Optimal. Leaf size=31 \[ -\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {x^3-1}}{x^2+x+1}\right ) \]
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Rubi [A] time = 0.07, antiderivative size = 27, normalized size of antiderivative = 0.87, number of steps used = 2, number of rules used = 2, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {2145, 206} \begin {gather*} \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} (1-x)}{\sqrt {x^3-1}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 2145
Rubi steps
\begin {align*} \int \frac {-2-2 x+x^2}{\left (3-x+x^2\right ) \sqrt {-1+x^3}} \, dx &=4 \operatorname {Subst}\left (\int \frac {1}{2-4 x^2} \, dx,x,\frac {1-x}{\sqrt {-1+x^3}}\right )\\ &=\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} (1-x)}{\sqrt {-1+x^3}}\right )\\ \end {align*}
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Mathematica [C] time = 0.72, size = 496, normalized size = 16.00 \begin {gather*} \frac {2 \sqrt {\frac {1-x}{1+\sqrt [3]{-1}}} \left (\frac {(-1)^{5/6} \left (1+\sqrt [3]{-1}\right ) \sqrt {x^2+x+1} \Pi \left (\frac {2 \sqrt {3}}{-2 i+\sqrt {3}-\sqrt {11}};\sin ^{-1}\left (\sqrt {\frac {1-(-1)^{2/3} x}{1+\sqrt [3]{-1}}}\right )|\sqrt [3]{-1}\right )}{-3+2 i \sqrt {3}+\sqrt {33}}-\frac {i \sqrt {11} \sqrt {x^2+x+1} \Pi \left (\frac {2 \sqrt {3}}{-2 i+\sqrt {3}-\sqrt {11}};\sin ^{-1}\left (\sqrt {\frac {1-(-1)^{2/3} x}{1+\sqrt [3]{-1}}}\right )|\sqrt [3]{-1}\right )}{2 i-\sqrt {3}+\sqrt {11}}-\frac {(-1)^{5/6} \left (1+\sqrt [3]{-1}\right ) \sqrt {x^2+x+1} \Pi \left (\frac {2 \sqrt {3}}{-2 i+\sqrt {3}+\sqrt {11}};\sin ^{-1}\left (\sqrt {\frac {1-(-1)^{2/3} x}{1+\sqrt [3]{-1}}}\right )|\sqrt [3]{-1}\right )}{3-2 i \sqrt {3}+\sqrt {33}}-\frac {i \sqrt {11} \sqrt {x^2+x+1} \Pi \left (\frac {2 \sqrt {3}}{-2 i+\sqrt {3}+\sqrt {11}};\sin ^{-1}\left (\sqrt {\frac {1-(-1)^{2/3} x}{1+\sqrt [3]{-1}}}\right )|\sqrt [3]{-1}\right )}{-2 i+\sqrt {3}+\sqrt {11}}+\frac {\left (x+\sqrt [3]{-1}\right ) \sqrt {\frac {(-1)^{2/3} x+\sqrt [3]{-1}}{1+\sqrt [3]{-1}}} F\left (\sin ^{-1}\left (\sqrt {\frac {1-(-1)^{2/3} x}{1+\sqrt [3]{-1}}}\right )|\sqrt [3]{-1}\right )}{\sqrt {\frac {1-(-1)^{2/3} x}{1+\sqrt [3]{-1}}}}\right )}{\sqrt {x^3-1}} \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [A] time = 1.05, size = 31, normalized size = 1.00 \begin {gather*} -\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {-1+x^3}}{1+x+x^2}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.45, size = 65, normalized size = 2.10 \begin {gather*} \frac {1}{4} \, \sqrt {2} \log \left (\frac {x^{4} + 14 \, x^{3} - 4 \, \sqrt {2} \sqrt {x^{3} - 1} {\left (x^{2} + 3 \, x - 1\right )} + 7 \, x^{2} - 6 \, x - 7}{x^{4} - 2 \, x^{3} + 7 \, x^{2} - 6 \, x + 9}\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^{2} - 2 \, x - 2}{\sqrt {x^{3} - 1} {\left (x^{2} - x + 3\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.52, size = 58, normalized size = 1.87
method | result | size |
trager | \(-\frac {\RootOf \left (\textit {\_Z}^{2}-2\right ) \ln \left (\frac {\RootOf \left (\textit {\_Z}^{2}-2\right ) x^{2}+3 \RootOf \left (\textit {\_Z}^{2}-2\right ) x -\RootOf \left (\textit {\_Z}^{2}-2\right )+4 \sqrt {x^{3}-1}}{x^{2}-x +3}\right )}{2}\) | \(58\) |
default | \(\frac {2 \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {-1+x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \EllipticF \left (\sqrt {\frac {-1+x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {x^{3}-1}}-\frac {2 \left (\frac {1}{2}-\frac {i \sqrt {11}}{2}\right ) \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {-1+x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \left (\frac {1}{2}+\frac {i \sqrt {11}}{2}\right ) \EllipticPi \left (\sqrt {\frac {-1+x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \frac {1}{4}+\frac {i \sqrt {11}}{4}+\frac {i \left (\frac {1}{2}+\frac {i \sqrt {11}}{2}\right ) \sqrt {3}}{6}, \sqrt {\frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{3 \sqrt {x^{3}-1}}-\frac {2 \left (\frac {1}{2}+\frac {i \sqrt {11}}{2}\right ) \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {-1+x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \left (\frac {1}{2}-\frac {i \sqrt {11}}{2}\right ) \EllipticPi \left (\sqrt {\frac {-1+x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \frac {1}{4}-\frac {i \sqrt {11}}{4}+\frac {i \left (\frac {1}{2}-\frac {i \sqrt {11}}{2}\right ) \sqrt {3}}{6}, \sqrt {\frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{3 \sqrt {x^{3}-1}}\) | \(423\) |
elliptic | \(\frac {2 \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {-1+x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \EllipticF \left (\sqrt {\frac {-1+x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {x^{3}-1}}+\frac {2 \left (-\frac {1}{2}+\frac {i \sqrt {11}}{2}\right ) \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {-1+x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \left (\frac {1}{2}+\frac {i \sqrt {11}}{2}\right ) \EllipticPi \left (\sqrt {\frac {-1+x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \frac {1}{4}+\frac {i \sqrt {11}}{4}+\frac {i \left (\frac {1}{2}+\frac {i \sqrt {11}}{2}\right ) \sqrt {3}}{6}, \sqrt {\frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{3 \sqrt {x^{3}-1}}+\frac {2 \left (-\frac {1}{2}-\frac {i \sqrt {11}}{2}\right ) \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {-1+x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \left (\frac {1}{2}-\frac {i \sqrt {11}}{2}\right ) \EllipticPi \left (\sqrt {\frac {-1+x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \frac {1}{4}-\frac {i \sqrt {11}}{4}+\frac {i \left (\frac {1}{2}-\frac {i \sqrt {11}}{2}\right ) \sqrt {3}}{6}, \sqrt {\frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{3 \sqrt {x^{3}-1}}\) | \(423\) |
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^{2} - 2 \, x - 2}{\sqrt {x^{3} - 1} {\left (x^{2} - x + 3\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.20, size = 275, normalized size = 8.87 \begin {gather*} \frac {\left (3+\sqrt {3}\,1{}\mathrm {i}\right )\,\sqrt {-\frac {x+\frac {1}{2}-\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {\frac {x+\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {-\frac {x-1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\left (-\mathrm {F}\left (\mathrm {asin}\left (\sqrt {-\frac {x-1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\right )\middle |-\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}\right )+\Pi \left (\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{\frac {1}{2}+\frac {\sqrt {11}\,1{}\mathrm {i}}{2}};\mathrm {asin}\left (\sqrt {-\frac {x-1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\right )\middle |-\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}\right )+\Pi \left (-\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {1}{2}+\frac {\sqrt {11}\,1{}\mathrm {i}}{2}};\mathrm {asin}\left (\sqrt {-\frac {x-1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\right )\middle |-\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}\right )\right )}{\sqrt {x^3+\left (-\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )-1\right )\,x+\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}} \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^{2} - 2 x - 2}{\sqrt {\left (x - 1\right ) \left (x^{2} + x + 1\right )} \left (x^{2} - x + 3\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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