\(\int \frac {(-3+2 \sqrt {2})^3-(-3+2 \sqrt {2})^2 x^2-(-3+2 \sqrt {2}) x^4+x^6}{577-408 \sqrt {2}+(328-232 \sqrt {2}) x^2+(78-56 \sqrt {2}) x^4+(8-8 \sqrt {2}) x^6+x^8} \, dx\) [100]

Optimal result

Integrand size = 99, antiderivative size = 22 \[ \int \frac {\left (-3+2 \sqrt {2}\right )^3-\left (-3+2 \sqrt {2}\right )^2 x^2-\left (-3+2 \sqrt {2}\right ) x^4+x^6}{577-408 \sqrt {2}+\left (328-232 \sqrt {2}\right ) x^2+\left (78-56 \sqrt {2}\right ) x^4+\left (8-8 \sqrt {2}\right ) x^6+x^8} \, dx=-\frac {1}{2} \text {arctanh}\left (\frac {2 x}{3-2 \sqrt {2}+x^2}\right ) \] Output:

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

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(179\) vs. \(2(22)=44\).

Time = 0.15 (sec) , antiderivative size = 179, normalized size of antiderivative = 8.14 \[ \int \frac {\left (-3+2 \sqrt {2}\right )^3-\left (-3+2 \sqrt {2}\right )^2 x^2-\left (-3+2 \sqrt {2}\right ) x^4+x^6}{577-408 \sqrt {2}+\left (328-232 \sqrt {2}\right ) x^2+\left (78-56 \sqrt {2}\right ) x^4+\left (8-8 \sqrt {2}\right ) x^6+x^8} \, dx=\frac {\left (-17+12 \sqrt {2}+\left (-2+4 \sqrt {2}\right ) x^2-x^4\right ) \left (99-70 \sqrt {2}+\left (17-12 \sqrt {2}\right ) x^2+\left (-3+2 \sqrt {2}\right ) x^4-x^6\right ) \left (\log \left (-3+2 \sqrt {2}-2 x-x^2\right )-\log \left (-3+2 \sqrt {2}+2 x-x^2\right )\right )}{4 \left (-3+2 \sqrt {2}+x^2\right ) \left (-577+408 \sqrt {2}+8 \left (-41+29 \sqrt {2}\right ) x^2+\left (-78+56 \sqrt {2}\right ) x^4+8 \left (-1+\sqrt {2}\right ) x^6-x^8\right )} \] Input:

Integrate[((-3 + 2*Sqrt[2])^3 - (-3 + 2*Sqrt[2])^2*x^2 - (-3 + 2*Sqrt[2])* 
x^4 + x^6)/(577 - 408*Sqrt[2] + (328 - 232*Sqrt[2])*x^2 + (78 - 56*Sqrt[2] 
)*x^4 + (8 - 8*Sqrt[2])*x^6 + x^8),x]
 

Output:

((-17 + 12*Sqrt[2] + (-2 + 4*Sqrt[2])*x^2 - x^4)*(99 - 70*Sqrt[2] + (17 - 
12*Sqrt[2])*x^2 + (-3 + 2*Sqrt[2])*x^4 - x^6)*(Log[-3 + 2*Sqrt[2] - 2*x - 
x^2] - Log[-3 + 2*Sqrt[2] + 2*x - x^2]))/(4*(-3 + 2*Sqrt[2] + x^2)*(-577 + 
 408*Sqrt[2] + 8*(-41 + 29*Sqrt[2])*x^2 + (-78 + 56*Sqrt[2])*x^4 + 8*(-1 + 
 Sqrt[2])*x^6 - x^8))
 

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.

\(\displaystyle \int \frac {x^6-\left (2 \sqrt {2}-3\right ) x^4-\left (2 \sqrt {2}-3\right )^2 x^2+\left (2 \sqrt {2}-3\right )^3}{x^8+\left (8-8 \sqrt {2}\right ) x^6+\left (78-56 \sqrt {2}\right ) x^4+\left (328-232 \sqrt {2}\right ) x^2-408 \sqrt {2}+577} \, dx\)

\(\Big \downarrow \) 7292

\(\displaystyle \int \frac {x^6-\left (2 \sqrt {2}-3\right ) x^4-\left (2 \sqrt {2}-3\right )^2 x^2+\left (2 \sqrt {2}-3\right )^3}{x^8+\left (8-8 \sqrt {2}\right ) x^6+\left (78-56 \sqrt {2}\right ) x^4+\left (328-232 \sqrt {2}\right ) x^2+577 \left (1-\frac {408 \sqrt {2}}{577}\right )}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {x^6}{x^8+8 \left (1-\sqrt {2}\right ) x^6+78 \left (1-\frac {28 \sqrt {2}}{39}\right ) x^4+328 \left (1-\frac {29 \sqrt {2}}{41}\right ) x^2+577 \left (1-\frac {408 \sqrt {2}}{577}\right )}+\frac {\left (3-2 \sqrt {2}\right ) x^4}{x^8+8 \left (1-\sqrt {2}\right ) x^6+78 \left (1-\frac {28 \sqrt {2}}{39}\right ) x^4+328 \left (1-\frac {29 \sqrt {2}}{41}\right ) x^2+577 \left (1-\frac {408 \sqrt {2}}{577}\right )}+\frac {\left (12 \sqrt {2}-17\right ) x^2}{x^8+8 \left (1-\sqrt {2}\right ) x^6+78 \left (1-\frac {28 \sqrt {2}}{39}\right ) x^4+328 \left (1-\frac {29 \sqrt {2}}{41}\right ) x^2+577 \left (1-\frac {408 \sqrt {2}}{577}\right )}+\frac {70 \sqrt {2}-99}{x^8+8 \left (1-\sqrt {2}\right ) x^6+78 \left (1-\frac {28 \sqrt {2}}{39}\right ) x^4+328 \left (1-\frac {29 \sqrt {2}}{41}\right ) x^2+577 \left (1-\frac {408 \sqrt {2}}{577}\right )}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\left (\left (99-70 \sqrt {2}\right ) \int \frac {1}{x^8+8 \left (1-\sqrt {2}\right ) x^6+78 \left (1-\frac {28 \sqrt {2}}{39}\right ) x^4+328 \left (1-\frac {29 \sqrt {2}}{41}\right ) x^2+577 \left (1-\frac {408 \sqrt {2}}{577}\right )}dx\right )-\left (17-12 \sqrt {2}\right ) \int \frac {x^2}{x^8+8 \left (1-\sqrt {2}\right ) x^6+78 \left (1-\frac {28 \sqrt {2}}{39}\right ) x^4+328 \left (1-\frac {29 \sqrt {2}}{41}\right ) x^2+577 \left (1-\frac {408 \sqrt {2}}{577}\right )}dx+\left (3-2 \sqrt {2}\right ) \int \frac {x^4}{x^8+8 \left (1-\sqrt {2}\right ) x^6+78 \left (1-\frac {28 \sqrt {2}}{39}\right ) x^4+328 \left (1-\frac {29 \sqrt {2}}{41}\right ) x^2+577 \left (1-\frac {408 \sqrt {2}}{577}\right )}dx+\int \frac {x^6}{x^8+8 \left (1-\sqrt {2}\right ) x^6+78 \left (1-\frac {28 \sqrt {2}}{39}\right ) x^4+328 \left (1-\frac {29 \sqrt {2}}{41}\right ) x^2+577 \left (1-\frac {408 \sqrt {2}}{577}\right )}dx\)

Input:

Int[((-3 + 2*Sqrt[2])^3 - (-3 + 2*Sqrt[2])^2*x^2 - (-3 + 2*Sqrt[2])*x^4 + 
x^6)/(577 - 408*Sqrt[2] + (328 - 232*Sqrt[2])*x^2 + (78 - 56*Sqrt[2])*x^4 
+ (8 - 8*Sqrt[2])*x^6 + x^8),x]
 

Output:

$Aborted
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 7292

Int[u_, x_Symbol] :> With[{v = NormalizeIntegrand[u, x]}, Int[v, x] /; v =! 
= u]
 

rule 7293

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v] 
]
 

Maple [A] (verified)

Time = 0.20 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.55

method	result	size
default	\(\frac {\ln \left (x^{2}-2 \sqrt {2}-2 x +3\right )}{4}-\frac {\ln \left (x^{2}-2 \sqrt {2}+2 x +3\right )}{4}\)	\(34\)
risch	\(\frac {\ln \left (x^{2}-2 \sqrt {2}-2 x +3\right )}{4}-\frac {\ln \left (x^{2}-2 \sqrt {2}+2 x +3\right )}{4}\)	\(34\)
parallelrisch	\(\frac {\ln \left (x^{2}-2 \sqrt {2}-2 x +3\right )}{4}-\frac {\ln \left (x^{2}-2 \sqrt {2}+2 x +3\right )}{4}\)	\(34\)

Input:

int(((-3+2*2^(1/2))^3-(-3+2*2^(1/2))^2*x^2-(-3+2*2^(1/2))*x^4+x^6)/(577-40 
8*2^(1/2)+(328-232*2^(1/2))*x^2+(78-56*2^(1/2))*x^4+(8-8*2^(1/2))*x^6+x^8) 
,x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.50 \[ \int \frac {\left (-3+2 \sqrt {2}\right )^3-\left (-3+2 \sqrt {2}\right )^2 x^2-\left (-3+2 \sqrt {2}\right ) x^4+x^6}{577-408 \sqrt {2}+\left (328-232 \sqrt {2}\right ) x^2+\left (78-56 \sqrt {2}\right ) x^4+\left (8-8 \sqrt {2}\right ) x^6+x^8} \, dx=-\frac {1}{4} \, \log \left (x^{2} + 2 \, x - 2 \, \sqrt {2} + 3\right ) + \frac {1}{4} \, \log \left (x^{2} - 2 \, x - 2 \, \sqrt {2} + 3\right ) \] Input:

integrate(((-3+2*2^(1/2))^3-(-3+2*2^(1/2))^2*x^2-(-3+2*2^(1/2))*x^4+x^6)/( 
577-408*2^(1/2)+(328-232*2^(1/2))*x^2+(78-56*2^(1/2))*x^4+(8-8*2^(1/2))*x^ 
6+x^8),x, algorithm="fricas")
 

Output:

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

Sympy [F(-2)]

Exception generated. \[ \int \frac {\left (-3+2 \sqrt {2}\right )^3-\left (-3+2 \sqrt {2}\right )^2 x^2-\left (-3+2 \sqrt {2}\right ) x^4+x^6}{577-408 \sqrt {2}+\left (328-232 \sqrt {2}\right ) x^2+\left (78-56 \sqrt {2}\right ) x^4+\left (8-8 \sqrt {2}\right ) x^6+x^8} \, dx=\text {Exception raised: PolynomialError} \] Input:

integrate(((-3+2*2**(1/2))**3-(-3+2*2**(1/2))**2*x**2-(-3+2*2**(1/2))*x**4 
+x**6)/(577-408*2**(1/2)+(328-232*2**(1/2))*x**2+(78-56*2**(1/2))*x**4+(8- 
8*2**(1/2))*x**6+x**8),x)
 

Output:

Exception raised: PolynomialError >> 1/(-489331912114255602061892417478047 
2498117708482611714912381696*_t**4 + 3460099133069698398004476359279702930 
052248019321310378430976*sqrt(2)*_t**4 - 159769239484575670917838951113184 
628965915778476
 

Maxima [F]

\[ \int \frac {\left (-3+2 \sqrt {2}\right )^3-\left (-3+2 \sqrt {2}\right )^2 x^2-\left (-3+2 \sqrt {2}\right ) x^4+x^6}{577-408 \sqrt {2}+\left (328-232 \sqrt {2}\right ) x^2+\left (78-56 \sqrt {2}\right ) x^4+\left (8-8 \sqrt {2}\right ) x^6+x^8} \, dx=\int { \frac {x^{6} - x^{4} {\left (2 \, \sqrt {2} - 3\right )} - x^{2} {\left (2 \, \sqrt {2} - 3\right )}^{2} + {\left (2 \, \sqrt {2} - 3\right )}^{3}}{x^{8} - 8 \, x^{6} {\left (\sqrt {2} - 1\right )} - 2 \, x^{4} {\left (28 \, \sqrt {2} - 39\right )} - 8 \, x^{2} {\left (29 \, \sqrt {2} - 41\right )} - 408 \, \sqrt {2} + 577} \,d x } \] Input:

integrate(((-3+2*2^(1/2))^3-(-3+2*2^(1/2))^2*x^2-(-3+2*2^(1/2))*x^4+x^6)/( 
577-408*2^(1/2)+(328-232*2^(1/2))*x^2+(78-56*2^(1/2))*x^4+(8-8*2^(1/2))*x^ 
6+x^8),x, algorithm="maxima")
 

Output:

integrate((x^6 - x^4*(2*sqrt(2) - 3) - x^2*(2*sqrt(2) - 3)^2 + (2*sqrt(2) 
- 3)^3)/(x^8 - 8*x^6*(sqrt(2) - 1) - 2*x^4*(28*sqrt(2) - 39) - 8*x^2*(29*s 
qrt(2) - 41) - 408*sqrt(2) + 577), x)
 

Giac [A] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.59 \[ \int \frac {\left (-3+2 \sqrt {2}\right )^3-\left (-3+2 \sqrt {2}\right )^2 x^2-\left (-3+2 \sqrt {2}\right ) x^4+x^6}{577-408 \sqrt {2}+\left (328-232 \sqrt {2}\right ) x^2+\left (78-56 \sqrt {2}\right ) x^4+\left (8-8 \sqrt {2}\right ) x^6+x^8} \, dx=-\frac {1}{4} \, \log \left ({\left | x^{2} + 2 \, x - 2 \, \sqrt {2} + 3 \right |}\right ) + \frac {1}{4} \, \log \left ({\left | x^{2} - 2 \, x - 2 \, \sqrt {2} + 3 \right |}\right ) \] Input:

integrate(((-3+2*2^(1/2))^3-(-3+2*2^(1/2))^2*x^2-(-3+2*2^(1/2))*x^4+x^6)/( 
577-408*2^(1/2)+(328-232*2^(1/2))*x^2+(78-56*2^(1/2))*x^4+(8-8*2^(1/2))*x^ 
6+x^8),x, algorithm="giac")
 

Output:

-1/4*log(abs(x^2 + 2*x - 2*sqrt(2) + 3)) + 1/4*log(abs(x^2 - 2*x - 2*sqrt( 
2) + 3))
 

Mupad [B] (verification not implemented)

Time = 22.84 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.55 \[ \int \frac {\left (-3+2 \sqrt {2}\right )^3-\left (-3+2 \sqrt {2}\right )^2 x^2-\left (-3+2 \sqrt {2}\right ) x^4+x^6}{577-408 \sqrt {2}+\left (328-232 \sqrt {2}\right ) x^2+\left (78-56 \sqrt {2}\right ) x^4+\left (8-8 \sqrt {2}\right ) x^6+x^8} \, dx=-\frac {\mathrm {atanh}\left (\frac {x\,\left (16\,\sqrt {2}-16\right )}{2\,\left (20\,\sqrt {2}+4\,\sqrt {2}\,x^2-4\,x^2-28\right )}\right )}{2} \] Input:

int((x^4*(2*2^(1/2) - 3) - (2*2^(1/2) - 3)^3 + x^2*(2*2^(1/2) - 3)^2 - x^6 
)/(x^6*(8*2^(1/2) - 8) + x^4*(56*2^(1/2) - 78) + x^2*(232*2^(1/2) - 328) + 
 408*2^(1/2) - x^8 - 577),x)
 

Output:

-atanh((x*(16*2^(1/2) - 16))/(2*(20*2^(1/2) + 4*2^(1/2)*x^2 - 4*x^2 - 28)) 
)/2
 

Reduce [F]

\[ \int \frac {\left (-3+2 \sqrt {2}\right )^3-\left (-3+2 \sqrt {2}\right )^2 x^2-\left (-3+2 \sqrt {2}\right ) x^4+x^6}{577-408 \sqrt {2}+\left (328-232 \sqrt {2}\right ) x^2+\left (78-56 \sqrt {2}\right ) x^4+\left (8-8 \sqrt {2}\right ) x^6+x^8} \, dx=6 \sqrt {2}\, \left (\int \frac {x^{4}}{x^{8}+4 x^{6}+6 x^{4}-124 x^{2}+1}d x \right )+4 \sqrt {2}\, \left (\int \frac {x^{2}}{x^{8}+4 x^{6}+6 x^{4}-124 x^{2}+1}d x \right )-2 \sqrt {2}\, \left (\int \frac {1}{x^{8}+4 x^{6}+6 x^{4}-124 x^{2}+1}d x \right )+\int \frac {x^{6}}{x^{8}+4 x^{6}+6 x^{4}-124 x^{2}+1}d x -\left (\int \frac {x^{4}}{x^{8}+4 x^{6}+6 x^{4}-124 x^{2}+1}d x \right )+27 \left (\int \frac {x^{2}}{x^{8}+4 x^{6}+6 x^{4}-124 x^{2}+1}d x \right )-3 \left (\int \frac {1}{x^{8}+4 x^{6}+6 x^{4}-124 x^{2}+1}d x \right ) \] Input:

int(((-3+2*2^(1/2))^3-(-3+2*2^(1/2))^2*x^2-(-3+2*2^(1/2))*x^4+x^6)/(577-40 
8*2^(1/2)+(328-232*2^(1/2))*x^2+(78-56*2^(1/2))*x^4+(8-8*2^(1/2))*x^6+x^8) 
,x)
 

Output:

6*sqrt(2)*int(x**4/(x**8 + 4*x**6 + 6*x**4 - 124*x**2 + 1),x) + 4*sqrt(2)* 
int(x**2/(x**8 + 4*x**6 + 6*x**4 - 124*x**2 + 1),x) - 2*sqrt(2)*int(1/(x** 
8 + 4*x**6 + 6*x**4 - 124*x**2 + 1),x) + int(x**6/(x**8 + 4*x**6 + 6*x**4 
- 124*x**2 + 1),x) - int(x**4/(x**8 + 4*x**6 + 6*x**4 - 124*x**2 + 1),x) + 
 27*int(x**2/(x**8 + 4*x**6 + 6*x**4 - 124*x**2 + 1),x) - 3*int(1/(x**8 + 
4*x**6 + 6*x**4 - 124*x**2 + 1),x)