Optimal. Leaf size=165 \[ \frac {\log \left (3 \sqrt [3]{x^2+x+1}+\sqrt [3]{3} x-\sqrt [3]{3}\right )}{3 \sqrt [3]{3}}-\frac {\log \left (3^{2/3} x^2+9 \left (x^2+x+1\right )^{2/3}+\left (3 \sqrt [3]{3}-3 \sqrt [3]{3} x\right ) \sqrt [3]{x^2+x+1}-2\ 3^{2/3} x+3^{2/3}\right )}{6 \sqrt [3]{3}}-\frac {\tan ^{-1}\left (\frac {\frac {\sqrt [3]{x^2+x+1}}{\sqrt {3}}-\frac {2 x}{3 \sqrt [6]{3}}+\frac {2}{3 \sqrt [6]{3}}}{\sqrt [3]{x^2+x+1}}\right )}{3^{5/6}} \]
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Rubi [A] time = 0.01, antiderivative size = 84, normalized size of antiderivative = 0.51, number of steps used = 1, number of rules used = 1, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {750} \begin {gather*} \frac {\log \left (-3^{2/3} \sqrt [3]{x^2+x+1}-x+1\right )}{2 \sqrt [3]{3}}-\frac {\tan ^{-1}\left (\frac {2 (1-x)}{3 \sqrt [6]{3} \sqrt [3]{x^2+x+1}}+\frac {1}{\sqrt {3}}\right )}{3^{5/6}}-\frac {\log (x+2)}{2 \sqrt [3]{3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 750
Rubi steps
\begin {align*} \int \frac {1}{(2+x) \sqrt [3]{1+x+x^2}} \, dx &=-\frac {\tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 (1-x)}{3 \sqrt [6]{3} \sqrt [3]{1+x+x^2}}\right )}{3^{5/6}}-\frac {\log (2+x)}{2 \sqrt [3]{3}}+\frac {\log \left (1-x-3^{2/3} \sqrt [3]{1+x+x^2}\right )}{2 \sqrt [3]{3}}\\ \end {align*}
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Mathematica [C] time = 0.06, size = 118, normalized size = 0.72 \begin {gather*} -\frac {3 \sqrt [3]{\frac {2 x-i \sqrt {3}+1}{x+2}} \sqrt [3]{\frac {2 x+i \sqrt {3}+1}{x+2}} F_1\left (\frac {2}{3};\frac {1}{3},\frac {1}{3};\frac {5}{3};\frac {3-i \sqrt {3}}{2 x+4},\frac {3+i \sqrt {3}}{2 x+4}\right )}{2\ 2^{2/3} \sqrt [3]{x^2+x+1}} \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [A] time = 0.20, size = 165, normalized size = 1.00 \begin {gather*} \frac {\log \left (3 \sqrt [3]{x^2+x+1}+\sqrt [3]{3} x-\sqrt [3]{3}\right )}{3 \sqrt [3]{3}}-\frac {\log \left (3^{2/3} x^2+9 \left (x^2+x+1\right )^{2/3}+\left (3 \sqrt [3]{3}-3 \sqrt [3]{3} x\right ) \sqrt [3]{x^2+x+1}-2\ 3^{2/3} x+3^{2/3}\right )}{6 \sqrt [3]{3}}-\frac {\tan ^{-1}\left (\frac {\frac {\sqrt [3]{x^2+x+1}}{\sqrt {3}}-\frac {2 x}{3 \sqrt [6]{3}}+\frac {2}{3 \sqrt [6]{3}}}{\sqrt [3]{x^2+x+1}}\right )}{3^{5/6}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 2.21, size = 165, normalized size = 1.00 \begin {gather*} -\frac {1}{18} \cdot 3^{\frac {2}{3}} \log \left (\frac {3 \cdot 3^{\frac {2}{3}} {\left (x^{2} + x + 1\right )}^{\frac {2}{3}} + 3^{\frac {1}{3}} {\left (x^{2} - 2 \, x + 1\right )} - 3 \, {\left (x^{2} + x + 1\right )}^{\frac {1}{3}} {\left (x - 1\right )}}{x^{2} + 4 \, x + 4}\right ) + \frac {1}{9} \cdot 3^{\frac {2}{3}} \log \left (\frac {3^{\frac {1}{3}} {\left (x - 1\right )} + 3 \, {\left (x^{2} + x + 1\right )}^{\frac {1}{3}}}{x + 2}\right ) - \frac {1}{3} \cdot 3^{\frac {1}{6}} \arctan \left (\frac {3^{\frac {1}{6}} {\left (6 \cdot 3^{\frac {2}{3}} {\left (x^{2} + x + 1\right )}^{\frac {2}{3}} {\left (x - 1\right )} + 3^{\frac {1}{3}} {\left (x^{3} + 6 \, x^{2} + 12 \, x + 8\right )} + 6 \, {\left (x^{2} + x + 1\right )}^{\frac {1}{3}} {\left (x^{2} - 2 \, x + 1\right )}\right )}}{3 \, {\left (x^{3} - 12 \, x^{2} - 6 \, x - 10\right )}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{{\left (x^{2} + x + 1\right )}^{\frac {1}{3}} {\left (x + 2\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 14.23, size = 1377, normalized size = 8.35 \begin {gather*} \text {Expression too large to display} \end {gather*}
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 \begin {gather*} \int \frac {1}{{\left (x^{2} + x + 1\right )}^{\frac {1}{3}} {\left (x + 2\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{\left (x+2\right )\,{\left (x^2+x+1\right )}^{1/3}} \,d x \end {gather*}
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 \begin {gather*} \int \frac {1}{\left (x + 2\right ) \sqrt [3]{x^{2} + x + 1}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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