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 = 0.0197251, antiderivative size = 43, normalized size of antiderivative = 1., 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 1175
Rule 377
Rule 206
Rule 203
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*}
Mathematica [C] time = 0.0454008, 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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Maple [B] time = 0.018, size = 70, normalized size = 1.6 \begin{align*} -{\frac{1}{2}\arctan \left ({x{\frac{1}{\sqrt{{x}^{2}+2}}}} \right ) }+{\frac{\sqrt{3}}{12}{\it Artanh} \left ({\frac{ \left ( 4-2\,x \right ) \sqrt{3}}{6}{\frac{1}{\sqrt{ \left ( 1+x \right ) ^{2}+1-2\,x}}}} \right ) }-{\frac{\sqrt{3}}{12}{\it Artanh} \left ({\frac{ \left ( 4+2\,x \right ) \sqrt{3}}{6}{\frac{1}{\sqrt{ \left ( -1+x \right ) ^{2}+1+2\,x}}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (x^{4} - 1\right )} \sqrt{x^{2} + 2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.02215, size = 189, normalized size = 4.4 \begin{align*} \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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right ) \sqrt{x^{2} + 2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.11023, size = 100, normalized size = 2.33 \begin{align*} -\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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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