Integrand size = 13, antiderivative size = 87 \[ \int \frac {1}{x^3 \left (1+x^2\right )^{2/3}} \, dx=-\frac {\sqrt [3]{1+x^2}}{2 x^2}+\frac {\arctan \left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{1+x^2}}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {1}{3} \log \left (-1+\sqrt [3]{1+x^2}\right )+\frac {1}{6} \log \left (1+\sqrt [3]{1+x^2}+\left (1+x^2\right )^{2/3}\right ) \]
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Time = 0.04 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.77, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {272, 44, 59, 632, 210, 31} \[ \int \frac {1}{x^3 \left (1+x^2\right )^{2/3}} \, dx=\frac {\arctan \left (\frac {2 \sqrt [3]{x^2+1}+1}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {\sqrt [3]{x^2+1}}{2 x^2}-\frac {1}{2} \log \left (1-\sqrt [3]{x^2+1}\right )+\frac {\log (x)}{3} \]
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Rule 31
Rule 44
Rule 59
Rule 210
Rule 272
Rule 632
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {1}{x^2 (1+x)^{2/3}} \, dx,x,x^2\right ) \\ & = -\frac {\sqrt [3]{1+x^2}}{2 x^2}-\frac {1}{3} \text {Subst}\left (\int \frac {1}{x (1+x)^{2/3}} \, dx,x,x^2\right ) \\ & = -\frac {\sqrt [3]{1+x^2}}{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 {\sqrt [3]{1+x^2}}{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 {\sqrt [3]{1+x^2}}{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 = 81, normalized size of antiderivative = 0.93 \[ \int \frac {1}{x^3 \left (1+x^2\right )^{2/3}} \, dx=\frac {1}{6} \left (-\frac {3 \sqrt [3]{1+x^2}}{x^2}+2 \sqrt {3} \arctan \left (\frac {1+2 \sqrt [3]{1+x^2}}{\sqrt {3}}\right )-2 \log \left (-1+\sqrt [3]{1+x^2}\right )+\log \left (1+\sqrt [3]{1+x^2}+\left (1+x^2\right )^{2/3}\right )\right ) \]
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Result contains higher order function than in optimal. Order 5 vs. order 3.
Time = 2.23 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.63
method | result | size |
meijerg | \(\frac {\frac {5 \Gamma \left (\frac {2}{3}\right ) x^{2} \operatorname {hypergeom}\left (\left [1, 1, \frac {8}{3}\right ], \left [2, 3\right ], -x^{2}\right )}{9}-\frac {2 \left (\frac {1}{2}+\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \left (3\right )}{2}+2 \ln \left (x \right )\right ) \Gamma \left (\frac {2}{3}\right )}{3}-\frac {\Gamma \left (\frac {2}{3}\right )}{x^{2}}}{2 \Gamma \left (\frac {2}{3}\right )}\) | \(55\) |
risch | \(-\frac {\left (x^{2}+1\right )^{\frac {1}{3}}}{2 x^{2}}-\frac {-\frac {2 \Gamma \left (\frac {2}{3}\right ) x^{2} \operatorname {hypergeom}\left (\left [1, 1, \frac {5}{3}\right ], \left [2, 2\right ], -x^{2}\right )}{3}+\left (\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \left (3\right )}{2}+2 \ln \left (x \right )\right ) \Gamma \left (\frac {2}{3}\right )}{3 \Gamma \left (\frac {2}{3}\right )}\) | \(59\) |
pseudoelliptic | \(\frac {2 \sqrt {3}\, \arctan \left (\frac {\left (1+2 \left (x^{2}+1\right )^{\frac {1}{3}}\right ) \sqrt {3}}{3}\right ) x^{2}+\ln \left (1+\left (x^{2}+1\right )^{\frac {1}{3}}+\left (x^{2}+1\right )^{\frac {2}{3}}\right ) x^{2}-2 \ln \left (-1+\left (x^{2}+1\right )^{\frac {1}{3}}\right ) x^{2}-3 \left (x^{2}+1\right )^{\frac {1}{3}}}{6 \left (-1+\left (x^{2}+1\right )^{\frac {1}{3}}\right ) \left (1+\left (x^{2}+1\right )^{\frac {1}{3}}+\left (x^{2}+1\right )^{\frac {2}{3}}\right )}\) | \(103\) |
trager | \(-\frac {\left (x^{2}+1\right )^{\frac {1}{3}}}{2 x^{2}}+\frac {\ln \left (-\frac {144 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right )^{2} x^{2}+90 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) \left (x^{2}+1\right )^{\frac {2}{3}}+54 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) x^{2}-144 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right )^{2}+90 \left (x^{2}+1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right )+9 \left (x^{2}+1\right )^{\frac {2}{3}}+2 x^{2}+114 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right )+9 \left (x^{2}+1\right )^{\frac {1}{3}}+5}{x^{2}}\right )}{3}-2 \ln \left (-\frac {144 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right )^{2} x^{2}+90 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) \left (x^{2}+1\right )^{\frac {2}{3}}+54 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) x^{2}-144 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right )^{2}+90 \left (x^{2}+1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right )+9 \left (x^{2}+1\right )^{\frac {2}{3}}+2 x^{2}+114 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right )+9 \left (x^{2}+1\right )^{\frac {1}{3}}+5}{x^{2}}\right ) \operatorname {RootOf}\left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right )+2 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) \ln \left (-\frac {144 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right )^{2} x^{2}-90 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) \left (x^{2}+1\right )^{\frac {2}{3}}-102 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) x^{2}-144 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right )^{2}-90 \left (x^{2}+1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right )+24 \left (x^{2}+1\right )^{\frac {2}{3}}+15 x^{2}-66 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right )+24 \left (x^{2}+1\right )^{\frac {1}{3}}+20}{x^{2}}\right )\) | \(441\) |
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Time = 0.25 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.90 \[ \int \frac {1}{x^3 \left (1+x^2\right )^{2/3}} \, dx=\frac {2 \, \sqrt {3} x^{2} \arctan \left (\frac {2}{3} \, \sqrt {3} {\left (x^{2} + 1\right )}^{\frac {1}{3}} + \frac {1}{3} \, \sqrt {3}\right ) + x^{2} \log \left ({\left (x^{2} + 1\right )}^{\frac {2}{3}} + {\left (x^{2} + 1\right )}^{\frac {1}{3}} + 1\right ) - 2 \, x^{2} \log \left ({\left (x^{2} + 1\right )}^{\frac {1}{3}} - 1\right ) - 3 \, {\left (x^{2} + 1\right )}^{\frac {1}{3}}}{6 \, x^{2}} \]
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Result contains complex when optimal does not.
Time = 0.63 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.37 \[ \int \frac {1}{x^3 \left (1+x^2\right )^{2/3}} \, dx=- \frac {\Gamma \left (\frac {5}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {2}{3}, \frac {5}{3} \\ \frac {8}{3} \end {matrix}\middle | {\frac {e^{i \pi }}{x^{2}}} \right )}}{2 x^{\frac {10}{3}} \Gamma \left (\frac {8}{3}\right )} \]
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Time = 0.28 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.76 \[ \int \frac {1}{x^3 \left (1+x^2\right )^{2/3}} \, dx=\frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (x^{2} + 1\right )}^{\frac {1}{3}} + 1\right )}\right ) - \frac {{\left (x^{2} + 1\right )}^{\frac {1}{3}}}{2 \, x^{2}} + \frac {1}{6} \, \log \left ({\left (x^{2} + 1\right )}^{\frac {2}{3}} + {\left (x^{2} + 1\right )}^{\frac {1}{3}} + 1\right ) - \frac {1}{3} \, \log \left ({\left (x^{2} + 1\right )}^{\frac {1}{3}} - 1\right ) \]
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Time = 0.27 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.76 \[ \int \frac {1}{x^3 \left (1+x^2\right )^{2/3}} \, dx=\frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (x^{2} + 1\right )}^{\frac {1}{3}} + 1\right )}\right ) - \frac {{\left (x^{2} + 1\right )}^{\frac {1}{3}}}{2 \, x^{2}} + \frac {1}{6} \, \log \left ({\left (x^{2} + 1\right )}^{\frac {2}{3}} + {\left (x^{2} + 1\right )}^{\frac {1}{3}} + 1\right ) - \frac {1}{3} \, \log \left ({\left (x^{2} + 1\right )}^{\frac {1}{3}} - 1\right ) \]
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Time = 5.54 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.92 \[ \int \frac {1}{x^3 \left (1+x^2\right )^{2/3}} \, dx=-\frac {\ln \left ({\left (x^2+1\right )}^{1/3}-1\right )}{3}-\frac {{\left (x^2+1\right )}^{1/3}}{2\,x^2}-\ln \left (3\,{\left (x^2+1\right )}^{1/3}+\frac {3}{2}-\frac {\sqrt {3}\,3{}\mathrm {i}}{2}\right )\,\left (-\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )+\ln \left (3\,{\left (x^2+1\right )}^{1/3}+\frac {3}{2}+\frac {\sqrt {3}\,3{}\mathrm {i}}{2}\right )\,\left (\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right ) \]
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