\(\int \frac {1}{x^5 \sqrt [3]{-1+x^4}} \, dx\) [1267]

Optimal result

Integrand size = 13, antiderivative size = 92 \[ \int \frac {1}{x^5 \sqrt [3]{-1+x^4}} \, dx=\frac {\left (-1+x^4\right )^{2/3}}{4 x^4}-\frac {\arctan \left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{-1+x^4}}{\sqrt {3}}\right )}{4 \sqrt {3}}-\frac {1}{12} \log \left (1+\sqrt [3]{-1+x^4}\right )+\frac {1}{24} \log \left (1-\sqrt [3]{-1+x^4}+\left (-1+x^4\right )^{2/3}\right ) \]

[Out]

1/4*(x^4-1)^(2/3)/x^4+1/12*3^(1/2)*arctan(-1/3*3^(1/2)+2/3*(x^4-1)^(1/3)*3^(1/2))-1/12*ln(1+(x^4-1)^(1/3))+1/2
4*ln(1-(x^4-1)^(1/3)+(x^4-1)^(2/3))

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.74, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {272, 44, 58, 632, 210, 31} \[ \int \frac {1}{x^5 \sqrt [3]{-1+x^4}} \, dx=-\frac {\arctan \left (\frac {1-2 \sqrt [3]{x^4-1}}{\sqrt {3}}\right )}{4 \sqrt {3}}+\frac {\left (x^4-1\right )^{2/3}}{4 x^4}-\frac {1}{8} \log \left (\sqrt [3]{x^4-1}+1\right )+\frac {\log (x)}{6} \]

[In]

Int[1/(x^5*(-1 + x^4)^(1/3)),x]

[Out]

(-1 + x^4)^(2/3)/(4*x^4) - ArcTan[(1 - 2*(-1 + x^4)^(1/3))/Sqrt[3]]/(4*Sqrt[3]) + Log[x]/6 - Log[1 + (-1 + x^4
)^(1/3)]/8

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

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 58

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(1/3)), x_Symbol] :> With[{q = Rt[-(b*c - a*d)/b, 3]}, Simp[L
og[RemoveContent[a + b*x, x]]/(2*b*q), x] + (Dist[3/(2*b), Subst[Int[1/(q^2 - q*x + x^2), x], x, (c + d*x)^(1/
3)], x] - Dist[3/(2*b*q), Subst[Int[1/(q + x), x], x, (c + d*x)^(1/3)], x])] /; FreeQ[{a, b, c, d}, x] && NegQ
[(b*c - a*d)/b]

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 632

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

Rubi steps \begin{align*} \text {integral}& = \frac {1}{4} \text {Subst}\left (\int \frac {1}{\sqrt [3]{-1+x} x^2} \, dx,x,x^4\right ) \\ & = \frac {\left (-1+x^4\right )^{2/3}}{4 x^4}+\frac {1}{12} \text {Subst}\left (\int \frac {1}{\sqrt [3]{-1+x} x} \, dx,x,x^4\right ) \\ & = \frac {\left (-1+x^4\right )^{2/3}}{4 x^4}+\frac {\log (x)}{6}-\frac {1}{8} \text {Subst}\left (\int \frac {1}{1+x} \, dx,x,\sqrt [3]{-1+x^4}\right )+\frac {1}{8} \text {Subst}\left (\int \frac {1}{1-x+x^2} \, dx,x,\sqrt [3]{-1+x^4}\right ) \\ & = \frac {\left (-1+x^4\right )^{2/3}}{4 x^4}+\frac {\log (x)}{6}-\frac {1}{8} \log \left (1+\sqrt [3]{-1+x^4}\right )-\frac {1}{4} \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,-1+2 \sqrt [3]{-1+x^4}\right ) \\ & = \frac {\left (-1+x^4\right )^{2/3}}{4 x^4}-\frac {\arctan \left (\frac {1-2 \sqrt [3]{-1+x^4}}{\sqrt {3}}\right )}{4 \sqrt {3}}+\frac {\log (x)}{6}-\frac {1}{8} \log \left (1+\sqrt [3]{-1+x^4}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.90 \[ \int \frac {1}{x^5 \sqrt [3]{-1+x^4}} \, dx=\frac {1}{24} \left (\frac {6 \left (-1+x^4\right )^{2/3}}{x^4}-2 \sqrt {3} \arctan \left (\frac {1-2 \sqrt [3]{-1+x^4}}{\sqrt {3}}\right )-2 \log \left (1+\sqrt [3]{-1+x^4}\right )+\log \left (1-\sqrt [3]{-1+x^4}+\left (-1+x^4\right )^{2/3}\right )\right ) \]

[In]

Integrate[1/(x^5*(-1 + x^4)^(1/3)),x]

[Out]

((6*(-1 + x^4)^(2/3))/x^4 - 2*Sqrt[3]*ArcTan[(1 - 2*(-1 + x^4)^(1/3))/Sqrt[3]] - 2*Log[1 + (-1 + x^4)^(1/3)] +
Log[1 - (-1 + x^4)^(1/3) + (-1 + x^4)^(2/3)])/24

Maple [C] (warning: unable to verify)

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

Time = 4.64 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.04

method	result	size
risch	\(\frac {\left (x^{4}-1\right )^{\frac {2}{3}}}{4 x^{4}}+\frac {\sqrt {3}\, \Gamma \left (\frac {2}{3}\right ) {\left (-\operatorname {signum}\left (x^{4}-1\right )\right )}^{\frac {1}{3}} \left (\frac {2 \pi \sqrt {3}\, x^{4} \operatorname {hypergeom}\left (\left [1, 1, \frac {4}{3}\right ], \left [2, 2\right ], x^{4}\right )}{9 \Gamma \left (\frac {2}{3}\right )}+\frac {2 \left (-\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \left (3\right )}{2}+4 \ln \left (x \right )+i \pi \right ) \pi \sqrt {3}}{3 \Gamma \left (\frac {2}{3}\right )}\right )}{24 \pi \operatorname {signum}\left (x^{4}-1\right )^{\frac {1}{3}}}\)	\(96\)
meijerg	\(-\frac {\sqrt {3}\, \Gamma \left (\frac {2}{3}\right ) {\left (-\operatorname {signum}\left (x^{4}-1\right )\right )}^{\frac {1}{3}} \left (-\frac {4 \pi \sqrt {3}\, x^{4} \operatorname {hypergeom}\left (\left [1, 1, \frac {7}{3}\right ], \left [2, 3\right ], x^{4}\right )}{27 \Gamma \left (\frac {2}{3}\right )}-\frac {2 \left (2-\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \left (3\right )}{2}+4 \ln \left (x \right )+i \pi \right ) \pi \sqrt {3}}{9 \Gamma \left (\frac {2}{3}\right )}+\frac {2 \pi \sqrt {3}}{3 \Gamma \left (\frac {2}{3}\right ) x^{4}}\right )}{8 \pi \operatorname {signum}\left (x^{4}-1\right )^{\frac {1}{3}}}\)	\(97\)
pseudoelliptic	\(\frac {2 \sqrt {3}\, \arctan \left (\frac {\left (2 \left (x^{4}-1\right )^{\frac {1}{3}}-1\right ) \sqrt {3}}{3}\right ) x^{4}-2 \ln \left (1+\left (x^{4}-1\right )^{\frac {1}{3}}\right ) x^{4}+\ln \left (1-\left (x^{4}-1\right )^{\frac {1}{3}}+\left (x^{4}-1\right )^{\frac {2}{3}}\right ) x^{4}+6 \left (x^{4}-1\right )^{\frac {2}{3}}}{24 \left (1+\left (x^{4}-1\right )^{\frac {1}{3}}\right ) \left (1-\left (x^{4}-1\right )^{\frac {1}{3}}+\left (x^{4}-1\right )^{\frac {2}{3}}\right )}\)	\(107\)
trager	\(\frac {\left (x^{4}-1\right )^{\frac {2}{3}}}{4 x^{4}}-\frac {\ln \left (-\frac {8852696853933772800 \operatorname {RootOf}\left (518400 \textit {\_Z}^{2}-720 \textit {\_Z} +1\right )^{2} x^{4}+181557103383026640 \operatorname {RootOf}\left (518400 \textit {\_Z}^{2}-720 \textit {\_Z} +1\right ) x^{4}[phone] x^{4}+74249290852441200 \operatorname {RootOf}\left (518400 \textit {\_Z}^{2}-720 \textit {\_Z} +1\right ) \left (x^{4}-1\right )^{\frac {2}{3}}-141643149662940364800 \operatorname {RootOf}\left (518400 \textit {\_Z}^{2}-720 \textit {\_Z} +1\right )^{2}-433058965335588240 \operatorname {RootOf}\left (518400 \textit {\_Z}^{2}-720 \textit {\_Z} +1\right ) \left (x^{4}-1\right )^{\frac {1}{3}}[phone] \left (x^{4}-1\right )^{\frac {2}{3}}-310581659433945600 \operatorname {RootOf}\left (518400 \textit {\_Z}^{2}-720 \textit {\_Z} +1\right )[phone] \left (x^{4}-1\right )^{\frac {1}{3}}-170107369307837}{x^{4}}\right )}{12}+\frac {\ln \left (-\frac {18286560000 \operatorname {RootOf}\left (518400 \textit {\_Z}^{2}-720 \textit {\_Z} +1\right )^{2} x^{4}-390343680 \operatorname {RootOf}\left (518400 \textit {\_Z}^{2}-720 \textit {\_Z} +1\right ) x^{4}+195285 x^{4}[phone] \operatorname {RootOf}\left (518400 \textit {\_Z}^{2}-720 \textit {\_Z} +1\right ) \left (x^{4}-1\right )^{\frac {2}{3}}-292584960000 \operatorname {RootOf}\left (518400 \textit {\_Z}^{2}-720 \textit {\_Z} +1\right )^{2}[phone] \operatorname {RootOf}\left (518400 \textit {\_Z}^{2}-720 \textit {\_Z} +1\right ) \left (x^{4}-1\right )^{\frac {1}{3}}-853728 \left (x^{4}-1\right )^{\frac {2}{3}}[phone] \operatorname {RootOf}\left (518400 \textit {\_Z}^{2}-720 \textit {\_Z} +1\right )-666879 \left (x^{4}-1\right )^{\frac {1}{3}}-377551}{x^{4}}\right )}{12}-60 \ln \left (-\frac {18286560000 \operatorname {RootOf}\left (518400 \textit {\_Z}^{2}-720 \textit {\_Z} +1\right )^{2} x^{4}-390343680 \operatorname {RootOf}\left (518400 \textit {\_Z}^{2}-720 \textit {\_Z} +1\right ) x^{4}+195285 x^{4}[phone] \operatorname {RootOf}\left (518400 \textit {\_Z}^{2}-720 \textit {\_Z} +1\right ) \left (x^{4}-1\right )^{\frac {2}{3}}-292584960000 \operatorname {RootOf}\left (518400 \textit {\_Z}^{2}-720 \textit {\_Z} +1\right )^{2}[phone] \operatorname {RootOf}\left (518400 \textit {\_Z}^{2}-720 \textit {\_Z} +1\right ) \left (x^{4}-1\right )^{\frac {1}{3}}-853728 \left (x^{4}-1\right )^{\frac {2}{3}}[phone] \operatorname {RootOf}\left (518400 \textit {\_Z}^{2}-720 \textit {\_Z} +1\right )-666879 \left (x^{4}-1\right )^{\frac {1}{3}}-377551}{x^{4}}\right ) \operatorname {RootOf}\left (518400 \textit {\_Z}^{2}-720 \textit {\_Z} +1\right )\)	\(430\)

[In]

int(1/x^5/(x^4-1)^(1/3),x,method=_RETURNVERBOSE)

[Out]

1/4*(x^4-1)^(2/3)/x^4+1/24/Pi*3^(1/2)*GAMMA(2/3)/signum(x^4-1)^(1/3)*(-signum(x^4-1))^(1/3)*(2/9*Pi*3^(1/2)/GA
MMA(2/3)*x^4*hypergeom([1,1,4/3],[2,2],x^4)+2/3*(-1/6*Pi*3^(1/2)-3/2*ln(3)+4*ln(x)+I*Pi)*Pi*3^(1/2)/GAMMA(2/3)
)

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.87 \[ \int \frac {1}{x^5 \sqrt [3]{-1+x^4}} \, dx=\frac {2 \, \sqrt {3} x^{4} \arctan \left (\frac {2}{3} \, \sqrt {3} {\left (x^{4} - 1\right )}^{\frac {1}{3}} - \frac {1}{3} \, \sqrt {3}\right ) + x^{4} \log \left ({\left (x^{4} - 1\right )}^{\frac {2}{3}} - {\left (x^{4} - 1\right )}^{\frac {1}{3}} + 1\right ) - 2 \, x^{4} \log \left ({\left (x^{4} - 1\right )}^{\frac {1}{3}} + 1\right ) + 6 \, {\left (x^{4} - 1\right )}^{\frac {2}{3}}}{24 \, x^{4}} \]

[In]

integrate(1/x^5/(x^4-1)^(1/3),x, algorithm="fricas")

[Out]

1/24*(2*sqrt(3)*x^4*arctan(2/3*sqrt(3)*(x^4 - 1)^(1/3) - 1/3*sqrt(3)) + x^4*log((x^4 - 1)^(2/3) - (x^4 - 1)^(1
/3) + 1) - 2*x^4*log((x^4 - 1)^(1/3) + 1) + 6*(x^4 - 1)^(2/3))/x^4

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.79 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.37 \[ \int \frac {1}{x^5 \sqrt [3]{-1+x^4}} \, dx=- \frac {\Gamma \left (\frac {4}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {4}{3} \\ \frac {7}{3} \end {matrix}\middle | {\frac {e^{2 i \pi }}{x^{4}}} \right )}}{4 x^{\frac {16}{3}} \Gamma \left (\frac {7}{3}\right )} \]

[In]

integrate(1/x**5/(x**4-1)**(1/3),x)

[Out]

-gamma(4/3)*hyper((1/3, 4/3), (7/3,), exp_polar(2*I*pi)/x**4)/(4*x**(16/3)*gamma(7/3))

Maxima [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.74 \[ \int \frac {1}{x^5 \sqrt [3]{-1+x^4}} \, dx=\frac {1}{12} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (x^{4} - 1\right )}^{\frac {1}{3}} - 1\right )}\right ) + \frac {{\left (x^{4} - 1\right )}^{\frac {2}{3}}}{4 \, x^{4}} + \frac {1}{24} \, \log \left ({\left (x^{4} - 1\right )}^{\frac {2}{3}} - {\left (x^{4} - 1\right )}^{\frac {1}{3}} + 1\right ) - \frac {1}{12} \, \log \left ({\left (x^{4} - 1\right )}^{\frac {1}{3}} + 1\right ) \]

[In]

integrate(1/x^5/(x^4-1)^(1/3),x, algorithm="maxima")

[Out]

1/12*sqrt(3)*arctan(1/3*sqrt(3)*(2*(x^4 - 1)^(1/3) - 1)) + 1/4*(x^4 - 1)^(2/3)/x^4 + 1/24*log((x^4 - 1)^(2/3)
- (x^4 - 1)^(1/3) + 1) - 1/12*log((x^4 - 1)^(1/3) + 1)

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.75 \[ \int \frac {1}{x^5 \sqrt [3]{-1+x^4}} \, dx=\frac {1}{12} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (x^{4} - 1\right )}^{\frac {1}{3}} - 1\right )}\right ) + \frac {{\left (x^{4} - 1\right )}^{\frac {2}{3}}}{4 \, x^{4}} + \frac {1}{24} \, \log \left ({\left (x^{4} - 1\right )}^{\frac {2}{3}} - {\left (x^{4} - 1\right )}^{\frac {1}{3}} + 1\right ) - \frac {1}{12} \, \log \left ({\left | {\left (x^{4} - 1\right )}^{\frac {1}{3}} + 1 \right |}\right ) \]

[In]

integrate(1/x^5/(x^4-1)^(1/3),x, algorithm="giac")

[Out]

1/12*sqrt(3)*arctan(1/3*sqrt(3)*(2*(x^4 - 1)^(1/3) - 1)) + 1/4*(x^4 - 1)^(2/3)/x^4 + 1/24*log((x^4 - 1)^(2/3)
- (x^4 - 1)^(1/3) + 1) - 1/12*log(abs((x^4 - 1)^(1/3) + 1))

Mupad [B] (verification not implemented)

Time = 5.86 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x^5 \sqrt [3]{-1+x^4}} \, dx=\frac {{\left (x^4-1\right )}^{2/3}}{4\,x^4}-\ln \left (9\,{\left (-\frac {1}{24}+\frac {\sqrt {3}\,1{}\mathrm {i}}{24}\right )}^2+\frac {{\left (x^4-1\right )}^{1/3}}{16}\right )\,\left (-\frac {1}{24}+\frac {\sqrt {3}\,1{}\mathrm {i}}{24}\right )+\ln \left (9\,{\left (\frac {1}{24}+\frac {\sqrt {3}\,1{}\mathrm {i}}{24}\right )}^2+\frac {{\left (x^4-1\right )}^{1/3}}{16}\right )\,\left (\frac {1}{24}+\frac {\sqrt {3}\,1{}\mathrm {i}}{24}\right )-\frac {\ln \left (\frac {{\left (x^4-1\right )}^{1/3}}{16}+\frac {1}{16}\right )}{12} \]

[In]

int(1/(x^5*(x^4 - 1)^(1/3)),x)

[Out]

log(9*((3^(1/2)*1i)/24 + 1/24)^2 + (x^4 - 1)^(1/3)/16)*((3^(1/2)*1i)/24 + 1/24) - log(9*((3^(1/2)*1i)/24 - 1/2
4)^2 + (x^4 - 1)^(1/3)/16)*((3^(1/2)*1i)/24 - 1/24) - log((x^4 - 1)^(1/3)/16 + 1/16)/12 + (x^4 - 1)^(2/3)/(4*x
^4)