \frac {1}{x \log ^2\left (\frac {x (x+1)}{-7 x^2+2 x+1}\right )}dx-\int \frac {1}{(x+1) \log ^2\left (\frac {x (x+1)}{-7 x^2+2 x+1}\right )}dx+\frac {7}{2} \left (4+\sqrt {2}\right ) \int \frac {1}{\left (14 x-4 \sqrt {2}-2\right ) \log ^2\left (\frac {x (x+1)}{-7 x^2+2 x+1}\right )}dx+\frac {7}{2} \left (4-\sqrt {2}\right ) \int \frac {1}{\left (14 x+4 \sqrt {2}-2\right ) \log ^2\left (\frac {x (x+1)}{-7 x^2+2 x+1}\right )}dx+\frac {7 \int \frac {1}{\left (14 x+4 \sqrt {2}-2\right ) \log ^2\left (\frac {x (x+1)}{-7 x^2+2 x+1}\right )}dx}{\sqrt {2}}+x\) |
Input:
Int[(1 + 2*x + 9*x^2 + (-x - 3*x^2 + 5*x^3 + 7*x^4)*Log[(-x - x^2)/(-1 - 2 *x + 7*x^2)]^2)/((-x - 3*x^2 + 5*x^3 + 7*x^4)*Log[(-x - x^2)/(-1 - 2*x + 7 *x^2)]^2),x]
Output:
$Aborted
Time = 0.24 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.32
method | result | size |
default | \(x +\frac {1}{\ln \left (\frac {\left (-1-x \right ) x}{7 x^{2}-2 x -1}\right )}\) | \(25\) |
parts | \(x +\frac {1}{\ln \left (\frac {\left (-1-x \right ) x}{7 x^{2}-2 x -1}\right )}\) | \(25\) |
risch | \(x +\frac {1}{\ln \left (\frac {-x^{2}-x}{7 x^{2}-2 x -1}\right )}\) | \(28\) |
norman | \(\frac {1+x \ln \left (\frac {-x^{2}-x}{7 x^{2}-2 x -1}\right )}{\ln \left (\frac {-x^{2}-x}{7 x^{2}-2 x -1}\right )}\) | \(54\) |
parallelrisch | \(-\frac {-49-49 x \ln \left (\frac {-x^{2}-x}{7 x^{2}-2 x -1}\right )+70 \ln \left (\frac {-x^{2}-x}{7 x^{2}-2 x -1}\right )}{49 \ln \left (\frac {-x^{2}-x}{7 x^{2}-2 x -1}\right )}\) | \(81\) |
Input:
int(((7*x^4+5*x^3-3*x^2-x)*ln((-x^2-x)/(7*x^2-2*x-1))^2+9*x^2+2*x+1)/(7*x^ 4+5*x^3-3*x^2-x)/ln((-x^2-x)/(7*x^2-2*x-1))^2,x,method=_RETURNVERBOSE)
Output:
x+1/ln((-1-x)*x/(7*x^2-2*x-1))
Leaf count of result is larger than twice the leaf count of optimal. 47 vs. \(2 (22) = 44\).
Time = 0.11 (sec) , antiderivative size = 47, normalized size of antiderivative = 2.47 \[ \int \frac {1+2 x+9 x^2+\left (-x-3 x^2+5 x^3+7 x^4\right ) \log ^2\left (\frac {-x-x^2}{-1-2 x+7 x^2}\right )}{\left (-x-3 x^2+5 x^3+7 x^4\right ) \log ^2\left (\frac {-x-x^2}{-1-2 x+7 x^2}\right )} \, dx=\frac {x \log \left (-\frac {x^{2} + x}{7 \, x^{2} - 2 \, x - 1}\right ) + 1}{\log \left (-\frac {x^{2} + x}{7 \, x^{2} - 2 \, x - 1}\right )} \] Input:
integrate(((7*x^4+5*x^3-3*x^2-x)*log((-x^2-x)/(7*x^2-2*x-1))^2+9*x^2+2*x+1 )/(7*x^4+5*x^3-3*x^2-x)/log((-x^2-x)/(7*x^2-2*x-1))^2,x, algorithm="fricas ")
Output:
(x*log(-(x^2 + x)/(7*x^2 - 2*x - 1)) + 1)/log(-(x^2 + x)/(7*x^2 - 2*x - 1) )
Time = 0.09 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.05 \[ \int \frac {1+2 x+9 x^2+\left (-x-3 x^2+5 x^3+7 x^4\right ) \log ^2\left (\frac {-x-x^2}{-1-2 x+7 x^2}\right )}{\left (-x-3 x^2+5 x^3+7 x^4\right ) \log ^2\left (\frac {-x-x^2}{-1-2 x+7 x^2}\right )} \, dx=x + \frac {1}{\log {\left (\frac {- x^{2} - x}{7 x^{2} - 2 x - 1} \right )}} \] Input:
integrate(((7*x**4+5*x**3-3*x**2-x)*ln((-x**2-x)/(7*x**2-2*x-1))**2+9*x**2 +2*x+1)/(7*x**4+5*x**3-3*x**2-x)/ln((-x**2-x)/(7*x**2-2*x-1))**2,x)
Output:
x + 1/log((-x**2 - x)/(7*x**2 - 2*x - 1))
Leaf count of result is larger than twice the leaf count of optimal. 52 vs. \(2 (22) = 44\).
Time = 0.08 (sec) , antiderivative size = 52, normalized size of antiderivative = 2.74 \[ \int \frac {1+2 x+9 x^2+\left (-x-3 x^2+5 x^3+7 x^4\right ) \log ^2\left (\frac {-x-x^2}{-1-2 x+7 x^2}\right )}{\left (-x-3 x^2+5 x^3+7 x^4\right ) \log ^2\left (\frac {-x-x^2}{-1-2 x+7 x^2}\right )} \, dx=\frac {x \log \left (-7 \, x^{2} + 2 \, x + 1\right ) - x \log \left (x + 1\right ) - x \log \left (x\right ) - 1}{\log \left (-7 \, x^{2} + 2 \, x + 1\right ) - \log \left (x + 1\right ) - \log \left (x\right )} \] Input:
integrate(((7*x^4+5*x^3-3*x^2-x)*log((-x^2-x)/(7*x^2-2*x-1))^2+9*x^2+2*x+1 )/(7*x^4+5*x^3-3*x^2-x)/log((-x^2-x)/(7*x^2-2*x-1))^2,x, algorithm="maxima ")
Output:
(x*log(-7*x^2 + 2*x + 1) - x*log(x + 1) - x*log(x) - 1)/(log(-7*x^2 + 2*x + 1) - log(x + 1) - log(x))
Time = 0.38 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.26 \[ \int \frac {1+2 x+9 x^2+\left (-x-3 x^2+5 x^3+7 x^4\right ) \log ^2\left (\frac {-x-x^2}{-1-2 x+7 x^2}\right )}{\left (-x-3 x^2+5 x^3+7 x^4\right ) \log ^2\left (\frac {-x-x^2}{-1-2 x+7 x^2}\right )} \, dx=x + \frac {1}{\log \left (-\frac {x^{2} + x}{7 \, x^{2} - 2 \, x - 1}\right )} \] Input:
integrate(((7*x^4+5*x^3-3*x^2-x)*log((-x^2-x)/(7*x^2-2*x-1))^2+9*x^2+2*x+1 )/(7*x^4+5*x^3-3*x^2-x)/log((-x^2-x)/(7*x^2-2*x-1))^2,x, algorithm="giac")
Output:
x + 1/log(-(x^2 + x)/(7*x^2 - 2*x - 1))
Time = 3.41 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.21 \[ \int \frac {1+2 x+9 x^2+\left (-x-3 x^2+5 x^3+7 x^4\right ) \log ^2\left (\frac {-x-x^2}{-1-2 x+7 x^2}\right )}{\left (-x-3 x^2+5 x^3+7 x^4\right ) \log ^2\left (\frac {-x-x^2}{-1-2 x+7 x^2}\right )} \, dx=x+\frac {1}{\ln \left (\frac {x^2+x}{-7\,x^2+2\,x+1}\right )} \] Input:
int(-(2*x - log((x + x^2)/(2*x - 7*x^2 + 1))^2*(x + 3*x^2 - 5*x^3 - 7*x^4) + 9*x^2 + 1)/(log((x + x^2)/(2*x - 7*x^2 + 1))^2*(x + 3*x^2 - 5*x^3 - 7*x ^4)),x)
Output:
x + 1/log((x + x^2)/(2*x - 7*x^2 + 1))
Time = 0.18 (sec) , antiderivative size = 53, normalized size of antiderivative = 2.79 \[ \int \frac {1+2 x+9 x^2+\left (-x-3 x^2+5 x^3+7 x^4\right ) \log ^2\left (\frac {-x-x^2}{-1-2 x+7 x^2}\right )}{\left (-x-3 x^2+5 x^3+7 x^4\right ) \log ^2\left (\frac {-x-x^2}{-1-2 x+7 x^2}\right )} \, dx=\frac {\mathrm {log}\left (\frac {-x^{2}-x}{7 x^{2}-2 x -1}\right ) x +1}{\mathrm {log}\left (\frac {-x^{2}-x}{7 x^{2}-2 x -1}\right )} \] Input:
int(((7*x^4+5*x^3-3*x^2-x)*log((-x^2-x)/(7*x^2-2*x-1))^2+9*x^2+2*x+1)/(7*x ^4+5*x^3-3*x^2-x)/log((-x^2-x)/(7*x^2-2*x-1))^2,x)
Output:
(log(( - x**2 - x)/(7*x**2 - 2*x - 1))*x + 1)/log(( - x**2 - x)/(7*x**2 - 2*x - 1))