Optimal. Leaf size=38 \[ -\frac{2\ 2^{2/3} \tanh ^{-1}\left (\frac{\sqrt{3} \left (1-\sqrt [3]{2} x\right )}{\sqrt{x^3-1}}\right )}{\sqrt{3}} \]
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Rubi [A] time = 0.112427, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {2137, 206} \[ -\frac{2\ 2^{2/3} \tanh ^{-1}\left (\frac{\sqrt{3} \left (1-\sqrt [3]{2} x\right )}{\sqrt{x^3-1}}\right )}{\sqrt{3}} \]
Antiderivative was successfully verified.
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Rule 2137
Rule 206
Rubi steps
\begin{align*} \int \frac{2^{2/3}+2 x}{\left (2^{2/3}-x\right ) \sqrt{-1+x^3}} \, dx &=-\left (\left (2\ 2^{2/3}\right ) \operatorname{Subst}\left (\int \frac{1}{1-3 x^2} \, dx,x,\frac{1-\sqrt [3]{2} x}{\sqrt{-1+x^3}}\right )\right )\\ &=-\frac{2\ 2^{2/3} \tanh ^{-1}\left (\frac{\sqrt{3} \left (1-\sqrt [3]{2} x\right )}{\sqrt{-1+x^3}}\right )}{\sqrt{3}}\\ \end{align*}
Mathematica [C] time = 0.304679, size = 325, normalized size = 8.55 \[ -\frac{4 \sqrt [6]{2} \sqrt{-\frac{i (x-1)}{\sqrt{3}+3 i}} \left (6 i \sqrt{3} \sqrt{2 i x+\sqrt{3}+i} \sqrt{x^2+x+1} \Pi \left (\frac{2 \sqrt{3}}{i+2 i 2^{2/3}+\sqrt{3}};\sin ^{-1}\left (\frac{\sqrt{2 i x+\sqrt{3}+i}}{\sqrt{2} \sqrt [4]{3}}\right )|\frac{2 \sqrt{3}}{3 i+\sqrt{3}}\right )+\sqrt{-2 i x+\sqrt{3}-i} \left (\left (-3 i \sqrt [3]{2}+4 \sqrt{3}+\sqrt [3]{2} \sqrt{3}\right ) x-\sqrt [3]{2} \sqrt{3}+2 \sqrt{3}-3 i \sqrt [3]{2}-6 i\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{2 i x+\sqrt{3}+i}}{\sqrt{2} \sqrt [4]{3}}\right )|\frac{2 \sqrt{3}}{3 i+\sqrt{3}}\right )\right )}{\sqrt{3} \left (1+2\ 2^{2/3}-i \sqrt{3}\right ) \sqrt{2 i x+\sqrt{3}+i} \sqrt{x^3-1}} \]
Warning: Unable to verify antiderivative.
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Maple [C] time = 0.029, size = 262, normalized size = 6.9 \begin{align*} -4\,{\frac{-3/2-i/2\sqrt{3}}{\sqrt{{x}^{3}-1}}\sqrt{{\frac{x-1}{-3/2-i/2\sqrt{3}}}}\sqrt{{\frac{x+1/2-i/2\sqrt{3}}{3/2-i/2\sqrt{3}}}}\sqrt{{\frac{x+1/2+i/2\sqrt{3}}{3/2+i/2\sqrt{3}}}}{\it EllipticF} \left ( \sqrt{{\frac{x-1}{-3/2-i/2\sqrt{3}}}},\sqrt{{\frac{3/2+i/2\sqrt{3}}{3/2-i/2\sqrt{3}}}} \right ) }-6\,{\frac{{2}^{2/3} \left ( -3/2-i/2\sqrt{3} \right ) }{\sqrt{{x}^{3}-1} \left ( -{2}^{2/3}+1 \right ) }\sqrt{{\frac{x-1}{-3/2-i/2\sqrt{3}}}}\sqrt{{\frac{x+1/2-i/2\sqrt{3}}{3/2-i/2\sqrt{3}}}}\sqrt{{\frac{x+1/2+i/2\sqrt{3}}{3/2+i/2\sqrt{3}}}}{\it EllipticPi} \left ( \sqrt{{\frac{x-1}{-3/2-i/2\sqrt{3}}}},{\frac{3/2+i/2\sqrt{3}}{-{2}^{2/3}+1}},\sqrt{{\frac{3/2+i/2\sqrt{3}}{3/2-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{2 \, x + 2^{\frac{2}{3}}}{\sqrt{x^{3} - 1}{\left (x - 2^{\frac{2}{3}}\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.42924, size = 717, normalized size = 18.87 \begin{align*} \frac{1}{6} \, \sqrt{6} 2^{\frac{1}{6}} \log \left (\frac{x^{18} + 1440 \, x^{15} + 17400 \, x^{12} - 21056 \, x^{9} - 10368 \, x^{6} + 15360 \, x^{3} + 2 \, \sqrt{6} 2^{\frac{1}{6}}{\left (126 \, x^{14} + 2664 \, x^{11} - 4608 \, x^{5} + 2304 \, x^{2} + 2^{\frac{2}{3}}{\left (x^{16} + 310 \, x^{13} + 2332 \, x^{10} - 2656 \, x^{7} - 256 \, x^{4} + 512 \, x\right )} + 2^{\frac{1}{3}}{\left (17 \, x^{15} + 1058 \, x^{12} + 2528 \, x^{9} - 5408 \, x^{6} + 2560 \, x^{3} - 512\right )}\right )} \sqrt{x^{3} - 1} + 24 \cdot 2^{\frac{2}{3}}{\left (x^{17} + 121 \, x^{14} + 478 \, x^{11} - 1144 \, x^{8} + 608 \, x^{5} - 64 \, x^{2}\right )} + 48 \cdot 2^{\frac{1}{3}}{\left (5 \, x^{16} + 176 \, x^{13} + 83 \, x^{10} - 680 \, x^{7} + 544 \, x^{4} - 128 \, x\right )} - 2048}{x^{18} - 24 \, x^{15} + 240 \, x^{12} - 1280 \, x^{9} + 3840 \, x^{6} - 6144 \, x^{3} + 4096}\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{2^{\frac{2}{3}}}{x \sqrt{x^{3} - 1} - 2^{\frac{2}{3}} \sqrt{x^{3} - 1}}\, dx - \int \frac{2 x}{x \sqrt{x^{3} - 1} - 2^{\frac{2}{3}} \sqrt{x^{3} - 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{2 \, x + 2^{\frac{2}{3}}}{\sqrt{x^{3} - 1}{\left (x - 2^{\frac{2}{3}}\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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