Optimal. Leaf size=63 \[ -\frac{1}{3} \sqrt{3+2 \sqrt{3}} \tan ^{-1}\left (\frac{\left (x+\sqrt{3}+1\right )^2}{\sqrt{3 \left (3+2 \sqrt{3}\right )} \sqrt{x^4-4 \sqrt{3} x^2-4}}\right ) \]
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Rubi [A] time = 0.140545, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {1740, 203} \[ -\frac{1}{3} \sqrt{3+2 \sqrt{3}} \tan ^{-1}\left (\frac{\left (x+\sqrt{3}+1\right )^2}{\sqrt{3 \left (3+2 \sqrt{3}\right )} \sqrt{x^4-4 \sqrt{3} x^2-4}}\right ) \]
Antiderivative was successfully verified.
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Rule 1740
Rule 203
Rubi steps
\begin{align*} \int \frac{1+\sqrt{3}+x}{\left (1-\sqrt{3}+x\right ) \sqrt{-4-4 \sqrt{3} x^2+x^4}} \, dx &=-\left (\left (4 \left (2+\sqrt{3}\right )\right ) \operatorname{Subst}\left (\int \frac{1}{6 \left (1-\sqrt{3}\right ) \left (1+\sqrt{3}\right )^3+3 \left (1+\sqrt{3}\right )^4+4 x^2} \, dx,x,\frac{\left (1+\sqrt{3}+x\right )^2}{\sqrt{-4-4 \sqrt{3} x^2+x^4}}\right )\right )\\ &=-\frac{1}{3} \sqrt{3+2 \sqrt{3}} \tan ^{-1}\left (\frac{\left (1+\sqrt{3}+x\right )^2}{\sqrt{3 \left (3+2 \sqrt{3}\right )} \sqrt{-4-4 \sqrt{3} x^2+x^4}}\right )\\ \end{align*}
Mathematica [C] time = 5.6137, size = 876, normalized size = 13.9 \[ -\frac{\sqrt{2} \sqrt{\frac{\sqrt{3}-1-\frac{4}{-x+\sqrt{3}+1}}{-3+\sqrt{3}-i \sqrt{4-2 \sqrt{3}}}} \left (-x+\sqrt{3}+1\right )^2 \left (\left (\frac{2 \left (2 i \sqrt{3} \sqrt{i \left (\sqrt{3}+1-\frac{8}{-x+\sqrt{3}+1}\right )+\sqrt{4-2 \sqrt{3}}}+\sqrt{6} \sqrt{2 \sqrt{4-2 \sqrt{3}}-\sqrt{12-6 \sqrt{3}}+i \sqrt{3}-i+\frac{8 i \left (-2+\sqrt{3}\right )}{-x+\sqrt{3}+1}}+\sqrt{-\frac{2 i \left (\left (-1+\sqrt{3}\right ) x-8 \sqrt{3}+14\right )}{-x+\sqrt{3}+1}+4 \sqrt{4-2 \sqrt{3}}-2 \sqrt{12-6 \sqrt{3}}}\right )}{x-\sqrt{3}-1}+i \sqrt{3} \sqrt{i \left (\sqrt{3}+1-\frac{8}{-x+\sqrt{3}+1}\right )+\sqrt{4-2 \sqrt{3}}}+i \sqrt{i \left (\sqrt{3}+1-\frac{8}{-x+\sqrt{3}+1}\right )+\sqrt{4-2 \sqrt{3}}}+\sqrt{-\frac{2 i \left (\left (-1+\sqrt{3}\right ) x-8 \sqrt{3}+14\right )}{-x+\sqrt{3}+1}+4 \sqrt{4-2 \sqrt{3}}-2 \sqrt{12-6 \sqrt{3}}}\right ) \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{\sqrt{4-2 \sqrt{3}}-i \left (\sqrt{3}+1-\frac{8}{-x+\sqrt{3}+1}\right )}}{2^{3/4} \sqrt [4]{2-\sqrt{3}}}\right ),\frac{2 \sqrt{4-2 \sqrt{3}}}{\sqrt{4-2 \sqrt{3}}+i \left (-3+\sqrt{3}\right )}\right )+2 \sqrt{6} \sqrt{\sqrt{4-2 \sqrt{3}}-i \left (\sqrt{3}+1-\frac{8}{-x+\sqrt{3}+1}\right )} \sqrt{\frac{x^2-2 \sqrt{3}+4}{\left (-x+\sqrt{3}+1\right )^2}} \Pi \left (\frac{2 \sqrt{4-2 \sqrt{3}}}{\sqrt{4-2 \sqrt{3}}-i \left (-3+\sqrt{3}\right )};\sin ^{-1}\left (\frac{\sqrt{\sqrt{4-2 \sqrt{3}}-i \left (\sqrt{3}+1-\frac{8}{-x+\sqrt{3}+1}\right )}}{2^{3/4} \sqrt [4]{2-\sqrt{3}}}\right )|\frac{2 \sqrt{4-2 \sqrt{3}}}{\sqrt{4-2 \sqrt{3}}+i \left (-3+\sqrt{3}\right )}\right )\right )}{\left (\sqrt{4-2 \sqrt{3}}-i \left (-3+\sqrt{3}\right )\right ) \sqrt{\sqrt{4-2 \sqrt{3}}-i \left (\sqrt{3}+1-\frac{8}{-x+\sqrt{3}+1}\right )} \sqrt{x^4-4 \sqrt{3} x^2-4}} \]
Warning: Unable to verify antiderivative.
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Maple [C] time = 0.138, size = 311, normalized size = 4.9 \begin{align*}{\frac{{\it EllipticF} \left ( x \left ({\frac{i}{2}}+{\frac{i}{2}}\sqrt{3} \right ) ,i\sqrt{1-4\,\sqrt{3} \left ( -1/2\,\sqrt{3}+1 \right ) } \right ) }{{\frac{i}{2}}+{\frac{i}{2}}\sqrt{3}}\sqrt{1- \left ( -1-{\frac{\sqrt{3}}{2}} \right ){x}^{2}}\sqrt{1- \left ( -{\frac{\sqrt{3}}{2}}+1 \right ){x}^{2}}{\frac{1}{\sqrt{-4+{x}^{4}-4\,{x}^{2}\sqrt{3}}}}}+2\,\sqrt{3} \left ( -1/2\,{\frac{1}{\sqrt{ \left ( \sqrt{3}-1 \right ) ^{4}-4\,\sqrt{3} \left ( \sqrt{3}-1 \right ) ^{2}-4}}{\it Artanh} \left ( 1/2\,{\frac{-4\,\sqrt{3} \left ( \sqrt{3}-1 \right ) ^{2}-8-4\,{x}^{2}\sqrt{3}+2\,{x}^{2} \left ( \sqrt{3}-1 \right ) ^{2}}{\sqrt{ \left ( \sqrt{3}-1 \right ) ^{4}-4\,\sqrt{3} \left ( \sqrt{3}-1 \right ) ^{2}-4}\sqrt{-4+{x}^{4}-4\,{x}^{2}\sqrt{3}}}} \right ) }-{\frac{\sqrt{1- \left ( -1-1/2\,\sqrt{3} \right ){x}^{2}}\sqrt{1- \left ( -1/2\,\sqrt{3}+1 \right ){x}^{2}}}{\sqrt{-1-1/2\,\sqrt{3}} \left ( \sqrt{3}-1 \right ) \sqrt{-4+{x}^{4}-4\,{x}^{2}\sqrt{3}}}{\it EllipticPi} \left ( \sqrt{-1-1/2\,\sqrt{3}}x,{\frac{1}{ \left ( -1-1/2\,\sqrt{3} \right ) \left ( \sqrt{3}-1 \right ) ^{2}}},{\frac{\sqrt{-1/2\,\sqrt{3}+1}}{\sqrt{-1-1/2\,\sqrt{3}}}} \right ) } \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 + \sqrt{3} + 1}{\sqrt{x^{4} - 4 \, \sqrt{3} x^{2} - 4}{\left (x - \sqrt{3} + 1\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.66524, size = 301, normalized size = 4.78 \begin{align*} \frac{1}{6} \, \sqrt{2 \, \sqrt{3} + 3} \arctan \left (-\frac{{\left (9 \, x^{4} - 30 \, x^{3} + 18 \, x^{2} - 2 \, \sqrt{3}{\left (2 \, x^{4} - 10 \, x^{3} + 3 \, x^{2} - 10 \, x + 2\right )} + 24\right )} \sqrt{x^{4} - 4 \, \sqrt{3} x^{2} - 4} \sqrt{2 \, \sqrt{3} + 3}}{11 \, x^{6} - 42 \, x^{5} + 66 \, x^{4} - 176 \, x^{3} - 132 \, x^{2} - 168 \, x - 88}\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 + 1 + \sqrt{3}}{\left (x - \sqrt{3} + 1\right ) \sqrt{x^{4} - 4 \sqrt{3} x^{2} - 4}}\, 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 + \sqrt{3} + 1}{\sqrt{x^{4} - 4 \, \sqrt{3} x^{2} - 4}{\left (x - \sqrt{3} + 1\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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