Optimal. Leaf size=8 \[ \sinh ^{-1}\left (\frac {2+x}{2}\right ) \]
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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 = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {633, 221}
\begin {gather*} \sinh ^{-1}\left (\frac {x+2}{2}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 221
Rule 633
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {8+4 x+x^2}} \, dx &=\frac {1}{4} \text {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{16}}} \, dx,x,4+2 x\right )\\ &=\sinh ^{-1}\left (\frac {2+x}{2}\right )\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(20\) vs. \(2(8)=16\).
time = 0.05, size = 20, normalized size = 2.50 \begin {gather*} -\log \left (-2-x+\sqrt {8+4 x+x^2}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.09, size = 7, normalized size = 0.88
method | result | size |
default | \(\arcsinh \left (1+\frac {x}{2}\right )\) | \(7\) |
trager | \(-\ln \left (\sqrt {x^{2}+4 x +8}-2-x \right )\) | \(19\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 2.24, size = 6, normalized size = 0.75 \begin {gather*} \operatorname {arsinh}\left (\frac {1}{2} \, x + 1\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. 18 vs.
\(2 (6) = 12\).
time = 0.60, size = 18, normalized size = 2.25 \begin {gather*} -\log \left (-x + \sqrt {x^{2} + 4 \, x + 8} - 2\right ) \end {gather*}
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 \begin {gather*} \int \frac {1}{\sqrt {x^{2} + 4 x + 8}}\, dx \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. 34 vs.
\(2 (6) = 12\).
time = 0.45, size = 34, normalized size = 4.25 \begin {gather*} \frac {1}{2} \, \sqrt {x^{2} + 4 \, x + 8} {\left (x + 2\right )} - 2 \, \log \left (-x + \sqrt {x^{2} + 4 \, x + 8} - 2\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.20, size = 14, normalized size = 1.75 \begin {gather*} \ln \left (x+\sqrt {x^2+4\,x+8}+2\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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