\(\int \frac {(3-2 \sqrt {2}+x^2)^2 (-3+2 \sqrt {2}+x^2)}{577-408 \sqrt {2}+(328-232 \sqrt {2}) x^2+(78-56 \sqrt {2}) x^4+(8-8 \sqrt {2}) x^6+x^8} \, dx\) [101]

Optimal result

Integrand size = 80, antiderivative size = 22 \[ \int \frac {\left (3-2 \sqrt {2}+x^2\right )^2 \left (-3+2 \sqrt {2}+x^2\right )}{577-408 \sqrt {2}+\left (328-232 \sqrt {2}\right ) x^2+\left (78-56 \sqrt {2}\right ) x^4+\left (8-8 \sqrt {2}\right ) x^6+x^8} \, dx=-\frac {1}{2} \text {arctanh}\left (\frac {2 x}{3-2 \sqrt {2}+x^2}\right ) \] 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. \(139\) vs. \(2(22)=44\).

Time = 0.09 (sec) , antiderivative size = 139, normalized size of antiderivative = 6.32 \[ \int \frac {\left (3-2 \sqrt {2}+x^2\right )^2 \left (-3+2 \sqrt {2}+x^2\right )}{577-408 \sqrt {2}+\left (328-232 \sqrt {2}\right ) x^2+\left (78-56 \sqrt {2}\right ) x^4+\left (8-8 \sqrt {2}\right ) x^6+x^8} \, dx=-\frac {\left (3-2 \sqrt {2}+x^2\right )^2 \left (-17+12 \sqrt {2}+\left (-2+4 \sqrt {2}\right ) x^2-x^4\right ) \left (\log \left (-3+2 \sqrt {2}-2 x-x^2\right )-\log \left (-3+2 \sqrt {2}+2 x-x^2\right )\right )}{4 \left (-577+408 \sqrt {2}+8 \left (-41+29 \sqrt {2}\right ) x^2+\left (-78+56 \sqrt {2}\right ) x^4+8 \left (-1+\sqrt {2}\right ) x^6-x^8\right )} \] Input:

Integrate[((3 - 2*Sqrt[2] + x^2)^2*(-3 + 2*Sqrt[2] + x^2))/(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:

-1/4*((3 - 2*Sqrt[2] + x^2)^2*(-17 + 12*Sqrt[2] + (-2 + 4*Sqrt[2])*x^2 - x 
^4)*(Log[-3 + 2*Sqrt[2] - 2*x - x^2] - Log[-3 + 2*Sqrt[2] + 2*x - 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 [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(103\) vs. \(2(22)=44\).

Time = 0.69 (sec) , antiderivative size = 103, normalized size of antiderivative = 4.68, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {2019, 2019, 1475, 1081, 2009}

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.

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

\(\Big \downarrow \) 2019

\(\displaystyle \int \frac {\left (x^2-2 \sqrt {2}+3\right ) \left (x^2+2 \sqrt {2}-3\right )}{x^6+\left (5-6 \sqrt {2}\right ) x^4+\left (39-28 \sqrt {2}\right ) x^2-70 \sqrt {2}+99}dx\)

\(\Big \downarrow \) 2019

\(\displaystyle \int \frac {x^2+2 \sqrt {2}-3}{x^4+\left (2-4 \sqrt {2}\right ) x^2-12 \sqrt {2}+17}dx\)

\(\Big \downarrow \) 1475

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

\(\Big \downarrow \) 1081

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{2} \left (\frac {1}{2} \log \left (-x+\sqrt {2 \left (\sqrt {2}-1\right )}+1\right )-\frac {1}{2} \log \left (x-\sqrt {2 \left (\sqrt {2}-1\right )}+1\right )\right )+\frac {1}{2} \left (\frac {1}{2} \log \left (-x-\sqrt {2 \left (\sqrt {2}-1\right )}+1\right )-\frac {1}{2} \log \left (x+\sqrt {2 \left (\sqrt {2}-1\right )}+1\right )\right )\)

Input:

Int[((3 - 2*Sqrt[2] + x^2)^2*(-3 + 2*Sqrt[2] + x^2))/(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:

(Log[1 + Sqrt[2*(-1 + Sqrt[2])] - x]/2 - Log[1 - Sqrt[2*(-1 + Sqrt[2])] + 
x]/2)/2 + (Log[1 - Sqrt[2*(-1 + Sqrt[2])] - x]/2 - Log[1 + Sqrt[2*(-1 + Sq 
rt[2])] + x]/2)/2
 

Defintions of rubi rules used

rule 1081

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = Rt[b^2 
- 4*a*c, 2]}, Simp[c   Int[ExpandIntegrand[1/((b/2 - q/2 + c*x)*(b/2 + q/2 
+ c*x)), x], x], x]] /; FreeQ[{a, b, c}, x] && NiceSqrtQ[b^2 - 4*a*c]
 

rule 1475

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] : 
> With[{q = Rt[2*(d/e) - b/c, 2]}, Simp[e/(2*c)   Int[1/Simp[d/e + q*x + x^ 
2, x], x], x] + Simp[e/(2*c)   Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; F 
reeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - a*e^2, 0] && 
 (GtQ[2*(d/e) - b/c, 0] || ( !LtQ[2*(d/e) - b/c, 0] && EqQ[d - e*Rt[a/c, 2] 
, 0]))
 

rule 2009

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

rule 2019

Int[(u_.)*(Px_)^(p_.)*(Qx_)^(q_.), x_Symbol] :> Int[u*PolynomialQuotient[Px 
, Qx, x]^p*Qx^(p + q), x] /; FreeQ[q, x] && PolyQ[Px, x] && PolyQ[Qx, x] && 
 EqQ[PolynomialRemainder[Px, Qx, x], 0] && IntegerQ[p] && LtQ[p*q, 0]
 

Maple [A] (verified)

Time = 0.23 (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)+x^2)^2*(-3+2*2^(1/2)+x^2)/(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,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 \[ \int \frac {\left (3-2 \sqrt {2}+x^2\right )^2 \left (-3+2 \sqrt {2}+x^2\right )}{577-408 \sqrt {2}+\left (328-232 \sqrt {2}\right ) x^2+\left (78-56 \sqrt {2}\right ) x^4+\left (8-8 \sqrt {2}\right ) x^6+x^8} \, dx=-\frac {1}{4} \, \log \left (x^{2} + 2 \, x - 2 \, \sqrt {2} + 3\right ) + \frac {1}{4} \, \log \left (x^{2} - 2 \, x - 2 \, \sqrt {2} + 3\right ) \] Input:

integrate((3-2*2^(1/2)+x^2)^2*(-3+2*2^(1/2)+x^2)/(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="fri 
cas")
 

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

integrate((3-2*2**(1/2)+x**2)**2*(-3+2*2**(1/2)+x**2)/(577-408*2**(1/2)+(3 
28-232*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]

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

integrate((3-2*2^(1/2)+x^2)^2*(-3+2*2^(1/2)+x^2)/(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="max 
ima")
 

Output:

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

Giac [A] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.59 \[ \int \frac {\left (3-2 \sqrt {2}+x^2\right )^2 \left (-3+2 \sqrt {2}+x^2\right )}{577-408 \sqrt {2}+\left (328-232 \sqrt {2}\right ) x^2+\left (78-56 \sqrt {2}\right ) x^4+\left (8-8 \sqrt {2}\right ) x^6+x^8} \, dx=-\frac {1}{4} \, \log \left ({\left | x^{2} + 2 \, x - 2 \, \sqrt {2} + 3 \right |}\right ) + \frac {1}{4} \, \log \left ({\left | x^{2} - 2 \, x - 2 \, \sqrt {2} + 3 \right |}\right ) \] Input:

integrate((3-2*2^(1/2)+x^2)^2*(-3+2*2^(1/2)+x^2)/(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="gia 
c")
 

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 = 22.65 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.55 \[ \int \frac {\left (3-2 \sqrt {2}+x^2\right )^2 \left (-3+2 \sqrt {2}+x^2\right )}{577-408 \sqrt {2}+\left (328-232 \sqrt {2}\right ) x^2+\left (78-56 \sqrt {2}\right ) x^4+\left (8-8 \sqrt {2}\right ) x^6+x^8} \, dx=-\frac {\mathrm {atanh}\left (\frac {x\,\left (16\,\sqrt {2}-16\right )}{2\,\left (20\,\sqrt {2}+4\,\sqrt {2}\,x^2-4\,x^2-28\right )}\right )}{2} \] Input:

int(-((x^2 - 2*2^(1/2) + 3)^2*(2*2^(1/2) + x^2 - 3))/(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 - 57 
7),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]

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

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

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)