Optimal. Leaf size=27 \[ \sqrt {1+x+x^2}-\frac {1}{2} \sinh ^{-1}\left (\frac {1+2 x}{\sqrt {3}}\right ) \]
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Rubi [A]
time = 0.01, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {654, 633, 221}
\begin {gather*} \sqrt {x^2+x+1}-\frac {1}{2} \sinh ^{-1}\left (\frac {2 x+1}{\sqrt {3}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 221
Rule 633
Rule 654
Rubi steps
\begin {align*} \int \frac {x}{\sqrt {1+x+x^2}} \, dx &=\sqrt {1+x+x^2}-\frac {1}{2} \int \frac {1}{\sqrt {1+x+x^2}} \, dx\\ &=\sqrt {1+x+x^2}-\frac {\text {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{3}}} \, dx,x,1+2 x\right )}{2 \sqrt {3}}\\ &=\sqrt {1+x+x^2}-\frac {1}{2} \sinh ^{-1}\left (\frac {1+2 x}{\sqrt {3}}\right )\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 33, normalized size = 1.22 \begin {gather*} \sqrt {1+x+x^2}+\frac {1}{2} \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 = 0.17, size = 21, normalized size = 0.78
method | result | size |
default | \(\sqrt {x^{2}+x +1}-\frac {\arcsinh \left (\frac {2 \sqrt {3}\, \left (x +\frac {1}{2}\right )}{3}\right )}{2}\) | \(21\) |
risch | \(\sqrt {x^{2}+x +1}-\frac {\arcsinh \left (\frac {2 \sqrt {3}\, \left (x +\frac {1}{2}\right )}{3}\right )}{2}\) | \(21\) |
trager | \(\sqrt {x^{2}+x +1}-\frac {\ln \left (2 x +1+2 \sqrt {x^{2}+x +1}\right )}{2}\) | \(28\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 4.10, size = 22, normalized size = 0.81 \begin {gather*} \sqrt {x^{2} + x + 1} - \frac {1}{2} \, \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 = 0.53, size = 27, normalized size = 1.00 \begin {gather*} \sqrt {x^{2} + x + 1} + \frac {1}{2} \, \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 {x}{\sqrt {x^{2} + x + 1}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.45, size = 27, normalized size = 1.00 \begin {gather*} \sqrt {x^{2} + x + 1} + \frac {1}{2} \, \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 = 0.05, size = 23, normalized size = 0.85 \begin {gather*} \sqrt {x^2+x+1}-\frac {\ln \left (x+\sqrt {x^2+x+1}+\frac {1}{2}\right )}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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