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

Optimal. Leaf size=92 \[ \frac {\left (x^4-1\right )^{2/3}}{4 x^4}-\frac {1}{12} \log \left (\sqrt [3]{x^4-1}+1\right )+\frac {1}{24} \log \left (\left (x^4-1\right )^{2/3}-\sqrt [3]{x^4-1}+1\right )-\frac {\tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{x^4-1}}{\sqrt {3}}\right )}{4 \sqrt {3}} \]

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Rubi [A]  time = 0.05, antiderivative size = 68, normalized size of antiderivative = 0.74, number of steps used = 6, number of rules used = 6, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {266, 51, 56, 618, 204, 31} \begin {gather*} \frac {\left (x^4-1\right )^{2/3}}{4 x^4}-\frac {1}{8} \log \left (\sqrt [3]{x^4-1}+1\right )-\frac {\tan ^{-1}\left (\frac {1-2 \sqrt [3]{x^4-1}}{\sqrt {3}}\right )}{4 \sqrt {3}}+\frac {\log (x)}{6} \end {gather*}

Antiderivative was successfully verified.

[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 51

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] && LtQ[m, -1] &&  !(LtQ[n, -1] && (EqQ[a, 0] || (NeQ[
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d, m, n, x]

Rule 56

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(1/3)), x_Symbol] :> With[{q = Rt[-((b*c - a*d)/b), 3]}, Simp
[Log[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] && Ne
gQ[(b*c - a*d)/b]

Rule 204

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

Rule 266

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 618

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*} \int \frac {1}{x^5 \sqrt [3]{-1+x^4}} \, dx &=\frac {1}{4} \operatorname {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} \operatorname {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} \operatorname {Subst}\left (\int \frac {1}{1+x} \, dx,x,\sqrt [3]{-1+x^4}\right )+\frac {1}{8} \operatorname {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} \operatorname {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 {\tan ^{-1}\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*}

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Mathematica [C]  time = 0.00, size = 28, normalized size = 0.30 \begin {gather*} \frac {3}{8} \left (x^4-1\right )^{2/3} \, _2F_1\left (\frac {2}{3},2;\frac {5}{3};1-x^4\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x^5*(-1 + x^4)^(1/3)),x]

[Out]

(3*(-1 + x^4)^(2/3)*Hypergeometric2F1[2/3, 2, 5/3, 1 - x^4])/8

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IntegrateAlgebraic [A]  time = 0.07, size = 92, normalized size = 1.00 \begin {gather*} \frac {\left (-1+x^4\right )^{2/3}}{4 x^4}-\frac {\tan ^{-1}\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 ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[1/(x^5*(-1 + x^4)^(1/3)),x]

[Out]

(-1 + x^4)^(2/3)/(4*x^4) - ArcTan[1/Sqrt[3] - (2*(-1 + x^4)^(1/3))/Sqrt[3]]/(4*Sqrt[3]) - Log[1 + (-1 + x^4)^(
1/3)]/12 + Log[1 - (-1 + x^4)^(1/3) + (-1 + x^4)^(2/3)]/24

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fricas [A]  time = 0.47, size = 80, normalized size = 0.87 \begin {gather*} \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}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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giac [A]  time = 0.43, size = 69, normalized size = 0.75 \begin {gather*} \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 ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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maple [C]  time = 4.36, size = 96, normalized size = 1.04

method	result	size
risch	\(\frac {\left (x^{4}-1\right )^{\frac {2}{3}}}{4 x^{4}}+\frac {\sqrt {3}\, \Gamma \left (\frac {2}{3}\right ) \left (-\mathrm {signum}\left (x^{4}-1\right )\right )^{\frac {1}{3}} \left (\frac {2 \pi \sqrt {3}\, x^{4} \hypergeom \left (\left [1, 1, \frac {4}{3}\right ], \left [2, 2\right ], x^{4}\right )}{9 \Gamma \left (\frac {2}{3}\right )}+\frac {2 \left (-\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \relax (3)}{2}+4 \ln \relax (x )+i \pi \right ) \pi \sqrt {3}}{3 \Gamma \left (\frac {2}{3}\right )}\right )}{24 \pi \mathrm {signum}\left (x^{4}-1\right )^{\frac {1}{3}}}\)	\(96\)
meijerg	\(-\frac {\sqrt {3}\, \Gamma \left (\frac {2}{3}\right ) \left (-\mathrm {signum}\left (x^{4}-1\right )\right )^{\frac {1}{3}} \left (-\frac {4 \pi \sqrt {3}\, x^{4} \hypergeom \left (\left [1, 1, \frac {7}{3}\right ], \left [2, 3\right ], x^{4}\right )}{27 \Gamma \left (\frac {2}{3}\right )}-\frac {2 \left (2-\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \relax (3)}{2}+4 \ln \relax (x )+i \pi \right ) \pi \sqrt {3}}{9 \Gamma \left (\frac {2}{3}\right )}+\frac {2 \pi \sqrt {3}}{3 \Gamma \left (\frac {2}{3}\right ) x^{4}}\right )}{8 \pi \mathrm {signum}\left (x^{4}-1\right )^{\frac {1}{3}}}\)	\(97\)
trager	\(\frac {\left (x^{4}-1\right )^{\frac {2}{3}}}{4 x^{4}}+\frac {\ln \left (\frac {6749049600 \RootOf \left (518400 \textit {\_Z}^{2}-720 \textit {\_Z} +1\right )^{2} x^{4}[phone] \RootOf \left (518400 \textit {\_Z}^{2}-720 \textit {\_Z} +1\right ) x^{4}-493850 x^{4}[phone] \RootOf \left (518400 \textit {\_Z}^{2}-720 \textit {\_Z} +1\right ) \left (x^{4}-1\right )^{\frac {2}{3}}+666879 \left (x^{4}-1\right )^{\frac {2}{3}}-134531280 \RootOf \left (518400 \textit {\_Z}^{2}-720 \textit {\_Z} +1\right ) \left (x^{4}-1\right )^{\frac {1}{3}}-107984793600 \RootOf \left (518400 \textit {\_Z}^{2}-720 \textit {\_Z} +1\right )^{2}-666879 \left (x^{4}-1\right )^{\frac {1}{3}}[phone] \RootOf \left (518400 \textit {\_Z}^{2}-720 \textit {\_Z} +1\right )+458575}{x^{4}}\right )}{12}-60 \ln \left (\frac {6749049600 \RootOf \left (518400 \textit {\_Z}^{2}-720 \textit {\_Z} +1\right )^{2} x^{4}[phone] \RootOf \left (518400 \textit {\_Z}^{2}-720 \textit {\_Z} +1\right ) x^{4}-493850 x^{4}[phone] \RootOf \left (518400 \textit {\_Z}^{2}-720 \textit {\_Z} +1\right ) \left (x^{4}-1\right )^{\frac {2}{3}}+666879 \left (x^{4}-1\right )^{\frac {2}{3}}-134531280 \RootOf \left (518400 \textit {\_Z}^{2}-720 \textit {\_Z} +1\right ) \left (x^{4}-1\right )^{\frac {1}{3}}-107984793600 \RootOf \left (518400 \textit {\_Z}^{2}-720 \textit {\_Z} +1\right )^{2}-666879 \left (x^{4}-1\right )^{\frac {1}{3}}[phone] \RootOf \left (518400 \textit {\_Z}^{2}-720 \textit {\_Z} +1\right )+458575}{x^{4}}\right ) \RootOf \left (518400 \textit {\_Z}^{2}-720 \textit {\_Z} +1\right )+60 \RootOf \left (518400 \textit {\_Z}^{2}-720 \textit {\_Z} +1\right ) \ln \left (-\frac {-421815600 \RootOf \left (518400 \textit {\_Z}^{2}-720 \textit {\_Z} +1\right )^{2} x^{4}+7786305 \RootOf \left (518400 \textit {\_Z}^{2}-720 \textit {\_Z} +1\right ) x^{4}+20865 x^{4}+8408205 \RootOf \left (518400 \textit {\_Z}^{2}-720 \textit {\_Z} +1\right ) \left (x^{4}-1\right )^{\frac {2}{3}}-53358 \left (x^{4}-1\right )^{\frac {2}{3}}-8408205 \RootOf \left (518400 \textit {\_Z}^{2}-720 \textit {\_Z} +1\right ) \left (x^{4}-1\right )^{\frac {1}{3}}[phone] \RootOf \left (518400 \textit {\_Z}^{2}-720 \textit {\_Z} +1\right )^{2}+53358 \left (x^{4}-1\right )^{\frac {1}{3}}-965475 \RootOf \left (518400 \textit {\_Z}^{2}-720 \textit {\_Z} +1\right )-40339}{x^{4}}\right )\)	\(439\)

Verification of antiderivative is not currently implemented for this CAS.

[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)
)

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maxima [A]  time = 0.50, size = 68, normalized size = 0.74 \begin {gather*} \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 ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [B]  time = 0.94, size = 92, normalized size = 1.00 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [C]  time = 0.99, size = 34, normalized size = 0.37 \begin {gather*} - \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 )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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