Optimal. Leaf size=46 \[ \frac{\tanh ^{-1}\left (\frac{2 x+\sqrt{3}}{\sqrt{7}}\right )}{\sqrt{7}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{3}-2 x}{\sqrt{7}}\right )}{\sqrt{7}} \]
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Rubi [A] time = 0.0351032, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {1161, 618, 206} \[ \frac{\tanh ^{-1}\left (\frac{2 x+\sqrt{3}}{\sqrt{7}}\right )}{\sqrt{7}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{3}-2 x}{\sqrt{7}}\right )}{\sqrt{7}} \]
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 &=-\left (\frac{1}{2} \int \frac{1}{-1-\sqrt{3} x+x^2} \, dx\right )-\frac{1}{2} \int \frac{1}{-1+\sqrt{3} x+x^2} \, dx\\ &=\operatorname{Subst}\left (\int \frac{1}{7-x^2} \, dx,x,-\sqrt{3}+2 x\right )+\operatorname{Subst}\left (\int \frac{1}{7-x^2} \, dx,x,\sqrt{3}+2 x\right )\\ &=-\frac{\tanh ^{-1}\left (\frac{\sqrt{3}-2 x}{\sqrt{7}}\right )}{\sqrt{7}}+\frac{\tanh ^{-1}\left (\frac{\sqrt{3}+2 x}{\sqrt{7}}\right )}{\sqrt{7}}\\ \end{align*}
Mathematica [A] time = 0.0142958, size = 40, normalized size = 0.87 \[ \frac{\log \left (x^2+\sqrt{7} x+1\right )-\log \left (-x^2+\sqrt{7} x-1\right )}{2 \sqrt{7}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.055, size = 82, normalized size = 1.8 \begin{align*}{\frac{ \left ( 6+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 ( -6+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 = 1.39183, size = 104, normalized size = 2.26 \begin{align*} \frac{1}{14} \, \sqrt{7} \log \left (\frac{x^{4} + 9 \, x^{2} + 2 \, \sqrt{7}{\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 = 0.101374, size = 39, normalized size = 0.85 \begin{align*} - \frac{\sqrt{7} \log{\left (x^{2} - \sqrt{7} x + 1 \right )}}{14} + \frac{\sqrt{7} \log{\left (x^{2} + \sqrt{7} x + 1 \right )}}{14} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14646, size = 53, normalized size = 1.15 \begin{align*} -\frac{1}{14} \, \sqrt{7} \log \left (\frac{{\left | 2 \, x - 2 \, \sqrt{7} + \frac{2}{x} \right |}}{{\left | 2 \, x + 2 \, \sqrt{7} + \frac{2}{x} \right |}}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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