\(\int \frac {\sqrt {-2+x^4} (2+x^4)}{4-6 x^4+x^8} \, dx\) [684]

Optimal result

Integrand size = 27, antiderivative size = 53 \[ \int \frac {\sqrt {-2+x^4} \left (2+x^4\right )}{4-6 x^4+x^8} \, dx=-\frac {\arctan \left (\frac {\sqrt [4]{2} x}{\sqrt {-2+x^4}}\right )}{2 \sqrt [4]{2}}-\frac {\text {arctanh}\left (\frac {\sqrt [4]{2} x}{\sqrt {-2+x^4}}\right )}{2 \sqrt [4]{2}} \]

[Out]

-1/4*arctan(2^(1/4)*x/(x^4-2)^(1/2))*2^(3/4)-1/4*arctanh(2^(1/4)*x/(x^4-2)^(1/2))*2^(3/4)

Rubi [C] (verified)

Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.

Time = 1.06 (sec) , antiderivative size = 647, normalized size of antiderivative = 12.21, number of steps used = 40, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.370, Rules used = {6860, 415, 229, 418, 1229, 1471, 554, 259, 552, 551} \[ \int \frac {\sqrt {-2+x^4} \left (2+x^4\right )}{4-6 x^4+x^8} \, dx=\frac {\left (1+\sqrt {5}\right ) \sqrt {\frac {x^2+\sqrt {2}}{\sqrt {2}-x^2}} \sqrt {\sqrt {2} x^2-2} \operatorname {EllipticF}\left (\arcsin \left (\frac {2^{3/4} x}{\sqrt {\sqrt {2} x^2-2}}\right ),\frac {1}{2}\right )}{4 \sqrt [4]{2} \sqrt {\frac {1}{2-\sqrt {2} x^2}} \sqrt {x^4-2}}+\frac {\left (1-\sqrt {5}\right ) \sqrt {\frac {x^2+\sqrt {2}}{\sqrt {2}-x^2}} \sqrt {\sqrt {2} x^2-2} \operatorname {EllipticF}\left (\arcsin \left (\frac {2^{3/4} x}{\sqrt {\sqrt {2} x^2-2}}\right ),\frac {1}{2}\right )}{4 \sqrt [4]{2} \sqrt {\frac {1}{2-\sqrt {2} x^2}} \sqrt {x^4-2}}-\frac {\left (2+\sqrt {2 \left (3+\sqrt {5}\right )}\right ) \sqrt {\sqrt {2}-x^2} \sqrt {\sqrt {2} x^2+2} \operatorname {EllipticPi}\left (-\sqrt {\frac {2}{3+\sqrt {5}}},\arcsin \left (\frac {x}{\sqrt [4]{2}}\right ),-1\right )}{4 \left (\sqrt {2}+\sqrt {3+\sqrt {5}}\right ) \sqrt {x^4-2}}-\frac {\left (2-\sqrt {2 \left (3+\sqrt {5}\right )}\right ) \sqrt {\sqrt {2}-x^2} \sqrt {\sqrt {2} x^2+2} \operatorname {EllipticPi}\left (\sqrt {\frac {2}{3+\sqrt {5}}},\arcsin \left (\frac {x}{\sqrt [4]{2}}\right ),-1\right )}{4 \left (\sqrt {2}-\sqrt {3+\sqrt {5}}\right ) \sqrt {x^4-2}}-\frac {\left (\sqrt {2}+\sqrt {3+\sqrt {5}}\right ) \sqrt {\sqrt {2}-x^2} \sqrt {\sqrt {2} x^2+2} \operatorname {EllipticPi}\left (-\sqrt {\frac {1}{2} \left (3+\sqrt {5}\right )},\arcsin \left (\frac {x}{\sqrt [4]{2}}\right ),-1\right )}{2 \left (2+\sqrt {2 \left (3+\sqrt {5}\right )}\right ) \sqrt {x^4-2}}-\frac {\left (\sqrt {2}-\sqrt {3+\sqrt {5}}\right ) \sqrt {\sqrt {2}-x^2} \sqrt {\sqrt {2} x^2+2} \operatorname {EllipticPi}\left (\sqrt {\frac {1}{2} \left (3+\sqrt {5}\right )},\arcsin \left (\frac {x}{\sqrt [4]{2}}\right ),-1\right )}{2 \left (2-\sqrt {2 \left (3+\sqrt {5}\right )}\right ) \sqrt {x^4-2}} \]

[In]

Int[(Sqrt[-2 + x^4]*(2 + x^4))/(4 - 6*x^4 + x^8),x]

[Out]

((1 - Sqrt[5])*Sqrt[(Sqrt[2] + x^2)/(Sqrt[2] - x^2)]*Sqrt[-2 + Sqrt[2]*x^2]*EllipticF[ArcSin[(2^(3/4)*x)/Sqrt[
-2 + Sqrt[2]*x^2]], 1/2])/(4*2^(1/4)*Sqrt[(2 - Sqrt[2]*x^2)^(-1)]*Sqrt[-2 + x^4]) + ((1 + Sqrt[5])*Sqrt[(Sqrt[
2] + x^2)/(Sqrt[2] - x^2)]*Sqrt[-2 + Sqrt[2]*x^2]*EllipticF[ArcSin[(2^(3/4)*x)/Sqrt[-2 + Sqrt[2]*x^2]], 1/2])/
(4*2^(1/4)*Sqrt[(2 - Sqrt[2]*x^2)^(-1)]*Sqrt[-2 + x^4]) - ((2 + Sqrt[2*(3 + Sqrt[5])])*Sqrt[Sqrt[2] - x^2]*Sqr
t[2 + Sqrt[2]*x^2]*EllipticPi[-Sqrt[2/(3 + Sqrt[5])], ArcSin[x/2^(1/4)], -1])/(4*(Sqrt[2] + Sqrt[3 + Sqrt[5]])
*Sqrt[-2 + x^4]) - ((2 - Sqrt[2*(3 + Sqrt[5])])*Sqrt[Sqrt[2] - x^2]*Sqrt[2 + Sqrt[2]*x^2]*EllipticPi[Sqrt[2/(3
+ Sqrt[5])], ArcSin[x/2^(1/4)], -1])/(4*(Sqrt[2] - Sqrt[3 + Sqrt[5]])*Sqrt[-2 + x^4]) - ((Sqrt[2] + Sqrt[3 +
Sqrt[5]])*Sqrt[Sqrt[2] - x^2]*Sqrt[2 + Sqrt[2]*x^2]*EllipticPi[-Sqrt[(3 + Sqrt[5])/2], ArcSin[x/2^(1/4)], -1])
/(2*(2 + Sqrt[2*(3 + Sqrt[5])])*Sqrt[-2 + x^4]) - ((Sqrt[2] - Sqrt[3 + Sqrt[5]])*Sqrt[Sqrt[2] - x^2]*Sqrt[2 +
Sqrt[2]*x^2]*EllipticPi[Sqrt[(3 + Sqrt[5])/2], ArcSin[x/2^(1/4)], -1])/(2*(2 - Sqrt[2*(3 + Sqrt[5])])*Sqrt[-2
+ x^4])

Rule 229

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[(-a)*b, 2]}, Simp[Sqrt[(a - q*x^2)/(a + q*x^2)]*(Sq
rt[(a + q*x^2)/q]/(Sqrt[2]*Sqrt[a + b*x^4]*Sqrt[a/(a + q*x^2)]))*EllipticF[ArcSin[x/Sqrt[(a + q*x^2)/(2*q)]],
1/2], x]] /; FreeQ[{a, b}, x] && LtQ[a, 0] && GtQ[b, 0]

Rule 259

Int[((a1_.) + (b1_.)*(x_)^(n_))^(p_)*((a2_.) + (b2_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[(a1 + b1*x^n)^FracPar
t[p]*((a2 + b2*x^n)^FracPart[p]/(a1*a2 + b1*b2*x^(2*n))^FracPart[p]), Int[(a1*a2 + b1*b2*x^(2*n))^p, x], x] /;
FreeQ[{a1, b1, a2, b2, n, p}, x] && EqQ[a2*b1 + a1*b2, 0] &&  !IntegerQ[p]

Rule 415

Int[Sqrt[(a_) + (b_.)*(x_)^4]/((c_) + (d_.)*(x_)^4), x_Symbol] :> Dist[b/d, Int[1/Sqrt[a + b*x^4], x], x] - Di
st[(b*c - a*d)/d, Int[1/(Sqrt[a + b*x^4]*(c + d*x^4)), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 418

Int[1/(Sqrt[(a_) + (b_.)*(x_)^4]*((c_) + (d_.)*(x_)^4)), x_Symbol] :> Dist[1/(2*c), Int[1/(Sqrt[a + b*x^4]*(1
- Rt[-d/c, 2]*x^2)), x], x] + Dist[1/(2*c), Int[1/(Sqrt[a + b*x^4]*(1 + Rt[-d/c, 2]*x^2)), x], x] /; FreeQ[{a,
b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 551

Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x_)^2]), x_Symbol] :> Simp[(1/(a*Sqr
t[c]*Sqrt[e]*Rt[-d/c, 2]))*EllipticPi[b*(c/(a*d)), ArcSin[Rt[-d/c, 2]*x], c*(f/(d*e))], x] /; FreeQ[{a, b, c,
d, e, f}, x] &&  !GtQ[d/c, 0] && GtQ[c, 0] && GtQ[e, 0] &&  !( !GtQ[f/e, 0] && SimplerSqrtQ[-f/e, -d/c])

Rule 552

Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x_)^2]), x_Symbol] :> Dist[Sqrt[1 +
(d/c)*x^2]/Sqrt[c + d*x^2], Int[1/((a + b*x^2)*Sqrt[1 + (d/c)*x^2]*Sqrt[e + f*x^2]), x], x] /; FreeQ[{a, b, c,
d, e, f}, x] &&  !GtQ[c, 0]

Rule 554

Int[Sqrt[(c_) + (d_.)*(x_)^2]/(((a_) + (b_.)*(x_)^2)*Sqrt[(e_) + (f_.)*(x_)^2]), x_Symbol] :> Dist[d/b, Int[1/
(Sqrt[c + d*x^2]*Sqrt[e + f*x^2]), x], x] + Dist[(b*c - a*d)/b, Int[1/((a + b*x^2)*Sqrt[c + d*x^2]*Sqrt[e + f*
x^2]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NegQ[d/c]

Rule 1229

Int[1/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> With[{q = Rt[(-a)*c, 2]}, Dist[c/(c*d +
e*q), Int[1/Sqrt[a + c*x^4], x], x] + Dist[e/(c*d + e*q), Int[(q - c*x^2)/((d + e*x^2)*Sqrt[a + c*x^4]), x], x
]] /; FreeQ[{a, c, d, e}, x] && GtQ[(-a)*c, 0] &&  !LtQ[c, 0]

Rule 1471

Int[((d_) + (e_.)*(x_)^(n_))^(q_.)*((f_) + (g_.)*(x_)^(n_))^(r_.)*((a_) + (c_.)*(x_)^(n2_))^(p_), x_Symbol] :>
Dist[(a + c*x^(2*n))^FracPart[p]/((d + e*x^n)^FracPart[p]*(a/d + (c*x^n)/e)^FracPart[p]), Int[(d + e*x^n)^(p
+ q)*(f + g*x^n)^r*(a/d + (c/e)*x^n)^p, x], x] /; FreeQ[{a, c, d, e, f, g, n, p, q, r}, x] && EqQ[n2, 2*n] &&
EqQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[p]

Rule 6860

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] /; SumQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {\left (1+\sqrt {5}\right ) \sqrt {-2+x^4}}{-6-2 \sqrt {5}+2 x^4}+\frac {\left (1-\sqrt {5}\right ) \sqrt {-2+x^4}}{-6+2 \sqrt {5}+2 x^4}\right ) \, dx \\ & = \left (1-\sqrt {5}\right ) \int \frac {\sqrt {-2+x^4}}{-6+2 \sqrt {5}+2 x^4} \, dx+\left (1+\sqrt {5}\right ) \int \frac {\sqrt {-2+x^4}}{-6-2 \sqrt {5}+2 x^4} \, dx \\ & = \frac {1}{2} \left (1-\sqrt {5}\right ) \int \frac {1}{\sqrt {-2+x^4}} \, dx+\left (2 \left (3-\sqrt {5}\right )\right ) \int \frac {1}{\sqrt {-2+x^4} \left (-6+2 \sqrt {5}+2 x^4\right )} \, dx+\frac {1}{2} \left (1+\sqrt {5}\right ) \int \frac {1}{\sqrt {-2+x^4}} \, dx+\left (2 \left (3+\sqrt {5}\right )\right ) \int \frac {1}{\sqrt {-2+x^4} \left (-6-2 \sqrt {5}+2 x^4\right )} \, dx \\ & = \frac {\left (1-\sqrt {5}\right ) \sqrt {\frac {\sqrt {2}+x^2}{\sqrt {2}-x^2}} \sqrt {-2+\sqrt {2} x^2} \operatorname {EllipticF}\left (\arcsin \left (\frac {2^{3/4} x}{\sqrt {-2+\sqrt {2} x^2}}\right ),\frac {1}{2}\right )}{4 \sqrt [4]{2} \sqrt {\frac {1}{2-\sqrt {2} x^2}} \sqrt {-2+x^4}}+\frac {\left (1+\sqrt {5}\right ) \sqrt {\frac {\sqrt {2}+x^2}{\sqrt {2}-x^2}} \sqrt {-2+\sqrt {2} x^2} \operatorname {EllipticF}\left (\arcsin \left (\frac {2^{3/4} x}{\sqrt {-2+\sqrt {2} x^2}}\right ),\frac {1}{2}\right )}{4 \sqrt [4]{2} \sqrt {\frac {1}{2-\sqrt {2} x^2}} \sqrt {-2+x^4}}-\frac {1}{2} \int \frac {1}{\left (1-\frac {x^2}{\sqrt {3+\sqrt {5}}}\right ) \sqrt {-2+x^4}} \, dx-\frac {1}{2} \int \frac {1}{\left (1+\frac {x^2}{\sqrt {3+\sqrt {5}}}\right ) \sqrt {-2+x^4}} \, dx-\frac {1}{2} \int \frac {1}{\left (1-\frac {1}{2} \sqrt {3+\sqrt {5}} x^2\right ) \sqrt {-2+x^4}} \, dx-\frac {1}{2} \int \frac {1}{\left (1+\frac {1}{2} \sqrt {3+\sqrt {5}} x^2\right ) \sqrt {-2+x^4}} \, dx \\ & = \frac {\left (1-\sqrt {5}\right ) \sqrt {\frac {\sqrt {2}+x^2}{\sqrt {2}-x^2}} \sqrt {-2+\sqrt {2} x^2} \operatorname {EllipticF}\left (\arcsin \left (\frac {2^{3/4} x}{\sqrt {-2+\sqrt {2} x^2}}\right ),\frac {1}{2}\right )}{4 \sqrt [4]{2} \sqrt {\frac {1}{2-\sqrt {2} x^2}} \sqrt {-2+x^4}}+\frac {\left (1+\sqrt {5}\right ) \sqrt {\frac {\sqrt {2}+x^2}{\sqrt {2}-x^2}} \sqrt {-2+\sqrt {2} x^2} \operatorname {EllipticF}\left (\arcsin \left (\frac {2^{3/4} x}{\sqrt {-2+\sqrt {2} x^2}}\right ),\frac {1}{2}\right )}{4 \sqrt [4]{2} \sqrt {\frac {1}{2-\sqrt {2} x^2}} \sqrt {-2+x^4}}-\frac {\int \frac {1}{\sqrt {-2+x^4}} \, dx}{2 \left (1-\sqrt {\frac {2}{3+\sqrt {5}}}\right )}-\frac {\int \frac {1}{\sqrt {-2+x^4}} \, dx}{2 \left (1+\sqrt {\frac {2}{3+\sqrt {5}}}\right )}-\frac {\int \frac {\sqrt {2}-x^2}{\left (1-\frac {x^2}{\sqrt {3+\sqrt {5}}}\right ) \sqrt {-2+x^4}} \, dx}{2 \left (\sqrt {2}-\sqrt {3+\sqrt {5}}\right )}-\frac {\int \frac {\sqrt {2}-x^2}{\left (1+\frac {x^2}{\sqrt {3+\sqrt {5}}}\right ) \sqrt {-2+x^4}} \, dx}{2 \left (\sqrt {2}+\sqrt {3+\sqrt {5}}\right )}-\frac {\int \frac {1}{\sqrt {-2+x^4}} \, dx}{2-\sqrt {2 \left (3+\sqrt {5}\right )}}+\frac {\sqrt {3+\sqrt {5}} \int \frac {\sqrt {2}-x^2}{\left (1-\frac {1}{2} \sqrt {3+\sqrt {5}} x^2\right ) \sqrt {-2+x^4}} \, dx}{2 \left (2-\sqrt {2 \left (3+\sqrt {5}\right )}\right )}-\frac {\int \frac {1}{\sqrt {-2+x^4}} \, dx}{2+\sqrt {2 \left (3+\sqrt {5}\right )}}-\frac {\sqrt {3+\sqrt {5}} \int \frac {\sqrt {2}-x^2}{\left (1+\frac {1}{2} \sqrt {3+\sqrt {5}} x^2\right ) \sqrt {-2+x^4}} \, dx}{2 \left (2+\sqrt {2 \left (3+\sqrt {5}\right )}\right )} \\ & = \frac {\left (1-\sqrt {5}\right ) \sqrt {\frac {\sqrt {2}+x^2}{\sqrt {2}-x^2}} \sqrt {-2+\sqrt {2} x^2} \operatorname {EllipticF}\left (\arcsin \left (\frac {2^{3/4} x}{\sqrt {-2+\sqrt {2} x^2}}\right ),\frac {1}{2}\right )}{4 \sqrt [4]{2} \sqrt {\frac {1}{2-\sqrt {2} x^2}} \sqrt {-2+x^4}}+\frac {\left (1+\sqrt {5}\right ) \sqrt {\frac {\sqrt {2}+x^2}{\sqrt {2}-x^2}} \sqrt {-2+\sqrt {2} x^2} \operatorname {EllipticF}\left (\arcsin \left (\frac {2^{3/4} x}{\sqrt {-2+\sqrt {2} x^2}}\right ),\frac {1}{2}\right )}{4 \sqrt [4]{2} \sqrt {\frac {1}{2-\sqrt {2} x^2}} \sqrt {-2+x^4}}-\frac {\sqrt {\frac {\sqrt {2}+x^2}{\sqrt {2}-x^2}} \sqrt {-2+\sqrt {2} x^2} \operatorname {EllipticF}\left (\arcsin \left (\frac {2^{3/4} x}{\sqrt {-2+\sqrt {2} x^2}}\right ),\frac {1}{2}\right )}{4 \sqrt [4]{2} \left (1-\sqrt {\frac {2}{3+\sqrt {5}}}\right ) \sqrt {\frac {1}{2-\sqrt {2} x^2}} \sqrt {-2+x^4}}-\frac {\sqrt {\frac {\sqrt {2}+x^2}{\sqrt {2}-x^2}} \sqrt {-2+\sqrt {2} x^2} \operatorname {EllipticF}\left (\arcsin \left (\frac {2^{3/4} x}{\sqrt {-2+\sqrt {2} x^2}}\right ),\frac {1}{2}\right )}{4 \sqrt [4]{2} \left (1+\sqrt {\frac {2}{3+\sqrt {5}}}\right ) \sqrt {\frac {1}{2-\sqrt {2} x^2}} \sqrt {-2+x^4}}-\frac {\sqrt {\frac {\sqrt {2}+x^2}{\sqrt {2}-x^2}} \sqrt {-2+\sqrt {2} x^2} \operatorname {EllipticF}\left (\arcsin \left (\frac {2^{3/4} x}{\sqrt {-2+\sqrt {2} x^2}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{2} \left (2-\sqrt {2 \left (3+\sqrt {5}\right )}\right ) \sqrt {\frac {1}{2-\sqrt {2} x^2}} \sqrt {-2+x^4}}-\frac {\sqrt {\frac {\sqrt {2}+x^2}{\sqrt {2}-x^2}} \sqrt {-2+\sqrt {2} x^2} \operatorname {EllipticF}\left (\arcsin \left (\frac {2^{3/4} x}{\sqrt {-2+\sqrt {2} x^2}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{2} \left (2+\sqrt {2 \left (3+\sqrt {5}\right )}\right ) \sqrt {\frac {1}{2-\sqrt {2} x^2}} \sqrt {-2+x^4}}-\frac {\left (\sqrt {-\sqrt {2}-x^2} \sqrt {\sqrt {2}-x^2}\right ) \int \frac {\sqrt {\sqrt {2}-x^2}}{\sqrt {-\sqrt {2}-x^2} \left (1-\frac {x^2}{\sqrt {3+\sqrt {5}}}\right )} \, dx}{2 \left (\sqrt {2}-\sqrt {3+\sqrt {5}}\right ) \sqrt {-2+x^4}}-\frac {\left (\sqrt {-\sqrt {2}-x^2} \sqrt {\sqrt {2}-x^2}\right ) \int \frac {\sqrt {\sqrt {2}-x^2}}{\sqrt {-\sqrt {2}-x^2} \left (1+\frac {x^2}{\sqrt {3+\sqrt {5}}}\right )} \, dx}{2 \left (\sqrt {2}+\sqrt {3+\sqrt {5}}\right ) \sqrt {-2+x^4}}+\frac {\left (\sqrt {3+\sqrt {5}} \sqrt {-\sqrt {2}-x^2} \sqrt {\sqrt {2}-x^2}\right ) \int \frac {\sqrt {\sqrt {2}-x^2}}{\sqrt {-\sqrt {2}-x^2} \left (1-\frac {1}{2} \sqrt {3+\sqrt {5}} x^2\right )} \, dx}{2 \left (2-\sqrt {2 \left (3+\sqrt {5}\right )}\right ) \sqrt {-2+x^4}}-\frac {\left (\sqrt {3+\sqrt {5}} \sqrt {-\sqrt {2}-x^2} \sqrt {\sqrt {2}-x^2}\right ) \int \frac {\sqrt {\sqrt {2}-x^2}}{\sqrt {-\sqrt {2}-x^2} \left (1+\frac {1}{2} \sqrt {3+\sqrt {5}} x^2\right )} \, dx}{2 \left (2+\sqrt {2 \left (3+\sqrt {5}\right )}\right ) \sqrt {-2+x^4}} \\ & = \text {Too large to display} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.30 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.83 \[ \int \frac {\sqrt {-2+x^4} \left (2+x^4\right )}{4-6 x^4+x^8} \, dx=-\frac {\arctan \left (\frac {\sqrt [4]{2} x}{\sqrt {-2+x^4}}\right )+\text {arctanh}\left (\frac {\sqrt [4]{2} x}{\sqrt {-2+x^4}}\right )}{2 \sqrt [4]{2}} \]

[In]

Integrate[(Sqrt[-2 + x^4]*(2 + x^4))/(4 - 6*x^4 + x^8),x]

[Out]

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

Maple [A] (verified)

Time = 4.69 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.77

method	result	size
pseudoelliptic	\(\frac {2^{\frac {3}{4}} \left (\arctan \left (\frac {2^{\frac {3}{4}} \sqrt {x^{4}-2}}{2 x}\right )-\operatorname {arctanh}\left (\frac {2^{\frac {3}{4}} \sqrt {x^{4}-2}}{2 x}\right )\right )}{4}\)	\(41\)
default	\(\frac {2^{\frac {3}{4}} \left (2 \arctan \left (\frac {2^{\frac {3}{4}} \sqrt {x^{4}-2}}{2 x}\right )-\ln \left (\frac {\frac {\sqrt {x^{4}-2}\, \sqrt {2}}{2 x}+\frac {2^{\frac {3}{4}}}{2}}{\frac {\sqrt {x^{4}-2}\, \sqrt {2}}{2 x}-\frac {2^{\frac {3}{4}}}{2}}\right )\right )}{8}\)	\(73\)
elliptic	\(\frac {2^{\frac {3}{4}} \left (2 \arctan \left (\frac {2^{\frac {3}{4}} \sqrt {x^{4}-2}}{2 x}\right )-\ln \left (\frac {\frac {\sqrt {x^{4}-2}\, \sqrt {2}}{2 x}+\frac {2^{\frac {3}{4}}}{2}}{\frac {\sqrt {x^{4}-2}\, \sqrt {2}}{2 x}-\frac {2^{\frac {3}{4}}}{2}}\right )\right )}{8}\)	\(73\)
trager	\(-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2}\right ) \ln \left (\frac {-\operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2}\right ) x^{2} \operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2}+2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2}\right ) x^{4}+8 \sqrt {x^{4}-2}\, x -4 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2}\right )}{\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2} x^{2}+2 x^{4}-4}\right )}{8}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right ) \ln \left (\frac {-\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{3} x^{2}-2 x^{4} \operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )+8 \sqrt {x^{4}-2}\, x +4 \operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )}{\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2} x^{2}-2 x^{4}+4}\right )}{8}\)	\(184\)

[In]

int((x^4-2)^(1/2)*(x^4+2)/(x^8-6*x^4+4),x,method=_RETURNVERBOSE)

[Out]

1/4*2^(3/4)*(arctan(1/2*2^(3/4)/x*(x^4-2)^(1/2))-arctanh(1/2*2^(3/4)/x*(x^4-2)^(1/2)))

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.34 (sec) , antiderivative size = 296, normalized size of antiderivative = 5.58 \[ \int \frac {\sqrt {-2+x^4} \left (2+x^4\right )}{4-6 x^4+x^8} \, dx=-\frac {1}{16} \cdot 2^{\frac {3}{4}} \log \left (\frac {2^{\frac {3}{4}} {\left (x^{8} - 2 \, x^{4} + 4\right )} + 4 \, {\left (x^{5} + \sqrt {2} x^{3} - 2 \, x\right )} \sqrt {x^{4} - 2} + 4 \cdot 2^{\frac {1}{4}} {\left (x^{6} - 2 \, x^{2}\right )}}{x^{8} - 6 \, x^{4} + 4}\right ) + \frac {1}{16} \cdot 2^{\frac {3}{4}} \log \left (-\frac {2^{\frac {3}{4}} {\left (x^{8} - 2 \, x^{4} + 4\right )} - 4 \, {\left (x^{5} + \sqrt {2} x^{3} - 2 \, x\right )} \sqrt {x^{4} - 2} + 4 \cdot 2^{\frac {1}{4}} {\left (x^{6} - 2 \, x^{2}\right )}}{x^{8} - 6 \, x^{4} + 4}\right ) - \frac {1}{16} i \cdot 2^{\frac {3}{4}} \log \left (\frac {2^{\frac {3}{4}} {\left (i \, x^{8} - 2 i \, x^{4} + 4 i\right )} + 4 \, {\left (x^{5} - \sqrt {2} x^{3} - 2 \, x\right )} \sqrt {x^{4} - 2} - 4 \cdot 2^{\frac {1}{4}} {\left (i \, x^{6} - 2 i \, x^{2}\right )}}{x^{8} - 6 \, x^{4} + 4}\right ) + \frac {1}{16} i \cdot 2^{\frac {3}{4}} \log \left (\frac {2^{\frac {3}{4}} {\left (-i \, x^{8} + 2 i \, x^{4} - 4 i\right )} + 4 \, {\left (x^{5} - \sqrt {2} x^{3} - 2 \, x\right )} \sqrt {x^{4} - 2} - 4 \cdot 2^{\frac {1}{4}} {\left (-i \, x^{6} + 2 i \, x^{2}\right )}}{x^{8} - 6 \, x^{4} + 4}\right ) \]

[In]

integrate((x^4-2)^(1/2)*(x^4+2)/(x^8-6*x^4+4),x, algorithm="fricas")

[Out]

-1/16*2^(3/4)*log((2^(3/4)*(x^8 - 2*x^4 + 4) + 4*(x^5 + sqrt(2)*x^3 - 2*x)*sqrt(x^4 - 2) + 4*2^(1/4)*(x^6 - 2*
x^2))/(x^8 - 6*x^4 + 4)) + 1/16*2^(3/4)*log(-(2^(3/4)*(x^8 - 2*x^4 + 4) - 4*(x^5 + sqrt(2)*x^3 - 2*x)*sqrt(x^4
- 2) + 4*2^(1/4)*(x^6 - 2*x^2))/(x^8 - 6*x^4 + 4)) - 1/16*I*2^(3/4)*log((2^(3/4)*(I*x^8 - 2*I*x^4 + 4*I) + 4*
(x^5 - sqrt(2)*x^3 - 2*x)*sqrt(x^4 - 2) - 4*2^(1/4)*(I*x^6 - 2*I*x^2))/(x^8 - 6*x^4 + 4)) + 1/16*I*2^(3/4)*log
((2^(3/4)*(-I*x^8 + 2*I*x^4 - 4*I) + 4*(x^5 - sqrt(2)*x^3 - 2*x)*sqrt(x^4 - 2) - 4*2^(1/4)*(-I*x^6 + 2*I*x^2))
/(x^8 - 6*x^4 + 4))

Sympy [F(-1)]

Timed out. \[ \int \frac {\sqrt {-2+x^4} \left (2+x^4\right )}{4-6 x^4+x^8} \, dx=\text {Timed out} \]

[In]

integrate((x**4-2)**(1/2)*(x**4+2)/(x**8-6*x**4+4),x)

[Out]

Timed out

Maxima [F]

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

[In]

integrate((x^4-2)^(1/2)*(x^4+2)/(x^8-6*x^4+4),x, algorithm="maxima")

[Out]

integrate((x^4 + 2)*sqrt(x^4 - 2)/(x^8 - 6*x^4 + 4), x)

Giac [F]

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

[In]

integrate((x^4-2)^(1/2)*(x^4+2)/(x^8-6*x^4+4),x, algorithm="giac")

[Out]

integrate((x^4 + 2)*sqrt(x^4 - 2)/(x^8 - 6*x^4 + 4), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {-2+x^4} \left (2+x^4\right )}{4-6 x^4+x^8} \, dx=\int \frac {\sqrt {x^4-2}\,\left (x^4+2\right )}{x^8-6\,x^4+4} \,d x \]

[In]

int(((x^4 - 2)^(1/2)*(x^4 + 2))/(x^8 - 6*x^4 + 4),x)

[Out]

int(((x^4 - 2)^(1/2)*(x^4 + 2))/(x^8 - 6*x^4 + 4), x)