Integrand size = 13, antiderivative size = 49 \[ \int \frac {1}{x^6 \sqrt {2+x^2}} \, dx=-\frac {\sqrt {2+x^2}}{10 x^5}+\frac {\sqrt {2+x^2}}{15 x^3}-\frac {\sqrt {2+x^2}}{15 x} \]
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Time = 0.01 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {277, 270} \[ \int \frac {1}{x^6 \sqrt {2+x^2}} \, dx=-\frac {\sqrt {x^2+2}}{15 x}-\frac {\sqrt {x^2+2}}{10 x^5}+\frac {\sqrt {x^2+2}}{15 x^3} \]
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Rule 270
Rule 277
Rubi steps \begin{align*} \text {integral}& = -\frac {\sqrt {2+x^2}}{10 x^5}-\frac {2}{5} \int \frac {1}{x^4 \sqrt {2+x^2}} \, dx \\ & = -\frac {\sqrt {2+x^2}}{10 x^5}+\frac {\sqrt {2+x^2}}{15 x^3}+\frac {2}{15} \int \frac {1}{x^2 \sqrt {2+x^2}} \, dx \\ & = -\frac {\sqrt {2+x^2}}{10 x^5}+\frac {\sqrt {2+x^2}}{15 x^3}-\frac {\sqrt {2+x^2}}{15 x} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.57 \[ \int \frac {1}{x^6 \sqrt {2+x^2}} \, dx=\frac {\sqrt {2+x^2} \left (-3+2 x^2-2 x^4\right )}{30 x^5} \]
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Time = 0.23 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.51
method | result | size |
gosper | \(-\frac {\sqrt {x^{2}+2}\, \left (2 x^{4}-2 x^{2}+3\right )}{30 x^{5}}\) | \(25\) |
trager | \(-\frac {\sqrt {x^{2}+2}\, \left (2 x^{4}-2 x^{2}+3\right )}{30 x^{5}}\) | \(25\) |
pseudoelliptic | \(-\frac {\sqrt {x^{2}+2}\, \left (2 x^{4}-2 x^{2}+3\right )}{30 x^{5}}\) | \(25\) |
meijerg | \(-\frac {\sqrt {2}\, \left (\frac {2}{3} x^{4}-\frac {2}{3} x^{2}+1\right ) \sqrt {1+\frac {x^{2}}{2}}}{10 x^{5}}\) | \(30\) |
risch | \(-\frac {2 x^{6}+2 x^{4}-x^{2}+6}{30 x^{5} \sqrt {x^{2}+2}}\) | \(30\) |
default | \(-\frac {\sqrt {x^{2}+2}}{10 x^{5}}+\frac {\sqrt {x^{2}+2}}{15 x^{3}}-\frac {\sqrt {x^{2}+2}}{15 x}\) | \(38\) |
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none
Time = 0.26 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.63 \[ \int \frac {1}{x^6 \sqrt {2+x^2}} \, dx=-\frac {2 \, x^{5} + {\left (2 \, x^{4} - 2 \, x^{2} + 3\right )} \sqrt {x^{2} + 2}}{30 \, x^{5}} \]
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Time = 1.07 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.84 \[ \int \frac {1}{x^6 \sqrt {2+x^2}} \, dx=- \frac {\sqrt {1 + \frac {2}{x^{2}}}}{15} + \frac {\sqrt {1 + \frac {2}{x^{2}}}}{15 x^{2}} - \frac {\sqrt {1 + \frac {2}{x^{2}}}}{10 x^{4}} \]
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none
Time = 0.28 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.76 \[ \int \frac {1}{x^6 \sqrt {2+x^2}} \, dx=-\frac {\sqrt {x^{2} + 2}}{15 \, x} + \frac {\sqrt {x^{2} + 2}}{15 \, x^{3}} - \frac {\sqrt {x^{2} + 2}}{10 \, x^{5}} \]
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none
Time = 0.29 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.04 \[ \int \frac {1}{x^6 \sqrt {2+x^2}} \, dx=\frac {32 \, {\left (5 \, {\left (x - \sqrt {x^{2} + 2}\right )}^{4} - 5 \, {\left (x - \sqrt {x^{2} + 2}\right )}^{2} + 2\right )}}{15 \, {\left ({\left (x - \sqrt {x^{2} + 2}\right )}^{2} - 2\right )}^{5}} \]
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Time = 0.03 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.51 \[ \int \frac {1}{x^6 \sqrt {2+x^2}} \, dx=-\sqrt {x^2+2}\,\left (\frac {1}{15\,x}-\frac {1}{15\,x^3}+\frac {1}{10\,x^5}\right ) \]
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