Optimal. Leaf size=32 \[ \frac{2 \tan ^{-1}\left (\frac{\sqrt{3} \left (\sqrt [3]{2} x+1\right )}{\sqrt{x^3+1}}\right )}{\sqrt{3}} \]
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Rubi [A] time = 0.0963157, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.069, Rules used = {2137, 203} \[ \frac{2 \tan ^{-1}\left (\frac{\sqrt{3} \left (\sqrt [3]{2} x+1\right )}{\sqrt{x^3+1}}\right )}{\sqrt{3}} \]
Antiderivative was successfully verified.
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Rule 2137
Rule 203
Rubi steps
\begin{align*} \int \frac{1-\sqrt [3]{2} x}{\left (2^{2/3}+x\right ) \sqrt{1+x^3}} \, dx &=2 \operatorname{Subst}\left (\int \frac{1}{1+3 x^2} \, dx,x,\frac{1+\sqrt [3]{2} x}{\sqrt{1+x^3}}\right )\\ &=\frac{2 \tan ^{-1}\left (\frac{\sqrt{3} \left (1+\sqrt [3]{2} x\right )}{\sqrt{1+x^3}}\right )}{\sqrt{3}}\\ \end{align*}
Mathematica [C] time = 0.448681, size = 323, normalized size = 10.09 \[ -\frac{2 \sqrt{\frac{2}{3}} \sqrt{\frac{i (x+1)}{\sqrt{3}+3 i}} \left (\sqrt{2 i x+\sqrt{3}-i} \left (\left (-3 i \sqrt [3]{2}+4 \sqrt{3}+\sqrt [3]{2} \sqrt{3}\right ) x+\sqrt [3]{2} \sqrt{3}-2 \sqrt{3}+3 i \sqrt [3]{2}+6 i\right ) \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{-2 i x+\sqrt{3}+i}}{\sqrt{2} \sqrt [4]{3}}\right ),\frac{2 \sqrt{3}}{\sqrt{3}+3 i}\right )-6 i \sqrt{3} \sqrt{-2 i x+\sqrt{3}+i} \sqrt{x^2-x+1} \Pi \left (\frac{2 \sqrt{3}}{i+2 i 2^{2/3}+\sqrt{3}};\sin ^{-1}\left (\frac{\sqrt{-2 i x+\sqrt{3}+i}}{\sqrt{2} \sqrt [4]{3}}\right )|\frac{2 \sqrt{3}}{3 i+\sqrt{3}}\right )\right )}{\left (1+2\ 2^{2/3}-i \sqrt{3}\right ) \sqrt{-2 i x+\sqrt{3}+i} \sqrt{x^3+1}} \]
Warning: Unable to verify antiderivative.
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Maple [C] time = 0.059, size = 258, normalized size = 8.1 \begin{align*} -2\,{\frac{\sqrt [3]{2} \left ( 3/2-i/2\sqrt{3} \right ) }{\sqrt{{x}^{3}+1}}\sqrt{{\frac{1+x}{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{1+x}{3/2-i/2\sqrt{3}}}},\sqrt{{\frac{-3/2+i/2\sqrt{3}}{-3/2-i/2\sqrt{3}}}} \right ) }+6\,{\frac{3/2-i/2\sqrt{3}}{\sqrt{{x}^{3}+1} \left ({2}^{2/3}-1 \right ) }\sqrt{{\frac{1+x}{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{1+x}{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^{\frac{1}{3}} x - 1}{\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 [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \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{\sqrt [3]{2} x}{x \sqrt{x^{3} + 1} + 2^{\frac{2}{3}} \sqrt{x^{3} + 1}}\, dx - \int - \frac{1}{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^{\frac{1}{3}} x - 1}{\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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