Optimal. Leaf size=47 \[ \frac {\sqrt {x^2+2} \operatorname {EllipticF}\left (\tan ^{-1}(x),\frac {1}{2}\right )}{\sqrt {2} \sqrt {x^2+1} \sqrt {\frac {x^2+2}{x^2+1}}} \]
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Rubi [A] time = 0.01, antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {418} \[ \frac {\sqrt {x^2+2} F\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{\sqrt {2} \sqrt {x^2+1} \sqrt {\frac {x^2+2}{x^2+1}}} \]
Antiderivative was successfully verified.
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Rule 418
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {1+x^2} \sqrt {2+x^2}} \, dx &=\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*}
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Mathematica [C] time = 0.02, size = 19, normalized size = 0.40 \[ -\frac {i \operatorname {EllipticF}\left (i \sinh ^{-1}(x),\frac {1}{2}\right )}{\sqrt {2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.65, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {x^{2} + 2} \sqrt {x^{2} + 1}}{x^{4} + 3 \, x^{2} + 2}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {x^{2} + 2} \sqrt {x^{2} + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.03, size = 15, normalized size = 0.32 \[ -i \EllipticF \left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {x^{2} + 2} \sqrt {x^{2} + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {1}{\sqrt {x^2+1}\,\sqrt {x^2+2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {x^{2} + 1} \sqrt {x^{2} + 2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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