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

Optimal. Leaf size=53 \[ -\frac {\tan ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt {x^4-2}}\right )}{2 \sqrt [4]{2}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt {x^4-2}}\right )}{2 \sqrt [4]{2}} \]

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Rubi [C]  time = 1.43, antiderivative size = 647, normalized size of antiderivative = 12.21, number of steps used = 40, number of rules used = 10, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.370, Rules used = {6728, 406, 223, 409, 1215, 1457, 540, 253, 538, 537} \begin {gather*} \frac {\left (1+\sqrt {5}\right ) \sqrt {\frac {x^2+\sqrt {2}}{\sqrt {2}-x^2}} \sqrt {\sqrt {2} x^2-2} F\left (\sin ^{-1}\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} F\left (\sin ^{-1}\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} \Pi \left (-\sqrt {\frac {2}{3+\sqrt {5}}};\left .\sin ^{-1}\left (\frac {x}{\sqrt [4]{2}}\right )\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} \Pi \left (\sqrt {\frac {2}{3+\sqrt {5}}};\left .\sin ^{-1}\left (\frac {x}{\sqrt [4]{2}}\right )\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} \Pi \left (-\sqrt {\frac {1}{2} \left (3+\sqrt {5}\right )};\left .\sin ^{-1}\left (\frac {x}{\sqrt [4]{2}}\right )\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} \Pi \left (\sqrt {\frac {1}{2} \left (3+\sqrt {5}\right )};\left .\sin ^{-1}\left (\frac {x}{\sqrt [4]{2}}\right )\right |-1\right )}{2 \left (2-\sqrt {2 \left (3+\sqrt {5}\right )}\right ) \sqrt {x^4-2}} \end {gather*}

Warning: Unable to verify antiderivative.

[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 223

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]*EllipticF[ArcSin[x/Sqrt[(a + q*x^2)/(2*q)]], 1/2])/(Sqrt[2]*Sqrt[a + b*x^4]*Sqrt[a/(a + q*x^
2)]), x]] /; FreeQ[{a, b}, x] && LtQ[a, 0] && GtQ[b, 0]

Rule 253

Int[((a1_.) + (b1_.)*(x_)^(n_))^(p_)*((a2_.) + (b2_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[((a1 + b1*x^n)^FracPa
rt[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 406

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 409

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 537

Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x_)^2]), x_Symbol] :> Simp[(1*Ellipt
icPi[(b*c)/(a*d), ArcSin[Rt[-(d/c), 2]*x], (c*f)/(d*e)])/(a*Sqrt[c]*Sqrt[e]*Rt[-(d/c), 2]), 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 538

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

Rule 540

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 1215

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 1457

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*x^n)/e)^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 6728

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*} \int \frac {\sqrt {-2+x^4} \left (2+x^4\right )}{4-6 x^4+x^8} \, dx &=\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} F\left (\sin ^{-1}\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} F\left (\sin ^{-1}\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} F\left (\sin ^{-1}\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} F\left (\sin ^{-1}\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} F\left (\sin ^{-1}\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} F\left (\sin ^{-1}\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} F\left (\sin ^{-1}\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} F\left (\sin ^{-1}\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} F\left (\sin ^{-1}\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} F\left (\sin ^{-1}\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}}\\ &=\frac {\left (1-\sqrt {5}\right ) \sqrt {\frac {\sqrt {2}+x^2}{\sqrt {2}-x^2}} \sqrt {-2+\sqrt {2} x^2} F\left (\sin ^{-1}\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} F\left (\sin ^{-1}\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} F\left (\sin ^{-1}\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} F\left (\sin ^{-1}\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} F\left (\sin ^{-1}\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} F\left (\sin ^{-1}\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 {1}{\sqrt {-\sqrt {2}-x^2} \sqrt {\sqrt {2}-x^2} \left (1+\frac {x^2}{\sqrt {3+\sqrt {5}}}\right )} \, dx}{2 \sqrt {-2+x^4}}-\frac {\left (\sqrt {3+\sqrt {5}} \sqrt {-\sqrt {2}-x^2} \sqrt {\sqrt {2}-x^2}\right ) \int \frac {1}{\sqrt {-\sqrt {2}-x^2} \sqrt {\sqrt {2}-x^2}} \, dx}{2 \left (\sqrt {2}-\sqrt {3+\sqrt {5}}\right ) \sqrt {-2+x^4}}+\frac {\left (\left (-\sqrt {2}+\sqrt {3+\sqrt {5}}\right ) \sqrt {-\sqrt {2}-x^2} \sqrt {\sqrt {2}-x^2}\right ) \int \frac {1}{\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 {1}{\sqrt {-\sqrt {2}-x^2} \sqrt {\sqrt {2}-x^2}} \, 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 {1}{\sqrt {-\sqrt {2}-x^2} \sqrt {\sqrt {2}-x^2}} \, dx}{\left (2-\sqrt {2 \left (3+\sqrt {5}\right )}\right ) \sqrt {-2+x^4}}-\frac {\left (\left (1-\sqrt {\frac {1}{2} \left (3+\sqrt {5}\right )}\right ) \sqrt {-\sqrt {2}-x^2} \sqrt {\sqrt {2}-x^2}\right ) \int \frac {1}{\sqrt {-\sqrt {2}-x^2} \sqrt {\sqrt {2}-x^2} \left (1-\frac {1}{2} \sqrt {3+\sqrt {5}} x^2\right )} \, dx}{\left (2-\sqrt {2 \left (3+\sqrt {5}\right )}\right ) \sqrt {-2+x^4}}+\frac {\left (\sqrt {-\sqrt {2}-x^2} \sqrt {\sqrt {2}-x^2}\right ) \int \frac {1}{\sqrt {-\sqrt {2}-x^2} \sqrt {\sqrt {2}-x^2}} \, dx}{\left (2+\sqrt {2 \left (3+\sqrt {5}\right )}\right ) \sqrt {-2+x^4}}-\frac {\left (\left (1+\sqrt {\frac {1}{2} \left (3+\sqrt {5}\right )}\right ) \sqrt {-\sqrt {2}-x^2} \sqrt {\sqrt {2}-x^2}\right ) \int \frac {1}{\sqrt {-\sqrt {2}-x^2} \sqrt {\sqrt {2}-x^2} \left (1+\frac {1}{2} \sqrt {3+\sqrt {5}} x^2\right )} \, dx}{\left (2+\sqrt {2 \left (3+\sqrt {5}\right )}\right ) \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} F\left (\sin ^{-1}\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} F\left (\sin ^{-1}\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} F\left (\sin ^{-1}\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} F\left (\sin ^{-1}\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} F\left (\sin ^{-1}\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} F\left (\sin ^{-1}\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 {3+\sqrt {5}} \int \frac {1}{\sqrt {-2+x^4}} \, dx}{2 \left (\sqrt {2}-\sqrt {3+\sqrt {5}}\right )}+\frac {\sqrt {3+\sqrt {5}} \int \frac {1}{\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 {\int \frac {1}{\sqrt {-2+x^4}} \, dx}{2+\sqrt {2 \left (3+\sqrt {5}\right )}}-\frac {\left (\sqrt {\sqrt {2}-x^2} \sqrt {1+\frac {x^2}{\sqrt {2}}}\right ) \int \frac {1}{\sqrt {\sqrt {2}-x^2} \sqrt {1+\frac {x^2}{\sqrt {2}}} \left (1+\frac {x^2}{\sqrt {3+\sqrt {5}}}\right )} \, dx}{2 \sqrt {-2+x^4}}+\frac {\left (\left (-\sqrt {2}+\sqrt {3+\sqrt {5}}\right ) \sqrt {\sqrt {2}-x^2} \sqrt {1+\frac {x^2}{\sqrt {2}}}\right ) \int \frac {1}{\sqrt {\sqrt {2}-x^2} \sqrt {1+\frac {x^2}{\sqrt {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 (\left (1-\sqrt {\frac {1}{2} \left (3+\sqrt {5}\right )}\right ) \sqrt {\sqrt {2}-x^2} \sqrt {1+\frac {x^2}{\sqrt {2}}}\right ) \int \frac {1}{\sqrt {\sqrt {2}-x^2} \sqrt {1+\frac {x^2}{\sqrt {2}}} \left (1-\frac {1}{2} \sqrt {3+\sqrt {5}} x^2\right )} \, dx}{\left (2-\sqrt {2 \left (3+\sqrt {5}\right )}\right ) \sqrt {-2+x^4}}-\frac {\left (\left (1+\sqrt {\frac {1}{2} \left (3+\sqrt {5}\right )}\right ) \sqrt {\sqrt {2}-x^2} \sqrt {1+\frac {x^2}{\sqrt {2}}}\right ) \int \frac {1}{\sqrt {\sqrt {2}-x^2} \sqrt {1+\frac {x^2}{\sqrt {2}}} \left (1+\frac {1}{2} \sqrt {3+\sqrt {5}} x^2\right )} \, dx}{\left (2+\sqrt {2 \left (3+\sqrt {5}\right )}\right ) \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} F\left (\sin ^{-1}\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} F\left (\sin ^{-1}\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} F\left (\sin ^{-1}\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} F\left (\sin ^{-1}\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 {3+\sqrt {5}} \sqrt {\frac {\sqrt {2}+x^2}{\sqrt {2}-x^2}} \sqrt {-2+\sqrt {2} x^2} F\left (\sin ^{-1}\left (\frac {2^{3/4} x}{\sqrt {-2+\sqrt {2} x^2}}\right )|\frac {1}{2}\right )}{4 \sqrt [4]{2} \left (\sqrt {2}-\sqrt {3+\sqrt {5}}\right ) \sqrt {\frac {1}{2-\sqrt {2} x^2}} \sqrt {-2+x^4}}+\frac {\sqrt {3+\sqrt {5}} \sqrt {\frac {\sqrt {2}+x^2}{\sqrt {2}-x^2}} \sqrt {-2+\sqrt {2} x^2} F\left (\sin ^{-1}\left (\frac {2^{3/4} x}{\sqrt {-2+\sqrt {2} x^2}}\right )|\frac {1}{2}\right )}{4 \sqrt [4]{2} \left (\sqrt {2}+\sqrt {3+\sqrt {5}}\right ) \sqrt {\frac {1}{2-\sqrt {2} x^2}} \sqrt {-2+x^4}}-\frac {\sqrt {\sqrt {2}-x^2} \sqrt {2+\sqrt {2} x^2} \Pi \left (-\sqrt {\frac {2}{3+\sqrt {5}}};\left .\sin ^{-1}\left (\frac {x}{\sqrt [4]{2}}\right )\right |-1\right )}{2 \sqrt {2} \sqrt {-2+x^4}}-\frac {\sqrt {\sqrt {2}-x^2} \sqrt {2+\sqrt {2} x^2} \Pi \left (\sqrt {\frac {2}{3+\sqrt {5}}};\left .\sin ^{-1}\left (\frac {x}{\sqrt [4]{2}}\right )\right |-1\right )}{2 \sqrt {2} \sqrt {-2+x^4}}-\frac {\sqrt {\sqrt {2}-x^2} \sqrt {2+\sqrt {2} x^2} \Pi \left (-\sqrt {\frac {1}{2} \left (3+\sqrt {5}\right )};\left .\sin ^{-1}\left (\frac {x}{\sqrt [4]{2}}\right )\right |-1\right )}{2 \sqrt {2} \sqrt {-2+x^4}}-\frac {\sqrt {\sqrt {2}-x^2} \sqrt {2+\sqrt {2} x^2} \Pi \left (\sqrt {\frac {1}{2} \left (3+\sqrt {5}\right )};\left .\sin ^{-1}\left (\frac {x}{\sqrt [4]{2}}\right )\right |-1\right )}{2 \sqrt {2} \sqrt {-2+x^4}}\\ \end {align*}

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Mathematica [C]  time = 0.28, size = 154, normalized size = 2.91 \begin {gather*} \frac {\sqrt {2-x^4} \left (2 F\left (\left .\sin ^{-1}\left (\frac {x}{\sqrt [4]{2}}\right )\right |-1\right )-\Pi \left (-\sqrt {\frac {2}{3+\sqrt {5}}};\left .\sin ^{-1}\left (\frac {x}{\sqrt [4]{2}}\right )\right |-1\right )-\Pi \left (\sqrt {\frac {2}{3+\sqrt {5}}};\left .\sin ^{-1}\left (\frac {x}{\sqrt [4]{2}}\right )\right |-1\right )-\Pi \left (-\sqrt {\frac {1}{2} \left (3+\sqrt {5}\right )};\left .\sin ^{-1}\left (\frac {x}{\sqrt [4]{2}}\right )\right |-1\right )-\Pi \left (\sqrt {\frac {1}{2} \left (3+\sqrt {5}\right )};\left .\sin ^{-1}\left (\frac {x}{\sqrt [4]{2}}\right )\right |-1\right )\right )}{2 \sqrt [4]{2} \sqrt {x^4-2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.32, size = 53, normalized size = 1.00 \begin {gather*} -\frac {\tan ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt {-2+x^4}}\right )}{2 \sqrt [4]{2}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt {-2+x^4}}\right )}{2 \sqrt [4]{2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

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fricas [B]  time = 0.77, size = 226, normalized size = 4.26 \begin {gather*} \frac {1}{4} \cdot 2^{\frac {3}{4}} \arctan \left (\frac {2^{\frac {3}{4}} {\left (2 \cdot 2^{\frac {3}{4}} {\left (x^{6} - 2 \, x^{2}\right )} + 2^{\frac {1}{4}} {\left (x^{8} - 2 \, x^{4} + 4\right )}\right )} - 4 \, \sqrt {x^{4} - 2} {\left (2^{\frac {3}{4}} x^{3} + 2^{\frac {1}{4}} {\left (x^{5} - 2 \, x\right )}\right )}}{2 \, {\left (x^{8} - 6 \, x^{4} + 4\right )}}\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} \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 ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Unab
le to convert to real 1/4 Error: Bad Argument ValueUnable to convert to real 1/4 Error: Bad Argument Value

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maple [B]  time = 1.15, size = 79, normalized size = 1.49

method	result	size
default	\(\frac {\left (\frac {2^{\frac {1}{4}} \arctan \left (\frac {2^{\frac {3}{4}} \sqrt {x^{4}-2}}{2 x}\right )}{2}-\frac {2^{\frac {1}{4}} \ln \left (\frac {\frac {\sqrt {x^{4}-2}\, \sqrt {2}}{2 x}+\frac {2^{\frac {3}{4}}}{2}}{\frac {\sqrt {x^{4}-2}\, \sqrt {2}}{2 x}-\frac {2^{\frac {3}{4}}}{2}}\right )}{4}\right ) \sqrt {2}}{2}\)	\(79\)
elliptic	\(\frac {\left (\frac {2^{\frac {1}{4}} \arctan \left (\frac {2^{\frac {3}{4}} \sqrt {x^{4}-2}}{2 x}\right )}{2}-\frac {2^{\frac {1}{4}} \ln \left (\frac {\frac {\sqrt {x^{4}-2}\, \sqrt {2}}{2 x}+\frac {2^{\frac {3}{4}}}{2}}{\frac {\sqrt {x^{4}-2}\, \sqrt {2}}{2 x}-\frac {2^{\frac {3}{4}}}{2}}\right )}{4}\right ) \sqrt {2}}{2}\)	\(79\)
trager	\(-\frac {\RootOf \left (\textit {\_Z}^{4}-8\right ) \ln \left (\frac {\RootOf \left (\textit {\_Z}^{4}-8\right )^{3} x^{2}+2 x^{4} \RootOf \left (\textit {\_Z}^{4}-8\right )+8 \sqrt {x^{4}-2}\, x -4 \RootOf \left (\textit {\_Z}^{4}-8\right )}{\RootOf \left (\textit {\_Z}^{4}-8\right )^{2} x^{2}-2 x^{4}+4}\right )}{8}+\frac {\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-8\right )^{2}\right ) \ln \left (\frac {\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-8\right )^{2}\right ) x^{2} \RootOf \left (\textit {\_Z}^{4}-8\right )^{2}-2 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-8\right )^{2}\right ) x^{4}+8 \sqrt {x^{4}-2}\, x +4 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-8\right )^{2}\right )}{\RootOf \left (\textit {\_Z}^{4}-8\right )^{2} x^{2}+2 x^{4}-4}\right )}{8}\)	\(182\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{4} + 2\right )} \sqrt {x^{4} - 2}}{x^{8} - 6 \, x^{4} + 4}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {\sqrt {x^4-2}\,\left (x^4+2\right )}{x^8-6\,x^4+4} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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