Integrand size = 13, antiderivative size = 87 \[ \int \frac {\left (1+x^2\right )^{2/3}}{x} \, dx=\frac {3}{4} \left (1+x^2\right )^{2/3}+\frac {1}{2} \sqrt {3} \arctan \left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{1+x^2}}{\sqrt {3}}\right )+\frac {1}{2} \log \left (-1+\sqrt [3]{1+x^2}\right )-\frac {1}{4} \log \left (1+\sqrt [3]{1+x^2}+\left (1+x^2\right )^{2/3}\right ) \]
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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, 52, 57, 632, 210, 31} \[ \int \frac {\left (1+x^2\right )^{2/3}}{x} \, dx=\frac {1}{2} \sqrt {3} \arctan \left (\frac {2 \sqrt [3]{x^2+1}+1}{\sqrt {3}}\right )+\frac {3}{4} \left (x^2+1\right )^{2/3}+\frac {3}{4} \log \left (1-\sqrt [3]{x^2+1}\right )-\frac {\log (x)}{2} \]
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Rule 31
Rule 52
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} \, dx,x,x^2\right ) \\ & = \frac {3}{4} \left (1+x^2\right )^{2/3}+\frac {1}{2} \text {Subst}\left (\int \frac {1}{x \sqrt [3]{1+x}} \, dx,x,x^2\right ) \\ & = \frac {3}{4} \left (1+x^2\right )^{2/3}-\frac {\log (x)}{2}-\frac {3}{4} \text {Subst}\left (\int \frac {1}{1-x} \, dx,x,\sqrt [3]{1+x^2}\right )+\frac {3}{4} \text {Subst}\left (\int \frac {1}{1+x+x^2} \, dx,x,\sqrt [3]{1+x^2}\right ) \\ & = \frac {3}{4} \left (1+x^2\right )^{2/3}-\frac {\log (x)}{2}+\frac {3}{4} \log \left (1-\sqrt [3]{1+x^2}\right )-\frac {3}{2} \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+2 \sqrt [3]{1+x^2}\right ) \\ & = \frac {3}{4} \left (1+x^2\right )^{2/3}+\frac {1}{2} \sqrt {3} \arctan \left (\frac {1+2 \sqrt [3]{1+x^2}}{\sqrt {3}}\right )-\frac {\log (x)}{2}+\frac {3}{4} \log \left (1-\sqrt [3]{1+x^2}\right ) \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.92 \[ \int \frac {\left (1+x^2\right )^{2/3}}{x} \, dx=\frac {1}{4} \left (3 \left (1+x^2\right )^{2/3}+2 \sqrt {3} \arctan \left (\frac {1+2 \sqrt [3]{1+x^2}}{\sqrt {3}}\right )+2 \log \left (-1+\sqrt [3]{1+x^2}\right )-\log \left (1+\sqrt [3]{1+x^2}+\left (1+x^2\right )^{2/3}\right )\right ) \]
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Result contains higher order function than in optimal. Order 5 vs. order 3.
Time = 1.72 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.74
method | result | size |
meijerg | \(-\frac {\sqrt {3}\, \Gamma \left (\frac {2}{3}\right ) \left (-\frac {2 \pi \sqrt {3}\, x^{2} \operatorname {hypergeom}\left (\left [\frac {1}{3}, 1, 1\right ], \left [2, 2\right ], -x^{2}\right )}{3 \Gamma \left (\frac {2}{3}\right )}-\frac {\left (\frac {3}{2}-\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \left (3\right )}{2}+2 \ln \left (x \right )\right ) \pi \sqrt {3}}{\Gamma \left (\frac {2}{3}\right )}\right )}{6 \pi }\) | \(64\) |
pseudoelliptic | \(\frac {3 \left (x^{2}+1\right )^{\frac {2}{3}}}{4}+\frac {\ln \left (-1+\left (x^{2}+1\right )^{\frac {1}{3}}\right )}{2}-\frac {\ln \left (1+\left (x^{2}+1\right )^{\frac {1}{3}}+\left (x^{2}+1\right )^{\frac {2}{3}}\right )}{4}+\frac {\sqrt {3}\, \arctan \left (\frac {\left (1+2 \left (x^{2}+1\right )^{\frac {1}{3}}\right ) \sqrt {3}}{3}\right )}{2}\) | \(64\) |
trager | \(\frac {3 \left (x^{2}+1\right )^{\frac {2}{3}}}{4}+\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 )}{2}-\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 )}{2}-\frac {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 )}{2}\) | \(425\) |
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Time = 0.27 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.75 \[ \int \frac {\left (1+x^2\right )^{2/3}}{x} \, dx=\frac {1}{2} \, \sqrt {3} \arctan \left (\frac {2}{3} \, \sqrt {3} {\left (x^{2} + 1\right )}^{\frac {1}{3}} + \frac {1}{3} \, \sqrt {3}\right ) + \frac {3}{4} \, {\left (x^{2} + 1\right )}^{\frac {2}{3}} - \frac {1}{4} \, \log \left ({\left (x^{2} + 1\right )}^{\frac {2}{3}} + {\left (x^{2} + 1\right )}^{\frac {1}{3}} + 1\right ) + \frac {1}{2} \, \log \left ({\left (x^{2} + 1\right )}^{\frac {1}{3}} - 1\right ) \]
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Result contains complex when optimal does not.
Time = 0.56 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.43 \[ \int \frac {\left (1+x^2\right )^{2/3}}{x} \, dx=- \frac {x^{\frac {4}{3}} \Gamma \left (- \frac {2}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{3}, - \frac {2}{3} \\ \frac {1}{3} \end {matrix}\middle | {\frac {e^{i \pi }}{x^{2}}} \right )}}{2 \Gamma \left (\frac {1}{3}\right )} \]
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Time = 0.27 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.72 \[ \int \frac {\left (1+x^2\right )^{2/3}}{x} \, dx=\frac {1}{2} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (x^{2} + 1\right )}^{\frac {1}{3}} + 1\right )}\right ) + \frac {3}{4} \, {\left (x^{2} + 1\right )}^{\frac {2}{3}} - \frac {1}{4} \, \log \left ({\left (x^{2} + 1\right )}^{\frac {2}{3}} + {\left (x^{2} + 1\right )}^{\frac {1}{3}} + 1\right ) + \frac {1}{2} \, \log \left ({\left (x^{2} + 1\right )}^{\frac {1}{3}} - 1\right ) \]
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Time = 0.28 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.72 \[ \int \frac {\left (1+x^2\right )^{2/3}}{x} \, dx=\frac {1}{2} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (x^{2} + 1\right )}^{\frac {1}{3}} + 1\right )}\right ) + \frac {3}{4} \, {\left (x^{2} + 1\right )}^{\frac {2}{3}} - \frac {1}{4} \, \log \left ({\left (x^{2} + 1\right )}^{\frac {2}{3}} + {\left (x^{2} + 1\right )}^{\frac {1}{3}} + 1\right ) + \frac {1}{2} \, \log \left ({\left (x^{2} + 1\right )}^{\frac {1}{3}} - 1\right ) \]
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Time = 5.49 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.02 \[ \int \frac {\left (1+x^2\right )^{2/3}}{x} \, dx=\frac {\ln \left (\frac {9\,{\left (x^2+1\right )}^{1/3}}{4}-\frac {9}{4}\right )}{2}+\ln \left (\frac {9\,{\left (x^2+1\right )}^{1/3}}{4}-9\,{\left (-\frac {1}{4}+\frac {\sqrt {3}\,1{}\mathrm {i}}{4}\right )}^2\right )\,\left (-\frac {1}{4}+\frac {\sqrt {3}\,1{}\mathrm {i}}{4}\right )-\ln \left (\frac {9\,{\left (x^2+1\right )}^{1/3}}{4}-9\,{\left (\frac {1}{4}+\frac {\sqrt {3}\,1{}\mathrm {i}}{4}\right )}^2\right )\,\left (\frac {1}{4}+\frac {\sqrt {3}\,1{}\mathrm {i}}{4}\right )+\frac {3\,{\left (x^2+1\right )}^{2/3}}{4} \]
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