Integrand size = 23, antiderivative size = 124 \[ \int \frac {1}{\sqrt [4]{2-b x^2} \left (4-b x^2\right )} \, dx=\frac {\arctan \left (\frac {2-\sqrt {2} \sqrt {2-b x^2}}{\sqrt [4]{2} \sqrt {b} x \sqrt [4]{2-b x^2}}\right )}{2\ 2^{3/4} \sqrt {b}}+\frac {\text {arctanh}\left (\frac {2+\sqrt {2} \sqrt {2-b x^2}}{\sqrt [4]{2} \sqrt {b} x \sqrt [4]{2-b x^2}}\right )}{2\ 2^{3/4} \sqrt {b}} \]
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Time = 0.01 (sec) , antiderivative size = 124, 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 = {406} \[ \int \frac {1}{\sqrt [4]{2-b x^2} \left (4-b x^2\right )} \, dx=\frac {\arctan \left (\frac {2-\sqrt {2} \sqrt {2-b x^2}}{\sqrt [4]{2} \sqrt {b} x \sqrt [4]{2-b x^2}}\right )}{2\ 2^{3/4} \sqrt {b}}+\frac {\text {arctanh}\left (\frac {\sqrt {2} \sqrt {2-b x^2}+2}{\sqrt [4]{2} \sqrt {b} x \sqrt [4]{2-b x^2}}\right )}{2\ 2^{3/4} \sqrt {b}} \]
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Rule 406
Rubi steps \begin{align*} \text {integral}& = \frac {\tan ^{-1}\left (\frac {2-\sqrt {2} \sqrt {2-b x^2}}{\sqrt [4]{2} \sqrt {b} x \sqrt [4]{2-b x^2}}\right )}{2\ 2^{3/4} \sqrt {b}}+\frac {\tanh ^{-1}\left (\frac {2+\sqrt {2} \sqrt {2-b x^2}}{\sqrt [4]{2} \sqrt {b} x \sqrt [4]{2-b x^2}}\right )}{2\ 2^{3/4} \sqrt {b}} \\ \end{align*}
Time = 0.35 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.99 \[ \int \frac {1}{\sqrt [4]{2-b x^2} \left (4-b x^2\right )} \, dx=\frac {\arctan \left (\frac {2^{3/4} b x^2-4 \sqrt [4]{2} \sqrt {2-b x^2}}{4 \sqrt {b} x \sqrt [4]{2-b x^2}}\right )+\text {arctanh}\left (\frac {2\ 2^{3/4} \sqrt {b} x \sqrt [4]{2-b x^2}}{\sqrt {2} b x^2+4 \sqrt {2-b x^2}}\right )}{4\ 2^{3/4} \sqrt {b}} \]
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\[\int \frac {1}{\left (-b \,x^{2}+2\right )^{\frac {1}{4}} \left (-b \,x^{2}+4\right )}d x\]
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Result contains complex when optimal does not.
Time = 5.25 (sec) , antiderivative size = 388, normalized size of antiderivative = 3.13 \[ \int \frac {1}{\sqrt [4]{2-b x^2} \left (4-b x^2\right )} \, dx=\frac {1}{8} \, \left (\frac {1}{2}\right )^{\frac {1}{4}} \left (-\frac {1}{b^{2}}\right )^{\frac {1}{4}} \log \left (-\frac {\left (\frac {1}{2}\right )^{\frac {3}{4}} \sqrt {-b x^{2} + 2} b^{2} x \left (-\frac {1}{b^{2}}\right )^{\frac {3}{4}} + \left (\frac {1}{2}\right )^{\frac {1}{4}} b x \left (-\frac {1}{b^{2}}\right )^{\frac {1}{4}} + 2 \, \sqrt {\frac {1}{2}} {\left (-b x^{2} + 2\right )}^{\frac {1}{4}} b \sqrt {-\frac {1}{b^{2}}} - {\left (-b x^{2} + 2\right )}^{\frac {3}{4}}}{b x^{2} - 4}\right ) - \frac {1}{8} \, \left (\frac {1}{2}\right )^{\frac {1}{4}} \left (-\frac {1}{b^{2}}\right )^{\frac {1}{4}} \log \left (\frac {\left (\frac {1}{2}\right )^{\frac {3}{4}} \sqrt {-b x^{2} + 2} b^{2} x \left (-\frac {1}{b^{2}}\right )^{\frac {3}{4}} + \left (\frac {1}{2}\right )^{\frac {1}{4}} b x \left (-\frac {1}{b^{2}}\right )^{\frac {1}{4}} - 2 \, \sqrt {\frac {1}{2}} {\left (-b x^{2} + 2\right )}^{\frac {1}{4}} b \sqrt {-\frac {1}{b^{2}}} + {\left (-b x^{2} + 2\right )}^{\frac {3}{4}}}{b x^{2} - 4}\right ) + \frac {1}{8} i \, \left (\frac {1}{2}\right )^{\frac {1}{4}} \left (-\frac {1}{b^{2}}\right )^{\frac {1}{4}} \log \left (\frac {i \, \left (\frac {1}{2}\right )^{\frac {3}{4}} \sqrt {-b x^{2} + 2} b^{2} x \left (-\frac {1}{b^{2}}\right )^{\frac {3}{4}} - i \, \left (\frac {1}{2}\right )^{\frac {1}{4}} b x \left (-\frac {1}{b^{2}}\right )^{\frac {1}{4}} + 2 \, \sqrt {\frac {1}{2}} {\left (-b x^{2} + 2\right )}^{\frac {1}{4}} b \sqrt {-\frac {1}{b^{2}}} + {\left (-b x^{2} + 2\right )}^{\frac {3}{4}}}{b x^{2} - 4}\right ) - \frac {1}{8} i \, \left (\frac {1}{2}\right )^{\frac {1}{4}} \left (-\frac {1}{b^{2}}\right )^{\frac {1}{4}} \log \left (\frac {-i \, \left (\frac {1}{2}\right )^{\frac {3}{4}} \sqrt {-b x^{2} + 2} b^{2} x \left (-\frac {1}{b^{2}}\right )^{\frac {3}{4}} + i \, \left (\frac {1}{2}\right )^{\frac {1}{4}} b x \left (-\frac {1}{b^{2}}\right )^{\frac {1}{4}} + 2 \, \sqrt {\frac {1}{2}} {\left (-b x^{2} + 2\right )}^{\frac {1}{4}} b \sqrt {-\frac {1}{b^{2}}} + {\left (-b x^{2} + 2\right )}^{\frac {3}{4}}}{b x^{2} - 4}\right ) \]
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\[ \int \frac {1}{\sqrt [4]{2-b x^2} \left (4-b x^2\right )} \, dx=- \int \frac {1}{b x^{2} \sqrt [4]{- b x^{2} + 2} - 4 \sqrt [4]{- b x^{2} + 2}}\, dx \]
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\[ \int \frac {1}{\sqrt [4]{2-b x^2} \left (4-b x^2\right )} \, dx=\int { -\frac {1}{{\left (b x^{2} - 4\right )} {\left (-b x^{2} + 2\right )}^{\frac {1}{4}}} \,d x } \]
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\[ \int \frac {1}{\sqrt [4]{2-b x^2} \left (4-b x^2\right )} \, dx=\int { -\frac {1}{{\left (b x^{2} - 4\right )} {\left (-b x^{2} + 2\right )}^{\frac {1}{4}}} \,d x } \]
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Timed out. \[ \int \frac {1}{\sqrt [4]{2-b x^2} \left (4-b x^2\right )} \, dx=-\int \frac {1}{{\left (2-b\,x^2\right )}^{1/4}\,\left (b\,x^2-4\right )} \,d x \]
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