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

Optimal. Leaf size=47 \[ \tanh ^{-1}\left (\frac {x}{\sqrt {x^4+2 x^2-1}}\right )-\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {x^4+2 x^2-1}}\right ) \]

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Rubi [C]  time = 2.61, antiderivative size = 1670, normalized size of antiderivative = 35.53, number of steps used = 52, number of rules used = 13, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.351, Rules used = {6725, 1208, 1187, 1098, 1184, 1214, 1456, 540, 421, 419, 538, 537, 6728}

result too large to display

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

Rule 419

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

Rule 421

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Dist[Sqrt[1 + (d*x^2)/c]/Sqrt[c + d*
x^2], Int[1/(Sqrt[a + b*x^2]*Sqrt[1 + (d*x^2)/c]), x], x] /; FreeQ[{a, b, c, d}, x] &&  !GtQ[c, 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 1098

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

Rule 1184

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

Rule 1187

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

Rule 1208

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

Rule 1214

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

Rule 1456

Int[((d_) + (e_.)*(x_)^(n_))^(q_.)*((f_) + (g_.)*(x_)^(n_))^(r_.)*((a_) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_))^
(p_), x_Symbol] :> Dist[(a + b*x^n + c*x^(2*n))^FracPart[p]/((d + e*x^n)^FracPart[p]*(a/d + (c*x^n)/e)^FracPar
t[p]), Int[(d + e*x^n)^(p + q)*(f + g*x^n)^r*(a/d + (c*x^n)/e)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p,
q, r}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] &&  !IntegerQ[p]

Rule 6725

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a + b*x^n), x]}, Int[v, x]
/; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ[n, 0]

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

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Mathematica [C]  time = 5.04, size = 1280, normalized size = 27.23 \begin {gather*} \frac {(x-1) (x+1) \left (x^2+1\right ) \left (x^4+x^2-1\right ) \left (-\frac {4 \left (\sqrt {-1+\sqrt {2}}+i \sqrt {1+\sqrt {2}}\right ) \sqrt {-\frac {x+i \sqrt {1+\sqrt {2}}}{i \sqrt {1+\sqrt {2}} x-\sqrt {-1+\sqrt {2}} x+\sqrt {2}-(1+i)}} \sqrt {\frac {i \sqrt {1+\sqrt {2}}-x}{-\left (\left (\sqrt {-1+\sqrt {2}}+i \sqrt {1+\sqrt {2}}\right ) x\right )+\sqrt {2}-(1-i)}} \sqrt {\frac {\left (\sqrt {-1+\sqrt {2}}-i \sqrt {1+\sqrt {2}}\right ) x+\sqrt {2}-(1+i)}{-\left (\left (\sqrt {-1+\sqrt {2}}+i \sqrt {1+\sqrt {2}}\right ) x\right )+\sqrt {2}-(1-i)}} \left (\sqrt {-1+\sqrt {2}} \left (\Pi \left (\frac {\left (-1+\sqrt {-1+\sqrt {2}}\right ) \left (\sqrt {-1+\sqrt {2}}+i \sqrt {1+\sqrt {2}}\right )}{\left (1+\sqrt {-1+\sqrt {2}}\right ) \left (\sqrt {-1+\sqrt {2}}-i \sqrt {1+\sqrt {2}}\right )};\left .\sin ^{-1}\left (\sqrt {\frac {-i \sqrt {1+\sqrt {2}} x+\sqrt {-1+\sqrt {2}} x+\sqrt {2}-(1+i)}{-\left (\left (\sqrt {-1+\sqrt {2}}+i \sqrt {1+\sqrt {2}}\right ) x\right )+\sqrt {2}-(1-i)}}\right )\right |-i\right )-\Pi \left (\frac {\left (1+\sqrt {-1+\sqrt {2}}\right ) \left (\sqrt {-1+\sqrt {2}}+i \sqrt {1+\sqrt {2}}\right )}{\left (-1+\sqrt {-1+\sqrt {2}}\right ) \left (\sqrt {-1+\sqrt {2}}-i \sqrt {1+\sqrt {2}}\right )};\left .\sin ^{-1}\left (\sqrt {\frac {-i \sqrt {1+\sqrt {2}} x+\sqrt {-1+\sqrt {2}} x+\sqrt {2}-(1+i)}{-\left (\left (\sqrt {-1+\sqrt {2}}+i \sqrt {1+\sqrt {2}}\right ) x\right )+\sqrt {2}-(1-i)}}\right )\right |-i\right )\right )-F\left (\left .\sin ^{-1}\left (\sqrt {\frac {\left (\sqrt {-1+\sqrt {2}}-i \sqrt {1+\sqrt {2}}\right ) x+\sqrt {2}-(1+i)}{-\left (\left (\sqrt {-1+\sqrt {2}}+i \sqrt {1+\sqrt {2}}\right ) x\right )+\sqrt {2}-(1-i)}}\right )\right |-i\right )\right ) \left (\sqrt {-1+\sqrt {2}}-x\right )^2}{\left (-2+\sqrt {2}\right ) \left (\sqrt {-1+\sqrt {2}}-i \sqrt {1+\sqrt {2}}\right )}-i \sqrt {1+\sqrt {2}} \sqrt {-x^4-2 x^2+1} F\left (i \sinh ^{-1}\left (\frac {x}{\sqrt {1+\sqrt {2}}}\right )|-3-2 \sqrt {2}\right )+2 i \sqrt {1+\sqrt {2}} \sqrt {-x^4-2 x^2+1} \Pi \left (1+\sqrt {2};i \sinh ^{-1}\left (\frac {x}{\sqrt {1+\sqrt {2}}}\right )|-3-2 \sqrt {2}\right )-\frac {i \sqrt {5 \left (1+\sqrt {2}\right )} \sqrt {-x^4-2 x^2+1} \Pi \left (-\frac {2 \left (1+\sqrt {2}\right )}{-1+\sqrt {5}};i \sinh ^{-1}\left (\frac {x}{\sqrt {1+\sqrt {2}}}\right )|-3-2 \sqrt {2}\right )}{-3+\sqrt {5}}+\frac {3 i \sqrt {1+\sqrt {2}} \sqrt {-x^4-2 x^2+1} \Pi \left (-\frac {2 \left (1+\sqrt {2}\right )}{-1+\sqrt {5}};i \sinh ^{-1}\left (\frac {x}{\sqrt {1+\sqrt {2}}}\right )|-3-2 \sqrt {2}\right )}{-3+\sqrt {5}}-\frac {i \sqrt {5 \left (1+\sqrt {2}\right )} \sqrt {-x^4-2 x^2+1} \Pi \left (\frac {2 \left (1+\sqrt {2}\right )}{1+\sqrt {5}};i \sinh ^{-1}\left (\frac {x}{\sqrt {1+\sqrt {2}}}\right )|-3-2 \sqrt {2}\right )}{3+\sqrt {5}}-\frac {3 i \sqrt {1+\sqrt {2}} \sqrt {-x^4-2 x^2+1} \Pi \left (\frac {2 \left (1+\sqrt {2}\right )}{1+\sqrt {5}};i \sinh ^{-1}\left (\frac {x}{\sqrt {1+\sqrt {2}}}\right )|-3-2 \sqrt {2}\right )}{3+\sqrt {5}}\right )}{\sqrt {x^4+2 x^2-1} \left (x^8+x^6-2 x^4-x^2+1\right )} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((-1 + x)*(1 + x)*(1 + x^2)*(-1 + x^2 + x^4)*((-I)*Sqrt[1 + Sqrt[2]]*Sqrt[1 - 2*x^2 - x^4]*EllipticF[I*ArcSinh
[x/Sqrt[1 + Sqrt[2]]], -3 - 2*Sqrt[2]] + (2*I)*Sqrt[1 + Sqrt[2]]*Sqrt[1 - 2*x^2 - x^4]*EllipticPi[1 + Sqrt[2],
I*ArcSinh[x/Sqrt[1 + Sqrt[2]]], -3 - 2*Sqrt[2]] + ((3*I)*Sqrt[1 + Sqrt[2]]*Sqrt[1 - 2*x^2 - x^4]*EllipticPi[(
-2*(1 + Sqrt[2]))/(-1 + Sqrt[5]), I*ArcSinh[x/Sqrt[1 + Sqrt[2]]], -3 - 2*Sqrt[2]])/(-3 + Sqrt[5]) - (I*Sqrt[5*
(1 + Sqrt[2])]*Sqrt[1 - 2*x^2 - x^4]*EllipticPi[(-2*(1 + Sqrt[2]))/(-1 + Sqrt[5]), I*ArcSinh[x/Sqrt[1 + Sqrt[2
]]], -3 - 2*Sqrt[2]])/(-3 + Sqrt[5]) - ((3*I)*Sqrt[1 + Sqrt[2]]*Sqrt[1 - 2*x^2 - x^4]*EllipticPi[(2*(1 + Sqrt[
2]))/(1 + Sqrt[5]), I*ArcSinh[x/Sqrt[1 + Sqrt[2]]], -3 - 2*Sqrt[2]])/(3 + Sqrt[5]) - (I*Sqrt[5*(1 + Sqrt[2])]*
Sqrt[1 - 2*x^2 - x^4]*EllipticPi[(2*(1 + Sqrt[2]))/(1 + Sqrt[5]), I*ArcSinh[x/Sqrt[1 + Sqrt[2]]], -3 - 2*Sqrt[
2]])/(3 + Sqrt[5]) - (4*(Sqrt[-1 + Sqrt[2]] + I*Sqrt[1 + Sqrt[2]])*(Sqrt[-1 + Sqrt[2]] - x)^2*Sqrt[-((I*Sqrt[1
+ Sqrt[2]] + x)/((-1 - I) + Sqrt[2] - Sqrt[-1 + Sqrt[2]]*x + I*Sqrt[1 + Sqrt[2]]*x))]*Sqrt[(I*Sqrt[1 + Sqrt[2
]] - x)/((-1 + I) + Sqrt[2] - (Sqrt[-1 + Sqrt[2]] + I*Sqrt[1 + Sqrt[2]])*x)]*Sqrt[((-1 - I) + Sqrt[2] + (Sqrt[
-1 + Sqrt[2]] - I*Sqrt[1 + Sqrt[2]])*x)/((-1 + I) + Sqrt[2] - (Sqrt[-1 + Sqrt[2]] + I*Sqrt[1 + Sqrt[2]])*x)]*(
-EllipticF[ArcSin[Sqrt[((-1 - I) + Sqrt[2] + (Sqrt[-1 + Sqrt[2]] - I*Sqrt[1 + Sqrt[2]])*x)/((-1 + I) + Sqrt[2]
- (Sqrt[-1 + Sqrt[2]] + I*Sqrt[1 + Sqrt[2]])*x)]], -I] + Sqrt[-1 + Sqrt[2]]*(EllipticPi[((-1 + Sqrt[-1 + Sqrt
[2]])*(Sqrt[-1 + Sqrt[2]] + I*Sqrt[1 + Sqrt[2]]))/((1 + Sqrt[-1 + Sqrt[2]])*(Sqrt[-1 + Sqrt[2]] - I*Sqrt[1 + S
qrt[2]])), ArcSin[Sqrt[((-1 - I) + Sqrt[2] + Sqrt[-1 + Sqrt[2]]*x - I*Sqrt[1 + Sqrt[2]]*x)/((-1 + I) + Sqrt[2]
- (Sqrt[-1 + Sqrt[2]] + I*Sqrt[1 + Sqrt[2]])*x)]], -I] - EllipticPi[((1 + Sqrt[-1 + Sqrt[2]])*(Sqrt[-1 + Sqrt
[2]] + I*Sqrt[1 + Sqrt[2]]))/((-1 + Sqrt[-1 + Sqrt[2]])*(Sqrt[-1 + Sqrt[2]] - I*Sqrt[1 + Sqrt[2]])), ArcSin[Sq
rt[((-1 - I) + Sqrt[2] + Sqrt[-1 + Sqrt[2]]*x - I*Sqrt[1 + Sqrt[2]]*x)/((-1 + I) + Sqrt[2] - (Sqrt[-1 + Sqrt[2
]] + I*Sqrt[1 + Sqrt[2]])*x)]], -I])))/((-2 + Sqrt[2])*(Sqrt[-1 + Sqrt[2]] - I*Sqrt[1 + Sqrt[2]]))))/(Sqrt[-1
+ 2*x^2 + x^4]*(1 - x^2 - 2*x^4 + x^6 + x^8))

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IntegrateAlgebraic [A]  time = 0.39, size = 47, normalized size = 1.00 \begin {gather*} \tanh ^{-1}\left (\frac {x}{\sqrt {-1+2 x^2+x^4}}\right )-\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {-1+2 x^2+x^4}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 0.53, size = 109, normalized size = 2.32 \begin {gather*} \frac {1}{4} \, \sqrt {2} \log \left (-\frac {x^{8} + 16 \, x^{6} + 30 \, x^{4} - 4 \, \sqrt {2} {\left (x^{5} + 4 \, x^{3} - x\right )} \sqrt {x^{4} + 2 \, x^{2} - 1} - 16 \, x^{2} + 1}{x^{8} - 2 \, x^{4} + 1}\right ) + \frac {1}{2} \, \log \left (\frac {x^{4} + 3 \, x^{2} + 2 \, \sqrt {x^{4} + 2 \, x^{2} - 1} x - 1}{x^{4} + x^{2} - 1}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*sqrt(2)*log(-(x^8 + 16*x^6 + 30*x^4 - 4*sqrt(2)*(x^5 + 4*x^3 - x)*sqrt(x^4 + 2*x^2 - 1) - 16*x^2 + 1)/(x^8
- 2*x^4 + 1)) + 1/2*log((x^4 + 3*x^2 + 2*sqrt(x^4 + 2*x^2 - 1)*x - 1)/(x^4 + x^2 - 1))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {x^{4} + 2 \, x^{2} - 1} {\left (x^{4} + 1\right )}}{{\left (x^{4} + x^{2} - 1\right )} {\left (x^{4} - 1\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.55, size = 76, normalized size = 1.62

method	result	size
elliptic	\(\frac {\left (-\ln \left (1+\frac {\sqrt {x^{4}+2 x^{2}-1}\, \sqrt {2}}{2 x}\right )+\sqrt {2}\, \arctanh \left (\frac {\sqrt {x^{4}+2 x^{2}-1}}{x}\right )+\ln \left (-1+\frac {\sqrt {x^{4}+2 x^{2}-1}\, \sqrt {2}}{2 x}\right )\right ) \sqrt {2}}{2}\)	\(76\)
trager	\(\frac {\ln \left (-\frac {x^{4}+2 \sqrt {x^{4}+2 x^{2}-1}\, x +3 x^{2}-1}{x^{4}+x^{2}-1}\right )}{2}+\frac {\RootOf \left (\textit {\_Z}^{2}-2\right ) \ln \left (-\frac {-\RootOf \left (\textit {\_Z}^{2}-2\right ) x^{4}-4 \RootOf \left (\textit {\_Z}^{2}-2\right ) x^{2}+4 \sqrt {x^{4}+2 x^{2}-1}\, x +\RootOf \left (\textit {\_Z}^{2}-2\right )}{\left (-1+x \right ) \left (1+x \right ) \left (x^{2}+1\right )}\right )}{2}\)	\(114\)
default	\(-\frac {2 \sqrt {1-\left (1-\sqrt {2}\right ) x^{2}}\, \sqrt {1-\left (1+\sqrt {2}\right ) x^{2}}\, \left (\EllipticF \left (x \sqrt {1-\sqrt {2}}, i+i \sqrt {2}\right )-\EllipticE \left (x \sqrt {1-\sqrt {2}}, i+i \sqrt {2}\right )\right )}{\sqrt {1-\sqrt {2}}\, \sqrt {x^{4}+2 x^{2}-1}\, \left (2+2 \sqrt {2}\right )}-\frac {2 \sqrt {1-\left (1-\sqrt {2}\right ) x^{2}}\, \sqrt {1-\left (1+\sqrt {2}\right ) x^{2}}\, \EllipticPi \left (x \sqrt {1-\sqrt {2}}, \frac {1}{1-\sqrt {2}}, \frac {\sqrt {1+\sqrt {2}}}{\sqrt {1-\sqrt {2}}}\right )}{\sqrt {1-\sqrt {2}}\, \sqrt {x^{4}+2 x^{2}-1}}+\frac {\sqrt {\sqrt {2}\, x^{2}-x^{2}+1}\, \sqrt {-\sqrt {2}\, x^{2}-x^{2}+1}\, \EllipticF \left (x \sqrt {1-\sqrt {2}}, i+i \sqrt {2}\right )}{\sqrt {1-\sqrt {2}}\, \sqrt {x^{4}+2 x^{2}-1}}+\frac {2 \sqrt {\sqrt {2}\, x^{2}-x^{2}+1}\, \sqrt {-\sqrt {2}\, x^{2}-x^{2}+1}\, \EllipticF \left (x \sqrt {1-\sqrt {2}}, i+i \sqrt {2}\right )}{\sqrt {1-\sqrt {2}}\, \sqrt {x^{4}+2 x^{2}-1}\, \left (2+2 \sqrt {2}\right )}-\frac {2 \sqrt {\sqrt {2}\, x^{2}-x^{2}+1}\, \sqrt {-\sqrt {2}\, x^{2}-x^{2}+1}\, \EllipticE \left (x \sqrt {1-\sqrt {2}}, i+i \sqrt {2}\right )}{\sqrt {1-\sqrt {2}}\, \sqrt {x^{4}+2 x^{2}-1}\, \left (2+2 \sqrt {2}\right )}-\frac {2 \sqrt {\sqrt {2}\, x^{2}-x^{2}+1}\, \sqrt {-\sqrt {2}\, x^{2}-x^{2}+1}\, \EllipticPi \left (x \sqrt {1-\sqrt {2}}, -\frac {1}{1-\sqrt {2}}, \frac {\sqrt {1+\sqrt {2}}}{\sqrt {1-\sqrt {2}}}\right )}{\sqrt {1-\sqrt {2}}\, \sqrt {x^{4}+2 x^{2}-1}}-\frac {\left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (\textit {\_Z}^{4}+\textit {\_Z}^{2}-1\right )}{\sum }\underline {\hspace {1.25 ex}}\alpha \left (-\frac {\arctanh \left (\frac {\left (\underline {\hspace {1.25 ex}}\alpha ^{2}+1\right ) \left (-2 \underline {\hspace {1.25 ex}}\alpha ^{2}+x^{2}+1\right )}{\sqrt {\underline {\hspace {1.25 ex}}\alpha ^{2}}\, \sqrt {x^{4}+2 x^{2}-1}}\right )}{\sqrt {\underline {\hspace {1.25 ex}}\alpha ^{2}}}-\frac {2 \left (\underline {\hspace {1.25 ex}}\alpha ^{3}+\underline {\hspace {1.25 ex}}\alpha \right ) \sqrt {\sqrt {2}\, x^{2}-x^{2}+1}\, \sqrt {-\sqrt {2}\, x^{2}-x^{2}+1}\, \EllipticPi \left (x \sqrt {1-\sqrt {2}}, -\underline {\hspace {1.25 ex}}\alpha ^{2} \sqrt {2}-\underline {\hspace {1.25 ex}}\alpha ^{2}-\sqrt {2}-1, \frac {\sqrt {1+\sqrt {2}}}{\sqrt {1-\sqrt {2}}}\right )}{\sqrt {1-\sqrt {2}}\, \sqrt {x^{4}+2 x^{2}-1}}\right )\right )}{4}\)	\(712\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {x^{4} + 2 \, x^{2} - 1} {\left (x^{4} + 1\right )}}{{\left (x^{4} + x^{2} - 1\right )} {\left (x^{4} - 1\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (x^{4} + 1\right ) \sqrt {x^{4} + 2 x^{2} - 1}}{\left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right ) \left (x^{4} + x^{2} - 1\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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