Optimal. Leaf size=40 \[ -\frac {2\ 2^{2/3} \tan ^{-1}\left (\frac {\sqrt {3} \left (1-\sqrt [3]{2} x\right )}{\sqrt {1-x^3}}\right )}{\sqrt {3}} \]
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Rubi [A] time = 0.12, antiderivative size = 40, normalized size of antiderivative = 1.00, 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 = {2137, 203} \[ -\frac {2\ 2^{2/3} \tan ^{-1}\left (\frac {\sqrt {3} \left (1-\sqrt [3]{2} x\right )}{\sqrt {1-x^3}}\right )}{\sqrt {3}} \]
Antiderivative was successfully verified.
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Rule 203
Rule 2137
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} \tan ^{-1}\left (\frac {\sqrt {3} \left (1-\sqrt [3]{2} x\right )}{\sqrt {1-x^3}}\right )}{\sqrt {3}}\\ \end {align*}
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Mathematica [C] time = 0.37, size = 327, normalized size = 8.18 \[ -\frac {4 \sqrt [6]{2} \sqrt {-\frac {i (x-1)}{\sqrt {3}+3 i}} \left (6 i \sqrt {3} \sqrt {2 i x+\sqrt {3}+i} \sqrt {x^2+x+1} \Pi \left (\frac {2 \sqrt {3}}{i+2 i 2^{2/3}+\sqrt {3}};\sin ^{-1}\left (\frac {\sqrt {2 i x+\sqrt {3}+i}}{\sqrt {2} \sqrt [4]{3}}\right )|\frac {2 \sqrt {3}}{3 i+\sqrt {3}}\right )+\sqrt {-2 i x+\sqrt {3}-i} \left (\left (-3 i \sqrt [3]{2}+4 \sqrt {3}+\sqrt [3]{2} \sqrt {3}\right ) x-\sqrt [3]{2} \sqrt {3}+2 \sqrt {3}-3 i \sqrt [3]{2}-6 i\right ) F\left (\sin ^{-1}\left (\frac {\sqrt {2 i x+\sqrt {3}+i}}{\sqrt {2} \sqrt [4]{3}}\right )|\frac {2 \sqrt {3}}{3 i+\sqrt {3}}\right )\right )}{\sqrt {3} \left (1+2\ 2^{2/3}-i \sqrt {3}\right ) \sqrt {2 i x+\sqrt {3}+i} \sqrt {1-x^3}} \]
Warning: Unable to verify antiderivative.
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fricas [B] time = 1.38, size = 76, normalized size = 1.90 \[ -\frac {1}{3} \, \sqrt {6} 2^{\frac {1}{6}} \arctan \left (\frac {\sqrt {6} 2^{\frac {1}{6}} {\left (2 \, x^{5} - 2 \, x^{2} + 2^{\frac {2}{3}} {\left (7 \, x^{4} - 4 \, x\right )} - 2^{\frac {1}{3}} {\left (5 \, x^{3} - 2\right )}\right )} \sqrt {-x^{3} + 1}}{12 \, {\left (2 \, x^{6} - 3 \, x^{3} + 1\right )}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.06, size = 253, normalized size = 6.32 \[ \frac {4 i \sqrt {3}\, \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {x -1}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {-i \left (x +\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \EllipticF \left (\frac {\sqrt {3}\, \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \sqrt {\frac {i \sqrt {3}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )}{3 \sqrt {-x^{3}+1}}+\frac {2 i 2^{\frac {2}{3}} \sqrt {3}\, \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {x -1}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {-i \left (x +\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \EllipticPi \left (\frac {\sqrt {3}\, \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \frac {i \sqrt {3}}{-\frac {1}{2}+\frac {i \sqrt {3}}{2}-2^{\frac {2}{3}}}, \sqrt {\frac {i \sqrt {3}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {-x^{3}+1}\, \left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}-2^{\frac {2}{3}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -\int \frac {2 \, x + 2^{\frac {2}{3}}}{\sqrt {-x^{3} + 1} {\left (x - 2^{\frac {2}{3}}\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.63, size = 74, normalized size = 1.85 \[ \frac {2^{2/3}\,\sqrt {3}\,\ln \left (\frac {\left (\sqrt {1-x^3}-\sqrt {3}\,1{}\mathrm {i}+2^{1/3}\,\sqrt {3}\,x\,1{}\mathrm {i}\right )\,{\left (\sqrt {3}\,1{}\mathrm {i}+\sqrt {1-x^3}-2^{1/3}\,\sqrt {3}\,x\,1{}\mathrm {i}\right )}^3}{{\left (x-2^{2/3}\right )}^6}\right )\,1{}\mathrm {i}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ - \int \frac {2^{\frac {2}{3}}}{x \sqrt {1 - x^{3}} - 2^{\frac {2}{3}} \sqrt {1 - x^{3}}}\, dx - \int \frac {2 x}{x \sqrt {1 - x^{3}} - 2^{\frac {2}{3}} \sqrt {1 - x^{3}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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