Integrand size = 73, antiderivative size = 26 \[ \int \frac {x+x^3-3 x^4-2 x \log (x)}{4 x^4-12 x^5+9 x^6+\left (4 x^2-6 x^3+4 x^4-6 x^5\right ) \log (x)+\left (1+2 x^2+x^4\right ) \log ^2(x)} \, dx=i \pi +\log (4)+\frac {1}{-2+3 x-\log (x)-\frac {\log (x)}{x^2}} \] Output:
2*ln(2)+I*Pi+1/(3*x-2-ln(x)-ln(x)/x^2)
Time = 0.22 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \frac {x+x^3-3 x^4-2 x \log (x)}{4 x^4-12 x^5+9 x^6+\left (4 x^2-6 x^3+4 x^4-6 x^5\right ) \log (x)+\left (1+2 x^2+x^4\right ) \log ^2(x)} \, dx=-\frac {x^2}{2 x^2-3 x^3+\log (x)+x^2 \log (x)} \] Input:
Integrate[(x + x^3 - 3*x^4 - 2*x*Log[x])/(4*x^4 - 12*x^5 + 9*x^6 + (4*x^2 - 6*x^3 + 4*x^4 - 6*x^5)*Log[x] + (1 + 2*x^2 + x^4)*Log[x]^2),x]
Output:
-(x^2/(2*x^2 - 3*x^3 + Log[x] + x^2*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.
\(\displaystyle \int \frac {-3 x^4+x^3+x-2 x \log (x)}{9 x^6-12 x^5+4 x^4+\left (x^4+2 x^2+1\right ) \log ^2(x)+\left (-6 x^5+4 x^4-6 x^3+4 x^2\right ) \log (x)} \, dx\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle \int \frac {-3 x^4+x^3+x-2 x \log (x)}{\left (x^2 (3 x-2)-\left (x^2+1\right ) \log (x)\right )^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {2 x}{\left (x^2+1\right ) \left (3 x^3-2 x^2-x^2 \log (x)-\log (x)\right )}+\frac {\left (-3 x^5+x^4-9 x^3+6 x^2+1\right ) x}{\left (x^2+1\right ) \left (-3 x^3+2 x^2+x^2 \log (x)+\log (x)\right )^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 6 \int \frac {1}{\left (3 x^3-\log (x) x^2-2 x^2-\log (x)\right )^2}dx+(2-3 i) \int \frac {1}{(i-x) \left (3 x^3-\log (x) x^2-2 x^2-\log (x)\right )^2}dx+5 \int \frac {x}{\left (3 x^3-\log (x) x^2-2 x^2-\log (x)\right )^2}dx-6 \int \frac {x^2}{\left (3 x^3-\log (x) x^2-2 x^2-\log (x)\right )^2}dx+\int \frac {x^3}{\left (3 x^3-\log (x) x^2-2 x^2-\log (x)\right )^2}dx-(2+3 i) \int \frac {1}{(x+i) \left (3 x^3-\log (x) x^2-2 x^2-\log (x)\right )^2}dx-\int \frac {1}{(i-x) \left (3 x^3-\log (x) x^2-2 x^2-\log (x)\right )}dx+\int \frac {1}{(x+i) \left (3 x^3-\log (x) x^2-2 x^2-\log (x)\right )}dx-3 \int \frac {x^4}{\left (3 x^3-\log (x) x^2-2 x^2-\log (x)\right )^2}dx\) |
Input:
Int[(x + x^3 - 3*x^4 - 2*x*Log[x])/(4*x^4 - 12*x^5 + 9*x^6 + (4*x^2 - 6*x^ 3 + 4*x^4 - 6*x^5)*Log[x] + (1 + 2*x^2 + x^4)*Log[x]^2),x]
Output:
$Aborted
Time = 0.24 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.04
method | result | size |
default | \(-\frac {x^{2}}{x^{2} \ln \left (x \right )-3 x^{3}+2 x^{2}+\ln \left (x \right )}\) | \(27\) |
norman | \(\frac {x^{2}}{3 x^{3}-x^{2} \ln \left (x \right )-2 x^{2}-\ln \left (x \right )}\) | \(29\) |
risch | \(\frac {x^{2}}{3 x^{3}-x^{2} \ln \left (x \right )-2 x^{2}-\ln \left (x \right )}\) | \(29\) |
parallelrisch | \(\frac {x^{2}}{3 x^{3}-x^{2} \ln \left (x \right )-2 x^{2}-\ln \left (x \right )}\) | \(29\) |
Input:
int((-2*x*ln(x)-3*x^4+x^3+x)/((x^4+2*x^2+1)*ln(x)^2+(-6*x^5+4*x^4-6*x^3+4* x^2)*ln(x)+9*x^6-12*x^5+4*x^4),x,method=_RETURNVERBOSE)
Output:
-x^2/(x^2*ln(x)-3*x^3+2*x^2+ln(x))
Time = 0.09 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \frac {x+x^3-3 x^4-2 x \log (x)}{4 x^4-12 x^5+9 x^6+\left (4 x^2-6 x^3+4 x^4-6 x^5\right ) \log (x)+\left (1+2 x^2+x^4\right ) \log ^2(x)} \, dx=\frac {x^{2}}{3 \, x^{3} - 2 \, x^{2} - {\left (x^{2} + 1\right )} \log \left (x\right )} \] Input:
integrate((-2*x*log(x)-3*x^4+x^3+x)/((x^4+2*x^2+1)*log(x)^2+(-6*x^5+4*x^4- 6*x^3+4*x^2)*log(x)+9*x^6-12*x^5+4*x^4),x, algorithm="fricas")
Output:
x^2/(3*x^3 - 2*x^2 - (x^2 + 1)*log(x))
Time = 0.11 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.85 \[ \int \frac {x+x^3-3 x^4-2 x \log (x)}{4 x^4-12 x^5+9 x^6+\left (4 x^2-6 x^3+4 x^4-6 x^5\right ) \log (x)+\left (1+2 x^2+x^4\right ) \log ^2(x)} \, dx=- \frac {x^{2}}{- 3 x^{3} + 2 x^{2} + \left (x^{2} + 1\right ) \log {\left (x \right )}} \] Input:
integrate((-2*x*ln(x)-3*x**4+x**3+x)/((x**4+2*x**2+1)*ln(x)**2+(-6*x**5+4* x**4-6*x**3+4*x**2)*ln(x)+9*x**6-12*x**5+4*x**4),x)
Output:
-x**2/(-3*x**3 + 2*x**2 + (x**2 + 1)*log(x))
Time = 0.07 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \frac {x+x^3-3 x^4-2 x \log (x)}{4 x^4-12 x^5+9 x^6+\left (4 x^2-6 x^3+4 x^4-6 x^5\right ) \log (x)+\left (1+2 x^2+x^4\right ) \log ^2(x)} \, dx=\frac {x^{2}}{3 \, x^{3} - 2 \, x^{2} - {\left (x^{2} + 1\right )} \log \left (x\right )} \] Input:
integrate((-2*x*log(x)-3*x^4+x^3+x)/((x^4+2*x^2+1)*log(x)^2+(-6*x^5+4*x^4- 6*x^3+4*x^2)*log(x)+9*x^6-12*x^5+4*x^4),x, algorithm="maxima")
Output:
x^2/(3*x^3 - 2*x^2 - (x^2 + 1)*log(x))
Time = 0.12 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.08 \[ \int \frac {x+x^3-3 x^4-2 x \log (x)}{4 x^4-12 x^5+9 x^6+\left (4 x^2-6 x^3+4 x^4-6 x^5\right ) \log (x)+\left (1+2 x^2+x^4\right ) \log ^2(x)} \, dx=\frac {x^{2}}{3 \, x^{3} - x^{2} \log \left (x\right ) - 2 \, x^{2} - \log \left (x\right )} \] Input:
integrate((-2*x*log(x)-3*x^4+x^3+x)/((x^4+2*x^2+1)*log(x)^2+(-6*x^5+4*x^4- 6*x^3+4*x^2)*log(x)+9*x^6-12*x^5+4*x^4),x, algorithm="giac")
Output:
x^2/(3*x^3 - x^2*log(x) - 2*x^2 - log(x))
Timed out. \[ \int \frac {x+x^3-3 x^4-2 x \log (x)}{4 x^4-12 x^5+9 x^6+\left (4 x^2-6 x^3+4 x^4-6 x^5\right ) \log (x)+\left (1+2 x^2+x^4\right ) \log ^2(x)} \, dx=\int \frac {x-2\,x\,\ln \left (x\right )+x^3-3\,x^4}{{\ln \left (x\right )}^2\,\left (x^4+2\,x^2+1\right )+\ln \left (x\right )\,\left (-6\,x^5+4\,x^4-6\,x^3+4\,x^2\right )+4\,x^4-12\,x^5+9\,x^6} \,d x \] Input:
int((x - 2*x*log(x) + x^3 - 3*x^4)/(log(x)^2*(2*x^2 + x^4 + 1) + log(x)*(4 *x^2 - 6*x^3 + 4*x^4 - 6*x^5) + 4*x^4 - 12*x^5 + 9*x^6),x)
Output:
int((x - 2*x*log(x) + x^3 - 3*x^4)/(log(x)^2*(2*x^2 + x^4 + 1) + log(x)*(4 *x^2 - 6*x^3 + 4*x^4 - 6*x^5) + 4*x^4 - 12*x^5 + 9*x^6), x)
Time = 0.20 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \frac {x+x^3-3 x^4-2 x \log (x)}{4 x^4-12 x^5+9 x^6+\left (4 x^2-6 x^3+4 x^4-6 x^5\right ) \log (x)+\left (1+2 x^2+x^4\right ) \log ^2(x)} \, dx=-\frac {x^{2}}{\mathrm {log}\left (x \right ) x^{2}+\mathrm {log}\left (x \right )-3 x^{3}+2 x^{2}} \] Input:
int((-2*x*log(x)-3*x^4+x^3+x)/((x^4+2*x^2+1)*log(x)^2+(-6*x^5+4*x^4-6*x^3+ 4*x^2)*log(x)+9*x^6-12*x^5+4*x^4),x)
Output:
( - x**2)/(log(x)*x**2 + log(x) - 3*x**3 + 2*x**2)