Integrand size = 25, antiderivative size = 113 \[ \int \frac {x^4}{\sqrt [4]{x^2+x^4} \left (1+x^4+x^8\right )} \, dx=-\frac {1}{4} \text {RootSum}\left [3-3 \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-\log (x)+\log \left (\sqrt [4]{x^2+x^4}-x \text {$\#$1}\right )}{-3 \text {$\#$1}+2 \text {$\#$1}^5}\&\right ]+\frac {1}{4} \text {RootSum}\left [1-\text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-\log (x)+\log \left (\sqrt [4]{x^2+x^4}-x \text {$\#$1}\right )}{-\text {$\#$1}+2 \text {$\#$1}^5}\&\right ] \] Output:
Unintegrable
Time = 0.35 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.31 \[ \int \frac {x^4}{\sqrt [4]{x^2+x^4} \left (1+x^4+x^8\right )} \, dx=\frac {\sqrt {x} \sqrt [4]{1+x^2} \left (-\text {RootSum}\left [3-3 \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-\log \left (\sqrt {x}\right )+\log \left (\sqrt [4]{1+x^2}-\sqrt {x} \text {$\#$1}\right )}{-3 \text {$\#$1}+2 \text {$\#$1}^5}\&\right ]+\text {RootSum}\left [1-\text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-\log \left (\sqrt {x}\right )+\log \left (\sqrt [4]{1+x^2}-\sqrt {x} \text {$\#$1}\right )}{-\text {$\#$1}+2 \text {$\#$1}^5}\&\right ]\right )}{4 \sqrt [4]{x^2+x^4}} \] Input:
Integrate[x^4/((x^2 + x^4)^(1/4)*(1 + x^4 + x^8)),x]
Output:
(Sqrt[x]*(1 + x^2)^(1/4)*(-RootSum[3 - 3*#1^4 + #1^8 & , (-Log[Sqrt[x]] + Log[(1 + x^2)^(1/4) - Sqrt[x]*#1])/(-3*#1 + 2*#1^5) & ] + RootSum[1 - #1^4 + #1^8 & , (-Log[Sqrt[x]] + Log[(1 + x^2)^(1/4) - Sqrt[x]*#1])/(-#1 + 2*# 1^5) & ]))/(4*(x^2 + x^4)^(1/4))
Result contains complex when optimal does not.
Time = 2.13 (sec) , antiderivative size = 884, normalized size of antiderivative = 7.82, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {2467, 2035, 7279, 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.
\(\displaystyle \int \frac {x^4}{\sqrt [4]{x^4+x^2} \left (x^8+x^4+1\right )} \, dx\) |
\(\Big \downarrow \) 2467 |
\(\displaystyle \frac {\sqrt {x} \sqrt [4]{x^2+1} \int \frac {x^{7/2}}{\sqrt [4]{x^2+1} \left (x^8+x^4+1\right )}dx}{\sqrt [4]{x^4+x^2}}\) |
\(\Big \downarrow \) 2035 |
\(\displaystyle \frac {2 \sqrt {x} \sqrt [4]{x^2+1} \int \frac {x^4}{\sqrt [4]{x^2+1} \left (x^8+x^4+1\right )}d\sqrt {x}}{\sqrt [4]{x^4+x^2}}\) |
\(\Big \downarrow \) 7279 |
\(\displaystyle \frac {2 \sqrt {x} \sqrt [4]{x^2+1} \int \left (\frac {1+\frac {i}{\sqrt {3}}}{\left (2 x^4-i \sqrt {3}+1\right ) \sqrt [4]{x^2+1}}+\frac {1-\frac {i}{\sqrt {3}}}{\left (2 x^4+i \sqrt {3}+1\right ) \sqrt [4]{x^2+1}}\right )d\sqrt {x}}{\sqrt [4]{x^4+x^2}}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {2 \sqrt {x} \sqrt [4]{x^2+1} \left (-\frac {\left (3-i \sqrt {3}\right ) \arctan \left (\frac {\sqrt [4]{-2+\sqrt {-2-2 i \sqrt {3}}} \sqrt {x}}{\sqrt [8]{2 \left (-1-i \sqrt {3}\right )} \sqrt [4]{x^2+1}}\right )}{6 \left (2 \left (-1-i \sqrt {3}\right )\right )^{7/8} \sqrt [4]{-2+\sqrt {-2-2 i \sqrt {3}}}}-\frac {\left (3-i \sqrt {3}\right ) \arctan \left (\frac {\sqrt [4]{2+\sqrt {-2-2 i \sqrt {3}}} \sqrt {x}}{\sqrt [8]{2 \left (-1-i \sqrt {3}\right )} \sqrt [4]{x^2+1}}\right )}{6 \left (2 \left (-1-i \sqrt {3}\right )\right )^{7/8} \sqrt [4]{2+\sqrt {-2-2 i \sqrt {3}}}}-\frac {\left (3+i \sqrt {3}\right ) \arctan \left (\frac {\sqrt [4]{-2+\sqrt {-2+2 i \sqrt {3}}} \sqrt {x}}{\sqrt [8]{2 \left (-1+i \sqrt {3}\right )} \sqrt [4]{x^2+1}}\right )}{6 \left (2 \left (-1+i \sqrt {3}\right )\right )^{7/8} \sqrt [4]{-2+\sqrt {-2+2 i \sqrt {3}}}}-\frac {\left (3+i \sqrt {3}\right ) \arctan \left (\frac {\sqrt [4]{2+\sqrt {-2+2 i \sqrt {3}}} \sqrt {x}}{\sqrt [8]{2 \left (-1+i \sqrt {3}\right )} \sqrt [4]{x^2+1}}\right )}{6 \left (2 \left (-1+i \sqrt {3}\right )\right )^{7/8} \sqrt [4]{2+\sqrt {-2+2 i \sqrt {3}}}}-\frac {\left (3-i \sqrt {3}\right ) \text {arctanh}\left (\frac {\sqrt [4]{-2+\sqrt {-2-2 i \sqrt {3}}} \sqrt {x}}{\sqrt [8]{2 \left (-1-i \sqrt {3}\right )} \sqrt [4]{x^2+1}}\right )}{6 \left (2 \left (-1-i \sqrt {3}\right )\right )^{7/8} \sqrt [4]{-2+\sqrt {-2-2 i \sqrt {3}}}}-\frac {\left (3-i \sqrt {3}\right ) \text {arctanh}\left (\frac {\sqrt [4]{2+\sqrt {-2-2 i \sqrt {3}}} \sqrt {x}}{\sqrt [8]{2 \left (-1-i \sqrt {3}\right )} \sqrt [4]{x^2+1}}\right )}{6 \left (2 \left (-1-i \sqrt {3}\right )\right )^{7/8} \sqrt [4]{2+\sqrt {-2-2 i \sqrt {3}}}}-\frac {\left (3+i \sqrt {3}\right ) \text {arctanh}\left (\frac {\sqrt [4]{-2+\sqrt {-2+2 i \sqrt {3}}} \sqrt {x}}{\sqrt [8]{2 \left (-1+i \sqrt {3}\right )} \sqrt [4]{x^2+1}}\right )}{6 \left (2 \left (-1+i \sqrt {3}\right )\right )^{7/8} \sqrt [4]{-2+\sqrt {-2+2 i \sqrt {3}}}}-\frac {\left (3+i \sqrt {3}\right ) \text {arctanh}\left (\frac {\sqrt [4]{2+\sqrt {-2+2 i \sqrt {3}}} \sqrt {x}}{\sqrt [8]{2 \left (-1+i \sqrt {3}\right )} \sqrt [4]{x^2+1}}\right )}{6 \left (2 \left (-1+i \sqrt {3}\right )\right )^{7/8} \sqrt [4]{2+\sqrt {-2+2 i \sqrt {3}}}}\right )}{\sqrt [4]{x^4+x^2}}\) |
Input:
Int[x^4/((x^2 + x^4)^(1/4)*(1 + x^4 + x^8)),x]
Output:
(2*Sqrt[x]*(1 + x^2)^(1/4)*(-1/6*((3 - I*Sqrt[3])*ArcTan[((-2 + Sqrt[-2 - (2*I)*Sqrt[3]])^(1/4)*Sqrt[x])/((2*(-1 - I*Sqrt[3]))^(1/8)*(1 + x^2)^(1/4) )])/((2*(-1 - I*Sqrt[3]))^(7/8)*(-2 + Sqrt[-2 - (2*I)*Sqrt[3]])^(1/4)) - ( (3 - I*Sqrt[3])*ArcTan[((2 + Sqrt[-2 - (2*I)*Sqrt[3]])^(1/4)*Sqrt[x])/((2* (-1 - I*Sqrt[3]))^(1/8)*(1 + x^2)^(1/4))])/(6*(2*(-1 - I*Sqrt[3]))^(7/8)*( 2 + Sqrt[-2 - (2*I)*Sqrt[3]])^(1/4)) - ((3 + I*Sqrt[3])*ArcTan[((-2 + Sqrt [-2 + (2*I)*Sqrt[3]])^(1/4)*Sqrt[x])/((2*(-1 + I*Sqrt[3]))^(1/8)*(1 + x^2) ^(1/4))])/(6*(2*(-1 + I*Sqrt[3]))^(7/8)*(-2 + Sqrt[-2 + (2*I)*Sqrt[3]])^(1 /4)) - ((3 + I*Sqrt[3])*ArcTan[((2 + Sqrt[-2 + (2*I)*Sqrt[3]])^(1/4)*Sqrt[ x])/((2*(-1 + I*Sqrt[3]))^(1/8)*(1 + x^2)^(1/4))])/(6*(2*(-1 + I*Sqrt[3])) ^(7/8)*(2 + Sqrt[-2 + (2*I)*Sqrt[3]])^(1/4)) - ((3 - I*Sqrt[3])*ArcTanh[(( -2 + Sqrt[-2 - (2*I)*Sqrt[3]])^(1/4)*Sqrt[x])/((2*(-1 - I*Sqrt[3]))^(1/8)* (1 + x^2)^(1/4))])/(6*(2*(-1 - I*Sqrt[3]))^(7/8)*(-2 + Sqrt[-2 - (2*I)*Sqr t[3]])^(1/4)) - ((3 - I*Sqrt[3])*ArcTanh[((2 + Sqrt[-2 - (2*I)*Sqrt[3]])^( 1/4)*Sqrt[x])/((2*(-1 - I*Sqrt[3]))^(1/8)*(1 + x^2)^(1/4))])/(6*(2*(-1 - I *Sqrt[3]))^(7/8)*(2 + Sqrt[-2 - (2*I)*Sqrt[3]])^(1/4)) - ((3 + I*Sqrt[3])* ArcTanh[((-2 + Sqrt[-2 + (2*I)*Sqrt[3]])^(1/4)*Sqrt[x])/((2*(-1 + I*Sqrt[3 ]))^(1/8)*(1 + x^2)^(1/4))])/(6*(2*(-1 + I*Sqrt[3]))^(7/8)*(-2 + Sqrt[-2 + (2*I)*Sqrt[3]])^(1/4)) - ((3 + I*Sqrt[3])*ArcTanh[((2 + Sqrt[-2 + (2*I)*S qrt[3]])^(1/4)*Sqrt[x])/((2*(-1 + I*Sqrt[3]))^(1/8)*(1 + x^2)^(1/4))])/...
Int[(Fx_)*(x_)^(m_), x_Symbol] :> With[{k = Denominator[m]}, Simp[k Subst [Int[x^(k*(m + 1) - 1)*SubstPower[Fx, x, k], x], x, x^(1/k)], x]] /; Fracti onQ[m] && AlgebraicFunctionQ[Fx, x]
Int[(Fx_.)*(Px_)^(p_), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Simp[Px^F racPart[p]/(x^(r*FracPart[p])*ExpandToSum[Px/x^r, x]^FracPart[p]) Int[x^( p*r)*ExpandToSum[Px/x^r, x]^p*Fx, x], x] /; IGtQ[r, 0]] /; FreeQ[p, x] && P olyQ[Px, x] && !IntegerQ[p] && !MonomialQ[Px, x] && !PolyQ[Fx, x]
Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[ {v = RationalFunctionExpand[u/(a + b*x^n + c*x^(2*n)), x]}, Int[v, x] /; Su mQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]
Time = 106.70 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.88
method | result | size |
pseudoelliptic | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}-\textit {\_Z}^{4}+1\right )}{\sum }\frac {\ln \left (\frac {-\textit {\_R} x +\left (x^{2} \left (x^{2}+1\right )\right )^{\frac {1}{4}}}{x}\right )}{2 \textit {\_R}^{5}-\textit {\_R}}\right )}{4}-\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}-3 \textit {\_Z}^{4}+3\right )}{\sum }\frac {\ln \left (\frac {-\textit {\_R} x +\left (x^{2} \left (x^{2}+1\right )\right )^{\frac {1}{4}}}{x}\right )}{2 \textit {\_R}^{5}-3 \textit {\_R}}\right )}{4}\) | \(100\) |
trager | \(\text {Expression too large to display}\) | \(5899\) |
Input:
int(x^4/(x^4+x^2)^(1/4)/(x^8+x^4+1),x,method=_RETURNVERBOSE)
Output:
1/4*sum(ln((-_R*x+(x^2*(x^2+1))^(1/4))/x)/(2*_R^5-_R),_R=RootOf(_Z^8-_Z^4+ 1))-1/4*sum(ln((-_R*x+(x^2*(x^2+1))^(1/4))/x)/(2*_R^5-3*_R),_R=RootOf(_Z^8 -3*_Z^4+3))
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 9.79 (sec) , antiderivative size = 2949, normalized size of antiderivative = 26.10 \[ \int \frac {x^4}{\sqrt [4]{x^2+x^4} \left (1+x^4+x^8\right )} \, dx=\text {Too large to display} \] Input:
integrate(x^4/(x^4+x^2)^(1/4)/(x^8+x^4+1),x, algorithm="fricas")
Output:
1/8*sqrt(1/3)*sqrt(-sqrt(3/2*sqrt(-1/3) + 1/2))*log(-(3*sqrt(1/3)*(x^5 - x ^3 + 2*sqrt(x^4 + x^2)*(x^3 + sqrt(-1/3)*(x^3 + 2*x))*sqrt(3/2*sqrt(-1/3) + 1/2) + sqrt(-1/3)*(3*x^5 + 5*x^3 + x) - x)*sqrt(-sqrt(3/2*sqrt(-1/3) + 1 /2)) + 2*(x^4 + x^2)^(3/4)*(2*x^2 + 3*sqrt(-1/3) + 1) - 2*(3*sqrt(-1/3)*x^ 4 - x^4 - 2*x^2)*(x^4 + x^2)^(1/4)*sqrt(3/2*sqrt(-1/3) + 1/2))/(x^5 + x^3 + x)) - 1/8*sqrt(1/3)*sqrt(-sqrt(3/2*sqrt(-1/3) + 1/2))*log((3*sqrt(1/3)*( x^5 - x^3 + 2*sqrt(x^4 + x^2)*(x^3 + sqrt(-1/3)*(x^3 + 2*x))*sqrt(3/2*sqrt (-1/3) + 1/2) + sqrt(-1/3)*(3*x^5 + 5*x^3 + x) - x)*sqrt(-sqrt(3/2*sqrt(-1 /3) + 1/2)) - 2*(x^4 + x^2)^(3/4)*(2*x^2 + 3*sqrt(-1/3) + 1) + 2*(3*sqrt(- 1/3)*x^4 - x^4 - 2*x^2)*(x^4 + x^2)^(1/4)*sqrt(3/2*sqrt(-1/3) + 1/2))/(x^5 + x^3 + x)) - 1/8*sqrt(1/3)*sqrt(-sqrt(1/2*sqrt(-1/3) + 1/2))*log(-(3*sqr t(1/3)*(21*x^5 + x^3 + 2*sqrt(x^4 + x^2)*(13*x^3 - 3*sqrt(-1/3)*(3*x^3 - 8 *x) - 2*x)*sqrt(1/2*sqrt(-1/3) + 1/2) - sqrt(-1/3)*(7*x^5 - 37*x^3 - 11*x) - 5*x)*sqrt(-sqrt(1/2*sqrt(-1/3) + 1/2)) + 2*(x^4 + x^2)^(3/4)*(2*x^2 - 3 *sqrt(-1/3)*(8*x^2 - 5) + 11) - 6*(3*x^4 - 8*x^2 + sqrt(-1/3)*(13*x^4 - 2* x^2))*(x^4 + x^2)^(1/4)*sqrt(1/2*sqrt(-1/3) + 1/2))/(x^5 - x^3 + x)) + 1/8 *sqrt(1/3)*sqrt(-sqrt(1/2*sqrt(-1/3) + 1/2))*log((3*sqrt(1/3)*(21*x^5 + x^ 3 + 2*sqrt(x^4 + x^2)*(13*x^3 - 3*sqrt(-1/3)*(3*x^3 - 8*x) - 2*x)*sqrt(1/2 *sqrt(-1/3) + 1/2) - sqrt(-1/3)*(7*x^5 - 37*x^3 - 11*x) - 5*x)*sqrt(-sqrt( 1/2*sqrt(-1/3) + 1/2)) - 2*(x^4 + x^2)^(3/4)*(2*x^2 - 3*sqrt(-1/3)*(8*x...
Not integrable
Time = 1.47 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.33 \[ \int \frac {x^4}{\sqrt [4]{x^2+x^4} \left (1+x^4+x^8\right )} \, dx=\int \frac {x^{4}}{\sqrt [4]{x^{2} \left (x^{2} + 1\right )} \left (x^{2} - x + 1\right ) \left (x^{2} + x + 1\right ) \left (x^{4} - x^{2} + 1\right )}\, dx \] Input:
integrate(x**4/(x**4+x**2)**(1/4)/(x**8+x**4+1),x)
Output:
Integral(x**4/((x**2*(x**2 + 1))**(1/4)*(x**2 - x + 1)*(x**2 + x + 1)*(x** 4 - x**2 + 1)), x)
Not integrable
Time = 0.16 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.88 \[ \int \frac {x^4}{\sqrt [4]{x^2+x^4} \left (1+x^4+x^8\right )} \, dx=\int { \frac {x^{4}}{{\left (x^{8} + x^{4} + 1\right )} {\left (x^{4} + x^{2}\right )}^{\frac {1}{4}}} \,d x } \] Input:
integrate(x^4/(x^4+x^2)^(1/4)/(x^8+x^4+1),x, algorithm="maxima")
Output:
2/21*(4*x^5 + x^3 - 3*x)*x^(7/2)/((x^8 + x^4 + 1)*(x^2 + 1)^(1/4)) - integ rate(8/21*((4*x^4 + x^2 - 3)*x^(15/2) + 2*(4*x^4 + x^2 - 3)*x^(7/2))/((x^1 6 + 2*x^12 + 3*x^8 + 2*x^4 + 1)*(x^2 + 1)^(1/4)), x)
Exception generated. \[ \int \frac {x^4}{\sqrt [4]{x^2+x^4} \left (1+x^4+x^8\right )} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(x^4/(x^4+x^2)^(1/4)/(x^8+x^4+1),x, algorithm="giac")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:proot error [1,0,0,0,1,0,0,0,1]proo t error [1,0,0,0,-1,0,0,0,1]proot error [1,0,-10,0,1]proot error [1,0,-10, 0,1]proot
Not integrable
Time = 7.84 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.22 \[ \int \frac {x^4}{\sqrt [4]{x^2+x^4} \left (1+x^4+x^8\right )} \, dx=\int \frac {x^4}{{\left (x^4+x^2\right )}^{1/4}\,\left (x^8+x^4+1\right )} \,d x \] Input:
int(x^4/((x^2 + x^4)^(1/4)*(x^4 + x^8 + 1)),x)
Output:
int(x^4/((x^2 + x^4)^(1/4)*(x^4 + x^8 + 1)), x)
Not integrable
Time = 2.23 (sec) , antiderivative size = 303, normalized size of antiderivative = 2.68 \[ \int \frac {x^4}{\sqrt [4]{x^2+x^4} \left (1+x^4+x^8\right )} \, dx=\frac {2 \sqrt {x}\, \left (x^{2}+1\right )^{\frac {1}{4}}-\sqrt {x^{2}+1}\, \left (\int \frac {\left (x^{2}+1\right )^{\frac {3}{4}}}{\sqrt {x}\, x^{12}+2 \sqrt {x}\, x^{10}+2 \sqrt {x}\, x^{8}+2 \sqrt {x}\, x^{6}+2 \sqrt {x}\, x^{4}+2 \sqrt {x}\, x^{2}+\sqrt {x}}d x \right )-2 \sqrt {x^{2}+1}\, \left (\int \frac {\sqrt {x}\, \left (x^{2}+1\right )^{\frac {3}{4}} x^{9}}{x^{12}+2 x^{10}+2 x^{8}+2 x^{6}+2 x^{4}+2 x^{2}+1}d x \right )-\sqrt {x^{2}+1}\, \left (\int \frac {\sqrt {x}\, \left (x^{2}+1\right )^{\frac {3}{4}} x^{7}}{x^{12}+2 x^{10}+2 x^{8}+2 x^{6}+2 x^{4}+2 x^{2}+1}d x \right )+\sqrt {x^{2}+1}\, \left (\int \frac {\sqrt {x}\, \left (x^{2}+1\right )^{\frac {3}{4}} x^{3}}{x^{12}+2 x^{10}+2 x^{8}+2 x^{6}+2 x^{4}+2 x^{2}+1}d x \right )-2 \sqrt {x^{2}+1}\, \left (\int \frac {\sqrt {x}\, \left (x^{2}+1\right )^{\frac {3}{4}} x}{x^{12}+2 x^{10}+2 x^{8}+2 x^{6}+2 x^{4}+2 x^{2}+1}d x \right )}{2 \sqrt {x^{2}+1}} \] Input:
int(x^4/(x^4+x^2)^(1/4)/(x^8+x^4+1),x)
Output:
(2*sqrt(x)*(x**2 + 1)**(1/4) - sqrt(x**2 + 1)*int((x**2 + 1)**(3/4)/(sqrt( x)*x**12 + 2*sqrt(x)*x**10 + 2*sqrt(x)*x**8 + 2*sqrt(x)*x**6 + 2*sqrt(x)*x **4 + 2*sqrt(x)*x**2 + sqrt(x)),x) - 2*sqrt(x**2 + 1)*int((sqrt(x)*(x**2 + 1)**(3/4)*x**9)/(x**12 + 2*x**10 + 2*x**8 + 2*x**6 + 2*x**4 + 2*x**2 + 1) ,x) - sqrt(x**2 + 1)*int((sqrt(x)*(x**2 + 1)**(3/4)*x**7)/(x**12 + 2*x**10 + 2*x**8 + 2*x**6 + 2*x**4 + 2*x**2 + 1),x) + sqrt(x**2 + 1)*int((sqrt(x) *(x**2 + 1)**(3/4)*x**3)/(x**12 + 2*x**10 + 2*x**8 + 2*x**6 + 2*x**4 + 2*x **2 + 1),x) - 2*sqrt(x**2 + 1)*int((sqrt(x)*(x**2 + 1)**(3/4)*x)/(x**12 + 2*x**10 + 2*x**8 + 2*x**6 + 2*x**4 + 2*x**2 + 1),x))/(2*sqrt(x**2 + 1))