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

Optimal result

Integrand size = 18, antiderivative size = 95 \[ \int \frac {-1+x^2}{x \left (-1+x^3\right )^{2/3}} \, dx=-\frac {\arctan \left (\frac {1+\frac {2 x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {\arctan \left (\frac {1-2 \sqrt [3]{-1+x^3}}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {\log (x)}{2}-\frac {1}{2} \log \left (x-\sqrt [3]{-1+x^3}\right )-\frac {1}{2} \log \left (1+\sqrt [3]{-1+x^3}\right ) \]

[Out] -1/3*arctan(1/3*(1+2*x/(x^3-1)^(1/3))*3^(1/2))*3^(1/2)+1/3*arctan(1/3* 
(1-2*(x^3-1)^(1/3))*3^(1/2))*3^(1/2)+1/2*ln(x)-1/2*ln(x-(x^3-1)^(1/3)) 
-1/2*ln(1+(x^3-1)^(1/3))
 

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.389, Rules used = {2385, 798, 70, 1083, 217, 16, 853} \[ \int \frac {-1+x^2}{x \left (-1+x^3\right )^{2/3}} \, dx=-\frac {\arctan \left (\frac {\frac {2 x}{\sqrt [3]{x^3-1}}+1}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {\arctan \left (\frac {1-2 \sqrt [3]{x^3-1}}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {1}{2} \log \left (x-\sqrt [3]{x^3-1}\right )-\frac {1}{2} \log \left (\sqrt [3]{x^3-1}+1\right )+\frac {\log (x)}{2} \]

[In] Int[(-1 + x^2)/(x*(-1 + x^3)^(2/3)),x]
 

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

Rule 16

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

Rule 70

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(2/3)), x_Symbol] :> With[ 
{q = Rt[-(b*c - a*d)/b, 3]}, Simp[-Log[RemoveContent[a + b*x, x]]/(2*b*q^2) 
, x] + (Simp[3/(2*b*q)   Subst[Int[1/(q^2 - q*x + x^2), x], x, (c + d*x)^(1 
/3)], x] + Simp[3/(2*b*q^2)   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 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 853

Int[(x_)/((a_) + (b_.)*(x_)^3)^(2/3), x_Symbol] :> With[{q = Rt[b, 3]}, Sim 
p[-ArcTan[(1 + 2*q*(x/(a + b*x^3)^(1/3)))/Sqrt[3]]/(Sqrt[3]*q^2), x] - Simp 
[Log[q*x - (a + b*x^3)^(1/3)]/(2*q^2), x]] /; FreeQ[{a, b}, x]
 

Rule 1083

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

Rule 2385

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a1_) + (b1_.)*(x_)^(n_.))^(p_.)*((a2_) + (b2 
_.)*(x_)^(n_.))^(p_.), x_Symbol] :> Int[(c*x)^m*Pq*(a1*a2 + b1*b2*x^(2*n))^ 
p, x] /; FreeQ[{a1, b1, a2, b2, c, m, n, p}, x] && PolyQ[Pq, x] && EqQ[a2*b 
1 + a1*b2, 0] && (IntegerQ[p] || (GtQ[a1, 0] && GtQ[a2, 0]))
 

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

Mathematica [A] (verified)

Time = 1.15 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.91 \[ \int \frac {-1+x^2}{x \left (-1+x^3\right )^{2/3}} \, dx=\frac {1}{2} \left (-2 \sqrt {3} \arctan \left (\frac {\sqrt {3} (-1+x)}{-1+x+2 \sqrt [3]{-1+x^3}}\right )-2 \log \left (1-x+\sqrt [3]{-1+x^3}\right )+\log \left (1-2 x+x^2+(-1+x) \sqrt [3]{-1+x^3}+\left (-1+x^3\right )^{2/3}\right )\right ) \]

[In] Integrate[(-1 + x^2)/(x*(-1 + x^3)^(2/3)),x]
 

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

Maple [C] (warning: unable to verify)

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

Time = 0.67 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.04

method	result	size
meijerg	\(\frac {{\left (-\operatorname {signum}\left (x^{3}-1\right )\right )}^{\frac {2}{3}} x^{2} \operatorname {hypergeom}\left (\left [\frac {2}{3}, \frac {2}{3}\right ], \left [\frac {5}{3}\right ], x^{3}\right )}{2 \operatorname {signum}\left (x^{3}-1\right )^{\frac {2}{3}}}-\frac {{\left (-\operatorname {signum}\left (x^{3}-1\right )\right )}^{\frac {2}{3}} \left (\left (\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \left (3\right )}{2}+3 \ln \left (x \right )+i \pi \right ) \Gamma \left (\frac {2}{3}\right )+\frac {2 \Gamma \left (\frac {2}{3}\right ) x^{3} \operatorname {hypergeom}\left (\left [1, 1, \frac {5}{3}\right ], \left [2, 2\right ], x^{3}\right )}{3}\right )}{3 \Gamma \left (\frac {2}{3}\right ) \operatorname {signum}\left (x^{3}-1\right )^{\frac {2}{3}}}\)	\(99\)
trager	\(-\ln \left (-\frac {58 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{2}-145 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x -99 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {2}{3}}-99 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {1}{3}} x -157 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{2}+58 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2}+99 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {1}{3}}+117 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x -28 \left (x^{3}-1\right )^{\frac {2}{3}}-28 \left (x^{3}-1\right )^{\frac {1}{3}} x +30 x^{2}-157 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )+28 \left (x^{3}-1\right )^{\frac {1}{3}}-18 x +30}{x}\right ) \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )+\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \ln \left (-\frac {58 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{2}-145 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x +99 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {2}{3}}+99 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {1}{3}} x +41 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{2}+58 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2}-99 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {1}{3}}+173 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x -127 \left (x^{3}-1\right )^{\frac {2}{3}}-127 \left (x^{3}-1\right )^{\frac {1}{3}} x -69 x^{2}+41 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )+127 \left (x^{3}-1\right )^{\frac {1}{3}}-46 x -69}{x}\right )+\ln \left (-\frac {58 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{2}-145 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x -99 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {2}{3}}-99 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {1}{3}} x -157 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{2}+58 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2}+99 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {1}{3}}+117 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x -28 \left (x^{3}-1\right )^{\frac {2}{3}}-28 \left (x^{3}-1\right )^{\frac {1}{3}} x +30 x^{2}-157 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )+28 \left (x^{3}-1\right )^{\frac {1}{3}}-18 x +30}{x}\right )\)	\(560\)

[In] int((x^2-1)/x/(x^3-1)^(2/3),x,method=_RETURNVERBOSE)
 

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

Fricas [A] (verification not implemented)

none

Time = 0.40 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.93 \[ \int \frac {-1+x^2}{x \left (-1+x^3\right )^{2/3}} \, dx=-\sqrt {3} \arctan \left (-\frac {4 \, \sqrt {3} {\left (x^{3} - 1\right )}^{\frac {1}{3}} {\left (x - 1\right )} + \sqrt {3} {\left (x^{2} + x + 1\right )} - 2 \, \sqrt {3} {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{3 \, {\left (3 \, x^{2} - 5 \, x + 3\right )}}\right ) - \frac {1}{2} \, \log \left (\frac {{\left (x^{3} - 1\right )}^{\frac {1}{3}} {\left (x - 1\right )} + x - {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{x}\right ) \]

[In] integrate((x^2-1)/x/(x^3-1)^(2/3),x, algorithm="fricas")
 

[Out] -sqrt(3)*arctan(-1/3*(4*sqrt(3)*(x^3 - 1)^(1/3)*(x - 1) + sqrt(3)*(x^2 
 + x + 1) - 2*sqrt(3)*(x^3 - 1)^(2/3))/(3*x^2 - 5*x + 3)) - 1/2*log((( 
x^3 - 1)^(1/3)*(x - 1) + x - (x^3 - 1)^(2/3))/x)
 

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 1.10 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.68 \[ \int \frac {-1+x^2}{x \left (-1+x^3\right )^{2/3}} \, dx=\frac {x^{2} e^{- \frac {2 i \pi }{3}} \Gamma \left (\frac {2}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {2}{3}, \frac {2}{3} \\ \frac {5}{3} \end {matrix}\middle | {x^{3}} \right )}}{3 \Gamma \left (\frac {5}{3}\right )} + \frac {\Gamma \left (\frac {2}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {2}{3}, \frac {2}{3} \\ \frac {5}{3} \end {matrix}\middle | {\frac {e^{2 i \pi }}{x^{3}}} \right )}}{3 x^{2} \Gamma \left (\frac {5}{3}\right )} \]

[In] integrate((x**2-1)/x/(x**3-1)**(2/3),x)
 

[Out] x**2*exp(-2*I*pi/3)*gamma(2/3)*hyper((2/3, 2/3), (5/3,), x**3)/(3*gamm 
a(5/3)) + gamma(2/3)*hyper((2/3, 2/3), (5/3,), exp_polar(2*I*pi)/x**3) 
/(3*x**2*gamma(5/3))
 

Maxima [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.31 \[ \int \frac {-1+x^2}{x \left (-1+x^3\right )^{2/3}} \, dx=-\frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (x^{3} - 1\right )}^{\frac {1}{3}} - 1\right )}\right ) + \frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (\frac {2 \, {\left (x^{3} - 1\right )}^{\frac {1}{3}}}{x} + 1\right )}\right ) + \frac {1}{6} \, \log \left ({\left (x^{3} - 1\right )}^{\frac {2}{3}} - {\left (x^{3} - 1\right )}^{\frac {1}{3}} + 1\right ) - \frac {1}{3} \, \log \left ({\left (x^{3} - 1\right )}^{\frac {1}{3}} + 1\right ) + \frac {1}{6} \, \log \left (\frac {{\left (x^{3} - 1\right )}^{\frac {1}{3}}}{x} + \frac {{\left (x^{3} - 1\right )}^{\frac {2}{3}}}{x^{2}} + 1\right ) - \frac {1}{3} \, \log \left (\frac {{\left (x^{3} - 1\right )}^{\frac {1}{3}}}{x} - 1\right ) \]

[In] integrate((x^2-1)/x/(x^3-1)^(2/3),x, algorithm="maxima")
 

[Out] -1/3*sqrt(3)*arctan(1/3*sqrt(3)*(2*(x^3 - 1)^(1/3) - 1)) + 1/3*sqrt(3) 
*arctan(1/3*sqrt(3)*(2*(x^3 - 1)^(1/3)/x + 1)) + 1/6*log((x^3 - 1)^(2/ 
3) - (x^3 - 1)^(1/3) + 1) - 1/3*log((x^3 - 1)^(1/3) + 1) + 1/6*log((x^ 
3 - 1)^(1/3)/x + (x^3 - 1)^(2/3)/x^2 + 1) - 1/3*log((x^3 - 1)^(1/3)/x 
- 1)
 

Giac [F]

\[ \int \frac {-1+x^2}{x \left (-1+x^3\right )^{2/3}} \, dx=\int { \frac {x^{2} - 1}{{\left (x^{3} - 1\right )}^{\frac {2}{3}} x} \,d x } \]

[In] integrate((x^2-1)/x/(x^3-1)^(2/3),x, algorithm="giac")
 

[Out] integrate((x^2 - 1)/((x^3 - 1)^(2/3)*x), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {-1+x^2}{x \left (-1+x^3\right )^{2/3}} \, dx=\int \frac {x^2-1}{x\,{\left (x^3-1\right )}^{2/3}} \,d x \]

[In] int((x^2 - 1)/(x*(x^3 - 1)^(2/3)),x)
 

[Out] int((x^2 - 1)/(x*(x^3 - 1)^(2/3)), x)