Optimal. Leaf size=22 \[ \frac {\arctan \left (\frac {\sqrt {-2+x^2}}{\sqrt {2}}\right )}{\sqrt {2}} \]
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Rubi [A]
time = 0.01, antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {272, 65, 209}
\begin {gather*} \frac {\arctan \left (\frac {\sqrt {x^2-2}}{\sqrt {2}}\right )}{\sqrt {2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 209
Rule 272
Rubi steps
\begin {gather*} \begin {aligned} \text {Integral} &=\frac {1}{2} \text {Subst}\left (\int \frac {1}{\sqrt {-2+x} x} \, dx,x,x^2\right )\\ &=\text {Subst}\left (\int \frac {1}{2+x^2} \, dx,x,\sqrt {-2+x^2}\right )\\ &=\frac {\arctan \left (\frac {\sqrt {-2+x^2}}{\sqrt {2}}\right )}{\sqrt {2}}\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.01, size = 22, normalized size = 1.00 \begin {gather*} \frac {\arctan \left (\frac {\sqrt {-2+x^2}}{\sqrt {2}}\right )}{\sqrt {2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.14, size = 18, normalized size = 0.82
method | result | size |
default | \(-\frac {\sqrt {2}\, \arctan \left (\frac {\sqrt {2}}{\sqrt {x^{2}-2}}\right )}{2}\) | \(18\) |
pseudoelliptic | \(\frac {\arctan \left (\frac {\sqrt {x^{2}-2}\, \sqrt {2}}{2}\right ) \sqrt {2}}{2}\) | \(19\) |
trager | \(\frac {\mathit {RootOf}\left (\textit {\_Z}^{2}+2\right ) \ln \left (\frac {\mathit {RootOf}\left (\textit {\_Z}^{2}+2\right )+\sqrt {x^{2}-2}}{x}\right )}{2}\) | \(28\) |
meijerg | \(\frac {\sqrt {2}\, \sqrt {-\mathrm {signum}\left (-1+\frac {x^{2}}{2}\right )}\, \left (\left (-3 \ln \left (2\right )+2 \ln \left (x \right )+i \pi \right ) \sqrt {\pi }-2 \sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {1-\frac {x^{2}}{2}}}{2}\right )\right )}{4 \sqrt {\pi }\, \sqrt {\mathrm {signum}\left (-1+\frac {x^{2}}{2}\right )}}\) | \(68\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.42, size = 14, normalized size = 0.64 \begin {gather*} -\frac {1}{2} \, \sqrt {2} \arcsin \left (\frac {\sqrt {2}}{{\left | x \right |}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.57, size = 24, normalized size = 1.09 \begin {gather*} \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} x + \frac {1}{2} \, \sqrt {2} \sqrt {x^{2} - 2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 0.62, size = 39, normalized size = 1.77 \begin {gather*} \begin {cases} \frac {\sqrt {2} i \operatorname {acosh}{\left (\frac {\sqrt {2}}{x} \right )}}{2} & \text {for}\: \frac {1}{\left |{x^{2}}\right |} > \frac {1}{2} \\- \frac {\sqrt {2} \operatorname {asin}{\left (\frac {\sqrt {2}}{x} \right )}}{2} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.46, size = 18, normalized size = 0.82 \begin {gather*} \frac {1}{2} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} \sqrt {x^{2} - 2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.12, size = 18, normalized size = 0.82 \begin {gather*} \frac {\sqrt {2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,\sqrt {x^2-2}}{2}\right )}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Chatgpt [F] Failed to verify
time = 1.00, size = 14, normalized size = 0.64 \begin {gather*} -\frac {\sqrt {2}\, \sqrt {-x^{2}+3}}{2} \end {gather*}
Warning: Unable to verify antiderivative.
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