Integrand size = 13, antiderivative size = 90 \[ \int \frac {\left (-1+x^2\right )^{2/3}}{x^3} \, dx=-\frac {\left (-1+x^2\right )^{2/3}}{2 x^2}-\frac {\arctan \left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{-1+x^2}}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {1}{3} \log \left (1+\sqrt [3]{-1+x^2}\right )+\frac {1}{6} \log \left (1-\sqrt [3]{-1+x^2}+\left (-1+x^2\right )^{2/3}\right ) \]
[Out]
Time = 0.04 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.73, 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} \[ \int \frac {\left (-1+x^2\right )^{2/3}}{x^3} \, dx=-\frac {\arctan \left (\frac {1-2 \sqrt [3]{x^2-1}}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {\left (x^2-1\right )^{2/3}}{2 x^2}-\frac {1}{2} \log \left (\sqrt [3]{x^2-1}+1\right )+\frac {\log (x)}{3} \]
[In]
[Out]
Rule 31
Rule 43
Rule 58
Rule 210
Rule 272
Rule 632
Rubi steps \begin{align*} \text {integral}& = \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 {\arctan \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*}
Time = 0.09 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.92 \[ \int \frac {\left (-1+x^2\right )^{2/3}}{x^3} \, dx=\frac {1}{6} \left (-\frac {3 \left (-1+x^2\right )^{2/3}}{x^2}-2 \sqrt {3} \arctan \left (\frac {1-2 \sqrt [3]{-1+x^2}}{\sqrt {3}}\right )-2 \log \left (1+\sqrt [3]{-1+x^2}\right )+\log \left (1-\sqrt [3]{-1+x^2}+\left (-1+x^2\right )^{2/3}\right )\right ) \]
[In]
[Out]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 2.94 (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 | \(\frac {2 \sqrt {3}\, \arctan \left (\frac {\left (2 \left (x^{2}-1\right )^{\frac {1}{3}}-1\right ) \sqrt {3}}{3}\right ) x^{2}-2 \ln \left (1+\left (x^{2}-1\right )^{\frac {1}{3}}\right ) x^{2}+\ln \left (1-\left (x^{2}-1\right )^{\frac {1}{3}}+\left (x^{2}-1\right )^{\frac {2}{3}}\right ) x^{2}-3 \left (x^{2}-1\right )^{\frac {2}{3}}}{6 \left (1+\left (x^{2}-1\right )^{\frac {1}{3}}\right ) \left (1-\left (x^{2}-1\right )^{\frac {1}{3}}+\left (x^{2}-1\right )^{\frac {2}{3}}\right )}\) | \(107\) |
trager | \(-\frac {\left (x^{2}-1\right )^{\frac {2}{3}}}{2 x^{2}}-\frac {\ln \left (\frac {-576 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}-12 \textit {\_Z} +1\right )^{2} x^{2}+180 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}-12 \textit {\_Z} +1\right ) \left (x^{2}-1\right )^{\frac {2}{3}}+204 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}-12 \textit {\_Z} +1\right ) x^{2}-24 \left (x^{2}-1\right )^{\frac {2}{3}}-108 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}-12 \textit {\_Z} +1\right ) \left (x^{2}-1\right )^{\frac {1}{3}}+2304 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}-12 \textit {\_Z} +1\right )^{2}-15 x^{2}-15 \left (x^{2}-1\right )^{\frac {1}{3}}-480 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}-12 \textit {\_Z} +1\right )+25}{x^{2}}\right )}{3}+4 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}-12 \textit {\_Z} +1\right ) \ln \left (-\frac {-720 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}-12 \textit {\_Z} +1\right )^{2} x^{2}+180 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}-12 \textit {\_Z} +1\right ) \left (x^{2}-1\right )^{\frac {2}{3}}-72 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}-12 \textit {\_Z} +1\right ) x^{2}+9 \left (x^{2}-1\right )^{\frac {2}{3}}+288 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}-12 \textit {\_Z} +1\right ) \left (x^{2}-1\right )^{\frac {1}{3}}+2880 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}-12 \textit {\_Z} +1\right )^{2}+8 x^{2}-15 \left (x^{2}-1\right )^{\frac {1}{3}}-132 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}-12 \textit {\_Z} +1\right )-4}{x^{2}}\right )\) | \(294\) |
[In]
[Out]
none
Time = 0.30 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.89 \[ \int \frac {\left (-1+x^2\right )^{2/3}}{x^3} \, dx=\frac {2 \, \sqrt {3} x^{2} \arctan \left (\frac {2}{3} \, \sqrt {3} {\left (x^{2} - 1\right )}^{\frac {1}{3}} - \frac {1}{3} \, \sqrt {3}\right ) + x^{2} \log \left ({\left (x^{2} - 1\right )}^{\frac {2}{3}} - {\left (x^{2} - 1\right )}^{\frac {1}{3}} + 1\right ) - 2 \, x^{2} \log \left ({\left (x^{2} - 1\right )}^{\frac {1}{3}} + 1\right ) - 3 \, {\left (x^{2} - 1\right )}^{\frac {2}{3}}}{6 \, x^{2}} \]
[In]
[Out]
Result contains complex when optimal does not.
Time = 0.61 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.40 \[ \int \frac {\left (-1+x^2\right )^{2/3}}{x^3} \, dx=- \frac {\Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{3}, \frac {1}{3} \\ \frac {4}{3} \end {matrix}\middle | {\frac {e^{2 i \pi }}{x^{2}}} \right )}}{2 x^{\frac {2}{3}} \Gamma \left (\frac {4}{3}\right )} \]
[In]
[Out]
none
Time = 0.28 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.76 \[ \int \frac {\left (-1+x^2\right )^{2/3}}{x^3} \, dx=\frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (x^{2} - 1\right )}^{\frac {1}{3}} - 1\right )}\right ) - \frac {{\left (x^{2} - 1\right )}^{\frac {2}{3}}}{2 \, x^{2}} + \frac {1}{6} \, \log \left ({\left (x^{2} - 1\right )}^{\frac {2}{3}} - {\left (x^{2} - 1\right )}^{\frac {1}{3}} + 1\right ) - \frac {1}{3} \, \log \left ({\left (x^{2} - 1\right )}^{\frac {1}{3}} + 1\right ) \]
[In]
[Out]
none
Time = 0.29 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.77 \[ \int \frac {\left (-1+x^2\right )^{2/3}}{x^3} \, dx=\frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (x^{2} - 1\right )}^{\frac {1}{3}} - 1\right )}\right ) - \frac {{\left (x^{2} - 1\right )}^{\frac {2}{3}}}{2 \, x^{2}} + \frac {1}{6} \, \log \left ({\left (x^{2} - 1\right )}^{\frac {2}{3}} - {\left (x^{2} - 1\right )}^{\frac {1}{3}} + 1\right ) - \frac {1}{3} \, \log \left ({\left | {\left (x^{2} - 1\right )}^{\frac {1}{3}} + 1 \right |}\right ) \]
[In]
[Out]
Time = 5.76 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.96 \[ \int \frac {\left (-1+x^2\right )^{2/3}}{x^3} \, dx=-\frac {\ln \left ({\left (x^2-1\right )}^{1/3}+1\right )}{3}-\ln \left (9\,{\left (-\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )}^2+{\left (x^2-1\right )}^{1/3}\right )\,\left (-\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )+\ln \left (9\,{\left (\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )}^2+{\left (x^2-1\right )}^{1/3}\right )\,\left (\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )-\frac {{\left (x^2-1\right )}^{2/3}}{2\,x^2} \]
[In]
[Out]