Optimal. Leaf size=22 \[ -\tanh ^{-1}\left (\frac {1-x}{2 \sqrt {1+x+x^2}}\right ) \]
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Rubi [A]
time = 0.01, antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {738, 212}
\begin {gather*} -\tanh ^{-1}\left (\frac {1-x}{2 \sqrt {x^2+x+1}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 738
Rubi steps
\begin {align*} \int \frac {1}{(1+x) \sqrt {1+x+x^2}} \, dx &=-\left (2 \text {Subst}\left (\int \frac {1}{4-x^2} \, dx,x,\frac {1-x}{\sqrt {1+x+x^2}}\right )\right )\\ &=-\tanh ^{-1}\left (\frac {1-x}{2 \sqrt {1+x+x^2}}\right )\\ \end {align*}
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Mathematica [A]
time = 0.08, size = 18, normalized size = 0.82 \begin {gather*} 2 \tanh ^{-1}\left (1+x-\sqrt {1+x+x^2}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.13, size = 22, normalized size = 1.00
method | result | size |
default | \(-\arctanh \left (\frac {1-x}{2 \sqrt {\left (1+x \right )^{2}-x}}\right )\) | \(22\) |
trager | \(-\ln \left (\frac {2 \sqrt {x^{2}+x +1}+1-x}{1+x}\right )\) | \(25\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 1.85, size = 25, normalized size = 1.14 \begin {gather*} \operatorname {arsinh}\left (\frac {\sqrt {3} x}{3 \, {\left | x + 1 \right |}} - \frac {\sqrt {3}}{3 \, {\left | x + 1 \right |}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.49, size = 30, normalized size = 1.36 \begin {gather*} -\log \left (-x + \sqrt {x^{2} + x + 1}\right ) + \log \left (-x + \sqrt {x^{2} + x + 1} - 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}{\left (x + 1\right ) \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 = 1.74, size = 32, normalized size = 1.45 \begin {gather*} -\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 = -0.05 \begin {gather*} \int \frac {1}{\left (x+1\right )\,\sqrt {x^2+x+1}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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