Integrand size = 26, antiderivative size = 92 \[ \int \frac {1}{x^2 \sqrt {a^2+2 a b x^2+b^2 x^4}} \, dx=-\frac {a+b x^2}{a x \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\sqrt {b} \left (a+b x^2\right ) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{3/2} \sqrt {a^2+2 a b x^2+b^2 x^4}} \]
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Time = 0.02 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {1126, 331, 211} \[ \int \frac {1}{x^2 \sqrt {a^2+2 a b x^2+b^2 x^4}} \, dx=-\frac {a+b x^2}{a x \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\sqrt {b} \left (a+b x^2\right ) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{3/2} \sqrt {a^2+2 a b x^2+b^2 x^4}} \]
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Rule 211
Rule 331
Rule 1126
Rubi steps \begin{align*} \text {integral}& = \frac {\left (a b+b^2 x^2\right ) \int \frac {1}{x^2 \left (a b+b^2 x^2\right )} \, dx}{\sqrt {a^2+2 a b x^2+b^2 x^4}} \\ & = -\frac {a+b x^2}{a x \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (b \left (a b+b^2 x^2\right )\right ) \int \frac {1}{a b+b^2 x^2} \, dx}{a \sqrt {a^2+2 a b x^2+b^2 x^4}} \\ & = -\frac {a+b x^2}{a x \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\sqrt {b} \left (a+b x^2\right ) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{3/2} \sqrt {a^2+2 a b x^2+b^2 x^4}} \\ \end{align*}
Time = 1.01 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.61 \[ \int \frac {1}{x^2 \sqrt {a^2+2 a b x^2+b^2 x^4}} \, dx=-\frac {\left (a+b x^2\right ) \left (\sqrt {a}+\sqrt {b} x \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right )}{a^{3/2} x \sqrt {\left (a+b x^2\right )^2}} \]
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Time = 0.81 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.54
method | result | size |
default | \(-\frac {\left (b \,x^{2}+a \right ) \left (b \arctan \left (\frac {b x}{\sqrt {a b}}\right ) x +\sqrt {a b}\right )}{\sqrt {\left (b \,x^{2}+a \right )^{2}}\, a \sqrt {a b}\, x}\) | \(50\) |
risch | \(-\frac {\sqrt {\left (b \,x^{2}+a \right )^{2}}}{\left (b \,x^{2}+a \right ) a x}+\frac {\sqrt {\left (b \,x^{2}+a \right )^{2}}\, \sqrt {-a b}\, \ln \left (-b x +\sqrt {-a b}\right )}{2 \left (b \,x^{2}+a \right ) a^{2}}-\frac {\sqrt {\left (b \,x^{2}+a \right )^{2}}\, \sqrt {-a b}\, \ln \left (-b x -\sqrt {-a b}\right )}{2 \left (b \,x^{2}+a \right ) a^{2}}\) | \(118\) |
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Time = 0.26 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.89 \[ \int \frac {1}{x^2 \sqrt {a^2+2 a b x^2+b^2 x^4}} \, dx=\left [\frac {x \sqrt {-\frac {b}{a}} \log \left (\frac {b x^{2} - 2 \, a x \sqrt {-\frac {b}{a}} - a}{b x^{2} + a}\right ) - 2}{2 \, a x}, -\frac {x \sqrt {\frac {b}{a}} \arctan \left (x \sqrt {\frac {b}{a}}\right ) + 1}{a x}\right ] \]
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\[ \int \frac {1}{x^2 \sqrt {a^2+2 a b x^2+b^2 x^4}} \, dx=\int \frac {1}{x^{2} \sqrt {\left (a + b x^{2}\right )^{2}}}\, dx \]
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Time = 0.29 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.32 \[ \int \frac {1}{x^2 \sqrt {a^2+2 a b x^2+b^2 x^4}} \, dx=-\frac {b \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} a} - \frac {1}{a x} \]
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Time = 0.28 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.40 \[ \int \frac {1}{x^2 \sqrt {a^2+2 a b x^2+b^2 x^4}} \, dx=-{\left (\frac {b \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} a} + \frac {1}{a x}\right )} \mathrm {sgn}\left (b x^{2} + a\right ) \]
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Timed out. \[ \int \frac {1}{x^2 \sqrt {a^2+2 a b x^2+b^2 x^4}} \, dx=\int \frac {1}{x^2\,\sqrt {{\left (b\,x^2+a\right )}^2}} \,d x \]
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