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

Optimal. Leaf size=520 \[ -\frac {\text {RootSum}\left [\text {$\#$1}^8-2 \text {$\#$1}^6-6 \text {$\#$1}^4+2 \text {$\#$1}^2+1\& ,\frac {-3 \text {$\#$1}^6 \log \left (\sqrt {x^4+1}+x^2-1\right )+3 \text {$\#$1}^6 \log \left (\text {$\#$1} \sqrt {x^4+1}+\text {$\#$1} x^2-\text {$\#$1}+\sqrt {2} \sqrt {\sqrt {x^4+1}+x^2} x\right )-\text {$\#$1}^4 \log \left (\sqrt {x^4+1}+x^2-1\right )+\text {$\#$1}^4 \log \left (\text {$\#$1} \sqrt {x^4+1}+\text {$\#$1} x^2-\text {$\#$1}+\sqrt {2} \sqrt {\sqrt {x^4+1}+x^2} x\right )+7 \text {$\#$1}^2 \log \left (\sqrt {x^4+1}+x^2-1\right )-7 \text {$\#$1}^2 \log \left (\text {$\#$1} \sqrt {x^4+1}+\text {$\#$1} x^2-\text {$\#$1}+\sqrt {2} \sqrt {\sqrt {x^4+1}+x^2} x\right )-\log \left (\text {$\#$1} \sqrt {x^4+1}+\text {$\#$1} x^2-\text {$\#$1}+\sqrt {2} \sqrt {\sqrt {x^4+1}+x^2} x\right )+\log \left (\sqrt {x^4+1}+x^2-1\right )}{2 \text {$\#$1}^7-3 \text {$\#$1}^5-6 \text {$\#$1}^3+\text {$\#$1}}\& \right ]}{5 \sqrt {2}}+\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} x \sqrt {\sqrt {x^4+1}+x^2}}{\sqrt {x^4+1}+x^2+1}\right )+\frac {-2 x \sqrt {x^4+1} \sqrt {\sqrt {x^4+1}+x^2} \left (2 x^4+3 x^2+2\right )-2 x \left (2 x^6+3 x^4+3 x^2+1\right ) \sqrt {\sqrt {x^4+1}+x^2}}{10 \sqrt {x^4+1} \left (x^4+x^2-1\right ) x^2+5 \left (x^4+x^2-1\right ) \left (2 x^4+1\right )} \]

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Rubi [C]  time = 9.54, antiderivative size = 2475, normalized size of antiderivative = 4.76, number of steps used = 112, number of rules used = 8, integrand size = 47, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.170, Rules used = {6742, 2132, 206, 2133, 731, 725, 6725, 6728}

result too large to display

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(((16*I)/5)*Sqrt[1 - I*x^2])/(((16 + 8*I) - 8*Sqrt[5])*(Sqrt[2*(-1 + Sqrt[5])] - 2*x)) + (((2*I)/5)*Sqrt[1 - I
*x^2])/(((2 + I) + Sqrt[5])*(I*Sqrt[2*(1 + Sqrt[5])] - 2*x)) + (((2*I)/5)*Sqrt[1 - I*x^2])/(((-2 - I) + Sqrt[5
])*(Sqrt[2*(-1 + Sqrt[5])] + 2*x)) - (((2*I)/5)*Sqrt[1 - I*x^2])/(((2 + I) + Sqrt[5])*(I*Sqrt[2*(1 + Sqrt[5])]
+ 2*x)) + (((2*I)/5)*Sqrt[1 + I*x^2])/(((-2 + I) + Sqrt[5])*(Sqrt[2*(-1 + Sqrt[5])] - 2*x)) - (((2*I)/5)*Sqrt
[1 + I*x^2])/(((2 - I) + Sqrt[5])*(I*Sqrt[2*(1 + Sqrt[5])] - 2*x)) - (((2*I)/5)*Sqrt[1 + I*x^2])/(((-2 + I) +
Sqrt[5])*(Sqrt[2*(-1 + Sqrt[5])] + 2*x)) + (((2*I)/5)*Sqrt[1 + I*x^2])/(((2 - I) + Sqrt[5])*(I*Sqrt[2*(1 + Sqr
t[5])] + 2*x)) - (((2*I)/5)*ArcTan[((1 + I)*(Sqrt[2] - I*Sqrt[-1 + Sqrt[5]]*x))/(Sqrt[2*((-1 - 2*I) + Sqrt[5])
]*Sqrt[1 + I*x^2])])/((1 - Sqrt[5])*Sqrt[(3 + I) - (1 + I)*Sqrt[5]]) - (((6*I)/5)*ArcTan[((1 + I)*(Sqrt[2] - I
*Sqrt[-1 + Sqrt[5]]*x))/(Sqrt[2*((-1 - 2*I) + Sqrt[5])]*Sqrt[1 + I*x^2])])/Sqrt[(-5 - 5*I)*((-2 + I) + Sqrt[5]
)] + (((2*I)/5)*ArcTan[((1 + I)*(Sqrt[2] + I*Sqrt[-1 + Sqrt[5]]*x))/(Sqrt[2*((-1 - 2*I) + Sqrt[5])]*Sqrt[1 + I
*x^2])])/((1 - Sqrt[5])*Sqrt[(3 + I) - (1 + I)*Sqrt[5]]) + (((6*I)/5)*ArcTan[((1 + I)*(Sqrt[2] + I*Sqrt[-1 + S
qrt[5]]*x))/(Sqrt[2*((-1 - 2*I) + Sqrt[5])]*Sqrt[1 + I*x^2])])/Sqrt[(-5 - 5*I)*((-2 + I) + Sqrt[5])] + (((2*I)
/5)*Sqrt[2/(-1 + Sqrt[5])]*ArcTan[((1/2 + I/2)*(2 - I*Sqrt[2*(-1 + Sqrt[5])]*x))/(Sqrt[(-1 - 2*I) + Sqrt[5]]*S
qrt[1 + I*x^2])])/((-1 - 2*I) + Sqrt[5])^(3/2) - (((2*I)/5)*Sqrt[2/(-1 + Sqrt[5])]*ArcTan[((1/2 + I/2)*(2 + I*
Sqrt[2*(-1 + Sqrt[5])]*x))/(Sqrt[(-1 - 2*I) + Sqrt[5]]*Sqrt[1 + I*x^2])])/((-1 - 2*I) + Sqrt[5])^(3/2) - (6*Ar
cTan[((1 + I)*(Sqrt[2] - Sqrt[1 + Sqrt[5]]*x))/(Sqrt[(-2 - 4*I) - 2*Sqrt[5]]*Sqrt[1 + I*x^2])])/(5*Sqrt[(-5 -
5*I)*((2 - I) + Sqrt[5])]) + (2*ArcTan[((1 + I)*(Sqrt[2] - Sqrt[1 + Sqrt[5]]*x))/(Sqrt[(-2 - 4*I) - 2*Sqrt[5]]
*Sqrt[1 + I*x^2])])/(5*(1 + Sqrt[5])*Sqrt[(-1 - I)*((2 - I) + Sqrt[5])]) + (6*ArcTan[((1 + I)*(Sqrt[2] + Sqrt[
1 + Sqrt[5]]*x))/(Sqrt[(-2 - 4*I) - 2*Sqrt[5]]*Sqrt[1 + I*x^2])])/(5*Sqrt[(-5 - 5*I)*((2 - I) + Sqrt[5])]) - (
2*ArcTan[((1 + I)*(Sqrt[2] + Sqrt[1 + Sqrt[5]]*x))/(Sqrt[(-2 - 4*I) - 2*Sqrt[5]]*Sqrt[1 + I*x^2])])/(5*(1 + Sq
rt[5])*Sqrt[(-1 - I)*((2 - I) + Sqrt[5])]) + (2*ArcTan[((1/2 + I/2)*(2 - Sqrt[2*(1 + Sqrt[5])]*x))/(Sqrt[(-1 -
2*I) - Sqrt[5]]*Sqrt[1 + I*x^2])])/(5*((1 + 2*I) + Sqrt[5])*Sqrt[(-1 - I)*((2 - I) + Sqrt[5])]) - (2*ArcTan[(
(1/2 + I/2)*(2 + Sqrt[2*(1 + Sqrt[5])]*x))/(Sqrt[(-1 - 2*I) - Sqrt[5]]*Sqrt[1 + I*x^2])])/(5*((1 + 2*I) + Sqrt
[5])*Sqrt[(-1 - I)*((2 - I) + Sqrt[5])]) - (I*Sqrt[(2 + 2*I)*((-2 - I) + Sqrt[5])]*ArcTanh[(Sqrt[2] - I*Sqrt[-
1 + Sqrt[5]]*x)/(Sqrt[(2 + I) - I*Sqrt[5]]*Sqrt[1 - I*x^2])])/((35 + 5*I) - (15 + 5*I)*Sqrt[5]) - ((3/5 - (3*I
)/5)*ArcTanh[(Sqrt[2] - I*Sqrt[-1 + Sqrt[5]]*x)/(Sqrt[(2 + I) - I*Sqrt[5]]*Sqrt[1 - I*x^2])])/Sqrt[(5/2 + (5*I
)/2)*((-2 - I) + Sqrt[5])] + (I*Sqrt[(2 + 2*I)*((-2 - I) + Sqrt[5])]*ArcTanh[(Sqrt[2] + I*Sqrt[-1 + Sqrt[5]]*x
)/(Sqrt[(2 + I) - I*Sqrt[5]]*Sqrt[1 - I*x^2])])/((35 + 5*I) - (15 + 5*I)*Sqrt[5]) + ((3/5 - (3*I)/5)*ArcTanh[(
Sqrt[2] + I*Sqrt[-1 + Sqrt[5]]*x)/(Sqrt[(2 + I) - I*Sqrt[5]]*Sqrt[1 - I*x^2])])/Sqrt[(5/2 + (5*I)/2)*((-2 - I)
+ Sqrt[5])] + ((1/25 - (3*I)/25)*Sqrt[(1/2 + I/2)*((-2 - I) + Sqrt[5])]*ArcTanh[(2 - I*Sqrt[2*(-1 + Sqrt[5])]
*x)/(Sqrt[(4 + 2*I) - (2*I)*Sqrt[5]]*Sqrt[1 - I*x^2])])/(3 - Sqrt[5]) - ((1/25 - (3*I)/25)*Sqrt[(1/2 + I/2)*((
-2 - I) + Sqrt[5])]*ArcTanh[(2 + I*Sqrt[2*(-1 + Sqrt[5])]*x)/(Sqrt[(4 + 2*I) - (2*I)*Sqrt[5]]*Sqrt[1 - I*x^2])
])/(3 - Sqrt[5]) + ((3/5 + (3*I)/5)*ArcTanh[(Sqrt[2] - Sqrt[1 + Sqrt[5]]*x)/(Sqrt[(2 + I) + I*Sqrt[5]]*Sqrt[1
- I*x^2])])/Sqrt[(5/2 + (5*I)/2)*((2 + I) + Sqrt[5])] - (Sqrt[(2 + 2*I)*((2 + I) + Sqrt[5])]*ArcTanh[(Sqrt[2]
- Sqrt[1 + Sqrt[5]]*x)/(Sqrt[(2 + I) + I*Sqrt[5]]*Sqrt[1 - I*x^2])])/((35 + 5*I) + (15 + 5*I)*Sqrt[5]) - ((3/5
+ (3*I)/5)*ArcTanh[(Sqrt[2] + Sqrt[1 + Sqrt[5]]*x)/(Sqrt[(2 + I) + I*Sqrt[5]]*Sqrt[1 - I*x^2])])/Sqrt[(5/2 +
(5*I)/2)*((2 + I) + Sqrt[5])] + (Sqrt[(2 + 2*I)*((2 + I) + Sqrt[5])]*ArcTanh[(Sqrt[2] + Sqrt[1 + Sqrt[5]]*x)/(
Sqrt[(2 + I) + I*Sqrt[5]]*Sqrt[1 - I*x^2])])/((35 + 5*I) + (15 + 5*I)*Sqrt[5]) - ((3/25 + I/25)*Sqrt[(1/2 + I/
2)*((2 + I) + Sqrt[5])]*ArcTanh[(2 - Sqrt[2*(1 + Sqrt[5])]*x)/(Sqrt[(4 + 2*I) + (2*I)*Sqrt[5]]*Sqrt[1 - I*x^2]
)])/(3 + Sqrt[5]) + ((3/25 + I/25)*Sqrt[(1/2 + I/2)*((2 + I) + Sqrt[5])]*ArcTanh[(2 + Sqrt[2*(1 + Sqrt[5])]*x)
/(Sqrt[(4 + 2*I) + (2*I)*Sqrt[5]]*Sqrt[1 - I*x^2])])/(3 + Sqrt[5]) + ArcTanh[(Sqrt[2]*x)/Sqrt[x^2 + Sqrt[1 + x
^4]]]/Sqrt[2]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 725

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

Rule 731

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(d + e*x)^(m + 1)*(a + c*x^2)^(p
+ 1))/((m + 1)*(c*d^2 + a*e^2)), x] + Dist[(c*d)/(c*d^2 + a*e^2), Int[(d + e*x)^(m + 1)*(a + c*x^2)^p, x], x]
/; FreeQ[{a, c, d, e, m, p}, x] && NeQ[c*d^2 + a*e^2, 0] && EqQ[m + 2*p + 3, 0]

Rule 2132

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

Rule 2133

Int[(((c_.) + (d_.)*(x_))^(m_.)*Sqrt[(b_.)*(x_)^2 + Sqrt[(a_) + (e_.)*(x_)^4]])/Sqrt[(a_) + (e_.)*(x_)^4], x_S
ymbol] :> Dist[(1 - I)/2, Int[(c + d*x)^m/Sqrt[Sqrt[a] - I*b*x^2], x], x] + Dist[(1 + I)/2, Int[(c + d*x)^m/Sq
rt[Sqrt[a] + I*b*x^2], x], x] /; FreeQ[{a, b, c, d, m}, x] && EqQ[e, b^2] && GtQ[a, 0]

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]

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

\begin {align*} \int \frac {\left (1+x^2+x^4\right )^2 \sqrt {x^2+\sqrt {1+x^4}}}{\sqrt {1+x^4} \left (-1+x^2+x^4\right )^2} \, dx &=\int \left (\frac {\sqrt {x^2+\sqrt {1+x^4}}}{\sqrt {1+x^4}}+\frac {4 \sqrt {x^2+\sqrt {1+x^4}}}{\sqrt {1+x^4} \left (-1+x^2+x^4\right )^2}+\frac {4 \sqrt {x^2+\sqrt {1+x^4}}}{\sqrt {1+x^4} \left (-1+x^2+x^4\right )}\right ) \, dx\\ &=4 \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{\sqrt {1+x^4} \left (-1+x^2+x^4\right )^2} \, dx+4 \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{\sqrt {1+x^4} \left (-1+x^2+x^4\right )} \, dx+\int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{\sqrt {1+x^4}} \, dx\\ &=4 \int \left (-\frac {2 \sqrt {x^2+\sqrt {1+x^4}}}{\sqrt {5} \left (-1+\sqrt {5}-2 x^2\right ) \sqrt {1+x^4}}-\frac {2 \sqrt {x^2+\sqrt {1+x^4}}}{\sqrt {5} \left (1+\sqrt {5}+2 x^2\right ) \sqrt {1+x^4}}\right ) \, dx+4 \int \left (\frac {4 \sqrt {x^2+\sqrt {1+x^4}}}{5 \left (-1+\sqrt {5}-2 x^2\right )^2 \sqrt {1+x^4}}+\frac {4 \sqrt {x^2+\sqrt {1+x^4}}}{5 \sqrt {5} \left (-1+\sqrt {5}-2 x^2\right ) \sqrt {1+x^4}}+\frac {4 \sqrt {x^2+\sqrt {1+x^4}}}{5 \left (1+\sqrt {5}+2 x^2\right )^2 \sqrt {1+x^4}}+\frac {4 \sqrt {x^2+\sqrt {1+x^4}}}{5 \sqrt {5} \left (1+\sqrt {5}+2 x^2\right ) \sqrt {1+x^4}}\right ) \, dx+\operatorname {Subst}\left (\int \frac {1}{1-2 x^2} \, dx,x,\frac {x}{\sqrt {x^2+\sqrt {1+x^4}}}\right )\\ &=\frac {\tanh ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {x^2+\sqrt {1+x^4}}}\right )}{\sqrt {2}}+\frac {16}{5} \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{\left (-1+\sqrt {5}-2 x^2\right )^2 \sqrt {1+x^4}} \, dx+\frac {16}{5} \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{\left (1+\sqrt {5}+2 x^2\right )^2 \sqrt {1+x^4}} \, dx+\frac {16 \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{\left (-1+\sqrt {5}-2 x^2\right ) \sqrt {1+x^4}} \, dx}{5 \sqrt {5}}+\frac {16 \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{\left (1+\sqrt {5}+2 x^2\right ) \sqrt {1+x^4}} \, dx}{5 \sqrt {5}}-\frac {8 \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{\left (-1+\sqrt {5}-2 x^2\right ) \sqrt {1+x^4}} \, dx}{\sqrt {5}}-\frac {8 \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{\left (1+\sqrt {5}+2 x^2\right ) \sqrt {1+x^4}} \, dx}{\sqrt {5}}\\ &=\frac {\tanh ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {x^2+\sqrt {1+x^4}}}\right )}{\sqrt {2}}+\frac {16}{5} \int \left (\frac {\sqrt {x^2+\sqrt {1+x^4}}}{2 \left (-1+\sqrt {5}\right ) \left (\sqrt {2 \left (-1+\sqrt {5}\right )}-2 x\right )^2 \sqrt {1+x^4}}+\frac {\sqrt {x^2+\sqrt {1+x^4}}}{2 \left (-1+\sqrt {5}\right ) \left (\sqrt {2 \left (-1+\sqrt {5}\right )}+2 x\right )^2 \sqrt {1+x^4}}+\frac {\sqrt {x^2+\sqrt {1+x^4}}}{\left (-1+\sqrt {5}\right ) \left (2 \left (-1+\sqrt {5}\right )-4 x^2\right ) \sqrt {1+x^4}}\right ) \, dx+\frac {16}{5} \int \left (-\frac {\sqrt {x^2+\sqrt {1+x^4}}}{2 \left (1+\sqrt {5}\right ) \left (i \sqrt {2 \left (1+\sqrt {5}\right )}-2 x\right )^2 \sqrt {1+x^4}}-\frac {\sqrt {x^2+\sqrt {1+x^4}}}{2 \left (1+\sqrt {5}\right ) \left (i \sqrt {2 \left (1+\sqrt {5}\right )}+2 x\right )^2 \sqrt {1+x^4}}-\frac {\sqrt {x^2+\sqrt {1+x^4}}}{\left (1+\sqrt {5}\right ) \left (-2 \left (1+\sqrt {5}\right )-4 x^2\right ) \sqrt {1+x^4}}\right ) \, dx+\frac {16 \int \left (\frac {\sqrt {x^2+\sqrt {1+x^4}}}{2 \sqrt {-1+\sqrt {5}} \left (\sqrt {-1+\sqrt {5}}-\sqrt {2} x\right ) \sqrt {1+x^4}}+\frac {\sqrt {x^2+\sqrt {1+x^4}}}{2 \sqrt {-1+\sqrt {5}} \left (\sqrt {-1+\sqrt {5}}+\sqrt {2} x\right ) \sqrt {1+x^4}}\right ) \, dx}{5 \sqrt {5}}+\frac {16 \int \left (\frac {i \sqrt {x^2+\sqrt {1+x^4}}}{2 \sqrt {1+\sqrt {5}} \left (i \sqrt {1+\sqrt {5}}-\sqrt {2} x\right ) \sqrt {1+x^4}}+\frac {i \sqrt {x^2+\sqrt {1+x^4}}}{2 \sqrt {1+\sqrt {5}} \left (i \sqrt {1+\sqrt {5}}+\sqrt {2} x\right ) \sqrt {1+x^4}}\right ) \, dx}{5 \sqrt {5}}-\frac {8 \int \left (\frac {\sqrt {x^2+\sqrt {1+x^4}}}{2 \sqrt {-1+\sqrt {5}} \left (\sqrt {-1+\sqrt {5}}-\sqrt {2} x\right ) \sqrt {1+x^4}}+\frac {\sqrt {x^2+\sqrt {1+x^4}}}{2 \sqrt {-1+\sqrt {5}} \left (\sqrt {-1+\sqrt {5}}+\sqrt {2} x\right ) \sqrt {1+x^4}}\right ) \, dx}{\sqrt {5}}-\frac {8 \int \left (\frac {i \sqrt {x^2+\sqrt {1+x^4}}}{2 \sqrt {1+\sqrt {5}} \left (i \sqrt {1+\sqrt {5}}-\sqrt {2} x\right ) \sqrt {1+x^4}}+\frac {i \sqrt {x^2+\sqrt {1+x^4}}}{2 \sqrt {1+\sqrt {5}} \left (i \sqrt {1+\sqrt {5}}+\sqrt {2} x\right ) \sqrt {1+x^4}}\right ) \, dx}{\sqrt {5}}\\ &=\frac {\tanh ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {x^2+\sqrt {1+x^4}}}\right )}{\sqrt {2}}+\frac {8 \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{\left (\sqrt {2 \left (-1+\sqrt {5}\right )}-2 x\right )^2 \sqrt {1+x^4}} \, dx}{5 \left (-1+\sqrt {5}\right )}+\frac {8 \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{\left (\sqrt {2 \left (-1+\sqrt {5}\right )}+2 x\right )^2 \sqrt {1+x^4}} \, dx}{5 \left (-1+\sqrt {5}\right )}+\frac {16 \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{\left (2 \left (-1+\sqrt {5}\right )-4 x^2\right ) \sqrt {1+x^4}} \, dx}{5 \left (-1+\sqrt {5}\right )}+\frac {8 \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{\left (\sqrt {-1+\sqrt {5}}-\sqrt {2} x\right ) \sqrt {1+x^4}} \, dx}{5 \sqrt {5 \left (-1+\sqrt {5}\right )}}+\frac {8 \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{\left (\sqrt {-1+\sqrt {5}}+\sqrt {2} x\right ) \sqrt {1+x^4}} \, dx}{5 \sqrt {5 \left (-1+\sqrt {5}\right )}}-\frac {4 \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{\left (\sqrt {-1+\sqrt {5}}-\sqrt {2} x\right ) \sqrt {1+x^4}} \, dx}{\sqrt {5 \left (-1+\sqrt {5}\right )}}-\frac {4 \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{\left (\sqrt {-1+\sqrt {5}}+\sqrt {2} x\right ) \sqrt {1+x^4}} \, dx}{\sqrt {5 \left (-1+\sqrt {5}\right )}}-\frac {8 \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{\left (i \sqrt {2 \left (1+\sqrt {5}\right )}-2 x\right )^2 \sqrt {1+x^4}} \, dx}{5 \left (1+\sqrt {5}\right )}-\frac {8 \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{\left (i \sqrt {2 \left (1+\sqrt {5}\right )}+2 x\right )^2 \sqrt {1+x^4}} \, dx}{5 \left (1+\sqrt {5}\right )}-\frac {16 \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{\left (-2 \left (1+\sqrt {5}\right )-4 x^2\right ) \sqrt {1+x^4}} \, dx}{5 \left (1+\sqrt {5}\right )}+\frac {(8 i) \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{\left (i \sqrt {1+\sqrt {5}}-\sqrt {2} x\right ) \sqrt {1+x^4}} \, dx}{5 \sqrt {5 \left (1+\sqrt {5}\right )}}+\frac {(8 i) \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{\left (i \sqrt {1+\sqrt {5}}+\sqrt {2} x\right ) \sqrt {1+x^4}} \, dx}{5 \sqrt {5 \left (1+\sqrt {5}\right )}}-\frac {(4 i) \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{\left (i \sqrt {1+\sqrt {5}}-\sqrt {2} x\right ) \sqrt {1+x^4}} \, dx}{\sqrt {5 \left (1+\sqrt {5}\right )}}-\frac {(4 i) \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{\left (i \sqrt {1+\sqrt {5}}+\sqrt {2} x\right ) \sqrt {1+x^4}} \, dx}{\sqrt {5 \left (1+\sqrt {5}\right )}}\\ &=\frac {\tanh ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {x^2+\sqrt {1+x^4}}}\right )}{\sqrt {2}}+-\frac {\left (\frac {4}{5}+\frac {4 i}{5}\right ) \int \frac {1}{\left (\sqrt {2 \left (-1+\sqrt {5}\right )}-2 x\right )^2 \sqrt {1+i x^2}} \, dx}{1-\sqrt {5}}+-\frac {\left (\frac {4}{5}+\frac {4 i}{5}\right ) \int \frac {1}{\left (\sqrt {2 \left (-1+\sqrt {5}\right )}+2 x\right )^2 \sqrt {1+i x^2}} \, dx}{1-\sqrt {5}}+-\frac {\left (\frac {4}{5}-\frac {4 i}{5}\right ) \int \frac {1}{\left (\sqrt {2 \left (-1+\sqrt {5}\right )}-2 x\right )^2 \sqrt {1-i x^2}} \, dx}{1-\sqrt {5}}+-\frac {\left (\frac {4}{5}-\frac {4 i}{5}\right ) \int \frac {1}{\left (\sqrt {2 \left (-1+\sqrt {5}\right )}+2 x\right )^2 \sqrt {1-i x^2}} \, dx}{1-\sqrt {5}}+\frac {16 \int \left (\frac {\sqrt {x^2+\sqrt {1+x^4}}}{4 \sqrt {-1+\sqrt {5}} \left (\sqrt {-1+\sqrt {5}}-\sqrt {2} x\right ) \sqrt {1+x^4}}+\frac {\sqrt {x^2+\sqrt {1+x^4}}}{4 \sqrt {-1+\sqrt {5}} \left (\sqrt {-1+\sqrt {5}}+\sqrt {2} x\right ) \sqrt {1+x^4}}\right ) \, dx}{5 \left (-1+\sqrt {5}\right )}+\frac {\left (\frac {4}{5}-\frac {4 i}{5}\right ) \int \frac {1}{\left (\sqrt {-1+\sqrt {5}}-\sqrt {2} x\right ) \sqrt {1-i x^2}} \, dx}{\sqrt {5 \left (-1+\sqrt {5}\right )}}+\frac {\left (\frac {4}{5}-\frac {4 i}{5}\right ) \int \frac {1}{\left (\sqrt {-1+\sqrt {5}}+\sqrt {2} x\right ) \sqrt {1-i x^2}} \, dx}{\sqrt {5 \left (-1+\sqrt {5}\right )}}+\frac {\left (\frac {4}{5}+\frac {4 i}{5}\right ) \int \frac {1}{\left (\sqrt {-1+\sqrt {5}}-\sqrt {2} x\right ) \sqrt {1+i x^2}} \, dx}{\sqrt {5 \left (-1+\sqrt {5}\right )}}+\frac {\left (\frac {4}{5}+\frac {4 i}{5}\right ) \int \frac {1}{\left (\sqrt {-1+\sqrt {5}}+\sqrt {2} x\right ) \sqrt {1+i x^2}} \, dx}{\sqrt {5 \left (-1+\sqrt {5}\right )}}-\frac {(2-2 i) \int \frac {1}{\left (\sqrt {-1+\sqrt {5}}-\sqrt {2} x\right ) \sqrt {1-i x^2}} \, dx}{\sqrt {5 \left (-1+\sqrt {5}\right )}}-\frac {(2-2 i) \int \frac {1}{\left (\sqrt {-1+\sqrt {5}}+\sqrt {2} x\right ) \sqrt {1-i x^2}} \, dx}{\sqrt {5 \left (-1+\sqrt {5}\right )}}-\frac {(2+2 i) \int \frac {1}{\left (\sqrt {-1+\sqrt {5}}-\sqrt {2} x\right ) \sqrt {1+i x^2}} \, dx}{\sqrt {5 \left (-1+\sqrt {5}\right )}}-\frac {(2+2 i) \int \frac {1}{\left (\sqrt {-1+\sqrt {5}}+\sqrt {2} x\right ) \sqrt {1+i x^2}} \, dx}{\sqrt {5 \left (-1+\sqrt {5}\right )}}-\frac {\left (\frac {4}{5}-\frac {4 i}{5}\right ) \int \frac {1}{\left (i \sqrt {2 \left (1+\sqrt {5}\right )}-2 x\right )^2 \sqrt {1-i x^2}} \, dx}{1+\sqrt {5}}-\frac {\left (\frac {4}{5}-\frac {4 i}{5}\right ) \int \frac {1}{\left (i \sqrt {2 \left (1+\sqrt {5}\right )}+2 x\right )^2 \sqrt {1-i x^2}} \, dx}{1+\sqrt {5}}-\frac {\left (\frac {4}{5}+\frac {4 i}{5}\right ) \int \frac {1}{\left (i \sqrt {2 \left (1+\sqrt {5}\right )}-2 x\right )^2 \sqrt {1+i x^2}} \, dx}{1+\sqrt {5}}-\frac {\left (\frac {4}{5}+\frac {4 i}{5}\right ) \int \frac {1}{\left (i \sqrt {2 \left (1+\sqrt {5}\right )}+2 x\right )^2 \sqrt {1+i x^2}} \, dx}{1+\sqrt {5}}-\frac {16 \int \left (-\frac {i \sqrt {x^2+\sqrt {1+x^4}}}{4 \sqrt {1+\sqrt {5}} \left (i \sqrt {1+\sqrt {5}}-\sqrt {2} x\right ) \sqrt {1+x^4}}-\frac {i \sqrt {x^2+\sqrt {1+x^4}}}{4 \sqrt {1+\sqrt {5}} \left (i \sqrt {1+\sqrt {5}}+\sqrt {2} x\right ) \sqrt {1+x^4}}\right ) \, dx}{5 \left (1+\sqrt {5}\right )}--\frac {(2-2 i) \int \frac {1}{\left (i \sqrt {1+\sqrt {5}}-\sqrt {2} x\right ) \sqrt {1+i x^2}} \, dx}{\sqrt {5 \left (1+\sqrt {5}\right )}}--\frac {(2-2 i) \int \frac {1}{\left (i \sqrt {1+\sqrt {5}}+\sqrt {2} x\right ) \sqrt {1+i x^2}} \, dx}{\sqrt {5 \left (1+\sqrt {5}\right )}}+-\frac {\left (\frac {4}{5}-\frac {4 i}{5}\right ) \int \frac {1}{\left (i \sqrt {1+\sqrt {5}}-\sqrt {2} x\right ) \sqrt {1+i x^2}} \, dx}{\sqrt {5 \left (1+\sqrt {5}\right )}}+-\frac {\left (\frac {4}{5}-\frac {4 i}{5}\right ) \int \frac {1}{\left (i \sqrt {1+\sqrt {5}}+\sqrt {2} x\right ) \sqrt {1+i x^2}} \, dx}{\sqrt {5 \left (1+\sqrt {5}\right )}}+\frac {\left (\frac {4}{5}+\frac {4 i}{5}\right ) \int \frac {1}{\left (i \sqrt {1+\sqrt {5}}-\sqrt {2} x\right ) \sqrt {1-i x^2}} \, dx}{\sqrt {5 \left (1+\sqrt {5}\right )}}+\frac {\left (\frac {4}{5}+\frac {4 i}{5}\right ) \int \frac {1}{\left (i \sqrt {1+\sqrt {5}}+\sqrt {2} x\right ) \sqrt {1-i x^2}} \, dx}{\sqrt {5 \left (1+\sqrt {5}\right )}}-\frac {(2+2 i) \int \frac {1}{\left (i \sqrt {1+\sqrt {5}}-\sqrt {2} x\right ) \sqrt {1-i x^2}} \, dx}{\sqrt {5 \left (1+\sqrt {5}\right )}}-\frac {(2+2 i) \int \frac {1}{\left (i \sqrt {1+\sqrt {5}}+\sqrt {2} x\right ) \sqrt {1-i x^2}} \, dx}{\sqrt {5 \left (1+\sqrt {5}\right )}}\\ &=\frac {2 i \sqrt {1-i x^2}}{\left ((10+5 i)-5 \sqrt {5}\right ) \left (\sqrt {2 \left (-1+\sqrt {5}\right )}-2 x\right )}+\frac {2 i \sqrt {1-i x^2}}{5 \left ((2+i)+\sqrt {5}\right ) \left (i \sqrt {2 \left (1+\sqrt {5}\right )}-2 x\right )}+\frac {2 i \sqrt {1-i x^2}}{5 \left ((-2-i)+\sqrt {5}\right ) \left (\sqrt {2 \left (-1+\sqrt {5}\right )}+2 x\right )}-\frac {2 i \sqrt {1-i x^2}}{5 \left ((2+i)+\sqrt {5}\right ) \left (i \sqrt {2 \left (1+\sqrt {5}\right )}+2 x\right )}+\frac {2 i \sqrt {1+i x^2}}{5 \left ((-2+i)+\sqrt {5}\right ) \left (\sqrt {2 \left (-1+\sqrt {5}\right )}-2 x\right )}-\frac {2 i \sqrt {1+i x^2}}{5 \left ((2-i)+\sqrt {5}\right ) \left (i \sqrt {2 \left (1+\sqrt {5}\right )}-2 x\right )}-\frac {2 i \sqrt {1+i x^2}}{5 \left ((-2+i)+\sqrt {5}\right ) \left (\sqrt {2 \left (-1+\sqrt {5}\right )}+2 x\right )}+\frac {2 i \sqrt {1+i x^2}}{5 \left ((2-i)+\sqrt {5}\right ) \left (i \sqrt {2 \left (1+\sqrt {5}\right )}+2 x\right )}+\frac {\tanh ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {x^2+\sqrt {1+x^4}}}\right )}{\sqrt {2}}+-\frac {\left ((2-2 i) \sqrt {\frac {2}{-1+\sqrt {5}}}\right ) \int \frac {1}{\left (\sqrt {2 \left (-1+\sqrt {5}\right )}-2 x\right ) \sqrt {1-i x^2}} \, dx}{(5-10 i)-5 \sqrt {5}}+-\frac {\left ((2-2 i) \sqrt {\frac {2}{-1+\sqrt {5}}}\right ) \int \frac {1}{\left (\sqrt {2 \left (-1+\sqrt {5}\right )}+2 x\right ) \sqrt {1-i x^2}} \, dx}{(5-10 i)-5 \sqrt {5}}+-\frac {\left ((2+2 i) \sqrt {\frac {2}{-1+\sqrt {5}}}\right ) \int \frac {1}{\left (\sqrt {2 \left (-1+\sqrt {5}\right )}-2 x\right ) \sqrt {1+i x^2}} \, dx}{(5+10 i)-5 \sqrt {5}}+-\frac {\left ((2+2 i) \sqrt {\frac {2}{-1+\sqrt {5}}}\right ) \int \frac {1}{\left (\sqrt {2 \left (-1+\sqrt {5}\right )}+2 x\right ) \sqrt {1+i x^2}} \, dx}{(5+10 i)-5 \sqrt {5}}+\frac {4 \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{\left (\sqrt {-1+\sqrt {5}}-\sqrt {2} x\right ) \sqrt {1+x^4}} \, dx}{5 \left (-1+\sqrt {5}\right )^{3/2}}+\frac {4 \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{\left (\sqrt {-1+\sqrt {5}}+\sqrt {2} x\right ) \sqrt {1+x^4}} \, dx}{5 \left (-1+\sqrt {5}\right )^{3/2}}--\frac {(2+2 i) \operatorname {Subst}\left (\int \frac {1}{2+i \left (-1+\sqrt {5}\right )-x^2} \, dx,x,\frac {-\sqrt {2}-i \sqrt {-1+\sqrt {5}} x}{\sqrt {1+i x^2}}\right )}{\sqrt {5 \left (-1+\sqrt {5}\right )}}--\frac {(2+2 i) \operatorname {Subst}\left (\int \frac {1}{2+i \left (-1+\sqrt {5}\right )-x^2} \, dx,x,\frac {\sqrt {2}-i \sqrt {-1+\sqrt {5}} x}{\sqrt {1+i x^2}}\right )}{\sqrt {5 \left (-1+\sqrt {5}\right )}}--\frac {(2-2 i) \operatorname {Subst}\left (\int \frac {1}{2-i \left (-1+\sqrt {5}\right )-x^2} \, dx,x,\frac {-\sqrt {2}+i \sqrt {-1+\sqrt {5}} x}{\sqrt {1-i x^2}}\right )}{\sqrt {5 \left (-1+\sqrt {5}\right )}}--\frac {(2-2 i) \operatorname {Subst}\left (\int \frac {1}{2-i \left (-1+\sqrt {5}\right )-x^2} \, dx,x,\frac {\sqrt {2}+i \sqrt {-1+\sqrt {5}} x}{\sqrt {1-i x^2}}\right )}{\sqrt {5 \left (-1+\sqrt {5}\right )}}+-\frac {\left (\frac {4}{5}+\frac {4 i}{5}\right ) \operatorname {Subst}\left (\int \frac {1}{2+i \left (-1+\sqrt {5}\right )-x^2} \, dx,x,\frac {-\sqrt {2}-i \sqrt {-1+\sqrt {5}} x}{\sqrt {1+i x^2}}\right )}{\sqrt {5 \left (-1+\sqrt {5}\right )}}+-\frac {\left (\frac {4}{5}+\frac {4 i}{5}\right ) \operatorname {Subst}\left (\int \frac {1}{2+i \left (-1+\sqrt {5}\right )-x^2} \, dx,x,\frac {\sqrt {2}-i \sqrt {-1+\sqrt {5}} x}{\sqrt {1+i x^2}}\right )}{\sqrt {5 \left (-1+\sqrt {5}\right )}}+-\frac {\left (\frac {4}{5}-\frac {4 i}{5}\right ) \operatorname {Subst}\left (\int \frac {1}{2-i \left (-1+\sqrt {5}\right )-x^2} \, dx,x,\frac {-\sqrt {2}+i \sqrt {-1+\sqrt {5}} x}{\sqrt {1-i x^2}}\right )}{\sqrt {5 \left (-1+\sqrt {5}\right )}}+-\frac {\left (\frac {4}{5}-\frac {4 i}{5}\right ) \operatorname {Subst}\left (\int \frac {1}{2-i \left (-1+\sqrt {5}\right )-x^2} \, dx,x,\frac {\sqrt {2}+i \sqrt {-1+\sqrt {5}} x}{\sqrt {1-i x^2}}\right )}{\sqrt {5 \left (-1+\sqrt {5}\right )}}+\frac {(4 i) \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{\left (i \sqrt {1+\sqrt {5}}-\sqrt {2} x\right ) \sqrt {1+x^4}} \, dx}{5 \left (1+\sqrt {5}\right )^{3/2}}+\frac {(4 i) \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{\left (i \sqrt {1+\sqrt {5}}+\sqrt {2} x\right ) \sqrt {1+x^4}} \, dx}{5 \left (1+\sqrt {5}\right )^{3/2}}--\frac {(2+2 i) \operatorname {Subst}\left (\int \frac {1}{2-i \left (-1-\sqrt {5}\right )-x^2} \, dx,x,\frac {-\sqrt {2}-\sqrt {1+\sqrt {5}} x}{\sqrt {1-i x^2}}\right )}{\sqrt {5 \left (1+\sqrt {5}\right )}}--\frac {(2+2 i) \operatorname {Subst}\left (\int \frac {1}{2-i \left (-1-\sqrt {5}\right )-x^2} \, dx,x,\frac {\sqrt {2}-\sqrt {1+\sqrt {5}} x}{\sqrt {1-i x^2}}\right )}{\sqrt {5 \left (1+\sqrt {5}\right )}}+-\frac {\left (\frac {4}{5}+\frac {4 i}{5}\right ) \operatorname {Subst}\left (\int \frac {1}{2-i \left (-1-\sqrt {5}\right )-x^2} \, dx,x,\frac {-\sqrt {2}-\sqrt {1+\sqrt {5}} x}{\sqrt {1-i x^2}}\right )}{\sqrt {5 \left (1+\sqrt {5}\right )}}+-\frac {\left (\frac {4}{5}+\frac {4 i}{5}\right ) \operatorname {Subst}\left (\int \frac {1}{2-i \left (-1-\sqrt {5}\right )-x^2} \, dx,x,\frac {\sqrt {2}-\sqrt {1+\sqrt {5}} x}{\sqrt {1-i x^2}}\right )}{\sqrt {5 \left (1+\sqrt {5}\right )}}+\frac {\left (\frac {4}{5}-\frac {4 i}{5}\right ) \operatorname {Subst}\left (\int \frac {1}{2+i \left (-1-\sqrt {5}\right )-x^2} \, dx,x,\frac {-\sqrt {2}+\sqrt {1+\sqrt {5}} x}{\sqrt {1+i x^2}}\right )}{\sqrt {5 \left (1+\sqrt {5}\right )}}+\frac {\left (\frac {4}{5}-\frac {4 i}{5}\right ) \operatorname {Subst}\left (\int \frac {1}{2+i \left (-1-\sqrt {5}\right )-x^2} \, dx,x,\frac {\sqrt {2}+\sqrt {1+\sqrt {5}} x}{\sqrt {1+i x^2}}\right )}{\sqrt {5 \left (1+\sqrt {5}\right )}}-\frac {(2-2 i) \operatorname {Subst}\left (\int \frac {1}{2+i \left (-1-\sqrt {5}\right )-x^2} \, dx,x,\frac {-\sqrt {2}+\sqrt {1+\sqrt {5}} x}{\sqrt {1+i x^2}}\right )}{\sqrt {5 \left (1+\sqrt {5}\right )}}-\frac {(2-2 i) \operatorname {Subst}\left (\int \frac {1}{2+i \left (-1-\sqrt {5}\right )-x^2} \, dx,x,\frac {\sqrt {2}+\sqrt {1+\sqrt {5}} x}{\sqrt {1+i x^2}}\right )}{\sqrt {5 \left (1+\sqrt {5}\right )}}--\frac {\left ((2+2 i) \sqrt {\frac {2}{1+\sqrt {5}}}\right ) \int \frac {1}{\left (i \sqrt {2 \left (1+\sqrt {5}\right )}-2 x\right ) \sqrt {1-i x^2}} \, dx}{(5-10 i)+5 \sqrt {5}}--\frac {\left ((2+2 i) \sqrt {\frac {2}{1+\sqrt {5}}}\right ) \int \frac {1}{\left (i \sqrt {2 \left (1+\sqrt {5}\right )}+2 x\right ) \sqrt {1-i x^2}} \, dx}{(5-10 i)+5 \sqrt {5}}-\frac {\left ((2-2 i) \sqrt {\frac {2}{1+\sqrt {5}}}\right ) \int \frac {1}{\left (i \sqrt {2 \left (1+\sqrt {5}\right )}-2 x\right ) \sqrt {1+i x^2}} \, dx}{(5+10 i)+5 \sqrt {5}}-\frac {\left ((2-2 i) \sqrt {\frac {2}{1+\sqrt {5}}}\right ) \int \frac {1}{\left (i \sqrt {2 \left (1+\sqrt {5}\right )}+2 x\right ) \sqrt {1+i x^2}} \, dx}{(5+10 i)+5 \sqrt {5}}\\ &=\frac {2 i \sqrt {1-i x^2}}{\left ((10+5 i)-5 \sqrt {5}\right ) \left (\sqrt {2 \left (-1+\sqrt {5}\right )}-2 x\right )}+\frac {2 i \sqrt {1-i x^2}}{5 \left ((2+i)+\sqrt {5}\right ) \left (i \sqrt {2 \left (1+\sqrt {5}\right )}-2 x\right )}+\frac {2 i \sqrt {1-i x^2}}{5 \left ((-2-i)+\sqrt {5}\right ) \left (\sqrt {2 \left (-1+\sqrt {5}\right )}+2 x\right )}-\frac {2 i \sqrt {1-i x^2}}{5 \left ((2+i)+\sqrt {5}\right ) \left (i \sqrt {2 \left (1+\sqrt {5}\right )}+2 x\right )}+\frac {2 i \sqrt {1+i x^2}}{5 \left ((-2+i)+\sqrt {5}\right ) \left (\sqrt {2 \left (-1+\sqrt {5}\right )}-2 x\right )}-\frac {2 i \sqrt {1+i x^2}}{5 \left ((2-i)+\sqrt {5}\right ) \left (i \sqrt {2 \left (1+\sqrt {5}\right )}-2 x\right )}-\frac {2 i \sqrt {1+i x^2}}{5 \left ((-2+i)+\sqrt {5}\right ) \left (\sqrt {2 \left (-1+\sqrt {5}\right )}+2 x\right )}+\frac {2 i \sqrt {1+i x^2}}{5 \left ((2-i)+\sqrt {5}\right ) \left (i \sqrt {2 \left (1+\sqrt {5}\right )}+2 x\right )}-\frac {6 i \tan ^{-1}\left (\frac {(1+i) \left (\sqrt {2}-i \sqrt {-1+\sqrt {5}} x\right )}{\sqrt {2 \left ((-1-2 i)+\sqrt {5}\right )} \sqrt {1+i x^2}}\right )}{5 \sqrt {(-5-5 i) \left ((-2+i)+\sqrt {5}\right )}}+\frac {6 i \tan ^{-1}\left (\frac {(1+i) \left (\sqrt {2}+i \sqrt {-1+\sqrt {5}} x\right )}{\sqrt {2 \left ((-1-2 i)+\sqrt {5}\right )} \sqrt {1+i x^2}}\right )}{5 \sqrt {(-5-5 i) \left ((-2+i)+\sqrt {5}\right )}}-\frac {6 \tan ^{-1}\left (\frac {(1+i) \left (\sqrt {2}-\sqrt {1+\sqrt {5}} x\right )}{\sqrt {(-2-4 i)-2 \sqrt {5}} \sqrt {1+i x^2}}\right )}{5 \sqrt {(-5-5 i) \left ((2-i)+\sqrt {5}\right )}}+\frac {6 \tan ^{-1}\left (\frac {(1+i) \left (\sqrt {2}+\sqrt {1+\sqrt {5}} x\right )}{\sqrt {(-2-4 i)-2 \sqrt {5}} \sqrt {1+i x^2}}\right )}{5 \sqrt {(-5-5 i) \left ((2-i)+\sqrt {5}\right )}}-\frac {\left (\frac {3}{5}-\frac {3 i}{5}\right ) \tanh ^{-1}\left (\frac {\sqrt {2}-i \sqrt {-1+\sqrt {5}} x}{\sqrt {(2+i)-i \sqrt {5}} \sqrt {1-i x^2}}\right )}{\sqrt {\left (\frac {5}{2}+\frac {5 i}{2}\right ) \left ((-2-i)+\sqrt {5}\right )}}+\frac {\left (\frac {3}{5}-\frac {3 i}{5}\right ) \tanh ^{-1}\left (\frac {\sqrt {2}+i \sqrt {-1+\sqrt {5}} x}{\sqrt {(2+i)-i \sqrt {5}} \sqrt {1-i x^2}}\right )}{\sqrt {\left (\frac {5}{2}+\frac {5 i}{2}\right ) \left ((-2-i)+\sqrt {5}\right )}}+\frac {\left (\frac {3}{5}+\frac {3 i}{5}\right ) \tanh ^{-1}\left (\frac {\sqrt {2}-\sqrt {1+\sqrt {5}} x}{\sqrt {(2+i)+i \sqrt {5}} \sqrt {1-i x^2}}\right )}{\sqrt {\left (\frac {5}{2}+\frac {5 i}{2}\right ) \left ((2+i)+\sqrt {5}\right )}}-\frac {\left (\frac {3}{5}+\frac {3 i}{5}\right ) \tanh ^{-1}\left (\frac {\sqrt {2}+\sqrt {1+\sqrt {5}} x}{\sqrt {(2+i)+i \sqrt {5}} \sqrt {1-i x^2}}\right )}{\sqrt {\left (\frac {5}{2}+\frac {5 i}{2}\right ) \left ((2+i)+\sqrt {5}\right )}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {x^2+\sqrt {1+x^4}}}\right )}{\sqrt {2}}+\frac {\left ((2-2 i) \sqrt {\frac {2}{-1+\sqrt {5}}}\right ) \operatorname {Subst}\left (\int \frac {1}{4-2 i \left (-1+\sqrt {5}\right )-x^2} \, dx,x,\frac {-2+i \sqrt {2 \left (-1+\sqrt {5}\right )} x}{\sqrt {1-i x^2}}\right )}{(5-10 i)-5 \sqrt {5}}+\frac {\left ((2-2 i) \sqrt {\frac {2}{-1+\sqrt {5}}}\right ) \operatorname {Subst}\left (\int \frac {1}{4-2 i \left (-1+\sqrt {5}\right )-x^2} \, dx,x,\frac {2+i \sqrt {2 \left (-1+\sqrt {5}\right )} x}{\sqrt {1-i x^2}}\right )}{(5-10 i)-5 \sqrt {5}}+\frac {\left ((2+2 i) \sqrt {\frac {2}{-1+\sqrt {5}}}\right ) \operatorname {Subst}\left (\int \frac {1}{4+2 i \left (-1+\sqrt {5}\right )-x^2} \, dx,x,\frac {-2-i \sqrt {2 \left (-1+\sqrt {5}\right )} x}{\sqrt {1+i x^2}}\right )}{(5+10 i)-5 \sqrt {5}}+\frac {\left ((2+2 i) \sqrt {\frac {2}{-1+\sqrt {5}}}\right ) \operatorname {Subst}\left (\int \frac {1}{4+2 i \left (-1+\sqrt {5}\right )-x^2} \, dx,x,\frac {2-i \sqrt {2 \left (-1+\sqrt {5}\right )} x}{\sqrt {1+i x^2}}\right )}{(5+10 i)-5 \sqrt {5}}+\frac {\left (\frac {2}{5}-\frac {2 i}{5}\right ) \int \frac {1}{\left (\sqrt {-1+\sqrt {5}}-\sqrt {2} x\right ) \sqrt {1-i x^2}} \, dx}{\left (-1+\sqrt {5}\right )^{3/2}}+\frac {\left (\frac {2}{5}-\frac {2 i}{5}\right ) \int \frac {1}{\left (\sqrt {-1+\sqrt {5}}+\sqrt {2} x\right ) \sqrt {1-i x^2}} \, dx}{\left (-1+\sqrt {5}\right )^{3/2}}+\frac {\left (\frac {2}{5}+\frac {2 i}{5}\right ) \int \frac {1}{\left (\sqrt {-1+\sqrt {5}}-\sqrt {2} x\right ) \sqrt {1+i x^2}} \, dx}{\left (-1+\sqrt {5}\right )^{3/2}}+\frac {\left (\frac {2}{5}+\frac {2 i}{5}\right ) \int \frac {1}{\left (\sqrt {-1+\sqrt {5}}+\sqrt {2} x\right ) \sqrt {1+i x^2}} \, dx}{\left (-1+\sqrt {5}\right )^{3/2}}+-\frac {\left (\frac {2}{5}-\frac {2 i}{5}\right ) \int \frac {1}{\left (i \sqrt {1+\sqrt {5}}-\sqrt {2} x\right ) \sqrt {1+i x^2}} \, dx}{\left (1+\sqrt {5}\right )^{3/2}}+-\frac {\left (\frac {2}{5}-\frac {2 i}{5}\right ) \int \frac {1}{\left (i \sqrt {1+\sqrt {5}}+\sqrt {2} x\right ) \sqrt {1+i x^2}} \, dx}{\left (1+\sqrt {5}\right )^{3/2}}+\frac {\left (\frac {2}{5}+\frac {2 i}{5}\right ) \int \frac {1}{\left (i \sqrt {1+\sqrt {5}}-\sqrt {2} x\right ) \sqrt {1-i x^2}} \, dx}{\left (1+\sqrt {5}\right )^{3/2}}+\frac {\left (\frac {2}{5}+\frac {2 i}{5}\right ) \int \frac {1}{\left (i \sqrt {1+\sqrt {5}}+\sqrt {2} x\right ) \sqrt {1-i x^2}} \, dx}{\left (1+\sqrt {5}\right )^{3/2}}-\frac {\left ((2+2 i) \sqrt {\frac {2}{1+\sqrt {5}}}\right ) \operatorname {Subst}\left (\int \frac {1}{4+2 i \left (1+\sqrt {5}\right )-x^2} \, dx,x,\frac {-2-\sqrt {2 \left (1+\sqrt {5}\right )} x}{\sqrt {1-i x^2}}\right )}{(5-10 i)+5 \sqrt {5}}-\frac {\left ((2+2 i) \sqrt {\frac {2}{1+\sqrt {5}}}\right ) \operatorname {Subst}\left (\int \frac {1}{4+2 i \left (1+\sqrt {5}\right )-x^2} \, dx,x,\frac {2-\sqrt {2 \left (1+\sqrt {5}\right )} x}{\sqrt {1-i x^2}}\right )}{(5-10 i)+5 \sqrt {5}}--\frac {\left ((2-2 i) \sqrt {\frac {2}{1+\sqrt {5}}}\right ) \operatorname {Subst}\left (\int \frac {1}{4-2 i \left (1+\sqrt {5}\right )-x^2} \, dx,x,\frac {-2+\sqrt {2 \left (1+\sqrt {5}\right )} x}{\sqrt {1+i x^2}}\right )}{(5+10 i)+5 \sqrt {5}}--\frac {\left ((2-2 i) \sqrt {\frac {2}{1+\sqrt {5}}}\right ) \operatorname {Subst}\left (\int \frac {1}{4-2 i \left (1+\sqrt {5}\right )-x^2} \, dx,x,\frac {2+\sqrt {2 \left (1+\sqrt {5}\right )} x}{\sqrt {1+i x^2}}\right )}{(5+10 i)+5 \sqrt {5}}\\ &=\frac {2 i \sqrt {1-i x^2}}{\left ((10+5 i)-5 \sqrt {5}\right ) \left (\sqrt {2 \left (-1+\sqrt {5}\right )}-2 x\right )}+\frac {2 i \sqrt {1-i x^2}}{5 \left ((2+i)+\sqrt {5}\right ) \left (i \sqrt {2 \left (1+\sqrt {5}\right )}-2 x\right )}+\frac {2 i \sqrt {1-i x^2}}{5 \left ((-2-i)+\sqrt {5}\right ) \left (\sqrt {2 \left (-1+\sqrt {5}\right )}+2 x\right )}-\frac {2 i \sqrt {1-i x^2}}{5 \left ((2+i)+\sqrt {5}\right ) \left (i \sqrt {2 \left (1+\sqrt {5}\right )}+2 x\right )}+\frac {2 i \sqrt {1+i x^2}}{5 \left ((-2+i)+\sqrt {5}\right ) \left (\sqrt {2 \left (-1+\sqrt {5}\right )}-2 x\right )}-\frac {2 i \sqrt {1+i x^2}}{5 \left ((2-i)+\sqrt {5}\right ) \left (i \sqrt {2 \left (1+\sqrt {5}\right )}-2 x\right )}-\frac {2 i \sqrt {1+i x^2}}{5 \left ((-2+i)+\sqrt {5}\right ) \left (\sqrt {2 \left (-1+\sqrt {5}\right )}+2 x\right )}+\frac {2 i \sqrt {1+i x^2}}{5 \left ((2-i)+\sqrt {5}\right ) \left (i \sqrt {2 \left (1+\sqrt {5}\right )}+2 x\right )}-\frac {6 i \tan ^{-1}\left (\frac {(1+i) \left (\sqrt {2}-i \sqrt {-1+\sqrt {5}} x\right )}{\sqrt {2 \left ((-1-2 i)+\sqrt {5}\right )} \sqrt {1+i x^2}}\right )}{5 \sqrt {(-5-5 i) \left ((-2+i)+\sqrt {5}\right )}}+\frac {6 i \tan ^{-1}\left (\frac {(1+i) \left (\sqrt {2}+i \sqrt {-1+\sqrt {5}} x\right )}{\sqrt {2 \left ((-1-2 i)+\sqrt {5}\right )} \sqrt {1+i x^2}}\right )}{5 \sqrt {(-5-5 i) \left ((-2+i)+\sqrt {5}\right )}}+\frac {2 i \tan ^{-1}\left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \left (2-i \sqrt {2 \left (-1+\sqrt {5}\right )} x\right )}{\sqrt {(-1-2 i)+\sqrt {5}} \sqrt {1+i x^2}}\right )}{5 \sqrt {(3+i)-(1+i) \sqrt {5}} \left ((-1-2 i)+\sqrt {5}\right )}-\frac {2 i \tan ^{-1}\left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \left (2+i \sqrt {2 \left (-1+\sqrt {5}\right )} x\right )}{\sqrt {(-1-2 i)+\sqrt {5}} \sqrt {1+i x^2}}\right )}{5 \sqrt {(3+i)-(1+i) \sqrt {5}} \left ((-1-2 i)+\sqrt {5}\right )}-\frac {6 \tan ^{-1}\left (\frac {(1+i) \left (\sqrt {2}-\sqrt {1+\sqrt {5}} x\right )}{\sqrt {(-2-4 i)-2 \sqrt {5}} \sqrt {1+i x^2}}\right )}{5 \sqrt {(-5-5 i) \left ((2-i)+\sqrt {5}\right )}}+\frac {6 \tan ^{-1}\left (\frac {(1+i) \left (\sqrt {2}+\sqrt {1+\sqrt {5}} x\right )}{\sqrt {(-2-4 i)-2 \sqrt {5}} \sqrt {1+i x^2}}\right )}{5 \sqrt {(-5-5 i) \left ((2-i)+\sqrt {5}\right )}}+\frac {2 \tan ^{-1}\left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \left (2-\sqrt {2 \left (1+\sqrt {5}\right )} x\right )}{\sqrt {(-1-2 i)-\sqrt {5}} \sqrt {1+i x^2}}\right )}{5 \left ((1+2 i)+\sqrt {5}\right ) \sqrt {(-1-i) \left ((2-i)+\sqrt {5}\right )}}-\frac {2 \tan ^{-1}\left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \left (2+\sqrt {2 \left (1+\sqrt {5}\right )} x\right )}{\sqrt {(-1-2 i)-\sqrt {5}} \sqrt {1+i x^2}}\right )}{5 \left ((1+2 i)+\sqrt {5}\right ) \sqrt {(-1-i) \left ((2-i)+\sqrt {5}\right )}}-\frac {\left (\frac {3}{5}-\frac {3 i}{5}\right ) \tanh ^{-1}\left (\frac {\sqrt {2}-i \sqrt {-1+\sqrt {5}} x}{\sqrt {(2+i)-i \sqrt {5}} \sqrt {1-i x^2}}\right )}{\sqrt {\left (\frac {5}{2}+\frac {5 i}{2}\right ) \left ((-2-i)+\sqrt {5}\right )}}+\frac {\left (\frac {3}{5}-\frac {3 i}{5}\right ) \tanh ^{-1}\left (\frac {\sqrt {2}+i \sqrt {-1+\sqrt {5}} x}{\sqrt {(2+i)-i \sqrt {5}} \sqrt {1-i x^2}}\right )}{\sqrt {\left (\frac {5}{2}+\frac {5 i}{2}\right ) \left ((-2-i)+\sqrt {5}\right )}}+\frac {\left (\frac {1}{25}-\frac {3 i}{25}\right ) \sqrt {\left (\frac {1}{2}+\frac {i}{2}\right ) \left ((-2-i)+\sqrt {5}\right )} \tanh ^{-1}\left (\frac {2-i \sqrt {2 \left (-1+\sqrt {5}\right )} x}{\sqrt {(4+2 i)-2 i \sqrt {5}} \sqrt {1-i x^2}}\right )}{3-\sqrt {5}}-\frac {\left (\frac {1}{25}-\frac {3 i}{25}\right ) \sqrt {\left (\frac {1}{2}+\frac {i}{2}\right ) \left ((-2-i)+\sqrt {5}\right )} \tanh ^{-1}\left (\frac {2+i \sqrt {2 \left (-1+\sqrt {5}\right )} x}{\sqrt {(4+2 i)-2 i \sqrt {5}} \sqrt {1-i x^2}}\right )}{3-\sqrt {5}}+\frac {\left (\frac {3}{5}+\frac {3 i}{5}\right ) \tanh ^{-1}\left (\frac {\sqrt {2}-\sqrt {1+\sqrt {5}} x}{\sqrt {(2+i)+i \sqrt {5}} \sqrt {1-i x^2}}\right )}{\sqrt {\left (\frac {5}{2}+\frac {5 i}{2}\right ) \left ((2+i)+\sqrt {5}\right )}}-\frac {\left (\frac {3}{5}+\frac {3 i}{5}\right ) \tanh ^{-1}\left (\frac {\sqrt {2}+\sqrt {1+\sqrt {5}} x}{\sqrt {(2+i)+i \sqrt {5}} \sqrt {1-i x^2}}\right )}{\sqrt {\left (\frac {5}{2}+\frac {5 i}{2}\right ) \left ((2+i)+\sqrt {5}\right )}}-\frac {\left (\frac {3}{25}+\frac {i}{25}\right ) \sqrt {\left (\frac {1}{2}+\frac {i}{2}\right ) \left ((2+i)+\sqrt {5}\right )} \tanh ^{-1}\left (\frac {2-\sqrt {2 \left (1+\sqrt {5}\right )} x}{\sqrt {(4+2 i)+2 i \sqrt {5}} \sqrt {1-i x^2}}\right )}{3+\sqrt {5}}+\frac {\left (\frac {3}{25}+\frac {i}{25}\right ) \sqrt {\left (\frac {1}{2}+\frac {i}{2}\right ) \left ((2+i)+\sqrt {5}\right )} \tanh ^{-1}\left (\frac {2+\sqrt {2 \left (1+\sqrt {5}\right )} x}{\sqrt {(4+2 i)+2 i \sqrt {5}} \sqrt {1-i x^2}}\right )}{3+\sqrt {5}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {x^2+\sqrt {1+x^4}}}\right )}{\sqrt {2}}+-\frac {\left (\frac {2}{5}+\frac {2 i}{5}\right ) \operatorname {Subst}\left (\int \frac {1}{2+i \left (-1+\sqrt {5}\right )-x^2} \, dx,x,\frac {-\sqrt {2}-i \sqrt {-1+\sqrt {5}} x}{\sqrt {1+i x^2}}\right )}{\left (-1+\sqrt {5}\right )^{3/2}}+-\frac {\left (\frac {2}{5}+\frac {2 i}{5}\right ) \operatorname {Subst}\left (\int \frac {1}{2+i \left (-1+\sqrt {5}\right )-x^2} \, dx,x,\frac {\sqrt {2}-i \sqrt {-1+\sqrt {5}} x}{\sqrt {1+i x^2}}\right )}{\left (-1+\sqrt {5}\right )^{3/2}}+-\frac {\left (\frac {2}{5}-\frac {2 i}{5}\right ) \operatorname {Subst}\left (\int \frac {1}{2-i \left (-1+\sqrt {5}\right )-x^2} \, dx,x,\frac {-\sqrt {2}+i \sqrt {-1+\sqrt {5}} x}{\sqrt {1-i x^2}}\right )}{\left (-1+\sqrt {5}\right )^{3/2}}+-\frac {\left (\frac {2}{5}-\frac {2 i}{5}\right ) \operatorname {Subst}\left (\int \frac {1}{2-i \left (-1+\sqrt {5}\right )-x^2} \, dx,x,\frac {\sqrt {2}+i \sqrt {-1+\sqrt {5}} x}{\sqrt {1-i x^2}}\right )}{\left (-1+\sqrt {5}\right )^{3/2}}+-\frac {\left (\frac {2}{5}+\frac {2 i}{5}\right ) \operatorname {Subst}\left (\int \frac {1}{2-i \left (-1-\sqrt {5}\right )-x^2} \, dx,x,\frac {-\sqrt {2}-\sqrt {1+\sqrt {5}} x}{\sqrt {1-i x^2}}\right )}{\left (1+\sqrt {5}\right )^{3/2}}+-\frac {\left (\frac {2}{5}+\frac {2 i}{5}\right ) \operatorname {Subst}\left (\int \frac {1}{2-i \left (-1-\sqrt {5}\right )-x^2} \, dx,x,\frac {\sqrt {2}-\sqrt {1+\sqrt {5}} x}{\sqrt {1-i x^2}}\right )}{\left (1+\sqrt {5}\right )^{3/2}}+\frac {\left (\frac {2}{5}-\frac {2 i}{5}\right ) \operatorname {Subst}\left (\int \frac {1}{2+i \left (-1-\sqrt {5}\right )-x^2} \, dx,x,\frac {-\sqrt {2}+\sqrt {1+\sqrt {5}} x}{\sqrt {1+i x^2}}\right )}{\left (1+\sqrt {5}\right )^{3/2}}+\frac {\left (\frac {2}{5}-\frac {2 i}{5}\right ) \operatorname {Subst}\left (\int \frac {1}{2+i \left (-1-\sqrt {5}\right )-x^2} \, dx,x,\frac {\sqrt {2}+\sqrt {1+\sqrt {5}} x}{\sqrt {1+i x^2}}\right )}{\left (1+\sqrt {5}\right )^{3/2}}\\ &=\frac {2 i \sqrt {1-i x^2}}{\left ((10+5 i)-5 \sqrt {5}\right ) \left (\sqrt {2 \left (-1+\sqrt {5}\right )}-2 x\right )}+\frac {2 i \sqrt {1-i x^2}}{5 \left ((2+i)+\sqrt {5}\right ) \left (i \sqrt {2 \left (1+\sqrt {5}\right )}-2 x\right )}+\frac {2 i \sqrt {1-i x^2}}{5 \left ((-2-i)+\sqrt {5}\right ) \left (\sqrt {2 \left (-1+\sqrt {5}\right )}+2 x\right )}-\frac {2 i \sqrt {1-i x^2}}{5 \left ((2+i)+\sqrt {5}\right ) \left (i \sqrt {2 \left (1+\sqrt {5}\right )}+2 x\right )}+\frac {2 i \sqrt {1+i x^2}}{5 \left ((-2+i)+\sqrt {5}\right ) \left (\sqrt {2 \left (-1+\sqrt {5}\right )}-2 x\right )}-\frac {2 i \sqrt {1+i x^2}}{5 \left ((2-i)+\sqrt {5}\right ) \left (i \sqrt {2 \left (1+\sqrt {5}\right )}-2 x\right )}-\frac {2 i \sqrt {1+i x^2}}{5 \left ((-2+i)+\sqrt {5}\right ) \left (\sqrt {2 \left (-1+\sqrt {5}\right )}+2 x\right )}+\frac {2 i \sqrt {1+i x^2}}{5 \left ((2-i)+\sqrt {5}\right ) \left (i \sqrt {2 \left (1+\sqrt {5}\right )}+2 x\right )}-\frac {6 i \tan ^{-1}\left (\frac {(1+i) \left (\sqrt {2}-i \sqrt {-1+\sqrt {5}} x\right )}{\sqrt {2 \left ((-1-2 i)+\sqrt {5}\right )} \sqrt {1+i x^2}}\right )}{5 \sqrt {(-5-5 i) \left ((-2+i)+\sqrt {5}\right )}}+\frac {2 i \tan ^{-1}\left (\frac {(1+i) \left (\sqrt {2}-i \sqrt {-1+\sqrt {5}} x\right )}{\sqrt {2 \left ((-1-2 i)+\sqrt {5}\right )} \sqrt {1+i x^2}}\right )}{5 \left (-1+\sqrt {5}\right )^{3/2} \sqrt {\frac {1}{2} \left ((-1-2 i)+\sqrt {5}\right )}}+\frac {6 i \tan ^{-1}\left (\frac {(1+i) \left (\sqrt {2}+i \sqrt {-1+\sqrt {5}} x\right )}{\sqrt {2 \left ((-1-2 i)+\sqrt {5}\right )} \sqrt {1+i x^2}}\right )}{5 \sqrt {(-5-5 i) \left ((-2+i)+\sqrt {5}\right )}}-\frac {2 i \tan ^{-1}\left (\frac {(1+i) \left (\sqrt {2}+i \sqrt {-1+\sqrt {5}} x\right )}{\sqrt {2 \left ((-1-2 i)+\sqrt {5}\right )} \sqrt {1+i x^2}}\right )}{5 \left (-1+\sqrt {5}\right )^{3/2} \sqrt {\frac {1}{2} \left ((-1-2 i)+\sqrt {5}\right )}}+\frac {2 i \tan ^{-1}\left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \left (2-i \sqrt {2 \left (-1+\sqrt {5}\right )} x\right )}{\sqrt {(-1-2 i)+\sqrt {5}} \sqrt {1+i x^2}}\right )}{5 \sqrt {(3+i)-(1+i) \sqrt {5}} \left ((-1-2 i)+\sqrt {5}\right )}-\frac {2 i \tan ^{-1}\left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \left (2+i \sqrt {2 \left (-1+\sqrt {5}\right )} x\right )}{\sqrt {(-1-2 i)+\sqrt {5}} \sqrt {1+i x^2}}\right )}{5 \sqrt {(3+i)-(1+i) \sqrt {5}} \left ((-1-2 i)+\sqrt {5}\right )}+\frac {4 \tan ^{-1}\left (\frac {(1+i) \left (\sqrt {2}-\sqrt {1+\sqrt {5}} x\right )}{\sqrt {(-2-4 i)-2 \sqrt {5}} \sqrt {1+i x^2}}\right )}{5 \sqrt {(-2-4 i)-2 \sqrt {5}} \left (1+\sqrt {5}\right )^{3/2}}-\frac {6 \tan ^{-1}\left (\frac {(1+i) \left (\sqrt {2}-\sqrt {1+\sqrt {5}} x\right )}{\sqrt {(-2-4 i)-2 \sqrt {5}} \sqrt {1+i x^2}}\right )}{5 \sqrt {(-5-5 i) \left ((2-i)+\sqrt {5}\right )}}-\frac {4 \tan ^{-1}\left (\frac {(1+i) \left (\sqrt {2}+\sqrt {1+\sqrt {5}} x\right )}{\sqrt {(-2-4 i)-2 \sqrt {5}} \sqrt {1+i x^2}}\right )}{5 \sqrt {(-2-4 i)-2 \sqrt {5}} \left (1+\sqrt {5}\right )^{3/2}}+\frac {6 \tan ^{-1}\left (\frac {(1+i) \left (\sqrt {2}+\sqrt {1+\sqrt {5}} x\right )}{\sqrt {(-2-4 i)-2 \sqrt {5}} \sqrt {1+i x^2}}\right )}{5 \sqrt {(-5-5 i) \left ((2-i)+\sqrt {5}\right )}}+\frac {2 \tan ^{-1}\left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \left (2-\sqrt {2 \left (1+\sqrt {5}\right )} x\right )}{\sqrt {(-1-2 i)-\sqrt {5}} \sqrt {1+i x^2}}\right )}{5 \left ((1+2 i)+\sqrt {5}\right ) \sqrt {(-1-i) \left ((2-i)+\sqrt {5}\right )}}-\frac {2 \tan ^{-1}\left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \left (2+\sqrt {2 \left (1+\sqrt {5}\right )} x\right )}{\sqrt {(-1-2 i)-\sqrt {5}} \sqrt {1+i x^2}}\right )}{5 \left ((1+2 i)+\sqrt {5}\right ) \sqrt {(-1-i) \left ((2-i)+\sqrt {5}\right )}}-\frac {\left (\frac {3}{5}-\frac {3 i}{5}\right ) \tanh ^{-1}\left (\frac {\sqrt {2}-i \sqrt {-1+\sqrt {5}} x}{\sqrt {(2+i)-i \sqrt {5}} \sqrt {1-i x^2}}\right )}{\sqrt {\left (\frac {5}{2}+\frac {5 i}{2}\right ) \left ((-2-i)+\sqrt {5}\right )}}+\frac {\left (\frac {2}{5}-\frac {2 i}{5}\right ) \tanh ^{-1}\left (\frac {\sqrt {2}-i \sqrt {-1+\sqrt {5}} x}{\sqrt {(2+i)-i \sqrt {5}} \sqrt {1-i x^2}}\right )}{\sqrt {(2+i)-i \sqrt {5}} \left (-1+\sqrt {5}\right )^{3/2}}+\frac {\left (\frac {3}{5}-\frac {3 i}{5}\right ) \tanh ^{-1}\left (\frac {\sqrt {2}+i \sqrt {-1+\sqrt {5}} x}{\sqrt {(2+i)-i \sqrt {5}} \sqrt {1-i x^2}}\right )}{\sqrt {\left (\frac {5}{2}+\frac {5 i}{2}\right ) \left ((-2-i)+\sqrt {5}\right )}}-\frac {\left (\frac {2}{5}-\frac {2 i}{5}\right ) \tanh ^{-1}\left (\frac {\sqrt {2}+i \sqrt {-1+\sqrt {5}} x}{\sqrt {(2+i)-i \sqrt {5}} \sqrt {1-i x^2}}\right )}{\sqrt {(2+i)-i \sqrt {5}} \left (-1+\sqrt {5}\right )^{3/2}}+\frac {\left (\frac {1}{25}-\frac {3 i}{25}\right ) \sqrt {\left (\frac {1}{2}+\frac {i}{2}\right ) \left ((-2-i)+\sqrt {5}\right )} \tanh ^{-1}\left (\frac {2-i \sqrt {2 \left (-1+\sqrt {5}\right )} x}{\sqrt {(4+2 i)-2 i \sqrt {5}} \sqrt {1-i x^2}}\right )}{3-\sqrt {5}}-\frac {\left (\frac {1}{25}-\frac {3 i}{25}\right ) \sqrt {\left (\frac {1}{2}+\frac {i}{2}\right ) \left ((-2-i)+\sqrt {5}\right )} \tanh ^{-1}\left (\frac {2+i \sqrt {2 \left (-1+\sqrt {5}\right )} x}{\sqrt {(4+2 i)-2 i \sqrt {5}} \sqrt {1-i x^2}}\right )}{3-\sqrt {5}}-\frac {\left (\frac {2}{5}+\frac {2 i}{5}\right ) \tanh ^{-1}\left (\frac {\sqrt {2}-\sqrt {1+\sqrt {5}} x}{\sqrt {(2+i)+i \sqrt {5}} \sqrt {1-i x^2}}\right )}{\sqrt {(2+i)+i \sqrt {5}} \left (1+\sqrt {5}\right )^{3/2}}+\frac {\left (\frac {3}{5}+\frac {3 i}{5}\right ) \tanh ^{-1}\left (\frac {\sqrt {2}-\sqrt {1+\sqrt {5}} x}{\sqrt {(2+i)+i \sqrt {5}} \sqrt {1-i x^2}}\right )}{\sqrt {\left (\frac {5}{2}+\frac {5 i}{2}\right ) \left ((2+i)+\sqrt {5}\right )}}+\frac {\left (\frac {2}{5}+\frac {2 i}{5}\right ) \tanh ^{-1}\left (\frac {\sqrt {2}+\sqrt {1+\sqrt {5}} x}{\sqrt {(2+i)+i \sqrt {5}} \sqrt {1-i x^2}}\right )}{\sqrt {(2+i)+i \sqrt {5}} \left (1+\sqrt {5}\right )^{3/2}}-\frac {\left (\frac {3}{5}+\frac {3 i}{5}\right ) \tanh ^{-1}\left (\frac {\sqrt {2}+\sqrt {1+\sqrt {5}} x}{\sqrt {(2+i)+i \sqrt {5}} \sqrt {1-i x^2}}\right )}{\sqrt {\left (\frac {5}{2}+\frac {5 i}{2}\right ) \left ((2+i)+\sqrt {5}\right )}}-\frac {\left (\frac {3}{25}+\frac {i}{25}\right ) \sqrt {\left (\frac {1}{2}+\frac {i}{2}\right ) \left ((2+i)+\sqrt {5}\right )} \tanh ^{-1}\left (\frac {2-\sqrt {2 \left (1+\sqrt {5}\right )} x}{\sqrt {(4+2 i)+2 i \sqrt {5}} \sqrt {1-i x^2}}\right )}{3+\sqrt {5}}+\frac {\left (\frac {3}{25}+\frac {i}{25}\right ) \sqrt {\left (\frac {1}{2}+\frac {i}{2}\right ) \left ((2+i)+\sqrt {5}\right )} \tanh ^{-1}\left (\frac {2+\sqrt {2 \left (1+\sqrt {5}\right )} x}{\sqrt {(4+2 i)+2 i \sqrt {5}} \sqrt {1-i x^2}}\right )}{3+\sqrt {5}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {x^2+\sqrt {1+x^4}}}\right )}{\sqrt {2}}\\ \end {align*}

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Mathematica [F]  time = 1.51, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (1+x^2+x^4\right )^2 \sqrt {x^2+\sqrt {1+x^4}}}{\sqrt {1+x^4} \left (-1+x^2+x^4\right )^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.00, size = 828, normalized size = 1.59 \begin {gather*} \frac {-2 x \sqrt {1+x^4} \left (2+3 x^2+2 x^4\right ) \sqrt {x^2+\sqrt {1+x^4}}-2 x \left (1+3 x^2+3 x^4+2 x^6\right ) \sqrt {x^2+\sqrt {1+x^4}}}{10 x^2 \sqrt {1+x^4} \left (-1+x^2+x^4\right )+5 \left (-1+x^2+x^4\right ) \left (1+2 x^4\right )}+\sqrt {2} \tanh ^{-1}\left (\frac {-\frac {1}{\sqrt {2}}+\frac {x^2}{\sqrt {2}}+\frac {\sqrt {1+x^4}}{\sqrt {2}}}{x \sqrt {x^2+\sqrt {1+x^4}}}\right )+2 \sqrt {2} \text {RootSum}\left [1-2 \text {$\#$1}^2-6 \text {$\#$1}^4+2 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {-8 \log \left (1+x^2+\sqrt {1+x^4}\right )+8 \log \left (\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}-\text {$\#$1}-x^2 \text {$\#$1}-\sqrt {1+x^4} \text {$\#$1}\right )+3 \log \left (1+x^2+\sqrt {1+x^4}\right ) \text {$\#$1}^2-3 \log \left (\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}-\text {$\#$1}-x^2 \text {$\#$1}-\sqrt {1+x^4} \text {$\#$1}\right ) \text {$\#$1}^2-\log \left (1+x^2+\sqrt {1+x^4}\right ) \text {$\#$1}^4+\log \left (\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}-\text {$\#$1}-x^2 \text {$\#$1}-\sqrt {1+x^4} \text {$\#$1}\right ) \text {$\#$1}^4}{-\text {$\#$1}-6 \text {$\#$1}^3+3 \text {$\#$1}^5+2 \text {$\#$1}^7}\&\right ]+\frac {\text {RootSum}\left [1-2 \text {$\#$1}^2-6 \text {$\#$1}^4+2 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {163 \log \left (1+x^2+\sqrt {1+x^4}\right )-163 \log \left (\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}-\text {$\#$1}-x^2 \text {$\#$1}-\sqrt {1+x^4} \text {$\#$1}\right )-59 \log \left (1+x^2+\sqrt {1+x^4}\right ) \text {$\#$1}^2+59 \log \left (\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}-\text {$\#$1}-x^2 \text {$\#$1}-\sqrt {1+x^4} \text {$\#$1}\right ) \text {$\#$1}^2+13 \log \left (1+x^2+\sqrt {1+x^4}\right ) \text {$\#$1}^4-13 \log \left (\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}-\text {$\#$1}-x^2 \text {$\#$1}-\sqrt {1+x^4} \text {$\#$1}\right ) \text {$\#$1}^4-\log \left (1+x^2+\sqrt {1+x^4}\right ) \text {$\#$1}^6+\log \left (\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}-\text {$\#$1}-x^2 \text {$\#$1}-\sqrt {1+x^4} \text {$\#$1}\right ) \text {$\#$1}^6}{-\text {$\#$1}-6 \text {$\#$1}^3+3 \text {$\#$1}^5+2 \text {$\#$1}^7}\&\right ]}{5 \sqrt {2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*x*Sqrt[1 + x^4]*(2 + 3*x^2 + 2*x^4)*Sqrt[x^2 + Sqrt[1 + x^4]] - 2*x*(1 + 3*x^2 + 3*x^4 + 2*x^6)*Sqrt[x^2 +
Sqrt[1 + x^4]])/(10*x^2*Sqrt[1 + x^4]*(-1 + x^2 + x^4) + 5*(-1 + x^2 + x^4)*(1 + 2*x^4)) + Sqrt[2]*ArcTanh[(-
(1/Sqrt[2]) + x^2/Sqrt[2] + Sqrt[1 + x^4]/Sqrt[2])/(x*Sqrt[x^2 + Sqrt[1 + x^4]])] + 2*Sqrt[2]*RootSum[1 - 2*#1
^2 - 6*#1^4 + 2*#1^6 + #1^8 & , (-8*Log[1 + x^2 + Sqrt[1 + x^4]] + 8*Log[Sqrt[2]*x*Sqrt[x^2 + Sqrt[1 + x^4]] -
#1 - x^2*#1 - Sqrt[1 + x^4]*#1] + 3*Log[1 + x^2 + Sqrt[1 + x^4]]*#1^2 - 3*Log[Sqrt[2]*x*Sqrt[x^2 + Sqrt[1 + x
^4]] - #1 - x^2*#1 - Sqrt[1 + x^4]*#1]*#1^2 - Log[1 + x^2 + Sqrt[1 + x^4]]*#1^4 + Log[Sqrt[2]*x*Sqrt[x^2 + Sqr
t[1 + x^4]] - #1 - x^2*#1 - Sqrt[1 + x^4]*#1]*#1^4)/(-#1 - 6*#1^3 + 3*#1^5 + 2*#1^7) & ] + RootSum[1 - 2*#1^2
- 6*#1^4 + 2*#1^6 + #1^8 & , (163*Log[1 + x^2 + Sqrt[1 + x^4]] - 163*Log[Sqrt[2]*x*Sqrt[x^2 + Sqrt[1 + x^4]] -
#1 - x^2*#1 - Sqrt[1 + x^4]*#1] - 59*Log[1 + x^2 + Sqrt[1 + x^4]]*#1^2 + 59*Log[Sqrt[2]*x*Sqrt[x^2 + Sqrt[1 +
x^4]] - #1 - x^2*#1 - Sqrt[1 + x^4]*#1]*#1^2 + 13*Log[1 + x^2 + Sqrt[1 + x^4]]*#1^4 - 13*Log[Sqrt[2]*x*Sqrt[x
^2 + Sqrt[1 + x^4]] - #1 - x^2*#1 - Sqrt[1 + x^4]*#1]*#1^4 - Log[1 + x^2 + Sqrt[1 + x^4]]*#1^6 + Log[Sqrt[2]*x
*Sqrt[x^2 + Sqrt[1 + x^4]] - #1 - x^2*#1 - Sqrt[1 + x^4]*#1]*#1^6)/(-#1 - 6*#1^3 + 3*#1^5 + 2*#1^7) & ]/(5*Sqr
t[2])

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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