\(\int \frac {4+x^2-2 x^4}{(1+x^2)^2 \sqrt {1+x^2+x^4}} \, dx\) [136]

Optimal result

Integrand size = 30, antiderivative size = 67 \[ \int \frac {4+x^2-2 x^4}{\left (1+x^2\right )^2 \sqrt {1+x^2+x^4}} \, dx=3 \arctan \left (\frac {x}{\sqrt {1+x^2+x^4}}\right )+\frac {\left (1+x^2\right ) \sqrt {\frac {1+x^2+x^4}{\left (1+x^2\right )^2}} E\left (2 \arctan (x)\left |\frac {1}{4}\right .\right )}{2 \sqrt {1+x^2+x^4}} \] Output:

3*arctan(x/(x^4+x^2+1)^(1/2))+1/2*(x^2+1)*((x^4+x^2+1)/(x^2+1)^2)^(1/2)*El 
lipticE(sin(2*arctan(x)),1/2)/(x^4+x^2+1)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 10.60 (sec) , antiderivative size = 226, normalized size of antiderivative = 3.37 \[ \int \frac {4+x^2-2 x^4}{\left (1+x^2\right )^2 \sqrt {1+x^2+x^4}} \, dx=\frac {\frac {x+x^3+x^5}{1+x^2}-5 (-1)^{2/3} \sqrt {1+\sqrt [3]{-1} x^2} \sqrt {1-(-1)^{2/3} x^2} \operatorname {EllipticF}\left (i \text {arcsinh}\left ((-1)^{5/6} x\right ),(-1)^{2/3}\right )+\sqrt [3]{-1} \sqrt {1+\sqrt [3]{-1} x^2} \sqrt {1-(-1)^{2/3} x^2} \left (-E\left (i \text {arcsinh}\left ((-1)^{5/6} x\right )|(-1)^{2/3}\right )+\operatorname {EllipticF}\left (i \text {arcsinh}\left ((-1)^{5/6} x\right ),(-1)^{2/3}\right )\right )+12 (-1)^{2/3} \sqrt {1+\sqrt [3]{-1} x^2} \sqrt {1-(-1)^{2/3} x^2} \operatorname {EllipticPi}\left (\sqrt [3]{-1},i \text {arcsinh}\left ((-1)^{5/6} x\right ),(-1)^{2/3}\right )}{2 \sqrt {1+x^2+x^4}} \] Input:

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

Output:

((x + x^3 + x^5)/(1 + x^2) - 5*(-1)^(2/3)*Sqrt[1 + (-1)^(1/3)*x^2]*Sqrt[1 
- (-1)^(2/3)*x^2]*EllipticF[I*ArcSinh[(-1)^(5/6)*x], (-1)^(2/3)] + (-1)^(1 
/3)*Sqrt[1 + (-1)^(1/3)*x^2]*Sqrt[1 - (-1)^(2/3)*x^2]*(-EllipticE[I*ArcSin 
h[(-1)^(5/6)*x], (-1)^(2/3)] + EllipticF[I*ArcSinh[(-1)^(5/6)*x], (-1)^(2/ 
3)]) + 12*(-1)^(2/3)*Sqrt[1 + (-1)^(1/3)*x^2]*Sqrt[1 - (-1)^(2/3)*x^2]*Ell 
ipticPi[(-1)^(1/3), I*ArcSinh[(-1)^(5/6)*x], (-1)^(2/3)])/(2*Sqrt[1 + x^2 
+ x^4])
 

Rubi [A] (verified)

Time = 0.44 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.72, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.233, Rules used = {2210, 25, 2230, 27, 1509, 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.

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

\(\Big \downarrow \) 2210

\(\displaystyle \frac {x \sqrt {x^4+x^2+1}}{2 \left (x^2+1\right )}-\frac {1}{2} \int -\frac {-x^4-6 x^2+7}{\left (x^2+1\right ) \sqrt {x^4+x^2+1}}dx\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {1}{2} \int \frac {-x^4-6 x^2+7}{\left (x^2+1\right ) \sqrt {x^4+x^2+1}}dx+\frac {\sqrt {x^4+x^2+1} x}{2 \left (x^2+1\right )}\)

\(\Big \downarrow \) 2230

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1509

\(\displaystyle \frac {1}{2} \left (6 \int \frac {1-x^2}{\left (x^2+1\right ) \sqrt {x^4+x^2+1}}dx+\frac {\left (x^2+1\right ) \sqrt {\frac {x^4+x^2+1}{\left (x^2+1\right )^2}} E\left (2 \arctan (x)\left |\frac {1}{4}\right .\right )}{\sqrt {x^4+x^2+1}}-\frac {\sqrt {x^4+x^2+1} x}{x^2+1}\right )+\frac {\sqrt {x^4+x^2+1} x}{2 \left (x^2+1\right )}\)

\(\Big \downarrow \) 2212

\(\displaystyle \frac {1}{2} \left (6 \int \frac {1}{\frac {x^2}{x^4+x^2+1}+1}d\frac {x}{\sqrt {x^4+x^2+1}}+\frac {\left (x^2+1\right ) \sqrt {\frac {x^4+x^2+1}{\left (x^2+1\right )^2}} E\left (2 \arctan (x)\left |\frac {1}{4}\right .\right )}{\sqrt {x^4+x^2+1}}-\frac {\sqrt {x^4+x^2+1} x}{x^2+1}\right )+\frac {\sqrt {x^4+x^2+1} x}{2 \left (x^2+1\right )}\)

\(\Big \downarrow \) 216

\(\displaystyle \frac {1}{2} \left (6 \arctan \left (\frac {x}{\sqrt {x^4+x^2+1}}\right )+\frac {\left (x^2+1\right ) \sqrt {\frac {x^4+x^2+1}{\left (x^2+1\right )^2}} E\left (2 \arctan (x)\left |\frac {1}{4}\right .\right )}{\sqrt {x^4+x^2+1}}-\frac {\sqrt {x^4+x^2+1} x}{x^2+1}\right )+\frac {\sqrt {x^4+x^2+1} x}{2 \left (x^2+1\right )}\)

Input:

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

Output:

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

Defintions of rubi rules used

rule 25

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

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

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 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 2210

Int[((P4x_)*((d_) + (e_.)*(x_)^2)^(q_))/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x 
_)^4], x_Symbol] :> With[{A = Coeff[P4x, x, 0], B = Coeff[P4x, x, 2], C = C 
oeff[P4x, x, 4]}, Simp[(-(C*d^2 - B*d*e + A*e^2))*x*(d + e*x^2)^(q + 1)*(Sq 
rt[a + b*x^2 + c*x^4]/(2*d*(q + 1)*(c*d^2 - b*d*e + a*e^2))), x] + Simp[1/( 
2*d*(q + 1)*(c*d^2 - b*d*e + a*e^2))   Int[((d + e*x^2)^(q + 1)/Sqrt[a + b* 
x^2 + c*x^4])*Simp[a*d*(C*d - B*e) + A*(a*e^2*(2*q + 3) + 2*d*(c*d - b*e)*( 
q + 1)) - 2*((B*d - A*e)*(b*e*(q + 2) - c*d*(q + 1)) - C*d*(b*d + a*e*(q + 
1)))*x^2 + c*(C*d^2 - B*d*e + A*e^2)*(2*q + 5)*x^4, x], x], x]] /; FreeQ[{a 
, b, c, d, e}, x] && PolyQ[P4x, x^2] && LeQ[Expon[P4x, x], 4] && ILtQ[q, -1 
]
 

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 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 = 1.51 (sec) , antiderivative size = 329, normalized size of antiderivative = 4.91

method	result	size
risch	\(\frac {x \sqrt {x^{4}+x^{2}+1}}{2 x^{2}+2}+\frac {2 \sqrt {1-\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) x^{2}}\, \sqrt {1-\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) x^{2}}\, \left (\operatorname {EllipticF}\left (\frac {x \sqrt {-2+2 i \sqrt {3}}}{2}, \frac {\sqrt {-2+2 i \sqrt {3}}}{2}\right )-\operatorname {EllipticE}\left (\frac {x \sqrt {-2+2 i \sqrt {3}}}{2}, \frac {\sqrt {-2+2 i \sqrt {3}}}{2}\right )\right )}{\sqrt {-2+2 i \sqrt {3}}\, \sqrt {x^{4}+x^{2}+1}\, \left (1+i \sqrt {3}\right )}-\frac {5 \sqrt {1-\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) x^{2}}\, \sqrt {1-\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) x^{2}}\, \operatorname {EllipticF}\left (\frac {x \sqrt {-2+2 i \sqrt {3}}}{2}, \frac {\sqrt {-2+2 i \sqrt {3}}}{2}\right )}{\sqrt {-2+2 i \sqrt {3}}\, \sqrt {x^{4}+x^{2}+1}}+\frac {6 \sqrt {1+\frac {x^{2}}{2}-\frac {i x^{2} \sqrt {3}}{2}}\, \sqrt {1+\frac {x^{2}}{2}+\frac {i x^{2} \sqrt {3}}{2}}\, \operatorname {EllipticPi}\left (\sqrt {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}\, x , -\frac {1}{-\frac {1}{2}+\frac {i \sqrt {3}}{2}}, \frac {\sqrt {-\frac {1}{2}-\frac {i \sqrt {3}}{2}}}{\sqrt {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}\, \sqrt {x^{4}+x^{2}+1}}\)	\(329\)
elliptic	\(\frac {x \sqrt {x^{4}+x^{2}+1}}{2 x^{2}+2}-\frac {5 \sqrt {1+\frac {x^{2}}{2}-\frac {i x^{2} \sqrt {3}}{2}}\, \sqrt {1+\frac {x^{2}}{2}+\frac {i x^{2} \sqrt {3}}{2}}\, \operatorname {EllipticF}\left (\frac {x \sqrt {-2+2 i \sqrt {3}}}{2}, \frac {\sqrt {-2+2 i \sqrt {3}}}{2}\right )}{\sqrt {-2+2 i \sqrt {3}}\, \sqrt {x^{4}+x^{2}+1}}+\frac {2 \sqrt {1+\frac {x^{2}}{2}-\frac {i x^{2} \sqrt {3}}{2}}\, \sqrt {1+\frac {x^{2}}{2}+\frac {i x^{2} \sqrt {3}}{2}}\, \operatorname {EllipticF}\left (\frac {x \sqrt {-2+2 i \sqrt {3}}}{2}, \frac {\sqrt {-2+2 i \sqrt {3}}}{2}\right )}{\sqrt {-2+2 i \sqrt {3}}\, \sqrt {x^{4}+x^{2}+1}\, \left (1+i \sqrt {3}\right )}-\frac {2 \sqrt {1+\frac {x^{2}}{2}-\frac {i x^{2} \sqrt {3}}{2}}\, \sqrt {1+\frac {x^{2}}{2}+\frac {i x^{2} \sqrt {3}}{2}}\, \operatorname {EllipticE}\left (\frac {x \sqrt {-2+2 i \sqrt {3}}}{2}, \frac {\sqrt {-2+2 i \sqrt {3}}}{2}\right )}{\sqrt {-2+2 i \sqrt {3}}\, \sqrt {x^{4}+x^{2}+1}\, \left (1+i \sqrt {3}\right )}+\frac {6 \sqrt {1+\frac {x^{2}}{2}-\frac {i x^{2} \sqrt {3}}{2}}\, \sqrt {1+\frac {x^{2}}{2}+\frac {i x^{2} \sqrt {3}}{2}}\, \operatorname {EllipticPi}\left (\sqrt {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}\, x , -\frac {1}{-\frac {1}{2}+\frac {i \sqrt {3}}{2}}, \frac {\sqrt {-\frac {1}{2}-\frac {i \sqrt {3}}{2}}}{\sqrt {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}\, \sqrt {x^{4}+x^{2}+1}}\)	\(398\)
default	\(-\frac {4 \sqrt {1-\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) x^{2}}\, \sqrt {1-\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) x^{2}}\, \operatorname {EllipticF}\left (\frac {x \sqrt {-2+2 i \sqrt {3}}}{2}, \frac {\sqrt {-2+2 i \sqrt {3}}}{2}\right )}{\sqrt {-2+2 i \sqrt {3}}\, \sqrt {x^{4}+x^{2}+1}}+\frac {x \sqrt {x^{4}+x^{2}+1}}{2 x^{2}+2}-\frac {\sqrt {1+\frac {x^{2}}{2}-\frac {i x^{2} \sqrt {3}}{2}}\, \sqrt {1+\frac {x^{2}}{2}+\frac {i x^{2} \sqrt {3}}{2}}\, \operatorname {EllipticF}\left (\frac {x \sqrt {-2+2 i \sqrt {3}}}{2}, \frac {\sqrt {-2+2 i \sqrt {3}}}{2}\right )}{\sqrt {-2+2 i \sqrt {3}}\, \sqrt {x^{4}+x^{2}+1}}+\frac {2 \sqrt {1+\frac {x^{2}}{2}-\frac {i x^{2} \sqrt {3}}{2}}\, \sqrt {1+\frac {x^{2}}{2}+\frac {i x^{2} \sqrt {3}}{2}}\, \operatorname {EllipticF}\left (\frac {x \sqrt {-2+2 i \sqrt {3}}}{2}, \frac {\sqrt {-2+2 i \sqrt {3}}}{2}\right )}{\sqrt {-2+2 i \sqrt {3}}\, \sqrt {x^{4}+x^{2}+1}\, \left (1+i \sqrt {3}\right )}-\frac {2 \sqrt {1+\frac {x^{2}}{2}-\frac {i x^{2} \sqrt {3}}{2}}\, \sqrt {1+\frac {x^{2}}{2}+\frac {i x^{2} \sqrt {3}}{2}}\, \operatorname {EllipticE}\left (\frac {x \sqrt {-2+2 i \sqrt {3}}}{2}, \frac {\sqrt {-2+2 i \sqrt {3}}}{2}\right )}{\sqrt {-2+2 i \sqrt {3}}\, \sqrt {x^{4}+x^{2}+1}\, \left (1+i \sqrt {3}\right )}+\frac {6 \sqrt {1+\frac {x^{2}}{2}-\frac {i x^{2} \sqrt {3}}{2}}\, \sqrt {1+\frac {x^{2}}{2}+\frac {i x^{2} \sqrt {3}}{2}}\, \operatorname {EllipticPi}\left (\sqrt {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}\, x , -\frac {1}{-\frac {1}{2}+\frac {i \sqrt {3}}{2}}, \frac {\sqrt {-\frac {1}{2}-\frac {i \sqrt {3}}{2}}}{\sqrt {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}\, \sqrt {x^{4}+x^{2}+1}}\)	\(480\)

Input:

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

Output:

1/2*x*(x^4+x^2+1)^(1/2)/(x^2+1)+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)/(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)))-5 
/(-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))+6/(-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))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 127 vs. \(2 (58) = 116\).

Time = 0.10 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.90 \[ \int \frac {4+x^2-2 x^4}{\left (1+x^2\right )^2 \sqrt {1+x^2+x^4}} \, dx=-\frac {2 \, \sqrt {-3} {\left (x^{2} + 1\right )} \sqrt {\frac {1}{2} \, \sqrt {-3} - \frac {1}{2}} F(\arcsin \left (x \sqrt {\frac {1}{2} \, \sqrt {-3} - \frac {1}{2}}\right )\,|\,\frac {1}{2} \, \sqrt {-3} - \frac {1}{2}) + {\left (x^{2} - \sqrt {-3} {\left (x^{2} + 1\right )} + 1\right )} \sqrt {\frac {1}{2} \, \sqrt {-3} - \frac {1}{2}} E(\arcsin \left (x \sqrt {\frac {1}{2} \, \sqrt {-3} - \frac {1}{2}}\right )\,|\,\frac {1}{2} \, \sqrt {-3} - \frac {1}{2}) - 12 \, {\left (x^{2} + 1\right )} \arctan \left (\frac {x}{\sqrt {x^{4} + x^{2} + 1}}\right ) - 2 \, \sqrt {x^{4} + x^{2} + 1} x}{4 \, {\left (x^{2} + 1\right )}} \] Input:

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

Output:

-1/4*(2*sqrt(-3)*(x^2 + 1)*sqrt(1/2*sqrt(-3) - 1/2)*elliptic_f(arcsin(x*sq 
rt(1/2*sqrt(-3) - 1/2)), 1/2*sqrt(-3) - 1/2) + (x^2 - sqrt(-3)*(x^2 + 1) + 
 1)*sqrt(1/2*sqrt(-3) - 1/2)*elliptic_e(arcsin(x*sqrt(1/2*sqrt(-3) - 1/2)) 
, 1/2*sqrt(-3) - 1/2) - 12*(x^2 + 1)*arctan(x/sqrt(x^4 + x^2 + 1)) - 2*sqr 
t(x^4 + x^2 + 1)*x)/(x^2 + 1)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

4*int(sqrt(x**4 + x**2 + 1)/(x**8 + 3*x**6 + 4*x**4 + 3*x**2 + 1),x) - 2*i 
nt((sqrt(x**4 + x**2 + 1)*x**4)/(x**8 + 3*x**6 + 4*x**4 + 3*x**2 + 1),x) + 
 int((sqrt(x**4 + x**2 + 1)*x**2)/(x**8 + 3*x**6 + 4*x**4 + 3*x**2 + 1),x)