Optimal. Leaf size=33 \[ -\frac {2 \tanh ^{-1}\left (\frac {\sqrt {x^3+1}}{\sqrt {3} \left (x^2-x+1\right )}\right )}{\sqrt {3}} \]
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Rubi [A] time = 0.07, antiderivative size = 26, normalized size of antiderivative = 0.79, number of steps used = 2, number of rules used = 2, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {2145, 206} \begin {gather*} -\frac {2 \tanh ^{-1}\left (\frac {x+1}{\sqrt {3} \sqrt {x^3+1}}\right )}{\sqrt {3}} \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 (2-4 x+3 x^2\right ) \sqrt {1+x^3}} \, dx &=-\left (4 \operatorname {Subst}\left (\int \frac {1}{6-2 x^2} \, dx,x,\frac {1+x}{\sqrt {1+x^3}}\right )\right )\\ &=-\frac {2 \tanh ^{-1}\left (\frac {1+x}{\sqrt {3} \sqrt {1+x^3}}\right )}{\sqrt {3}}\\ \end {align*}
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Mathematica [C] time = 1.45, size = 394, normalized size = 11.94 \begin {gather*} \frac {2 \sqrt {\frac {x+1}{1+\sqrt [3]{-1}}} \sqrt {x^2-x+1} \left (-\frac {\sqrt {3} \left (1+\sqrt [3]{-1}\right ) \left (\sqrt [3]{-1}-x\right ) F\left (\sin ^{-1}\left (\sqrt {\frac {(-1)^{2/3} x+1}{1+\sqrt [3]{-1}}}\right )|\sqrt [3]{-1}\right )}{(-1)^{2/3} x+1}+\frac {3 i \sqrt {2} \Pi \left (\frac {6 \sqrt {3}}{i-2 \sqrt {2}+3 \sqrt {3}};\sin ^{-1}\left (\sqrt {\frac {(-1)^{2/3} x+1}{1+\sqrt [3]{-1}}}\right )|\sqrt [3]{-1}\right )}{-2 i+3 (-1)^{5/6}+\sqrt {2}}+\frac {15 \Pi \left (\frac {6 \sqrt {3}}{i-2 \sqrt {2}+3 \sqrt {3}};\sin ^{-1}\left (\sqrt {\frac {\left (i+\sqrt {3}\right ) x-2 i}{-3 i+\sqrt {3}}}\right )|\frac {1}{2} \left (1+i \sqrt {3}\right )\right )}{-2 i+3 (-1)^{5/6}+\sqrt {2}}+\frac {6 i \left (\sqrt {2}+5 i\right ) \Pi \left (\frac {6 \sqrt {3}}{i+2 \sqrt {2}+3 \sqrt {3}};\sin ^{-1}\left (\sqrt {\frac {\left (i+\sqrt {3}\right ) x-2 i}{-3 i+\sqrt {3}}}\right )|\frac {1}{2} \left (1+i \sqrt {3}\right )\right )}{i+2 \sqrt {2}+3 \sqrt {3}}\right )}{9 \sqrt {x^3+1}} \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [A] time = 1.09, size = 33, normalized size = 1.00 \begin {gather*} -\frac {2 \tanh ^{-1}\left (\frac {\sqrt {1+x^3}}{\sqrt {3} \left (1-x+x^2\right )}\right )}{\sqrt {3}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.51, size = 66, normalized size = 2.00 \begin {gather*} \frac {1}{6} \, \sqrt {3} \log \left (\frac {9 \, x^{4} - 4 \, \sqrt {3} \sqrt {x^{3} + 1} {\left (3 \, x^{2} - 2 \, x + 4\right )} + 28 \, x^{2} - 16 \, x + 28}{9 \, x^{4} - 24 \, x^{3} + 28 \, x^{2} - 16 \, x + 4}\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 (3 \, x^{2} - 4 \, x + 2\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.60, size = 61, normalized size = 1.85
method | result | size |
trager | \(\frac {\RootOf \left (\textit {\_Z}^{2}-3\right ) \ln \left (\frac {3 \RootOf \left (\textit {\_Z}^{2}-3\right ) x^{2}-2 \RootOf \left (\textit {\_Z}^{2}-3\right ) x +4 \RootOf \left (\textit {\_Z}^{2}-3\right )-6 \sqrt {x^{3}+1}}{3 x^{2}-4 x +2}\right )}{3}\) | \(61\) |
default | \(\frac {2 \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \EllipticF \left (\sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{3 \sqrt {x^{3}+1}}+\frac {4 \left (\frac {5}{6}+\frac {i \sqrt {2}}{6}\right ) \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \left (-\frac {5}{9}+\frac {i \sqrt {2}}{9}\right ) \EllipticPi \left (\sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \frac {5}{6}-\frac {i \sqrt {2}}{6}-\frac {7 i \sqrt {3}}{18}+\frac {i \left (\frac {2}{3}+\frac {i \sqrt {2}}{3}\right ) \sqrt {3}}{6}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{3 \sqrt {x^{3}+1}}+\frac {4 \left (\frac {5}{6}-\frac {i \sqrt {2}}{6}\right ) \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \left (-\frac {5}{9}-\frac {i \sqrt {2}}{9}\right ) \EllipticPi \left (\sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \frac {5}{6}+\frac {i \sqrt {2}}{6}-\frac {7 i \sqrt {3}}{18}+\frac {i \left (\frac {2}{3}-\frac {i \sqrt {2}}{3}\right ) \sqrt {3}}{6}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{3 \sqrt {x^{3}+1}}\) | \(435\) |
elliptic | \(\frac {2 \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \EllipticF \left (\sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{3 \sqrt {x^{3}+1}}+\frac {2 \left (\frac {5}{9}+\frac {i \sqrt {2}}{9}\right ) \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \left (-\frac {5}{9}+\frac {i \sqrt {2}}{9}\right ) \EllipticPi \left (\sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \frac {5}{6}-\frac {i \sqrt {2}}{6}-\frac {7 i \sqrt {3}}{18}+\frac {i \left (\frac {2}{3}+\frac {i \sqrt {2}}{3}\right ) \sqrt {3}}{6}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {x^{3}+1}}+\frac {2 \left (\frac {5}{9}-\frac {i \sqrt {2}}{9}\right ) \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \left (-\frac {5}{9}-\frac {i \sqrt {2}}{9}\right ) \EllipticPi \left (\sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \frac {5}{6}+\frac {i \sqrt {2}}{6}-\frac {7 i \sqrt {3}}{18}+\frac {i \left (\frac {2}{3}-\frac {i \sqrt {2}}{3}\right ) \sqrt {3}}{6}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {x^{3}+1}}\) | \(435\) |
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 (3 \, x^{2} - 4 \, x + 2\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.21, size = 274, normalized size = 8.30 \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+1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {\frac {\frac {1}{2}-x+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{\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 {5}{3}+\frac {\sqrt {2}\,1{}\mathrm {i}}{3}};\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 {5}{3}+\frac {\sqrt {2}\,1{}\mathrm {i}}{3}};\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 )}{3\,\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 (3 x^{2} - 4 x + 2\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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