Integrand size = 14, antiderivative size = 62 \[ \int \frac {1}{x^2 \left (1+x+x^2\right )^{3/2}} \, dx=\frac {2 (1-x)}{3 x \sqrt {1+x+x^2}}-\frac {5 \sqrt {1+x+x^2}}{3 x}+\frac {3}{2} \text {arctanh}\left (\frac {2+x}{2 \sqrt {1+x+x^2}}\right ) \]
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Time = 0.02 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {754, 820, 738, 212} \[ \int \frac {1}{x^2 \left (1+x+x^2\right )^{3/2}} \, dx=\frac {3}{2} \text {arctanh}\left (\frac {x+2}{2 \sqrt {x^2+x+1}}\right )+\frac {2 (1-x)}{3 x \sqrt {x^2+x+1}}-\frac {5 \sqrt {x^2+x+1}}{3 x} \]
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Rule 212
Rule 738
Rule 754
Rule 820
Rubi steps \begin{align*} \text {integral}& = \frac {2 (1-x)}{3 x \sqrt {1+x+x^2}}+\frac {2}{3} \int \frac {\frac {5}{2}-x}{x^2 \sqrt {1+x+x^2}} \, dx \\ & = \frac {2 (1-x)}{3 x \sqrt {1+x+x^2}}-\frac {5 \sqrt {1+x+x^2}}{3 x}-\frac {3}{2} \int \frac {1}{x \sqrt {1+x+x^2}} \, dx \\ & = \frac {2 (1-x)}{3 x \sqrt {1+x+x^2}}-\frac {5 \sqrt {1+x+x^2}}{3 x}+3 \text {Subst}\left (\int \frac {1}{4-x^2} \, dx,x,\frac {2+x}{\sqrt {1+x+x^2}}\right ) \\ & = \frac {2 (1-x)}{3 x \sqrt {1+x+x^2}}-\frac {5 \sqrt {1+x+x^2}}{3 x}+\frac {3}{2} \text {arctanh}\left (\frac {2+x}{2 \sqrt {1+x+x^2}}\right ) \\ \end{align*}
Time = 0.19 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.73 \[ \int \frac {1}{x^2 \left (1+x+x^2\right )^{3/2}} \, dx=\frac {-3-7 x-5 x^2}{3 x \sqrt {1+x+x^2}}-3 \text {arctanh}\left (x-\sqrt {1+x+x^2}\right ) \]
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Time = 0.44 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.66
method | result | size |
risch | \(-\frac {5 x^{2}+7 x +3}{3 x \sqrt {x^{2}+x +1}}+\frac {3 \,\operatorname {arctanh}\left (\frac {2+x}{2 \sqrt {x^{2}+x +1}}\right )}{2}\) | \(41\) |
trager | \(-\frac {5 x^{2}+7 x +3}{3 x \sqrt {x^{2}+x +1}}-\frac {3 \ln \left (\frac {-2-x +2 \sqrt {x^{2}+x +1}}{x}\right )}{2}\) | \(47\) |
default | \(-\frac {1}{x \sqrt {x^{2}+x +1}}-\frac {3}{2 \sqrt {x^{2}+x +1}}-\frac {5 \left (1+2 x \right )}{6 \sqrt {x^{2}+x +1}}+\frac {3 \,\operatorname {arctanh}\left (\frac {2+x}{2 \sqrt {x^{2}+x +1}}\right )}{2}\) | \(56\) |
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Leaf count of result is larger than twice the leaf count of optimal. 94 vs. \(2 (46) = 92\).
Time = 0.25 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.52 \[ \int \frac {1}{x^2 \left (1+x+x^2\right )^{3/2}} \, dx=-\frac {10 \, x^{3} + 10 \, x^{2} - 9 \, {\left (x^{3} + x^{2} + x\right )} \log \left (-x + \sqrt {x^{2} + x + 1} + 1\right ) + 9 \, {\left (x^{3} + x^{2} + x\right )} \log \left (-x + \sqrt {x^{2} + x + 1} - 1\right ) + 2 \, {\left (5 \, x^{2} + 7 \, x + 3\right )} \sqrt {x^{2} + x + 1} + 10 \, x}{6 \, {\left (x^{3} + x^{2} + x\right )}} \]
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\[ \int \frac {1}{x^2 \left (1+x+x^2\right )^{3/2}} \, dx=\int \frac {1}{x^{2} \left (x^{2} + x + 1\right )^{\frac {3}{2}}}\, dx \]
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none
Time = 0.27 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.94 \[ \int \frac {1}{x^2 \left (1+x+x^2\right )^{3/2}} \, dx=-\frac {5 \, x}{3 \, \sqrt {x^{2} + x + 1}} - \frac {7}{3 \, \sqrt {x^{2} + x + 1}} - \frac {1}{\sqrt {x^{2} + x + 1} x} + \frac {3}{2} \, \operatorname {arsinh}\left (\frac {\sqrt {3} x}{3 \, {\left | x \right |}} + \frac {2 \, \sqrt {3}}{3 \, {\left | x \right |}}\right ) \]
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none
Time = 0.30 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.29 \[ \int \frac {1}{x^2 \left (1+x+x^2\right )^{3/2}} \, dx=-\frac {2 \, {\left (x + 2\right )}}{3 \, \sqrt {x^{2} + x + 1}} + \frac {x - \sqrt {x^{2} + x + 1} + 2}{{\left (x - \sqrt {x^{2} + x + 1}\right )}^{2} - 1} + \frac {3}{2} \, \log \left ({\left | -x + \sqrt {x^{2} + x + 1} + 1 \right |}\right ) - \frac {3}{2} \, \log \left ({\left | -x + \sqrt {x^{2} + x + 1} - 1 \right |}\right ) \]
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Timed out. \[ \int \frac {1}{x^2 \left (1+x+x^2\right )^{3/2}} \, dx=\int \frac {1}{x^2\,{\left (x^2+x+1\right )}^{3/2}} \,d x \]
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