\(\int \frac {(-1+x^5)^{2/3}}{x^6} \, dx\) [1273]

Optimal result

Integrand size = 13, antiderivative size = 92 \[ \int \frac {\left (-1+x^5\right )^{2/3}}{x^6} \, dx=-\frac {\left (-1+x^5\right )^{2/3}}{5 x^5}-\frac {2 \arctan \left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{-1+x^5}}{\sqrt {3}}\right )}{5 \sqrt {3}}-\frac {2}{15} \log \left (1+\sqrt [3]{-1+x^5}\right )+\frac {1}{15} \log \left (1-\sqrt [3]{-1+x^5}+\left (-1+x^5\right )^{2/3}\right ) \]

[Out]

-1/5*(x^5-1)^(2/3)/x^5+2/15*3^(1/2)*arctan(-1/3*3^(1/2)+2/3*(x^5-1)^(1/3)*3^(1/2))-2/15*ln(1+(x^5-1)^(1/3))+1/
15*ln(1-(x^5-1)^(1/3)+(x^5-1)^(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, 43, 58, 632, 210, 31} \[ \int \frac {\left (-1+x^5\right )^{2/3}}{x^6} \, dx=-\frac {2 \arctan \left (\frac {1-2 \sqrt [3]{x^5-1}}{\sqrt {3}}\right )}{5 \sqrt {3}}-\frac {\left (x^5-1\right )^{2/3}}{5 x^5}-\frac {1}{5} \log \left (\sqrt [3]{x^5-1}+1\right )+\frac {\log (x)}{3} \]

[In]

Int[(-1 + x^5)^(2/3)/x^6,x]

[Out]

-1/5*(-1 + x^5)^(2/3)/x^5 - (2*ArcTan[(1 - 2*(-1 + x^5)^(1/3))/Sqrt[3]])/(5*Sqrt[3]) + Log[x]/3 - Log[1 + (-1
+ x^5)^(1/3)]/5

Rule 31

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

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^n/(b*(
m + 1))), x] - Dist[d*(n/(b*(m + 1))), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d, n
}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, -1] &&  !IntegerQ[n] && GtQ[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}{5} \text {Subst}\left (\int \frac {(-1+x)^{2/3}}{x^2} \, dx,x,x^5\right ) \\ & = -\frac {\left (-1+x^5\right )^{2/3}}{5 x^5}+\frac {2}{15} \text {Subst}\left (\int \frac {1}{\sqrt [3]{-1+x} x} \, dx,x,x^5\right ) \\ & = -\frac {\left (-1+x^5\right )^{2/3}}{5 x^5}+\frac {\log (x)}{3}-\frac {1}{5} \text {Subst}\left (\int \frac {1}{1+x} \, dx,x,\sqrt [3]{-1+x^5}\right )+\frac {1}{5} \text {Subst}\left (\int \frac {1}{1-x+x^2} \, dx,x,\sqrt [3]{-1+x^5}\right ) \\ & = -\frac {\left (-1+x^5\right )^{2/3}}{5 x^5}+\frac {\log (x)}{3}-\frac {1}{5} \log \left (1+\sqrt [3]{-1+x^5}\right )-\frac {2}{5} \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,-1+2 \sqrt [3]{-1+x^5}\right ) \\ & = -\frac {\left (-1+x^5\right )^{2/3}}{5 x^5}-\frac {2 \arctan \left (\frac {1-2 \sqrt [3]{-1+x^5}}{\sqrt {3}}\right )}{5 \sqrt {3}}+\frac {\log (x)}{3}-\frac {1}{5} \log \left (1+\sqrt [3]{-1+x^5}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.90 \[ \int \frac {\left (-1+x^5\right )^{2/3}}{x^6} \, dx=\frac {1}{15} \left (-\frac {3 \left (-1+x^5\right )^{2/3}}{x^5}-2 \sqrt {3} \arctan \left (\frac {1-2 \sqrt [3]{-1+x^5}}{\sqrt {3}}\right )-2 \log \left (1+\sqrt [3]{-1+x^5}\right )+\log \left (1-\sqrt [3]{-1+x^5}+\left (-1+x^5\right )^{2/3}\right )\right ) \]

[In]

Integrate[(-1 + x^5)^(2/3)/x^6,x]

[Out]

((-3*(-1 + x^5)^(2/3))/x^5 - 2*Sqrt[3]*ArcTan[(1 - 2*(-1 + x^5)^(1/3))/Sqrt[3]] - 2*Log[1 + (-1 + x^5)^(1/3)]
+ Log[1 - (-1 + x^5)^(1/3) + (-1 + x^5)^(2/3)])/15

Maple [C] (warning: unable to verify)

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

Time = 6.77 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.04

method	result	size
risch	\(-\frac {\left (x^{5}-1\right )^{\frac {2}{3}}}{5 x^{5}}+\frac {\sqrt {3}\, \Gamma \left (\frac {2}{3}\right ) {\left (-\operatorname {signum}\left (x^{5}-1\right )\right )}^{\frac {1}{3}} \left (\frac {2 \pi \sqrt {3}\, x^{5} \operatorname {hypergeom}\left (\left [1, 1, \frac {4}{3}\right ], \left [2, 2\right ], x^{5}\right )}{9 \Gamma \left (\frac {2}{3}\right )}+\frac {2 \left (-\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \left (3\right )}{2}+5 \ln \left (x \right )+i \pi \right ) \pi \sqrt {3}}{3 \Gamma \left (\frac {2}{3}\right )}\right )}{15 \pi \operatorname {signum}\left (x^{5}-1\right )^{\frac {1}{3}}}\)	\(96\)
meijerg	\(\frac {\sqrt {3}\, \Gamma \left (\frac {2}{3}\right ) \operatorname {signum}\left (x^{5}-1\right )^{\frac {2}{3}} \left (-\frac {\pi \sqrt {3}\, x^{5} \operatorname {hypergeom}\left (\left [1, 1, \frac {4}{3}\right ], \left [2, 3\right ], x^{5}\right )}{9 \Gamma \left (\frac {2}{3}\right )}-\frac {2 \left (-\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \left (3\right )}{2}-1+5 \ln \left (x \right )+i \pi \right ) \pi \sqrt {3}}{3 \Gamma \left (\frac {2}{3}\right )}-\frac {\pi \sqrt {3}}{\Gamma \left (\frac {2}{3}\right ) x^{5}}\right )}{15 \pi {\left (-\operatorname {signum}\left (x^{5}-1\right )\right )}^{\frac {2}{3}}}\)	\(97\)
pseudoelliptic	\(\frac {2 \sqrt {3}\, \arctan \left (\frac {\left (2 \left (x^{5}-1\right )^{\frac {1}{3}}-1\right ) \sqrt {3}}{3}\right ) x^{5}-2 \ln \left (1+\left (x^{5}-1\right )^{\frac {1}{3}}\right ) x^{5}+\ln \left (1-\left (x^{5}-1\right )^{\frac {1}{3}}+\left (x^{5}-1\right )^{\frac {2}{3}}\right ) x^{5}-3 \left (x^{5}-1\right )^{\frac {2}{3}}}{15 \left (1+\left (x^{5}-1\right )^{\frac {1}{3}}\right ) \left (1-\left (x^{5}-1\right )^{\frac {1}{3}}+\left (x^{5}-1\right )^{\frac {2}{3}}\right )}\)	\(107\)
trager	\(-\frac {\left (x^{5}-1\right )^{\frac {2}{3}}}{5 x^{5}}-\frac {2 \ln \left (\frac {30885108600167424 \operatorname {RootOf}\left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right )^{2} x^{5}-10233787081119648 \operatorname {RootOf}\left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right ) x^{5}[phone] x^{5}-12071260649204448 \left (x^{5}-1\right )^{\frac {2}{3}} \operatorname {RootOf}\left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right )-988323475205357568 \operatorname {RootOf}\left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right )^{2}+5593680301194816 \operatorname {RootOf}\left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right ) \left (x^{5}-1\right )^{\frac {1}{3}}[phone] \left (x^{5}-1\right )^{\frac {2}{3}}+27959977150455072 \operatorname {RootOf}\left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right )[phone] \left (x^{5}-1\right )^{\frac {1}{3}}-165507463554694}{x^{5}}\right )}{15}+\frac {64 \operatorname {RootOf}\left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right ) \ln \left (-\frac {25005193182296064 \operatorname {RootOf}\left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right )^{2} x^{5}+7492402988215776 \operatorname {RootOf}\left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right ) x^{5}-100537462891170 x^{5}-12071260649204448 \left (x^{5}-1\right )^{\frac {2}{3}} \operatorname {RootOf}\left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right )-800166181833474048 \operatorname {RootOf}\left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right )^{2}-17664940950399264 \operatorname {RootOf}\left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right ) \left (x^{5}-1\right )^{\frac {1}{3}}-58267503137446 \left (x^{5}-1\right )^{\frac {2}{3}}+2741384092903872 \operatorname {RootOf}\left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right )[phone] \left (x^{5}-1\right )^{\frac {1}{3}}[phone]}{x^{5}}\right )}{5}\)	\(294\)

[In]

int((x^5-1)^(2/3)/x^6,x,method=_RETURNVERBOSE)

[Out]

-1/5*(x^5-1)^(2/3)/x^5+1/15/Pi*3^(1/2)*GAMMA(2/3)/signum(x^5-1)^(1/3)*(-signum(x^5-1))^(1/3)*(2/9*Pi*3^(1/2)/G
AMMA(2/3)*x^5*hypergeom([1,1,4/3],[2,2],x^5)+2/3*(-1/6*Pi*3^(1/2)-3/2*ln(3)+5*ln(x)+I*Pi)*Pi*3^(1/2)/GAMMA(2/3
))

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.87 \[ \int \frac {\left (-1+x^5\right )^{2/3}}{x^6} \, dx=\frac {2 \, \sqrt {3} x^{5} \arctan \left (\frac {2}{3} \, \sqrt {3} {\left (x^{5} - 1\right )}^{\frac {1}{3}} - \frac {1}{3} \, \sqrt {3}\right ) + x^{5} \log \left ({\left (x^{5} - 1\right )}^{\frac {2}{3}} - {\left (x^{5} - 1\right )}^{\frac {1}{3}} + 1\right ) - 2 \, x^{5} \log \left ({\left (x^{5} - 1\right )}^{\frac {1}{3}} + 1\right ) - 3 \, {\left (x^{5} - 1\right )}^{\frac {2}{3}}}{15 \, x^{5}} \]

[In]

integrate((x^5-1)^(2/3)/x^6,x, algorithm="fricas")

[Out]

1/15*(2*sqrt(3)*x^5*arctan(2/3*sqrt(3)*(x^5 - 1)^(1/3) - 1/3*sqrt(3)) + x^5*log((x^5 - 1)^(2/3) - (x^5 - 1)^(1
/3) + 1) - 2*x^5*log((x^5 - 1)^(1/3) + 1) - 3*(x^5 - 1)^(2/3))/x^5

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.85 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.39 \[ \int \frac {\left (-1+x^5\right )^{2/3}}{x^6} \, dx=- \frac {\Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{3}, \frac {1}{3} \\ \frac {4}{3} \end {matrix}\middle | {\frac {e^{2 i \pi }}{x^{5}}} \right )}}{5 x^{\frac {5}{3}} \Gamma \left (\frac {4}{3}\right )} \]

[In]

integrate((x**5-1)**(2/3)/x**6,x)

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.74 \[ \int \frac {\left (-1+x^5\right )^{2/3}}{x^6} \, dx=\frac {2}{15} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (x^{5} - 1\right )}^{\frac {1}{3}} - 1\right )}\right ) - \frac {{\left (x^{5} - 1\right )}^{\frac {2}{3}}}{5 \, x^{5}} + \frac {1}{15} \, \log \left ({\left (x^{5} - 1\right )}^{\frac {2}{3}} - {\left (x^{5} - 1\right )}^{\frac {1}{3}} + 1\right ) - \frac {2}{15} \, \log \left ({\left (x^{5} - 1\right )}^{\frac {1}{3}} + 1\right ) \]

[In]

integrate((x^5-1)^(2/3)/x^6,x, algorithm="maxima")

[Out]

2/15*sqrt(3)*arctan(1/3*sqrt(3)*(2*(x^5 - 1)^(1/3) - 1)) - 1/5*(x^5 - 1)^(2/3)/x^5 + 1/15*log((x^5 - 1)^(2/3)
- (x^5 - 1)^(1/3) + 1) - 2/15*log((x^5 - 1)^(1/3) + 1)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.75 \[ \int \frac {\left (-1+x^5\right )^{2/3}}{x^6} \, dx=\frac {2}{15} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (x^{5} - 1\right )}^{\frac {1}{3}} - 1\right )}\right ) - \frac {{\left (x^{5} - 1\right )}^{\frac {2}{3}}}{5 \, x^{5}} + \frac {1}{15} \, \log \left ({\left (x^{5} - 1\right )}^{\frac {2}{3}} - {\left (x^{5} - 1\right )}^{\frac {1}{3}} + 1\right ) - \frac {2}{15} \, \log \left ({\left | {\left (x^{5} - 1\right )}^{\frac {1}{3}} + 1 \right |}\right ) \]

[In]

integrate((x^5-1)^(2/3)/x^6,x, algorithm="giac")

[Out]

2/15*sqrt(3)*arctan(1/3*sqrt(3)*(2*(x^5 - 1)^(1/3) - 1)) - 1/5*(x^5 - 1)^(2/3)/x^5 + 1/15*log((x^5 - 1)^(2/3)
- (x^5 - 1)^(1/3) + 1) - 2/15*log(abs((x^5 - 1)^(1/3) + 1))

Mupad [B] (verification not implemented)

Time = 5.86 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.00 \[ \int \frac {\left (-1+x^5\right )^{2/3}}{x^6} \, dx=-\frac {2\,\ln \left (\frac {4\,{\left (x^5-1\right )}^{1/3}}{25}+\frac {4}{25}\right )}{15}-\ln \left (9\,{\left (-\frac {1}{15}+\frac {\sqrt {3}\,1{}\mathrm {i}}{15}\right )}^2+\frac {4\,{\left (x^5-1\right )}^{1/3}}{25}\right )\,\left (-\frac {1}{15}+\frac {\sqrt {3}\,1{}\mathrm {i}}{15}\right )+\ln \left (9\,{\left (\frac {1}{15}+\frac {\sqrt {3}\,1{}\mathrm {i}}{15}\right )}^2+\frac {4\,{\left (x^5-1\right )}^{1/3}}{25}\right )\,\left (\frac {1}{15}+\frac {\sqrt {3}\,1{}\mathrm {i}}{15}\right )-\frac {{\left (x^5-1\right )}^{2/3}}{5\,x^5} \]

[In]

int((x^5 - 1)^(2/3)/x^6,x)

[Out]

log(9*((3^(1/2)*1i)/15 + 1/15)^2 + (4*(x^5 - 1)^(1/3))/25)*((3^(1/2)*1i)/15 + 1/15) - log(9*((3^(1/2)*1i)/15 -
1/15)^2 + (4*(x^5 - 1)^(1/3))/25)*((3^(1/2)*1i)/15 - 1/15) - (2*log((4*(x^5 - 1)^(1/3))/25 + 4/25))/15 - (x^5
- 1)^(2/3)/(5*x^5)