Optimal. Leaf size=41 \[ \tanh ^{-1}\left (\frac {x}{\sqrt {x^2-2}}\right )-\sqrt {\frac {3}{2}} \tanh ^{-1}\left (\frac {\sqrt {\frac {2}{3}} x}{\sqrt {x^2-2}}\right ) \]
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Rubi [A] time = 0.02, antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {483, 217, 206, 377, 207} \[ \tanh ^{-1}\left (\frac {x}{\sqrt {x^2-2}}\right )-\sqrt {\frac {3}{2}} \tanh ^{-1}\left (\frac {\sqrt {\frac {2}{3}} x}{\sqrt {x^2-2}}\right ) \]
Antiderivative was successfully verified.
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Rule 206
Rule 207
Rule 217
Rule 377
Rule 483
Rubi steps
\begin {align*} \int \frac {x^2}{\left (-6+x^2\right ) \sqrt {-2+x^2}} \, dx &=6 \int \frac {1}{\left (-6+x^2\right ) \sqrt {-2+x^2}} \, dx+\int \frac {1}{\sqrt {-2+x^2}} \, dx\\ &=6 \operatorname {Subst}\left (\int \frac {1}{-6+4 x^2} \, dx,x,\frac {x}{\sqrt {-2+x^2}}\right )+\operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {x}{\sqrt {-2+x^2}}\right )\\ &=\tanh ^{-1}\left (\frac {x}{\sqrt {-2+x^2}}\right )-\sqrt {\frac {3}{2}} \tanh ^{-1}\left (\frac {\sqrt {\frac {2}{3}} x}{\sqrt {-2+x^2}}\right )\\ \end {align*}
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Mathematica [A] time = 0.03, size = 41, normalized size = 1.00 \[ \log \left (\sqrt {x^2-2}+x\right )-\sqrt {\frac {3}{2}} \tanh ^{-1}\left (\frac {\sqrt {\frac {2}{3}} x}{\sqrt {x^2-2}}\right ) \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.12, size = 66, normalized size = 1.61 \[ -\log \left (\sqrt {x^2-2}-x\right )-\sqrt {\frac {3}{2}} \tanh ^{-1}\left (-\frac {x^2}{2 \sqrt {6}}+\frac {\sqrt {x^2-2} x}{2 \sqrt {6}}+\sqrt {\frac {3}{2}}\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.67, size = 77, normalized size = 1.88 \[ \frac {1}{4} \, \sqrt {3} \sqrt {2} \log \left (-\frac {2 \, \sqrt {3} \sqrt {2} {\left (5 \, x^{2} - 6\right )} - 25 \, x^{2} + 2 \, {\left (5 \, \sqrt {3} \sqrt {2} x - 12 \, x\right )} \sqrt {x^{2} - 2} + 30}{x^{2} - 6}\right ) - \log \left (-x + \sqrt {x^{2} - 2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.66, size = 72, normalized size = 1.76 \[ -\frac {1}{4} \, \sqrt {6} \log \left (\frac {{\left | 2 \, {\left (x - \sqrt {x^{2} - 2}\right )}^{2} - 8 \, \sqrt {6} - 20 \right |}}{{\left | 2 \, {\left (x - \sqrt {x^{2} - 2}\right )}^{2} + 8 \, \sqrt {6} - 20 \right |}}\right ) - \frac {1}{2} \, \log \left ({\left (x - \sqrt {x^{2} - 2}\right )}^{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.31, size = 64, normalized size = 1.56
method | result | size |
trager | \(-\ln \left (x -\sqrt {x^{2}-2}\right )-\frac {\RootOf \left (\textit {\_Z}^{2}-6\right ) \ln \left (-\frac {5 \RootOf \left (\textit {\_Z}^{2}-6\right ) x^{2}+12 \sqrt {x^{2}-2}\, x -6 \RootOf \left (\textit {\_Z}^{2}-6\right )}{x^{2}-6}\right )}{4}\) | \(64\) |
default | \(\ln \left (x +\sqrt {x^{2}-2}\right )-\frac {\sqrt {6}\, \arctanh \left (\frac {8+2 \sqrt {6}\, \left (x -\sqrt {6}\right )}{4 \sqrt {\left (x -\sqrt {6}\right )^{2}+2 \sqrt {6}\, \left (x -\sqrt {6}\right )+4}}\right )}{4}+\frac {\sqrt {6}\, \arctanh \left (\frac {8-2 \sqrt {6}\, \left (x +\sqrt {6}\right )}{4 \sqrt {\left (x +\sqrt {6}\right )^{2}-2 \sqrt {6}\, \left (x +\sqrt {6}\right )+4}}\right )}{4}\) | \(100\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.46, size = 107, normalized size = 2.61 \[ \frac {1}{12} \, \sqrt {6} {\left (2 \, \sqrt {6} \log \left (x + \sqrt {x^{2} - 2}\right ) - 3 \, \log \left (\sqrt {6} + \frac {4 \, \sqrt {x^{2} - 2}}{{\left | 2 \, x - 2 \, \sqrt {6} \right |}} + \frac {8}{{\left | 2 \, x - 2 \, \sqrt {6} \right |}}\right ) + 3 \, \log \left (-\sqrt {6} + \frac {4 \, \sqrt {x^{2} - 2}}{{\left | 2 \, x + 2 \, \sqrt {6} \right |}} + \frac {8}{{\left | 2 \, x + 2 \, \sqrt {6} \right |}}\right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {x^2}{\sqrt {x^2-2}\,\left (x^2-6\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2}}{\left (x^{2} - 6\right ) \sqrt {x^{2} - 2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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