Integrand size = 24, antiderivative size = 61 \[ \int \frac {x^2}{\left (-2+3 x^2\right ) \left (-1+3 x^2\right )^{3/4}} \, dx=\frac {\arctan \left (\frac {\sqrt {\frac {3}{2}} x}{\sqrt [4]{-1+3 x^2}}\right )}{3 \sqrt {6}}-\frac {\text {arctanh}\left (\frac {\sqrt {\frac {3}{2}} x}{\sqrt [4]{-1+3 x^2}}\right )}{3 \sqrt {6}} \]
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Time = 0.01 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {453} \[ \int \frac {x^2}{\left (-2+3 x^2\right ) \left (-1+3 x^2\right )^{3/4}} \, dx=\frac {\arctan \left (\frac {\sqrt {\frac {3}{2}} x}{\sqrt [4]{3 x^2-1}}\right )}{3 \sqrt {6}}-\frac {\text {arctanh}\left (\frac {\sqrt {\frac {3}{2}} x}{\sqrt [4]{3 x^2-1}}\right )}{3 \sqrt {6}} \]
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Rule 453
Rubi steps \begin{align*} \text {integral}& = \frac {\tan ^{-1}\left (\frac {\sqrt {\frac {3}{2}} x}{\sqrt [4]{-1+3 x^2}}\right )}{3 \sqrt {6}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {\frac {3}{2}} x}{\sqrt [4]{-1+3 x^2}}\right )}{3 \sqrt {6}} \\ \end{align*}
Time = 1.61 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.89 \[ \int \frac {x^2}{\left (-2+3 x^2\right ) \left (-1+3 x^2\right )^{3/4}} \, dx=\frac {\arctan \left (\frac {\sqrt {\frac {3}{2}} x}{\sqrt [4]{-1+3 x^2}}\right )-\text {arctanh}\left (\frac {\sqrt {\frac {3}{2}} x}{\sqrt [4]{-1+3 x^2}}\right )}{3 \sqrt {6}} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 3.20 (sec) , antiderivative size = 139, normalized size of antiderivative = 2.28
method | result | size |
trager | \(\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-6\right ) \ln \left (-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-6\right ) \left (3 x^{2}-1\right )^{\frac {3}{4}}-3 \sqrt {3 x^{2}-1}\, x +\operatorname {RootOf}\left (\textit {\_Z}^{2}-6\right ) \left (3 x^{2}-1\right )^{\frac {1}{4}}-3 x}{3 x^{2}-2}\right )}{18}-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+6\right ) \ln \left (-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+6\right ) \left (3 x^{2}-1\right )^{\frac {3}{4}}-3 \sqrt {3 x^{2}-1}\, x -\operatorname {RootOf}\left (\textit {\_Z}^{2}+6\right ) \left (3 x^{2}-1\right )^{\frac {1}{4}}+3 x}{3 x^{2}-2}\right )}{18}\) | \(139\) |
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Leaf count of result is larger than twice the leaf count of optimal. 104 vs. \(2 (43) = 86\).
Time = 0.27 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.70 \[ \int \frac {x^2}{\left (-2+3 x^2\right ) \left (-1+3 x^2\right )^{3/4}} \, dx=-\frac {1}{18} \, \sqrt {6} \arctan \left (\frac {\sqrt {6} {\left (3 \, x^{2} - 1\right )}^{\frac {1}{4}}}{3 \, x}\right ) + \frac {1}{36} \, \sqrt {6} \log \left (-\frac {9 \, x^{4} - 6 \, \sqrt {6} {\left (3 \, x^{2} - 1\right )}^{\frac {1}{4}} x^{3} + 12 \, \sqrt {3 \, x^{2} - 1} x^{2} - 4 \, \sqrt {6} {\left (3 \, x^{2} - 1\right )}^{\frac {3}{4}} x + 12 \, x^{2} - 4}{9 \, x^{4} - 12 \, x^{2} + 4}\right ) \]
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\[ \int \frac {x^2}{\left (-2+3 x^2\right ) \left (-1+3 x^2\right )^{3/4}} \, dx=\int \frac {x^{2}}{\left (3 x^{2} - 2\right ) \left (3 x^{2} - 1\right )^{\frac {3}{4}}}\, dx \]
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\[ \int \frac {x^2}{\left (-2+3 x^2\right ) \left (-1+3 x^2\right )^{3/4}} \, dx=\int { \frac {x^{2}}{{\left (3 \, x^{2} - 1\right )}^{\frac {3}{4}} {\left (3 \, x^{2} - 2\right )}} \,d x } \]
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\[ \int \frac {x^2}{\left (-2+3 x^2\right ) \left (-1+3 x^2\right )^{3/4}} \, dx=\int { \frac {x^{2}}{{\left (3 \, x^{2} - 1\right )}^{\frac {3}{4}} {\left (3 \, x^{2} - 2\right )}} \,d x } \]
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Timed out. \[ \int \frac {x^2}{\left (-2+3 x^2\right ) \left (-1+3 x^2\right )^{3/4}} \, dx=\int \frac {x^2}{{\left (3\,x^2-1\right )}^{3/4}\,\left (3\,x^2-2\right )} \,d x \]
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