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, 209}
\begin {gather*} \frac {\text {ArcTan}(x)}{\sqrt {2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 22
Rule 209
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 \begin {gather*} \frac {\tan ^{-1}(x)}{\sqrt {2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 8, normalized size = 1.00
method | result | size |
meijerg | \(\frac {\arctan \left (x \right ) \sqrt {2}}{2}\) | \(8\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 34 vs.
\(2 (7) = 14\).
time = 1.20, size = 34, normalized size = 4.25 \begin {gather*} -\frac {1}{4} \, \sqrt {2} \arctan \left (\frac {\sqrt {2} \sqrt {2 \, x^{2} + 2} \sqrt {x^{2} + 1} x}{x^{4} - 1}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.95, size = 8, normalized size = 1.00 \begin {gather*} \frac {\sqrt {2} \operatorname {atan}{\left (x \right )}}{2} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {1}{\sqrt {x^2+1}\,\sqrt {2\,x^2+2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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