Optimal. Leaf size=47 \[ \frac {1}{24 x^3 \sqrt {-8+x^2}}+\frac {1}{48 x \sqrt {-8+x^2}}-\frac {x}{192 \sqrt {-8+x^2}} \]
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Rubi [A]
time = 0.01, antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {277, 197}
\begin {gather*} -\frac {x}{192 \sqrt {x^2-8}}+\frac {1}{48 \sqrt {x^2-8} x}+\frac {1}{24 \sqrt {x^2-8} x^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 197
Rule 277
Rubi steps
\begin {align*} \int \frac {1}{x^4 \left (-8+x^2\right )^{3/2}} \, dx &=\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*}
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Mathematica [A]
time = 0.03, size = 28, normalized size = 0.60 \begin {gather*} \frac {8+4 x^2-x^4}{192 x^3 \sqrt {-8+x^2}} \end {gather*}
Antiderivative was successfully verified.
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Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 2 in
optimal.
time = 3.85, size = 127, normalized size = 2.70 \begin {gather*} \text {Piecewise}\left [\left \{\left \{\frac {I \left (8+4 x^2-x^4\right ) \sqrt {\frac {8-x^2}{x^2}}}{192 x^2 \left (-8+x^2\right )},\frac {1}{\text {Abs}\left [x^2\right ]}>\frac {1}{8}\right \}\right \},\frac {4 x^2 \sqrt {1-\frac {8}{x^2}}}{-1536 x^2+192 x^4}-\frac {x^4 \sqrt {1-\frac {8}{x^2}}}{-1536 x^2+192 x^4}+\frac {8 \sqrt {1-\frac {8}{x^2}}}{-1536 x^2+192 x^4}\right ] \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.08, size = 36, normalized size = 0.77
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 | \(-\frac {x^{4}-4 x^{2}-8}{192 x^{3} \sqrt {x^{2}-8}}\) | \(23\) |
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 | \(-\frac {\sqrt {2}\, \left (-\mathrm {signum}\left (-1+\frac {x^{2}}{8}\right )\right )^{\frac {3}{2}} \left (-\frac {1}{8} x^{4}+\frac {1}{2} x^{2}+1\right )}{96 \mathrm {signum}\left (-1+\frac {x^{2}}{8}\right )^{\frac {3}{2}} x^{3} \sqrt {1-\frac {x^{2}}{8}}}\) | \(52\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.33, size = 35, normalized size = 0.74 \begin {gather*} -\frac {x}{192 \, \sqrt {x^{2} - 8}} + \frac {1}{48 \, \sqrt {x^{2} - 8} x} + \frac {1}{24 \, \sqrt {x^{2} - 8} x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.31, size = 40, normalized size = 0.85 \begin {gather*} -\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 )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 1.72, size = 153, normalized size = 3.26 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.00, size = 81, normalized size = 1.72 \begin {gather*} -\frac {2 x \sqrt {x^{2}-8}}{1024 \left (x^{2}-8\right )}+\frac {2 \left (-3 \left (\sqrt {x^{2}-8}-x\right )^{4}-96 \left (\sqrt {x^{2}-8}-x\right )^{2}-320\right )}{192 \left (\left (\sqrt {x^{2}-8}-x\right )^{2}+8\right )^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.45, size = 24, normalized size = 0.51 \begin {gather*} \frac {-x^4+4\,x^2+8}{192\,x^3\,\sqrt {x^2-8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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