\(\int \frac {1}{x^3 (-1+x^2)^{3/4}} \, dx\) [1279]

Optimal result

Integrand size = 13, antiderivative size = 93 \[ \int \frac {1}{x^3 \left (-1+x^2\right )^{3/4}} \, dx=\frac {\sqrt [4]{-1+x^2}}{2 x^2}-\frac {3 \arctan \left (\frac {\sqrt {2} \sqrt [4]{-1+x^2}}{-1+\sqrt {-1+x^2}}\right )}{4 \sqrt {2}}+\frac {3 \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{-1+x^2}}{1+\sqrt {-1+x^2}}\right )}{4 \sqrt {2}} \]

[Out]

1/2*(x^2-1)^(1/4)/x^2-3/8*arctan(2^(1/2)*(x^2-1)^(1/4)/(-1+(x^2-1)^(1/2)))*2^(1/2)+3/8*arctanh(2^(1/2)*(x^2-1)
^(1/4)/(1+(x^2-1)^(1/2)))*2^(1/2)

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.56, number of steps used = 12, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.692, Rules used = {272, 44, 65, 217, 1179, 642, 1176, 631, 210} \[ \int \frac {1}{x^3 \left (-1+x^2\right )^{3/4}} \, dx=-\frac {3 \arctan \left (1-\sqrt {2} \sqrt [4]{x^2-1}\right )}{4 \sqrt {2}}+\frac {3 \arctan \left (\sqrt {2} \sqrt [4]{x^2-1}+1\right )}{4 \sqrt {2}}+\frac {\sqrt [4]{x^2-1}}{2 x^2}-\frac {3 \log \left (\sqrt {x^2-1}-\sqrt {2} \sqrt [4]{x^2-1}+1\right )}{8 \sqrt {2}}+\frac {3 \log \left (\sqrt {x^2-1}+\sqrt {2} \sqrt [4]{x^2-1}+1\right )}{8 \sqrt {2}} \]

[In]

Int[1/(x^3*(-1 + x^2)^(3/4)),x]

[Out]

(-1 + x^2)^(1/4)/(2*x^2) - (3*ArcTan[1 - Sqrt[2]*(-1 + x^2)^(1/4)])/(4*Sqrt[2]) + (3*ArcTan[1 + Sqrt[2]*(-1 +
x^2)^(1/4)])/(4*Sqrt[2]) - (3*Log[1 - Sqrt[2]*(-1 + x^2)^(1/4) + Sqrt[-1 + x^2]])/(8*Sqrt[2]) + (3*Log[1 + Sqr
t[2]*(-1 + x^2)^(1/4) + Sqrt[-1 + x^2]])/(8*Sqrt[2])

Rule 44

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n + 1
)/((b*c - a*d)*(m + 1))), x] - Dist[d*((m + n + 2)/((b*c - a*d)*(m + 1))), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, -1] &&  !IntegerQ[n] && LtQ[n, 0]

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 210

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

Rule 217

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]}, Di
st[1/(2*r), Int[(r - s*x^2)/(a + b*x^4), x], x] + Dist[1/(2*r), Int[(r + s*x^2)/(a + b*x^4), x], x]] /; FreeQ[
{a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] && AtomQ[SplitProduct[SumBaseQ, b
]]))

Rule 272

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 631

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[a*(c/b^2)]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 642

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[d*(Log[RemoveContent[a + b*x +
c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 1176

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

Rule 1179

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

Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {1}{(-1+x)^{3/4} x^2} \, dx,x,x^2\right ) \\ & = \frac {\sqrt [4]{-1+x^2}}{2 x^2}+\frac {3}{8} \text {Subst}\left (\int \frac {1}{(-1+x)^{3/4} x} \, dx,x,x^2\right ) \\ & = \frac {\sqrt [4]{-1+x^2}}{2 x^2}+\frac {3}{2} \text {Subst}\left (\int \frac {1}{1+x^4} \, dx,x,\sqrt [4]{-1+x^2}\right ) \\ & = \frac {\sqrt [4]{-1+x^2}}{2 x^2}+\frac {3}{4} \text {Subst}\left (\int \frac {1-x^2}{1+x^4} \, dx,x,\sqrt [4]{-1+x^2}\right )+\frac {3}{4} \text {Subst}\left (\int \frac {1+x^2}{1+x^4} \, dx,x,\sqrt [4]{-1+x^2}\right ) \\ & = \frac {\sqrt [4]{-1+x^2}}{2 x^2}+\frac {3}{8} \text {Subst}\left (\int \frac {1}{1-\sqrt {2} x+x^2} \, dx,x,\sqrt [4]{-1+x^2}\right )+\frac {3}{8} \text {Subst}\left (\int \frac {1}{1+\sqrt {2} x+x^2} \, dx,x,\sqrt [4]{-1+x^2}\right )-\frac {3 \text {Subst}\left (\int \frac {\sqrt {2}+2 x}{-1-\sqrt {2} x-x^2} \, dx,x,\sqrt [4]{-1+x^2}\right )}{8 \sqrt {2}}-\frac {3 \text {Subst}\left (\int \frac {\sqrt {2}-2 x}{-1+\sqrt {2} x-x^2} \, dx,x,\sqrt [4]{-1+x^2}\right )}{8 \sqrt {2}} \\ & = \frac {\sqrt [4]{-1+x^2}}{2 x^2}-\frac {3 \log \left (1-\sqrt {2} \sqrt [4]{-1+x^2}+\sqrt {-1+x^2}\right )}{8 \sqrt {2}}+\frac {3 \log \left (1+\sqrt {2} \sqrt [4]{-1+x^2}+\sqrt {-1+x^2}\right )}{8 \sqrt {2}}+\frac {3 \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\sqrt {2} \sqrt [4]{-1+x^2}\right )}{4 \sqrt {2}}-\frac {3 \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\sqrt {2} \sqrt [4]{-1+x^2}\right )}{4 \sqrt {2}} \\ & = \frac {\sqrt [4]{-1+x^2}}{2 x^2}-\frac {3 \arctan \left (1-\sqrt {2} \sqrt [4]{-1+x^2}\right )}{4 \sqrt {2}}+\frac {3 \arctan \left (1+\sqrt {2} \sqrt [4]{-1+x^2}\right )}{4 \sqrt {2}}-\frac {3 \log \left (1-\sqrt {2} \sqrt [4]{-1+x^2}+\sqrt {-1+x^2}\right )}{8 \sqrt {2}}+\frac {3 \log \left (1+\sqrt {2} \sqrt [4]{-1+x^2}+\sqrt {-1+x^2}\right )}{8 \sqrt {2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.15 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.96 \[ \int \frac {1}{x^3 \left (-1+x^2\right )^{3/4}} \, dx=\frac {1}{8} \left (\frac {4 \sqrt [4]{-1+x^2}}{x^2}+3 \sqrt {2} \arctan \left (\frac {-1+\sqrt {-1+x^2}}{\sqrt {2} \sqrt [4]{-1+x^2}}\right )+3 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{-1+x^2}}{1+\sqrt {-1+x^2}}\right )\right ) \]

[In]

Integrate[1/(x^3*(-1 + x^2)^(3/4)),x]

[Out]

((4*(-1 + x^2)^(1/4))/x^2 + 3*Sqrt[2]*ArcTan[(-1 + Sqrt[-1 + x^2])/(Sqrt[2]*(-1 + x^2)^(1/4))] + 3*Sqrt[2]*Arc
Tanh[(Sqrt[2]*(-1 + x^2)^(1/4))/(1 + Sqrt[-1 + x^2])])/8

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 3.79 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.76

method	result	size
meijerg	\(-\frac {{\left (-\operatorname {signum}\left (x^{2}-1\right )\right )}^{\frac {3}{4}} \left (-\frac {21 \Gamma \left (\frac {3}{4}\right ) x^{2} \operatorname {hypergeom}\left (\left [1, 1, \frac {11}{4}\right ], \left [2, 3\right ], x^{2}\right )}{32}-\frac {3 \left (\frac {1}{3}-3 \ln \left (2\right )+\frac {\pi }{2}+2 \ln \left (x \right )+i \pi \right ) \Gamma \left (\frac {3}{4}\right )}{4}+\frac {\Gamma \left (\frac {3}{4}\right )}{x^{2}}\right )}{2 \Gamma \left (\frac {3}{4}\right ) \operatorname {signum}\left (x^{2}-1\right )^{\frac {3}{4}}}\)	\(71\)
risch	\(\frac {\left (x^{2}-1\right )^{\frac {1}{4}}}{2 x^{2}}+\frac {3 {\left (-\operatorname {signum}\left (x^{2}-1\right )\right )}^{\frac {3}{4}} \left (\frac {3 \Gamma \left (\frac {3}{4}\right ) x^{2} \operatorname {hypergeom}\left (\left [1, 1, \frac {7}{4}\right ], \left [2, 2\right ], x^{2}\right )}{4}+\left (-3 \ln \left (2\right )+\frac {\pi }{2}+2 \ln \left (x \right )+i \pi \right ) \Gamma \left (\frac {3}{4}\right )\right )}{8 \Gamma \left (\frac {3}{4}\right ) \operatorname {signum}\left (x^{2}-1\right )^{\frac {3}{4}}}\)	\(76\)
pseudoelliptic	\(\frac {3 \ln \left (\frac {-\sqrt {2}\, \left (x^{2}-1\right )^{\frac {1}{4}}-\sqrt {x^{2}-1}-1}{\sqrt {2}\, \left (x^{2}-1\right )^{\frac {1}{4}}-\sqrt {x^{2}-1}-1}\right ) \sqrt {2}\, x^{2}+6 \arctan \left (\sqrt {2}\, \left (x^{2}-1\right )^{\frac {1}{4}}+1\right ) \sqrt {2}\, x^{2}+6 \arctan \left (\sqrt {2}\, \left (x^{2}-1\right )^{\frac {1}{4}}-1\right ) \sqrt {2}\, x^{2}+8 \left (x^{2}-1\right )^{\frac {1}{4}}}{16 x^{2}}\)	\(117\)
trager	\(\frac {\left (x^{2}-1\right )^{\frac {1}{4}}}{2 x^{2}}-\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) \ln \left (-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} x^{2}+2 \left (x^{2}-1\right )^{\frac {3}{4}}-2 \sqrt {x^{2}-1}\, \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )+2 \left (x^{2}-1\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2}-2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3}}{x^{2}}\right )}{8}+\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} \ln \left (-\frac {2 \sqrt {x^{2}-1}\, \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3}-2 \left (x^{2}-1\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2}-\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) x^{2}+2 \left (x^{2}-1\right )^{\frac {3}{4}}+2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )}{x^{2}}\right )}{8}\)	\(171\)

[In]

int(1/x^3/(x^2-1)^(3/4),x,method=_RETURNVERBOSE)

[Out]

-1/2/GAMMA(3/4)/signum(x^2-1)^(3/4)*(-signum(x^2-1))^(3/4)*(-21/32*GAMMA(3/4)*x^2*hypergeom([1,1,11/4],[2,3],x
^2)-3/4*(1/3-3*ln(2)+1/2*Pi+2*ln(x)+I*Pi)*GAMMA(3/4)+GAMMA(3/4)/x^2)

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.25 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.19 \[ \int \frac {1}{x^3 \left (-1+x^2\right )^{3/4}} \, dx=\frac {\left (3 i + 3\right ) \, \sqrt {2} x^{2} \log \left (\left (i + 1\right ) \, \sqrt {2} + 2 \, {\left (x^{2} - 1\right )}^{\frac {1}{4}}\right ) - \left (3 i - 3\right ) \, \sqrt {2} x^{2} \log \left (-\left (i - 1\right ) \, \sqrt {2} + 2 \, {\left (x^{2} - 1\right )}^{\frac {1}{4}}\right ) + \left (3 i - 3\right ) \, \sqrt {2} x^{2} \log \left (\left (i - 1\right ) \, \sqrt {2} + 2 \, {\left (x^{2} - 1\right )}^{\frac {1}{4}}\right ) - \left (3 i + 3\right ) \, \sqrt {2} x^{2} \log \left (-\left (i + 1\right ) \, \sqrt {2} + 2 \, {\left (x^{2} - 1\right )}^{\frac {1}{4}}\right ) + 8 \, {\left (x^{2} - 1\right )}^{\frac {1}{4}}}{16 \, x^{2}} \]

[In]

integrate(1/x^3/(x^2-1)^(3/4),x, algorithm="fricas")

[Out]

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

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.75 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.37 \[ \int \frac {1}{x^3 \left (-1+x^2\right )^{3/4}} \, dx=- \frac {\Gamma \left (\frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{4}, \frac {7}{4} \\ \frac {11}{4} \end {matrix}\middle | {\frac {e^{2 i \pi }}{x^{2}}} \right )}}{2 x^{\frac {7}{2}} \Gamma \left (\frac {11}{4}\right )} \]

[In]

integrate(1/x**3/(x**2-1)**(3/4),x)

[Out]

-gamma(7/4)*hyper((3/4, 7/4), (11/4,), exp_polar(2*I*pi)/x**2)/(2*x**(7/2)*gamma(11/4))

Maxima [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.23 \[ \int \frac {1}{x^3 \left (-1+x^2\right )^{3/4}} \, dx=\frac {3}{8} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, {\left (x^{2} - 1\right )}^{\frac {1}{4}}\right )}\right ) + \frac {3}{8} \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, {\left (x^{2} - 1\right )}^{\frac {1}{4}}\right )}\right ) + \frac {3}{16} \, \sqrt {2} \log \left (\sqrt {2} {\left (x^{2} - 1\right )}^{\frac {1}{4}} + \sqrt {x^{2} - 1} + 1\right ) - \frac {3}{16} \, \sqrt {2} \log \left (-\sqrt {2} {\left (x^{2} - 1\right )}^{\frac {1}{4}} + \sqrt {x^{2} - 1} + 1\right ) + \frac {{\left (x^{2} - 1\right )}^{\frac {1}{4}}}{2 \, x^{2}} \]

[In]

integrate(1/x^3/(x^2-1)^(3/4),x, algorithm="maxima")

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.23 \[ \int \frac {1}{x^3 \left (-1+x^2\right )^{3/4}} \, dx=\frac {3}{8} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, {\left (x^{2} - 1\right )}^{\frac {1}{4}}\right )}\right ) + \frac {3}{8} \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, {\left (x^{2} - 1\right )}^{\frac {1}{4}}\right )}\right ) + \frac {3}{16} \, \sqrt {2} \log \left (\sqrt {2} {\left (x^{2} - 1\right )}^{\frac {1}{4}} + \sqrt {x^{2} - 1} + 1\right ) - \frac {3}{16} \, \sqrt {2} \log \left (-\sqrt {2} {\left (x^{2} - 1\right )}^{\frac {1}{4}} + \sqrt {x^{2} - 1} + 1\right ) + \frac {{\left (x^{2} - 1\right )}^{\frac {1}{4}}}{2 \, x^{2}} \]

[In]

integrate(1/x^3/(x^2-1)^(3/4),x, algorithm="giac")

[Out]

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

Mupad [B] (verification not implemented)

Time = 5.85 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.61 \[ \int \frac {1}{x^3 \left (-1+x^2\right )^{3/4}} \, dx=\frac {{\left (x^2-1\right )}^{1/4}}{2\,x^2}+\sqrt {2}\,\mathrm {atan}\left (\sqrt {2}\,{\left (x^2-1\right )}^{1/4}\,\left (\frac {1}{2}-\frac {1}{2}{}\mathrm {i}\right )\right )\,\left (\frac {3}{8}+\frac {3}{8}{}\mathrm {i}\right )+\sqrt {2}\,\mathrm {atan}\left (\sqrt {2}\,{\left (x^2-1\right )}^{1/4}\,\left (\frac {1}{2}+\frac {1}{2}{}\mathrm {i}\right )\right )\,\left (\frac {3}{8}-\frac {3}{8}{}\mathrm {i}\right ) \]

[In]

int(1/(x^3*(x^2 - 1)^(3/4)),x)

[Out]

2^(1/2)*atan(2^(1/2)*(x^2 - 1)^(1/4)*(1/2 - 1i/2))*(3/8 + 3i/8) + 2^(1/2)*atan(2^(1/2)*(x^2 - 1)^(1/4)*(1/2 +
1i/2))*(3/8 - 3i/8) + (x^2 - 1)^(1/4)/(2*x^2)