Integrand size = 13, antiderivative size = 87 \[ \int \frac {\left (1+x^2\right )^{2/3}}{x^3} \, dx=-\frac {\left (1+x^2\right )^{2/3}}{2 x^2}+\frac {\arctan \left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{1+x^2}}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {1}{3} \log \left (-1+\sqrt [3]{1+x^2}\right )-\frac {1}{6} \log \left (1+\sqrt [3]{1+x^2}+\left (1+x^2\right )^{2/3}\right ) \]
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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, 43, 57, 632, 210, 31} \[ \int \frac {\left (1+x^2\right )^{2/3}}{x^3} \, dx=\frac {\arctan \left (\frac {2 \sqrt [3]{x^2+1}+1}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {\left (x^2+1\right )^{2/3}}{2 x^2}+\frac {1}{2} \log \left (1-\sqrt [3]{x^2+1}\right )-\frac {\log (x)}{3} \]
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Rule 31
Rule 43
Rule 57
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}{x \sqrt [3]{1+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 = 0.07 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.95 \[ \int \frac {\left (1+x^2\right )^{2/3}}{x^3} \, dx=\frac {1}{6} \left (-\frac {3 \left (1+x^2\right )^{2/3}}{x^2}+2 \sqrt {3} \arctan \left (\frac {1+2 \sqrt [3]{1+x^2}}{\sqrt {3}}\right )+2 \log \left (-1+\sqrt [3]{1+x^2}\right )-\log \left (1+\sqrt [3]{1+x^2}+\left (1+x^2\right )^{2/3}\right )\right ) \]
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Result contains higher order function than in optimal. Order 5 vs. order 3.
Time = 2.26 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.87
method | result | size |
meijerg | \(-\frac {\sqrt {3}\, \Gamma \left (\frac {2}{3}\right ) \left (\frac {\pi \sqrt {3}\, x^{2} \operatorname {hypergeom}\left (\left [1, 1, \frac {4}{3}\right ], \left [2, 3\right ], -x^{2}\right )}{9 \Gamma \left (\frac {2}{3}\right )}-\frac {2 \left (-\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \left (3\right )}{2}-1+2 \ln \left (x \right )\right ) \pi \sqrt {3}}{3 \Gamma \left (\frac {2}{3}\right )}+\frac {\pi \sqrt {3}}{\Gamma \left (\frac {2}{3}\right ) x^{2}}\right )}{6 \pi }\) | \(76\) |
risch | \(-\frac {\left (x^{2}+1\right )^{\frac {2}{3}}}{2 x^{2}}+\frac {\sqrt {3}\, \Gamma \left (\frac {2}{3}\right ) \left (-\frac {2 \pi \sqrt {3}\, x^{2} \operatorname {hypergeom}\left (\left [1, 1, \frac {4}{3}\right ], \left [2, 2\right ], -x^{2}\right )}{9 \Gamma \left (\frac {2}{3}\right )}+\frac {2 \left (-\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \left (3\right )}{2}+2 \ln \left (x \right )\right ) \pi \sqrt {3}}{3 \Gamma \left (\frac {2}{3}\right )}\right )}{6 \pi }\) | \(76\) |
pseudoelliptic | \(\frac {2 \sqrt {3}\, \arctan \left (\frac {\left (1+2 \left (x^{2}+1\right )^{\frac {1}{3}}\right ) \sqrt {3}}{3}\right ) x^{2}+2 \ln \left (-1+\left (x^{2}+1\right )^{\frac {1}{3}}\right ) x^{2}-\ln \left (1+\left (x^{2}+1\right )^{\frac {1}{3}}+\left (x^{2}+1\right )^{\frac {2}{3}}\right ) x^{2}-3 \left (x^{2}+1\right )^{\frac {2}{3}}}{6 \left (-1+\left (x^{2}+1\right )^{\frac {1}{3}}\right ) \left (1+\left (x^{2}+1\right )^{\frac {1}{3}}+\left (x^{2}+1\right )^{\frac {2}{3}}\right )}\) | \(104\) |
trager | \(-\frac {\left (x^{2}+1\right )^{\frac {2}{3}}}{2 x^{2}}+\frac {\ln \left (-\frac {9 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{2}+45 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{2}+1\right )^{\frac {2}{3}}-3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{2}+24 \left (x^{2}+1\right )^{\frac {2}{3}}-72 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{2}+1\right )^{\frac {1}{3}}-9 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2}-12 x^{2}-9 \left (x^{2}+1\right )^{\frac {1}{3}}+24 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )-16}{x^{2}}\right )}{3}-\frac {\ln \left (\frac {45 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{2}+45 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{2}+1\right )^{\frac {2}{3}}-18 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{2}-9 \left (x^{2}+1\right )^{\frac {2}{3}}+27 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{2}+1\right )^{\frac {1}{3}}-45 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2}-8 x^{2}+24 \left (x^{2}+1\right )^{\frac {1}{3}}-87 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )-20}{x^{2}}\right )}{3}-\ln \left (\frac {45 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{2}+45 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{2}+1\right )^{\frac {2}{3}}-18 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{2}-9 \left (x^{2}+1\right )^{\frac {2}{3}}+27 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{2}+1\right )^{\frac {1}{3}}-45 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2}-8 x^{2}+24 \left (x^{2}+1\right )^{\frac {1}{3}}-87 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )-20}{x^{2}}\right ) \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )\) | \(428\) |
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Time = 0.26 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.91 \[ \int \frac {\left (1+x^2\right )^{2/3}}{x^3} \, dx=\frac {2 \, \sqrt {3} x^{2} \arctan \left (\frac {2}{3} \, \sqrt {3} {\left (x^{2} + 1\right )}^{\frac {1}{3}} + \frac {1}{3} \, \sqrt {3}\right ) - x^{2} \log \left ({\left (x^{2} + 1\right )}^{\frac {2}{3}} + {\left (x^{2} + 1\right )}^{\frac {1}{3}} + 1\right ) + 2 \, x^{2} \log \left ({\left (x^{2} + 1\right )}^{\frac {1}{3}} - 1\right ) - 3 \, {\left (x^{2} + 1\right )}^{\frac {2}{3}}}{6 \, x^{2}} \]
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Result contains complex when optimal does not.
Time = 0.60 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.39 \[ \int \frac {\left (1+x^2\right )^{2/3}}{x^3} \, dx=- \frac {\Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{3}, \frac {1}{3} \\ \frac {4}{3} \end {matrix}\middle | {\frac {e^{i \pi }}{x^{2}}} \right )}}{2 x^{\frac {2}{3}} \Gamma \left (\frac {4}{3}\right )} \]
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Time = 0.28 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.76 \[ \int \frac {\left (1+x^2\right )^{2/3}}{x^3} \, dx=\frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (x^{2} + 1\right )}^{\frac {1}{3}} + 1\right )}\right ) - \frac {{\left (x^{2} + 1\right )}^{\frac {2}{3}}}{2 \, x^{2}} - \frac {1}{6} \, \log \left ({\left (x^{2} + 1\right )}^{\frac {2}{3}} + {\left (x^{2} + 1\right )}^{\frac {1}{3}} + 1\right ) + \frac {1}{3} \, \log \left ({\left (x^{2} + 1\right )}^{\frac {1}{3}} - 1\right ) \]
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Time = 0.27 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.76 \[ \int \frac {\left (1+x^2\right )^{2/3}}{x^3} \, dx=\frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (x^{2} + 1\right )}^{\frac {1}{3}} + 1\right )}\right ) - \frac {{\left (x^{2} + 1\right )}^{\frac {2}{3}}}{2 \, x^{2}} - \frac {1}{6} \, \log \left ({\left (x^{2} + 1\right )}^{\frac {2}{3}} + {\left (x^{2} + 1\right )}^{\frac {1}{3}} + 1\right ) + \frac {1}{3} \, \log \left ({\left (x^{2} + 1\right )}^{\frac {1}{3}} - 1\right ) \]
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Time = 5.68 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.99 \[ \int \frac {\left (1+x^2\right )^{2/3}}{x^3} \, dx=\frac {\ln \left ({\left (x^2+1\right )}^{1/3}-1\right )}{3}+\ln \left ({\left (x^2+1\right )}^{1/3}-9\,{\left (-\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )}^2\right )\,\left (-\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )-\ln \left ({\left (x^2+1\right )}^{1/3}-9\,{\left (\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )}^2\right )\,\left (\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )-\frac {{\left (x^2+1\right )}^{2/3}}{2\,x^2} \]
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