Optimal. Leaf size=26 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{3} x}{\sqrt{x^4+x^2+1}}\right )}{\sqrt{3}} \]
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Rubi [A] time = 0.0432189, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.074, Rules used = {1698, 206} \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{3} x}{\sqrt{x^4+x^2+1}}\right )}{\sqrt{3}} \]
Antiderivative was successfully verified.
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Rule 1698
Rule 206
Rubi steps
\begin{align*} \int \frac{1+x^2}{\left (1-x^2\right ) \sqrt{1+x^2+x^4}} \, dx &=\operatorname{Subst}\left (\int \frac{1}{1-3 x^2} \, dx,x,\frac{x}{\sqrt{1+x^2+x^4}}\right )\\ &=\frac{\tanh ^{-1}\left (\frac{\sqrt{3} x}{\sqrt{1+x^2+x^4}}\right )}{\sqrt{3}}\\ \end{align*}
Mathematica [C] time = 1.23863, size = 522, normalized size = 20.08 \[ \frac{(-1)^{2/3} \left (-\sqrt [3]{-1} \left (\sqrt [3]{-1}-1\right )^2 \left (1+\sqrt [3]{-1}\right ) \sqrt{\sqrt [3]{-1} x^2+1} \sqrt{1-(-1)^{2/3} x^2} \text{EllipticF}\left (i \sinh ^{-1}\left ((-1)^{5/6} x\right ),(-1)^{2/3}\right )-2 i \sqrt{3} \sqrt{\frac{(-1)^{2/3}-x}{\left (1+\sqrt [3]{-1}\right ) \left (\sqrt [3]{-1}-x\right )}} \sqrt{\frac{x+(-1)^{2/3}}{-\sqrt [3]{-1} x+x-1}} \sqrt{\frac{-\sqrt [3]{-1} x+x+1}{\left (1+\sqrt [3]{-1}\right ) \left (\sqrt [3]{-1}-x\right )}} \left (\sqrt [3]{-1}-x\right )^2 \left (\left (1+\sqrt [3]{-1}\right ) \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{-\sqrt [3]{-1} x+x+1}{\left (1+\sqrt [3]{-1}\right ) \left (\sqrt [3]{-1}-x\right )}}\right ),-3\right )-2 \sqrt [3]{-1} \Pi \left (-1;\left .\sin ^{-1}\left (\sqrt{\frac{-\sqrt [3]{-1} x+x+1}{\left (1+\sqrt [3]{-1}\right ) \left (\sqrt [3]{-1}-x\right )}}\right )\right |-3\right )\right )+2 i \sqrt{3} \sqrt{\frac{(-1)^{2/3}-x}{\left (1+\sqrt [3]{-1}\right ) \left (\sqrt [3]{-1}-x\right )}} \sqrt{\frac{x+(-1)^{2/3}}{-\sqrt [3]{-1} x+x-1}} \sqrt{\frac{-\sqrt [3]{-1} x+x+1}{\left (1+\sqrt [3]{-1}\right ) \left (\sqrt [3]{-1}-x\right )}} \left (\sqrt [3]{-1}-x\right )^2 \left (\left (\sqrt [3]{-1}-1\right ) \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{-\sqrt [3]{-1} x+x+1}{\left (1+\sqrt [3]{-1}\right ) \left (\sqrt [3]{-1}-x\right )}}\right ),-3\right )-2 \sqrt [3]{-1} \Pi \left (3;\left .\sin ^{-1}\left (\sqrt{\frac{-\sqrt [3]{-1} x+x+1}{\left (1+\sqrt [3]{-1}\right ) \left (\sqrt [3]{-1}-x\right )}}\right )\right |-3\right )\right )\right )}{\left (1-(-1)^{2/3}\right ) \sqrt{x^4+x^2+1}} \]
Warning: Unable to verify antiderivative.
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Maple [C] time = 0.157, size = 184, normalized size = 7.1 \begin{align*} -2\,{\frac{\sqrt{1- \left ( -1/2+i/2\sqrt{3} \right ){x}^{2}}\sqrt{1- \left ( -1/2-i/2\sqrt{3} \right ){x}^{2}}{\it EllipticF} \left ( 1/2\,x\sqrt{-2+2\,i\sqrt{3}},1/2\,\sqrt{-2+2\,i\sqrt{3}} \right ) }{\sqrt{-2+2\,i\sqrt{3}}\sqrt{{x}^{4}+{x}^{2}+1}}}+2\,{\frac{\sqrt{1- \left ( -1/2+i/2\sqrt{3} \right ){x}^{2}}\sqrt{1- \left ( -1/2-i/2\sqrt{3} \right ){x}^{2}}}{\sqrt{-1/2+i/2\sqrt{3}}\sqrt{{x}^{4}+{x}^{2}+1}}{\it EllipticPi} \left ( \sqrt{-1/2+i/2\sqrt{3}}x, \left ( -1/2+i/2\sqrt{3} \right ) ^{-1},{\frac{\sqrt{-1/2-i/2\sqrt{3}}}{\sqrt{-1/2+i/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}{\sqrt{x^{4} + x^{2} + 1}{\left (x^{2} - 1\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.73039, size = 119, normalized size = 4.58 \begin{align*} \frac{1}{6} \, \sqrt{3} \log \left (\frac{x^{4} + 2 \, \sqrt{3} \sqrt{x^{4} + x^{2} + 1} x + 4 \, x^{2} + 1}{x^{4} - 2 \, x^{2} + 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{x^{2}}{x^{2} \sqrt{x^{4} + x^{2} + 1} - \sqrt{x^{4} + x^{2} + 1}}\, dx - \int \frac{1}{x^{2} \sqrt{x^{4} + x^{2} + 1} - \sqrt{x^{4} + x^{2} + 1}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int -\frac{x^{2} + 1}{\sqrt{x^{4} + x^{2} + 1}{\left (x^{2} - 1\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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