Optimal. Leaf size=28 \[ \frac {\sqrt {x^2+1} \tan ^{-1}(x)}{\sqrt {2} \sqrt {-x^2-1}} \]
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Rubi [A] time = 0.00, antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {23, 203} \[ \frac {\sqrt {x^2+1} \tan ^{-1}(x)}{\sqrt {2} \sqrt {-x^2-1}} \]
Antiderivative was successfully verified.
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Rule 23
Rule 203
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {-1-x^2} \sqrt {2+2 x^2}} \, dx &=\frac {\sqrt {2+2 x^2} \int \frac {1}{2+2 x^2} \, dx}{\sqrt {-1-x^2}}\\ &=\frac {\sqrt {1+x^2} \tan ^{-1}(x)}{\sqrt {2} \sqrt {-1-x^2}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 26, normalized size = 0.93 \[ \frac {\left (x^2+1\right ) \tan ^{-1}(x)}{\sqrt {2} \sqrt {-\left (x^2+1\right )^2}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.65, size = 104, normalized size = 3.71 \[ \frac {1}{8} \, \sqrt {2} \log \left (\frac {2 \, {\left (2 \, \sqrt {2 \, x^{2} + 2} \sqrt {-x^{2} - 1} x + \sqrt {2} {\left (x^{4} - 1\right )}\right )}}{x^{4} + 2 \, x^{2} + 1}\right ) - \frac {1}{8} \, \sqrt {2} \log \left (\frac {2 \, {\left (2 \, \sqrt {2 \, x^{2} + 2} \sqrt {-x^{2} - 1} x - \sqrt {2} {\left (x^{4} - 1\right )}\right )}}{x^{4} + 2 \, x^{2} + 1}\right ) \]
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 \[ \int \frac {1}{\sqrt {2 \, x^{2} + 2} \sqrt {-x^{2} - 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 24, normalized size = 0.86 \[ -\frac {\sqrt {-x^{2}-1}\, \sqrt {2}\, \arctan \relax (x )}{2 \sqrt {x^{2}+1}} \]
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 = -0.04 \[ \int \frac {1}{\sqrt {-x^2-1}\,\sqrt {2\,x^2+2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\sqrt {2} \int \frac {1}{\sqrt {- x^{2} - 1} \sqrt {x^{2} + 1}}\, dx}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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