Optimal. Leaf size=8 \[ \frac {\tan ^{-1}(x)}{\sqrt {2}} \]
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Rubi [A] time = 0.00, antiderivative size = 8, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {22, 203} \[ \frac {\tan ^{-1}(x)}{\sqrt {2}} \]
Antiderivative was successfully verified.
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Rule 22
Rule 203
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {1+x^2} \sqrt {2+2 x^2}} \, dx &=\sqrt {2} \int \frac {1}{2+2 x^2} \, dx\\ &=\frac {\tan ^{-1}(x)}{\sqrt {2}}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 8, normalized size = 1.00 \[ \frac {\tan ^{-1}(x)}{\sqrt {2}} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.04, size = 34, normalized size = 4.25 \[ -\frac {1}{4} \, \sqrt {2} \arctan \left (\frac {\sqrt {2} \sqrt {2 \, x^{2} + 2} \sqrt {x^{2} + 1} x}{x^{4} - 1}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.57, size = 26, normalized size = 3.25 \[ \frac {1}{4} \, \sqrt {2} i \log \left (i x - 1\right ) - \frac {1}{4} \, \sqrt {2} i \log \left (-i x - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.31, size = 8, normalized size = 1.00 \[ \frac {\sqrt {2}\, \arctan \relax (x )}{2} \]
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 \[ \int \frac {1}{\sqrt {2 \, x^{2} + 2} \sqrt {x^{2} + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.12 \[ \int \frac {1}{\sqrt {x^2+1}\,\sqrt {2\,x^2+2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.49, size = 8, normalized size = 1.00 \[ \frac {\sqrt {2} \operatorname {atan}{\relax (x )}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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