Optimal. Leaf size=45 \[ \sqrt {x^2+x+1}+2 \log \left (\sqrt {x^2+x+1}+x\right )-x-\frac {3}{2} \sinh ^{-1}\left (\frac {2 x+1}{\sqrt {3}}\right ) \]
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Rubi [A] time = 0.03, 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 = {2116, 893} \[ \frac {3}{2 \left (2 \left (\sqrt {x^2+x+1}+x\right )+1\right )}+2 \log \left (\sqrt {x^2+x+1}+x\right )-\frac {3}{2} \log \left (2 \left (\sqrt {x^2+x+1}+x\right )+1\right ) \]
Antiderivative was successfully verified.
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Rule 893
Rule 2116
Rubi steps
\begin {align*} \int \frac {1}{x+\sqrt {1+x+x^2}} \, dx &=2 \operatorname {Subst}\left (\int \frac {1+x+x^2}{x (1+2 x)^2} \, dx,x,x+\sqrt {1+x+x^2}\right )\\ &=2 \operatorname {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 = 0.04, size = 59, normalized size = 1.31 \[ \frac {3}{2 \left (2 \left (\sqrt {x^2+x+1}+x\right )+1\right )}+2 \log \left (\sqrt {x^2+x+1}+x\right )-\frac {3}{2} \log \left (2 \left (\sqrt {x^2+x+1}+x\right )+1\right ) \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.12, size = 54, normalized size = 1.20 \[ \sqrt {x^2+x+1}+2 \log \left (\sqrt {x^2+x+1}-x-2\right )-\frac {1}{2} \log \left (2 \sqrt {x^2+x+1}-2 x-1\right )-x \]
Antiderivative was successfully verified.
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fricas [A] time = 0.78, size = 63, normalized size = 1.40 \[ -x + \sqrt {x^{2} + x + 1} + \log \left (x + 1\right ) - \log \left (-x + \sqrt {x^{2} + x + 1}\right ) + \log \left (-x + \sqrt {x^{2} + x + 1} - 2\right ) + \frac {1}{2} \, \log \left (-2 \, x + 2 \, \sqrt {x^{2} + x + 1} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.65, size = 66, normalized size = 1.47 \[ -x + \sqrt {x^{2} + x + 1} + \frac {1}{2} \, \log \left (-2 \, x + 2 \, \sqrt {x^{2} + x + 1} - 1\right ) + \log \left ({\left | x + 1 \right |}\right ) - \log \left ({\left | -x + \sqrt {x^{2} + x + 1} \right |}\right ) + \log \left ({\left | -x + \sqrt {x^{2} + x + 1} - 2 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.12, size = 52, normalized size = 1.16
method | result | size |
default | \(\sqrt {\left (1+x \right )^{2}-x}-\frac {\arcsinh \left (\frac {2 \left (\frac {1}{2}+x \right ) \sqrt {3}}{3}\right )}{2}-\arctanh \left (\frac {1-x}{2 \sqrt {\left (1+x \right )^{2}-x}}\right )-x +\ln \left (1+x \right )\) | \(52\) |
trager | \(\sqrt {x^{2}+x +1}-x -\frac {\ln \left (\frac {2 x^{2} \sqrt {x^{2}+x +1}+2 x^{3}+8 \sqrt {x^{2}+x +1}\, x +9 x^{2}+14 \sqrt {x^{2}+x +1}+12 x +13}{\left (1+x \right )^{4}}\right )}{2}\) | \(71\) |
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 \[ \int \frac {1}{x + \sqrt {x^{2} + x + 1}}\,{d x} \]
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 \[ \ln \left (x+1\right )-x+\int \frac {\sqrt {x^2+x+1}}{x+1} \,d x \]
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 \[ \int \frac {1}{x + \sqrt {x^{2} + x + 1}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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