\(\int \frac {x^2 (2+3 x^2)}{(1+x^2) \sqrt {1+x^2+x^4}} \, dx\) [100]

Optimal result

Integrand size = 30, antiderivative size = 138 \[ \int \frac {x^2 \left (2+3 x^2\right )}{\left (1+x^2\right ) \sqrt {1+x^2+x^4}} \, dx=\frac {3 x \sqrt {1+x^2+x^4}}{1+x^2}+\frac {1}{2} \arctan \left (\frac {x}{\sqrt {1+x^2+x^4}}\right )-\frac {3 \left (1+x^2\right ) \sqrt {\frac {1+x^2+x^4}{\left (1+x^2\right )^2}} E\left (2 \arctan (x)\left |\frac {1}{4}\right .\right )}{\sqrt {1+x^2+x^4}}+\frac {5 \left (1+x^2\right ) \sqrt {\frac {1+x^2+x^4}{\left (1+x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan (x),\frac {1}{4}\right )}{4 \sqrt {1+x^2+x^4}} \] Output:

3*x*(x^4+x^2+1)^(1/2)/(x^2+1)+1/2*arctan(x/(x^4+x^2+1)^(1/2))-3*(x^2+1)*(( 
x^4+x^2+1)/(x^2+1)^2)^(1/2)*EllipticE(sin(2*arctan(x)),1/2)/(x^4+x^2+1)^(1 
/2)+5/4*(x^2+1)*((x^4+x^2+1)/(x^2+1)^2)^(1/2)*InverseJacobiAM(2*arctan(x), 
1/2)/(x^4+x^2+1)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 10.24 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.91 \[ \int \frac {x^2 \left (2+3 x^2\right )}{\left (1+x^2\right ) \sqrt {1+x^2+x^4}} \, dx=\frac {\sqrt [3]{-1} \sqrt {1+\sqrt [3]{-1} x^2} \sqrt {1-(-1)^{2/3} x^2} \left (3 E\left (i \text {arcsinh}\left ((-1)^{5/6} x\right )|(-1)^{2/3}\right )-\left (3+\sqrt [3]{-1}\right ) \operatorname {EllipticF}\left (i \text {arcsinh}\left ((-1)^{5/6} x\right ),(-1)^{2/3}\right )+\sqrt [3]{-1} \operatorname {EllipticPi}\left (\sqrt [3]{-1},i \text {arcsinh}\left ((-1)^{5/6} x\right ),(-1)^{2/3}\right )\right )}{\sqrt {1+x^2+x^4}} \] Input:

Integrate[(x^2*(2 + 3*x^2))/((1 + x^2)*Sqrt[1 + x^2 + x^4]),x]
 

Output:

((-1)^(1/3)*Sqrt[1 + (-1)^(1/3)*x^2]*Sqrt[1 - (-1)^(2/3)*x^2]*(3*EllipticE 
[I*ArcSinh[(-1)^(5/6)*x], (-1)^(2/3)] - (3 + (-1)^(1/3))*EllipticF[I*ArcSi 
nh[(-1)^(5/6)*x], (-1)^(2/3)] + (-1)^(1/3)*EllipticPi[(-1)^(1/3), I*ArcSin 
h[(-1)^(5/6)*x], (-1)^(2/3)]))/Sqrt[1 + x^2 + x^4]
 

Rubi [A] (verified)

Time = 0.49 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.01, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2230, 1509, 2214, 1416, 2212, 216}

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.

\(\displaystyle \int \frac {x^2 \left (3 x^2+2\right )}{\left (x^2+1\right ) \sqrt {x^4+x^2+1}} \, dx\)

\(\Big \downarrow \) 2230

\(\displaystyle \int \frac {2 x^2+3}{\left (x^2+1\right ) \sqrt {x^4+x^2+1}}dx-3 \int \frac {1-x^2}{\sqrt {x^4+x^2+1}}dx\)

\(\Big \downarrow \) 1509

\(\displaystyle \int \frac {2 x^2+3}{\left (x^2+1\right ) \sqrt {x^4+x^2+1}}dx-3 \left (\frac {\left (x^2+1\right ) \sqrt {\frac {x^4+x^2+1}{\left (x^2+1\right )^2}} E\left (2 \arctan (x)\left |\frac {1}{4}\right .\right )}{\sqrt {x^4+x^2+1}}-\frac {x \sqrt {x^4+x^2+1}}{x^2+1}\right )\)

\(\Big \downarrow \) 2214

\(\displaystyle \frac {5}{2} \int \frac {1}{\sqrt {x^4+x^2+1}}dx+\frac {1}{2} \int \frac {1-x^2}{\left (x^2+1\right ) \sqrt {x^4+x^2+1}}dx-3 \left (\frac {\left (x^2+1\right ) \sqrt {\frac {x^4+x^2+1}{\left (x^2+1\right )^2}} E\left (2 \arctan (x)\left |\frac {1}{4}\right .\right )}{\sqrt {x^4+x^2+1}}-\frac {x \sqrt {x^4+x^2+1}}{x^2+1}\right )\)

\(\Big \downarrow \) 1416

\(\displaystyle \frac {1}{2} \int \frac {1-x^2}{\left (x^2+1\right ) \sqrt {x^4+x^2+1}}dx+\frac {5 \left (x^2+1\right ) \sqrt {\frac {x^4+x^2+1}{\left (x^2+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan (x),\frac {1}{4}\right )}{4 \sqrt {x^4+x^2+1}}-3 \left (\frac {\left (x^2+1\right ) \sqrt {\frac {x^4+x^2+1}{\left (x^2+1\right )^2}} E\left (2 \arctan (x)\left |\frac {1}{4}\right .\right )}{\sqrt {x^4+x^2+1}}-\frac {x \sqrt {x^4+x^2+1}}{x^2+1}\right )\)

\(\Big \downarrow \) 2212

\(\displaystyle \frac {1}{2} \int \frac {1}{\frac {x^2}{x^4+x^2+1}+1}d\frac {x}{\sqrt {x^4+x^2+1}}+\frac {5 \left (x^2+1\right ) \sqrt {\frac {x^4+x^2+1}{\left (x^2+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan (x),\frac {1}{4}\right )}{4 \sqrt {x^4+x^2+1}}-3 \left (\frac {\left (x^2+1\right ) \sqrt {\frac {x^4+x^2+1}{\left (x^2+1\right )^2}} E\left (2 \arctan (x)\left |\frac {1}{4}\right .\right )}{\sqrt {x^4+x^2+1}}-\frac {x \sqrt {x^4+x^2+1}}{x^2+1}\right )\)

\(\Big \downarrow \) 216

\(\displaystyle \frac {1}{2} \arctan \left (\frac {x}{\sqrt {x^4+x^2+1}}\right )+\frac {5 \left (x^2+1\right ) \sqrt {\frac {x^4+x^2+1}{\left (x^2+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan (x),\frac {1}{4}\right )}{4 \sqrt {x^4+x^2+1}}-3 \left (\frac {\left (x^2+1\right ) \sqrt {\frac {x^4+x^2+1}{\left (x^2+1\right )^2}} E\left (2 \arctan (x)\left |\frac {1}{4}\right .\right )}{\sqrt {x^4+x^2+1}}-\frac {x \sqrt {x^4+x^2+1}}{x^2+1}\right )\)

Input:

Int[(x^2*(2 + 3*x^2))/((1 + x^2)*Sqrt[1 + x^2 + x^4]),x]
 

Output:

ArcTan[x/Sqrt[1 + x^2 + x^4]]/2 - 3*(-((x*Sqrt[1 + x^2 + x^4])/(1 + x^2)) 
+ ((1 + x^2)*Sqrt[(1 + x^2 + x^4)/(1 + x^2)^2]*EllipticE[2*ArcTan[x], 1/4] 
)/Sqrt[1 + x^2 + x^4]) + (5*(1 + x^2)*Sqrt[(1 + x^2 + x^4)/(1 + x^2)^2]*El 
lipticF[2*ArcTan[x], 1/4])/(4*Sqrt[1 + x^2 + x^4])
 

Defintions of rubi rules used

rule 216

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A 
rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a 
, 0] || GtQ[b, 0])
 

rule 1416

Int[1/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c 
/a, 4]}, Simp[(1 + q^2*x^2)*(Sqrt[(a + b*x^2 + c*x^4)/(a*(1 + q^2*x^2)^2)]/ 
(2*q*Sqrt[a + b*x^2 + c*x^4]))*EllipticF[2*ArcTan[q*x], 1/2 - b*(q^2/(4*c)) 
], x]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]
 

rule 1509

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbo 
l] :> With[{q = Rt[c/a, 4]}, Simp[(-d)*x*(Sqrt[a + b*x^2 + c*x^4]/(a*(1 + q 
^2*x^2))), x] + Simp[d*(1 + q^2*x^2)*(Sqrt[(a + b*x^2 + c*x^4)/(a*(1 + q^2* 
x^2)^2)]/(q*Sqrt[a + b*x^2 + c*x^4]))*EllipticE[2*ArcTan[q*x], 1/2 - b*(q^2 
/(4*c))], x] /; EqQ[e + d*q^2, 0]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 
- 4*a*c, 0] && PosQ[c/a]
 

rule 2212

Int[((A_) + (B_.)*(x_)^2)/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (b_.)*(x_)^2 + 
 (c_.)*(x_)^4]), x_Symbol] :> Simp[A   Subst[Int[1/(d - (b*d - 2*a*e)*x^2), 
 x], x, x/Sqrt[a + b*x^2 + c*x^4]], x] /; FreeQ[{a, b, c, d, e, A, B}, x] & 
& EqQ[c*d^2 - a*e^2, 0] && EqQ[B*d + A*e, 0]
 

rule 2214

Int[((A_) + (B_.)*(x_)^2)/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (b_.)*(x_)^2 + 
 (c_.)*(x_)^4]), x_Symbol] :> Simp[(B*d + A*e)/(2*d*e)   Int[1/Sqrt[a + b*x 
^2 + c*x^4], x], x] - Simp[(B*d - A*e)/(2*d*e)   Int[(d - e*x^2)/((d + e*x^ 
2)*Sqrt[a + b*x^2 + c*x^4]), x], x] /; FreeQ[{a, b, c, d, e, A, B}, x] && E 
qQ[c*d^2 - a*e^2, 0] && NeQ[B*d + A*e, 0]
 

rule 2230

Int[(P4x_)/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4]) 
, x_Symbol] :> With[{A = Coeff[P4x, x, 0], B = Coeff[P4x, x, 2], C = Coeff[ 
P4x, x, 4]}, Simp[-C/e^2   Int[(d - e*x^2)/Sqrt[a + b*x^2 + c*x^4], x], x] 
+ Simp[1/e^2   Int[(C*d^2 + A*e^2 + B*e^2*x^2)/((d + e*x^2)*Sqrt[a + b*x^2 
+ c*x^4]), x], x]] /; FreeQ[{a, b, c, d, e}, x] && PolyQ[P4x, x^2, 2] && Eq 
Q[c*d^2 - a*e^2, 0]
 

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 0.57 (sec) , antiderivative size = 308, normalized size of antiderivative = 2.23

method	result	size
default	\(\frac {\sqrt {1+\frac {x^{2}}{2}-\frac {i x^{2} \sqrt {3}}{2}}\, \sqrt {1+\frac {x^{2}}{2}+\frac {i x^{2} \sqrt {3}}{2}}\, \operatorname {EllipticPi}\left (\sqrt {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}\, x , -\frac {1}{-\frac {1}{2}+\frac {i \sqrt {3}}{2}}, \frac {\sqrt {-\frac {1}{2}-\frac {i \sqrt {3}}{2}}}{\sqrt {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}\, \sqrt {x^{4}+x^{2}+1}}-\frac {2 \sqrt {1-\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) x^{2}}\, \sqrt {1-\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) x^{2}}\, \operatorname {EllipticF}\left (\frac {x \sqrt {-2+2 i \sqrt {3}}}{2}, \frac {\sqrt {-2+2 i \sqrt {3}}}{2}\right )}{\sqrt {-2+2 i \sqrt {3}}\, \sqrt {x^{4}+x^{2}+1}}-\frac {12 \sqrt {1-\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) x^{2}}\, \sqrt {1-\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) x^{2}}\, \left (\operatorname {EllipticF}\left (\frac {x \sqrt {-2+2 i \sqrt {3}}}{2}, \frac {\sqrt {-2+2 i \sqrt {3}}}{2}\right )-\operatorname {EllipticE}\left (\frac {x \sqrt {-2+2 i \sqrt {3}}}{2}, \frac {\sqrt {-2+2 i \sqrt {3}}}{2}\right )\right )}{\sqrt {-2+2 i \sqrt {3}}\, \sqrt {x^{4}+x^{2}+1}\, \left (1+i \sqrt {3}\right )}\)	\(308\)
elliptic	\(-\frac {2 \sqrt {1+\frac {x^{2}}{2}-\frac {i x^{2} \sqrt {3}}{2}}\, \sqrt {1+\frac {x^{2}}{2}+\frac {i x^{2} \sqrt {3}}{2}}\, \operatorname {EllipticF}\left (\frac {x \sqrt {-2+2 i \sqrt {3}}}{2}, \frac {\sqrt {-2+2 i \sqrt {3}}}{2}\right )}{\sqrt {-2+2 i \sqrt {3}}\, \sqrt {x^{4}+x^{2}+1}}-\frac {12 \sqrt {1+\frac {x^{2}}{2}-\frac {i x^{2} \sqrt {3}}{2}}\, \sqrt {1+\frac {x^{2}}{2}+\frac {i x^{2} \sqrt {3}}{2}}\, \operatorname {EllipticF}\left (\frac {x \sqrt {-2+2 i \sqrt {3}}}{2}, \frac {\sqrt {-2+2 i \sqrt {3}}}{2}\right )}{\sqrt {-2+2 i \sqrt {3}}\, \sqrt {x^{4}+x^{2}+1}\, \left (1+i \sqrt {3}\right )}+\frac {12 \sqrt {1+\frac {x^{2}}{2}-\frac {i x^{2} \sqrt {3}}{2}}\, \sqrt {1+\frac {x^{2}}{2}+\frac {i x^{2} \sqrt {3}}{2}}\, \operatorname {EllipticE}\left (\frac {x \sqrt {-2+2 i \sqrt {3}}}{2}, \frac {\sqrt {-2+2 i \sqrt {3}}}{2}\right )}{\sqrt {-2+2 i \sqrt {3}}\, \sqrt {x^{4}+x^{2}+1}\, \left (1+i \sqrt {3}\right )}+\frac {\sqrt {1+\frac {x^{2}}{2}-\frac {i x^{2} \sqrt {3}}{2}}\, \sqrt {1+\frac {x^{2}}{2}+\frac {i x^{2} \sqrt {3}}{2}}\, \operatorname {EllipticPi}\left (\sqrt {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}\, x , -\frac {1}{-\frac {1}{2}+\frac {i \sqrt {3}}{2}}, \frac {\sqrt {-\frac {1}{2}-\frac {i \sqrt {3}}{2}}}{\sqrt {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}\, \sqrt {x^{4}+x^{2}+1}}\)	\(377\)

Input:

int(x^2*(3*x^2+2)/(x^2+1)/(x^4+x^2+1)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

1/(-1/2+1/2*I*3^(1/2))^(1/2)*(1+1/2*x^2-1/2*I*x^2*3^(1/2))^(1/2)*(1+1/2*x^ 
2+1/2*I*x^2*3^(1/2))^(1/2)/(x^4+x^2+1)^(1/2)*EllipticPi((-1/2+1/2*I*3^(1/2 
))^(1/2)*x,-1/(-1/2+1/2*I*3^(1/2)),(-1/2-1/2*I*3^(1/2))^(1/2)/(-1/2+1/2*I* 
3^(1/2))^(1/2))-2/(-2+2*I*3^(1/2))^(1/2)*(1-(-1/2+1/2*I*3^(1/2))*x^2)^(1/2 
)*(1-(-1/2-1/2*I*3^(1/2))*x^2)^(1/2)/(x^4+x^2+1)^(1/2)*EllipticF(1/2*x*(-2 
+2*I*3^(1/2))^(1/2),1/2*(-2+2*I*3^(1/2))^(1/2))-12/(-2+2*I*3^(1/2))^(1/2)* 
(1-(-1/2+1/2*I*3^(1/2))*x^2)^(1/2)*(1-(-1/2-1/2*I*3^(1/2))*x^2)^(1/2)/(x^4 
+x^2+1)^(1/2)/(1+I*3^(1/2))*(EllipticF(1/2*x*(-2+2*I*3^(1/2))^(1/2),1/2*(- 
2+2*I*3^(1/2))^(1/2))-EllipticE(1/2*x*(-2+2*I*3^(1/2))^(1/2),1/2*(-2+2*I*3 
^(1/2))^(1/2)))
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.86 \[ \int \frac {x^2 \left (2+3 x^2\right )}{\left (1+x^2\right ) \sqrt {1+x^2+x^4}} \, dx=\frac {6 \, {\left (\sqrt {-3} x - x\right )} \sqrt {\frac {1}{2} \, \sqrt {-3} - \frac {1}{2}} E(\arcsin \left (\frac {\sqrt {\frac {1}{2} \, \sqrt {-3} - \frac {1}{2}}}{x}\right )\,|\,\frac {1}{2} \, \sqrt {-3} - \frac {1}{2}) - {\left (7 \, \sqrt {-3} x - 5 \, x\right )} \sqrt {\frac {1}{2} \, \sqrt {-3} - \frac {1}{2}} F(\arcsin \left (\frac {\sqrt {\frac {1}{2} \, \sqrt {-3} - \frac {1}{2}}}{x}\right )\,|\,\frac {1}{2} \, \sqrt {-3} - \frac {1}{2}) + 2 \, x \arctan \left (\frac {x}{\sqrt {x^{4} + x^{2} + 1}}\right ) + 12 \, \sqrt {x^{4} + x^{2} + 1}}{4 \, x} \] Input:

integrate(x^2*(3*x^2+2)/(x^2+1)/(x^4+x^2+1)^(1/2),x, algorithm="fricas")
 

Output:

1/4*(6*(sqrt(-3)*x - x)*sqrt(1/2*sqrt(-3) - 1/2)*elliptic_e(arcsin(sqrt(1/ 
2*sqrt(-3) - 1/2)/x), 1/2*sqrt(-3) - 1/2) - (7*sqrt(-3)*x - 5*x)*sqrt(1/2* 
sqrt(-3) - 1/2)*elliptic_f(arcsin(sqrt(1/2*sqrt(-3) - 1/2)/x), 1/2*sqrt(-3 
) - 1/2) + 2*x*arctan(x/sqrt(x^4 + x^2 + 1)) + 12*sqrt(x^4 + x^2 + 1))/x
 

Sympy [F]

\[ \int \frac {x^2 \left (2+3 x^2\right )}{\left (1+x^2\right ) \sqrt {1+x^2+x^4}} \, dx=\int \frac {x^{2} \cdot \left (3 x^{2} + 2\right )}{\sqrt {\left (x^{2} - x + 1\right ) \left (x^{2} + x + 1\right )} \left (x^{2} + 1\right )}\, dx \] Input:

integrate(x**2*(3*x**2+2)/(x**2+1)/(x**4+x**2+1)**(1/2),x)
 

Output:

Integral(x**2*(3*x**2 + 2)/(sqrt((x**2 - x + 1)*(x**2 + x + 1))*(x**2 + 1) 
), x)
 

Maxima [F]

\[ \int \frac {x^2 \left (2+3 x^2\right )}{\left (1+x^2\right ) \sqrt {1+x^2+x^4}} \, dx=\int { \frac {{\left (3 \, x^{2} + 2\right )} x^{2}}{\sqrt {x^{4} + x^{2} + 1} {\left (x^{2} + 1\right )}} \,d x } \] Input:

integrate(x^2*(3*x^2+2)/(x^2+1)/(x^4+x^2+1)^(1/2),x, algorithm="maxima")
 

Output:

integrate((3*x^2 + 2)*x^2/(sqrt(x^4 + x^2 + 1)*(x^2 + 1)), x)
 

Giac [F]

\[ \int \frac {x^2 \left (2+3 x^2\right )}{\left (1+x^2\right ) \sqrt {1+x^2+x^4}} \, dx=\int { \frac {{\left (3 \, x^{2} + 2\right )} x^{2}}{\sqrt {x^{4} + x^{2} + 1} {\left (x^{2} + 1\right )}} \,d x } \] Input:

integrate(x^2*(3*x^2+2)/(x^2+1)/(x^4+x^2+1)^(1/2),x, algorithm="giac")
 

Output:

integrate((3*x^2 + 2)*x^2/(sqrt(x^4 + x^2 + 1)*(x^2 + 1)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^2 \left (2+3 x^2\right )}{\left (1+x^2\right ) \sqrt {1+x^2+x^4}} \, dx=\int \frac {x^2\,\left (3\,x^2+2\right )}{\left (x^2+1\right )\,\sqrt {x^4+x^2+1}} \,d x \] Input:

int((x^2*(3*x^2 + 2))/((x^2 + 1)*(x^2 + x^4 + 1)^(1/2)),x)
 

Output:

int((x^2*(3*x^2 + 2))/((x^2 + 1)*(x^2 + x^4 + 1)^(1/2)), x)
                                                                                    
                                                                                    
 

Reduce [F]

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

int(x^2*(3*x^2+2)/(x^2+1)/(x^4+x^2+1)^(1/2),x)
 

Output:

3*int((sqrt(x**4 + x**2 + 1)*x**4)/(x**6 + 2*x**4 + 2*x**2 + 1),x) + 2*int 
((sqrt(x**4 + x**2 + 1)*x**2)/(x**6 + 2*x**4 + 2*x**2 + 1),x)