Integrand size = 117, antiderivative size = 28 \[ \int \frac {-x+2 x^2-3 x^3+4 x^4-32 x^5+12 x^6+\left (2 x-4 x^2+10 x^3-6 x^4\right ) \log (x)}{49 x^4-70 x^5-3 x^6+20 x^7+4 x^8+\left (-14 x^2+24 x^3-6 x^4-4 x^5\right ) \log (x)+\left (1-2 x+x^2\right ) \log ^2(x)} \, dx=\frac {2 x+\frac {1+x}{-1+x}}{7+2 x-\frac {\log (x)}{x^2}} \] Output:
(2*x+(1+x)/(-1+x))/(2*x+7-ln(x)/x^2)
Time = 0.24 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.25 \[ \int \frac {-x+2 x^2-3 x^3+4 x^4-32 x^5+12 x^6+\left (2 x-4 x^2+10 x^3-6 x^4\right ) \log (x)}{49 x^4-70 x^5-3 x^6+20 x^7+4 x^8+\left (-14 x^2+24 x^3-6 x^4-4 x^5\right ) \log (x)+\left (1-2 x+x^2\right ) \log ^2(x)} \, dx=-\frac {x^2 \left (1-x+2 x^2\right )}{(-1+x) \left (-7 x^2-2 x^3+\log (x)\right )} \] Input:
Integrate[(-x + 2*x^2 - 3*x^3 + 4*x^4 - 32*x^5 + 12*x^6 + (2*x - 4*x^2 + 1 0*x^3 - 6*x^4)*Log[x])/(49*x^4 - 70*x^5 - 3*x^6 + 20*x^7 + 4*x^8 + (-14*x^ 2 + 24*x^3 - 6*x^4 - 4*x^5)*Log[x] + (1 - 2*x + x^2)*Log[x]^2),x]
Output:
-((x^2*(1 - x + 2*x^2))/((-1 + x)*(-7*x^2 - 2*x^3 + 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.
\(\displaystyle \int \frac {12 x^6-32 x^5+4 x^4-3 x^3+2 x^2+\left (-6 x^4+10 x^3-4 x^2+2 x\right ) \log (x)-x}{4 x^8+20 x^7-3 x^6-70 x^5+49 x^4+\left (x^2-2 x+1\right ) \log ^2(x)+\left (-4 x^5-6 x^4+24 x^3-14 x^2\right ) \log (x)} \, dx\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle \int \frac {x \left (12 x^5-32 x^4+4 x^3-3 x^2+\left (-6 x^3+10 x^2-4 x+2\right ) \log (x)+2 x-1\right )}{(1-x)^2 \left (x^2 (2 x+7)-\log (x)\right )^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {2 x \left (3 x^3-5 x^2+2 x-1\right )}{(x-1)^2 \left (2 x^3+7 x^2-\log (x)\right )}-\frac {x \left (12 x^5+22 x^4-8 x^3+12 x^2+x-1\right )}{(x-1) \left (2 x^3+7 x^2-\log (x)\right )^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -38 \int \frac {1}{\left (2 x^3+7 x^2-\log (x)\right )^2}dx-38 \int \frac {1}{(x-1) \left (2 x^3+7 x^2-\log (x)\right )^2}dx-39 \int \frac {x}{\left (2 x^3+7 x^2-\log (x)\right )^2}dx-38 \int \frac {x^2}{\left (2 x^3+7 x^2-\log (x)\right )^2}dx-26 \int \frac {x^3}{\left (2 x^3+7 x^2-\log (x)\right )^2}dx+2 \int \frac {1}{2 x^3+7 x^2-\log (x)}dx-2 \int \frac {1}{(x-1)^2 \left (2 x^3+7 x^2-\log (x)\right )}dx+2 \int \frac {x}{2 x^3+7 x^2-\log (x)}dx+6 \int \frac {x^2}{2 x^3+7 x^2-\log (x)}dx-12 \int \frac {x^5}{\left (2 x^3+7 x^2-\log (x)\right )^2}dx-34 \int \frac {x^4}{\left (2 x^3+7 x^2-\log (x)\right )^2}dx\) |
Input:
Int[(-x + 2*x^2 - 3*x^3 + 4*x^4 - 32*x^5 + 12*x^6 + (2*x - 4*x^2 + 10*x^3 - 6*x^4)*Log[x])/(49*x^4 - 70*x^5 - 3*x^6 + 20*x^7 + 4*x^8 + (-14*x^2 + 24 *x^3 - 6*x^4 - 4*x^5)*Log[x] + (1 - 2*x + x^2)*Log[x]^2),x]
Output:
$Aborted
Time = 1.11 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.32
method | result | size |
default | \(-\frac {2 x^{4}-x^{3}+x^{2}}{\left (-1+x \right ) \left (-2 x^{3}-7 x^{2}+\ln \left (x \right )\right )}\) | \(37\) |
risch | \(\frac {\left (2 x^{2}-x +1\right ) x^{2}}{\left (-1+x \right ) \left (2 x^{3}+7 x^{2}-\ln \left (x \right )\right )}\) | \(37\) |
parallelrisch | \(\frac {2 x^{4}-x^{3}+x^{2}}{2 x^{4}+5 x^{3}-7 x^{2}-x \ln \left (x \right )+\ln \left (x \right )}\) | \(41\) |
Input:
int(((-6*x^4+10*x^3-4*x^2+2*x)*ln(x)+12*x^6-32*x^5+4*x^4-3*x^3+2*x^2-x)/(( x^2-2*x+1)*ln(x)^2+(-4*x^5-6*x^4+24*x^3-14*x^2)*ln(x)+4*x^8+20*x^7-3*x^6-7 0*x^5+49*x^4),x,method=_RETURNVERBOSE)
Output:
-(2*x^4-x^3+x^2)/(-1+x)/(-2*x^3-7*x^2+ln(x))
Time = 0.08 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.43 \[ \int \frac {-x+2 x^2-3 x^3+4 x^4-32 x^5+12 x^6+\left (2 x-4 x^2+10 x^3-6 x^4\right ) \log (x)}{49 x^4-70 x^5-3 x^6+20 x^7+4 x^8+\left (-14 x^2+24 x^3-6 x^4-4 x^5\right ) \log (x)+\left (1-2 x+x^2\right ) \log ^2(x)} \, dx=\frac {2 \, x^{4} - x^{3} + x^{2}}{2 \, x^{4} + 5 \, x^{3} - 7 \, x^{2} - {\left (x - 1\right )} \log \left (x\right )} \] Input:
integrate(((-6*x^4+10*x^3-4*x^2+2*x)*log(x)+12*x^6-32*x^5+4*x^4-3*x^3+2*x^ 2-x)/((x^2-2*x+1)*log(x)^2+(-4*x^5-6*x^4+24*x^3-14*x^2)*log(x)+4*x^8+20*x^ 7-3*x^6-70*x^5+49*x^4),x, algorithm="fricas")
Output:
(2*x^4 - x^3 + x^2)/(2*x^4 + 5*x^3 - 7*x^2 - (x - 1)*log(x))
Time = 0.07 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.14 \[ \int \frac {-x+2 x^2-3 x^3+4 x^4-32 x^5+12 x^6+\left (2 x-4 x^2+10 x^3-6 x^4\right ) \log (x)}{49 x^4-70 x^5-3 x^6+20 x^7+4 x^8+\left (-14 x^2+24 x^3-6 x^4-4 x^5\right ) \log (x)+\left (1-2 x+x^2\right ) \log ^2(x)} \, dx=\frac {- 2 x^{4} + x^{3} - x^{2}}{- 2 x^{4} - 5 x^{3} + 7 x^{2} + \left (x - 1\right ) \log {\left (x \right )}} \] Input:
integrate(((-6*x**4+10*x**3-4*x**2+2*x)*ln(x)+12*x**6-32*x**5+4*x**4-3*x** 3+2*x**2-x)/((x**2-2*x+1)*ln(x)**2+(-4*x**5-6*x**4+24*x**3-14*x**2)*ln(x)+ 4*x**8+20*x**7-3*x**6-70*x**5+49*x**4),x)
Output:
(-2*x**4 + x**3 - x**2)/(-2*x**4 - 5*x**3 + 7*x**2 + (x - 1)*log(x))
Time = 0.07 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.43 \[ \int \frac {-x+2 x^2-3 x^3+4 x^4-32 x^5+12 x^6+\left (2 x-4 x^2+10 x^3-6 x^4\right ) \log (x)}{49 x^4-70 x^5-3 x^6+20 x^7+4 x^8+\left (-14 x^2+24 x^3-6 x^4-4 x^5\right ) \log (x)+\left (1-2 x+x^2\right ) \log ^2(x)} \, dx=\frac {2 \, x^{4} - x^{3} + x^{2}}{2 \, x^{4} + 5 \, x^{3} - 7 \, x^{2} - {\left (x - 1\right )} \log \left (x\right )} \] Input:
integrate(((-6*x^4+10*x^3-4*x^2+2*x)*log(x)+12*x^6-32*x^5+4*x^4-3*x^3+2*x^ 2-x)/((x^2-2*x+1)*log(x)^2+(-4*x^5-6*x^4+24*x^3-14*x^2)*log(x)+4*x^8+20*x^ 7-3*x^6-70*x^5+49*x^4),x, algorithm="maxima")
Output:
(2*x^4 - x^3 + x^2)/(2*x^4 + 5*x^3 - 7*x^2 - (x - 1)*log(x))
Time = 0.12 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.43 \[ \int \frac {-x+2 x^2-3 x^3+4 x^4-32 x^5+12 x^6+\left (2 x-4 x^2+10 x^3-6 x^4\right ) \log (x)}{49 x^4-70 x^5-3 x^6+20 x^7+4 x^8+\left (-14 x^2+24 x^3-6 x^4-4 x^5\right ) \log (x)+\left (1-2 x+x^2\right ) \log ^2(x)} \, dx=\frac {2 \, x^{4} - x^{3} + x^{2}}{2 \, x^{4} + 5 \, x^{3} - 7 \, x^{2} - x \log \left (x\right ) + \log \left (x\right )} \] Input:
integrate(((-6*x^4+10*x^3-4*x^2+2*x)*log(x)+12*x^6-32*x^5+4*x^4-3*x^3+2*x^ 2-x)/((x^2-2*x+1)*log(x)^2+(-4*x^5-6*x^4+24*x^3-14*x^2)*log(x)+4*x^8+20*x^ 7-3*x^6-70*x^5+49*x^4),x, algorithm="giac")
Output:
(2*x^4 - x^3 + x^2)/(2*x^4 + 5*x^3 - 7*x^2 - x*log(x) + log(x))
Timed out. \[ \int \frac {-x+2 x^2-3 x^3+4 x^4-32 x^5+12 x^6+\left (2 x-4 x^2+10 x^3-6 x^4\right ) \log (x)}{49 x^4-70 x^5-3 x^6+20 x^7+4 x^8+\left (-14 x^2+24 x^3-6 x^4-4 x^5\right ) \log (x)+\left (1-2 x+x^2\right ) \log ^2(x)} \, dx=\int \frac {\ln \left (x\right )\,\left (-6\,x^4+10\,x^3-4\,x^2+2\,x\right )-x+2\,x^2-3\,x^3+4\,x^4-32\,x^5+12\,x^6}{{\ln \left (x\right )}^2\,\left (x^2-2\,x+1\right )-\ln \left (x\right )\,\left (4\,x^5+6\,x^4-24\,x^3+14\,x^2\right )+49\,x^4-70\,x^5-3\,x^6+20\,x^7+4\,x^8} \,d x \] Input:
int((log(x)*(2*x - 4*x^2 + 10*x^3 - 6*x^4) - x + 2*x^2 - 3*x^3 + 4*x^4 - 3 2*x^5 + 12*x^6)/(log(x)^2*(x^2 - 2*x + 1) - log(x)*(14*x^2 - 24*x^3 + 6*x^ 4 + 4*x^5) + 49*x^4 - 70*x^5 - 3*x^6 + 20*x^7 + 4*x^8),x)
Output:
int((log(x)*(2*x - 4*x^2 + 10*x^3 - 6*x^4) - x + 2*x^2 - 3*x^3 + 4*x^4 - 3 2*x^5 + 12*x^6)/(log(x)^2*(x^2 - 2*x + 1) - log(x)*(14*x^2 - 24*x^3 + 6*x^ 4 + 4*x^5) + 49*x^4 - 70*x^5 - 3*x^6 + 20*x^7 + 4*x^8), x)
Time = 0.15 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.61 \[ \int \frac {-x+2 x^2-3 x^3+4 x^4-32 x^5+12 x^6+\left (2 x-4 x^2+10 x^3-6 x^4\right ) \log (x)}{49 x^4-70 x^5-3 x^6+20 x^7+4 x^8+\left (-14 x^2+24 x^3-6 x^4-4 x^5\right ) \log (x)+\left (1-2 x+x^2\right ) \log ^2(x)} \, dx=\frac {-\mathrm {log}\left (x \right ) x +\mathrm {log}\left (x \right )+6 x^{3}-8 x^{2}}{\mathrm {log}\left (x \right ) x -\mathrm {log}\left (x \right )-2 x^{4}-5 x^{3}+7 x^{2}} \] Input:
int(((-6*x^4+10*x^3-4*x^2+2*x)*log(x)+12*x^6-32*x^5+4*x^4-3*x^3+2*x^2-x)/( (x^2-2*x+1)*log(x)^2+(-4*x^5-6*x^4+24*x^3-14*x^2)*log(x)+4*x^8+20*x^7-3*x^ 6-70*x^5+49*x^4),x)
Output:
( - log(x)*x + log(x) + 6*x**3 - 8*x**2)/(log(x)*x - log(x) - 2*x**4 - 5*x **3 + 7*x**2)