Optimal. Leaf size=43 \[ -\frac {1}{2} \tan ^{-1}\left (\frac {x}{\sqrt {x^2+2}}\right )-\frac {\tanh ^{-1}\left (\frac {\sqrt {3} x}{\sqrt {x^2+2}}\right )}{2 \sqrt {3}} \]
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Rubi [A] time = 0.02, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {1175, 377, 206, 203} \[ -\frac {1}{2} \tan ^{-1}\left (\frac {x}{\sqrt {x^2+2}}\right )-\frac {\tanh ^{-1}\left (\frac {\sqrt {3} x}{\sqrt {x^2+2}}\right )}{2 \sqrt {3}} \]
Antiderivative was successfully verified.
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Rule 203
Rule 206
Rule 377
Rule 1175
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {2+x^2} \left (-1+x^4\right )} \, dx &=-\left (\frac {1}{2} \int \frac {1}{\left (1-x^2\right ) \sqrt {2+x^2}} \, dx\right )-\frac {1}{2} \int \frac {1}{\left (1+x^2\right ) \sqrt {2+x^2}} \, dx\\ &=-\left (\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{1-3 x^2} \, dx,x,\frac {x}{\sqrt {2+x^2}}\right )\right )-\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {x}{\sqrt {2+x^2}}\right )\\ &=-\frac {1}{2} \tan ^{-1}\left (\frac {x}{\sqrt {2+x^2}}\right )-\frac {\tanh ^{-1}\left (\frac {\sqrt {3} x}{\sqrt {2+x^2}}\right )}{2 \sqrt {3}}\\ \end {align*}
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Mathematica [C] time = 0.05, size = 96, normalized size = 2.23 \[ \frac {1}{12} \left (-3 \tan ^{-1}\left (\frac {-x+2 i}{\sqrt {x^2+2}}\right )+3 \tan ^{-1}\left (\frac {x+2 i}{\sqrt {x^2+2}}\right )+\sqrt {3} \tanh ^{-1}\left (\frac {2-x}{\sqrt {3} \sqrt {x^2+2}}\right )-\sqrt {3} \tanh ^{-1}\left (\frac {x+2}{\sqrt {3} \sqrt {x^2+2}}\right )\right ) \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.09, size = 65, normalized size = 1.51 \[ \frac {1}{2} \tan ^{-1}\left (x^2-\sqrt {x^2+2} x+1\right )-\frac {\tanh ^{-1}\left (-\frac {x^2}{\sqrt {3}}+\frac {\sqrt {x^2+2} x}{\sqrt {3}}+\frac {1}{\sqrt {3}}\right )}{2 \sqrt {3}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.99, size = 72, normalized size = 1.67 \[ \frac {1}{12} \, \sqrt {3} \log \left (\frac {4 \, x^{2} - \sqrt {3} {\left (2 \, x^{2} + 1\right )} - \sqrt {x^{2} + 2} {\left (2 \, \sqrt {3} x - 3 \, x\right )} + 2}{x^{2} - 1}\right ) - \frac {1}{2} \, \arctan \left (-x^{2} + \sqrt {x^{2} + 2} x - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.65, size = 74, normalized size = 1.72 \[ -\frac {1}{12} \, \sqrt {3} \log \left (\frac {{\left | 2 \, {\left (x - \sqrt {x^{2} + 2}\right )}^{2} - 4 \, \sqrt {3} - 8 \right |}}{{\left | 2 \, {\left (x - \sqrt {x^{2} + 2}\right )}^{2} + 4 \, \sqrt {3} - 8 \right |}}\right ) + \frac {1}{2} \, \arctan \left (\frac {1}{2} \, {\left (x - \sqrt {x^{2} + 2}\right )}^{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.28, size = 70, normalized size = 1.63
method | result | size |
default | \(-\frac {\arctan \left (\frac {x}{\sqrt {x^{2}+2}}\right )}{2}-\frac {\sqrt {3}\, \arctanh \left (\frac {\left (4+2 x \right ) \sqrt {3}}{6 \sqrt {\left (-1+x \right )^{2}+1+2 x}}\right )}{12}+\frac {\sqrt {3}\, \arctanh \left (\frac {\left (4-2 x \right ) \sqrt {3}}{6 \sqrt {\left (1+x \right )^{2}+1-2 x}}\right )}{12}\) | \(70\) |
trager | \(\frac {\RootOf \left (\textit {\_Z}^{2}-3\right ) \ln \left (-\frac {-2 \RootOf \left (\textit {\_Z}^{2}-3\right ) x^{2}+3 \sqrt {x^{2}+2}\, x -\RootOf \left (\textit {\_Z}^{2}-3\right )}{\left (1+x \right ) \left (-1+x \right )}\right )}{12}-\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {\sqrt {x^{2}+2}\, x +\RootOf \left (\textit {\_Z}^{2}+1\right )}{x^{2}+1}\right )}{4}\) | \(86\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (x^{4} - 1\right )} \sqrt {x^{2} + 2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.11, size = 107, normalized size = 2.49 \[ \frac {\sqrt {3}\,\left (\ln \left (x-1\right )-\ln \left (x+\sqrt {3}\,\sqrt {x^2+2}+2\right )\right )}{12}-\frac {\sqrt {3}\,\left (\ln \left (x+1\right )-\ln \left (\sqrt {3}\,\sqrt {x^2+2}-x+2\right )\right )}{12}+\frac {\ln \left (\sqrt {x^2+2}+2-x\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{4}-\frac {\ln \left (\sqrt {x^2+2}+2+x\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{4}+\frac {\ln \left (x-\mathrm {i}\right )\,1{}\mathrm {i}}{4}-\frac {\ln \left (x+1{}\mathrm {i}\right )\,1{}\mathrm {i}}{4} \]
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 \[ \int \frac {1}{\left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right ) \sqrt {x^{2} + 2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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