Optimal. Leaf size=44 \[ -\frac{2 \tan ^{-1}\left (\frac{\sqrt{2 \sqrt{3}-3} (x+1)}{\sqrt{-x^3-1}}\right )}{\sqrt{2 \sqrt{3}-3}} \]
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Rubi [A] time = 0.0936181, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062, Rules used = {2140, 203} \[ -\frac{2 \tan ^{-1}\left (\frac{\sqrt{2 \sqrt{3}-3} (x+1)}{\sqrt{-x^3-1}}\right )}{\sqrt{2 \sqrt{3}-3}} \]
Antiderivative was successfully verified.
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Rule 2140
Rule 203
Rubi steps
\begin{align*} \int \frac{1+\sqrt{3}+x}{\left (1-\sqrt{3}+x\right ) \sqrt{-1-x^3}} \, dx &=-\left (2 \operatorname{Subst}\left (\int \frac{1}{1-\left (3-2 \sqrt{3}\right ) x^2} \, dx,x,\frac{1+x}{\sqrt{-1-x^3}}\right )\right )\\ &=-\frac{2 \tan ^{-1}\left (\frac{\sqrt{-3+2 \sqrt{3}} (1+x)}{\sqrt{-1-x^3}}\right )}{\sqrt{-3+2 \sqrt{3}}}\\ \end{align*}
Mathematica [C] time = 0.3728, size = 269, normalized size = 6.11 \[ -\frac{2 \sqrt{6} \sqrt{\frac{i (x+1)}{\sqrt{3}+3 i}} \left (4 i \sqrt{-2 i x+\sqrt{3}+i} \sqrt{x^2-x+1} \Pi \left (\frac{2 i \sqrt{3}}{-3+(2+i) \sqrt{3}};\sin ^{-1}\left (\frac{\sqrt{-2 i x+\sqrt{3}+i}}{\sqrt{2} \sqrt [4]{3}}\right )|\frac{2 \sqrt{3}}{3 i+\sqrt{3}}\right )+\sqrt{2 i x+\sqrt{3}-i} \left (\left (\sqrt{3}+(-2-i)\right ) x-i \sqrt{3}+(1+2 i)\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{-2 i x+\sqrt{3}+i}}{\sqrt{2} \sqrt [4]{3}}\right )|\frac{2 \sqrt{3}}{3 i+\sqrt{3}}\right )\right )}{\left (-3+(2+i) \sqrt{3}\right ) \sqrt{-2 i x+\sqrt{3}+i} \sqrt{-x^3-1}} \]
Warning: Unable to verify antiderivative.
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Maple [C] time = 0.046, size = 247, normalized size = 5.6 \begin{align*}{-{\frac{2\,i}{3}}\sqrt{3}\sqrt{i \left ( x-{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}\sqrt{{\frac{1+x}{{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}}}\sqrt{-i \left ( x-{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}{\it EllipticF} \left ({\frac{\sqrt{3}}{3}\sqrt{i \left ( x-{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}},\sqrt{{\frac{i\sqrt{3}}{{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}}} \right ){\frac{1}{\sqrt{-{x}^{3}-1}}}}-{\frac{4\,i}{{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}-\sqrt{3}}\sqrt{i \left ( x-{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}\sqrt{{\frac{1+x}{{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}}}\sqrt{-i \left ( x-{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}{\it EllipticPi} \left ({\frac{\sqrt{3}}{3}\sqrt{i \left ( x-{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}},{\frac{i\sqrt{3}}{{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}-\sqrt{3}}},\sqrt{{\frac{i\sqrt{3}}{{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}}} \right ){\frac{1}{\sqrt{-{x}^{3}-1}}}} \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^{3} - 1}{\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 [A] time = 1.60785, size = 170, normalized size = 3.86 \begin{align*} \frac{1}{3} \, \sqrt{3} \sqrt{2 \, \sqrt{3} + 3} \arctan \left (\frac{\sqrt{-x^{3} - 1}{\left (\sqrt{3}{\left (x^{2} - 4 \, x - 2\right )} + 6 \, x + 6\right )} \sqrt{2 \, \sqrt{3} + 3}}{6 \,{\left (x^{3} + 1\right )}}\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}}{\sqrt{- \left (x + 1\right ) \left (x^{2} - x + 1\right )} \left (x - \sqrt{3} + 1\right )}\, 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^{3} - 1}{\left (x - \sqrt{3} + 1\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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