Integrand size = 28, antiderivative size = 65 \[ \int \frac {\sqrt {a+b x^2}}{\sqrt {a^2-b^2 x^4}} \, dx=\frac {\sqrt {a-b x^2} \sqrt {a+b x^2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a-b x^2}}\right )}{\sqrt {b} \sqrt {a^2-b^2 x^4}} \]
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Time = 0.01 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.107, Rules used = {1166, 223, 209} \[ \int \frac {\sqrt {a+b x^2}}{\sqrt {a^2-b^2 x^4}} \, dx=\frac {\sqrt {a-b x^2} \sqrt {a+b x^2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a-b x^2}}\right )}{\sqrt {b} \sqrt {a^2-b^2 x^4}} \]
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Rule 209
Rule 223
Rule 1166
Rubi steps \begin{align*} \text {integral}& = \frac {\left (\sqrt {a-b x^2} \sqrt {a+b x^2}\right ) \int \frac {1}{\sqrt {a-b x^2}} \, dx}{\sqrt {a^2-b^2 x^4}} \\ & = \frac {\left (\sqrt {a-b x^2} \sqrt {a+b x^2}\right ) \text {Subst}\left (\int \frac {1}{1+b x^2} \, dx,x,\frac {x}{\sqrt {a-b x^2}}\right )}{\sqrt {a^2-b^2 x^4}} \\ & = \frac {\sqrt {a-b x^2} \sqrt {a+b x^2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a-b x^2}}\right )}{\sqrt {b} \sqrt {a^2-b^2 x^4}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.96 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.77 \[ \int \frac {\sqrt {a+b x^2}}{\sqrt {a^2-b^2 x^4}} \, dx=\frac {i \log \left (-2 i \sqrt {b} x+\frac {2 \sqrt {a^2-b^2 x^4}}{\sqrt {a+b x^2}}\right )}{\sqrt {b}} \]
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Time = 0.23 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.83
method | result | size |
default | \(\frac {\sqrt {-b^{2} x^{4}+a^{2}}\, \arctan \left (\frac {\sqrt {b}\, x}{\sqrt {-b \,x^{2}+a}}\right )}{\sqrt {b \,x^{2}+a}\, \sqrt {-b \,x^{2}+a}\, \sqrt {b}}\) | \(54\) |
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none
Time = 0.27 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.86 \[ \int \frac {\sqrt {a+b x^2}}{\sqrt {a^2-b^2 x^4}} \, dx=\left [-\frac {\sqrt {-b} \log \left (-\frac {2 \, b^{2} x^{4} + a b x^{2} - 2 \, \sqrt {-b^{2} x^{4} + a^{2}} \sqrt {b x^{2} + a} \sqrt {-b} x - a^{2}}{b x^{2} + a}\right )}{2 \, b}, -\frac {\arctan \left (\frac {\sqrt {-b^{2} x^{4} + a^{2}} \sqrt {b x^{2} + a} \sqrt {b}}{b^{2} x^{3} + a b x}\right )}{\sqrt {b}}\right ] \]
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\[ \int \frac {\sqrt {a+b x^2}}{\sqrt {a^2-b^2 x^4}} \, dx=\int \frac {\sqrt {a + b x^{2}}}{\sqrt {- \left (- a + b x^{2}\right ) \left (a + b x^{2}\right )}}\, dx \]
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\[ \int \frac {\sqrt {a+b x^2}}{\sqrt {a^2-b^2 x^4}} \, dx=\int { \frac {\sqrt {b x^{2} + a}}{\sqrt {-b^{2} x^{4} + a^{2}}} \,d x } \]
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\[ \int \frac {\sqrt {a+b x^2}}{\sqrt {a^2-b^2 x^4}} \, dx=\int { \frac {\sqrt {b x^{2} + a}}{\sqrt {-b^{2} x^{4} + a^{2}}} \,d x } \]
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Timed out. \[ \int \frac {\sqrt {a+b x^2}}{\sqrt {a^2-b^2 x^4}} \, dx=\int \frac {\sqrt {b\,x^2+a}}{\sqrt {a^2-b^2\,x^4}} \,d x \]
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