Integrand size = 105, antiderivative size = 22 \[ \int \frac {-6+8 x+\left (-16 x-8 x^2\right ) \log \left (\frac {1}{4} (-3+4 x)\right )+(6-8 x) \log ^2\left (\frac {1}{4} (-3+4 x)\right )}{-3 x^2+4 x^3+\left (6 x^2-8 x^3\right ) \log ^2\left (\frac {1}{4} (-3+4 x)\right )+\left (-3 x^2+4 x^3\right ) \log ^4\left (\frac {1}{4} (-3+4 x)\right )} \, dx=3-\frac {2+x}{x-x \log ^2\left (-\frac {3}{4}+x\right )} \] Output:
3-1/(x-ln(x-3/4)^2*x)*(2+x)
Time = 0.16 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {-6+8 x+\left (-16 x-8 x^2\right ) \log \left (\frac {1}{4} (-3+4 x)\right )+(6-8 x) \log ^2\left (\frac {1}{4} (-3+4 x)\right )}{-3 x^2+4 x^3+\left (6 x^2-8 x^3\right ) \log ^2\left (\frac {1}{4} (-3+4 x)\right )+\left (-3 x^2+4 x^3\right ) \log ^4\left (\frac {1}{4} (-3+4 x)\right )} \, dx=-\frac {-2-x}{x \left (-1+\log ^2\left (-\frac {3}{4}+x\right )\right )} \] Input:
Integrate[(-6 + 8*x + (-16*x - 8*x^2)*Log[(-3 + 4*x)/4] + (6 - 8*x)*Log[(- 3 + 4*x)/4]^2)/(-3*x^2 + 4*x^3 + (6*x^2 - 8*x^3)*Log[(-3 + 4*x)/4]^2 + (-3 *x^2 + 4*x^3)*Log[(-3 + 4*x)/4]^4),x]
Output:
-((-2 - x)/(x*(-1 + Log[-3/4 + 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.
\(\displaystyle \int \frac {\left (-8 x^2-16 x\right ) \log \left (\frac {1}{4} (4 x-3)\right )+8 x+(6-8 x) \log ^2\left (\frac {1}{4} (4 x-3)\right )-6}{4 x^3-3 x^2+\left (4 x^3-3 x^2\right ) \log ^4\left (\frac {1}{4} (4 x-3)\right )+\left (6 x^2-8 x^3\right ) \log ^2\left (\frac {1}{4} (4 x-3)\right )} \, dx\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle \int \frac {2 \left (-4 x+(4 x-3) \log ^2\left (x-\frac {3}{4}\right )+4 x (x+2) \log \left (x-\frac {3}{4}\right )+3\right )}{(3-4 x) x^2 \left (1-\log ^2\left (x-\frac {3}{4}\right )\right )^2}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle 2 \int \frac {-\left ((3-4 x) \log ^2\left (x-\frac {3}{4}\right )\right )+4 x (x+2) \log \left (x-\frac {3}{4}\right )-4 x+3}{(3-4 x) x^2 \left (1-\log ^2\left (x-\frac {3}{4}\right )\right )^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle 2 \int \left (\frac {-x-2}{x (4 x-3) \left (\log \left (x-\frac {3}{4}\right )-1\right )^2}-\frac {1}{2 x^2 \left (\log \left (x-\frac {3}{4}\right )-1\right )}+\frac {1}{2 x^2 \left (\log \left (x-\frac {3}{4}\right )+1\right )}+\frac {x+2}{x (4 x-3) \left (\log \left (x-\frac {3}{4}\right )+1\right )^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 2 \left (-\frac {1}{2} \int \frac {1}{x^2 \left (\log \left (x-\frac {3}{4}\right )-1\right )}dx+\frac {1}{2} \int \frac {1}{x^2 \left (\log \left (x-\frac {3}{4}\right )+1\right )}dx+\frac {2}{3} \int \frac {1}{x \left (\log \left (x-\frac {3}{4}\right )-1\right )^2}dx-\frac {2}{3} \int \frac {1}{x \left (\log \left (x-\frac {3}{4}\right )+1\right )^2}dx-\frac {11}{12 \left (1-\log \left (x-\frac {3}{4}\right )\right )}-\frac {11}{12 \left (\log \left (x-\frac {3}{4}\right )+1\right )}\right )\) |
Input:
Int[(-6 + 8*x + (-16*x - 8*x^2)*Log[(-3 + 4*x)/4] + (6 - 8*x)*Log[(-3 + 4* x)/4]^2)/(-3*x^2 + 4*x^3 + (6*x^2 - 8*x^3)*Log[(-3 + 4*x)/4]^2 + (-3*x^2 + 4*x^3)*Log[(-3 + 4*x)/4]^4),x]
Output:
$Aborted
Time = 0.58 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.82
method | result | size |
norman | \(\frac {2+x}{x \left (\ln \left (x -\frac {3}{4}\right )^{2}-1\right )}\) | \(18\) |
risch | \(\frac {2+x}{x \left (\ln \left (x -\frac {3}{4}\right )^{2}-1\right )}\) | \(18\) |
derivativedivides | \(-\frac {-2 x -4}{2 x \left (\ln \left (x -\frac {3}{4}\right )^{2}-1\right )}\) | \(21\) |
default | \(-\frac {-2 x -4}{2 x \left (\ln \left (x -\frac {3}{4}\right )^{2}-1\right )}\) | \(21\) |
parallelrisch | \(\frac {32+16 x}{16 \left (\ln \left (x -\frac {3}{4}\right )^{2}-1\right ) x}\) | \(21\) |
Input:
int(((-8*x+6)*ln(x-3/4)^2+(-8*x^2-16*x)*ln(x-3/4)+8*x-6)/((4*x^3-3*x^2)*ln (x-3/4)^4+(-8*x^3+6*x^2)*ln(x-3/4)^2+4*x^3-3*x^2),x,method=_RETURNVERBOSE)
Output:
(2+x)/x/(ln(x-3/4)^2-1)
Time = 0.09 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.82 \[ \int \frac {-6+8 x+\left (-16 x-8 x^2\right ) \log \left (\frac {1}{4} (-3+4 x)\right )+(6-8 x) \log ^2\left (\frac {1}{4} (-3+4 x)\right )}{-3 x^2+4 x^3+\left (6 x^2-8 x^3\right ) \log ^2\left (\frac {1}{4} (-3+4 x)\right )+\left (-3 x^2+4 x^3\right ) \log ^4\left (\frac {1}{4} (-3+4 x)\right )} \, dx=\frac {x + 2}{x \log \left (x - \frac {3}{4}\right )^{2} - x} \] Input:
integrate(((-8*x+6)*log(x-3/4)^2+(-8*x^2-16*x)*log(x-3/4)+8*x-6)/((4*x^3-3 *x^2)*log(x-3/4)^4+(-8*x^3+6*x^2)*log(x-3/4)^2+4*x^3-3*x^2),x, algorithm=" fricas")
Output:
(x + 2)/(x*log(x - 3/4)^2 - x)
Time = 0.06 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.64 \[ \int \frac {-6+8 x+\left (-16 x-8 x^2\right ) \log \left (\frac {1}{4} (-3+4 x)\right )+(6-8 x) \log ^2\left (\frac {1}{4} (-3+4 x)\right )}{-3 x^2+4 x^3+\left (6 x^2-8 x^3\right ) \log ^2\left (\frac {1}{4} (-3+4 x)\right )+\left (-3 x^2+4 x^3\right ) \log ^4\left (\frac {1}{4} (-3+4 x)\right )} \, dx=\frac {x + 2}{x \log {\left (x - \frac {3}{4} \right )}^{2} - x} \] Input:
integrate(((-8*x+6)*ln(x-3/4)**2+(-8*x**2-16*x)*ln(x-3/4)+8*x-6)/((4*x**3- 3*x**2)*ln(x-3/4)**4+(-8*x**3+6*x**2)*ln(x-3/4)**2+4*x**3-3*x**2),x)
Output:
(x + 2)/(x*log(x - 3/4)**2 - x)
Leaf count of result is larger than twice the leaf count of optimal. 41 vs. \(2 (20) = 40\).
Time = 0.16 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.86 \[ \int \frac {-6+8 x+\left (-16 x-8 x^2\right ) \log \left (\frac {1}{4} (-3+4 x)\right )+(6-8 x) \log ^2\left (\frac {1}{4} (-3+4 x)\right )}{-3 x^2+4 x^3+\left (6 x^2-8 x^3\right ) \log ^2\left (\frac {1}{4} (-3+4 x)\right )+\left (-3 x^2+4 x^3\right ) \log ^4\left (\frac {1}{4} (-3+4 x)\right )} \, dx=-\frac {x + 2}{4 \, x \log \left (2\right ) \log \left (4 \, x - 3\right ) - x \log \left (4 \, x - 3\right )^{2} - {\left (4 \, \log \left (2\right )^{2} - 1\right )} x} \] Input:
integrate(((-8*x+6)*log(x-3/4)^2+(-8*x^2-16*x)*log(x-3/4)+8*x-6)/((4*x^3-3 *x^2)*log(x-3/4)^4+(-8*x^3+6*x^2)*log(x-3/4)^2+4*x^3-3*x^2),x, algorithm=" maxima")
Output:
-(x + 2)/(4*x*log(2)*log(4*x - 3) - x*log(4*x - 3)^2 - (4*log(2)^2 - 1)*x)
Time = 0.13 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.82 \[ \int \frac {-6+8 x+\left (-16 x-8 x^2\right ) \log \left (\frac {1}{4} (-3+4 x)\right )+(6-8 x) \log ^2\left (\frac {1}{4} (-3+4 x)\right )}{-3 x^2+4 x^3+\left (6 x^2-8 x^3\right ) \log ^2\left (\frac {1}{4} (-3+4 x)\right )+\left (-3 x^2+4 x^3\right ) \log ^4\left (\frac {1}{4} (-3+4 x)\right )} \, dx=\frac {x + 2}{x \log \left (x - \frac {3}{4}\right )^{2} - x} \] Input:
integrate(((-8*x+6)*log(x-3/4)^2+(-8*x^2-16*x)*log(x-3/4)+8*x-6)/((4*x^3-3 *x^2)*log(x-3/4)^4+(-8*x^3+6*x^2)*log(x-3/4)^2+4*x^3-3*x^2),x, algorithm=" giac")
Output:
(x + 2)/(x*log(x - 3/4)^2 - x)
Time = 2.58 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.77 \[ \int \frac {-6+8 x+\left (-16 x-8 x^2\right ) \log \left (\frac {1}{4} (-3+4 x)\right )+(6-8 x) \log ^2\left (\frac {1}{4} (-3+4 x)\right )}{-3 x^2+4 x^3+\left (6 x^2-8 x^3\right ) \log ^2\left (\frac {1}{4} (-3+4 x)\right )+\left (-3 x^2+4 x^3\right ) \log ^4\left (\frac {1}{4} (-3+4 x)\right )} \, dx=\frac {x+2}{x\,\left ({\ln \left (x-\frac {3}{4}\right )}^2-1\right )} \] Input:
int((log(x - 3/4)*(16*x + 8*x^2) - 8*x + log(x - 3/4)^2*(8*x - 6) + 6)/(lo g(x - 3/4)^4*(3*x^2 - 4*x^3) - log(x - 3/4)^2*(6*x^2 - 8*x^3) + 3*x^2 - 4* x^3),x)
Output:
(x + 2)/(x*(log(x - 3/4)^2 - 1))
Time = 0.24 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {-6+8 x+\left (-16 x-8 x^2\right ) \log \left (\frac {1}{4} (-3+4 x)\right )+(6-8 x) \log ^2\left (\frac {1}{4} (-3+4 x)\right )}{-3 x^2+4 x^3+\left (6 x^2-8 x^3\right ) \log ^2\left (\frac {1}{4} (-3+4 x)\right )+\left (-3 x^2+4 x^3\right ) \log ^4\left (\frac {1}{4} (-3+4 x)\right )} \, dx=\frac {\mathrm {log}\left (x -\frac {3}{4}\right )^{2} x +2}{x \left (\mathrm {log}\left (x -\frac {3}{4}\right )^{2}-1\right )} \] Input:
int(((-8*x+6)*log(x-3/4)^2+(-8*x^2-16*x)*log(x-3/4)+8*x-6)/((4*x^3-3*x^2)* log(x-3/4)^4+(-8*x^3+6*x^2)*log(x-3/4)^2+4*x^3-3*x^2),x)
Output:
(log((4*x - 3)/4)**2*x + 2)/(x*(log((4*x - 3)/4)**2 - 1))