Integrand size = 14, antiderivative size = 79 \[ \int \frac {1}{x^3 \left (1+x+x^2\right )^{3/2}} \, dx=\frac {2 (1-x)}{3 x^2 \sqrt {1+x+x^2}}-\frac {7 \sqrt {1+x+x^2}}{6 x^2}+\frac {37 \sqrt {1+x+x^2}}{12 x}-\frac {3}{8} \text {arctanh}\left (\frac {2+x}{2 \sqrt {1+x+x^2}}\right ) \]
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Time = 0.04 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {754, 848, 820, 738, 212} \[ \int \frac {1}{x^3 \left (1+x+x^2\right )^{3/2}} \, dx=-\frac {3}{8} \text {arctanh}\left (\frac {x+2}{2 \sqrt {x^2+x+1}}\right )+\frac {2 (1-x)}{3 x^2 \sqrt {x^2+x+1}}+\frac {37 \sqrt {x^2+x+1}}{12 x}-\frac {7 \sqrt {x^2+x+1}}{6 x^2} \]
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Rule 212
Rule 738
Rule 754
Rule 820
Rule 848
Rubi steps \begin{align*} \text {integral}& = \frac {2 (1-x)}{3 x^2 \sqrt {1+x+x^2}}+\frac {2}{3} \int \frac {\frac {7}{2}-2 x}{x^3 \sqrt {1+x+x^2}} \, dx \\ & = \frac {2 (1-x)}{3 x^2 \sqrt {1+x+x^2}}-\frac {7 \sqrt {1+x+x^2}}{6 x^2}-\frac {1}{3} \int \frac {\frac {37}{4}+\frac {7 x}{2}}{x^2 \sqrt {1+x+x^2}} \, dx \\ & = \frac {2 (1-x)}{3 x^2 \sqrt {1+x+x^2}}-\frac {7 \sqrt {1+x+x^2}}{6 x^2}+\frac {37 \sqrt {1+x+x^2}}{12 x}+\frac {3}{8} \int \frac {1}{x \sqrt {1+x+x^2}} \, dx \\ & = \frac {2 (1-x)}{3 x^2 \sqrt {1+x+x^2}}-\frac {7 \sqrt {1+x+x^2}}{6 x^2}+\frac {37 \sqrt {1+x+x^2}}{12 x}-\frac {3}{4} \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^2 \sqrt {1+x+x^2}}-\frac {7 \sqrt {1+x+x^2}}{6 x^2}+\frac {37 \sqrt {1+x+x^2}}{12 x}-\frac {3}{8} \text {arctanh}\left (\frac {2+x}{2 \sqrt {1+x+x^2}}\right ) \\ \end{align*}
Time = 0.20 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.66 \[ \int \frac {1}{x^3 \left (1+x+x^2\right )^{3/2}} \, dx=\frac {-6+15 x+23 x^2+37 x^3}{12 x^2 \sqrt {1+x+x^2}}+\frac {3}{4} \text {arctanh}\left (x-\sqrt {1+x+x^2}\right ) \]
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Time = 0.52 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.58
method | result | size |
risch | \(\frac {37 x^{3}+23 x^{2}+15 x -6}{12 \sqrt {x^{2}+x +1}\, x^{2}}-\frac {3 \,\operatorname {arctanh}\left (\frac {2+x}{2 \sqrt {x^{2}+x +1}}\right )}{8}\) | \(46\) |
trager | \(\frac {37 x^{3}+23 x^{2}+15 x -6}{12 \sqrt {x^{2}+x +1}\, x^{2}}-\frac {3 \ln \left (\frac {2 \sqrt {x^{2}+x +1}+2+x}{x}\right )}{8}\) | \(50\) |
default | \(-\frac {1}{2 x^{2} \sqrt {x^{2}+x +1}}+\frac {5}{4 x \sqrt {x^{2}+x +1}}+\frac {3}{8 \sqrt {x^{2}+x +1}}+\frac {\frac {37}{24}+\frac {37 x}{12}}{\sqrt {x^{2}+x +1}}-\frac {3 \,\operatorname {arctanh}\left (\frac {2+x}{2 \sqrt {x^{2}+x +1}}\right )}{8}\) | \(69\) |
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Time = 0.24 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.35 \[ \int \frac {1}{x^3 \left (1+x+x^2\right )^{3/2}} \, dx=\frac {74 \, x^{4} + 74 \, x^{3} + 74 \, x^{2} - 9 \, {\left (x^{4} + x^{3} + x^{2}\right )} \log \left (-x + \sqrt {x^{2} + x + 1} + 1\right ) + 9 \, {\left (x^{4} + x^{3} + x^{2}\right )} \log \left (-x + \sqrt {x^{2} + x + 1} - 1\right ) + 2 \, {\left (37 \, x^{3} + 23 \, x^{2} + 15 \, x - 6\right )} \sqrt {x^{2} + x + 1}}{24 \, {\left (x^{4} + x^{3} + x^{2}\right )}} \]
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\[ \int \frac {1}{x^3 \left (1+x+x^2\right )^{3/2}} \, dx=\int \frac {1}{x^{3} \left (x^{2} + x + 1\right )^{\frac {3}{2}}}\, dx \]
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Time = 0.28 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.90 \[ \int \frac {1}{x^3 \left (1+x+x^2\right )^{3/2}} \, dx=\frac {37 \, x}{12 \, \sqrt {x^{2} + x + 1}} + \frac {23}{12 \, \sqrt {x^{2} + x + 1}} + \frac {5}{4 \, \sqrt {x^{2} + x + 1} x} - \frac {1}{2 \, \sqrt {x^{2} + x + 1} x^{2}} - \frac {3}{8} \, \operatorname {arsinh}\left (\frac {\sqrt {3} x}{3 \, {\left | x \right |}} + \frac {2 \, \sqrt {3}}{3 \, {\left | x \right |}}\right ) \]
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Time = 0.29 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.48 \[ \int \frac {1}{x^3 \left (1+x+x^2\right )^{3/2}} \, dx=\frac {2 \, {\left (2 \, x + 1\right )}}{3 \, \sqrt {x^{2} + x + 1}} - \frac {3 \, {\left (x - \sqrt {x^{2} + x + 1}\right )}^{3} + 8 \, {\left (x - \sqrt {x^{2} + x + 1}\right )}^{2} - 13 \, x + 13 \, \sqrt {x^{2} + x + 1} - 16}{4 \, {\left ({\left (x - \sqrt {x^{2} + x + 1}\right )}^{2} - 1\right )}^{2}} - \frac {3}{8} \, \log \left ({\left | -x + \sqrt {x^{2} + x + 1} + 1 \right |}\right ) + \frac {3}{8} \, \log \left ({\left | -x + \sqrt {x^{2} + x + 1} - 1 \right |}\right ) \]
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Timed out. \[ \int \frac {1}{x^3 \left (1+x+x^2\right )^{3/2}} \, dx=\int \frac {1}{x^3\,{\left (x^2+x+1\right )}^{3/2}} \,d x \]
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