Integrand size = 15, antiderivative size = 28 \[ \int \frac {1}{\sqrt {a+\frac {b}{x^2}} x^2} \, dx=-\frac {\text {arctanh}\left (\frac {\sqrt {b}}{\sqrt {a+\frac {b}{x^2}} x}\right )}{\sqrt {b}} \]
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Time = 0.01 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {342, 223, 212} \[ \int \frac {1}{\sqrt {a+\frac {b}{x^2}} x^2} \, dx=-\frac {\text {arctanh}\left (\frac {\sqrt {b}}{x \sqrt {a+\frac {b}{x^2}}}\right )}{\sqrt {b}} \]
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Rule 212
Rule 223
Rule 342
Rubi steps \begin{align*} \text {integral}& = -\text {Subst}\left (\int \frac {1}{\sqrt {a+b x^2}} \, dx,x,\frac {1}{x}\right ) \\ & = -\text {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {1}{\sqrt {a+\frac {b}{x^2}} x}\right ) \\ & = -\frac {\tanh ^{-1}\left (\frac {\sqrt {b}}{\sqrt {a+\frac {b}{x^2}} x}\right )}{\sqrt {b}} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.79 \[ \int \frac {1}{\sqrt {a+\frac {b}{x^2}} x^2} \, dx=-\frac {\sqrt {b+a x^2} \text {arctanh}\left (\frac {\sqrt {b+a x^2}}{\sqrt {b}}\right )}{\sqrt {b} \sqrt {a+\frac {b}{x^2}} x} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(51\) vs. \(2(22)=44\).
Time = 0.02 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.86
method | result | size |
default | \(-\frac {\sqrt {a \,x^{2}+b}\, \ln \left (\frac {2 b +2 \sqrt {b}\, \sqrt {a \,x^{2}+b}}{x}\right )}{\sqrt {\frac {a \,x^{2}+b}{x^{2}}}\, x \sqrt {b}}\) | \(52\) |
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none
Time = 0.31 (sec) , antiderivative size = 79, normalized size of antiderivative = 2.82 \[ \int \frac {1}{\sqrt {a+\frac {b}{x^2}} x^2} \, dx=\left [\frac {\log \left (-\frac {a x^{2} - 2 \, \sqrt {b} x \sqrt {\frac {a x^{2} + b}{x^{2}}} + 2 \, b}{x^{2}}\right )}{2 \, \sqrt {b}}, \frac {\sqrt {-b} \arctan \left (\frac {\sqrt {-b} x \sqrt {\frac {a x^{2} + b}{x^{2}}}}{a x^{2} + b}\right )}{b}\right ] \]
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Time = 0.61 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.68 \[ \int \frac {1}{\sqrt {a+\frac {b}{x^2}} x^2} \, dx=- \frac {\operatorname {asinh}{\left (\frac {\sqrt {b}}{\sqrt {a} x} \right )}}{\sqrt {b}} \]
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none
Time = 0.26 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.46 \[ \int \frac {1}{\sqrt {a+\frac {b}{x^2}} x^2} \, dx=\frac {\log \left (\frac {\sqrt {a + \frac {b}{x^{2}}} x - \sqrt {b}}{\sqrt {a + \frac {b}{x^{2}}} x + \sqrt {b}}\right )}{2 \, \sqrt {b}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 46 vs. \(2 (22) = 44\).
Time = 0.30 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.64 \[ \int \frac {1}{\sqrt {a+\frac {b}{x^2}} x^2} \, dx=-\frac {\arctan \left (\frac {\sqrt {b}}{\sqrt {-b}}\right ) \mathrm {sgn}\left (x\right )}{\sqrt {-b}} + \frac {\arctan \left (\frac {\sqrt {a x^{2} + b}}{\sqrt {-b}}\right )}{\sqrt {-b} \mathrm {sgn}\left (x\right )} \]
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Time = 6.55 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.82 \[ \int \frac {1}{\sqrt {a+\frac {b}{x^2}} x^2} \, dx=-\frac {\ln \left (\sqrt {a+\frac {b}{x^2}}+\frac {\sqrt {b}}{x}\right )}{\sqrt {b}} \]
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