Integrand size = 101, antiderivative size = 24 \[ \int \frac {2-26 x^2+22 x^3-3 x^4+x^2 \log (x)+\left (-2-27 x^2+11 x^3-x^4+x^2 \log (x)\right ) \log \left (\frac {-2-27 x^2+11 x^3-x^4+x^2 \log (x)}{x^2}\right )}{-2-27 x^2+11 x^3-x^4+x^2 \log (x)} \, dx=x+x \log \left (-2-(5-x)^2-\frac {2}{x^2}+x+\log (x)\right ) \] Output:
x+ln(-2+x-2/x^2+ln(x)-(5-x)^2)*x
Time = 0.04 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92 \[ \int \frac {2-26 x^2+22 x^3-3 x^4+x^2 \log (x)+\left (-2-27 x^2+11 x^3-x^4+x^2 \log (x)\right ) \log \left (\frac {-2-27 x^2+11 x^3-x^4+x^2 \log (x)}{x^2}\right )}{-2-27 x^2+11 x^3-x^4+x^2 \log (x)} \, dx=x+x \log \left (-27-\frac {2}{x^2}+11 x-x^2+\log (x)\right ) \] Input:
Integrate[(2 - 26*x^2 + 22*x^3 - 3*x^4 + x^2*Log[x] + (-2 - 27*x^2 + 11*x^ 3 - x^4 + x^2*Log[x])*Log[(-2 - 27*x^2 + 11*x^3 - x^4 + x^2*Log[x])/x^2])/ (-2 - 27*x^2 + 11*x^3 - x^4 + x^2*Log[x]),x]
Output:
x + x*Log[-27 - 2/x^2 + 11*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.
\(\displaystyle \int \frac {-3 x^4+22 x^3-26 x^2+x^2 \log (x)+\left (-x^4+11 x^3-27 x^2+x^2 \log (x)-2\right ) \log \left (\frac {-x^4+11 x^3-27 x^2+x^2 \log (x)-2}{x^2}\right )+2}{-x^4+11 x^3-27 x^2+x^2 \log (x)-2} \, dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\log \left (-x^2-\frac {2}{x^2}+11 x+\log (x)-27\right )+\frac {3 x^4}{x^4-11 x^3+27 x^2-x^2 \log (x)+2}-\frac {22 x^3}{x^4-11 x^3+27 x^2-x^2 \log (x)+2}-\frac {x^2 \log (x)}{x^4-11 x^3+27 x^2-x^2 \log (x)+2}+\frac {26 x^2}{x^4-11 x^3+27 x^2-x^2 \log (x)+2}-\frac {2}{x^4-11 x^3+27 x^2-x^2 \log (x)+2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \int \log \left (-x^2+11 x+\log (x)-27-\frac {2}{x^2}\right )dx-4 \int \frac {1}{x^4-11 x^3-\log (x) x^2+27 x^2+2}dx-\int \frac {x^2}{x^4-11 x^3-\log (x) x^2+27 x^2+2}dx-11 \int \frac {x^3}{x^4-11 x^3-\log (x) x^2+27 x^2+2}dx+2 \int \frac {x^4}{x^4-11 x^3-\log (x) x^2+27 x^2+2}dx+x\) |
Input:
Int[(2 - 26*x^2 + 22*x^3 - 3*x^4 + x^2*Log[x] + (-2 - 27*x^2 + 11*x^3 - x^ 4 + x^2*Log[x])*Log[(-2 - 27*x^2 + 11*x^3 - x^4 + x^2*Log[x])/x^2])/(-2 - 27*x^2 + 11*x^3 - x^4 + x^2*Log[x]),x]
Output:
$Aborted
Time = 0.87 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.38
method | result | size |
default | \(x +\ln \left (\frac {x^{2} \ln \left (x \right )-x^{4}+11 x^{3}-27 x^{2}-2}{x^{2}}\right ) x\) | \(33\) |
parallelrisch | \(-\frac {733}{594}+\ln \left (\frac {x^{2} \ln \left (x \right )-x^{4}+11 x^{3}-27 x^{2}-2}{x^{2}}\right ) x +\frac {54 \ln \left (x \right )}{11}-\frac {27 \ln \left (-x^{2} \ln \left (x \right )+x^{4}-11 x^{3}+27 x^{2}+2\right )}{11}+x +\frac {27 \ln \left (\frac {x^{2} \ln \left (x \right )-x^{4}+11 x^{3}-27 x^{2}-2}{x^{2}}\right )}{11}\) | \(93\) |
risch | \(x \ln \left (2+x^{4}-11 x^{3}+\left (-\ln \left (x \right )+27\right ) x^{2}\right )-2 x \ln \left (x \right )+\frac {i \pi x \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right )}{2}-i \pi x \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{2}+\frac {i \pi x \operatorname {csgn}\left (i x^{2}\right )^{3}}{2}-\frac {i \pi x \,\operatorname {csgn}\left (\frac {i}{x^{2}}\right ) \operatorname {csgn}\left (i \left (-2-x^{4}+11 x^{3}-\left (-\ln \left (x \right )+27\right ) x^{2}\right )\right ) \operatorname {csgn}\left (\frac {i \left (-2-x^{4}+11 x^{3}-\left (-\ln \left (x \right )+27\right ) x^{2}\right )}{x^{2}}\right )}{2}+\frac {i \pi x \,\operatorname {csgn}\left (\frac {i}{x^{2}}\right ) {\operatorname {csgn}\left (\frac {i \left (-2-x^{4}+11 x^{3}-\left (-\ln \left (x \right )+27\right ) x^{2}\right )}{x^{2}}\right )}^{2}}{2}-\frac {i \pi x \,\operatorname {csgn}\left (i \left (-2-x^{4}+11 x^{3}-\left (-\ln \left (x \right )+27\right ) x^{2}\right )\right ) {\operatorname {csgn}\left (\frac {i \left (-2-x^{4}+11 x^{3}-\left (-\ln \left (x \right )+27\right ) x^{2}\right )}{x^{2}}\right )}^{2}}{2}-\frac {i \pi x {\operatorname {csgn}\left (\frac {i \left (-2-x^{4}+11 x^{3}-\left (-\ln \left (x \right )+27\right ) x^{2}\right )}{x^{2}}\right )}^{3}}{2}-i \pi x {\operatorname {csgn}\left (\frac {i \left (-2-x^{4}+11 x^{3}-\left (-\ln \left (x \right )+27\right ) x^{2}\right )}{x^{2}}\right )}^{2}+i \pi x +x\) | \(339\) |
Input:
int(((x^2*ln(x)-x^4+11*x^3-27*x^2-2)*ln((x^2*ln(x)-x^4+11*x^3-27*x^2-2)/x^ 2)+x^2*ln(x)-3*x^4+22*x^3-26*x^2+2)/(x^2*ln(x)-x^4+11*x^3-27*x^2-2),x,meth od=_RETURNVERBOSE)
Output:
x+ln((x^2*ln(x)-x^4+11*x^3-27*x^2-2)/x^2)*x
Time = 0.10 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.33 \[ \int \frac {2-26 x^2+22 x^3-3 x^4+x^2 \log (x)+\left (-2-27 x^2+11 x^3-x^4+x^2 \log (x)\right ) \log \left (\frac {-2-27 x^2+11 x^3-x^4+x^2 \log (x)}{x^2}\right )}{-2-27 x^2+11 x^3-x^4+x^2 \log (x)} \, dx=x \log \left (-\frac {x^{4} - 11 \, x^{3} - x^{2} \log \left (x\right ) + 27 \, x^{2} + 2}{x^{2}}\right ) + x \] Input:
integrate(((x^2*log(x)-x^4+11*x^3-27*x^2-2)*log((x^2*log(x)-x^4+11*x^3-27* x^2-2)/x^2)+x^2*log(x)-3*x^4+22*x^3-26*x^2+2)/(x^2*log(x)-x^4+11*x^3-27*x^ 2-2),x, algorithm="fricas")
Output:
x*log(-(x^4 - 11*x^3 - x^2*log(x) + 27*x^2 + 2)/x^2) + x
Time = 0.17 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.21 \[ \int \frac {2-26 x^2+22 x^3-3 x^4+x^2 \log (x)+\left (-2-27 x^2+11 x^3-x^4+x^2 \log (x)\right ) \log \left (\frac {-2-27 x^2+11 x^3-x^4+x^2 \log (x)}{x^2}\right )}{-2-27 x^2+11 x^3-x^4+x^2 \log (x)} \, dx=x \log {\left (\frac {- x^{4} + 11 x^{3} + x^{2} \log {\left (x \right )} - 27 x^{2} - 2}{x^{2}} \right )} + x \] Input:
integrate(((x**2*ln(x)-x**4+11*x**3-27*x**2-2)*ln((x**2*ln(x)-x**4+11*x**3 -27*x**2-2)/x**2)+x**2*ln(x)-3*x**4+22*x**3-26*x**2+2)/(x**2*ln(x)-x**4+11 *x**3-27*x**2-2),x)
Output:
x*log((-x**4 + 11*x**3 + x**2*log(x) - 27*x**2 - 2)/x**2) + x
Time = 0.06 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.38 \[ \int \frac {2-26 x^2+22 x^3-3 x^4+x^2 \log (x)+\left (-2-27 x^2+11 x^3-x^4+x^2 \log (x)\right ) \log \left (\frac {-2-27 x^2+11 x^3-x^4+x^2 \log (x)}{x^2}\right )}{-2-27 x^2+11 x^3-x^4+x^2 \log (x)} \, dx=x \log \left (-x^{4} + 11 \, x^{3} + x^{2} \log \left (x\right ) - 27 \, x^{2} - 2\right ) - 2 \, x \log \left (x\right ) + x \] Input:
integrate(((x^2*log(x)-x^4+11*x^3-27*x^2-2)*log((x^2*log(x)-x^4+11*x^3-27* x^2-2)/x^2)+x^2*log(x)-3*x^4+22*x^3-26*x^2+2)/(x^2*log(x)-x^4+11*x^3-27*x^ 2-2),x, algorithm="maxima")
Output:
x*log(-x^4 + 11*x^3 + x^2*log(x) - 27*x^2 - 2) - 2*x*log(x) + x
Time = 0.15 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.38 \[ \int \frac {2-26 x^2+22 x^3-3 x^4+x^2 \log (x)+\left (-2-27 x^2+11 x^3-x^4+x^2 \log (x)\right ) \log \left (\frac {-2-27 x^2+11 x^3-x^4+x^2 \log (x)}{x^2}\right )}{-2-27 x^2+11 x^3-x^4+x^2 \log (x)} \, dx=x \log \left (-x^{4} + 11 \, x^{3} + x^{2} \log \left (x\right ) - 27 \, x^{2} - 2\right ) - 2 \, x \log \left (x\right ) + x \] Input:
integrate(((x^2*log(x)-x^4+11*x^3-27*x^2-2)*log((x^2*log(x)-x^4+11*x^3-27* x^2-2)/x^2)+x^2*log(x)-3*x^4+22*x^3-26*x^2+2)/(x^2*log(x)-x^4+11*x^3-27*x^ 2-2),x, algorithm="giac")
Output:
x*log(-x^4 + 11*x^3 + x^2*log(x) - 27*x^2 - 2) - 2*x*log(x) + x
Time = 3.30 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.33 \[ \int \frac {2-26 x^2+22 x^3-3 x^4+x^2 \log (x)+\left (-2-27 x^2+11 x^3-x^4+x^2 \log (x)\right ) \log \left (\frac {-2-27 x^2+11 x^3-x^4+x^2 \log (x)}{x^2}\right )}{-2-27 x^2+11 x^3-x^4+x^2 \log (x)} \, dx=x\,\left (\ln \left (-\frac {27\,x^2-x^2\,\ln \left (x\right )-11\,x^3+x^4+2}{x^2}\right )+1\right ) \] Input:
int(-(x^2*log(x) - log(-(27*x^2 - x^2*log(x) - 11*x^3 + x^4 + 2)/x^2)*(27* x^2 - x^2*log(x) - 11*x^3 + x^4 + 2) - 26*x^2 + 22*x^3 - 3*x^4 + 2)/(27*x^ 2 - x^2*log(x) - 11*x^3 + x^4 + 2),x)
Output:
x*(log(-(27*x^2 - x^2*log(x) - 11*x^3 + x^4 + 2)/x^2) + 1)
Time = 0.15 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.33 \[ \int \frac {2-26 x^2+22 x^3-3 x^4+x^2 \log (x)+\left (-2-27 x^2+11 x^3-x^4+x^2 \log (x)\right ) \log \left (\frac {-2-27 x^2+11 x^3-x^4+x^2 \log (x)}{x^2}\right )}{-2-27 x^2+11 x^3-x^4+x^2 \log (x)} \, dx=x \left (\mathrm {log}\left (\frac {\mathrm {log}\left (x \right ) x^{2}-x^{4}+11 x^{3}-27 x^{2}-2}{x^{2}}\right )+1\right ) \] Input:
int(((x^2*log(x)-x^4+11*x^3-27*x^2-2)*log((x^2*log(x)-x^4+11*x^3-27*x^2-2) /x^2)+x^2*log(x)-3*x^4+22*x^3-26*x^2+2)/(x^2*log(x)-x^4+11*x^3-27*x^2-2),x )
Output:
x*(log((log(x)*x**2 - x**4 + 11*x**3 - 27*x**2 - 2)/x**2) + 1)