Integrand size = 21, antiderivative size = 61 \[ \int \frac {1}{\left (-2-3 x^2\right ) \sqrt [4]{-1-3 x^2}} \, dx=-\frac {\arctan \left (\frac {\sqrt {\frac {3}{2}} x}{\sqrt [4]{-1-3 x^2}}\right )}{2 \sqrt {6}}-\frac {\text {arctanh}\left (\frac {\sqrt {\frac {3}{2}} x}{\sqrt [4]{-1-3 x^2}}\right )}{2 \sqrt {6}} \]
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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, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {407} \[ \int \frac {1}{\left (-2-3 x^2\right ) \sqrt [4]{-1-3 x^2}} \, dx=-\frac {\arctan \left (\frac {\sqrt {\frac {3}{2}} x}{\sqrt [4]{-3 x^2-1}}\right )}{2 \sqrt {6}}-\frac {\text {arctanh}\left (\frac {\sqrt {\frac {3}{2}} x}{\sqrt [4]{-3 x^2-1}}\right )}{2 \sqrt {6}} \]
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Rule 407
Rubi steps \begin{align*} \text {integral}& = -\frac {\tan ^{-1}\left (\frac {\sqrt {\frac {3}{2}} x}{\sqrt [4]{-1-3 x^2}}\right )}{2 \sqrt {6}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {\frac {3}{2}} x}{\sqrt [4]{-1-3 x^2}}\right )}{2 \sqrt {6}} \\ \end{align*}
Time = 0.16 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.92 \[ \int \frac {1}{\left (-2-3 x^2\right ) \sqrt [4]{-1-3 x^2}} \, dx=-\frac {-\arctan \left (\frac {\sqrt {\frac {2}{3}} \sqrt [4]{-1-3 x^2}}{x}\right )+\text {arctanh}\left (\frac {\sqrt {\frac {3}{2}} x}{\sqrt [4]{-1-3 x^2}}\right )}{2 \sqrt {6}} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 2.50 (sec) , antiderivative size = 138, normalized size of antiderivative = 2.26
method | result | size |
trager | \(\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+6\right ) \ln \left (-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+6\right ) \left (-3 x^{2}-1\right )^{\frac {3}{4}}-3 \sqrt {-3 x^{2}-1}\, x +\operatorname {RootOf}\left (\textit {\_Z}^{2}+6\right ) \left (-3 x^{2}-1\right )^{\frac {1}{4}}-3 x}{3 x^{2}+2}\right )}{12}-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-6\right ) \ln \left (\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-6\right ) \left (-3 x^{2}-1\right )^{\frac {3}{4}}+3 \sqrt {-3 x^{2}-1}\, x -\operatorname {RootOf}\left (\textit {\_Z}^{2}-6\right ) \left (-3 x^{2}-1\right )^{\frac {1}{4}}-3 x}{3 x^{2}+2}\right )}{12}\) | \(138\) |
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Result contains complex when optimal does not.
Time = 2.07 (sec) , antiderivative size = 243, normalized size of antiderivative = 3.98 \[ \int \frac {1}{\left (-2-3 x^2\right ) \sqrt [4]{-1-3 x^2}} \, dx=-\frac {1}{24} \, \sqrt {6} \log \left (\frac {\sqrt {6} \sqrt {-3 \, x^{2} - 1} x - \sqrt {6} x + 2 \, {\left (-3 \, x^{2} - 1\right )}^{\frac {3}{4}} - 2 \, {\left (-3 \, x^{2} - 1\right )}^{\frac {1}{4}}}{3 \, {\left (3 \, x^{2} + 2\right )}}\right ) + \frac {1}{24} \, \sqrt {6} \log \left (-\frac {\sqrt {6} \sqrt {-3 \, x^{2} - 1} x - \sqrt {6} x - 2 \, {\left (-3 \, x^{2} - 1\right )}^{\frac {3}{4}} + 2 \, {\left (-3 \, x^{2} - 1\right )}^{\frac {1}{4}}}{3 \, {\left (3 \, x^{2} + 2\right )}}\right ) + \frac {1}{24} i \, \sqrt {6} \log \left (\frac {i \, \sqrt {6} \sqrt {-3 \, x^{2} - 1} x + i \, \sqrt {6} x + 2 \, {\left (-3 \, x^{2} - 1\right )}^{\frac {3}{4}} + 2 \, {\left (-3 \, x^{2} - 1\right )}^{\frac {1}{4}}}{3 \, {\left (3 \, x^{2} + 2\right )}}\right ) - \frac {1}{24} i \, \sqrt {6} \log \left (\frac {-i \, \sqrt {6} \sqrt {-3 \, x^{2} - 1} x - i \, \sqrt {6} x + 2 \, {\left (-3 \, x^{2} - 1\right )}^{\frac {3}{4}} + 2 \, {\left (-3 \, x^{2} - 1\right )}^{\frac {1}{4}}}{3 \, {\left (3 \, x^{2} + 2\right )}}\right ) \]
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\[ \int \frac {1}{\left (-2-3 x^2\right ) \sqrt [4]{-1-3 x^2}} \, dx=- \int \frac {1}{3 x^{2} \sqrt [4]{- 3 x^{2} - 1} + 2 \sqrt [4]{- 3 x^{2} - 1}}\, dx \]
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\[ \int \frac {1}{\left (-2-3 x^2\right ) \sqrt [4]{-1-3 x^2}} \, dx=\int { -\frac {1}{{\left (3 \, x^{2} + 2\right )} {\left (-3 \, x^{2} - 1\right )}^{\frac {1}{4}}} \,d x } \]
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\[ \int \frac {1}{\left (-2-3 x^2\right ) \sqrt [4]{-1-3 x^2}} \, dx=\int { -\frac {1}{{\left (3 \, x^{2} + 2\right )} {\left (-3 \, x^{2} - 1\right )}^{\frac {1}{4}}} \,d x } \]
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Timed out. \[ \int \frac {1}{\left (-2-3 x^2\right ) \sqrt [4]{-1-3 x^2}} \, dx=-\int \frac {1}{{\left (-3\,x^2-1\right )}^{1/4}\,\left (3\,x^2+2\right )} \,d x \]
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