Optimal. Leaf size=106 \[ \frac {\left ((x+1)^2\right )^{2/3} \left (\frac {\log \left (3^{2/3} \sqrt [3]{x+1}-3\right )}{3^{2/3}}-\frac {\log \left (\sqrt [3]{3} (x+1)^{2/3}+3^{2/3} \sqrt [3]{x+1}+3\right )}{2\ 3^{2/3}}-\frac {\tan ^{-1}\left (\frac {2 \sqrt [3]{x+1}}{3^{5/6}}+\frac {1}{\sqrt {3}}\right )}{\sqrt [6]{3}}\right )}{(x+1)^{4/3}} \]
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Rubi [A] time = 0.05, antiderivative size = 127, normalized size of antiderivative = 1.20, number of steps used = 5, number of rules used = 5, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {646, 57, 617, 204, 31} \begin {gather*} -\frac {(x+1)^{2/3} \log (2-x)}{2\ 3^{2/3} \sqrt [3]{x^2+2 x+1}}+\frac {\sqrt [3]{3} (x+1)^{2/3} \log \left (\sqrt [3]{3}-\sqrt [3]{x+1}\right )}{2 \sqrt [3]{x^2+2 x+1}}-\frac {(x+1)^{2/3} \tan ^{-1}\left (\frac {2 \sqrt [3]{x+1}+\sqrt [3]{3}}{3^{5/6}}\right )}{\sqrt [6]{3} \sqrt [3]{x^2+2 x+1}} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 57
Rule 204
Rule 617
Rule 646
Rubi steps
\begin {align*} \int \frac {1}{(-2+x) \sqrt [3]{1+2 x+x^2}} \, dx &=\frac {(1+x)^{2/3} \int \frac {1}{(-2+x) (1+x)^{2/3}} \, dx}{\sqrt [3]{1+2 x+x^2}}\\ &=-\frac {(1+x)^{2/3} \log (2-x)}{2\ 3^{2/3} \sqrt [3]{1+2 x+x^2}}-\frac {\left (\sqrt [3]{3} (1+x)^{2/3}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{3}-x} \, dx,x,\sqrt [3]{1+x}\right )}{2 \sqrt [3]{1+2 x+x^2}}-\frac {\left (3^{2/3} (1+x)^{2/3}\right ) \operatorname {Subst}\left (\int \frac {1}{3^{2/3}+\sqrt [3]{3} x+x^2} \, dx,x,\sqrt [3]{1+x}\right )}{2 \sqrt [3]{1+2 x+x^2}}\\ &=-\frac {(1+x)^{2/3} \log (2-x)}{2\ 3^{2/3} \sqrt [3]{1+2 x+x^2}}+\frac {\sqrt [3]{3} (1+x)^{2/3} \log \left (\sqrt [3]{3}-\sqrt [3]{1+x}\right )}{2 \sqrt [3]{1+2 x+x^2}}+\frac {\left (\sqrt [3]{3} (1+x)^{2/3}\right ) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2 \sqrt [3]{1+x}}{\sqrt [3]{3}}\right )}{\sqrt [3]{1+2 x+x^2}}\\ &=-\frac {(1+x)^{2/3} \tan ^{-1}\left (\frac {1}{3} \left (\sqrt {3}+2 \sqrt [6]{3} \sqrt [3]{1+x}\right )\right )}{\sqrt [6]{3} \sqrt [3]{1+2 x+x^2}}-\frac {(1+x)^{2/3} \log (2-x)}{2\ 3^{2/3} \sqrt [3]{1+2 x+x^2}}+\frac {\sqrt [3]{3} (1+x)^{2/3} \log \left (\sqrt [3]{3}-\sqrt [3]{1+x}\right )}{2 \sqrt [3]{1+2 x+x^2}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 100, normalized size = 0.94 \begin {gather*} -\frac {(x+1)^{2/3} \left (-2 \log \left (\sqrt [3]{3}-\sqrt [3]{x+1}\right )+\log \left ((x+1)^{2/3}+\sqrt [3]{3} \sqrt [3]{x+1}+3^{2/3}\right )+2 \sqrt {3} \tan ^{-1}\left (\frac {2 \sqrt [3]{x+1}+\sqrt [3]{3}}{3^{5/6}}\right )\right )}{2\ 3^{2/3} \sqrt [3]{(x+1)^2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.22, size = 153, normalized size = 1.44 \begin {gather*} \frac {\log \left (-3 \sqrt [3]{x^2+2 x+1}+3^{2/3} x+3^{2/3}\right )}{3^{2/3}}-\frac {\log \left (\sqrt [3]{3} \left (x^2+2 x+1\right )^{2/3}+3 \sqrt [3]{x^2+2 x+1}+3^{2/3} x+3^{2/3}\right )}{2\ 3^{2/3}}+\frac {\tan ^{-1}\left (\frac {3^{5/6} \sqrt [3]{x^2+2 x+1}}{\sqrt [3]{3} \sqrt [3]{x^2+2 x+1}+2 x+2}\right )}{\sqrt [6]{3}}-\frac {\log (x+1)}{3\ 3^{2/3}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 139, normalized size = 1.31 \begin {gather*} \frac {1}{3} \cdot 9^{\frac {1}{6}} \sqrt {3} \arctan \left (\frac {9^{\frac {1}{6}} {\left (9^{\frac {1}{3}} \sqrt {3} {\left (x + 1\right )} + 6 \, \sqrt {3} {\left (x^{2} + 2 \, x + 1\right )}^{\frac {1}{3}}\right )}}{9 \, {\left (x + 1\right )}}\right ) - \frac {1}{18} \cdot 9^{\frac {2}{3}} \log \left (\frac {9^{\frac {2}{3}} {\left (x^{2} + 2 \, x + 1\right )} + 3 \cdot 9^{\frac {1}{3}} {\left (x^{2} + 2 \, x + 1\right )}^{\frac {1}{3}} {\left (x + 1\right )} + 9 \, {\left (x^{2} + 2 \, x + 1\right )}^{\frac {2}{3}}}{x^{2} + 2 \, x + 1}\right ) + \frac {1}{9} \cdot 9^{\frac {2}{3}} \log \left (-\frac {9^{\frac {1}{3}} {\left (x + 1\right )} - 3 \, {\left (x^{2} + 2 \, x + 1\right )}^{\frac {1}{3}}}{x + 1}\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} + 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 = 2.69, size = 942, normalized size = 8.89
result too large to display
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} + 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+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]{\left (x + 1\right )^{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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