Integrand size = 95, antiderivative size = 99 \[ \int \frac {\left (-1+2 x^2\right ) \left (-1+4 x-4 x^2+4 x^4\right )}{\sqrt {\frac {1-2 x^2}{1+2 x^2}} \left (1+2 x^2\right ) \left (-1-8 x+32 x^2-40 x^3+46 x^4-64 x^5+56 x^6-32 x^7+8 x^8\right )} \, dx=\frac {\arctan \left (\frac {\sqrt [4]{\frac {3}{2}} \sqrt {\frac {1-2 x^2}{1+2 x^2}}}{-1+x}\right )}{2 \sqrt [4]{2} 3^{3/4}}-\frac {\text {arctanh}\left (\frac {\sqrt [4]{\frac {3}{2}} \sqrt {\frac {1-2 x^2}{1+2 x^2}}}{-1+x}\right )}{2 \sqrt [4]{2} 3^{3/4}} \] Output:
1/12*arctan(1/2*3^(1/4)*2^(3/4)*((-2*x^2+1)/(2*x^2+1))^(1/2)/(-1+x))*2^(3/ 4)*3^(1/4)-1/12*arctanh(1/2*3^(1/4)*2^(3/4)*((-2*x^2+1)/(2*x^2+1))^(1/2)/( -1+x))*2^(3/4)*3^(1/4)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 74.44 (sec) , antiderivative size = 64371, normalized size of antiderivative = 650.21 \[ \int \frac {\left (-1+2 x^2\right ) \left (-1+4 x-4 x^2+4 x^4\right )}{\sqrt {\frac {1-2 x^2}{1+2 x^2}} \left (1+2 x^2\right ) \left (-1-8 x+32 x^2-40 x^3+46 x^4-64 x^5+56 x^6-32 x^7+8 x^8\right )} \, dx=\text {Result too large to show} \] Input:
Integrate[((-1 + 2*x^2)*(-1 + 4*x - 4*x^2 + 4*x^4))/(Sqrt[(1 - 2*x^2)/(1 + 2*x^2)]*(1 + 2*x^2)*(-1 - 8*x + 32*x^2 - 40*x^3 + 46*x^4 - 64*x^5 + 56*x^ 6 - 32*x^7 + 8*x^8)),x]
Output:
Result too large to show
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.
| \(\displaystyle \int \frac {\left (2 x^2-1\right ) \left (4 x^4-4 x^2+4 x-1\right )}{\sqrt {\frac {1-2 x^2}{2 x^2+1}} \left (2 x^2+1\right ) \left (8 x^8-32 x^7+56 x^6-64 x^5+46 x^4-40 x^3+32 x^2-8 x-1\right )} \, dx\) |
| \(\Big \downarrow \) 2058 |
| \(\displaystyle \frac {\sqrt {1-2 x^2} \int -\frac {\sqrt {1-2 x^2} \left (-4 x^4+4 x^2-4 x+1\right )}{\sqrt {2 x^2+1} \left (-8 x^8+32 x^7-56 x^6+64 x^5-46 x^4+40 x^3-32 x^2+8 x+1\right )}dx}{\sqrt {\frac {1-2 x^2}{2 x^2+1}} \sqrt {2 x^2+1}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\sqrt {1-2 x^2} \int \frac {\sqrt {1-2 x^2} \left (-4 x^4+4 x^2-4 x+1\right )}{\sqrt {2 x^2+1} \left (-8 x^8+32 x^7-56 x^6+64 x^5-46 x^4+40 x^3-32 x^2+8 x+1\right )}dx}{\sqrt {\frac {1-2 x^2}{2 x^2+1}} \sqrt {2 x^2+1}}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {\sqrt {1-2 x^2} \int \left (\frac {4 \sqrt {1-2 x^2} x^4}{\sqrt {2 x^2+1} \left (8 x^8-32 x^7+56 x^6-64 x^5+46 x^4-40 x^3+32 x^2-8 x-1\right )}-\frac {4 \sqrt {1-2 x^2} x^2}{\sqrt {2 x^2+1} \left (8 x^8-32 x^7+56 x^6-64 x^5+46 x^4-40 x^3+32 x^2-8 x-1\right )}+\frac {4 \sqrt {1-2 x^2} x}{\sqrt {2 x^2+1} \left (8 x^8-32 x^7+56 x^6-64 x^5+46 x^4-40 x^3+32 x^2-8 x-1\right )}+\frac {\sqrt {1-2 x^2}}{\sqrt {2 x^2+1} \left (-8 x^8+32 x^7-56 x^6+64 x^5-46 x^4+40 x^3-32 x^2+8 x+1\right )}\right )dx}{\sqrt {\frac {1-2 x^2}{2 x^2+1}} \sqrt {2 x^2+1}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\sqrt {1-2 x^2} \left (\int \frac {\sqrt {1-2 x^2}}{\sqrt {2 x^2+1} \left (-8 x^8+32 x^7-56 x^6+64 x^5-46 x^4+40 x^3-32 x^2+8 x+1\right )}dx+4 \int \frac {x \sqrt {1-2 x^2}}{\sqrt {2 x^2+1} \left (8 x^8-32 x^7+56 x^6-64 x^5+46 x^4-40 x^3+32 x^2-8 x-1\right )}dx-4 \int \frac {x^2 \sqrt {1-2 x^2}}{\sqrt {2 x^2+1} \left (8 x^8-32 x^7+56 x^6-64 x^5+46 x^4-40 x^3+32 x^2-8 x-1\right )}dx+4 \int \frac {x^4 \sqrt {1-2 x^2}}{\sqrt {2 x^2+1} \left (8 x^8-32 x^7+56 x^6-64 x^5+46 x^4-40 x^3+32 x^2-8 x-1\right )}dx\right )}{\sqrt {\frac {1-2 x^2}{2 x^2+1}} \sqrt {2 x^2+1}}\) |
Input:
Int[((-1 + 2*x^2)*(-1 + 4*x - 4*x^2 + 4*x^4))/(Sqrt[(1 - 2*x^2)/(1 + 2*x^2 )]*(1 + 2*x^2)*(-1 - 8*x + 32*x^2 - 40*x^3 + 46*x^4 - 64*x^5 + 56*x^6 - 32 *x^7 + 8*x^8)),x]
Output:
$Aborted
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 4.48 (sec) , antiderivative size = 867, normalized size of antiderivative = 8.76
\[\text {Expression too large to display}\]
Input:
int((2*x^2-1)*(4*x^4-4*x^2+4*x-1)/((-2*x^2+1)/(2*x^2+1))^(1/2)/(2*x^2+1)/( 8*x^8-32*x^7+56*x^6-64*x^5+46*x^4-40*x^3+32*x^2-8*x-1),x)
Output:
1/24*(2*x^2-1)*sum((2*_alpha^3-2*_alpha^2+_alpha-1)*(8*2^(1/2)*(-4*_alpha^ 4+1)^(1/2)*(-2*x^2+1)^(1/2)*(2*x^2+1)^(1/2)*EllipticPi(x*2^(1/2),-32*_alph a^7+132*_alpha^6-240*_alpha^5+284*_alpha^4-216*_alpha^3+183*_alpha^2-148*_ alpha+48,1/2*(-2)^(1/2)*2^(1/2))*_alpha^7-32*2^(1/2)*(-4*_alpha^4+1)^(1/2) *(-2*x^2+1)^(1/2)*(2*x^2+1)^(1/2)*EllipticPi(x*2^(1/2),-32*_alpha^7+132*_a lpha^6-240*_alpha^5+284*_alpha^4-216*_alpha^3+183*_alpha^2-148*_alpha+48,1 /2*(-2)^(1/2)*2^(1/2))*_alpha^6+56*2^(1/2)*(-4*_alpha^4+1)^(1/2)*(-2*x^2+1 )^(1/2)*(2*x^2+1)^(1/2)*EllipticPi(x*2^(1/2),-32*_alpha^7+132*_alpha^6-240 *_alpha^5+284*_alpha^4-216*_alpha^3+183*_alpha^2-148*_alpha+48,1/2*(-2)^(1 /2)*2^(1/2))*_alpha^5-64*2^(1/2)*(-4*_alpha^4+1)^(1/2)*(-2*x^2+1)^(1/2)*(2 *x^2+1)^(1/2)*EllipticPi(x*2^(1/2),-32*_alpha^7+132*_alpha^6-240*_alpha^5+ 284*_alpha^4-216*_alpha^3+183*_alpha^2-148*_alpha+48,1/2*(-2)^(1/2)*2^(1/2 ))*_alpha^4+46*2^(1/2)*(-4*_alpha^4+1)^(1/2)*(-2*x^2+1)^(1/2)*(2*x^2+1)^(1 /2)*EllipticPi(x*2^(1/2),-32*_alpha^7+132*_alpha^6-240*_alpha^5+284*_alpha ^4-216*_alpha^3+183*_alpha^2-148*_alpha+48,1/2*(-2)^(1/2)*2^(1/2))*_alpha^ 3-40*2^(1/2)*(-4*_alpha^4+1)^(1/2)*(-2*x^2+1)^(1/2)*(2*x^2+1)^(1/2)*Ellipt icPi(x*2^(1/2),-32*_alpha^7+132*_alpha^6-240*_alpha^5+284*_alpha^4-216*_al pha^3+183*_alpha^2-148*_alpha+48,1/2*(-2)^(1/2)*2^(1/2))*_alpha^2+32*2^(1/ 2)*(-4*_alpha^4+1)^(1/2)*(-2*x^2+1)^(1/2)*(2*x^2+1)^(1/2)*EllipticPi(x*2^( 1/2),-32*_alpha^7+132*_alpha^6-240*_alpha^5+284*_alpha^4-216*_alpha^3+1...
Leaf count of result is larger than twice the leaf count of optimal. 602 vs. \(2 (85) = 170\).
Time = 0.40 (sec) , antiderivative size = 602, normalized size of antiderivative = 6.08 \[ \int \frac {\left (-1+2 x^2\right ) \left (-1+4 x-4 x^2+4 x^4\right )}{\sqrt {\frac {1-2 x^2}{1+2 x^2}} \left (1+2 x^2\right ) \left (-1-8 x+32 x^2-40 x^3+46 x^4-64 x^5+56 x^6-32 x^7+8 x^8\right )} \, dx=\frac {1}{216} \cdot 54^{\frac {3}{4}} \arctan \left (\frac {2 \, {\left (54^{\frac {3}{4}} {\left (4 \, x^{7} - 12 \, x^{6} + 16 \, x^{5} - 16 \, x^{4} + 13 \, x^{3} - 7 \, x^{2} + 3 \, x - 1\right )} - 9 \cdot 54^{\frac {1}{4}} {\left (4 \, x^{5} - 4 \, x^{4} - x + 1\right )}\right )} \sqrt {-\frac {2 \, x^{2} - 1}{2 \, x^{2} + 1}}}{9 \, {\left (8 \, x^{8} - 32 \, x^{7} + 56 \, x^{6} - 64 \, x^{5} + 46 \, x^{4} - 40 \, x^{3} + 32 \, x^{2} - 8 \, x - 1\right )}}\right ) - \frac {1}{432} \cdot 54^{\frac {3}{4}} \log \left (-\frac {54^{\frac {3}{4}} {\left (8 \, x^{8} - 32 \, x^{7} + 8 \, x^{6} + 32 \, x^{5} + 22 \, x^{4} - 40 \, x^{3} + 20 \, x^{2} - 32 \, x + 17\right )} + 36 \, {\left (24 \, x^{7} - 72 \, x^{6} + 84 \, x^{5} - 84 \, x^{4} + 78 \, x^{3} - 42 \, x^{2} + \sqrt {6} {\left (4 \, x^{7} - 12 \, x^{6} + 4 \, x^{5} - 4 \, x^{4} + 13 \, x^{3} - 7 \, x^{2} + 6 \, x - 4\right )} + 21 \, x - 9\right )} \sqrt {-\frac {2 \, x^{2} - 1}{2 \, x^{2} + 1}} + 18 \cdot 54^{\frac {1}{4}} {\left (8 \, x^{8} - 32 \, x^{7} + 48 \, x^{6} - 48 \, x^{5} + 62 \, x^{4} - 40 \, x^{3} + 10 \, x^{2} - 12 \, x + 7\right )}}{8 \, x^{8} - 32 \, x^{7} + 56 \, x^{6} - 64 \, x^{5} + 46 \, x^{4} - 40 \, x^{3} + 32 \, x^{2} - 8 \, x - 1}\right ) + \frac {1}{432} \cdot 54^{\frac {3}{4}} \log \left (\frac {54^{\frac {3}{4}} {\left (8 \, x^{8} - 32 \, x^{7} + 8 \, x^{6} + 32 \, x^{5} + 22 \, x^{4} - 40 \, x^{3} + 20 \, x^{2} - 32 \, x + 17\right )} - 36 \, {\left (24 \, x^{7} - 72 \, x^{6} + 84 \, x^{5} - 84 \, x^{4} + 78 \, x^{3} - 42 \, x^{2} + \sqrt {6} {\left (4 \, x^{7} - 12 \, x^{6} + 4 \, x^{5} - 4 \, x^{4} + 13 \, x^{3} - 7 \, x^{2} + 6 \, x - 4\right )} + 21 \, x - 9\right )} \sqrt {-\frac {2 \, x^{2} - 1}{2 \, x^{2} + 1}} + 18 \cdot 54^{\frac {1}{4}} {\left (8 \, x^{8} - 32 \, x^{7} + 48 \, x^{6} - 48 \, x^{5} + 62 \, x^{4} - 40 \, x^{3} + 10 \, x^{2} - 12 \, x + 7\right )}}{8 \, x^{8} - 32 \, x^{7} + 56 \, x^{6} - 64 \, x^{5} + 46 \, x^{4} - 40 \, x^{3} + 32 \, x^{2} - 8 \, x - 1}\right ) \] Input:
integrate((2*x^2-1)*(4*x^4-4*x^2+4*x-1)/((-2*x^2+1)/(2*x^2+1))^(1/2)/(2*x^ 2+1)/(8*x^8-32*x^7+56*x^6-64*x^5+46*x^4-40*x^3+32*x^2-8*x-1),x, algorithm= "fricas")
Output:
1/216*54^(3/4)*arctan(2/9*(54^(3/4)*(4*x^7 - 12*x^6 + 16*x^5 - 16*x^4 + 13 *x^3 - 7*x^2 + 3*x - 1) - 9*54^(1/4)*(4*x^5 - 4*x^4 - x + 1))*sqrt(-(2*x^2 - 1)/(2*x^2 + 1))/(8*x^8 - 32*x^7 + 56*x^6 - 64*x^5 + 46*x^4 - 40*x^3 + 3 2*x^2 - 8*x - 1)) - 1/432*54^(3/4)*log(-(54^(3/4)*(8*x^8 - 32*x^7 + 8*x^6 + 32*x^5 + 22*x^4 - 40*x^3 + 20*x^2 - 32*x + 17) + 36*(24*x^7 - 72*x^6 + 8 4*x^5 - 84*x^4 + 78*x^3 - 42*x^2 + sqrt(6)*(4*x^7 - 12*x^6 + 4*x^5 - 4*x^4 + 13*x^3 - 7*x^2 + 6*x - 4) + 21*x - 9)*sqrt(-(2*x^2 - 1)/(2*x^2 + 1)) + 18*54^(1/4)*(8*x^8 - 32*x^7 + 48*x^6 - 48*x^5 + 62*x^4 - 40*x^3 + 10*x^2 - 12*x + 7))/(8*x^8 - 32*x^7 + 56*x^6 - 64*x^5 + 46*x^4 - 40*x^3 + 32*x^2 - 8*x - 1)) + 1/432*54^(3/4)*log((54^(3/4)*(8*x^8 - 32*x^7 + 8*x^6 + 32*x^5 + 22*x^4 - 40*x^3 + 20*x^2 - 32*x + 17) - 36*(24*x^7 - 72*x^6 + 84*x^5 - 84*x^4 + 78*x^3 - 42*x^2 + sqrt(6)*(4*x^7 - 12*x^6 + 4*x^5 - 4*x^4 + 13*x^ 3 - 7*x^2 + 6*x - 4) + 21*x - 9)*sqrt(-(2*x^2 - 1)/(2*x^2 + 1)) + 18*54^(1 /4)*(8*x^8 - 32*x^7 + 48*x^6 - 48*x^5 + 62*x^4 - 40*x^3 + 10*x^2 - 12*x + 7))/(8*x^8 - 32*x^7 + 56*x^6 - 64*x^5 + 46*x^4 - 40*x^3 + 32*x^2 - 8*x - 1 ))
Timed out. \[ \int \frac {\left (-1+2 x^2\right ) \left (-1+4 x-4 x^2+4 x^4\right )}{\sqrt {\frac {1-2 x^2}{1+2 x^2}} \left (1+2 x^2\right ) \left (-1-8 x+32 x^2-40 x^3+46 x^4-64 x^5+56 x^6-32 x^7+8 x^8\right )} \, dx=\text {Timed out} \] Input:
integrate((2*x**2-1)*(4*x**4-4*x**2+4*x-1)/((-2*x**2+1)/(2*x**2+1))**(1/2) /(2*x**2+1)/(8*x**8-32*x**7+56*x**6-64*x**5+46*x**4-40*x**3+32*x**2-8*x-1) ,x)
Output:
Timed out
\[ \int \frac {\left (-1+2 x^2\right ) \left (-1+4 x-4 x^2+4 x^4\right )}{\sqrt {\frac {1-2 x^2}{1+2 x^2}} \left (1+2 x^2\right ) \left (-1-8 x+32 x^2-40 x^3+46 x^4-64 x^5+56 x^6-32 x^7+8 x^8\right )} \, dx=\int { \frac {{\left (4 \, x^{4} - 4 \, x^{2} + 4 \, x - 1\right )} {\left (2 \, x^{2} - 1\right )}}{{\left (8 \, x^{8} - 32 \, x^{7} + 56 \, x^{6} - 64 \, x^{5} + 46 \, x^{4} - 40 \, x^{3} + 32 \, x^{2} - 8 \, x - 1\right )} {\left (2 \, x^{2} + 1\right )} \sqrt {-\frac {2 \, x^{2} - 1}{2 \, x^{2} + 1}}} \,d x } \] Input:
integrate((2*x^2-1)*(4*x^4-4*x^2+4*x-1)/((-2*x^2+1)/(2*x^2+1))^(1/2)/(2*x^ 2+1)/(8*x^8-32*x^7+56*x^6-64*x^5+46*x^4-40*x^3+32*x^2-8*x-1),x, algorithm= "maxima")
Output:
integrate((4*x^4 - 4*x^2 + 4*x - 1)*(2*x^2 - 1)/((8*x^8 - 32*x^7 + 56*x^6 - 64*x^5 + 46*x^4 - 40*x^3 + 32*x^2 - 8*x - 1)*(2*x^2 + 1)*sqrt(-(2*x^2 - 1)/(2*x^2 + 1))), x)
\[ \int \frac {\left (-1+2 x^2\right ) \left (-1+4 x-4 x^2+4 x^4\right )}{\sqrt {\frac {1-2 x^2}{1+2 x^2}} \left (1+2 x^2\right ) \left (-1-8 x+32 x^2-40 x^3+46 x^4-64 x^5+56 x^6-32 x^7+8 x^8\right )} \, dx=\int { \frac {{\left (4 \, x^{4} - 4 \, x^{2} + 4 \, x - 1\right )} {\left (2 \, x^{2} - 1\right )}}{{\left (8 \, x^{8} - 32 \, x^{7} + 56 \, x^{6} - 64 \, x^{5} + 46 \, x^{4} - 40 \, x^{3} + 32 \, x^{2} - 8 \, x - 1\right )} {\left (2 \, x^{2} + 1\right )} \sqrt {-\frac {2 \, x^{2} - 1}{2 \, x^{2} + 1}}} \,d x } \] Input:
integrate((2*x^2-1)*(4*x^4-4*x^2+4*x-1)/((-2*x^2+1)/(2*x^2+1))^(1/2)/(2*x^ 2+1)/(8*x^8-32*x^7+56*x^6-64*x^5+46*x^4-40*x^3+32*x^2-8*x-1),x, algorithm= "giac")
Output:
integrate((4*x^4 - 4*x^2 + 4*x - 1)*(2*x^2 - 1)/((8*x^8 - 32*x^7 + 56*x^6 - 64*x^5 + 46*x^4 - 40*x^3 + 32*x^2 - 8*x - 1)*(2*x^2 + 1)*sqrt(-(2*x^2 - 1)/(2*x^2 + 1))), x)
Timed out. \[ \int \frac {\left (-1+2 x^2\right ) \left (-1+4 x-4 x^2+4 x^4\right )}{\sqrt {\frac {1-2 x^2}{1+2 x^2}} \left (1+2 x^2\right ) \left (-1-8 x+32 x^2-40 x^3+46 x^4-64 x^5+56 x^6-32 x^7+8 x^8\right )} \, dx=\int -\frac {\left (2\,x^2-1\right )\,\left (4\,x^4-4\,x^2+4\,x-1\right )}{\left (2\,x^2+1\right )\,\sqrt {-\frac {2\,x^2-1}{2\,x^2+1}}\,\left (-8\,x^8+32\,x^7-56\,x^6+64\,x^5-46\,x^4+40\,x^3-32\,x^2+8\,x+1\right )} \,d x \] Input:
int(-((2*x^2 - 1)*(4*x - 4*x^2 + 4*x^4 - 1))/((2*x^2 + 1)*(-(2*x^2 - 1)/(2 *x^2 + 1))^(1/2)*(8*x - 32*x^2 + 40*x^3 - 46*x^4 + 64*x^5 - 56*x^6 + 32*x^ 7 - 8*x^8 + 1)),x)
Output:
int(-((2*x^2 - 1)*(4*x - 4*x^2 + 4*x^4 - 1))/((2*x^2 + 1)*(-(2*x^2 - 1)/(2 *x^2 + 1))^(1/2)*(8*x - 32*x^2 + 40*x^3 - 46*x^4 + 64*x^5 - 56*x^6 + 32*x^ 7 - 8*x^8 + 1)), x)
\[ \int \frac {\left (-1+2 x^2\right ) \left (-1+4 x-4 x^2+4 x^4\right )}{\sqrt {\frac {1-2 x^2}{1+2 x^2}} \left (1+2 x^2\right ) \left (-1-8 x+32 x^2-40 x^3+46 x^4-64 x^5+56 x^6-32 x^7+8 x^8\right )} \, dx=-4 \left (\int \frac {\sqrt {2 x^{2}+1}\, \sqrt {-2 x^{2}+1}\, x^{4}}{16 x^{10}-64 x^{9}+120 x^{8}-160 x^{7}+148 x^{6}-144 x^{5}+110 x^{4}-56 x^{3}+30 x^{2}-8 x -1}d x \right )+4 \left (\int \frac {\sqrt {2 x^{2}+1}\, \sqrt {-2 x^{2}+1}\, x^{2}}{16 x^{10}-64 x^{9}+120 x^{8}-160 x^{7}+148 x^{6}-144 x^{5}+110 x^{4}-56 x^{3}+30 x^{2}-8 x -1}d x \right )-4 \left (\int \frac {\sqrt {2 x^{2}+1}\, \sqrt {-2 x^{2}+1}\, x}{16 x^{10}-64 x^{9}+120 x^{8}-160 x^{7}+148 x^{6}-144 x^{5}+110 x^{4}-56 x^{3}+30 x^{2}-8 x -1}d x \right )+\int \frac {\sqrt {2 x^{2}+1}\, \sqrt {-2 x^{2}+1}}{16 x^{10}-64 x^{9}+120 x^{8}-160 x^{7}+148 x^{6}-144 x^{5}+110 x^{4}-56 x^{3}+30 x^{2}-8 x -1}d x \] Input:
int((2*x^2-1)*(4*x^4-4*x^2+4*x-1)/((-2*x^2+1)/(2*x^2+1))^(1/2)/(2*x^2+1)/( 8*x^8-32*x^7+56*x^6-64*x^5+46*x^4-40*x^3+32*x^2-8*x-1),x)
Output:
- 4*int((sqrt(2*x**2 + 1)*sqrt( - 2*x**2 + 1)*x**4)/(16*x**10 - 64*x**9 + 120*x**8 - 160*x**7 + 148*x**6 - 144*x**5 + 110*x**4 - 56*x**3 + 30*x**2 - 8*x - 1),x) + 4*int((sqrt(2*x**2 + 1)*sqrt( - 2*x**2 + 1)*x**2)/(16*x**1 0 - 64*x**9 + 120*x**8 - 160*x**7 + 148*x**6 - 144*x**5 + 110*x**4 - 56*x* *3 + 30*x**2 - 8*x - 1),x) - 4*int((sqrt(2*x**2 + 1)*sqrt( - 2*x**2 + 1)*x )/(16*x**10 - 64*x**9 + 120*x**8 - 160*x**7 + 148*x**6 - 144*x**5 + 110*x* *4 - 56*x**3 + 30*x**2 - 8*x - 1),x) + int((sqrt(2*x**2 + 1)*sqrt( - 2*x** 2 + 1))/(16*x**10 - 64*x**9 + 120*x**8 - 160*x**7 + 148*x**6 - 144*x**5 + 110*x**4 - 56*x**3 + 30*x**2 - 8*x - 1),x)