Integrand size = 29, antiderivative size = 124 \[ \int \frac {1}{\left (a-b x^2\right )^{3/2} \sqrt {a^2-b^2 x^4}} \, dx=\frac {x \left (a+b x^2\right )}{4 a^2 \sqrt {a-b x^2} \sqrt {a^2-b^2 x^4}}+\frac {3 \sqrt {a-b x^2} \sqrt {a+b x^2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {b} x}{\sqrt {a+b x^2}}\right )}{4 \sqrt {2} a^2 \sqrt {b} \sqrt {a^2-b^2 x^4}} \]
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Time = 0.03 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {1166, 390, 385, 214} \[ \int \frac {1}{\left (a-b x^2\right )^{3/2} \sqrt {a^2-b^2 x^4}} \, dx=\frac {3 \sqrt {a-b x^2} \sqrt {a+b x^2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {b} x}{\sqrt {a+b x^2}}\right )}{4 \sqrt {2} a^2 \sqrt {b} \sqrt {a^2-b^2 x^4}}+\frac {x \left (a+b x^2\right )}{4 a^2 \sqrt {a-b x^2} \sqrt {a^2-b^2 x^4}} \]
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Rule 214
Rule 385
Rule 390
Rule 1166
Rubi steps \begin{align*} \text {integral}& = \frac {\left (\sqrt {a-b x^2} \sqrt {a+b x^2}\right ) \int \frac {1}{\left (a-b x^2\right )^2 \sqrt {a+b x^2}} \, dx}{\sqrt {a^2-b^2 x^4}} \\ & = \frac {x \left (a+b x^2\right )}{4 a^2 \sqrt {a-b x^2} \sqrt {a^2-b^2 x^4}}+\frac {\left (3 \sqrt {a-b x^2} \sqrt {a+b x^2}\right ) \int \frac {1}{\left (a-b x^2\right ) \sqrt {a+b x^2}} \, dx}{4 a \sqrt {a^2-b^2 x^4}} \\ & = \frac {x \left (a+b x^2\right )}{4 a^2 \sqrt {a-b x^2} \sqrt {a^2-b^2 x^4}}+\frac {\left (3 \sqrt {a-b x^2} \sqrt {a+b x^2}\right ) \text {Subst}\left (\int \frac {1}{a-2 a b x^2} \, dx,x,\frac {x}{\sqrt {a+b x^2}}\right )}{4 a \sqrt {a^2-b^2 x^4}} \\ & = \frac {x \left (a+b x^2\right )}{4 a^2 \sqrt {a-b x^2} \sqrt {a^2-b^2 x^4}}+\frac {3 \sqrt {a-b x^2} \sqrt {a+b x^2} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {b} x}{\sqrt {a+b x^2}}\right )}{4 \sqrt {2} a^2 \sqrt {b} \sqrt {a^2-b^2 x^4}} \\ \end{align*}
Time = 2.35 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.89 \[ \int \frac {1}{\left (a-b x^2\right )^{3/2} \sqrt {a^2-b^2 x^4}} \, dx=\frac {\sqrt {a^2-b^2 x^4} \left (2 \sqrt {b} x \sqrt {a+b x^2}+3 \sqrt {2} \left (a-b x^2\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {b} x}{\sqrt {a+b x^2}}\right )\right )}{8 a^2 \sqrt {b} \left (a-b x^2\right )^{3/2} \sqrt {a+b x^2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(498\) vs. \(2(100)=200\).
Time = 0.23 (sec) , antiderivative size = 499, normalized size of antiderivative = 4.02
method | result | size |
default | \(-\frac {\sqrt {-b^{2} x^{4}+a^{2}}\, b^{\frac {5}{2}} \left (-3 \sqrt {2}\, \ln \left (\frac {2 b \left (\sqrt {2}\, \sqrt {a}\, \sqrt {b \,x^{2}+a}+x \sqrt {a b}+a \right )}{b x -\sqrt {a b}}\right ) b^{\frac {3}{2}} x^{2} \sqrt {a}+3 \sqrt {2}\, \ln \left (\frac {2 b \left (\sqrt {2}\, \sqrt {a}\, \sqrt {b \,x^{2}+a}-x \sqrt {a b}+a \right )}{b x +\sqrt {a b}}\right ) b^{\frac {3}{2}} x^{2} \sqrt {a}+3 \sqrt {2}\, \ln \left (\frac {2 b \left (\sqrt {2}\, \sqrt {a}\, \sqrt {b \,x^{2}+a}+x \sqrt {a b}+a \right )}{b x -\sqrt {a b}}\right ) a^{\frac {3}{2}} \sqrt {b}-3 \sqrt {2}\, \ln \left (\frac {2 b \left (\sqrt {2}\, \sqrt {a}\, \sqrt {b \,x^{2}+a}-x \sqrt {a b}+a \right )}{b x +\sqrt {a b}}\right ) a^{\frac {3}{2}} \sqrt {b}+4 \ln \left (\frac {\sqrt {b}\, \sqrt {b \,x^{2}+a}+b x}{\sqrt {b}}\right ) b \,x^{2} \sqrt {a b}-4 \ln \left (\frac {\sqrt {b}\, \sqrt {-\frac {\left (-b x +\sqrt {-a b}\right ) \left (b x +\sqrt {-a b}\right )}{b}}+b x}{\sqrt {b}}\right ) b \,x^{2} \sqrt {a b}+4 \sqrt {b \,x^{2}+a}\, \sqrt {b}\, \sqrt {a b}\, x -4 \ln \left (\frac {\sqrt {b}\, \sqrt {b \,x^{2}+a}+b x}{\sqrt {b}}\right ) a \sqrt {a b}+4 \ln \left (\frac {\sqrt {b}\, \sqrt {-\frac {\left (-b x +\sqrt {-a b}\right ) \left (b x +\sqrt {-a b}\right )}{b}}+b x}{\sqrt {b}}\right ) a \sqrt {a b}\right )}{4 \sqrt {-b \,x^{2}+a}\, \sqrt {b \,x^{2}+a}\, \left (-\sqrt {-a b}+\sqrt {a b}\right )^{2} \left (\sqrt {-a b}+\sqrt {a b}\right )^{2} \left (b x -\sqrt {a b}\right ) \left (b x +\sqrt {a b}\right ) \sqrt {a b}}\) | \(499\) |
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none
Time = 0.25 (sec) , antiderivative size = 302, normalized size of antiderivative = 2.44 \[ \int \frac {1}{\left (a-b x^2\right )^{3/2} \sqrt {a^2-b^2 x^4}} \, dx=\left [\frac {4 \, \sqrt {-b^{2} x^{4} + a^{2}} \sqrt {-b x^{2} + a} b x + 3 \, \sqrt {2} {\left (b^{2} x^{4} - 2 \, a b x^{2} + a^{2}\right )} \sqrt {b} \log \left (-\frac {3 \, b^{2} x^{4} - 2 \, a b x^{2} - 2 \, \sqrt {2} \sqrt {-b^{2} x^{4} + a^{2}} \sqrt {-b x^{2} + a} \sqrt {b} x - a^{2}}{b^{2} x^{4} - 2 \, a b x^{2} + a^{2}}\right )}{16 \, {\left (a^{2} b^{3} x^{4} - 2 \, a^{3} b^{2} x^{2} + a^{4} b\right )}}, \frac {2 \, \sqrt {-b^{2} x^{4} + a^{2}} \sqrt {-b x^{2} + a} b x + 3 \, \sqrt {2} {\left (b^{2} x^{4} - 2 \, a b x^{2} + a^{2}\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {2} \sqrt {-b^{2} x^{4} + a^{2}} \sqrt {-b x^{2} + a} \sqrt {-b}}{2 \, {\left (b^{2} x^{3} - a b x\right )}}\right )}{8 \, {\left (a^{2} b^{3} x^{4} - 2 \, a^{3} b^{2} x^{2} + a^{4} b\right )}}\right ] \]
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\[ \int \frac {1}{\left (a-b x^2\right )^{3/2} \sqrt {a^2-b^2 x^4}} \, dx=\int \frac {1}{\sqrt {- \left (- a + b x^{2}\right ) \left (a + b x^{2}\right )} \left (a - b x^{2}\right )^{\frac {3}{2}}}\, dx \]
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\[ \int \frac {1}{\left (a-b x^2\right )^{3/2} \sqrt {a^2-b^2 x^4}} \, dx=\int { \frac {1}{\sqrt {-b^{2} x^{4} + a^{2}} {\left (-b x^{2} + a\right )}^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {1}{\left (a-b x^2\right )^{3/2} \sqrt {a^2-b^2 x^4}} \, dx=\int { \frac {1}{\sqrt {-b^{2} x^{4} + a^{2}} {\left (-b x^{2} + a\right )}^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {1}{\left (a-b x^2\right )^{3/2} \sqrt {a^2-b^2 x^4}} \, dx=\int \frac {1}{\sqrt {a^2-b^2\,x^4}\,{\left (a-b\,x^2\right )}^{3/2}} \,d x \]
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