Integrand size = 59, antiderivative size = 57 \[ \int \frac {\sqrt {-1+x^2-2 x^3+x^4} \left (1-x^3+x^4\right )}{\left (-1-2 x^3+x^4\right ) \left (-2-x^2-4 x^3+2 x^4\right )} \, dx=\text {arctanh}\left (\frac {x}{\sqrt {-1+x^2-2 x^3+x^4}}\right )-\sqrt {\frac {3}{2}} \text {arctanh}\left (\frac {\sqrt {\frac {3}{2}} x}{\sqrt {-1+x^2-2 x^3+x^4}}\right ) \] Output:
arctanh(x/(x^4-2*x^3+x^2-1)^(1/2))-1/2*6^(1/2)*arctanh(1/2*6^(1/2)*x/(x^4- 2*x^3+x^2-1)^(1/2))
Time = 0.84 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {-1+x^2-2 x^3+x^4} \left (1-x^3+x^4\right )}{\left (-1-2 x^3+x^4\right ) \left (-2-x^2-4 x^3+2 x^4\right )} \, dx=\text {arctanh}\left (\frac {x}{\sqrt {-1+x^2-2 x^3+x^4}}\right )-\sqrt {\frac {3}{2}} \text {arctanh}\left (\frac {\sqrt {\frac {3}{2}} x}{\sqrt {-1+x^2-2 x^3+x^4}}\right ) \] Input:
Integrate[(Sqrt[-1 + x^2 - 2*x^3 + x^4]*(1 - x^3 + x^4))/((-1 - 2*x^3 + x^ 4)*(-2 - x^2 - 4*x^3 + 2*x^4)),x]
Output:
ArcTanh[x/Sqrt[-1 + x^2 - 2*x^3 + x^4]] - Sqrt[3/2]*ArcTanh[(Sqrt[3/2]*x)/ Sqrt[-1 + x^2 - 2*x^3 + x^4]]
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.
| \(\displaystyle \int \frac {\sqrt {x^4-2 x^3+x^2-1} \left (x^4-x^3+1\right )}{\left (x^4-2 x^3-1\right ) \left (2 x^4-4 x^3-x^2-2\right )} \, dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {\left (4 x^2-6 x-1\right ) \sqrt {x^4-2 x^3+x^2-1}}{2 x^4-4 x^3-x^2-2}-\frac {x (2 x-3) \sqrt {x^4-2 x^3+x^2-1}}{x^4-2 x^3-1}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \int \frac {\sqrt {x^4-2 x^3+x^2-1}}{-2 x^4+4 x^3+x^2+2}dx+3 \int \frac {x \sqrt {x^4-2 x^3+x^2-1}}{x^4-2 x^3-1}dx-2 \int \frac {x^2 \sqrt {x^4-2 x^3+x^2-1}}{x^4-2 x^3-1}dx-6 \int \frac {x \sqrt {x^4-2 x^3+x^2-1}}{2 x^4-4 x^3-x^2-2}dx+4 \int \frac {x^2 \sqrt {x^4-2 x^3+x^2-1}}{2 x^4-4 x^3-x^2-2}dx\) |
Input:
Int[(Sqrt[-1 + x^2 - 2*x^3 + x^4]*(1 - x^3 + x^4))/((-1 - 2*x^3 + x^4)*(-2 - x^2 - 4*x^3 + 2*x^4)),x]
Output:
$Aborted
Time = 1.99 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.42
| method | result | size |
| default | \(-\frac {\sqrt {6}\, \operatorname {arctanh}\left (\frac {\sqrt {x^{4}-2 x^{3}+x^{2}-1}\, \sqrt {6}}{3 x}\right )}{2}-\frac {\ln \left (\frac {\sqrt {x^{4}-2 x^{3}+x^{2}-1}-x}{x}\right )}{2}+\frac {\ln \left (\frac {\sqrt {x^{4}-2 x^{3}+x^{2}-1}+x}{x}\right )}{2}\) | \(81\) |
| pseudoelliptic | \(-\frac {\sqrt {6}\, \operatorname {arctanh}\left (\frac {\sqrt {x^{4}-2 x^{3}+x^{2}-1}\, \sqrt {6}}{3 x}\right )}{2}-\frac {\ln \left (\frac {\sqrt {x^{4}-2 x^{3}+x^{2}-1}-x}{x}\right )}{2}+\frac {\ln \left (\frac {\sqrt {x^{4}-2 x^{3}+x^{2}-1}+x}{x}\right )}{2}\) | \(81\) |
| trager | \(-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-6\right ) \ln \left (-\frac {2 \operatorname {RootOf}\left (\textit {\_Z}^{2}-6\right ) x^{4}-4 \operatorname {RootOf}\left (\textit {\_Z}^{2}-6\right ) x^{3}+5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-6\right ) x^{2}+12 \sqrt {x^{4}-2 x^{3}+x^{2}-1}\, x -2 \operatorname {RootOf}\left (\textit {\_Z}^{2}-6\right )}{2 x^{4}-4 x^{3}-x^{2}-2}\right )}{4}+\frac {\ln \left (-\frac {x^{4}-2 x^{3}+2 \sqrt {x^{4}-2 x^{3}+x^{2}-1}\, x +2 x^{2}-1}{x^{4}-2 x^{3}-1}\right )}{2}\) | \(142\) |
| elliptic | \(\text {Expression too large to display}\) | \(15410\) |
Input:
int((x^4-2*x^3+x^2-1)^(1/2)*(x^4-x^3+1)/(x^4-2*x^3-1)/(2*x^4-4*x^3-x^2-2), x,method=_RETURNVERBOSE)
Output:
-1/2*6^(1/2)*arctanh(1/3*(x^4-2*x^3+x^2-1)^(1/2)/x*6^(1/2))-1/2*ln(((x^4-2 *x^3+x^2-1)^(1/2)-x)/x)+1/2*ln(((x^4-2*x^3+x^2-1)^(1/2)+x)/x)
Leaf count of result is larger than twice the leaf count of optimal. 174 vs. \(2 (46) = 92\).
Time = 0.21 (sec) , antiderivative size = 174, normalized size of antiderivative = 3.05 \[ \int \frac {\sqrt {-1+x^2-2 x^3+x^4} \left (1-x^3+x^4\right )}{\left (-1-2 x^3+x^4\right ) \left (-2-x^2-4 x^3+2 x^4\right )} \, dx=\frac {1}{4} \, \sqrt {\frac {3}{2}} \log \left (-\frac {4 \, x^{8} - 16 \, x^{7} + 60 \, x^{6} - 88 \, x^{5} + 41 \, x^{4} + 16 \, x^{3} - 8 \, \sqrt {\frac {3}{2}} {\left (2 \, x^{5} - 4 \, x^{4} + 5 \, x^{3} - 2 \, x\right )} \sqrt {x^{4} - 2 \, x^{3} + x^{2} - 1} - 44 \, x^{2} + 4}{4 \, x^{8} - 16 \, x^{7} + 12 \, x^{6} + 8 \, x^{5} - 7 \, x^{4} + 16 \, x^{3} + 4 \, x^{2} + 4}\right ) + \frac {1}{2} \, \log \left (-\frac {x^{4} - 2 \, x^{3} + 2 \, x^{2} + 2 \, \sqrt {x^{4} - 2 \, x^{3} + x^{2} - 1} x - 1}{x^{4} - 2 \, x^{3} - 1}\right ) \] Input:
integrate((x^4-2*x^3+x^2-1)^(1/2)*(x^4-x^3+1)/(x^4-2*x^3-1)/(2*x^4-4*x^3-x ^2-2),x, algorithm="fricas")
Output:
1/4*sqrt(3/2)*log(-(4*x^8 - 16*x^7 + 60*x^6 - 88*x^5 + 41*x^4 + 16*x^3 - 8 *sqrt(3/2)*(2*x^5 - 4*x^4 + 5*x^3 - 2*x)*sqrt(x^4 - 2*x^3 + x^2 - 1) - 44* x^2 + 4)/(4*x^8 - 16*x^7 + 12*x^6 + 8*x^5 - 7*x^4 + 16*x^3 + 4*x^2 + 4)) + 1/2*log(-(x^4 - 2*x^3 + 2*x^2 + 2*sqrt(x^4 - 2*x^3 + x^2 - 1)*x - 1)/(x^4 - 2*x^3 - 1))
Timed out. \[ \int \frac {\sqrt {-1+x^2-2 x^3+x^4} \left (1-x^3+x^4\right )}{\left (-1-2 x^3+x^4\right ) \left (-2-x^2-4 x^3+2 x^4\right )} \, dx=\text {Timed out} \] Input:
integrate((x**4-2*x**3+x**2-1)**(1/2)*(x**4-x**3+1)/(x**4-2*x**3-1)/(2*x** 4-4*x**3-x**2-2),x)
Output:
Timed out
\[ \int \frac {\sqrt {-1+x^2-2 x^3+x^4} \left (1-x^3+x^4\right )}{\left (-1-2 x^3+x^4\right ) \left (-2-x^2-4 x^3+2 x^4\right )} \, dx=\int { \frac {{\left (x^{4} - x^{3} + 1\right )} \sqrt {x^{4} - 2 \, x^{3} + x^{2} - 1}}{{\left (2 \, x^{4} - 4 \, x^{3} - x^{2} - 2\right )} {\left (x^{4} - 2 \, x^{3} - 1\right )}} \,d x } \] Input:
integrate((x^4-2*x^3+x^2-1)^(1/2)*(x^4-x^3+1)/(x^4-2*x^3-1)/(2*x^4-4*x^3-x ^2-2),x, algorithm="maxima")
Output:
integrate((x^4 - x^3 + 1)*sqrt(x^4 - 2*x^3 + x^2 - 1)/((2*x^4 - 4*x^3 - x^ 2 - 2)*(x^4 - 2*x^3 - 1)), x)
\[ \int \frac {\sqrt {-1+x^2-2 x^3+x^4} \left (1-x^3+x^4\right )}{\left (-1-2 x^3+x^4\right ) \left (-2-x^2-4 x^3+2 x^4\right )} \, dx=\int { \frac {{\left (x^{4} - x^{3} + 1\right )} \sqrt {x^{4} - 2 \, x^{3} + x^{2} - 1}}{{\left (2 \, x^{4} - 4 \, x^{3} - x^{2} - 2\right )} {\left (x^{4} - 2 \, x^{3} - 1\right )}} \,d x } \] Input:
integrate((x^4-2*x^3+x^2-1)^(1/2)*(x^4-x^3+1)/(x^4-2*x^3-1)/(2*x^4-4*x^3-x ^2-2),x, algorithm="giac")
Output:
integrate((x^4 - x^3 + 1)*sqrt(x^4 - 2*x^3 + x^2 - 1)/((2*x^4 - 4*x^3 - x^ 2 - 2)*(x^4 - 2*x^3 - 1)), x)
Timed out. \[ \int \frac {\sqrt {-1+x^2-2 x^3+x^4} \left (1-x^3+x^4\right )}{\left (-1-2 x^3+x^4\right ) \left (-2-x^2-4 x^3+2 x^4\right )} \, dx=\int \frac {\left (x^4-x^3+1\right )\,\sqrt {x^4-2\,x^3+x^2-1}}{\left (-x^4+2\,x^3+1\right )\,\left (-2\,x^4+4\,x^3+x^2+2\right )} \,d x \] Input:
int(((x^4 - x^3 + 1)*(x^2 - 2*x^3 + x^4 - 1)^(1/2))/((2*x^3 - x^4 + 1)*(x^ 2 + 4*x^3 - 2*x^4 + 2)),x)
Output:
int(((x^4 - x^3 + 1)*(x^2 - 2*x^3 + x^4 - 1)^(1/2))/((2*x^3 - x^4 + 1)*(x^ 2 + 4*x^3 - 2*x^4 + 2)), x)
\[ \int \frac {\sqrt {-1+x^2-2 x^3+x^4} \left (1-x^3+x^4\right )}{\left (-1-2 x^3+x^4\right ) \left (-2-x^2-4 x^3+2 x^4\right )} \, dx=\int \frac {\sqrt {x^{4}-2 x^{3}+x^{2}-1}}{2 x^{8}-8 x^{7}+7 x^{6}+2 x^{5}-4 x^{4}+8 x^{3}+x^{2}+2}d x +\int \frac {\sqrt {x^{4}-2 x^{3}+x^{2}-1}\, x^{4}}{2 x^{8}-8 x^{7}+7 x^{6}+2 x^{5}-4 x^{4}+8 x^{3}+x^{2}+2}d x -\left (\int \frac {\sqrt {x^{4}-2 x^{3}+x^{2}-1}\, x^{3}}{2 x^{8}-8 x^{7}+7 x^{6}+2 x^{5}-4 x^{4}+8 x^{3}+x^{2}+2}d x \right ) \] Input:
int((x^4-2*x^3+x^2-1)^(1/2)*(x^4-x^3+1)/(x^4-2*x^3-1)/(2*x^4-4*x^3-x^2-2), x)
Output:
int(sqrt(x**4 - 2*x**3 + x**2 - 1)/(2*x**8 - 8*x**7 + 7*x**6 + 2*x**5 - 4* x**4 + 8*x**3 + x**2 + 2),x) + int((sqrt(x**4 - 2*x**3 + x**2 - 1)*x**4)/( 2*x**8 - 8*x**7 + 7*x**6 + 2*x**5 - 4*x**4 + 8*x**3 + x**2 + 2),x) - int(( sqrt(x**4 - 2*x**3 + x**2 - 1)*x**3)/(2*x**8 - 8*x**7 + 7*x**6 + 2*x**5 - 4*x**4 + 8*x**3 + x**2 + 2),x)