\(\int \frac {(-1+x^2)^{2/3}}{x^3} \, dx\) [1226]

Optimal result

Integrand size = 13, antiderivative size = 90 \[ \int \frac {\left (-1+x^2\right )^{2/3}}{x^3} \, dx=-\frac {\left (-1+x^2\right )^{2/3}}{2 x^2}-\frac {\arctan \left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{-1+x^2}}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {1}{3} \log \left (1+\sqrt [3]{-1+x^2}\right )+\frac {1}{6} \log \left (1-\sqrt [3]{-1+x^2}+\left (-1+x^2\right )^{2/3}\right ) \] Output:

-1/2*(x^2-1)^(2/3)/x^2+1/3*3^(1/2)*arctan(-1/3*3^(1/2)+2/3*(x^2-1)^(1/3)*3 
^(1/2))-1/3*ln(1+(x^2-1)^(1/3))+1/6*ln(1-(x^2-1)^(1/3)+(x^2-1)^(2/3))
 

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.92 \[ \int \frac {\left (-1+x^2\right )^{2/3}}{x^3} \, dx=\frac {1}{6} \left (-\frac {3 \left (-1+x^2\right )^{2/3}}{x^2}-2 \sqrt {3} \arctan \left (\frac {1-2 \sqrt [3]{-1+x^2}}{\sqrt {3}}\right )-2 \log \left (1+\sqrt [3]{-1+x^2}\right )+\log \left (1-\sqrt [3]{-1+x^2}+\left (-1+x^2\right )^{2/3}\right )\right ) \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.19 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.82, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {243, 51, 68, 16, 1083, 217}

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.

\(\displaystyle \int \frac {\left (x^2-1\right )^{2/3}}{x^3} \, dx\)

\(\Big \downarrow \) 243

\(\displaystyle \frac {1}{2} \int \frac {\left (x^2-1\right )^{2/3}}{x^4}dx^2\)

\(\Big \downarrow \) 51

\(\displaystyle \frac {1}{2} \left (\frac {2}{3} \int \frac {1}{x^2 \sqrt [3]{x^2-1}}dx^2-\frac {\left (x^2-1\right )^{2/3}}{x^2}\right )\)

\(\Big \downarrow \) 68

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

\(\Big \downarrow \) 16

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

\(\Big \downarrow \) 1083

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

\(\Big \downarrow \) 217

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

Input:

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

Output:

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

Defintions of rubi rules used

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 51

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] - Simp[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 
] && ILtQ[m, -1] && FractionQ[n] && GtQ[n, 0]
 

rule 68

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 
] + (Simp[3/(2*b)   Subst[Int[1/(q^2 - q*x + x^2), x], x, (c + d*x)^(1/3)], 
 x] - Simp[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 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 243

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2   Subst[In 
t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I 
ntegerQ[(m - 1)/2]
 

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]
 

Maple [C] (warning: unable to verify)

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

Time = 1.28 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.07

method	result	size
risch	\(-\frac {\left (x^{2}-1\right )^{\frac {2}{3}}}{2 x^{2}}+\frac {\sqrt {3}\, \Gamma \left (\frac {2}{3}\right ) {\left (-\operatorname {signum}\left (x^{2}-1\right )\right )}^{\frac {1}{3}} \left (\frac {2 \pi \sqrt {3}\, x^{2} \operatorname {hypergeom}\left (\left [1, 1, \frac {4}{3}\right ], \left [2, 2\right ], x^{2}\right )}{9 \Gamma \left (\frac {2}{3}\right )}+\frac {2 \left (-\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \left (3\right )}{2}+2 \ln \left (x \right )+i \pi \right ) \pi \sqrt {3}}{3 \Gamma \left (\frac {2}{3}\right )}\right )}{6 \pi \operatorname {signum}\left (x^{2}-1\right )^{\frac {1}{3}}}\)	\(96\)
meijerg	\(\frac {\sqrt {3}\, \Gamma \left (\frac {2}{3}\right ) \operatorname {signum}\left (x^{2}-1\right )^{\frac {2}{3}} \left (-\frac {\pi \sqrt {3}\, x^{2} \operatorname {hypergeom}\left (\left [1, 1, \frac {4}{3}\right ], \left [2, 3\right ], x^{2}\right )}{9 \Gamma \left (\frac {2}{3}\right )}-\frac {2 \left (-\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \left (3\right )}{2}-1+2 \ln \left (x \right )+i \pi \right ) \pi \sqrt {3}}{3 \Gamma \left (\frac {2}{3}\right )}-\frac {\pi \sqrt {3}}{\Gamma \left (\frac {2}{3}\right ) x^{2}}\right )}{6 \pi {\left (-\operatorname {signum}\left (x^{2}-1\right )\right )}^{\frac {2}{3}}}\)	\(97\)
pseudoelliptic	\(\frac {2 \sqrt {3}\, \arctan \left (\frac {\left (2 \left (x^{2}-1\right )^{\frac {1}{3}}-1\right ) \sqrt {3}}{3}\right ) x^{2}+\ln \left (1-\left (x^{2}-1\right )^{\frac {1}{3}}+\left (x^{2}-1\right )^{\frac {2}{3}}\right ) x^{2}-2 \ln \left (1+\left (x^{2}-1\right )^{\frac {1}{3}}\right ) x^{2}-3 \left (x^{2}-1\right )^{\frac {2}{3}}}{6 \left (1-\left (x^{2}-1\right )^{\frac {1}{3}}+\left (x^{2}-1\right )^{\frac {2}{3}}\right ) \left (1+\left (x^{2}-1\right )^{\frac {1}{3}}\right )}\)	\(107\)
trager	\(-\frac {\left (x^{2}-1\right )^{\frac {2}{3}}}{2 x^{2}}+4 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}-12 \textit {\_Z} +1\right ) \ln \left (\frac {-36 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}-12 \textit {\_Z} +1\right )^{2} x^{2}+45 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}-12 \textit {\_Z} +1\right ) \left (x^{2}-1\right )^{\frac {2}{3}}-3 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}-12 \textit {\_Z} +1\right ) x^{2}-6 \left (x^{2}-1\right )^{\frac {2}{3}}-45 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}-12 \textit {\_Z} +1\right ) \left (x^{2}-1\right )^{\frac {1}{3}}+144 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}-12 \textit {\_Z} +1\right )^{2}+3 x^{2}+6 \left (x^{2}-1\right )^{\frac {1}{3}}+33 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}-12 \textit {\_Z} +1\right )-5}{x^{2}}\right )+\frac {\ln \left (-\frac {144 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}-12 \textit {\_Z} +1\right )^{2} x^{2}+180 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}-12 \textit {\_Z} +1\right ) \left (x^{2}-1\right )^{\frac {2}{3}}-36 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}-12 \textit {\_Z} +1\right ) x^{2}+9 \left (x^{2}-1\right )^{\frac {2}{3}}-180 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}-12 \textit {\_Z} +1\right ) \left (x^{2}-1\right )^{\frac {1}{3}}-576 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}-12 \textit {\_Z} +1\right )^{2}-10 x^{2}-9 \left (x^{2}-1\right )^{\frac {1}{3}}+228 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}-12 \textit {\_Z} +1\right )+5}{x^{2}}\right )}{3}-4 \ln \left (-\frac {144 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}-12 \textit {\_Z} +1\right )^{2} x^{2}+180 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}-12 \textit {\_Z} +1\right ) \left (x^{2}-1\right )^{\frac {2}{3}}-36 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}-12 \textit {\_Z} +1\right ) x^{2}+9 \left (x^{2}-1\right )^{\frac {2}{3}}-180 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}-12 \textit {\_Z} +1\right ) \left (x^{2}-1\right )^{\frac {1}{3}}-576 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}-12 \textit {\_Z} +1\right )^{2}-10 x^{2}-9 \left (x^{2}-1\right )^{\frac {1}{3}}+228 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}-12 \textit {\_Z} +1\right )+5}{x^{2}}\right ) \operatorname {RootOf}\left (144 \textit {\_Z}^{2}-12 \textit {\_Z} +1\right )\)	\(440\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.89 \[ \int \frac {\left (-1+x^2\right )^{2/3}}{x^3} \, dx=\frac {2 \, \sqrt {3} x^{2} \arctan \left (\frac {2}{3} \, \sqrt {3} {\left (x^{2} - 1\right )}^{\frac {1}{3}} - \frac {1}{3} \, \sqrt {3}\right ) + x^{2} \log \left ({\left (x^{2} - 1\right )}^{\frac {2}{3}} - {\left (x^{2} - 1\right )}^{\frac {1}{3}} + 1\right ) - 2 \, x^{2} \log \left ({\left (x^{2} - 1\right )}^{\frac {1}{3}} + 1\right ) - 3 \, {\left (x^{2} - 1\right )}^{\frac {2}{3}}}{6 \, x^{2}} \] Input:

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

Output:

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

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.67 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.40 \[ \int \frac {\left (-1+x^2\right )^{2/3}}{x^3} \, dx=- \frac {\Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{3}, \frac {1}{3} \\ \frac {4}{3} \end {matrix}\middle | {\frac {e^{2 i \pi }}{x^{2}}} \right )}}{2 x^{\frac {2}{3}} \Gamma \left (\frac {4}{3}\right )} \] Input:

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.76 \[ \int \frac {\left (-1+x^2\right )^{2/3}}{x^3} \, dx=\frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (x^{2} - 1\right )}^{\frac {1}{3}} - 1\right )}\right ) - \frac {{\left (x^{2} - 1\right )}^{\frac {2}{3}}}{2 \, x^{2}} + \frac {1}{6} \, \log \left ({\left (x^{2} - 1\right )}^{\frac {2}{3}} - {\left (x^{2} - 1\right )}^{\frac {1}{3}} + 1\right ) - \frac {1}{3} \, \log \left ({\left (x^{2} - 1\right )}^{\frac {1}{3}} + 1\right ) \] Input:

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

Output:

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

Giac [A] (verification not implemented)

Time = 0.33 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.77 \[ \int \frac {\left (-1+x^2\right )^{2/3}}{x^3} \, dx=\frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (x^{2} - 1\right )}^{\frac {1}{3}} - 1\right )}\right ) - \frac {{\left (x^{2} - 1\right )}^{\frac {2}{3}}}{2 \, x^{2}} + \frac {1}{6} \, \log \left ({\left (x^{2} - 1\right )}^{\frac {2}{3}} - {\left (x^{2} - 1\right )}^{\frac {1}{3}} + 1\right ) - \frac {1}{3} \, \log \left ({\left | {\left (x^{2} - 1\right )}^{\frac {1}{3}} + 1 \right |}\right ) \] Input:

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

Output:

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

Mupad [B] (verification not implemented)

Time = 8.98 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.96 \[ \int \frac {\left (-1+x^2\right )^{2/3}}{x^3} \, dx=-\frac {\ln \left ({\left (x^2-1\right )}^{1/3}+1\right )}{3}-\ln \left (9\,{\left (-\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )}^2+{\left (x^2-1\right )}^{1/3}\right )\,\left (-\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )+\ln \left (9\,{\left (\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )}^2+{\left (x^2-1\right )}^{1/3}\right )\,\left (\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )-\frac {{\left (x^2-1\right )}^{2/3}}{2\,x^2} \] Input:

int((x^2 - 1)^(2/3)/x^3,x)
                                                                                    
                                                                                    
 

Output:

log(9*((3^(1/2)*1i)/6 + 1/6)^2 + (x^2 - 1)^(1/3))*((3^(1/2)*1i)/6 + 1/6) - 
 log(9*((3^(1/2)*1i)/6 - 1/6)^2 + (x^2 - 1)^(1/3))*((3^(1/2)*1i)/6 - 1/6) 
- log((x^2 - 1)^(1/3) + 1)/3 - (x^2 - 1)^(2/3)/(2*x^2)
 

Reduce [F]

\[ \int \frac {\left (-1+x^2\right )^{2/3}}{x^3} \, dx=\frac {-3 \left (x^{2}-1\right )^{\frac {2}{3}}+4 \left (\int \frac {\left (x^{2}-1\right )^{\frac {2}{3}}}{x^{3}-x}d x \right ) x^{2}}{6 x^{2}} \] Input:

int((x^2-1)^(2/3)/x^3,x)
 

Output:

( - 3*(x**2 - 1)**(2/3) + 4*int((x**2 - 1)**(2/3)/(x**3 - x),x)*x**2)/(6*x 
**2)