Integrand size = 16, antiderivative size = 22 \[ \int \frac {1}{(1+x) \sqrt {1+x+x^2}} \, dx=-\text {arctanh}\left (\frac {1-x}{2 \sqrt {1+x+x^2}}\right ) \]
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Time = 0.01 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {738, 212} \[ \int \frac {1}{(1+x) \sqrt {1+x+x^2}} \, dx=-\text {arctanh}\left (\frac {1-x}{2 \sqrt {x^2+x+1}}\right ) \]
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Rule 212
Rule 738
Rubi steps \begin{align*} \text {integral}& = -\left (2 \text {Subst}\left (\int \frac {1}{4-x^2} \, dx,x,\frac {1-x}{\sqrt {1+x+x^2}}\right )\right ) \\ & = -\text {arctanh}\left (\frac {1-x}{2 \sqrt {1+x+x^2}}\right ) \\ \end{align*}
Time = 0.12 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.82 \[ \int \frac {1}{(1+x) \sqrt {1+x+x^2}} \, dx=2 \text {arctanh}\left (1+x-\sqrt {1+x+x^2}\right ) \]
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Time = 0.42 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00
method | result | size |
default | \(-\operatorname {arctanh}\left (\frac {1-x}{2 \sqrt {\left (1+x \right )^{2}-x}}\right )\) | \(22\) |
trager | \(-\ln \left (\frac {2 \sqrt {x^{2}+x +1}+1-x}{1+x}\right )\) | \(25\) |
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none
Time = 0.24 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.36 \[ \int \frac {1}{(1+x) \sqrt {1+x+x^2}} \, dx=-\log \left (-x + \sqrt {x^{2} + x + 1}\right ) + \log \left (-x + \sqrt {x^{2} + x + 1} - 2\right ) \]
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\[ \int \frac {1}{(1+x) \sqrt {1+x+x^2}} \, dx=\int \frac {1}{\left (x + 1\right ) \sqrt {x^{2} + x + 1}}\, dx \]
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none
Time = 0.27 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.14 \[ \int \frac {1}{(1+x) \sqrt {1+x+x^2}} \, dx=\operatorname {arsinh}\left (\frac {\sqrt {3} x}{3 \, {\left | x + 1 \right |}} - \frac {\sqrt {3}}{3 \, {\left | x + 1 \right |}}\right ) \]
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none
Time = 0.31 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.45 \[ \int \frac {1}{(1+x) \sqrt {1+x+x^2}} \, dx=-\log \left ({\left | -x + \sqrt {x^{2} + x + 1} \right |}\right ) + \log \left ({\left | -x + \sqrt {x^{2} + x + 1} - 2 \right |}\right ) \]
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Timed out. \[ \int \frac {1}{(1+x) \sqrt {1+x+x^2}} \, dx=\int \frac {1}{\left (x+1\right )\,\sqrt {x^2+x+1}} \,d x \]
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