Integrand size = 19, antiderivative size = 47 \[ \int \frac {1}{\sqrt {1+x^2} \sqrt {2+x^2}} \, dx=\frac {\sqrt {2+x^2} \operatorname {EllipticF}\left (\arctan (x),\frac {1}{2}\right )}{\sqrt {2} \sqrt {1+x^2} \sqrt {\frac {2+x^2}{1+x^2}}} \]
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Time = 0.01 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {429} \[ \int \frac {1}{\sqrt {1+x^2} \sqrt {2+x^2}} \, dx=\frac {\sqrt {x^2+2} \operatorname {EllipticF}\left (\arctan (x),\frac {1}{2}\right )}{\sqrt {2} \sqrt {x^2+1} \sqrt {\frac {x^2+2}{x^2+1}}} \]
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Rule 429
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {2+x^2} F\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{\sqrt {2} \sqrt {1+x^2} \sqrt {\frac {2+x^2}{1+x^2}}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.18 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.40 \[ \int \frac {1}{\sqrt {1+x^2} \sqrt {2+x^2}} \, dx=-\frac {i \operatorname {EllipticF}\left (i \text {arcsinh}(x),\frac {1}{2}\right )}{\sqrt {2}} \]
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Result contains complex when optimal does not.
Time = 2.51 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.32
method | result | size |
default | \(-i F\left (\frac {i x \sqrt {2}}{2}, \sqrt {2}\right )\) | \(15\) |
elliptic | \(-\frac {i \sqrt {\left (x^{2}+1\right ) \left (x^{2}+2\right )}\, \sqrt {2}\, \sqrt {2 x^{2}+4}\, F\left (\frac {i x \sqrt {2}}{2}, \sqrt {2}\right )}{2 \sqrt {x^{2}+2}\, \sqrt {x^{4}+3 x^{2}+2}}\) | \(59\) |
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Result contains complex when optimal does not.
Time = 0.09 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.23 \[ \int \frac {1}{\sqrt {1+x^2} \sqrt {2+x^2}} \, dx=-i \, F(\arcsin \left (\frac {1}{2} i \, \sqrt {2} x\right )\,|\,2) \]
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\[ \int \frac {1}{\sqrt {1+x^2} \sqrt {2+x^2}} \, dx=\int \frac {1}{\sqrt {x^{2} + 1} \sqrt {x^{2} + 2}}\, dx \]
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\[ \int \frac {1}{\sqrt {1+x^2} \sqrt {2+x^2}} \, dx=\int { \frac {1}{\sqrt {x^{2} + 2} \sqrt {x^{2} + 1}} \,d x } \]
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\[ \int \frac {1}{\sqrt {1+x^2} \sqrt {2+x^2}} \, dx=\int { \frac {1}{\sqrt {x^{2} + 2} \sqrt {x^{2} + 1}} \,d x } \]
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Timed out. \[ \int \frac {1}{\sqrt {1+x^2} \sqrt {2+x^2}} \, dx=\int \frac {1}{\sqrt {x^2+1}\,\sqrt {x^2+2}} \,d x \]
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