Optimal. Leaf size=46 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{7}-2 x}{\sqrt{3}}\right )}{\sqrt{3}}-\frac{\tanh ^{-1}\left (\frac{2 x+\sqrt{7}}{\sqrt{3}}\right )}{\sqrt{3}} \]
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Rubi [A] time = 0.0360198, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {1161, 618, 206} \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{7}-2 x}{\sqrt{3}}\right )}{\sqrt{3}}-\frac{\tanh ^{-1}\left (\frac{2 x+\sqrt{7}}{\sqrt{3}}\right )}{\sqrt{3}} \]
Antiderivative was successfully verified.
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Rule 1161
Rule 618
Rule 206
Rubi steps
\begin{align*} \int \frac{1+x^2}{1-5 x^2+x^4} \, dx &=\frac{1}{2} \int \frac{1}{1-\sqrt{7} x+x^2} \, dx+\frac{1}{2} \int \frac{1}{1+\sqrt{7} x+x^2} \, dx\\ &=-\operatorname{Subst}\left (\int \frac{1}{3-x^2} \, dx,x,-\sqrt{7}+2 x\right )-\operatorname{Subst}\left (\int \frac{1}{3-x^2} \, dx,x,\sqrt{7}+2 x\right )\\ &=\frac{\tanh ^{-1}\left (\frac{\sqrt{7}-2 x}{\sqrt{3}}\right )}{\sqrt{3}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{7}+2 x}{\sqrt{3}}\right )}{\sqrt{3}}\\ \end{align*}
Mathematica [A] time = 0.0120748, size = 40, normalized size = 0.87 \[ \frac{\log \left (-x^2+\sqrt{3} x+1\right )-\log \left (x^2+\sqrt{3} x-1\right )}{2 \sqrt{3}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.073, size = 82, normalized size = 1.8 \begin{align*} -{\frac{ \left ( 14+2\,\sqrt{21} \right ) \sqrt{21}}{42\,\sqrt{7}+42\,\sqrt{3}}{\it Artanh} \left ( 4\,{\frac{x}{2\,\sqrt{7}+2\,\sqrt{3}}} \right ) }-{\frac{ \left ( -14+2\,\sqrt{21} \right ) \sqrt{21}}{42\,\sqrt{7}-42\,\sqrt{3}}{\it Artanh} \left ( 4\,{\frac{x}{2\,\sqrt{7}-2\,\sqrt{3}}} \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{x^{2} + 1}{x^{4} - 5 \, x^{2} + 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.33009, size = 100, normalized size = 2.17 \begin{align*} \frac{1}{6} \, \sqrt{3} \log \left (\frac{x^{4} + x^{2} - 2 \, \sqrt{3}{\left (x^{3} - x\right )} + 1}{x^{4} - 5 \, x^{2} + 1}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.101554, size = 39, normalized size = 0.85 \begin{align*} \frac{\sqrt{3} \log{\left (x^{2} - \sqrt{3} x - 1 \right )}}{6} - \frac{\sqrt{3} \log{\left (x^{2} + \sqrt{3} x - 1 \right )}}{6} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15542, size = 53, normalized size = 1.15 \begin{align*} \frac{1}{6} \, \sqrt{3} \log \left (\frac{{\left | 2 \, x - 2 \, \sqrt{3} - \frac{2}{x} \right |}}{{\left | 2 \, x + 2 \, \sqrt{3} - \frac{2}{x} \right |}}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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