Optimal. Leaf size=125 \[ -\frac {\log \left (\sqrt [3]{2} \sqrt [3]{x^2-2 x-3}+2\right )}{2\ 2^{2/3}}+\frac {\log \left (2^{2/3} \left (x^2-2 x-3\right )^{2/3}-2 \sqrt [3]{2} \sqrt [3]{x^2-2 x-3}+4\right )}{4\ 2^{2/3}}-\frac {\sqrt {3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {\sqrt [3]{2} \sqrt [3]{x^2-2 x-3}}{\sqrt {3}}\right )}{2\ 2^{2/3}} \]
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Rubi [A] time = 0.06, antiderivative size = 84, normalized size of antiderivative = 0.67, number of steps used = 6, number of rules used = 6, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {694, 266, 56, 617, 204, 31} \begin {gather*} -\frac {3 \log \left (\sqrt [3]{(x-1)^2-4}+2^{2/3}\right )}{4\ 2^{2/3}}+\frac {\log (1-x)}{2\ 2^{2/3}}-\frac {\sqrt {3} \tan ^{-1}\left (\frac {1-\sqrt [3]{2} \sqrt [3]{(x-1)^2-4}}{\sqrt {3}}\right )}{2\ 2^{2/3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 56
Rule 204
Rule 266
Rule 617
Rule 694
Rubi steps
\begin {align*} \int \frac {1}{(-1+x) \sqrt [3]{-3-2 x+x^2}} \, dx &=\operatorname {Subst}\left (\int \frac {1}{x \sqrt [3]{-4+x^2}} \, dx,x,-1+x\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{-4+x} x} \, dx,x,(-1+x)^2\right )\\ &=\frac {\log (1-x)}{2\ 2^{2/3}}+\frac {3}{4} \operatorname {Subst}\left (\int \frac {1}{2 \sqrt [3]{2}-2^{2/3} x+x^2} \, dx,x,\sqrt [3]{-4+(-1+x)^2}\right )-\frac {3 \operatorname {Subst}\left (\int \frac {1}{2^{2/3}+x} \, dx,x,\sqrt [3]{-4+(-1+x)^2}\right )}{4\ 2^{2/3}}\\ &=-\frac {3 \log \left (2^{2/3}+\sqrt [3]{-4+(-1+x)^2}\right )}{4\ 2^{2/3}}+\frac {\log (1-x)}{2\ 2^{2/3}}+\frac {3 \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\sqrt [3]{2} \sqrt [3]{-4+(-1+x)^2}\right )}{2\ 2^{2/3}}\\ &=-\frac {\sqrt {3} \tan ^{-1}\left (\frac {1-\sqrt [3]{2} \sqrt [3]{-4+(-1+x)^2}}{\sqrt {3}}\right )}{2\ 2^{2/3}}-\frac {3 \log \left (2^{2/3}+\sqrt [3]{-4+(-1+x)^2}\right )}{4\ 2^{2/3}}+\frac {\log (1-x)}{2\ 2^{2/3}}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 36, normalized size = 0.29 \begin {gather*} \frac {3}{16} \left ((x-1)^2-4\right )^{2/3} \, _2F_1\left (\frac {2}{3},1;\frac {5}{3};\frac {1}{4} \left (4-(x-1)^2\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.22, size = 125, normalized size = 1.00 \begin {gather*} -\frac {\log \left (\sqrt [3]{2} \sqrt [3]{x^2-2 x-3}+2\right )}{2\ 2^{2/3}}+\frac {\log \left (2^{2/3} \left (x^2-2 x-3\right )^{2/3}-2 \sqrt [3]{2} \sqrt [3]{x^2-2 x-3}+4\right )}{4\ 2^{2/3}}-\frac {\sqrt {3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {\sqrt [3]{2} \sqrt [3]{x^2-2 x-3}}{\sqrt {3}}\right )}{2\ 2^{2/3}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 113, normalized size = 0.90 \begin {gather*} \frac {1}{4} \cdot 4^{\frac {1}{6}} \sqrt {3} \left (-1\right )^{\frac {1}{3}} \arctan \left (\frac {1}{6} \cdot 4^{\frac {1}{6}} \sqrt {3} {\left (2 \, \left (-1\right )^{\frac {1}{3}} {\left (x^{2} - 2 \, x - 3\right )}^{\frac {1}{3}} + 4^{\frac {1}{3}}\right )}\right ) - \frac {1}{16} \cdot 4^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} \log \left (-4^{\frac {1}{3}} \left (-1\right )^{\frac {2}{3}} {\left (x^{2} - 2 \, x - 3\right )}^{\frac {1}{3}} - 4^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} + {\left (x^{2} - 2 \, x - 3\right )}^{\frac {2}{3}}\right ) + \frac {1}{8} \cdot 4^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} \log \left (4^{\frac {1}{3}} \left (-1\right )^{\frac {2}{3}} + {\left (x^{2} - 2 \, x - 3\right )}^{\frac {1}{3}}\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 - 3\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 = 8.00, size = 1199, normalized size = 9.59 \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} - 2 \, x - 3\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 = -0.01 \begin {gather*} \int \frac {1}{\left (x-1\right )\,{\left (x^2-2\,x-3\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}{\sqrt [3]{\left (x - 3\right ) \left (x + 1\right )} \left (x - 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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