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

Optimal. Leaf size=90 \[ -\frac {\left (-1+x^2\right )^{2/3}}{2 x^2}-\frac {\text {ArcTan}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{-1+x^2}}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {1}{3} \log \left (1+\sqrt [3]{-1+x^2}\right )+\frac {1}{6} \log \left (1-\sqrt [3]{-1+x^2}+\left (-1+x^2\right )^{2/3}\right ) \]

[Out]

-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))

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Rubi [A]
time = 0.03, antiderivative size = 66, normalized size of antiderivative = 0.73, 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 = {272, 43, 58, 632, 210, 31} \begin {gather*} -\frac {\text {ArcTan}\left (\frac {1-2 \sqrt [3]{x^2-1}}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {\left (x^2-1\right )^{2/3}}{2 x^2}-\frac {1}{2} \log \left (\sqrt [3]{x^2-1}+1\right )+\frac {\log (x)}{3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-1 + x^2)^(2/3)/x^3,x]

[Out]

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

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

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Mathematica [A]
time = 0.09, size = 83, normalized size = 0.92 \begin {gather*} \frac {1}{6} \left (-\frac {3 \left (-1+x^2\right )^{2/3}}{x^2}-2 \sqrt {3} \text {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 ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-1 + x^2)^(2/3)/x^3,x]

[Out]

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

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 1.75, size = 96, normalized size = 1.07

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(x^2-1)^(2/3)/x^2+1/6/Pi*3^(1/2)*GAMMA(2/3)/signum(x^2-1)^(1/3)*(-signum(x^2-1))^(1/3)*(2/9*Pi*3^(1/2)/GA
MMA(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)
)

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Maxima [A]
time = 0.47, size = 68, normalized size = 0.76 \begin {gather*} \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 ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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)

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Fricas [A]
time = 0.47, size = 80, normalized size = 0.89 \begin {gather*} \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}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [C] Result contains complex when optimal does not.
time = 0.58, size = 36, normalized size = 0.40 \begin {gather*} - \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 )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 0.40, size = 69, normalized size = 0.77 \begin {gather*} \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 ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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))

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Mupad [B]
time = 0.88, size = 86, normalized size = 0.96 \begin {gather*} -\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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2 - 1)^(2/3)/x^3,x)

[Out]

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)

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