Optimal. Leaf size=12 \[ \sinh ^{-1}\left (\frac {1+2 x}{\sqrt {3}}\right ) \]
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Rubi [A]
time = 0.01, antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {633, 221}
\begin {gather*} \sinh ^{-1}\left (\frac {2 x+1}{\sqrt {3}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 221
Rule 633
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {1+x+x^2}} \, dx &=\frac {\text {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{3}}} \, dx,x,1+2 x\right )}{\sqrt {3}}\\ &=\sinh ^{-1}\left (\frac {1+2 x}{\sqrt {3}}\right )\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 20, normalized size = 1.67 \begin {gather*} -\log \left (-1-2 x+2 \sqrt {1+x+x^2}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.13, size = 10, normalized size = 0.83
method | result | size |
default | \(\arcsinh \left (\frac {2 \sqrt {3}\, \left (x +\frac {1}{2}\right )}{3}\right )\) | \(10\) |
trager | \(-\ln \left (2 \sqrt {x^{2}+x +1}-1-2 x \right )\) | \(19\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 1.73, size = 11, normalized size = 0.92 \begin {gather*} \operatorname {arsinh}\left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x + 1\right )}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.45, size = 18, normalized size = 1.50 \begin {gather*} -\log \left (-2 \, x + 2 \, \sqrt {x^{2} + x + 1} - 1\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} + x + 1}}\, 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.
\(2 (11) = 22\).
time = 1.41, size = 34, normalized size = 2.83 \begin {gather*} \frac {1}{4} \, \sqrt {x^{2} + x + 1} {\left (2 \, x + 1\right )} - \frac {3}{8} \, \log \left (-2 \, x + 2 \, \sqrt {x^{2} + x + 1} - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.24, size = 12, normalized size = 1.00 \begin {gather*} \ln \left (x+\sqrt {x^2+x+1}+\frac {1}{2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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