\(\int \frac {\sqrt [4]{-1+x^6}}{x} \, dx\) [1245]

Optimal result

Integrand size = 13, antiderivative size = 90 \[ \int \frac {\sqrt [4]{-1+x^6}}{x} \, dx=\frac {2}{3} \sqrt [4]{-1+x^6}+\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{-1+x^6}}{-1+\sqrt {-1+x^6}}\right )}{3 \sqrt {2}}-\frac {\text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{-1+x^6}}{1+\sqrt {-1+x^6}}\right )}{3 \sqrt {2}} \]

[Out]

2/3*(x^6-1)^(1/4)+1/6*arctan(2^(1/2)*(x^6-1)^(1/4)/(-1+(x^6-1)^(1/2)))*2^(1/2)-1/6*arctanh(2^(1/2)*(x^6-1)^(1/
4)/(1+(x^6-1)^(1/2)))*2^(1/2)

Rubi [A] (verified)

Time = 0.10 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.58, number of steps used = 12, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.692, Rules used = {272, 52, 65, 217, 1179, 642, 1176, 631, 210} \[ \int \frac {\sqrt [4]{-1+x^6}}{x} \, dx=\frac {\arctan \left (1-\sqrt {2} \sqrt [4]{x^6-1}\right )}{3 \sqrt {2}}-\frac {\arctan \left (\sqrt {2} \sqrt [4]{x^6-1}+1\right )}{3 \sqrt {2}}+\frac {2}{3} \sqrt [4]{x^6-1}+\frac {\log \left (\sqrt {x^6-1}-\sqrt {2} \sqrt [4]{x^6-1}+1\right )}{6 \sqrt {2}}-\frac {\log \left (\sqrt {x^6-1}+\sqrt {2} \sqrt [4]{x^6-1}+1\right )}{6 \sqrt {2}} \]

[In]

Int[(-1 + x^6)^(1/4)/x,x]

[Out]

(2*(-1 + x^6)^(1/4))/3 + ArcTan[1 - Sqrt[2]*(-1 + x^6)^(1/4)]/(3*Sqrt[2]) - ArcTan[1 + Sqrt[2]*(-1 + x^6)^(1/4
)]/(3*Sqrt[2]) + Log[1 - Sqrt[2]*(-1 + x^6)^(1/4) + Sqrt[-1 + x^6]]/(6*Sqrt[2]) - Log[1 + Sqrt[2]*(-1 + x^6)^(
1/4) + Sqrt[-1 + x^6]]/(6*Sqrt[2])

Rule 52

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

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}{6} \text {Subst}\left (\int \frac {\sqrt [4]{-1+x}}{x} \, dx,x,x^6\right ) \\ & = \frac {2}{3} \sqrt [4]{-1+x^6}-\frac {1}{6} \text {Subst}\left (\int \frac {1}{(-1+x)^{3/4} x} \, dx,x,x^6\right ) \\ & = \frac {2}{3} \sqrt [4]{-1+x^6}-\frac {2}{3} \text {Subst}\left (\int \frac {1}{1+x^4} \, dx,x,\sqrt [4]{-1+x^6}\right ) \\ & = \frac {2}{3} \sqrt [4]{-1+x^6}-\frac {1}{3} \text {Subst}\left (\int \frac {1-x^2}{1+x^4} \, dx,x,\sqrt [4]{-1+x^6}\right )-\frac {1}{3} \text {Subst}\left (\int \frac {1+x^2}{1+x^4} \, dx,x,\sqrt [4]{-1+x^6}\right ) \\ & = \frac {2}{3} \sqrt [4]{-1+x^6}-\frac {1}{6} \text {Subst}\left (\int \frac {1}{1-\sqrt {2} x+x^2} \, dx,x,\sqrt [4]{-1+x^6}\right )-\frac {1}{6} \text {Subst}\left (\int \frac {1}{1+\sqrt {2} x+x^2} \, dx,x,\sqrt [4]{-1+x^6}\right )+\frac {\text {Subst}\left (\int \frac {\sqrt {2}+2 x}{-1-\sqrt {2} x-x^2} \, dx,x,\sqrt [4]{-1+x^6}\right )}{6 \sqrt {2}}+\frac {\text {Subst}\left (\int \frac {\sqrt {2}-2 x}{-1+\sqrt {2} x-x^2} \, dx,x,\sqrt [4]{-1+x^6}\right )}{6 \sqrt {2}} \\ & = \frac {2}{3} \sqrt [4]{-1+x^6}+\frac {\log \left (1-\sqrt {2} \sqrt [4]{-1+x^6}+\sqrt {-1+x^6}\right )}{6 \sqrt {2}}-\frac {\log \left (1+\sqrt {2} \sqrt [4]{-1+x^6}+\sqrt {-1+x^6}\right )}{6 \sqrt {2}}-\frac {\text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\sqrt {2} \sqrt [4]{-1+x^6}\right )}{3 \sqrt {2}}+\frac {\text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\sqrt {2} \sqrt [4]{-1+x^6}\right )}{3 \sqrt {2}} \\ & = \frac {2}{3} \sqrt [4]{-1+x^6}+\frac {\arctan \left (1-\sqrt {2} \sqrt [4]{-1+x^6}\right )}{3 \sqrt {2}}-\frac {\arctan \left (1+\sqrt {2} \sqrt [4]{-1+x^6}\right )}{3 \sqrt {2}}+\frac {\log \left (1-\sqrt {2} \sqrt [4]{-1+x^6}+\sqrt {-1+x^6}\right )}{6 \sqrt {2}}-\frac {\log \left (1+\sqrt {2} \sqrt [4]{-1+x^6}+\sqrt {-1+x^6}\right )}{6 \sqrt {2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.96 \[ \int \frac {\sqrt [4]{-1+x^6}}{x} \, dx=\frac {1}{6} \left (4 \sqrt [4]{-1+x^6}-\sqrt {2} \arctan \left (\frac {-1+\sqrt {-1+x^6}}{\sqrt {2} \sqrt [4]{-1+x^6}}\right )-\sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{-1+x^6}}{1+\sqrt {-1+x^6}}\right )\right ) \]

[In]

Integrate[(-1 + x^6)^(1/4)/x,x]

[Out]

(4*(-1 + x^6)^(1/4) - Sqrt[2]*ArcTan[(-1 + Sqrt[-1 + x^6])/(Sqrt[2]*(-1 + x^6)^(1/4))] - Sqrt[2]*ArcTanh[(Sqrt
[2]*(-1 + x^6)^(1/4))/(1 + Sqrt[-1 + x^6])])/6

Maple [C] (warning: unable to verify)

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

Time = 10.16 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.71

method	result	size
meijerg	\(-\frac {\operatorname {signum}\left (x^{6}-1\right )^{\frac {1}{4}} \left (\Gamma \left (\frac {3}{4}\right ) x^{6} \operatorname {hypergeom}\left (\left [\frac {3}{4}, 1, 1\right ], \left [2, 2\right ], x^{6}\right )-4 \left (4-3 \ln \left (2\right )+\frac {\pi }{2}+6 \ln \left (x \right )+i \pi \right ) \Gamma \left (\frac {3}{4}\right )\right )}{24 \Gamma \left (\frac {3}{4}\right ) {\left (-\operatorname {signum}\left (x^{6}-1\right )\right )}^{\frac {1}{4}}}\)	\(64\)
pseudoelliptic	\(\frac {2 \left (x^{6}-1\right )^{\frac {1}{4}}}{3}-\frac {\ln \left (\frac {\sqrt {x^{6}-1}+\left (x^{6}-1\right )^{\frac {1}{4}} \sqrt {2}+1}{\sqrt {x^{6}-1}-\left (x^{6}-1\right )^{\frac {1}{4}} \sqrt {2}+1}\right ) \sqrt {2}}{12}-\frac {\arctan \left (\left (x^{6}-1\right )^{\frac {1}{4}} \sqrt {2}+1\right ) \sqrt {2}}{6}-\frac {\arctan \left (\left (x^{6}-1\right )^{\frac {1}{4}} \sqrt {2}-1\right ) \sqrt {2}}{6}\)	\(99\)
trager	\(\frac {2 \left (x^{6}-1\right )^{\frac {1}{4}}}{3}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) \ln \left (-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} x^{6}+2 \left (x^{6}-1\right )^{\frac {3}{4}}-2 \sqrt {x^{6}-1}\, \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )+2 \left (x^{6}-1\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2}-2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3}}{x^{6}}\right )}{6}-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} \ln \left (-\frac {-\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) x^{6}+2 \sqrt {x^{6}-1}\, \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3}+2 \left (x^{6}-1\right )^{\frac {3}{4}}-2 \left (x^{6}-1\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2}+2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )}{x^{6}}\right )}{6}\)	\(168\)

[In]

int((x^6-1)^(1/4)/x,x,method=_RETURNVERBOSE)

[Out]

-1/24/GAMMA(3/4)*signum(x^6-1)^(1/4)/(-signum(x^6-1))^(1/4)*(GAMMA(3/4)*x^6*hypergeom([3/4,1,1],[2,2],x^6)-4*(
4-3*ln(2)+1/2*Pi+6*ln(x)+I*Pi)*GAMMA(3/4))

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.27 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.04 \[ \int \frac {\sqrt [4]{-1+x^6}}{x} \, dx=-\left (\frac {1}{12} i + \frac {1}{12}\right ) \, \sqrt {2} \log \left (\left (i + 1\right ) \, \sqrt {2} + 2 \, {\left (x^{6} - 1\right )}^{\frac {1}{4}}\right ) + \left (\frac {1}{12} i - \frac {1}{12}\right ) \, \sqrt {2} \log \left (-\left (i - 1\right ) \, \sqrt {2} + 2 \, {\left (x^{6} - 1\right )}^{\frac {1}{4}}\right ) - \left (\frac {1}{12} i - \frac {1}{12}\right ) \, \sqrt {2} \log \left (\left (i - 1\right ) \, \sqrt {2} + 2 \, {\left (x^{6} - 1\right )}^{\frac {1}{4}}\right ) + \left (\frac {1}{12} i + \frac {1}{12}\right ) \, \sqrt {2} \log \left (-\left (i + 1\right ) \, \sqrt {2} + 2 \, {\left (x^{6} - 1\right )}^{\frac {1}{4}}\right ) + \frac {2}{3} \, {\left (x^{6} - 1\right )}^{\frac {1}{4}} \]

[In]

integrate((x^6-1)^(1/4)/x,x, algorithm="fricas")

[Out]

-(1/12*I + 1/12)*sqrt(2)*log((I + 1)*sqrt(2) + 2*(x^6 - 1)^(1/4)) + (1/12*I - 1/12)*sqrt(2)*log(-(I - 1)*sqrt(
2) + 2*(x^6 - 1)^(1/4)) - (1/12*I - 1/12)*sqrt(2)*log((I - 1)*sqrt(2) + 2*(x^6 - 1)^(1/4)) + (1/12*I + 1/12)*s
qrt(2)*log(-(I + 1)*sqrt(2) + 2*(x^6 - 1)^(1/4)) + 2/3*(x^6 - 1)^(1/4)

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.58 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.43 \[ \int \frac {\sqrt [4]{-1+x^6}}{x} \, dx=- \frac {x^{\frac {3}{2}} \Gamma \left (- \frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{4}, - \frac {1}{4} \\ \frac {3}{4} \end {matrix}\middle | {\frac {e^{2 i \pi }}{x^{6}}} \right )}}{6 \Gamma \left (\frac {3}{4}\right )} \]

[In]

integrate((x**6-1)**(1/4)/x,x)

[Out]

-x**(3/2)*gamma(-1/4)*hyper((-1/4, -1/4), (3/4,), exp_polar(2*I*pi)/x**6)/(6*gamma(3/4))

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.23 \[ \int \frac {\sqrt [4]{-1+x^6}}{x} \, dx=-\frac {1}{6} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, {\left (x^{6} - 1\right )}^{\frac {1}{4}}\right )}\right ) - \frac {1}{6} \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, {\left (x^{6} - 1\right )}^{\frac {1}{4}}\right )}\right ) - \frac {1}{12} \, \sqrt {2} \log \left (\sqrt {2} {\left (x^{6} - 1\right )}^{\frac {1}{4}} + \sqrt {x^{6} - 1} + 1\right ) + \frac {1}{12} \, \sqrt {2} \log \left (-\sqrt {2} {\left (x^{6} - 1\right )}^{\frac {1}{4}} + \sqrt {x^{6} - 1} + 1\right ) + \frac {2}{3} \, {\left (x^{6} - 1\right )}^{\frac {1}{4}} \]

[In]

integrate((x^6-1)^(1/4)/x,x, algorithm="maxima")

[Out]

-1/6*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2) + 2*(x^6 - 1)^(1/4))) - 1/6*sqrt(2)*arctan(-1/2*sqrt(2)*(sqrt(2) - 2*
(x^6 - 1)^(1/4))) - 1/12*sqrt(2)*log(sqrt(2)*(x^6 - 1)^(1/4) + sqrt(x^6 - 1) + 1) + 1/12*sqrt(2)*log(-sqrt(2)*
(x^6 - 1)^(1/4) + sqrt(x^6 - 1) + 1) + 2/3*(x^6 - 1)^(1/4)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.23 \[ \int \frac {\sqrt [4]{-1+x^6}}{x} \, dx=-\frac {1}{6} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, {\left (x^{6} - 1\right )}^{\frac {1}{4}}\right )}\right ) - \frac {1}{6} \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, {\left (x^{6} - 1\right )}^{\frac {1}{4}}\right )}\right ) - \frac {1}{12} \, \sqrt {2} \log \left (\sqrt {2} {\left (x^{6} - 1\right )}^{\frac {1}{4}} + \sqrt {x^{6} - 1} + 1\right ) + \frac {1}{12} \, \sqrt {2} \log \left (-\sqrt {2} {\left (x^{6} - 1\right )}^{\frac {1}{4}} + \sqrt {x^{6} - 1} + 1\right ) + \frac {2}{3} \, {\left (x^{6} - 1\right )}^{\frac {1}{4}} \]

[In]

integrate((x^6-1)^(1/4)/x,x, algorithm="giac")

[Out]

-1/6*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2) + 2*(x^6 - 1)^(1/4))) - 1/6*sqrt(2)*arctan(-1/2*sqrt(2)*(sqrt(2) - 2*
(x^6 - 1)^(1/4))) - 1/12*sqrt(2)*log(sqrt(2)*(x^6 - 1)^(1/4) + sqrt(x^6 - 1) + 1) + 1/12*sqrt(2)*log(-sqrt(2)*
(x^6 - 1)^(1/4) + sqrt(x^6 - 1) + 1) + 2/3*(x^6 - 1)^(1/4)

Mupad [B] (verification not implemented)

Time = 5.70 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.60 \[ \int \frac {\sqrt [4]{-1+x^6}}{x} \, dx=\frac {2\,{\left (x^6-1\right )}^{1/4}}{3}+\sqrt {2}\,\mathrm {atan}\left (\sqrt {2}\,{\left (x^6-1\right )}^{1/4}\,\left (\frac {1}{2}-\frac {1}{2}{}\mathrm {i}\right )\right )\,\left (-\frac {1}{6}-\frac {1}{6}{}\mathrm {i}\right )+\sqrt {2}\,\mathrm {atan}\left (\sqrt {2}\,{\left (x^6-1\right )}^{1/4}\,\left (\frac {1}{2}+\frac {1}{2}{}\mathrm {i}\right )\right )\,\left (-\frac {1}{6}+\frac {1}{6}{}\mathrm {i}\right ) \]

[In]

int((x^6 - 1)^(1/4)/x,x)

[Out]

(2*(x^6 - 1)^(1/4))/3 - 2^(1/2)*atan(2^(1/2)*(x^6 - 1)^(1/4)*(1/2 + 1i/2))*(1/6 - 1i/6) - 2^(1/2)*atan(2^(1/2)
*(x^6 - 1)^(1/4)*(1/2 - 1i/2))*(1/6 + 1i/6)