Optimal. Leaf size=125 \[ \frac {\sqrt [3]{x-1} (x+1)^{2/3} \left (-\log \left (\sqrt [3]{x-1}-\sqrt [3]{x+1}\right )+\frac {1}{2} \log \left ((x-1)^{2/3}+\sqrt [3]{x+1} \sqrt [3]{x-1}+(x+1)^{2/3}\right )+\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{x+1}}{2 \sqrt [3]{x-1}+\sqrt [3]{x+1}}\right )\right )}{\sqrt [6]{(x-1)^2 (x+1)^4}} \]
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Rubi [A] time = 0.04, antiderivative size = 159, normalized size of antiderivative = 1.27, number of steps used = 3, number of rules used = 3, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.094, Rules used = {6688, 6719, 59} \begin {gather*} -\frac {\sqrt [3]{x-1} (x+1)^{2/3} \log (x+1)}{2 \sqrt [6]{(1-x)^2 (x+1)^4}}-\frac {3 \sqrt [3]{x-1} (x+1)^{2/3} \log \left (\frac {\sqrt [3]{x-1}}{\sqrt [3]{x+1}}-1\right )}{2 \sqrt [6]{(1-x)^2 (x+1)^4}}-\frac {\sqrt {3} \sqrt [3]{x-1} (x+1)^{2/3} \tan ^{-1}\left (\frac {2 \sqrt [3]{x-1}}{\sqrt {3} \sqrt [3]{x+1}}+\frac {1}{\sqrt {3}}\right )}{\sqrt [6]{(1-x)^2 (x+1)^4}} \end {gather*}
Antiderivative was successfully verified.
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Rule 59
Rule 6688
Rule 6719
Rubi steps
\begin {align*} \int \frac {1}{\sqrt [6]{1+2 x-x^2-4 x^3-x^4+2 x^5+x^6}} \, dx &=\int \frac {1}{\sqrt [6]{(-1+x)^2 (1+x)^4}} \, dx\\ &=\frac {\left (\sqrt [3]{-1+x} (1+x)^{2/3}\right ) \int \frac {1}{\sqrt [3]{-1+x} (1+x)^{2/3}} \, dx}{\sqrt [6]{(-1+x)^2 (1+x)^4}}\\ &=-\frac {\sqrt {3} \sqrt [3]{-1+x} (1+x)^{2/3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{-1+x}}{\sqrt {3} \sqrt [3]{1+x}}\right )}{\sqrt [6]{(1-x)^2 (1+x)^4}}-\frac {\sqrt [3]{-1+x} (1+x)^{2/3} \log (1+x)}{2 \sqrt [6]{(1-x)^2 (1+x)^4}}-\frac {3 \sqrt [3]{-1+x} (1+x)^{2/3} \log \left (-1+\frac {\sqrt [3]{-1+x}}{\sqrt [3]{1+x}}\right )}{2 \sqrt [6]{(1-x)^2 (1+x)^4}}\\ \end {align*}
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Mathematica [C] time = 0.02, size = 53, normalized size = 0.42 \begin {gather*} \frac {3 (x-1) (x+1)^{2/3} \, _2F_1\left (\frac {2}{3},\frac {2}{3};\frac {5}{3};\frac {1-x}{2}\right )}{2\ 2^{2/3} \sqrt [6]{(x-1)^2 (x+1)^4}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 11.53, size = 122, normalized size = 0.98 \begin {gather*} \frac {\sqrt [3]{x-1} (x+1)^{2/3} \left (-\log \left (\frac {\sqrt [3]{x+1}}{\sqrt [3]{x-1}}-1\right )+\frac {1}{2} \log \left (\frac {(x+1)^{2/3}}{(x-1)^{2/3}}+\frac {\sqrt [3]{x+1}}{\sqrt [3]{x-1}}+1\right )+\sqrt {3} \tan ^{-1}\left (\frac {2 \sqrt [3]{x+1}}{\sqrt {3} \sqrt [3]{x-1}}+\frac {1}{\sqrt {3}}\right )\right )}{\sqrt [6]{(x-1)^2 (x+1)^4}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.39, size = 188, normalized size = 1.50 \begin {gather*} -\sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (x + 1\right )} + 2 \, \sqrt {3} {\left (x^{6} + 2 \, x^{5} - x^{4} - 4 \, x^{3} - x^{2} + 2 \, x + 1\right )}^{\frac {1}{6}}}{3 \, {\left (x + 1\right )}}\right ) + \frac {1}{2} \, \log \left (\frac {x^{2} + {\left (x^{6} + 2 \, x^{5} - x^{4} - 4 \, x^{3} - x^{2} + 2 \, x + 1\right )}^{\frac {1}{6}} {\left (x + 1\right )} + 2 \, x + {\left (x^{6} + 2 \, x^{5} - x^{4} - 4 \, x^{3} - x^{2} + 2 \, x + 1\right )}^{\frac {1}{3}} + 1}{x^{2} + 2 \, x + 1}\right ) - \log \left (-\frac {x - {\left (x^{6} + 2 \, x^{5} - x^{4} - 4 \, x^{3} - x^{2} + 2 \, x + 1\right )}^{\frac {1}{6}} + 1}{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^{6} + 2 \, x^{5} - x^{4} - 4 \, x^{3} - x^{2} + 2 \, x + 1\right )}^{\frac {1}{6}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.03, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (x^{6}+2 x^{5}-x^{4}-4 x^{3}-x^{2}+2 x +1\right )^{\frac {1}{6}}}\, dx \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^{6} + 2 \, x^{5} - x^{4} - 4 \, x^{3} - x^{2} + 2 \, x + 1\right )}^{\frac {1}{6}}}\,{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^6+2\,x^5-x^4-4\,x^3-x^2+2\,x+1\right )}^{1/6}} \,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}{\sqrt [6]{x^{6} + 2 x^{5} - x^{4} - 4 x^{3} - x^{2} + 2 x + 1}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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