Optimal. Leaf size=243 \[ \frac {\log \left (-2 \sqrt [3]{x^3-1}+\sqrt [3]{2} x-\sqrt [3]{2}\right )}{\sqrt [3]{2}}-\frac {1}{3} \log \left (\sqrt [3]{x^3-1}-x\right )+\frac {\tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{x^3-1}+x}\right )}{\sqrt {3}}-\frac {\sqrt {3} \tan ^{-1}\left (\frac {\frac {\sqrt [3]{x^3-1}}{\sqrt {3}}+\frac {\sqrt [3]{2} x}{\sqrt {3}}-\frac {\sqrt [3]{2}}{\sqrt {3}}}{\sqrt [3]{x^3-1}}\right )}{\sqrt [3]{2}}+\frac {1}{6} \log \left (\sqrt [3]{x^3-1} x+\left (x^3-1\right )^{2/3}+x^2\right )-\frac {\log \left (4 \left (x^3-1\right )^{2/3}+\left (2 \sqrt [3]{2} x-2 \sqrt [3]{2}\right ) \sqrt [3]{x^3-1}+2^{2/3} x^2-2\ 2^{2/3} x+2^{2/3}\right )}{2 \sqrt [3]{2}} \]
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Rubi [A] time = 0.10, antiderivative size = 139, normalized size of antiderivative = 0.57, number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2152, 239, 2148} \begin {gather*} -\frac {1}{2} \log \left (\sqrt [3]{x^3-1}-x\right )+\frac {3 \log \left (2^{2/3} \sqrt [3]{x^3-1}-x+1\right )}{2 \sqrt [3]{2}}-\frac {\sqrt {3} \tan ^{-1}\left (\frac {1-\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{x^3-1}}}{\sqrt {3}}\right )}{\sqrt [3]{2}}+\frac {\tan ^{-1}\left (\frac {\frac {2 x}{\sqrt [3]{x^3-1}}+1}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {\log \left ((1-x) (x+1)^2\right )}{2 \sqrt [3]{2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 239
Rule 2148
Rule 2152
Rubi steps
\begin {align*} \int \frac {-1+x}{(1+x) \sqrt [3]{-1+x^3}} \, dx &=-\left (2 \int \frac {1}{(1+x) \sqrt [3]{-1+x^3}} \, dx\right )+\int \frac {1}{\sqrt [3]{-1+x^3}} \, dx\\ &=-\frac {\sqrt {3} \tan ^{-1}\left (\frac {1-\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{\sqrt [3]{2}}+\frac {\tan ^{-1}\left (\frac {1+\frac {2 x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {\log \left ((1-x) (1+x)^2\right )}{2 \sqrt [3]{2}}-\frac {1}{2} \log \left (-x+\sqrt [3]{-1+x^3}\right )+\frac {3 \log \left (1-x+2^{2/3} \sqrt [3]{-1+x^3}\right )}{2 \sqrt [3]{2}}\\ \end {align*}
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Mathematica [F] time = 0.19, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {-1+x}{(1+x) \sqrt [3]{-1+x^3}} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 5.43, size = 243, normalized size = 1.00 \begin {gather*} \frac {\tan ^{-1}\left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{-1+x^3}}\right )}{\sqrt {3}}-\frac {\sqrt {3} \tan ^{-1}\left (\frac {-\frac {\sqrt [3]{2}}{\sqrt {3}}+\frac {\sqrt [3]{2} x}{\sqrt {3}}+\frac {\sqrt [3]{-1+x^3}}{\sqrt {3}}}{\sqrt [3]{-1+x^3}}\right )}{\sqrt [3]{2}}+\frac {\log \left (-\sqrt [3]{2}+\sqrt [3]{2} x-2 \sqrt [3]{-1+x^3}\right )}{\sqrt [3]{2}}-\frac {1}{3} \log \left (-x+\sqrt [3]{-1+x^3}\right )+\frac {1}{6} \log \left (x^2+x \sqrt [3]{-1+x^3}+\left (-1+x^3\right )^{2/3}\right )-\frac {\log \left (2^{2/3}-2\ 2^{2/3} x+2^{2/3} x^2+\left (-2 \sqrt [3]{2}+2 \sqrt [3]{2} x\right ) \sqrt [3]{-1+x^3}+4 \left (-1+x^3\right )^{2/3}\right )}{2 \sqrt [3]{2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 2.50, size = 370, normalized size = 1.52 \begin {gather*} \frac {1}{6} \cdot 4^{\frac {1}{3}} \sqrt {3} \arctan \left (\frac {4 \cdot 4^{\frac {2}{3}} \sqrt {3} {\left (x^{4} + 2 \, x^{3} + 2 \, x^{2} + 2 \, x + 1\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}} + 2 \cdot 4^{\frac {1}{3}} \sqrt {3} {\left (5 \, x^{5} - 5 \, x^{4} + 6 \, x^{3} - 6 \, x^{2} + 5 \, x - 5\right )} {\left (x^{3} - 1\right )}^{\frac {1}{3}} + \sqrt {3} {\left (13 \, x^{6} + 2 \, x^{5} + 19 \, x^{4} - 4 \, x^{3} + 19 \, x^{2} + 2 \, x + 13\right )}}{3 \, {\left (3 \, x^{6} - 18 \, x^{5} - 3 \, x^{4} - 28 \, x^{3} - 3 \, x^{2} - 18 \, x + 3\right )}}\right ) + \frac {1}{3} \, \sqrt {3} \arctan \left (-\frac {4 \, \sqrt {3} {\left (x^{3} - 1\right )}^{\frac {1}{3}} x^{2} - 2 \, \sqrt {3} {\left (x^{3} - 1\right )}^{\frac {2}{3}} x + \sqrt {3} {\left (x^{3} - 1\right )}}{9 \, x^{3} - 1}\right ) - \frac {1}{12} \cdot 4^{\frac {1}{3}} \log \left (\frac {8 \cdot 4^{\frac {1}{3}} {\left (x^{3} - 1\right )}^{\frac {2}{3}} {\left (x^{2} + 1\right )} + 4^{\frac {2}{3}} {\left (5 \, x^{4} + 6 \, x^{2} + 5\right )} + 4 \, {\left (3 \, x^{3} - x^{2} + x - 3\right )} {\left (x^{3} - 1\right )}^{\frac {1}{3}}}{x^{4} + 4 \, x^{3} + 6 \, x^{2} + 4 \, x + 1}\right ) + \frac {1}{6} \cdot 4^{\frac {1}{3}} \log \left (\frac {4^{\frac {2}{3}} {\left (x^{3} - 1\right )}^{\frac {1}{3}} {\left (x - 1\right )} + 4^{\frac {1}{3}} {\left (x^{2} + 2 \, x + 1\right )} - 4 \, {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{x^{2} + 2 \, x + 1}\right ) - \frac {1}{6} \, \log \left (-3 \, {\left (x^{3} - 1\right )}^{\frac {1}{3}} x^{2} + 3 \, {\left (x^{3} - 1\right )}^{\frac {2}{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 {x - 1}{{\left (x^{3} - 1\right )}^{\frac {1}{3}} {\left (x + 1\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 21.08, size = 1639, normalized size = 6.74
method | result | size |
trager | \(\text {Expression too large to display}\) | \(1639\) |
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 {x - 1}{{\left (x^{3} - 1\right )}^{\frac {1}{3}} {\left (x + 1\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 = -0.00 \begin {gather*} \int \frac {x-1}{{\left (x^3-1\right )}^{1/3}\,\left (x+1\right )} \,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 {x - 1}{\sqrt [3]{\left (x - 1\right ) \left (x^{2} + x + 1\right )} \left (x + 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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