Integrand size = 19, antiderivative size = 169 \[ \int \frac {1}{\left (3+x^2\right ) \sqrt [3]{1+3 x^2}} \, dx=\frac {\arctan \left (\frac {\sqrt {3} x}{1+2 \sqrt [3]{1+3 x^2}}\right )}{4 \sqrt {3}}-\frac {i \text {arctanh}\left (\frac {2 i \sqrt {3} x-2 i \sqrt {3} x \sqrt [3]{1+3 x^2}}{-1+3 x^2+2 \sqrt [3]{1+3 x^2}-4 \left (1+3 x^2\right )^{2/3}}\right )}{8 \sqrt {3}}+\frac {1}{8} \text {arctanh}\left (\frac {6 x \sqrt [3]{1+3 x^2}}{1+3 x^2-2 \sqrt [3]{1+3 x^2}+4 \left (1+3 x^2\right )^{2/3}}\right ) \]
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Time = 0.01 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.48, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {403} \[ \int \frac {1}{\left (3+x^2\right ) \sqrt [3]{1+3 x^2}} \, dx=\frac {\arctan \left (\frac {\left (1-\sqrt [3]{3 x^2+1}\right )^2}{3 \sqrt {3} x}\right )}{4 \sqrt {3}}+\frac {\arctan \left (\frac {x}{\sqrt {3}}\right )}{4 \sqrt {3}}-\frac {1}{4} \text {arctanh}\left (\frac {1-\sqrt [3]{3 x^2+1}}{x}\right ) \]
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Rule 403
Rubi steps \begin{align*} \text {integral}& = \frac {\arctan \left (\frac {x}{\sqrt {3}}\right )}{4 \sqrt {3}}+\frac {\arctan \left (\frac {\left (1-\sqrt [3]{1+3 x^2}\right )^2}{3 \sqrt {3} x}\right )}{4 \sqrt {3}}-\frac {1}{4} \text {arctanh}\left (\frac {1-\sqrt [3]{1+3 x^2}}{x}\right ) \\ \end{align*}
Result contains higher order function than in optimal. Order 6 vs. order 3 in optimal.
Time = 4.21 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.75 \[ \int \frac {1}{\left (3+x^2\right ) \sqrt [3]{1+3 x^2}} \, dx=-\frac {9 x \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{3},1,\frac {3}{2},-3 x^2,-\frac {x^2}{3}\right )}{\left (3+x^2\right ) \sqrt [3]{1+3 x^2} \left (-9 \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{3},1,\frac {3}{2},-3 x^2,-\frac {x^2}{3}\right )+2 x^2 \left (\operatorname {AppellF1}\left (\frac {3}{2},\frac {1}{3},2,\frac {5}{2},-3 x^2,-\frac {x^2}{3}\right )+3 \operatorname {AppellF1}\left (\frac {3}{2},\frac {4}{3},1,\frac {5}{2},-3 x^2,-\frac {x^2}{3}\right )\right )\right )} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 1.70 (sec) , antiderivative size = 445, normalized size of antiderivative = 2.63
method | result | size |
trager | \(\frac {\ln \left (-\frac {12 \operatorname {RootOf}\left (48 \textit {\_Z}^{2}-12 \textit {\_Z} +1\right ) \left (3 x^{2}+1\right )^{\frac {1}{3}} x +6 \operatorname {RootOf}\left (48 \textit {\_Z}^{2}-12 \textit {\_Z} +1\right ) x^{2}+\left (3 x^{2}+1\right )^{\frac {2}{3}}-12 \operatorname {RootOf}\left (48 \textit {\_Z}^{2}-12 \textit {\_Z} +1\right ) \left (3 x^{2}+1\right )^{\frac {1}{3}}-\left (3 x^{2}+1\right )^{\frac {1}{3}} x -24 \operatorname {RootOf}\left (48 \textit {\_Z}^{2}-12 \textit {\_Z} +1\right ) x -x^{2}+\left (3 x^{2}+1\right )^{\frac {1}{3}}-6 \operatorname {RootOf}\left (48 \textit {\_Z}^{2}-12 \textit {\_Z} +1\right )+4 x +1}{x^{2}+3}\right )}{4}-\ln \left (-\frac {12 \operatorname {RootOf}\left (48 \textit {\_Z}^{2}-12 \textit {\_Z} +1\right ) \left (3 x^{2}+1\right )^{\frac {1}{3}} x +6 \operatorname {RootOf}\left (48 \textit {\_Z}^{2}-12 \textit {\_Z} +1\right ) x^{2}+\left (3 x^{2}+1\right )^{\frac {2}{3}}-12 \operatorname {RootOf}\left (48 \textit {\_Z}^{2}-12 \textit {\_Z} +1\right ) \left (3 x^{2}+1\right )^{\frac {1}{3}}-\left (3 x^{2}+1\right )^{\frac {1}{3}} x -24 \operatorname {RootOf}\left (48 \textit {\_Z}^{2}-12 \textit {\_Z} +1\right ) x -x^{2}+\left (3 x^{2}+1\right )^{\frac {1}{3}}-6 \operatorname {RootOf}\left (48 \textit {\_Z}^{2}-12 \textit {\_Z} +1\right )+4 x +1}{x^{2}+3}\right ) \operatorname {RootOf}\left (48 \textit {\_Z}^{2}-12 \textit {\_Z} +1\right )+\operatorname {RootOf}\left (48 \textit {\_Z}^{2}-12 \textit {\_Z} +1\right ) \ln \left (-\frac {-24 \operatorname {RootOf}\left (48 \textit {\_Z}^{2}-12 \textit {\_Z} +1\right ) \left (3 x^{2}+1\right )^{\frac {1}{3}} x -12 \operatorname {RootOf}\left (48 \textit {\_Z}^{2}-12 \textit {\_Z} +1\right ) x^{2}+2 \left (3 x^{2}+1\right )^{\frac {2}{3}}+24 \operatorname {RootOf}\left (48 \textit {\_Z}^{2}-12 \textit {\_Z} +1\right ) \left (3 x^{2}+1\right )^{\frac {1}{3}}+4 \left (3 x^{2}+1\right )^{\frac {1}{3}} x +48 \operatorname {RootOf}\left (48 \textit {\_Z}^{2}-12 \textit {\_Z} +1\right ) x +x^{2}-4 \left (3 x^{2}+1\right )^{\frac {1}{3}}+12 \operatorname {RootOf}\left (48 \textit {\_Z}^{2}-12 \textit {\_Z} +1\right )-4 x -1}{x^{2}+3}\right )\) | \(445\) |
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Leaf count of result is larger than twice the leaf count of optimal. 345 vs. \(2 (134) = 268\).
Time = 1.02 (sec) , antiderivative size = 345, normalized size of antiderivative = 2.04 \[ \int \frac {1}{\left (3+x^2\right ) \sqrt [3]{1+3 x^2}} \, dx=\frac {1}{36} \, \sqrt {3} \arctan \left (\frac {4 \, \sqrt {3} {\left (3 \, x^{4} - 10 \, x^{3} - 36 \, x^{2} + 18 \, x + 9\right )} {\left (3 \, x^{2} + 1\right )}^{\frac {2}{3}} - 4 \, \sqrt {3} {\left (x^{5} + 15 \, x^{4} - 26 \, x^{3} - 54 \, x^{2} + 9 \, x - 9\right )} {\left (3 \, x^{2} + 1\right )}^{\frac {1}{3}} + \sqrt {3} {\left (x^{6} - 2 \, x^{5} - 105 \, x^{4} - 28 \, x^{3} + 63 \, x^{2} + 126 \, x + 9\right )}}{x^{6} + 126 \, x^{5} - 225 \, x^{4} - 828 \, x^{3} - 81 \, x^{2} - 162 \, x + 81}\right ) - \frac {1}{36} \, \sqrt {3} \arctan \left (\frac {2 \, {\left (2 \, \sqrt {3} {\left (23 \, x^{3} + 9 \, x\right )} {\left (3 \, x^{2} + 1\right )}^{\frac {2}{3}} + \sqrt {3} {\left (x^{5} - 80 \, x^{3} - 9 \, x\right )} {\left (3 \, x^{2} + 1\right )}^{\frac {1}{3}} + \sqrt {3} {\left (11 \, x^{5} + 10 \, x^{3} - 9 \, x\right )}\right )}}{x^{6} - 657 \, x^{4} - 189 \, x^{2} - 27}\right ) + \frac {1}{24} \, \log \left (\frac {x^{6} + 108 \, x^{5} + 549 \, x^{4} + 360 \, x^{3} + 99 \, x^{2} + 6 \, {\left (3 \, x^{4} + 32 \, x^{3} + 42 \, x^{2} + 3\right )} {\left (3 \, x^{2} + 1\right )}^{\frac {2}{3}} + 6 \, {\left (x^{5} + 27 \, x^{4} + 70 \, x^{3} + 18 \, x^{2} + 9 \, x + 3\right )} {\left (3 \, x^{2} + 1\right )}^{\frac {1}{3}} + 108 \, x - 9}{x^{6} + 9 \, x^{4} + 27 \, x^{2} + 27}\right ) \]
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\[ \int \frac {1}{\left (3+x^2\right ) \sqrt [3]{1+3 x^2}} \, dx=\int \frac {1}{\left (x^{2} + 3\right ) \sqrt [3]{3 x^{2} + 1}}\, dx \]
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\[ \int \frac {1}{\left (3+x^2\right ) \sqrt [3]{1+3 x^2}} \, dx=\int { \frac {1}{{\left (3 \, x^{2} + 1\right )}^{\frac {1}{3}} {\left (x^{2} + 3\right )}} \,d x } \]
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\[ \int \frac {1}{\left (3+x^2\right ) \sqrt [3]{1+3 x^2}} \, dx=\int { \frac {1}{{\left (3 \, x^{2} + 1\right )}^{\frac {1}{3}} {\left (x^{2} + 3\right )}} \,d x } \]
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Timed out. \[ \int \frac {1}{\left (3+x^2\right ) \sqrt [3]{1+3 x^2}} \, dx=\int \frac {1}{\left (x^2+3\right )\,{\left (3\,x^2+1\right )}^{1/3}} \,d x \]
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