\(\int \frac {1}{x \sqrt [4]{-1+x^6}} \, dx\) [1019]

Optimal result

Integrand size = 13, antiderivative size = 77 \[ \int \frac {1}{x \sqrt [4]{-1+x^6}} \, dx=-\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{-1+x^6}}{-1+\sqrt {-1+x^6}}\right )}{3 \sqrt {2}}-\frac {\text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{-1+x^6}}{1+\sqrt {-1+x^6}}\right )}{3 \sqrt {2}} \] Output:

-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)
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.88 \[ \int \frac {1}{x \sqrt [4]{-1+x^6}} \, dx=\frac {\arctan \left (\frac {-1+\sqrt {-1+x^6}}{\sqrt {2} \sqrt [4]{-1+x^6}}\right )-\text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{-1+x^6}}{1+\sqrt {-1+x^6}}\right )}{3 \sqrt {2}} \] Input:

Integrate[1/(x*(-1 + x^6)^(1/4)),x]
 

Output:

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

Rubi [A] (warning: unable to verify)

Time = 0.31 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.64, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.769, Rules used = {798, 73, 826, 1476, 1082, 217, 1479, 25, 27, 1103}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 798

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

\(\Big \downarrow \) 73

\(\displaystyle \frac {2}{3} \int \frac {x^{12}}{x^{24}+1}d\sqrt [4]{x^6-1}\)

\(\Big \downarrow \) 826

\(\displaystyle \frac {2}{3} \left (\frac {1}{2} \int \frac {x^{12}+1}{x^{24}+1}d\sqrt [4]{x^6-1}-\frac {1}{2} \int \frac {1-x^{12}}{x^{24}+1}d\sqrt [4]{x^6-1}\right )\)

\(\Big \downarrow \) 1476

\(\displaystyle \frac {2}{3} \left (\frac {1}{2} \left (\frac {1}{2} \int \frac {1}{x^{12}-\sqrt {2} \sqrt [4]{x^6-1}+1}d\sqrt [4]{x^6-1}+\frac {1}{2} \int \frac {1}{x^{12}+\sqrt {2} \sqrt [4]{x^6-1}+1}d\sqrt [4]{x^6-1}\right )-\frac {1}{2} \int \frac {1-x^{12}}{x^{24}+1}d\sqrt [4]{x^6-1}\right )\)

\(\Big \downarrow \) 1082

\(\displaystyle \frac {2}{3} \left (\frac {1}{2} \left (\frac {\int \frac {1}{-x^{12}-1}d\left (1-\sqrt {2} \sqrt [4]{x^6-1}\right )}{\sqrt {2}}-\frac {\int \frac {1}{-x^{12}-1}d\left (\sqrt {2} \sqrt [4]{x^6-1}+1\right )}{\sqrt {2}}\right )-\frac {1}{2} \int \frac {1-x^{12}}{x^{24}+1}d\sqrt [4]{x^6-1}\right )\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {2}{3} \left (\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt [4]{x^6-1}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt [4]{x^6-1}\right )}{\sqrt {2}}\right )-\frac {1}{2} \int \frac {1-x^{12}}{x^{24}+1}d\sqrt [4]{x^6-1}\right )\)

\(\Big \downarrow \) 1479

\(\displaystyle \frac {2}{3} \left (\frac {1}{2} \left (\frac {\int -\frac {\sqrt {2}-2 \sqrt [4]{x^6-1}}{x^{12}-\sqrt {2} \sqrt [4]{x^6-1}+1}d\sqrt [4]{x^6-1}}{2 \sqrt {2}}+\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{x^6-1}+1\right )}{x^{12}+\sqrt {2} \sqrt [4]{x^6-1}+1}d\sqrt [4]{x^6-1}}{2 \sqrt {2}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt [4]{x^6-1}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt [4]{x^6-1}\right )}{\sqrt {2}}\right )\right )\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {2}{3} \left (\frac {1}{2} \left (-\frac {\int \frac {\sqrt {2}-2 \sqrt [4]{x^6-1}}{x^{12}-\sqrt {2} \sqrt [4]{x^6-1}+1}d\sqrt [4]{x^6-1}}{2 \sqrt {2}}-\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{x^6-1}+1\right )}{x^{12}+\sqrt {2} \sqrt [4]{x^6-1}+1}d\sqrt [4]{x^6-1}}{2 \sqrt {2}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt [4]{x^6-1}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt [4]{x^6-1}\right )}{\sqrt {2}}\right )\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2}{3} \left (\frac {1}{2} \left (-\frac {\int \frac {\sqrt {2}-2 \sqrt [4]{x^6-1}}{x^{12}-\sqrt {2} \sqrt [4]{x^6-1}+1}d\sqrt [4]{x^6-1}}{2 \sqrt {2}}-\frac {1}{2} \int \frac {\sqrt {2} \sqrt [4]{x^6-1}+1}{x^{12}+\sqrt {2} \sqrt [4]{x^6-1}+1}d\sqrt [4]{x^6-1}\right )+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt [4]{x^6-1}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt [4]{x^6-1}\right )}{\sqrt {2}}\right )\right )\)

\(\Big \downarrow \) 1103

\(\displaystyle \frac {2}{3} \left (\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt [4]{x^6-1}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt [4]{x^6-1}\right )}{\sqrt {2}}\right )+\frac {1}{2} \left (\frac {\log \left (x^{12}-\sqrt {2} \sqrt [4]{x^6-1}+1\right )}{2 \sqrt {2}}-\frac {\log \left (x^{12}+\sqrt {2} \sqrt [4]{x^6-1}+1\right )}{2 \sqrt {2}}\right )\right )\)

Input:

Int[1/(x*(-1 + x^6)^(1/4)),x]
 

Output:

(2*((-(ArcTan[1 - Sqrt[2]*(-1 + x^6)^(1/4)]/Sqrt[2]) + ArcTan[1 + Sqrt[2]* 
(-1 + x^6)^(1/4)]/Sqrt[2])/2 + (Log[1 + x^12 - Sqrt[2]*(-1 + x^6)^(1/4)]/( 
2*Sqrt[2]) - Log[1 + x^12 + Sqrt[2]*(-1 + x^6)^(1/4)]/(2*Sqrt[2]))/2))/3
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 73

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ 
{p = Denominator[m]}, Simp[p/b   Subst[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] && Lt 
Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL 
inearQ[a, b, c, d, m, n, x]
 

rule 217

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 798

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[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 826

Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[a/b, 
2]], s = Denominator[Rt[a/b, 2]]}, Simp[1/(2*s)   Int[(r + s*x^2)/(a + b*x^ 
4), x], x] - Simp[1/(2*s)   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 1082

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S 
implify[a*(c/b^2)]}, Simp[-2/b   Subst[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])] /; Fre 
eQ[{a, b, c}, x]
 

rule 1103

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S 
imp[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 1476

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 
2*(d/e), 2]}, Simp[e/(2*c)   Int[1/Simp[d/e + q*x + x^2, x], x], x] + Simp[ 
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 1479

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

Maple [C] (warning: unable to verify)

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

Time = 8.46 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.03

method	result	size
meijerg	\(\frac {\sqrt {2}\, \Gamma \left (\frac {3}{4}\right ) {\left (-\operatorname {signum}\left (x^{6}-1\right )\right )}^{\frac {1}{4}} \left (\frac {\left (-3 \ln \left (2\right )-\frac {\pi }{2}+6 \ln \left (x \right )+i \pi \right ) \pi \sqrt {2}}{\Gamma \left (\frac {3}{4}\right )}+\frac {\pi \sqrt {2}\, x^{6} \operatorname {hypergeom}\left (\left [1, 1, \frac {5}{4}\right ], \left [2, 2\right ], x^{6}\right )}{4 \Gamma \left (\frac {3}{4}\right )}\right )}{12 \pi \operatorname {signum}\left (x^{6}-1\right )^{\frac {1}{4}}}\)	\(79\)
pseudoelliptic	\(\frac {\sqrt {2}\, \left (\ln \left (\frac {\sqrt {x^{6}-1}-\left (x^{6}-1\right )^{\frac {1}{4}} \sqrt {2}+1}{\sqrt {x^{6}-1}+\left (x^{6}-1\right )^{\frac {1}{4}} \sqrt {2}+1}\right )+2 \arctan \left (\left (x^{6}-1\right )^{\frac {1}{4}} \sqrt {2}+1\right )+2 \arctan \left (\left (x^{6}-1\right )^{\frac {1}{4}} \sqrt {2}-1\right )\right )}{12}\)	\(84\)
trager	\(-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} \ln \left (-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} x^{6}+2 \left (x^{6}-1\right )^{\frac {3}{4}}-2 \sqrt {x^{6}-1}\, \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )+2 \left (x^{6}-1\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2}-2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3}}{x^{6}}\right )}{6}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) \ln \left (-\frac {-\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) x^{6}+2 \sqrt {x^{6}-1}\, \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3}+2 \left (x^{6}-1\right )^{\frac {3}{4}}-2 \left (x^{6}-1\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2}+2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )}{x^{6}}\right )}{6}\)	\(159\)

Input:

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

Output:

1/12/Pi*2^(1/2)*GAMMA(3/4)/signum(x^6-1)^(1/4)*(-signum(x^6-1))^(1/4)*((-3 
*ln(2)-1/2*Pi+6*ln(x)+I*Pi)*Pi*2^(1/2)/GAMMA(3/4)+1/4*Pi*2^(1/2)/GAMMA(3/4 
)*x^6*hypergeom([1,1,5/4],[2,2],x^6))
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.19 \[ \int \frac {1}{x \sqrt [4]{-1+x^6}} \, dx=\frac {1}{6} \, \sqrt {2} \arctan \left (\sqrt {2} {\left (x^{6} - 1\right )}^{\frac {1}{4}} + 1\right ) + \frac {1}{6} \, \sqrt {2} \arctan \left (\sqrt {2} {\left (x^{6} - 1\right )}^{\frac {1}{4}} - 1\right ) - \frac {1}{12} \, \sqrt {2} \log \left (\sqrt {2} {\left (x^{6} - 1\right )}^{\frac {1}{4}} + \sqrt {x^{6} - 1} + 1\right ) + \frac {1}{12} \, \sqrt {2} \log \left (-\sqrt {2} {\left (x^{6} - 1\right )}^{\frac {1}{4}} + \sqrt {x^{6} - 1} + 1\right ) \] Input:

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

Output:

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

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.51 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.44 \[ \int \frac {1}{x \sqrt [4]{-1+x^6}} \, dx=- \frac {\Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {1}{4} \\ \frac {5}{4} \end {matrix}\middle | {\frac {e^{2 i \pi }}{x^{6}}} \right )}}{6 x^{\frac {3}{2}} \Gamma \left (\frac {5}{4}\right )} \] Input:

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

Output:

-gamma(1/4)*hyper((1/4, 1/4), (5/4,), exp_polar(2*I*pi)/x**6)/(6*x**(3/2)* 
gamma(5/4))
 

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.32 \[ \int \frac {1}{x \sqrt [4]{-1+x^6}} \, dx=\frac {1}{6} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, {\left (x^{6} - 1\right )}^{\frac {1}{4}}\right )}\right ) + \frac {1}{6} \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, {\left (x^{6} - 1\right )}^{\frac {1}{4}}\right )}\right ) - \frac {1}{12} \, \sqrt {2} \log \left (\sqrt {2} {\left (x^{6} - 1\right )}^{\frac {1}{4}} + \sqrt {x^{6} - 1} + 1\right ) + \frac {1}{12} \, \sqrt {2} \log \left (-\sqrt {2} {\left (x^{6} - 1\right )}^{\frac {1}{4}} + \sqrt {x^{6} - 1} + 1\right ) \] Input:

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

Output:

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(sq 
rt(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)
 

Giac [A] (verification not implemented)

Time = 0.32 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.32 \[ \int \frac {1}{x \sqrt [4]{-1+x^6}} \, dx=\frac {1}{6} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, {\left (x^{6} - 1\right )}^{\frac {1}{4}}\right )}\right ) + \frac {1}{6} \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, {\left (x^{6} - 1\right )}^{\frac {1}{4}}\right )}\right ) - \frac {1}{12} \, \sqrt {2} \log \left (\sqrt {2} {\left (x^{6} - 1\right )}^{\frac {1}{4}} + \sqrt {x^{6} - 1} + 1\right ) + \frac {1}{12} \, \sqrt {2} \log \left (-\sqrt {2} {\left (x^{6} - 1\right )}^{\frac {1}{4}} + \sqrt {x^{6} - 1} + 1\right ) \] Input:

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

Output:

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(sq 
rt(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)
 

Mupad [B] (verification not implemented)

Time = 7.95 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.58 \[ \int \frac {1}{x \sqrt [4]{-1+x^6}} \, dx=\sqrt {2}\,\mathrm {atan}\left (\sqrt {2}\,{\left (x^6-1\right )}^{1/4}\,\left (\frac {1}{2}-\frac {1}{2}{}\mathrm {i}\right )\right )\,\left (\frac {1}{6}-\frac {1}{6}{}\mathrm {i}\right )+\sqrt {2}\,\mathrm {atan}\left (\sqrt {2}\,{\left (x^6-1\right )}^{1/4}\,\left (\frac {1}{2}+\frac {1}{2}{}\mathrm {i}\right )\right )\,\left (\frac {1}{6}+\frac {1}{6}{}\mathrm {i}\right ) \] Input:

int(1/(x*(x^6 - 1)^(1/4)),x)
 

Output:

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)
 

Reduce [F]

\[ \int \frac {1}{x \sqrt [4]{-1+x^6}} \, dx=\int \frac {1}{\left (x^{6}-1\right )^{\frac {1}{4}} x}d x \] Input:

int(1/x/(x^6-1)^(1/4),x)
 

Output:

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