Optimal. Leaf size=52 \[ -\frac {1}{6 \left (-2+x^2\right )^{3/2}}+\frac {1}{4 \sqrt {-2+x^2}}+\frac {\tan ^{-1}\left (\frac {\sqrt {-2+x^2}}{\sqrt {2}}\right )}{4 \sqrt {2}} \]
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Rubi [A]
time = 0.01, antiderivative size = 52, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {272, 53, 65,
209} \begin {gather*} \frac {1}{4 \sqrt {x^2-2}}-\frac {1}{6 \left (x^2-2\right )^{3/2}}+\frac {\tan ^{-1}\left (\frac {\sqrt {x^2-2}}{\sqrt {2}}\right )}{4 \sqrt {2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 53
Rule 65
Rule 209
Rule 272
Rubi steps
\begin {align*} \int \frac {1}{x \left (-2+x^2\right )^{5/2}} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {1}{(-2+x)^{5/2} x} \, dx,x,x^2\right )\\ &=-\frac {1}{6 \left (-2+x^2\right )^{3/2}}-\frac {1}{4} \text {Subst}\left (\int \frac {1}{(-2+x)^{3/2} x} \, dx,x,x^2\right )\\ &=-\frac {1}{6 \left (-2+x^2\right )^{3/2}}+\frac {1}{4 \sqrt {-2+x^2}}+\frac {1}{8} \text {Subst}\left (\int \frac {1}{\sqrt {-2+x} x} \, dx,x,x^2\right )\\ &=-\frac {1}{6 \left (-2+x^2\right )^{3/2}}+\frac {1}{4 \sqrt {-2+x^2}}+\frac {1}{4} \text {Subst}\left (\int \frac {1}{2+x^2} \, dx,x,\sqrt {-2+x^2}\right )\\ &=-\frac {1}{6 \left (-2+x^2\right )^{3/2}}+\frac {1}{4 \sqrt {-2+x^2}}+\frac {\tan ^{-1}\left (\frac {\sqrt {-2+x^2}}{\sqrt {2}}\right )}{4 \sqrt {2}}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 46, normalized size = 0.88 \begin {gather*} \frac {-8+3 x^2}{12 \left (-2+x^2\right )^{3/2}}+\frac {\tan ^{-1}\left (\frac {\sqrt {-2+x^2}}{\sqrt {2}}\right )}{4 \sqrt {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 3 in
optimal.
time = 6.71, size = 613, normalized size = 11.79 \begin {gather*} \text {Piecewise}\left [\left \{\left \{\frac {12 I \sqrt {2} x^2 \text {Log}\left [x^2\right ]+12 x^2 \sqrt {-2+x^2}-3 I \sqrt {2} x^4 \text {Log}\left [x^2\right ]+6 \sqrt {2} x^4 \left (-\text {ArcSin}\left [\frac {\sqrt {2}}{x}\right ]+I \text {Log}\left [x\right ]\right )-32 \sqrt {-2+x^2}-12 I \sqrt {2} \text {Log}\left [x^2\right ]+24 \sqrt {2} \left (-I x^2 \text {Log}\left [x\right ]+x^2 \text {ArcSin}\left [\frac {\sqrt {2}}{x}\right ]-\text {ArcSin}\left [\frac {\sqrt {2}}{x}\right ]+I \text {Log}\left [x\right ]\right )}{48 \left (4-4 x^2+x^4\right )},\text {Abs}\left [x^2\right ]>2\right \}\right \},\frac {-24 I x^2 \text {Log}\left [1+\sqrt {1-\frac {x^2}{2}}\right ]}{96 \sqrt {2}-96 \sqrt {2} x^2+24 \sqrt {2} x^4}-\frac {12 I x^2 \text {Log}\left [2\right ]}{96 \sqrt {2}-96 \sqrt {2} x^2+24 \sqrt {2} x^4}+\frac {I 6 \sqrt {2} x^2 \sqrt {2-x^2}}{96 \sqrt {2}-96 \sqrt {2} x^2+24 \sqrt {2} x^4}+\frac {I 12 x^2 \text {Log}\left [x^2\right ]}{96 \sqrt {2}-96 \sqrt {2} x^2+24 \sqrt {2} x^4}-\frac {3 I x^4 \text {Log}\left [x^2\right ]}{96 \sqrt {2}-96 \sqrt {2} x^2+24 \sqrt {2} x^4}+\frac {I 3 x^4 \text {Log}\left [2\right ]}{96 \sqrt {2}-96 \sqrt {2} x^2+24 \sqrt {2} x^4}+\frac {I 6 x^4 \text {Log}\left [1+\sqrt {1-\frac {x^2}{2}}\right ]}{96 \sqrt {2}-96 \sqrt {2} x^2+24 \sqrt {2} x^4}-\frac {12 \text {Pi}}{96 \sqrt {2}-96 \sqrt {2} x^2+24 \sqrt {2} x^4}+\frac {12 \text {Pi} x^2}{96 \sqrt {2}-96 \sqrt {2} x^2+24 \sqrt {2} x^4}-\frac {3 \text {Pi} x^4}{96 \sqrt {2}-96 \sqrt {2} x^2+24 \sqrt {2} x^4}-\frac {16 I \sqrt {2} \sqrt {2-x^2}}{96 \sqrt {2}-96 \sqrt {2} x^2+24 \sqrt {2} x^4}-\frac {12 I \text {Log}\left [x^2\right ]}{96 \sqrt {2}-96 \sqrt {2} x^2+24 \sqrt {2} x^4}+\frac {I 12 \text {Log}\left [2\right ]}{96 \sqrt {2}-96 \sqrt {2} x^2+24 \sqrt {2} x^4}+\frac {I 24 \text {Log}\left [1+\sqrt {1-\frac {x^2}{2}}\right ]}{96 \sqrt {2}-96 \sqrt {2} x^2+24 \sqrt {2} x^4}\right ] \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.11, size = 37, normalized size = 0.71
method | result | size |
risch | \(\frac {3 x^{2}-8}{12 \left (x^{2}-2\right )^{\frac {3}{2}}}-\frac {\sqrt {2}\, \arctan \left (\frac {\sqrt {2}}{\sqrt {x^{2}-2}}\right )}{8}\) | \(35\) |
default | \(-\frac {1}{6 \left (x^{2}-2\right )^{\frac {3}{2}}}+\frac {1}{4 \sqrt {x^{2}-2}}-\frac {\sqrt {2}\, \arctan \left (\frac {\sqrt {2}}{\sqrt {x^{2}-2}}\right )}{8}\) | \(37\) |
trager | \(\frac {3 x^{2}-8}{12 \left (x^{2}-2\right )^{\frac {3}{2}}}-\frac {\RootOf \left (\textit {\_Z}^{2}+2\right ) \ln \left (\frac {\sqrt {x^{2}-2}-\RootOf \left (\textit {\_Z}^{2}+2\right )}{x}\right )}{8}\) | \(47\) |
meijerg | \(\frac {\sqrt {2}\, \left (-\mathrm {signum}\left (-1+\frac {x^{2}}{2}\right )\right )^{\frac {5}{2}} \left (\frac {3 \left (\frac {8}{3}-3 \ln \left (2\right )+2 \ln \left (x \right )+i \pi \right ) \sqrt {\pi }}{4}-2 \sqrt {\pi }+\frac {\sqrt {\pi }\, \left (-6 x^{2}+16\right )}{8 \left (-\frac {x^{2}}{2}+1\right )^{\frac {3}{2}}}-\frac {3 \sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {-\frac {x^{2}}{2}+1}}{2}\right )}{2}\right )}{12 \sqrt {\pi }\, \mathrm {signum}\left (-1+\frac {x^{2}}{2}\right )^{\frac {5}{2}}}\) | \(96\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.33, size = 33, normalized size = 0.63 \begin {gather*} -\frac {1}{8} \, \sqrt {2} \arcsin \left (\frac {\sqrt {2}}{{\left | x \right |}}\right ) + \frac {1}{4 \, \sqrt {x^{2} - 2}} - \frac {1}{6 \, {\left (x^{2} - 2\right )}^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.31, size = 65, normalized size = 1.25 \begin {gather*} \frac {3 \, \sqrt {2} {\left (x^{4} - 4 \, x^{2} + 4\right )} \arctan \left (-\frac {1}{2} \, \sqrt {2} x + \frac {1}{2} \, \sqrt {2} \sqrt {x^{2} - 2}\right ) + {\left (3 \, x^{2} - 8\right )} \sqrt {x^{2} - 2}}{12 \, {\left (x^{4} - 4 \, x^{2} + 4\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 2.09, size = 984, normalized size = 18.92 \begin {gather*} \begin {cases} \frac {6 i x^{4} \log {\left (x \right )}}{24 \sqrt {2} x^{4} - 96 \sqrt {2} x^{2} + 96 \sqrt {2}} - \frac {3 i x^{4} \log {\left (x^{2} \right )}}{24 \sqrt {2} x^{4} - 96 \sqrt {2} x^{2} + 96 \sqrt {2}} - \frac {6 x^{4} \operatorname {asin}{\left (\frac {\sqrt {2}}{x} \right )}}{24 \sqrt {2} x^{4} - 96 \sqrt {2} x^{2} + 96 \sqrt {2}} + \frac {6 \sqrt {2} x^{2} \sqrt {x^{2} - 2}}{24 \sqrt {2} x^{4} - 96 \sqrt {2} x^{2} + 96 \sqrt {2}} - \frac {24 i x^{2} \log {\left (x \right )}}{24 \sqrt {2} x^{4} - 96 \sqrt {2} x^{2} + 96 \sqrt {2}} + \frac {12 i x^{2} \log {\left (x^{2} \right )}}{24 \sqrt {2} x^{4} - 96 \sqrt {2} x^{2} + 96 \sqrt {2}} + \frac {24 x^{2} \operatorname {asin}{\left (\frac {\sqrt {2}}{x} \right )}}{24 \sqrt {2} x^{4} - 96 \sqrt {2} x^{2} + 96 \sqrt {2}} - \frac {16 \sqrt {2} \sqrt {x^{2} - 2}}{24 \sqrt {2} x^{4} - 96 \sqrt {2} x^{2} + 96 \sqrt {2}} + \frac {24 i \log {\left (x \right )}}{24 \sqrt {2} x^{4} - 96 \sqrt {2} x^{2} + 96 \sqrt {2}} - \frac {12 i \log {\left (x^{2} \right )}}{24 \sqrt {2} x^{4} - 96 \sqrt {2} x^{2} + 96 \sqrt {2}} - \frac {24 \operatorname {asin}{\left (\frac {\sqrt {2}}{x} \right )}}{24 \sqrt {2} x^{4} - 96 \sqrt {2} x^{2} + 96 \sqrt {2}} & \text {for}\: \left |{x^{2}}\right | > 2 \\- \frac {3 i x^{4} \log {\left (x^{2} \right )}}{24 \sqrt {2} x^{4} - 96 \sqrt {2} x^{2} + 96 \sqrt {2}} + \frac {6 i x^{4} \log {\left (\sqrt {1 - \frac {x^{2}}{2}} + 1 \right )}}{24 \sqrt {2} x^{4} - 96 \sqrt {2} x^{2} + 96 \sqrt {2}} - \frac {3 \pi x^{4}}{24 \sqrt {2} x^{4} - 96 \sqrt {2} x^{2} + 96 \sqrt {2}} + \frac {3 i x^{4} \log {\left (2 \right )}}{24 \sqrt {2} x^{4} - 96 \sqrt {2} x^{2} + 96 \sqrt {2}} + \frac {6 \sqrt {2} i x^{2} \sqrt {2 - x^{2}}}{24 \sqrt {2} x^{4} - 96 \sqrt {2} x^{2} + 96 \sqrt {2}} + \frac {12 i x^{2} \log {\left (x^{2} \right )}}{24 \sqrt {2} x^{4} - 96 \sqrt {2} x^{2} + 96 \sqrt {2}} - \frac {24 i x^{2} \log {\left (\sqrt {1 - \frac {x^{2}}{2}} + 1 \right )}}{24 \sqrt {2} x^{4} - 96 \sqrt {2} x^{2} + 96 \sqrt {2}} + \frac {12 \pi x^{2}}{24 \sqrt {2} x^{4} - 96 \sqrt {2} x^{2} + 96 \sqrt {2}} - \frac {12 i x^{2} \log {\left (2 \right )}}{24 \sqrt {2} x^{4} - 96 \sqrt {2} x^{2} + 96 \sqrt {2}} - \frac {16 \sqrt {2} i \sqrt {2 - x^{2}}}{24 \sqrt {2} x^{4} - 96 \sqrt {2} x^{2} + 96 \sqrt {2}} - \frac {12 i \log {\left (x^{2} \right )}}{24 \sqrt {2} x^{4} - 96 \sqrt {2} x^{2} + 96 \sqrt {2}} + \frac {24 i \log {\left (\sqrt {1 - \frac {x^{2}}{2}} + 1 \right )}}{24 \sqrt {2} x^{4} - 96 \sqrt {2} x^{2} + 96 \sqrt {2}} - \frac {12 \pi }{24 \sqrt {2} x^{4} - 96 \sqrt {2} x^{2} + 96 \sqrt {2}} + \frac {12 i \log {\left (2 \right )}}{24 \sqrt {2} x^{4} - 96 \sqrt {2} x^{2} + 96 \sqrt {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 = 55, normalized size = 1.06 \begin {gather*} \frac {3 \left (x^{2}-2\right )-2}{12 \sqrt {x^{2}-2} \left (x^{2}-2\right )}+\frac {\arctan \left (\frac {\sqrt {x^{2}-2}}{\sqrt {2}}\right )}{4 \sqrt {2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.47, size = 34, normalized size = 0.65 \begin {gather*} \frac {\sqrt {2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,\sqrt {x^2-2}}{2}\right )}{8}+\frac {\frac {x^2}{4}-\frac {2}{3}}{{\left (x^2-2\right )}^{3/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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