Optimal. Leaf size=14 \[ -\tanh ^{-1}\left (\frac {\sqrt {2+x}}{2}\right ) \]
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Rubi [A]
time = 0.00, antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {65, 213}
\begin {gather*} -\tanh ^{-1}\left (\frac {\sqrt {x+2}}{2}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 213
Rubi steps
\begin {align*} \int \frac {1}{(-2+x) \sqrt {2+x}} \, dx &=2 \text {Subst}\left (\int \frac {1}{-4+x^2} \, dx,x,\sqrt {2+x}\right )\\ &=-\tanh ^{-1}\left (\frac {\sqrt {2+x}}{2}\right )\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 14, normalized size = 1.00 \begin {gather*} -\tanh ^{-1}\left (\frac {\sqrt {2+x}}{2}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(21\) vs.
\(2(10)=20\).
time = 0.16, size = 22, normalized size = 1.57
method | result | size |
trager | \(-\frac {\ln \left (-\frac {6+x +4 \sqrt {2+x}}{-2+x}\right )}{2}\) | \(21\) |
derivativedivides | \(\frac {\ln \left (\sqrt {2+x}-2\right )}{2}-\frac {\ln \left (\sqrt {2+x}+2\right )}{2}\) | \(22\) |
default | \(\frac {\ln \left (\sqrt {2+x}-2\right )}{2}-\frac {\ln \left (\sqrt {2+x}+2\right )}{2}\) | \(22\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 21 vs.
\(2 (10) = 20\).
time = 0.28, size = 21, normalized size = 1.50 \begin {gather*} -\frac {1}{2} \, \log \left (\sqrt {x + 2} + 2\right ) + \frac {1}{2} \, \log \left (\sqrt {x + 2} - 2\right ) \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. 21 vs.
\(2 (10) = 20\).
time = 0.81, size = 21, normalized size = 1.50 \begin {gather*} -\frac {1}{2} \, \log \left (\sqrt {x + 2} + 2\right ) + \frac {1}{2} \, \log \left (\sqrt {x + 2} - 2\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 26 vs.
\(2 (10) = 20\).
time = 0.29, size = 26, normalized size = 1.86 \begin {gather*} \begin {cases} - \operatorname {acoth}{\left (\frac {\sqrt {x + 2}}{2} \right )} & \text {for}\: \left |{x + 2}\right | > 4 \\- \operatorname {atanh}{\left (\frac {\sqrt {x + 2}}{2} \right )} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 22 vs.
\(2 (10) = 20\).
time = 0.92, size = 22, normalized size = 1.57 \begin {gather*} -\frac {1}{2} \, \log \left (\sqrt {x + 2} + 2\right ) + \frac {1}{2} \, \log \left ({\left | \sqrt {x + 2} - 2 \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.05, size = 10, normalized size = 0.71 \begin {gather*} -\mathrm {atanh}\left (\frac {\sqrt {x+2}}{2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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