\(\int \frac {-1+x^4}{(1+x^2+x^4) \sqrt [4]{x^2+x^6}} \, dx\) [1335]

Optimal result

Integrand size = 27, antiderivative size = 96 \[ \int \frac {-1+x^4}{\left (1+x^2+x^4\right ) \sqrt [4]{x^2+x^6}} \, dx=-\frac {\arctan \left (\frac {\sqrt {2} x \sqrt [4]{x^2+x^6}}{-x^2+\sqrt {x^2+x^6}}\right )}{\sqrt {2}}-\frac {\text {arctanh}\left (\frac {\frac {x^2}{\sqrt {2}}+\frac {\sqrt {x^2+x^6}}{\sqrt {2}}}{x \sqrt [4]{x^2+x^6}}\right )}{\sqrt {2}} \] Output:

-1/2*arctan(2^(1/2)*x*(x^6+x^2)^(1/4)/(-x^2+(x^6+x^2)^(1/2)))*2^(1/2)-1/2* 
arctanh((1/2*2^(1/2)*x^2+1/2*(x^6+x^2)^(1/2)*2^(1/2))/x/(x^6+x^2)^(1/4))*2 
^(1/2)
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.07 \[ \int \frac {-1+x^4}{\left (1+x^2+x^4\right ) \sqrt [4]{x^2+x^6}} \, dx=-\frac {\sqrt {x} \sqrt [4]{1+x^4} \left (\arctan \left (\frac {\sqrt {2} \sqrt {x} \sqrt [4]{1+x^4}}{-x+\sqrt {1+x^4}}\right )+\text {arctanh}\left (\frac {\sqrt {2} \sqrt {x} \sqrt [4]{1+x^4}}{x+\sqrt {1+x^4}}\right )\right )}{\sqrt {2} \sqrt [4]{x^2+x^6}} \] Input:

Integrate[(-1 + x^4)/((1 + x^2 + x^4)*(x^2 + x^6)^(1/4)),x]
 

Output:

-((Sqrt[x]*(1 + x^4)^(1/4)*(ArcTan[(Sqrt[2]*Sqrt[x]*(1 + x^4)^(1/4))/(-x + 
 Sqrt[1 + x^4])] + ArcTanh[(Sqrt[2]*Sqrt[x]*(1 + x^4)^(1/4))/(x + Sqrt[1 + 
 x^4])]))/(Sqrt[2]*(x^2 + x^6)^(1/4)))
 

Rubi [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 6 vs. order 3 in optimal.

Time = 0.80 (sec) , antiderivative size = 260, normalized size of antiderivative = 2.71, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {2467, 25, 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.

\(\displaystyle \int \frac {x^4-1}{\left (x^4+x^2+1\right ) \sqrt [4]{x^6+x^2}} \, dx\)

\(\Big \downarrow \) 2467

\(\displaystyle \frac {\sqrt {x} \sqrt [4]{x^4+1} \int -\frac {1-x^4}{\sqrt {x} \sqrt [4]{x^4+1} \left (x^4+x^2+1\right )}dx}{\sqrt [4]{x^6+x^2}}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {\sqrt {x} \sqrt [4]{x^4+1} \int \frac {1-x^4}{\sqrt {x} \sqrt [4]{x^4+1} \left (x^4+x^2+1\right )}dx}{\sqrt [4]{x^6+x^2}}\)

\(\Big \downarrow \) 2035

\(\displaystyle -\frac {2 \sqrt {x} \sqrt [4]{x^4+1} \int \frac {1-x^4}{\sqrt [4]{x^4+1} \left (x^4+x^2+1\right )}d\sqrt {x}}{\sqrt [4]{x^6+x^2}}\)

\(\Big \downarrow \) 7279

\(\displaystyle -\frac {2 \sqrt {x} \sqrt [4]{x^4+1} \int \left (\frac {x^2+2}{\sqrt [4]{x^4+1} \left (x^4+x^2+1\right )}-\frac {1}{\sqrt [4]{x^4+1}}\right )d\sqrt {x}}{\sqrt [4]{x^6+x^2}}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {2 \sqrt {x} \sqrt [4]{x^4+1} \left (\sqrt {x} \operatorname {AppellF1}\left (\frac {1}{8},1,\frac {1}{4},\frac {9}{8},-\frac {2 x^4}{1-i \sqrt {3}},-x^4\right )+\sqrt {x} \operatorname {AppellF1}\left (\frac {1}{8},1,\frac {1}{4},\frac {9}{8},-\frac {2 x^4}{1+i \sqrt {3}},-x^4\right )+\frac {\left (-\sqrt {3}+i\right ) x^{5/2} \operatorname {AppellF1}\left (\frac {5}{8},\frac {1}{4},1,\frac {13}{8},-x^4,-\frac {2 x^4}{1-i \sqrt {3}}\right )}{5 \left (\sqrt {3}+i\right )}+\frac {\left (\sqrt {3}+i\right ) x^{5/2} \operatorname {AppellF1}\left (\frac {5}{8},\frac {1}{4},1,\frac {13}{8},-x^4,-\frac {2 x^4}{1+i \sqrt {3}}\right )}{5 \left (-\sqrt {3}+i\right )}-\sqrt {x} \operatorname {Hypergeometric2F1}\left (\frac {1}{8},\frac {1}{4},\frac {9}{8},-x^4\right )\right )}{\sqrt [4]{x^6+x^2}}\)

Input:

Int[(-1 + x^4)/((1 + x^2 + x^4)*(x^2 + x^6)^(1/4)),x]
 

Output:

(-2*Sqrt[x]*(1 + x^4)^(1/4)*(Sqrt[x]*AppellF1[1/8, 1, 1/4, 9/8, (-2*x^4)/( 
1 - I*Sqrt[3]), -x^4] + Sqrt[x]*AppellF1[1/8, 1, 1/4, 9/8, (-2*x^4)/(1 + I 
*Sqrt[3]), -x^4] + ((I - Sqrt[3])*x^(5/2)*AppellF1[5/8, 1/4, 1, 13/8, -x^4 
, (-2*x^4)/(1 - I*Sqrt[3])])/(5*(I + Sqrt[3])) + ((I + Sqrt[3])*x^(5/2)*Ap 
pellF1[5/8, 1/4, 1, 13/8, -x^4, (-2*x^4)/(1 + I*Sqrt[3])])/(5*(I - Sqrt[3] 
)) - Sqrt[x]*Hypergeometric2F1[1/8, 1/4, 9/8, -x^4]))/(x^2 + x^6)^(1/4)
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 2009

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

rule 2035

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]
 

rule 2467

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]
 

rule 7279

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]
 

Maple [A] (verified)

Time = 3.04 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.29

method	result	size
pseudoelliptic	\(\frac {\sqrt {2}\, \left (\ln \left (\frac {-\left (x^{2} \left (x^{4}+1\right )\right )^{\frac {1}{4}} \sqrt {2}\, x +x^{2}+\sqrt {x^{2} \left (x^{4}+1\right )}}{\left (x^{2} \left (x^{4}+1\right )\right )^{\frac {1}{4}} \sqrt {2}\, x +x^{2}+\sqrt {x^{2} \left (x^{4}+1\right )}}\right )+2 \arctan \left (\frac {\left (x^{2} \left (x^{4}+1\right )\right )^{\frac {1}{4}} \sqrt {2}+x}{x}\right )+2 \arctan \left (\frac {\left (x^{2} \left (x^{4}+1\right )\right )^{\frac {1}{4}} \sqrt {2}-x}{x}\right )\right )}{4}\)	\(124\)
trager	\(\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} \ln \left (-\frac {-\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} x^{5}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} x^{3}+2 \left (x^{6}+x^{2}\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x^{2}+2 \sqrt {x^{6}+x^{2}}\, \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) x -\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} x +2 \left (x^{6}+x^{2}\right )^{\frac {3}{4}}}{\left (x^{2}+x +1\right ) x \left (x^{2}-x +1\right )}\right )}{2}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) \ln \left (-\frac {2 \sqrt {x^{6}+x^{2}}\, \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} x -x^{5} \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )-2 \left (x^{6}+x^{2}\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) x^{3}+2 \left (x^{6}+x^{2}\right )^{\frac {3}{4}}-\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) x}{\left (x^{2}+x +1\right ) x \left (x^{2}-x +1\right )}\right )}{2}\)	\(240\)

Input:

int((x^4-1)/(x^4+x^2+1)/(x^6+x^2)^(1/4),x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 382 vs. \(2 (80) = 160\).

Time = 34.43 (sec) , antiderivative size = 382, normalized size of antiderivative = 3.98 \[ \int \frac {-1+x^4}{\left (1+x^2+x^4\right ) \sqrt [4]{x^2+x^6}} \, dx=-\frac {1}{4} \, \sqrt {2} \arctan \left (\frac {x^{9} + 2 \, x^{7} + 3 \, x^{5} + 2 \, x^{3} + 2 \, \sqrt {2} {\left (x^{6} + x^{2}\right )}^{\frac {3}{4}} {\left (x^{4} - 3 \, x^{2} + 1\right )} + 2 \, \sqrt {2} {\left (3 \, x^{6} - x^{4} + 3 \, x^{2}\right )} {\left (x^{6} + x^{2}\right )}^{\frac {1}{4}} + 4 \, \sqrt {x^{6} + x^{2}} {\left (x^{5} + x^{3} + x\right )} + x}{x^{9} - 14 \, x^{7} + 3 \, x^{5} - 14 \, x^{3} + x}\right ) - \frac {1}{4} \, \sqrt {2} \arctan \left (-\frac {x^{9} + 2 \, x^{7} + 3 \, x^{5} + 2 \, x^{3} - 2 \, \sqrt {2} {\left (x^{6} + x^{2}\right )}^{\frac {3}{4}} {\left (x^{4} - 3 \, x^{2} + 1\right )} - 2 \, \sqrt {2} {\left (3 \, x^{6} - x^{4} + 3 \, x^{2}\right )} {\left (x^{6} + x^{2}\right )}^{\frac {1}{4}} + 4 \, \sqrt {x^{6} + x^{2}} {\left (x^{5} + x^{3} + x\right )} + x}{x^{9} - 14 \, x^{7} + 3 \, x^{5} - 14 \, x^{3} + x}\right ) - \frac {1}{8} \, \sqrt {2} \log \left (\frac {x^{5} + x^{3} + 2 \, \sqrt {2} {\left (x^{6} + x^{2}\right )}^{\frac {1}{4}} x^{2} + 4 \, \sqrt {x^{6} + x^{2}} x + 2 \, \sqrt {2} {\left (x^{6} + x^{2}\right )}^{\frac {3}{4}} + x}{x^{5} + x^{3} + x}\right ) + \frac {1}{8} \, \sqrt {2} \log \left (\frac {x^{5} + x^{3} - 2 \, \sqrt {2} {\left (x^{6} + x^{2}\right )}^{\frac {1}{4}} x^{2} + 4 \, \sqrt {x^{6} + x^{2}} x - 2 \, \sqrt {2} {\left (x^{6} + x^{2}\right )}^{\frac {3}{4}} + x}{x^{5} + x^{3} + x}\right ) \] Input:

integrate((x^4-1)/(x^4+x^2+1)/(x^6+x^2)^(1/4),x, algorithm="fricas")
 

Output:

-1/4*sqrt(2)*arctan((x^9 + 2*x^7 + 3*x^5 + 2*x^3 + 2*sqrt(2)*(x^6 + x^2)^( 
3/4)*(x^4 - 3*x^2 + 1) + 2*sqrt(2)*(3*x^6 - x^4 + 3*x^2)*(x^6 + x^2)^(1/4) 
 + 4*sqrt(x^6 + x^2)*(x^5 + x^3 + x) + x)/(x^9 - 14*x^7 + 3*x^5 - 14*x^3 + 
 x)) - 1/4*sqrt(2)*arctan(-(x^9 + 2*x^7 + 3*x^5 + 2*x^3 - 2*sqrt(2)*(x^6 + 
 x^2)^(3/4)*(x^4 - 3*x^2 + 1) - 2*sqrt(2)*(3*x^6 - x^4 + 3*x^2)*(x^6 + x^2 
)^(1/4) + 4*sqrt(x^6 + x^2)*(x^5 + x^3 + x) + x)/(x^9 - 14*x^7 + 3*x^5 - 1 
4*x^3 + x)) - 1/8*sqrt(2)*log((x^5 + x^3 + 2*sqrt(2)*(x^6 + x^2)^(1/4)*x^2 
 + 4*sqrt(x^6 + x^2)*x + 2*sqrt(2)*(x^6 + x^2)^(3/4) + x)/(x^5 + x^3 + x)) 
 + 1/8*sqrt(2)*log((x^5 + x^3 - 2*sqrt(2)*(x^6 + x^2)^(1/4)*x^2 + 4*sqrt(x 
^6 + x^2)*x - 2*sqrt(2)*(x^6 + x^2)^(3/4) + x)/(x^5 + x^3 + x))
 

Sympy [F]

\[ \int \frac {-1+x^4}{\left (1+x^2+x^4\right ) \sqrt [4]{x^2+x^6}} \, dx=\int \frac {\left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right )}{\sqrt [4]{x^{2} \left (x^{4} + 1\right )} \left (x^{2} - x + 1\right ) \left (x^{2} + x + 1\right )}\, dx \] Input:

integrate((x**4-1)/(x**4+x**2+1)/(x**6+x**2)**(1/4),x)
 

Output:

Integral((x - 1)*(x + 1)*(x**2 + 1)/((x**2*(x**4 + 1))**(1/4)*(x**2 - x + 
1)*(x**2 + x + 1)), x)
 

Maxima [F]

\[ \int \frac {-1+x^4}{\left (1+x^2+x^4\right ) \sqrt [4]{x^2+x^6}} \, dx=\int { \frac {x^{4} - 1}{{\left (x^{6} + x^{2}\right )}^{\frac {1}{4}} {\left (x^{4} + x^{2} + 1\right )}} \,d x } \] Input:

integrate((x^4-1)/(x^4+x^2+1)/(x^6+x^2)^(1/4),x, algorithm="maxima")
 

Output:

integrate((x^4 - 1)/((x^6 + x^2)^(1/4)*(x^4 + x^2 + 1)), x)
 

Giac [F]

\[ \int \frac {-1+x^4}{\left (1+x^2+x^4\right ) \sqrt [4]{x^2+x^6}} \, dx=\int { \frac {x^{4} - 1}{{\left (x^{6} + x^{2}\right )}^{\frac {1}{4}} {\left (x^{4} + x^{2} + 1\right )}} \,d x } \] Input:

integrate((x^4-1)/(x^4+x^2+1)/(x^6+x^2)^(1/4),x, algorithm="giac")
 

Output:

integrate((x^4 - 1)/((x^6 + x^2)^(1/4)*(x^4 + x^2 + 1)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {-1+x^4}{\left (1+x^2+x^4\right ) \sqrt [4]{x^2+x^6}} \, dx=\int \frac {x^4-1}{{\left (x^6+x^2\right )}^{1/4}\,\left (x^4+x^2+1\right )} \,d x \] Input:

int((x^4 - 1)/((x^2 + x^6)^(1/4)*(x^2 + x^4 + 1)),x)
 

Output:

int((x^4 - 1)/((x^2 + x^6)^(1/4)*(x^2 + x^4 + 1)), x)
 

Reduce [F]

\[ \int \frac {-1+x^4}{\left (1+x^2+x^4\right ) \sqrt [4]{x^2+x^6}} \, dx=\frac {6 \sqrt {x}\, \left (x^{4}+1\right )^{\frac {5}{4}}+2 \sqrt {x}\, \left (x^{4}+1\right )^{\frac {1}{4}} x^{4}+2 \sqrt {x}\, \left (x^{4}+1\right )^{\frac {1}{4}}-7 \sqrt {x^{4}+1}\, \left (\int \frac {\left (x^{4}+1\right )^{\frac {3}{4}}}{\sqrt {x}\, x^{12}+\sqrt {x}\, x^{10}+3 \sqrt {x}\, x^{8}+2 \sqrt {x}\, x^{6}+3 \sqrt {x}\, x^{4}+\sqrt {x}\, x^{2}+\sqrt {x}}d x \right ) x^{4}-7 \sqrt {x^{4}+1}\, \left (\int \frac {\left (x^{4}+1\right )^{\frac {3}{4}}}{\sqrt {x}\, x^{12}+\sqrt {x}\, x^{10}+3 \sqrt {x}\, x^{8}+2 \sqrt {x}\, x^{6}+3 \sqrt {x}\, x^{4}+\sqrt {x}\, x^{2}+\sqrt {x}}d x \right )+3 \sqrt {x^{4}+1}\, \left (\int \frac {\sqrt {x}\, \left (x^{4}+1\right )^{\frac {3}{4}} x^{7}}{x^{12}+x^{10}+3 x^{8}+2 x^{6}+3 x^{4}+x^{2}+1}d x \right ) x^{4}+3 \sqrt {x^{4}+1}\, \left (\int \frac {\sqrt {x}\, \left (x^{4}+1\right )^{\frac {3}{4}} x^{7}}{x^{12}+x^{10}+3 x^{8}+2 x^{6}+3 x^{4}+x^{2}+1}d x \right )-4 \sqrt {x^{4}+1}\, \left (\int \frac {\sqrt {x}\, \left (x^{4}+1\right )^{\frac {3}{4}} x^{3}}{x^{12}+x^{10}+3 x^{8}+2 x^{6}+3 x^{4}+x^{2}+1}d x \right ) x^{4}-4 \sqrt {x^{4}+1}\, \left (\int \frac {\sqrt {x}\, \left (x^{4}+1\right )^{\frac {3}{4}} x^{3}}{x^{12}+x^{10}+3 x^{8}+2 x^{6}+3 x^{4}+x^{2}+1}d x \right )-4 \sqrt {x^{4}+1}\, \left (\int \frac {\sqrt {x}\, \left (x^{4}+1\right )^{\frac {3}{4}} x}{x^{12}+x^{10}+3 x^{8}+2 x^{6}+3 x^{4}+x^{2}+1}d x \right ) x^{4}-4 \sqrt {x^{4}+1}\, \left (\int \frac {\sqrt {x}\, \left (x^{4}+1\right )^{\frac {3}{4}} x}{x^{12}+x^{10}+3 x^{8}+2 x^{6}+3 x^{4}+x^{2}+1}d x \right )}{3 \sqrt {x^{4}+1}\, \left (x^{4}+1\right )} \] Input:

int((x^4-1)/(x^4+x^2+1)/(x^6+x^2)^(1/4),x)
 

Output:

(6*sqrt(x)*(x**4 + 1)**(5/4) + 2*sqrt(x)*(x**4 + 1)**(1/4)*x**4 + 2*sqrt(x 
)*(x**4 + 1)**(1/4) - 7*sqrt(x**4 + 1)*int((x**4 + 1)**(3/4)/(sqrt(x)*x**1 
2 + sqrt(x)*x**10 + 3*sqrt(x)*x**8 + 2*sqrt(x)*x**6 + 3*sqrt(x)*x**4 + sqr 
t(x)*x**2 + sqrt(x)),x)*x**4 - 7*sqrt(x**4 + 1)*int((x**4 + 1)**(3/4)/(sqr 
t(x)*x**12 + sqrt(x)*x**10 + 3*sqrt(x)*x**8 + 2*sqrt(x)*x**6 + 3*sqrt(x)*x 
**4 + sqrt(x)*x**2 + sqrt(x)),x) + 3*sqrt(x**4 + 1)*int((sqrt(x)*(x**4 + 1 
)**(3/4)*x**7)/(x**12 + x**10 + 3*x**8 + 2*x**6 + 3*x**4 + x**2 + 1),x)*x* 
*4 + 3*sqrt(x**4 + 1)*int((sqrt(x)*(x**4 + 1)**(3/4)*x**7)/(x**12 + x**10 
+ 3*x**8 + 2*x**6 + 3*x**4 + x**2 + 1),x) - 4*sqrt(x**4 + 1)*int((sqrt(x)* 
(x**4 + 1)**(3/4)*x**3)/(x**12 + x**10 + 3*x**8 + 2*x**6 + 3*x**4 + x**2 + 
 1),x)*x**4 - 4*sqrt(x**4 + 1)*int((sqrt(x)*(x**4 + 1)**(3/4)*x**3)/(x**12 
 + x**10 + 3*x**8 + 2*x**6 + 3*x**4 + x**2 + 1),x) - 4*sqrt(x**4 + 1)*int( 
(sqrt(x)*(x**4 + 1)**(3/4)*x)/(x**12 + x**10 + 3*x**8 + 2*x**6 + 3*x**4 + 
x**2 + 1),x)*x**4 - 4*sqrt(x**4 + 1)*int((sqrt(x)*(x**4 + 1)**(3/4)*x)/(x* 
*12 + x**10 + 3*x**8 + 2*x**6 + 3*x**4 + x**2 + 1),x))/(3*sqrt(x**4 + 1)*( 
x**4 + 1))