\(\int \frac {1}{x \sqrt [4]{-1+x^4}} \, dx\) [1014]

Optimal result

Integrand size = 13, antiderivative size = 77 \[ \int \frac {1}{x \sqrt [4]{-1+x^4}} \, dx=-\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{-1+x^4}}{-1+\sqrt {-1+x^4}}\right )}{2 \sqrt {2}}-\frac {\text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{-1+x^4}}{1+\sqrt {-1+x^4}}\right )}{2 \sqrt {2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.13 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.68, number of steps used = 11, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.615, Rules used = {272, 65, 303, 1176, 631, 210, 1179, 642} \[ \int \frac {1}{x \sqrt [4]{-1+x^4}} \, dx=-\frac {\arctan \left (1-\sqrt {2} \sqrt [4]{x^4-1}\right )}{2 \sqrt {2}}+\frac {\arctan \left (\sqrt {2} \sqrt [4]{x^4-1}+1\right )}{2 \sqrt {2}}+\frac {\log \left (\sqrt {x^4-1}-\sqrt {2} \sqrt [4]{x^4-1}+1\right )}{4 \sqrt {2}}-\frac {\log \left (\sqrt {x^4-1}+\sqrt {2} \sqrt [4]{x^4-1}+1\right )}{4 \sqrt {2}} \]

[In]

Int[1/(x*(-1 + x^4)^(1/4)),x]

[Out]

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

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 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 303

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

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

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.88 \[ \int \frac {1}{x \sqrt [4]{-1+x^4}} \, dx=\frac {\arctan \left (\frac {-1+\sqrt {-1+x^4}}{\sqrt {2} \sqrt [4]{-1+x^4}}\right )-\text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{-1+x^4}}{1+\sqrt {-1+x^4}}\right )}{2 \sqrt {2}} \]

[In]

Integrate[1/(x*(-1 + x^4)^(1/4)),x]

[Out]

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

Maple [C] (warning: unable to verify)

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

Time = 6.66 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.03

method	result	size
meijerg	\(\frac {\sqrt {2}\, \Gamma \left (\frac {3}{4}\right ) {\left (-\operatorname {signum}\left (x^{4}-1\right )\right )}^{\frac {1}{4}} \left (\frac {\pi \sqrt {2}\, x^{4} \operatorname {hypergeom}\left (\left [1, 1, \frac {5}{4}\right ], \left [2, 2\right ], x^{4}\right )}{4 \Gamma \left (\frac {3}{4}\right )}+\frac {\left (-3 \ln \left (2\right )-\frac {\pi }{2}+4 \ln \left (x \right )+i \pi \right ) \pi \sqrt {2}}{\Gamma \left (\frac {3}{4}\right )}\right )}{8 \pi \operatorname {signum}\left (x^{4}-1\right )^{\frac {1}{4}}}\)	\(79\)
pseudoelliptic	\(\frac {\sqrt {2}\, \left (\ln \left (\frac {\sqrt {x^{4}-1}-\left (x^{4}-1\right )^{\frac {1}{4}} \sqrt {2}+1}{\sqrt {x^{4}-1}+\left (x^{4}-1\right )^{\frac {1}{4}} \sqrt {2}+1}\right )+2 \arctan \left (\left (x^{4}-1\right )^{\frac {1}{4}} \sqrt {2}+1\right )+2 \arctan \left (\left (x^{4}-1\right )^{\frac {1}{4}} \sqrt {2}-1\right )\right )}{8}\)	\(84\)
trager	\(\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} \ln \left (-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} x^{4}-2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3}-2 \left (x^{4}-1\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2}-2 \sqrt {x^{4}-1}\, \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )-2 \left (x^{4}-1\right )^{\frac {3}{4}}}{x^{4}}\right )}{4}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) \ln \left (\frac {2 \sqrt {x^{4}-1}\, \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3}-\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) x^{4}-2 \left (x^{4}-1\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2}+2 \left (x^{4}-1\right )^{\frac {3}{4}}+2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )}{x^{4}}\right )}{4}\)	\(158\)

[In]

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

[Out]

1/8/Pi*2^(1/2)*GAMMA(3/4)/signum(x^4-1)^(1/4)*(-signum(x^4-1))^(1/4)*(1/4*Pi*2^(1/2)/GAMMA(3/4)*x^4*hypergeom(
[1,1,5/4],[2,2],x^4)+(-3*ln(2)-1/2*Pi+4*ln(x)+I*Pi)*Pi*2^(1/2)/GAMMA(3/4))

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.26 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.10 \[ \int \frac {1}{x \sqrt [4]{-1+x^4}} \, dx=\left (\frac {1}{8} i - \frac {1}{8}\right ) \, \sqrt {2} \log \left (\left (i + 1\right ) \, \sqrt {2} + 2 \, {\left (x^{4} - 1\right )}^{\frac {1}{4}}\right ) - \left (\frac {1}{8} i + \frac {1}{8}\right ) \, \sqrt {2} \log \left (-\left (i - 1\right ) \, \sqrt {2} + 2 \, {\left (x^{4} - 1\right )}^{\frac {1}{4}}\right ) + \left (\frac {1}{8} i + \frac {1}{8}\right ) \, \sqrt {2} \log \left (\left (i - 1\right ) \, \sqrt {2} + 2 \, {\left (x^{4} - 1\right )}^{\frac {1}{4}}\right ) - \left (\frac {1}{8} i - \frac {1}{8}\right ) \, \sqrt {2} \log \left (-\left (i + 1\right ) \, \sqrt {2} + 2 \, {\left (x^{4} - 1\right )}^{\frac {1}{4}}\right ) \]

[In]

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

[Out]

(1/8*I - 1/8)*sqrt(2)*log((I + 1)*sqrt(2) + 2*(x^4 - 1)^(1/4)) - (1/8*I + 1/8)*sqrt(2)*log(-(I - 1)*sqrt(2) +
2*(x^4 - 1)^(1/4)) + (1/8*I + 1/8)*sqrt(2)*log((I - 1)*sqrt(2) + 2*(x^4 - 1)^(1/4)) - (1/8*I - 1/8)*sqrt(2)*lo
g(-(I + 1)*sqrt(2) + 2*(x^4 - 1)^(1/4))

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.48 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.40 \[ \int \frac {1}{x \sqrt [4]{-1+x^4}} \, dx=- \frac {\Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {1}{4} \\ \frac {5}{4} \end {matrix}\middle | {\frac {e^{2 i \pi }}{x^{4}}} \right )}}{4 x \Gamma \left (\frac {5}{4}\right )} \]

[In]

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

[Out]

-gamma(1/4)*hyper((1/4, 1/4), (5/4,), exp_polar(2*I*pi)/x**4)/(4*x*gamma(5/4))

Maxima [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.32 \[ \int \frac {1}{x \sqrt [4]{-1+x^4}} \, dx=\frac {1}{4} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, {\left (x^{4} - 1\right )}^{\frac {1}{4}}\right )}\right ) + \frac {1}{4} \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, {\left (x^{4} - 1\right )}^{\frac {1}{4}}\right )}\right ) - \frac {1}{8} \, \sqrt {2} \log \left (\sqrt {2} {\left (x^{4} - 1\right )}^{\frac {1}{4}} + \sqrt {x^{4} - 1} + 1\right ) + \frac {1}{8} \, \sqrt {2} \log \left (-\sqrt {2} {\left (x^{4} - 1\right )}^{\frac {1}{4}} + \sqrt {x^{4} - 1} + 1\right ) \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.32 \[ \int \frac {1}{x \sqrt [4]{-1+x^4}} \, dx=\frac {1}{4} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, {\left (x^{4} - 1\right )}^{\frac {1}{4}}\right )}\right ) + \frac {1}{4} \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, {\left (x^{4} - 1\right )}^{\frac {1}{4}}\right )}\right ) - \frac {1}{8} \, \sqrt {2} \log \left (\sqrt {2} {\left (x^{4} - 1\right )}^{\frac {1}{4}} + \sqrt {x^{4} - 1} + 1\right ) + \frac {1}{8} \, \sqrt {2} \log \left (-\sqrt {2} {\left (x^{4} - 1\right )}^{\frac {1}{4}} + \sqrt {x^{4} - 1} + 1\right ) \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 5.95 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.58 \[ \int \frac {1}{x \sqrt [4]{-1+x^4}} \, dx=\sqrt {2}\,\mathrm {atan}\left (\sqrt {2}\,{\left (x^4-1\right )}^{1/4}\,\left (\frac {1}{2}-\frac {1}{2}{}\mathrm {i}\right )\right )\,\left (\frac {1}{4}-\frac {1}{4}{}\mathrm {i}\right )+\sqrt {2}\,\mathrm {atan}\left (\sqrt {2}\,{\left (x^4-1\right )}^{1/4}\,\left (\frac {1}{2}+\frac {1}{2}{}\mathrm {i}\right )\right )\,\left (\frac {1}{4}+\frac {1}{4}{}\mathrm {i}\right ) \]

[In]

int(1/(x*(x^4 - 1)^(1/4)),x)

[Out]

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