\(\int \frac {\sqrt {-2+x^2-2 x^3+2 x^4} (2-x^3+2 x^4)}{(-1-x^3+x^4) (-2-x^2-2 x^3+2 x^4)} \, dx\) [761]

Optimal result

Integrand size = 63, antiderivative size = 59 \[ \int \frac {\sqrt {-2+x^2-2 x^3+2 x^4} \left (2-x^3+2 x^4\right )}{\left (-1-x^3+x^4\right ) \left (-2-x^2-2 x^3+2 x^4\right )} \, dx=2 \text {arctanh}\left (\frac {x}{\sqrt {-2+x^2-2 x^3+2 x^4}}\right )-2 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} x}{\sqrt {-2+x^2-2 x^3+2 x^4}}\right ) \] Output:

2*arctanh(x/(2*x^4-2*x^3+x^2-2)^(1/2))-2*2^(1/2)*arctanh(2^(1/2)*x/(2*x^4- 
2*x^3+x^2-2)^(1/2))
 

Mathematica [A] (verified)

Time = 0.19 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.95 \[ \int \frac {\sqrt {-2+x^2-2 x^3+2 x^4} \left (2-x^3+2 x^4\right )}{\left (-1-x^3+x^4\right ) \left (-2-x^2-2 x^3+2 x^4\right )} \, dx=-2 \sqrt {2} \text {arctanh}\left (\frac {x}{\sqrt {-1+\frac {x^2}{2}-x^3+x^4}}\right )+2 \text {arctanh}\left (\frac {x}{\sqrt {-2+x^2-2 x^3+2 x^4}}\right ) \] Input:

Integrate[(Sqrt[-2 + x^2 - 2*x^3 + 2*x^4]*(2 - x^3 + 2*x^4))/((-1 - x^3 + 
x^4)*(-2 - x^2 - 2*x^3 + 2*x^4)),x]
 

Output:

-2*Sqrt[2]*ArcTanh[x/Sqrt[-1 + x^2/2 - x^3 + x^4]] + 2*ArcTanh[x/Sqrt[-2 + 
 x^2 - 2*x^3 + 2*x^4]]
 

Rubi [F]

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.

\(\displaystyle \int \frac {\sqrt {2 x^4-2 x^3+x^2-2} \left (2 x^4-x^3+2\right )}{\left (x^4-x^3-1\right ) \left (2 x^4-2 x^3-x^2-2\right )} \, dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {2 \left (4 x^2-3 x-1\right ) \sqrt {2 x^4-2 x^3+x^2-2}}{2 x^4-2 x^3-x^2-2}-\frac {x (4 x-3) \sqrt {2 x^4-2 x^3+x^2-2}}{x^4-x^3-1}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle 2 \int \frac {\sqrt {2 x^4-2 x^3+x^2-2}}{-2 x^4+2 x^3+x^2+2}dx+3 \int \frac {x \sqrt {2 x^4-2 x^3+x^2-2}}{x^4-x^3-1}dx-4 \int \frac {x^2 \sqrt {2 x^4-2 x^3+x^2-2}}{x^4-x^3-1}dx-6 \int \frac {x \sqrt {2 x^4-2 x^3+x^2-2}}{2 x^4-2 x^3-x^2-2}dx+8 \int \frac {x^2 \sqrt {2 x^4-2 x^3+x^2-2}}{2 x^4-2 x^3-x^2-2}dx\)

Input:

Int[(Sqrt[-2 + x^2 - 2*x^3 + 2*x^4]*(2 - x^3 + 2*x^4))/((-1 - x^3 + x^4)*( 
-2 - x^2 - 2*x^3 + 2*x^4)),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 12.48 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.44

method	result	size
default	\(\ln \left (\frac {\sqrt {2 x^{4}-2 x^{3}+x^{2}-2}+x}{x}\right )-2 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {2 x^{4}-2 x^{3}+x^{2}-2}\, \sqrt {2}}{2 x}\right )-\ln \left (\frac {\sqrt {2 x^{4}-2 x^{3}+x^{2}-2}-x}{x}\right )\)	\(85\)
pseudoelliptic	\(\ln \left (\frac {\sqrt {2 x^{4}-2 x^{3}+x^{2}-2}+x}{x}\right )-2 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {2 x^{4}-2 x^{3}+x^{2}-2}\, \sqrt {2}}{2 x}\right )-\ln \left (\frac {\sqrt {2 x^{4}-2 x^{3}+x^{2}-2}-x}{x}\right )\)	\(85\)
trager	\(\ln \left (-\frac {x^{4}-x^{3}+\sqrt {2 x^{4}-2 x^{3}+x^{2}-2}\, x +x^{2}-1}{x^{4}-x^{3}-1}\right )+\operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) \ln \left (\frac {2 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) x^{4}-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) x^{3}+3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) x^{2}-4 \sqrt {2 x^{4}-2 x^{3}+x^{2}-2}\, x -2 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right )}{2 x^{4}-2 x^{3}-x^{2}-2}\right )\)	\(139\)
elliptic	\(\text {Expression too large to display}\)	\(1159679\)

Input:

int((2*x^4-2*x^3+x^2-2)^(1/2)*(2*x^4-x^3+2)/(x^4-x^3-1)/(2*x^4-2*x^3-x^2-2 
),x,method=_RETURNVERBOSE)
 

Output:

ln(((2*x^4-2*x^3+x^2-2)^(1/2)+x)/x)-2*2^(1/2)*arctanh(1/2*(2*x^4-2*x^3+x^2 
-2)^(1/2)/x*2^(1/2))-ln(((2*x^4-2*x^3+x^2-2)^(1/2)-x)/x)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 168 vs. \(2 (51) = 102\).

Time = 0.17 (sec) , antiderivative size = 168, normalized size of antiderivative = 2.85 \[ \int \frac {\sqrt {-2+x^2-2 x^3+2 x^4} \left (2-x^3+2 x^4\right )}{\left (-1-x^3+x^4\right ) \left (-2-x^2-2 x^3+2 x^4\right )} \, dx=\frac {1}{2} \, \sqrt {2} \log \left (-\frac {4 \, x^{8} - 8 \, x^{7} + 32 \, x^{6} - 28 \, x^{5} + 9 \, x^{4} + 8 \, x^{3} - 4 \, \sqrt {2} {\left (2 \, x^{5} - 2 \, x^{4} + 3 \, x^{3} - 2 \, x\right )} \sqrt {2 \, x^{4} - 2 \, x^{3} + x^{2} - 2} - 28 \, x^{2} + 4}{4 \, x^{8} - 8 \, x^{7} + 4 \, x^{5} - 7 \, x^{4} + 8 \, x^{3} + 4 \, x^{2} + 4}\right ) + \log \left (-\frac {x^{4} - x^{3} + x^{2} + \sqrt {2 \, x^{4} - 2 \, x^{3} + x^{2} - 2} x - 1}{x^{4} - x^{3} - 1}\right ) \] Input:

integrate((2*x^4-2*x^3+x^2-2)^(1/2)*(2*x^4-x^3+2)/(x^4-x^3-1)/(2*x^4-2*x^3 
-x^2-2),x, algorithm="fricas")
 

Output:

1/2*sqrt(2)*log(-(4*x^8 - 8*x^7 + 32*x^6 - 28*x^5 + 9*x^4 + 8*x^3 - 4*sqrt 
(2)*(2*x^5 - 2*x^4 + 3*x^3 - 2*x)*sqrt(2*x^4 - 2*x^3 + x^2 - 2) - 28*x^2 + 
 4)/(4*x^8 - 8*x^7 + 4*x^5 - 7*x^4 + 8*x^3 + 4*x^2 + 4)) + log(-(x^4 - x^3 
 + x^2 + sqrt(2*x^4 - 2*x^3 + x^2 - 2)*x - 1)/(x^4 - x^3 - 1))
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\sqrt {-2+x^2-2 x^3+2 x^4} \left (2-x^3+2 x^4\right )}{\left (-1-x^3+x^4\right ) \left (-2-x^2-2 x^3+2 x^4\right )} \, dx=\text {Timed out} \] Input:

integrate((2*x**4-2*x**3+x**2-2)**(1/2)*(2*x**4-x**3+2)/(x**4-x**3-1)/(2*x 
**4-2*x**3-x**2-2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {\sqrt {-2+x^2-2 x^3+2 x^4} \left (2-x^3+2 x^4\right )}{\left (-1-x^3+x^4\right ) \left (-2-x^2-2 x^3+2 x^4\right )} \, dx=\int { \frac {{\left (2 \, x^{4} - x^{3} + 2\right )} \sqrt {2 \, x^{4} - 2 \, x^{3} + x^{2} - 2}}{{\left (2 \, x^{4} - 2 \, x^{3} - x^{2} - 2\right )} {\left (x^{4} - x^{3} - 1\right )}} \,d x } \] Input:

integrate((2*x^4-2*x^3+x^2-2)^(1/2)*(2*x^4-x^3+2)/(x^4-x^3-1)/(2*x^4-2*x^3 
-x^2-2),x, algorithm="maxima")
 

Output:

integrate((2*x^4 - x^3 + 2)*sqrt(2*x^4 - 2*x^3 + x^2 - 2)/((2*x^4 - 2*x^3 
- x^2 - 2)*(x^4 - x^3 - 1)), x)
 

Giac [F]

\[ \int \frac {\sqrt {-2+x^2-2 x^3+2 x^4} \left (2-x^3+2 x^4\right )}{\left (-1-x^3+x^4\right ) \left (-2-x^2-2 x^3+2 x^4\right )} \, dx=\int { \frac {{\left (2 \, x^{4} - x^{3} + 2\right )} \sqrt {2 \, x^{4} - 2 \, x^{3} + x^{2} - 2}}{{\left (2 \, x^{4} - 2 \, x^{3} - x^{2} - 2\right )} {\left (x^{4} - x^{3} - 1\right )}} \,d x } \] Input:

integrate((2*x^4-2*x^3+x^2-2)^(1/2)*(2*x^4-x^3+2)/(x^4-x^3-1)/(2*x^4-2*x^3 
-x^2-2),x, algorithm="giac")
 

Output:

integrate((2*x^4 - x^3 + 2)*sqrt(2*x^4 - 2*x^3 + x^2 - 2)/((2*x^4 - 2*x^3 
- x^2 - 2)*(x^4 - x^3 - 1)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {-2+x^2-2 x^3+2 x^4} \left (2-x^3+2 x^4\right )}{\left (-1-x^3+x^4\right ) \left (-2-x^2-2 x^3+2 x^4\right )} \, dx=\int \frac {\left (2\,x^4-x^3+2\right )\,\sqrt {2\,x^4-2\,x^3+x^2-2}}{\left (-x^4+x^3+1\right )\,\left (-2\,x^4+2\,x^3+x^2+2\right )} \,d x \] Input:

int(((2*x^4 - x^3 + 2)*(x^2 - 2*x^3 + 2*x^4 - 2)^(1/2))/((x^3 - x^4 + 1)*( 
x^2 + 2*x^3 - 2*x^4 + 2)),x)
 

Output:

int(((2*x^4 - x^3 + 2)*(x^2 - 2*x^3 + 2*x^4 - 2)^(1/2))/((x^3 - x^4 + 1)*( 
x^2 + 2*x^3 - 2*x^4 + 2)), x)
 

Reduce [F]

\[ \int \frac {\sqrt {-2+x^2-2 x^3+2 x^4} \left (2-x^3+2 x^4\right )}{\left (-1-x^3+x^4\right ) \left (-2-x^2-2 x^3+2 x^4\right )} \, dx=2 \left (\int \frac {\sqrt {2 x^{4}-2 x^{3}+x^{2}-2}}{2 x^{8}-4 x^{7}+x^{6}+x^{5}-4 x^{4}+4 x^{3}+x^{2}+2}d x \right )+2 \left (\int \frac {\sqrt {2 x^{4}-2 x^{3}+x^{2}-2}\, x^{4}}{2 x^{8}-4 x^{7}+x^{6}+x^{5}-4 x^{4}+4 x^{3}+x^{2}+2}d x \right )-\left (\int \frac {\sqrt {2 x^{4}-2 x^{3}+x^{2}-2}\, x^{3}}{2 x^{8}-4 x^{7}+x^{6}+x^{5}-4 x^{4}+4 x^{3}+x^{2}+2}d x \right ) \] Input:

int((2*x^4-2*x^3+x^2-2)^(1/2)*(2*x^4-x^3+2)/(x^4-x^3-1)/(2*x^4-2*x^3-x^2-2 
),x)
 

Output:

2*int(sqrt(2*x**4 - 2*x**3 + x**2 - 2)/(2*x**8 - 4*x**7 + x**6 + x**5 - 4* 
x**4 + 4*x**3 + x**2 + 2),x) + 2*int((sqrt(2*x**4 - 2*x**3 + x**2 - 2)*x** 
4)/(2*x**8 - 4*x**7 + x**6 + x**5 - 4*x**4 + 4*x**3 + x**2 + 2),x) - int(( 
sqrt(2*x**4 - 2*x**3 + x**2 - 2)*x**3)/(2*x**8 - 4*x**7 + x**6 + x**5 - 4* 
x**4 + 4*x**3 + x**2 + 2),x)