Integrand size = 16, antiderivative size = 147 \[ \int \frac {x^2}{2-2 x^2+x^4} \, dx=-\frac {1}{2} \sqrt {\frac {1}{2} \left (1+\sqrt {2}\right )} \arctan \left (\frac {\sqrt {2 \left (1+\sqrt {2}\right )}-2 x}{\sqrt {2 \left (-1+\sqrt {2}\right )}}\right )+\frac {1}{2} \sqrt {\frac {1}{2} \left (1+\sqrt {2}\right )} \arctan \left (\frac {\sqrt {2 \left (1+\sqrt {2}\right )}+2 x}{\sqrt {2 \left (-1+\sqrt {2}\right )}}\right )-\frac {\text {arctanh}\left (\frac {\sqrt {2 \left (1+\sqrt {2}\right )} x}{\sqrt {2}+x^2}\right )}{2 \sqrt {2 \left (1+\sqrt {2}\right )}} \] Output:
-1/4*(2+2*2^(1/2))^(1/2)*arctan(((2+2*2^(1/2))^(1/2)-2*x)/(-2+2*2^(1/2))^( 1/2))+1/4*(2+2*2^(1/2))^(1/2)*arctan(((2+2*2^(1/2))^(1/2)+2*x)/(-2+2*2^(1/ 2))^(1/2))-1/2*arctanh((2+2*2^(1/2))^(1/2)*x/(x^2+2^(1/2)))/(2+2*2^(1/2))^ (1/2)
Result contains complex when optimal does not.
Time = 0.02 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.27 \[ \int \frac {x^2}{2-2 x^2+x^4} \, dx=-\frac {\arctan \left (\frac {x}{\sqrt {-1-i}}\right )}{(-1-i)^{3/2}}-\frac {\arctan \left (\frac {x}{\sqrt {-1+i}}\right )}{(-1+i)^{3/2}} \] Input:
Integrate[x^2/(2 - 2*x^2 + x^4),x]
Output:
-(ArcTan[x/Sqrt[-1 - I]]/(-1 - I)^(3/2)) - ArcTan[x/Sqrt[-1 + I]]/(-1 + I) ^(3/2)
Time = 0.67 (sec) , antiderivative size = 190, normalized size of antiderivative = 1.29, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.438, Rules used = {1447, 1475, 1083, 217, 1478, 25, 1103}
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.
| \(\displaystyle \int \frac {x^2}{x^4-2 x^2+2} \, dx\) |
| \(\Big \downarrow \) 1447 |
| \(\displaystyle \frac {1}{2} \int \frac {x^2+\sqrt {2}}{x^4-2 x^2+2}dx-\frac {1}{2} \int \frac {\sqrt {2}-x^2}{x^4-2 x^2+2}dx\) |
| \(\Big \downarrow \) 1475 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{2} \int \frac {1}{x^2-\sqrt {2 \left (1+\sqrt {2}\right )} x+\sqrt {2}}dx+\frac {1}{2} \int \frac {1}{x^2+\sqrt {2 \left (1+\sqrt {2}\right )} x+\sqrt {2}}dx\right )-\frac {1}{2} \int \frac {\sqrt {2}-x^2}{x^4-2 x^2+2}dx\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle \frac {1}{2} \left (-\int \frac {1}{2 \left (1-\sqrt {2}\right )-\left (2 x-\sqrt {2 \left (1+\sqrt {2}\right )}\right )^2}d\left (2 x-\sqrt {2 \left (1+\sqrt {2}\right )}\right )-\int \frac {1}{2 \left (1-\sqrt {2}\right )-\left (2 x+\sqrt {2 \left (1+\sqrt {2}\right )}\right )^2}d\left (2 x+\sqrt {2 \left (1+\sqrt {2}\right )}\right )\right )-\frac {1}{2} \int \frac {\sqrt {2}-x^2}{x^4-2 x^2+2}dx\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {1}{2} \left (\frac {\arctan \left (\frac {2 x-\sqrt {2 \left (1+\sqrt {2}\right )}}{\sqrt {2 \left (\sqrt {2}-1\right )}}\right )}{\sqrt {2 \left (\sqrt {2}-1\right )}}+\frac {\arctan \left (\frac {2 x+\sqrt {2 \left (1+\sqrt {2}\right )}}{\sqrt {2 \left (\sqrt {2}-1\right )}}\right )}{\sqrt {2 \left (\sqrt {2}-1\right )}}\right )-\frac {1}{2} \int \frac {\sqrt {2}-x^2}{x^4-2 x^2+2}dx\) |
| \(\Big \downarrow \) 1478 |
| \(\displaystyle \frac {1}{2} \left (\frac {\int -\frac {\sqrt {2 \left (1+\sqrt {2}\right )}-2 x}{x^2-\sqrt {2 \left (1+\sqrt {2}\right )} x+\sqrt {2}}dx}{2 \sqrt {2 \left (1+\sqrt {2}\right )}}+\frac {\int -\frac {2 x+\sqrt {2 \left (1+\sqrt {2}\right )}}{x^2+\sqrt {2 \left (1+\sqrt {2}\right )} x+\sqrt {2}}dx}{2 \sqrt {2 \left (1+\sqrt {2}\right )}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\frac {2 x-\sqrt {2 \left (1+\sqrt {2}\right )}}{\sqrt {2 \left (\sqrt {2}-1\right )}}\right )}{\sqrt {2 \left (\sqrt {2}-1\right )}}+\frac {\arctan \left (\frac {2 x+\sqrt {2 \left (1+\sqrt {2}\right )}}{\sqrt {2 \left (\sqrt {2}-1\right )}}\right )}{\sqrt {2 \left (\sqrt {2}-1\right )}}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{2} \left (-\frac {\int \frac {\sqrt {2 \left (1+\sqrt {2}\right )}-2 x}{x^2-\sqrt {2 \left (1+\sqrt {2}\right )} x+\sqrt {2}}dx}{2 \sqrt {2 \left (1+\sqrt {2}\right )}}-\frac {\int \frac {2 x+\sqrt {2 \left (1+\sqrt {2}\right )}}{x^2+\sqrt {2 \left (1+\sqrt {2}\right )} x+\sqrt {2}}dx}{2 \sqrt {2 \left (1+\sqrt {2}\right )}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\frac {2 x-\sqrt {2 \left (1+\sqrt {2}\right )}}{\sqrt {2 \left (\sqrt {2}-1\right )}}\right )}{\sqrt {2 \left (\sqrt {2}-1\right )}}+\frac {\arctan \left (\frac {2 x+\sqrt {2 \left (1+\sqrt {2}\right )}}{\sqrt {2 \left (\sqrt {2}-1\right )}}\right )}{\sqrt {2 \left (\sqrt {2}-1\right )}}\right )\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {1}{2} \left (\frac {\arctan \left (\frac {2 x-\sqrt {2 \left (1+\sqrt {2}\right )}}{\sqrt {2 \left (\sqrt {2}-1\right )}}\right )}{\sqrt {2 \left (\sqrt {2}-1\right )}}+\frac {\arctan \left (\frac {2 x+\sqrt {2 \left (1+\sqrt {2}\right )}}{\sqrt {2 \left (\sqrt {2}-1\right )}}\right )}{\sqrt {2 \left (\sqrt {2}-1\right )}}\right )+\frac {1}{2} \left (\frac {\log \left (x^2-\sqrt {2 \left (1+\sqrt {2}\right )} x+\sqrt {2}\right )}{2 \sqrt {2 \left (1+\sqrt {2}\right )}}-\frac {\log \left (x^2+\sqrt {2 \left (1+\sqrt {2}\right )} x+\sqrt {2}\right )}{2 \sqrt {2 \left (1+\sqrt {2}\right )}}\right )\) |
Input:
Int[x^2/(2 - 2*x^2 + x^4),x]
Output:
(ArcTan[(-Sqrt[2*(1 + Sqrt[2])] + 2*x)/Sqrt[2*(-1 + Sqrt[2])]]/Sqrt[2*(-1 + Sqrt[2])] + ArcTan[(Sqrt[2*(1 + Sqrt[2])] + 2*x)/Sqrt[2*(-1 + Sqrt[2])]] /Sqrt[2*(-1 + Sqrt[2])])/2 + (Log[Sqrt[2] - Sqrt[2*(1 + Sqrt[2])]*x + x^2] /(2*Sqrt[2*(1 + Sqrt[2])]) - Log[Sqrt[2] + Sqrt[2*(1 + Sqrt[2])]*x + x^2]/ (2*Sqrt[2*(1 + Sqrt[2])]))/2
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2 Subst[I nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Int[(x_)^2/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ a/c, 2]}, Simp[1/2 Int[(q + x^2)/(a + b*x^2 + c*x^4), x], x] - Simp[1/2 Int[(q - x^2)/(a + b*x^2 + c*x^4), x], x]] /; FreeQ[{a, b, c}, x] && LtQ[b ^2 - 4*a*c, 0] && PosQ[a*c]
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]))
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*q) Int[(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Simp[e/(2*c*q) Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ [c*d^2 - a*e^2, 0] && !GtQ[b^2 - 4*a*c, 0]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.19 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.24
| method | result | size |
| risch | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4}-2 \textit {\_Z}^{2}+2\right )}{\sum }\frac {\textit {\_R}^{2} \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{3}-\textit {\_R}}\right )}{4}\) | \(36\) |
| default | \(-\frac {\sqrt {2+2 \sqrt {2}}\, \left (\sqrt {2}-1\right ) \left (\frac {\ln \left (x^{2}+x \sqrt {2+2 \sqrt {2}}+\sqrt {2}\right )}{2}-\frac {\sqrt {2+2 \sqrt {2}}\, \arctan \left (\frac {\sqrt {2+2 \sqrt {2}}+2 x}{\sqrt {-2+2 \sqrt {2}}}\right )}{\sqrt {-2+2 \sqrt {2}}}\right )}{4}-\frac {\sqrt {2+2 \sqrt {2}}\, \left (\sqrt {2}-1\right ) \left (-\frac {\ln \left (x^{2}-x \sqrt {2+2 \sqrt {2}}+\sqrt {2}\right )}{2}-\frac {\sqrt {2+2 \sqrt {2}}\, \arctan \left (\frac {2 x -\sqrt {2+2 \sqrt {2}}}{\sqrt {-2+2 \sqrt {2}}}\right )}{\sqrt {-2+2 \sqrt {2}}}\right )}{4}\) | \(169\) |
Input:
int(x^2/(x^4-2*x^2+2),x,method=_RETURNVERBOSE)
Output:
1/4*sum(_R^2/(_R^3-_R)*ln(x-_R),_R=RootOf(_Z^4-2*_Z^2+2))
Time = 0.08 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.07 \[ \int \frac {x^2}{2-2 x^2+x^4} \, dx=\frac {1}{2} \, \sqrt {\frac {1}{2} \, \sqrt {2} + \frac {1}{2}} \arctan \left (2 \, {\left ({\left (\sqrt {2} + 1\right )} \sqrt {\frac {1}{2} \, \sqrt {2} - \frac {1}{2}} + x\right )} \sqrt {\frac {1}{2} \, \sqrt {2} + \frac {1}{2}}\right ) - \frac {1}{2} \, \sqrt {\frac {1}{2} \, \sqrt {2} + \frac {1}{2}} \arctan \left (2 \, {\left ({\left (\sqrt {2} + 1\right )} \sqrt {\frac {1}{2} \, \sqrt {2} - \frac {1}{2}} - x\right )} \sqrt {\frac {1}{2} \, \sqrt {2} + \frac {1}{2}}\right ) - \frac {1}{4} \, \sqrt {\frac {1}{2} \, \sqrt {2} - \frac {1}{2}} \log \left (x^{2} + 2 \, {\left (\sqrt {2} x + x\right )} \sqrt {\frac {1}{2} \, \sqrt {2} - \frac {1}{2}} + \sqrt {2}\right ) + \frac {1}{4} \, \sqrt {\frac {1}{2} \, \sqrt {2} - \frac {1}{2}} \log \left (x^{2} - 2 \, {\left (\sqrt {2} x + x\right )} \sqrt {\frac {1}{2} \, \sqrt {2} - \frac {1}{2}} + \sqrt {2}\right ) \] Input:
integrate(x^2/(x^4-2*x^2+2),x, algorithm="fricas")
Output:
1/2*sqrt(1/2*sqrt(2) + 1/2)*arctan(2*((sqrt(2) + 1)*sqrt(1/2*sqrt(2) - 1/2 ) + x)*sqrt(1/2*sqrt(2) + 1/2)) - 1/2*sqrt(1/2*sqrt(2) + 1/2)*arctan(2*((s qrt(2) + 1)*sqrt(1/2*sqrt(2) - 1/2) - x)*sqrt(1/2*sqrt(2) + 1/2)) - 1/4*sq rt(1/2*sqrt(2) - 1/2)*log(x^2 + 2*(sqrt(2)*x + x)*sqrt(1/2*sqrt(2) - 1/2) + sqrt(2)) + 1/4*sqrt(1/2*sqrt(2) - 1/2)*log(x^2 - 2*(sqrt(2)*x + x)*sqrt( 1/2*sqrt(2) - 1/2) + sqrt(2))
Time = 0.27 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.16 \[ \int \frac {x^2}{2-2 x^2+x^4} \, dx=\operatorname {RootSum} {\left (128 t^{4} + 16 t^{2} + 1, \left ( t \mapsto t \log {\left (64 t^{3} + 4 t + x \right )} \right )\right )} \] Input:
integrate(x**2/(x**4-2*x**2+2),x)
Output:
RootSum(128*_t**4 + 16*_t**2 + 1, Lambda(_t, _t*log(64*_t**3 + 4*_t + x)))
\[ \int \frac {x^2}{2-2 x^2+x^4} \, dx=\int { \frac {x^{2}}{x^{4} - 2 \, x^{2} + 2} \,d x } \] Input:
integrate(x^2/(x^4-2*x^2+2),x, algorithm="maxima")
Output:
integrate(x^2/(x^4 - 2*x^2 + 2), x)
Time = 0.27 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.00 \[ \int \frac {x^2}{2-2 x^2+x^4} \, dx=\frac {1}{4} \, \sqrt {2 \, \sqrt {2} + 2} \arctan \left (\frac {2^{\frac {3}{4}} {\left (2 \, x + 2^{\frac {1}{4}} \sqrt {\sqrt {2} + 2}\right )}}{2 \, \sqrt {-\sqrt {2} + 2}}\right ) + \frac {1}{4} \, \sqrt {2 \, \sqrt {2} + 2} \arctan \left (\frac {2^{\frac {3}{4}} {\left (2 \, x - 2^{\frac {1}{4}} \sqrt {\sqrt {2} + 2}\right )}}{2 \, \sqrt {-\sqrt {2} + 2}}\right ) - \frac {1}{8} \, \sqrt {2 \, \sqrt {2} - 2} \log \left (x^{2} + 2^{\frac {1}{4}} x \sqrt {\sqrt {2} + 2} + \sqrt {2}\right ) + \frac {1}{8} \, \sqrt {2 \, \sqrt {2} - 2} \log \left (x^{2} - 2^{\frac {1}{4}} x \sqrt {\sqrt {2} + 2} + \sqrt {2}\right ) \] Input:
integrate(x^2/(x^4-2*x^2+2),x, algorithm="giac")
Output:
1/4*sqrt(2*sqrt(2) + 2)*arctan(1/2*2^(3/4)*(2*x + 2^(1/4)*sqrt(sqrt(2) + 2 ))/sqrt(-sqrt(2) + 2)) + 1/4*sqrt(2*sqrt(2) + 2)*arctan(1/2*2^(3/4)*(2*x - 2^(1/4)*sqrt(sqrt(2) + 2))/sqrt(-sqrt(2) + 2)) - 1/8*sqrt(2*sqrt(2) - 2)* log(x^2 + 2^(1/4)*x*sqrt(sqrt(2) + 2) + sqrt(2)) + 1/8*sqrt(2*sqrt(2) - 2) *log(x^2 - 2^(1/4)*x*sqrt(sqrt(2) + 2) + sqrt(2))
Time = 0.14 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.69 \[ \int \frac {x^2}{2-2 x^2+x^4} \, dx=\mathrm {atanh}\left (32\,x\,{\left (\sqrt {-\frac {\sqrt {2}}{32}-\frac {1}{32}}+\sqrt {\frac {\sqrt {2}}{32}-\frac {1}{32}}\right )}^3\right )\,\left (2\,\sqrt {-\frac {\sqrt {2}}{32}-\frac {1}{32}}+2\,\sqrt {\frac {\sqrt {2}}{32}-\frac {1}{32}}\right )+\mathrm {atanh}\left (32\,x\,{\left (\sqrt {-\frac {\sqrt {2}}{32}-\frac {1}{32}}-\sqrt {\frac {\sqrt {2}}{32}-\frac {1}{32}}\right )}^3\right )\,\left (2\,\sqrt {-\frac {\sqrt {2}}{32}-\frac {1}{32}}-2\,\sqrt {\frac {\sqrt {2}}{32}-\frac {1}{32}}\right ) \] Input:
int(x^2/(x^4 - 2*x^2 + 2),x)
Output:
atanh(32*x*((- 2^(1/2)/32 - 1/32)^(1/2) + (2^(1/2)/32 - 1/32)^(1/2))^3)*(2 *(- 2^(1/2)/32 - 1/32)^(1/2) + 2*(2^(1/2)/32 - 1/32)^(1/2)) + atanh(32*x*( (- 2^(1/2)/32 - 1/32)^(1/2) - (2^(1/2)/32 - 1/32)^(1/2))^3)*(2*(- 2^(1/2)/ 32 - 1/32)^(1/2) - 2*(2^(1/2)/32 - 1/32)^(1/2))
Time = 0.15 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.57 \[ \int \frac {x^2}{2-2 x^2+x^4} \, dx=-\frac {\sqrt {\sqrt {2}-1}\, \sqrt {2}\, \mathit {atan} \left (\frac {\sqrt {\sqrt {2}+1}\, \sqrt {2}-2 x}{\sqrt {\sqrt {2}-1}\, \sqrt {2}}\right )}{4}-\frac {\sqrt {\sqrt {2}-1}\, \mathit {atan} \left (\frac {\sqrt {\sqrt {2}+1}\, \sqrt {2}-2 x}{\sqrt {\sqrt {2}-1}\, \sqrt {2}}\right )}{2}+\frac {\sqrt {\sqrt {2}-1}\, \sqrt {2}\, \mathit {atan} \left (\frac {\sqrt {\sqrt {2}+1}\, \sqrt {2}+2 x}{\sqrt {\sqrt {2}-1}\, \sqrt {2}}\right )}{4}+\frac {\sqrt {\sqrt {2}-1}\, \mathit {atan} \left (\frac {\sqrt {\sqrt {2}+1}\, \sqrt {2}+2 x}{\sqrt {\sqrt {2}-1}\, \sqrt {2}}\right )}{2}-\frac {\sqrt {\sqrt {2}+1}\, \sqrt {2}\, \mathrm {log}\left (-\sqrt {\sqrt {2}+1}\, \sqrt {2}\, x +\sqrt {2}+x^{2}\right )}{8}+\frac {\sqrt {\sqrt {2}+1}\, \sqrt {2}\, \mathrm {log}\left (\sqrt {\sqrt {2}+1}\, \sqrt {2}\, x +\sqrt {2}+x^{2}\right )}{8}+\frac {\sqrt {\sqrt {2}+1}\, \mathrm {log}\left (-\sqrt {\sqrt {2}+1}\, \sqrt {2}\, x +\sqrt {2}+x^{2}\right )}{4}-\frac {\sqrt {\sqrt {2}+1}\, \mathrm {log}\left (\sqrt {\sqrt {2}+1}\, \sqrt {2}\, x +\sqrt {2}+x^{2}\right )}{4} \] Input:
int(x^2/(x^4-2*x^2+2),x)
Output:
( - 2*sqrt(sqrt(2) - 1)*sqrt(2)*atan((sqrt(sqrt(2) + 1)*sqrt(2) - 2*x)/(sq rt(sqrt(2) - 1)*sqrt(2))) - 4*sqrt(sqrt(2) - 1)*atan((sqrt(sqrt(2) + 1)*sq rt(2) - 2*x)/(sqrt(sqrt(2) - 1)*sqrt(2))) + 2*sqrt(sqrt(2) - 1)*sqrt(2)*at an((sqrt(sqrt(2) + 1)*sqrt(2) + 2*x)/(sqrt(sqrt(2) - 1)*sqrt(2))) + 4*sqrt (sqrt(2) - 1)*atan((sqrt(sqrt(2) + 1)*sqrt(2) + 2*x)/(sqrt(sqrt(2) - 1)*sq rt(2))) - sqrt(sqrt(2) + 1)*sqrt(2)*log( - sqrt(sqrt(2) + 1)*sqrt(2)*x + s qrt(2) + x**2) + sqrt(sqrt(2) + 1)*sqrt(2)*log(sqrt(sqrt(2) + 1)*sqrt(2)*x + sqrt(2) + x**2) + 2*sqrt(sqrt(2) + 1)*log( - sqrt(sqrt(2) + 1)*sqrt(2)* x + sqrt(2) + x**2) - 2*sqrt(sqrt(2) + 1)*log(sqrt(sqrt(2) + 1)*sqrt(2)*x + sqrt(2) + x**2))/8