Integrand size = 18, antiderivative size = 95 \[ \int \frac {-1+x^2}{x \left (-1+x^3\right )^{2/3}} \, dx=-\frac {\arctan \left (\frac {1+\frac {2 x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {\arctan \left (\frac {1-2 \sqrt [3]{-1+x^3}}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {\log (x)}{2}-\frac {1}{2} \log \left (x-\sqrt [3]{-1+x^3}\right )-\frac {1}{2} \log \left (1+\sqrt [3]{-1+x^3}\right ) \] Output:
-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))
Time = 1.39 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.91 \[ \int \frac {-1+x^2}{x \left (-1+x^3\right )^{2/3}} \, dx=\frac {1}{2} \left (-2 \sqrt {3} \arctan \left (\frac {\sqrt {3} (-1+x)}{-1+x+2 \sqrt [3]{-1+x^3}}\right )-2 \log \left (1-x+\sqrt [3]{-1+x^3}\right )+\log \left (1-2 x+x^2+(-1+x) \sqrt [3]{-1+x^3}+\left (-1+x^3\right )^{2/3}\right )\right ) \] Input:
Integrate[(-1 + x^2)/(x*(-1 + x^3)^(2/3)),x]
Output:
(-2*Sqrt[3]*ArcTan[(Sqrt[3]*(-1 + x))/(-1 + x + 2*(-1 + x^3)^(1/3))] - 2*L og[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
Time = 0.26 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2383, 2009}
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.
\(\displaystyle \int \frac {x^2-1}{x \left (x^3-1\right )^{2/3}} \, dx\) |
\(\Big \downarrow \) 2383 |
\(\displaystyle \int \left (\frac {x}{\left (x^3-1\right )^{2/3}}-\frac {1}{x \left (x^3-1\right )^{2/3}}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {\arctan \left (\frac {\frac {2 x}{\sqrt [3]{x^3-1}}+1}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {\arctan \left (\frac {1-2 \sqrt [3]{x^3-1}}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {1}{2} \log \left (x-\sqrt [3]{x^3-1}\right )-\frac {1}{2} \log \left (\sqrt [3]{x^3-1}+1\right )+\frac {\log (x)}{2}\) |
Input:
Int[(-1 + x^2)/(x*(-1 + x^3)^(2/3)),x]
Output:
-(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
Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> I nt[ExpandIntegrand[(c*x)^m*Pq*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, m, n , p}, x] && (PolyQ[Pq, x] || PolyQ[Pq, x^n]) && !IGtQ[m, 0]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.81 (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 {12 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{2}-30 \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}}+127 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {1}{3}} x -40 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{2}+12 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2}-127 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {1}{3}}+157 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x +127 \left (x^{3}-1\right )^{\frac {2}{3}}-28 \left (x^{3}-1\right )^{\frac {1}{3}} x -87 x^{2}-40 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )+28 \left (x^{3}-1\right )^{\frac {1}{3}}-58 x -87}{x}\right )+\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \ln \left (\frac {46 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{2}-115 \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}}+28 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {1}{3}} x -173 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{2}+46 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2}-28 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {1}{3}}+214 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x +28 \left (x^{3}-1\right )^{\frac {2}{3}}-127 \left (x^{3}-1\right )^{\frac {1}{3}} x +145 x^{2}-173 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )+127 \left (x^{3}-1\right )^{\frac {1}{3}}-87 x +145}{x}\right )\) | \(370\) |
Input:
int((x^2-1)/x/(x^3-1)^(2/3),x,method=_RETURNVERBOSE)
Output:
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))
Time = 0.31 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.93 \[ \int \frac {-1+x^2}{x \left (-1+x^3\right )^{2/3}} \, dx=-\sqrt {3} \arctan \left (-\frac {4 \, \sqrt {3} {\left (x^{3} - 1\right )}^{\frac {1}{3}} {\left (x - 1\right )} + \sqrt {3} {\left (x^{2} + x + 1\right )} - 2 \, \sqrt {3} {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{3 \, {\left (3 \, x^{2} - 5 \, x + 3\right )}}\right ) - \frac {1}{2} \, \log \left (\frac {{\left (x^{3} - 1\right )}^{\frac {1}{3}} {\left (x - 1\right )} + x - {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{x}\right ) \] Input:
integrate((x^2-1)/x/(x^3-1)^(2/3),x, algorithm="fricas")
Output:
-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)
Result contains complex when optimal does not.
Time = 1.21 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.68 \[ \int \frac {-1+x^2}{x \left (-1+x^3\right )^{2/3}} \, dx=\frac {x^{2} e^{- \frac {2 i \pi }{3}} \Gamma \left (\frac {2}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {2}{3}, \frac {2}{3} \\ \frac {5}{3} \end {matrix}\middle | {x^{3}} \right )}}{3 \Gamma \left (\frac {5}{3}\right )} + \frac {\Gamma \left (\frac {2}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {2}{3}, \frac {2}{3} \\ \frac {5}{3} \end {matrix}\middle | {\frac {e^{2 i \pi }}{x^{3}}} \right )}}{3 x^{2} \Gamma \left (\frac {5}{3}\right )} \] Input:
integrate((x**2-1)/x/(x**3-1)**(2/3),x)
Output:
x**2*exp(-2*I*pi/3)*gamma(2/3)*hyper((2/3, 2/3), (5/3,), x**3)/(3*gamma(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))
Time = 0.12 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.31 \[ \int \frac {-1+x^2}{x \left (-1+x^3\right )^{2/3}} \, dx=-\frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (x^{3} - 1\right )}^{\frac {1}{3}} - 1\right )}\right ) + \frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (\frac {2 \, {\left (x^{3} - 1\right )}^{\frac {1}{3}}}{x} + 1\right )}\right ) + \frac {1}{6} \, \log \left ({\left (x^{3} - 1\right )}^{\frac {2}{3}} - {\left (x^{3} - 1\right )}^{\frac {1}{3}} + 1\right ) - \frac {1}{3} \, \log \left ({\left (x^{3} - 1\right )}^{\frac {1}{3}} + 1\right ) + \frac {1}{6} \, \log \left (\frac {{\left (x^{3} - 1\right )}^{\frac {1}{3}}}{x} + \frac {{\left (x^{3} - 1\right )}^{\frac {2}{3}}}{x^{2}} + 1\right ) - \frac {1}{3} \, \log \left (\frac {{\left (x^{3} - 1\right )}^{\frac {1}{3}}}{x} - 1\right ) \] Input:
integrate((x^2-1)/x/(x^3-1)^(2/3),x, algorithm="maxima")
Output:
-1/3*sqrt(3)*arctan(1/3*sqrt(3)*(2*(x^3 - 1)^(1/3) - 1)) + 1/3*sqrt(3)*arc tan(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)
\[ \int \frac {-1+x^2}{x \left (-1+x^3\right )^{2/3}} \, dx=\int { \frac {x^{2} - 1}{{\left (x^{3} - 1\right )}^{\frac {2}{3}} x} \,d x } \] Input:
integrate((x^2-1)/x/(x^3-1)^(2/3),x, algorithm="giac")
Output:
integrate((x^2 - 1)/((x^3 - 1)^(2/3)*x), x)
Timed out. \[ \int \frac {-1+x^2}{x \left (-1+x^3\right )^{2/3}} \, dx=\int \frac {x^2-1}{x\,{\left (x^3-1\right )}^{2/3}} \,d x \] Input:
int((x^2 - 1)/(x*(x^3 - 1)^(2/3)),x)
Output:
int((x^2 - 1)/(x*(x^3 - 1)^(2/3)), x)
\[ \int \frac {-1+x^2}{x \left (-1+x^3\right )^{2/3}} \, dx=\int \frac {x}{\left (x^{3}-1\right )^{\frac {2}{3}}}d x -\left (\int \frac {1}{\left (x^{3}-1\right )^{\frac {2}{3}} x}d x \right ) \] Input:
int((x^2-1)/x/(x^3-1)^(2/3),x)
Output:
int(x/(x**3 - 1)**(2/3),x) - int(1/((x**3 - 1)**(2/3)*x),x)