Integrand size = 13, antiderivative size = 47 \[ \int \frac {1}{x^4 \left (-8+x^2\right )^{3/2}} \, dx=\frac {1}{24 x^3 \sqrt {-8+x^2}}+\frac {1}{48 x \sqrt {-8+x^2}}-\frac {x}{192 \sqrt {-8+x^2}} \]
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Time = 0.01 (sec) , antiderivative size = 47, 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, 197} \[ \int \frac {1}{x^4 \left (-8+x^2\right )^{3/2}} \, dx=-\frac {x}{192 \sqrt {x^2-8}}+\frac {1}{48 \sqrt {x^2-8} x}+\frac {1}{24 \sqrt {x^2-8} x^3} \]
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Rule 197
Rule 277
Rubi steps \begin{align*} \text {integral}& = \frac {1}{24 x^3 \sqrt {-8+x^2}}+\frac {1}{6} \int \frac {1}{x^2 \left (-8+x^2\right )^{3/2}} \, dx \\ & = \frac {1}{24 x^3 \sqrt {-8+x^2}}+\frac {1}{48 x \sqrt {-8+x^2}}+\frac {1}{24} \int \frac {1}{\left (-8+x^2\right )^{3/2}} \, dx \\ & = \frac {1}{24 x^3 \sqrt {-8+x^2}}+\frac {1}{48 x \sqrt {-8+x^2}}-\frac {x}{192 \sqrt {-8+x^2}} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.60 \[ \int \frac {1}{x^4 \left (-8+x^2\right )^{3/2}} \, dx=\frac {8+4 x^2-x^4}{192 x^3 \sqrt {-8+x^2}} \]
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Time = 0.29 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.49
method | result | size |
gosper | \(-\frac {x^{4}-4 x^{2}-8}{192 x^{3} \sqrt {x^{2}-8}}\) | \(23\) |
trager | \(-\frac {x^{4}-4 x^{2}-8}{192 x^{3} \sqrt {x^{2}-8}}\) | \(23\) |
risch | \(-\frac {x^{4}-4 x^{2}-8}{192 x^{3} \sqrt {x^{2}-8}}\) | \(23\) |
pseudoelliptic | \(-\frac {x^{4}-4 x^{2}-8}{192 x^{3} \sqrt {x^{2}-8}}\) | \(23\) |
default | \(\frac {1}{24 x^{3} \sqrt {x^{2}-8}}+\frac {1}{48 x \sqrt {x^{2}-8}}-\frac {x}{192 \sqrt {x^{2}-8}}\) | \(36\) |
meijerg | \(-\frac {\sqrt {2}\, {\left (-\operatorname {signum}\left (-1+\frac {x^{2}}{8}\right )\right )}^{\frac {3}{2}} \left (-\frac {1}{8} x^{4}+\frac {1}{2} x^{2}+1\right )}{96 \operatorname {signum}\left (-1+\frac {x^{2}}{8}\right )^{\frac {3}{2}} x^{3} \sqrt {1-\frac {x^{2}}{8}}}\) | \(52\) |
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none
Time = 0.25 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.85 \[ \int \frac {1}{x^4 \left (-8+x^2\right )^{3/2}} \, dx=-\frac {x^{5} - 8 \, x^{3} + {\left (x^{4} - 4 \, x^{2} - 8\right )} \sqrt {x^{2} - 8}}{192 \, {\left (x^{5} - 8 \, x^{3}\right )}} \]
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Result contains complex when optimal does not.
Time = 1.18 (sec) , antiderivative size = 153, normalized size of antiderivative = 3.26 \[ \int \frac {1}{x^4 \left (-8+x^2\right )^{3/2}} \, dx=\begin {cases} - \frac {i x^{4} \sqrt {-1 + \frac {8}{x^{2}}}}{192 x^{4} - 1536 x^{2}} + \frac {4 i x^{2} \sqrt {-1 + \frac {8}{x^{2}}}}{192 x^{4} - 1536 x^{2}} + \frac {8 i \sqrt {-1 + \frac {8}{x^{2}}}}{192 x^{4} - 1536 x^{2}} & \text {for}\: \frac {1}{\left |{x^{2}}\right |} > \frac {1}{8} \\- \frac {x^{4} \sqrt {1 - \frac {8}{x^{2}}}}{192 x^{4} - 1536 x^{2}} + \frac {4 x^{2} \sqrt {1 - \frac {8}{x^{2}}}}{192 x^{4} - 1536 x^{2}} + \frac {8 \sqrt {1 - \frac {8}{x^{2}}}}{192 x^{4} - 1536 x^{2}} & \text {otherwise} \end {cases} \]
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Time = 0.28 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.74 \[ \int \frac {1}{x^4 \left (-8+x^2\right )^{3/2}} \, dx=-\frac {x}{192 \, \sqrt {x^{2} - 8}} + \frac {1}{48 \, \sqrt {x^{2} - 8} x} + \frac {1}{24 \, \sqrt {x^{2} - 8} x^{3}} \]
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Time = 0.28 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.32 \[ \int \frac {1}{x^4 \left (-8+x^2\right )^{3/2}} \, dx=-\frac {x}{512 \, \sqrt {x^{2} - 8}} - \frac {3 \, {\left (x - \sqrt {x^{2} - 8}\right )}^{4} + 96 \, {\left (x - \sqrt {x^{2} - 8}\right )}^{2} + 320}{96 \, {\left ({\left (x - \sqrt {x^{2} - 8}\right )}^{2} + 8\right )}^{3}} \]
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Time = 0.46 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.51 \[ \int \frac {1}{x^4 \left (-8+x^2\right )^{3/2}} \, dx=\frac {-x^4+4\,x^2+8}{192\,x^3\,\sqrt {x^2-8}} \]
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