Integrand size = 23, antiderivative size = 79 \[ \int \frac {1}{\left (-2-b x^2\right ) \sqrt [4]{-1-b x^2}} \, dx=-\frac {\arctan \left (\frac {\sqrt {b} x}{\sqrt {2} \sqrt [4]{-1-b x^2}}\right )}{2 \sqrt {2} \sqrt {b}}-\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {2} \sqrt [4]{-1-b x^2}}\right )}{2 \sqrt {2} \sqrt {b}} \]
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Time = 0.01 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {407} \[ \int \frac {1}{\left (-2-b x^2\right ) \sqrt [4]{-1-b x^2}} \, dx=-\frac {\arctan \left (\frac {\sqrt {b} x}{\sqrt {2} \sqrt [4]{-b x^2-1}}\right )}{2 \sqrt {2} \sqrt {b}}-\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {2} \sqrt [4]{-b x^2-1}}\right )}{2 \sqrt {2} \sqrt {b}} \]
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Rule 407
Rubi steps \begin{align*} \text {integral}& = -\frac {\tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {2} \sqrt [4]{-1-b x^2}}\right )}{2 \sqrt {2} \sqrt {b}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {2} \sqrt [4]{-1-b x^2}}\right )}{2 \sqrt {2} \sqrt {b}} \\ \end{align*}
Time = 0.19 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.87 \[ \int \frac {1}{\left (-2-b x^2\right ) \sqrt [4]{-1-b x^2}} \, dx=-\frac {-\arctan \left (\frac {\sqrt {2} \sqrt [4]{-1-b x^2}}{\sqrt {b} x}\right )+\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {2} \sqrt [4]{-1-b x^2}}\right )}{2 \sqrt {2} \sqrt {b}} \]
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\[\int \frac {1}{\left (-b \,x^{2}-2\right ) \left (-b \,x^{2}-1\right )^{\frac {1}{4}}}d x\]
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Leaf count of result is larger than twice the leaf count of optimal. 142 vs. \(2 (57) = 114\).
Time = 5.87 (sec) , antiderivative size = 273, normalized size of antiderivative = 3.46 \[ \int \frac {1}{\left (-2-b x^2\right ) \sqrt [4]{-1-b x^2}} \, dx=\left [\frac {2 \, \sqrt {2} \sqrt {b} \arctan \left (\frac {\sqrt {2} {\left (-b x^{2} - 1\right )}^{\frac {1}{4}}}{\sqrt {b} x}\right ) + \sqrt {2} \sqrt {b} \log \left (-\frac {b^{2} x^{4} + 4 \, \sqrt {-b x^{2} - 1} b x^{2} - 4 \, b x^{2} - 2 \, \sqrt {2} {\left ({\left (-b x^{2} - 1\right )}^{\frac {1}{4}} b x^{3} + 2 \, {\left (-b x^{2} - 1\right )}^{\frac {3}{4}} x\right )} \sqrt {b} - 4}{b^{2} x^{4} + 4 \, b x^{2} + 4}\right )}{8 \, b}, \frac {2 \, \sqrt {2} \sqrt {-b} \arctan \left (\frac {\sqrt {2} {\left (-b x^{2} - 1\right )}^{\frac {1}{4}} \sqrt {-b}}{b x}\right ) - \sqrt {2} \sqrt {-b} \log \left (-\frac {b^{2} x^{4} - 4 \, \sqrt {-b x^{2} - 1} b x^{2} - 4 \, b x^{2} + 2 \, \sqrt {2} {\left ({\left (-b x^{2} - 1\right )}^{\frac {1}{4}} b x^{3} - 2 \, {\left (-b x^{2} - 1\right )}^{\frac {3}{4}} x\right )} \sqrt {-b} - 4}{b^{2} x^{4} + 4 \, b x^{2} + 4}\right )}{8 \, b}\right ] \]
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\[ \int \frac {1}{\left (-2-b x^2\right ) \sqrt [4]{-1-b x^2}} \, dx=- \int \frac {1}{b x^{2} \sqrt [4]{- b x^{2} - 1} + 2 \sqrt [4]{- b x^{2} - 1}}\, dx \]
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\[ \int \frac {1}{\left (-2-b x^2\right ) \sqrt [4]{-1-b x^2}} \, dx=\int { -\frac {1}{{\left (b x^{2} + 2\right )} {\left (-b x^{2} - 1\right )}^{\frac {1}{4}}} \,d x } \]
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\[ \int \frac {1}{\left (-2-b x^2\right ) \sqrt [4]{-1-b x^2}} \, dx=\int { -\frac {1}{{\left (b x^{2} + 2\right )} {\left (-b x^{2} - 1\right )}^{\frac {1}{4}}} \,d x } \]
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Timed out. \[ \int \frac {1}{\left (-2-b x^2\right ) \sqrt [4]{-1-b x^2}} \, dx=-\int \frac {1}{{\left (-b\,x^2-1\right )}^{1/4}\,\left (b\,x^2+2\right )} \,d x \]
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