Integrand size = 24, antiderivative size = 30 \[ \int \frac {1}{\sqrt {2+2 a-2 (1+a)+b x^2+c x^4}} \, dx=-\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+c x^4}}\right )}{\sqrt {b}} \]
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Time = 0.01 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {3, 2033, 212} \[ \int \frac {1}{\sqrt {2+2 a-2 (1+a)+b x^2+c x^4}} \, dx=-\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+c x^4}}\right )}{\sqrt {b}} \]
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Rule 3
Rule 212
Rule 2033
Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{\sqrt {b x^2+c x^4}} \, dx \\ & = -\text {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {x}{\sqrt {b x^2+c x^4}}\right ) \\ & = -\frac {\tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+c x^4}}\right )}{\sqrt {b}} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.73 \[ \int \frac {1}{\sqrt {2+2 a-2 (1+a)+b x^2+c x^4}} \, dx=-\frac {x \sqrt {b+c x^2} \text {arctanh}\left (\frac {\sqrt {b+c x^2}}{\sqrt {b}}\right )}{\sqrt {b} \sqrt {x^2 \left (b+c x^2\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(49\) vs. \(2(24)=48\).
Time = 0.24 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.67
method | result | size |
default | \(-\frac {x \sqrt {c \,x^{2}+b}\, \ln \left (\frac {2 b +2 \sqrt {b}\, \sqrt {c \,x^{2}+b}}{x}\right )}{\sqrt {c \,x^{4}+b \,x^{2}}\, \sqrt {b}}\) | \(50\) |
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none
Time = 0.28 (sec) , antiderivative size = 80, normalized size of antiderivative = 2.67 \[ \int \frac {1}{\sqrt {2+2 a-2 (1+a)+b x^2+c x^4}} \, dx=\left [\frac {\log \left (-\frac {c x^{3} + 2 \, b x - 2 \, \sqrt {c x^{4} + b x^{2}} \sqrt {b}}{x^{3}}\right )}{2 \, \sqrt {b}}, \frac {\sqrt {-b} \arctan \left (\frac {\sqrt {c x^{4} + b x^{2}} \sqrt {-b}}{c x^{3} + b x}\right )}{b}\right ] \]
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\[ \int \frac {1}{\sqrt {2+2 a-2 (1+a)+b x^2+c x^4}} \, dx=\int \frac {1}{\sqrt {b x^{2} + c x^{4}}}\, dx \]
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\[ \int \frac {1}{\sqrt {2+2 a-2 (1+a)+b x^2+c x^4}} \, dx=\int { \frac {1}{\sqrt {c x^{4} + b x^{2}}} \,d x } \]
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none
Time = 0.29 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.53 \[ \int \frac {1}{\sqrt {2+2 a-2 (1+a)+b x^2+c x^4}} \, dx=-\frac {\arctan \left (\frac {\sqrt {b}}{\sqrt {-b}}\right ) \mathrm {sgn}\left (x\right )}{\sqrt {-b}} + \frac {\arctan \left (\frac {\sqrt {c x^{2} + b}}{\sqrt {-b}}\right )}{\sqrt {-b} \mathrm {sgn}\left (x\right )} \]
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Timed out. \[ \int \frac {1}{\sqrt {2+2 a-2 (1+a)+b x^2+c x^4}} \, dx=\int \frac {1}{\sqrt {c\,x^4+b\,x^2}} \,d x \]
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