Optimal. Leaf size=45 \[ -x+\sqrt {1+x+x^2}-\frac {3}{2} \sinh ^{-1}\left (\frac {1+2 x}{\sqrt {3}}\right )+2 \log \left (x+\sqrt {1+x+x^2}\right ) \]
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Rubi [A]
time = 0.02, antiderivative size = 59, normalized size of antiderivative = 1.31, number of steps
used = 3, number of rules used = 2, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2141, 907}
\begin {gather*} \frac {3}{2 \left (2 \left (\sqrt {x^2+x+1}+x\right )+1\right )}+2 \log \left (\sqrt {x^2+x+1}+x\right )-\frac {3}{2} \log \left (2 \left (\sqrt {x^2+x+1}+x\right )+1\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 907
Rule 2141
Rubi steps
\begin {align*} \int \frac {1}{x+\sqrt {1+x+x^2}} \, dx &=2 \text {Subst}\left (\int \frac {1+x+x^2}{x (1+2 x)^2} \, dx,x,x+\sqrt {1+x+x^2}\right )\\ &=2 \text {Subst}\left (\int \left (\frac {1}{x}-\frac {3}{2 (1+2 x)^2}-\frac {3}{2 (1+2 x)}\right ) \, dx,x,x+\sqrt {1+x+x^2}\right )\\ &=\frac {3}{2 \left (1+2 \left (x+\sqrt {1+x+x^2}\right )\right )}+2 \log \left (x+\sqrt {1+x+x^2}\right )-\frac {3}{2} \log \left (1+2 \left (x+\sqrt {1+x+x^2}\right )\right )\\ \end {align*}
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Mathematica [A]
time = 0.08, size = 54, normalized size = 1.20 \begin {gather*} -x+\sqrt {1+x+x^2}+2 \log \left (-2-x+\sqrt {1+x+x^2}\right )-\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 = 0.06, size = 52, normalized size = 1.16
method | result | size |
default | \(\sqrt {\left (1+x \right )^{2}-x}-\frac {\arcsinh \left (\frac {2 \sqrt {3}\, \left (x +\frac {1}{2}\right )}{3}\right )}{2}-\arctanh \left (\frac {1-x}{2 \sqrt {\left (1+x \right )^{2}-x}}\right )-x +\ln \left (1+x \right )\) | \(52\) |
trager | \(\sqrt {x^{2}+x +1}-x +\frac {\ln \left (2 x^{2} \sqrt {x^{2}+x +1}-2 x^{3}+8 x \sqrt {x^{2}+x +1}-9 x^{2}+14 \sqrt {x^{2}+x +1}-12 x -13\right )}{2}\) | \(65\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.23, size = 63, normalized size = 1.40 \begin {gather*} -x + \sqrt {x^{2} + x + 1} + \log \left (x + 1\right ) - \log \left (-x + \sqrt {x^{2} + x + 1}\right ) + \log \left (-x + \sqrt {x^{2} + x + 1} - 2\right ) + \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 {1}{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 = 0.80, size = 66, normalized size = 1.47 \begin {gather*} -x + \sqrt {x^{2} + x + 1} + \frac {1}{2} \, \log \left (-2 \, x + 2 \, \sqrt {x^{2} + x + 1} - 1\right ) + \log \left ({\left | x + 1 \right |}\right ) - \log \left ({\left | -x + \sqrt {x^{2} + x + 1} \right |}\right ) + \log \left ({\left | -x + \sqrt {x^{2} + x + 1} - 2 \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \ln \left (x+1\right )-x+\int \frac {\sqrt {x^2+x+1}}{x+1} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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