Integrand size = 14, antiderivative size = 38 \[ \int \frac {1}{x^2 \sqrt {1+x+x^2}} \, dx=-\frac {\sqrt {1+x+x^2}}{x}+\frac {1}{2} \text {arctanh}\left (\frac {2+x}{2 \sqrt {1+x+x^2}}\right ) \]
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Time = 0.01 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {744, 738, 212} \[ \int \frac {1}{x^2 \sqrt {1+x+x^2}} \, dx=\frac {1}{2} \text {arctanh}\left (\frac {x+2}{2 \sqrt {x^2+x+1}}\right )-\frac {\sqrt {x^2+x+1}}{x} \]
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Rule 212
Rule 738
Rule 744
Rubi steps \begin{align*} \text {integral}& = -\frac {\sqrt {1+x+x^2}}{x}-\frac {1}{2} \int \frac {1}{x \sqrt {1+x+x^2}} \, dx \\ & = -\frac {\sqrt {1+x+x^2}}{x}+\text {Subst}\left (\int \frac {1}{4-x^2} \, dx,x,\frac {2+x}{\sqrt {1+x+x^2}}\right ) \\ & = -\frac {\sqrt {1+x+x^2}}{x}+\frac {1}{2} \text {arctanh}\left (\frac {2+x}{2 \sqrt {1+x+x^2}}\right ) \\ \end{align*}
Time = 0.11 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.87 \[ \int \frac {1}{x^2 \sqrt {1+x+x^2}} \, dx=-\frac {\sqrt {1+x+x^2}}{x}-\text {arctanh}\left (x-\sqrt {1+x+x^2}\right ) \]
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Time = 0.43 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.82
method | result | size |
default | \(\frac {\operatorname {arctanh}\left (\frac {2+x}{2 \sqrt {x^{2}+x +1}}\right )}{2}-\frac {\sqrt {x^{2}+x +1}}{x}\) | \(31\) |
risch | \(\frac {\operatorname {arctanh}\left (\frac {2+x}{2 \sqrt {x^{2}+x +1}}\right )}{2}-\frac {\sqrt {x^{2}+x +1}}{x}\) | \(31\) |
trager | \(-\frac {\sqrt {x^{2}+x +1}}{x}+\frac {\ln \left (\frac {2 \sqrt {x^{2}+x +1}+2+x}{x}\right )}{2}\) | \(35\) |
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Time = 0.24 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.37 \[ \int \frac {1}{x^2 \sqrt {1+x+x^2}} \, dx=\frac {x \log \left (-x + \sqrt {x^{2} + x + 1} + 1\right ) - x \log \left (-x + \sqrt {x^{2} + x + 1} - 1\right ) - 2 \, x - 2 \, \sqrt {x^{2} + x + 1}}{2 \, x} \]
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\[ \int \frac {1}{x^2 \sqrt {1+x+x^2}} \, dx=\int \frac {1}{x^{2} \sqrt {x^{2} + x + 1}}\, dx \]
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Time = 0.28 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.97 \[ \int \frac {1}{x^2 \sqrt {1+x+x^2}} \, dx=-\frac {\sqrt {x^{2} + x + 1}}{x} + \frac {1}{2} \, \operatorname {arsinh}\left (\frac {\sqrt {3} x}{3 \, {\left | x \right |}} + \frac {2 \, \sqrt {3}}{3 \, {\left | x \right |}}\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 67 vs. \(2 (30) = 60\).
Time = 0.29 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.76 \[ \int \frac {1}{x^2 \sqrt {1+x+x^2}} \, dx=\frac {x - \sqrt {x^{2} + x + 1} + 2}{{\left (x - \sqrt {x^{2} + x + 1}\right )}^{2} - 1} + \frac {1}{2} \, \log \left ({\left | -x + \sqrt {x^{2} + x + 1} + 1 \right |}\right ) - \frac {1}{2} \, \log \left ({\left | -x + \sqrt {x^{2} + x + 1} - 1 \right |}\right ) \]
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Time = 0.03 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.82 \[ \int \frac {1}{x^2 \sqrt {1+x+x^2}} \, dx=\frac {\mathrm {atanh}\left (\frac {\frac {x}{2}+1}{\sqrt {x^2+x+1}}\right )}{2}-\frac {\sqrt {x^2+x+1}}{x} \]
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