3.3.31 \(\int \frac {\sqrt {1+x^2+x^4}}{(1+x^2)^4} \, dx\) [231]

3.3.31.1 Optimal result

Integrand size = 20, antiderivative size = 166 \[ \int \frac {\sqrt {1+x^2+x^4}}{\left (1+x^2\right )^4} \, dx=\frac {x \sqrt {1+x^2+x^4}}{6 \left (1+x^2\right )^3}+\frac {x \sqrt {1+x^2+x^4}}{6 \left (1+x^2\right )^2}+\frac {1}{4} \arctan \left (\frac {x}{\sqrt {1+x^2+x^4}}\right )+\frac {\left (1+x^2\right ) \sqrt {\frac {1+x^2+x^4}{\left (1+x^2\right )^2}} E\left (2 \arctan (x)\left |\frac {1}{4}\right .\right )}{3 \sqrt {1+x^2+x^4}}-\frac {\left (1+x^2\right ) \sqrt {\frac {1+x^2+x^4}{\left (1+x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan (x),\frac {1}{4}\right )}{8 \sqrt {1+x^2+x^4}} \]

output

1/4*arctan(x/(x^4+x^2+1)^(1/2))+1/6*x*(x^4+x^2+1)^(1/2)/(x^2+1)^3+1/6*x*(x 
^4+x^2+1)^(1/2)/(x^2+1)^2+1/3*(x^2+1)*(cos(2*arctan(x))^2)^(1/2)/cos(2*arc 
tan(x))*EllipticE(sin(2*arctan(x)),1/2)*((x^4+x^2+1)/(x^2+1)^2)^(1/2)/(x^4 
+x^2+1)^(1/2)-1/8*(x^2+1)*(cos(2*arctan(x))^2)^(1/2)/cos(2*arctan(x))*Elli 
pticF(sin(2*arctan(x)),1/2)*((x^4+x^2+1)/(x^2+1)^2)^(1/2)/(x^4+x^2+1)^(1/2 
)
 

3.3.31.2 Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 10.31 (sec) , antiderivative size = 240, normalized size of antiderivative = 1.45 \[ \int \frac {\sqrt {1+x^2+x^4}}{\left (1+x^2\right )^4} \, dx=\frac {\frac {x \left (1+x^2+x^4\right ) \left (4+5 x^2+2 x^4\right )}{\left (1+x^2\right )^3}-2 \sqrt [3]{-1} \sqrt {1+\sqrt [3]{-1} x^2} \sqrt {1-(-1)^{2/3} x^2} \left (E\left (i \text {arcsinh}\left ((-1)^{5/6} x\right )|(-1)^{2/3}\right )-\operatorname {EllipticF}\left (i \text {arcsinh}\left ((-1)^{5/6} x\right ),(-1)^{2/3}\right )\right )-(-1)^{2/3} \sqrt {1+\sqrt [3]{-1} x^2} \sqrt {1-(-1)^{2/3} x^2} \operatorname {EllipticF}\left (i \text {arcsinh}\left ((-1)^{5/6} x\right ),(-1)^{2/3}\right )+3 (-1)^{2/3} \sqrt {1+\sqrt [3]{-1} x^2} \sqrt {1-(-1)^{2/3} x^2} \operatorname {EllipticPi}\left (\sqrt [3]{-1},i \text {arcsinh}\left ((-1)^{5/6} x\right ),(-1)^{2/3}\right )}{6 \sqrt {1+x^2+x^4}} \]

input

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

output

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

3.3.31.3 Rubi [A] (verified)

Time = 0.67 (sec) , antiderivative size = 215, normalized size of antiderivative = 1.30, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.650, Rules used = {1554, 25, 2210, 27, 2210, 27, 2230, 27, 1509, 1654, 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.

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

\(\Big \downarrow \) 1554

\(\displaystyle \frac {x \sqrt {x^4+x^2+1}}{6 \left (x^2+1\right )^3}-\frac {1}{6} \int -\frac {3 x^4+4 x^2+5}{\left (x^2+1\right )^3 \sqrt {x^4+x^2+1}}dx\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {1}{6} \int \frac {3 x^4+4 x^2+5}{\left (x^2+1\right )^3 \sqrt {x^4+x^2+1}}dx+\frac {\sqrt {x^4+x^2+1} x}{6 \left (x^2+1\right )^3}\)

\(\Big \downarrow \) 2210

\(\displaystyle \frac {1}{6} \left (\frac {x \sqrt {x^4+x^2+1}}{\left (x^2+1\right )^2}-\frac {1}{4} \int -\frac {4 \left (x^4+x^2+4\right )}{\left (x^2+1\right )^2 \sqrt {x^4+x^2+1}}dx\right )+\frac {\sqrt {x^4+x^2+1} x}{6 \left (x^2+1\right )^3}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{6} \left (\int \frac {x^4+x^2+4}{\left (x^2+1\right )^2 \sqrt {x^4+x^2+1}}dx+\frac {\sqrt {x^4+x^2+1} x}{\left (x^2+1\right )^2}\right )+\frac {\sqrt {x^4+x^2+1} x}{6 \left (x^2+1\right )^3}\)

\(\Big \downarrow \) 2210

\(\displaystyle \frac {1}{6} \left (-\frac {1}{2} \int -\frac {2 \left (-2 x^4-3 x^2+2\right )}{\left (x^2+1\right ) \sqrt {x^4+x^2+1}}dx+\frac {2 \sqrt {x^4+x^2+1} x}{x^2+1}+\frac {\sqrt {x^4+x^2+1} x}{\left (x^2+1\right )^2}\right )+\frac {\sqrt {x^4+x^2+1} x}{6 \left (x^2+1\right )^3}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{6} \left (\int \frac {-2 x^4-3 x^2+2}{\left (x^2+1\right ) \sqrt {x^4+x^2+1}}dx+\frac {2 \sqrt {x^4+x^2+1} x}{x^2+1}+\frac {\sqrt {x^4+x^2+1} x}{\left (x^2+1\right )^2}\right )+\frac {\sqrt {x^4+x^2+1} x}{6 \left (x^2+1\right )^3}\)

\(\Big \downarrow \) 2230

\(\displaystyle \frac {1}{6} \left (2 \int \frac {1-x^2}{\sqrt {x^4+x^2+1}}dx+\int -\frac {3 x^2}{\left (x^2+1\right ) \sqrt {x^4+x^2+1}}dx+\frac {2 \sqrt {x^4+x^2+1} x}{x^2+1}+\frac {\sqrt {x^4+x^2+1} x}{\left (x^2+1\right )^2}\right )+\frac {\sqrt {x^4+x^2+1} x}{6 \left (x^2+1\right )^3}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{6} \left (2 \int \frac {1-x^2}{\sqrt {x^4+x^2+1}}dx-3 \int \frac {x^2}{\left (x^2+1\right ) \sqrt {x^4+x^2+1}}dx+\frac {2 \sqrt {x^4+x^2+1} x}{x^2+1}+\frac {\sqrt {x^4+x^2+1} x}{\left (x^2+1\right )^2}\right )+\frac {\sqrt {x^4+x^2+1} x}{6 \left (x^2+1\right )^3}\)

\(\Big \downarrow \) 1509

\(\displaystyle \frac {1}{6} \left (-3 \int \frac {x^2}{\left (x^2+1\right ) \sqrt {x^4+x^2+1}}dx+2 \left (\frac {\left (x^2+1\right ) \sqrt {\frac {x^4+x^2+1}{\left (x^2+1\right )^2}} E\left (2 \arctan (x)\left |\frac {1}{4}\right .\right )}{\sqrt {x^4+x^2+1}}-\frac {x \sqrt {x^4+x^2+1}}{x^2+1}\right )+\frac {2 \sqrt {x^4+x^2+1} x}{x^2+1}+\frac {\sqrt {x^4+x^2+1} x}{\left (x^2+1\right )^2}\right )+\frac {\sqrt {x^4+x^2+1} x}{6 \left (x^2+1\right )^3}\)

\(\Big \downarrow \) 1654

\(\displaystyle \frac {1}{6} \left (-3 \left (\frac {1}{2} \int \frac {1}{\sqrt {x^4+x^2+1}}dx-\frac {1}{2} \int \frac {1-x^2}{\left (x^2+1\right ) \sqrt {x^4+x^2+1}}dx\right )+2 \left (\frac {\left (x^2+1\right ) \sqrt {\frac {x^4+x^2+1}{\left (x^2+1\right )^2}} E\left (2 \arctan (x)\left |\frac {1}{4}\right .\right )}{\sqrt {x^4+x^2+1}}-\frac {x \sqrt {x^4+x^2+1}}{x^2+1}\right )+\frac {2 \sqrt {x^4+x^2+1} x}{x^2+1}+\frac {\sqrt {x^4+x^2+1} x}{\left (x^2+1\right )^2}\right )+\frac {\sqrt {x^4+x^2+1} x}{6 \left (x^2+1\right )^3}\)

\(\Big \downarrow \) 1416

\(\displaystyle \frac {1}{6} \left (-3 \left (\frac {\left (x^2+1\right ) \sqrt {\frac {x^4+x^2+1}{\left (x^2+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan (x),\frac {1}{4}\right )}{4 \sqrt {x^4+x^2+1}}-\frac {1}{2} \int \frac {1-x^2}{\left (x^2+1\right ) \sqrt {x^4+x^2+1}}dx\right )+2 \left (\frac {\left (x^2+1\right ) \sqrt {\frac {x^4+x^2+1}{\left (x^2+1\right )^2}} E\left (2 \arctan (x)\left |\frac {1}{4}\right .\right )}{\sqrt {x^4+x^2+1}}-\frac {x \sqrt {x^4+x^2+1}}{x^2+1}\right )+\frac {2 \sqrt {x^4+x^2+1} x}{x^2+1}+\frac {\sqrt {x^4+x^2+1} x}{\left (x^2+1\right )^2}\right )+\frac {\sqrt {x^4+x^2+1} x}{6 \left (x^2+1\right )^3}\)

\(\Big \downarrow \) 2212

\(\displaystyle \frac {1}{6} \left (-3 \left (\frac {\left (x^2+1\right ) \sqrt {\frac {x^4+x^2+1}{\left (x^2+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan (x),\frac {1}{4}\right )}{4 \sqrt {x^4+x^2+1}}-\frac {1}{2} \int \frac {1}{\frac {x^2}{x^4+x^2+1}+1}d\frac {x}{\sqrt {x^4+x^2+1}}\right )+2 \left (\frac {\left (x^2+1\right ) \sqrt {\frac {x^4+x^2+1}{\left (x^2+1\right )^2}} E\left (2 \arctan (x)\left |\frac {1}{4}\right .\right )}{\sqrt {x^4+x^2+1}}-\frac {x \sqrt {x^4+x^2+1}}{x^2+1}\right )+\frac {2 \sqrt {x^4+x^2+1} x}{x^2+1}+\frac {\sqrt {x^4+x^2+1} x}{\left (x^2+1\right )^2}\right )+\frac {\sqrt {x^4+x^2+1} x}{6 \left (x^2+1\right )^3}\)

\(\Big \downarrow \) 216

\(\displaystyle \frac {1}{6} \left (-3 \left (\frac {\left (x^2+1\right ) \sqrt {\frac {x^4+x^2+1}{\left (x^2+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan (x),\frac {1}{4}\right )}{4 \sqrt {x^4+x^2+1}}-\frac {1}{2} \arctan \left (\frac {x}{\sqrt {x^4+x^2+1}}\right )\right )+2 \left (\frac {\left (x^2+1\right ) \sqrt {\frac {x^4+x^2+1}{\left (x^2+1\right )^2}} E\left (2 \arctan (x)\left |\frac {1}{4}\right .\right )}{\sqrt {x^4+x^2+1}}-\frac {x \sqrt {x^4+x^2+1}}{x^2+1}\right )+\frac {2 \sqrt {x^4+x^2+1} x}{x^2+1}+\frac {\sqrt {x^4+x^2+1} x}{\left (x^2+1\right )^2}\right )+\frac {\sqrt {x^4+x^2+1} x}{6 \left (x^2+1\right )^3}\)

input

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

output

(x*Sqrt[1 + x^2 + x^4])/(6*(1 + x^2)^3) + ((x*Sqrt[1 + x^2 + x^4])/(1 + x^ 
2)^2 + (2*x*Sqrt[1 + x^2 + x^4])/(1 + x^2) + 2*(-((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*ArcT 
an[x], 1/4])/Sqrt[1 + x^2 + x^4]) - 3*(-1/2*ArcTan[x/Sqrt[1 + x^2 + x^4]] 
+ ((1 + x^2)*Sqrt[(1 + x^2 + x^4)/(1 + x^2)^2]*EllipticF[2*ArcTan[x], 1/4] 
)/(4*Sqrt[1 + x^2 + x^4])))/6
 

3.3.31.3.1 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 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 1554

Int[((d_) + (e_.)*(x_)^2)^(q_)*Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_ 
Symbol] :> Simp[(-x)*(d + e*x^2)^(q + 1)*(Sqrt[a + b*x^2 + c*x^4]/(2*d*(q + 
 1))), x] + Simp[1/(2*d*(q + 1))   Int[(d + e*x^2)^(q + 1)*((a*(2*q + 3) + 
2*b*(q + 2)*x^2 + c*(2*q + 5)*x^4)/Sqrt[a + b*x^2 + c*x^4]), x], x] /; Free 
Q[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 
0] && ILtQ[q, -1]
 

rule 1654

Int[(x_)^2/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4]) 
, x_Symbol] :> Simp[1/(2*e)   Int[1/Sqrt[a + b*x^2 + c*x^4], x], x] - Simp[ 
1/(2*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}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a] && EqQ[c*d^2 
- a*e^2, 0]
 

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]
 

3.3.31.4 Maple [C] (verified)

Result contains complex when optimal does not.

Time = 0.87 (sec) , antiderivative size = 341, normalized size of antiderivative = 2.05

method	result	size
risch	\(\frac {\sqrt {x^{4}+x^{2}+1}\, x \left (2 x^{4}+5 x^{2}+4\right )}{6 \left (x^{2}+1\right )^{3}}-\frac {\sqrt {1-\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) x^{2}}\, \sqrt {1-\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) x^{2}}\, F\left (\frac {x \sqrt {-2+2 i \sqrt {3}}}{2}, \frac {\sqrt {-2+2 i \sqrt {3}}}{2}\right )}{3 \sqrt {-2+2 i \sqrt {3}}\, \sqrt {x^{4}+x^{2}+1}}+\frac {4 \sqrt {1-\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) x^{2}}\, \sqrt {1-\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) x^{2}}\, \left (F\left (\frac {x \sqrt {-2+2 i \sqrt {3}}}{2}, \frac {\sqrt {-2+2 i \sqrt {3}}}{2}\right )-E\left (\frac {x \sqrt {-2+2 i \sqrt {3}}}{2}, \frac {\sqrt {-2+2 i \sqrt {3}}}{2}\right )\right )}{3 \sqrt {-2+2 i \sqrt {3}}\, \sqrt {x^{4}+x^{2}+1}\, \left (1+i \sqrt {3}\right )}+\frac {\sqrt {1+\frac {x^{2}}{2}-\frac {i x^{2} \sqrt {3}}{2}}\, \sqrt {1+\frac {x^{2}}{2}+\frac {i x^{2} \sqrt {3}}{2}}\, \Pi \left (\sqrt {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}\, x , -\frac {1}{-\frac {1}{2}+\frac {i \sqrt {3}}{2}}, \frac {\sqrt {-\frac {1}{2}-\frac {i \sqrt {3}}{2}}}{\sqrt {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}}\right )}{2 \sqrt {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}\, \sqrt {x^{4}+x^{2}+1}}\)	\(341\)
default	\(\frac {x \sqrt {x^{4}+x^{2}+1}}{6 \left (x^{2}+1\right )^{3}}+\frac {x \sqrt {x^{4}+x^{2}+1}}{6 \left (x^{2}+1\right )^{2}}+\frac {x \sqrt {x^{4}+x^{2}+1}}{3 x^{2}+3}-\frac {\sqrt {1+\frac {x^{2}}{2}-\frac {i x^{2} \sqrt {3}}{2}}\, \sqrt {1+\frac {x^{2}}{2}+\frac {i x^{2} \sqrt {3}}{2}}\, F\left (\frac {x \sqrt {-2+2 i \sqrt {3}}}{2}, \frac {\sqrt {-2+2 i \sqrt {3}}}{2}\right )}{3 \sqrt {-2+2 i \sqrt {3}}\, \sqrt {x^{4}+x^{2}+1}}+\frac {4 \sqrt {1+\frac {x^{2}}{2}-\frac {i x^{2} \sqrt {3}}{2}}\, \sqrt {1+\frac {x^{2}}{2}+\frac {i x^{2} \sqrt {3}}{2}}\, F\left (\frac {x \sqrt {-2+2 i \sqrt {3}}}{2}, \frac {\sqrt {-2+2 i \sqrt {3}}}{2}\right )}{3 \sqrt {-2+2 i \sqrt {3}}\, \sqrt {x^{4}+x^{2}+1}\, \left (1+i \sqrt {3}\right )}-\frac {4 \sqrt {1+\frac {x^{2}}{2}-\frac {i x^{2} \sqrt {3}}{2}}\, \sqrt {1+\frac {x^{2}}{2}+\frac {i x^{2} \sqrt {3}}{2}}\, E\left (\frac {x \sqrt {-2+2 i \sqrt {3}}}{2}, \frac {\sqrt {-2+2 i \sqrt {3}}}{2}\right )}{3 \sqrt {-2+2 i \sqrt {3}}\, \sqrt {x^{4}+x^{2}+1}\, \left (1+i \sqrt {3}\right )}+\frac {\sqrt {1+\frac {x^{2}}{2}-\frac {i x^{2} \sqrt {3}}{2}}\, \sqrt {1+\frac {x^{2}}{2}+\frac {i x^{2} \sqrt {3}}{2}}\, \Pi \left (\sqrt {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}\, x , -\frac {1}{-\frac {1}{2}+\frac {i \sqrt {3}}{2}}, \frac {\sqrt {-\frac {1}{2}-\frac {i \sqrt {3}}{2}}}{\sqrt {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}}\right )}{2 \sqrt {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}\, \sqrt {x^{4}+x^{2}+1}}\)	\(438\)
elliptic	\(\frac {x \sqrt {x^{4}+x^{2}+1}}{6 \left (x^{2}+1\right )^{3}}+\frac {x \sqrt {x^{4}+x^{2}+1}}{6 \left (x^{2}+1\right )^{2}}+\frac {x \sqrt {x^{4}+x^{2}+1}}{3 x^{2}+3}-\frac {\sqrt {1+\frac {x^{2}}{2}-\frac {i x^{2} \sqrt {3}}{2}}\, \sqrt {1+\frac {x^{2}}{2}+\frac {i x^{2} \sqrt {3}}{2}}\, F\left (\frac {x \sqrt {-2+2 i \sqrt {3}}}{2}, \frac {\sqrt {-2+2 i \sqrt {3}}}{2}\right )}{3 \sqrt {-2+2 i \sqrt {3}}\, \sqrt {x^{4}+x^{2}+1}}+\frac {4 \sqrt {1+\frac {x^{2}}{2}-\frac {i x^{2} \sqrt {3}}{2}}\, \sqrt {1+\frac {x^{2}}{2}+\frac {i x^{2} \sqrt {3}}{2}}\, F\left (\frac {x \sqrt {-2+2 i \sqrt {3}}}{2}, \frac {\sqrt {-2+2 i \sqrt {3}}}{2}\right )}{3 \sqrt {-2+2 i \sqrt {3}}\, \sqrt {x^{4}+x^{2}+1}\, \left (1+i \sqrt {3}\right )}-\frac {4 \sqrt {1+\frac {x^{2}}{2}-\frac {i x^{2} \sqrt {3}}{2}}\, \sqrt {1+\frac {x^{2}}{2}+\frac {i x^{2} \sqrt {3}}{2}}\, E\left (\frac {x \sqrt {-2+2 i \sqrt {3}}}{2}, \frac {\sqrt {-2+2 i \sqrt {3}}}{2}\right )}{3 \sqrt {-2+2 i \sqrt {3}}\, \sqrt {x^{4}+x^{2}+1}\, \left (1+i \sqrt {3}\right )}+\frac {\sqrt {1+\frac {x^{2}}{2}-\frac {i x^{2} \sqrt {3}}{2}}\, \sqrt {1+\frac {x^{2}}{2}+\frac {i x^{2} \sqrt {3}}{2}}\, \Pi \left (\sqrt {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}\, x , -\frac {1}{-\frac {1}{2}+\frac {i \sqrt {3}}{2}}, \frac {\sqrt {-\frac {1}{2}-\frac {i \sqrt {3}}{2}}}{\sqrt {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}}\right )}{2 \sqrt {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}\, \sqrt {x^{4}+x^{2}+1}}\)	\(438\)

input

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

output

1/6*(x^4+x^2+1)^(1/2)*x*(2*x^4+5*x^2+4)/(x^2+1)^3-1/3/(-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))+4/3/(-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))*(Elliptic 
F(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)))+1/2/(-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))
 

3.3.31.5 Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 216, normalized size of antiderivative = 1.30 \[ \int \frac {\sqrt {1+x^2+x^4}}{\left (1+x^2\right )^4} \, dx=-\frac {4 \, \sqrt {2} {\left (x^{6} + 3 \, x^{4} + 3 \, x^{2} - \sqrt {-3} {\left (x^{6} + 3 \, x^{4} + 3 \, x^{2} + 1\right )} + 1\right )} \sqrt {\sqrt {-3} - 1} E(\arcsin \left (\frac {1}{2} \, \sqrt {2} x \sqrt {\sqrt {-3} - 1}\right )\,|\,\frac {1}{2} \, \sqrt {-3} - \frac {1}{2}) - \sqrt {2} {\left (3 \, x^{6} + 9 \, x^{4} + 9 \, x^{2} - 5 \, \sqrt {-3} {\left (x^{6} + 3 \, x^{4} + 3 \, x^{2} + 1\right )} + 3\right )} \sqrt {\sqrt {-3} - 1} F(\arcsin \left (\frac {1}{2} \, \sqrt {2} x \sqrt {\sqrt {-3} - 1}\right )\,|\,\frac {1}{2} \, \sqrt {-3} - \frac {1}{2}) - 12 \, {\left (x^{6} + 3 \, x^{4} + 3 \, x^{2} + 1\right )} \arctan \left (\frac {x}{\sqrt {x^{4} + x^{2} + 1}}\right ) - 8 \, {\left (2 \, x^{5} + 5 \, x^{3} + 4 \, x\right )} \sqrt {x^{4} + x^{2} + 1}}{48 \, {\left (x^{6} + 3 \, x^{4} + 3 \, x^{2} + 1\right )}} \]

input

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

output

-1/48*(4*sqrt(2)*(x^6 + 3*x^4 + 3*x^2 - sqrt(-3)*(x^6 + 3*x^4 + 3*x^2 + 1) 
 + 1)*sqrt(sqrt(-3) - 1)*elliptic_e(arcsin(1/2*sqrt(2)*x*sqrt(sqrt(-3) - 1 
)), 1/2*sqrt(-3) - 1/2) - sqrt(2)*(3*x^6 + 9*x^4 + 9*x^2 - 5*sqrt(-3)*(x^6 
 + 3*x^4 + 3*x^2 + 1) + 3)*sqrt(sqrt(-3) - 1)*elliptic_f(arcsin(1/2*sqrt(2 
)*x*sqrt(sqrt(-3) - 1)), 1/2*sqrt(-3) - 1/2) - 12*(x^6 + 3*x^4 + 3*x^2 + 1 
)*arctan(x/sqrt(x^4 + x^2 + 1)) - 8*(2*x^5 + 5*x^3 + 4*x)*sqrt(x^4 + x^2 + 
 1))/(x^6 + 3*x^4 + 3*x^2 + 1)
 

3.3.31.6 Sympy [F]

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

input

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

output

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

3.3.31.7 Maxima [F]

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

input

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

output

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

3.3.31.8 Giac [F]

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

input

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

output

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

3.3.31.9 Mupad [F(-1)]

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

input

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

output

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