Optimal. Leaf size=89 \[ -\frac {\sqrt {2} \tan ^{-1}\left (\frac {\sqrt {2 \sqrt {3}-3} \sqrt {x^3-1}}{x^2+x+1}\right )}{3^{3/4}}-\frac {\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {3+2 \sqrt {3}} \sqrt {x^3-1}}{x^2+x+1}\right )}{3^{3/4}} \]
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Rubi [C] time = 0.94, antiderivative size = 433, normalized size of antiderivative = 4.87, number of steps used = 13, number of rules used = 6, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {6728, 219, 2135, 2140, 206, 203} \begin {gather*} \frac {\sqrt {2} \tan ^{-1}\left (\frac {\sqrt {2 \sqrt {3}-3} (1-x)}{\sqrt {x^3-1}}\right )}{3^{3/4}}+\frac {\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {3+2 \sqrt {3}} (1-x)}{\sqrt {x^3-1}}\right )}{3^{3/4}}-\frac {2 \sqrt {2-\sqrt {3}} (1-x) \sqrt {\frac {x^2+x+1}{\left (-x-\sqrt {3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac {-x+\sqrt {3}+1}{-x-\sqrt {3}+1}\right )|-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {-\frac {1-x}{\left (-x-\sqrt {3}+1\right )^2}} \sqrt {x^3-1}}+\frac {\sqrt {2 \left (7-4 \sqrt {3}\right )} (1-x) \sqrt {\frac {x^2+x+1}{\left (-x-\sqrt {3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac {-x+\sqrt {3}+1}{-x-\sqrt {3}+1}\right )|-7+4 \sqrt {3}\right )}{3^{3/4} \sqrt {-\frac {1-x}{\left (-x-\sqrt {3}+1\right )^2}} \sqrt {x^3-1}}+\frac {\sqrt {2} (1-x) \sqrt {\frac {x^2+x+1}{\left (-x-\sqrt {3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac {-x+\sqrt {3}+1}{-x-\sqrt {3}+1}\right )|-7+4 \sqrt {3}\right )}{3^{3/4} \sqrt {-\frac {1-x}{\left (-x-\sqrt {3}+1\right )^2}} \sqrt {x^3-1}} \end {gather*}
Warning: Unable to verify antiderivative.
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Rule 203
Rule 206
Rule 219
Rule 2135
Rule 2140
Rule 6728
Rubi steps
\begin {align*} \int \frac {2+x^2}{\left (-2-2 x+x^2\right ) \sqrt {-1+x^3}} \, dx &=\int \left (\frac {1}{\sqrt {-1+x^3}}+\frac {2 (2+x)}{\left (-2-2 x+x^2\right ) \sqrt {-1+x^3}}\right ) \, dx\\ &=2 \int \frac {2+x}{\left (-2-2 x+x^2\right ) \sqrt {-1+x^3}} \, dx+\int \frac {1}{\sqrt {-1+x^3}} \, dx\\ &=-\frac {2 \sqrt {2-\sqrt {3}} (1-x) \sqrt {\frac {1+x+x^2}{\left (1-\sqrt {3}-x\right )^2}} F\left (\sin ^{-1}\left (\frac {1+\sqrt {3}-x}{1-\sqrt {3}-x}\right )|-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {-\frac {1-x}{\left (1-\sqrt {3}-x\right )^2}} \sqrt {-1+x^3}}+2 \int \left (\frac {1+\sqrt {3}}{\left (-2-2 \sqrt {3}+2 x\right ) \sqrt {-1+x^3}}+\frac {1-\sqrt {3}}{\left (-2+2 \sqrt {3}+2 x\right ) \sqrt {-1+x^3}}\right ) \, dx\\ &=-\frac {2 \sqrt {2-\sqrt {3}} (1-x) \sqrt {\frac {1+x+x^2}{\left (1-\sqrt {3}-x\right )^2}} F\left (\sin ^{-1}\left (\frac {1+\sqrt {3}-x}{1-\sqrt {3}-x}\right )|-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {-\frac {1-x}{\left (1-\sqrt {3}-x\right )^2}} \sqrt {-1+x^3}}+\left (2 \left (1-\sqrt {3}\right )\right ) \int \frac {1}{\left (-2+2 \sqrt {3}+2 x\right ) \sqrt {-1+x^3}} \, dx+\left (2 \left (1+\sqrt {3}\right )\right ) \int \frac {1}{\left (-2-2 \sqrt {3}+2 x\right ) \sqrt {-1+x^3}} \, dx\\ &=-\frac {2 \sqrt {2-\sqrt {3}} (1-x) \sqrt {\frac {1+x+x^2}{\left (1-\sqrt {3}-x\right )^2}} F\left (\sin ^{-1}\left (\frac {1+\sqrt {3}-x}{1-\sqrt {3}-x}\right )|-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {-\frac {1-x}{\left (1-\sqrt {3}-x\right )^2}} \sqrt {-1+x^3}}+\frac {1}{288} \left (-3+\sqrt {3}\right ) \int \frac {96 \left (1+\sqrt {3}\right )-96 x}{\left (-2+2 \sqrt {3}+2 x\right ) \sqrt {-1+x^3}} \, dx+\frac {1}{6} \left (-3+\sqrt {3}\right ) \int \frac {1}{\sqrt {-1+x^3}} \, dx-\frac {1}{288} \left (3+\sqrt {3}\right ) \int \frac {96 \left (1-\sqrt {3}\right )-96 x}{\left (-2-2 \sqrt {3}+2 x\right ) \sqrt {-1+x^3}} \, dx-\frac {1}{6} \left (3+\sqrt {3}\right ) \int \frac {1}{\sqrt {-1+x^3}} \, dx\\ &=\frac {\sqrt {2} (1-x) \sqrt {\frac {1+x+x^2}{\left (1-\sqrt {3}-x\right )^2}} F\left (\sin ^{-1}\left (\frac {1+\sqrt {3}-x}{1-\sqrt {3}-x}\right )|-7+4 \sqrt {3}\right )}{3^{3/4} \sqrt {-\frac {1-x}{\left (1-\sqrt {3}-x\right )^2}} \sqrt {-1+x^3}}+\frac {\sqrt {14-8 \sqrt {3}} (1-x) \sqrt {\frac {1+x+x^2}{\left (1-\sqrt {3}-x\right )^2}} F\left (\sin ^{-1}\left (\frac {1+\sqrt {3}-x}{1-\sqrt {3}-x}\right )|-7+4 \sqrt {3}\right )}{3^{3/4} \sqrt {-\frac {1-x}{\left (1-\sqrt {3}-x\right )^2}} \sqrt {-1+x^3}}-\frac {2 \sqrt {2-\sqrt {3}} (1-x) \sqrt {\frac {1+x+x^2}{\left (1-\sqrt {3}-x\right )^2}} F\left (\sin ^{-1}\left (\frac {1+\sqrt {3}-x}{1-\sqrt {3}-x}\right )|-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {-\frac {1-x}{\left (1-\sqrt {3}-x\right )^2}} \sqrt {-1+x^3}}+\frac {1}{3} \left (3-\sqrt {3}\right ) \operatorname {Subst}\left (\int \frac {1}{1-\left (3-2 \sqrt {3}\right ) x^2} \, dx,x,\frac {1+\frac {2 \left (1-\sqrt {3}\right ) x}{-2+2 \sqrt {3}}}{\sqrt {-1+x^3}}\right )+\frac {1}{3} \left (3+\sqrt {3}\right ) \operatorname {Subst}\left (\int \frac {1}{1-\left (3+2 \sqrt {3}\right ) x^2} \, dx,x,\frac {1+\frac {2 \left (1+\sqrt {3}\right ) x}{-2-2 \sqrt {3}}}{\sqrt {-1+x^3}}\right )\\ &=\frac {\sqrt {2} \tan ^{-1}\left (\frac {\sqrt {-3+2 \sqrt {3}} (1-x)}{\sqrt {-1+x^3}}\right )}{3^{3/4}}+\frac {\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {3+2 \sqrt {3}} (1-x)}{\sqrt {-1+x^3}}\right )}{3^{3/4}}+\frac {\sqrt {2} (1-x) \sqrt {\frac {1+x+x^2}{\left (1-\sqrt {3}-x\right )^2}} F\left (\sin ^{-1}\left (\frac {1+\sqrt {3}-x}{1-\sqrt {3}-x}\right )|-7+4 \sqrt {3}\right )}{3^{3/4} \sqrt {-\frac {1-x}{\left (1-\sqrt {3}-x\right )^2}} \sqrt {-1+x^3}}+\frac {\sqrt {14-8 \sqrt {3}} (1-x) \sqrt {\frac {1+x+x^2}{\left (1-\sqrt {3}-x\right )^2}} F\left (\sin ^{-1}\left (\frac {1+\sqrt {3}-x}{1-\sqrt {3}-x}\right )|-7+4 \sqrt {3}\right )}{3^{3/4} \sqrt {-\frac {1-x}{\left (1-\sqrt {3}-x\right )^2}} \sqrt {-1+x^3}}-\frac {2 \sqrt {2-\sqrt {3}} (1-x) \sqrt {\frac {1+x+x^2}{\left (1-\sqrt {3}-x\right )^2}} F\left (\sin ^{-1}\left (\frac {1+\sqrt {3}-x}{1-\sqrt {3}-x}\right )|-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {-\frac {1-x}{\left (1-\sqrt {3}-x\right )^2}} \sqrt {-1+x^3}}\\ \end {align*}
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Mathematica [C] time = 0.80, size = 279, normalized size = 3.13 \begin {gather*} \frac {2 \sqrt {\frac {1-x}{1+\sqrt [3]{-1}}} \left ((1+i) \left (3-i \sqrt {3}\right ) \sqrt {x^2+x+1} \left (\Pi \left (\frac {2 \sqrt {3}}{-3 i+(1+2 i) \sqrt {3}};\sin ^{-1}\left (\sqrt {\frac {1-(-1)^{2/3} x}{1+\sqrt [3]{-1}}}\right )|\sqrt [3]{-1}\right )-i \Pi \left (\frac {2 i \sqrt {3}}{3+(2+i) \sqrt {3}};\sin ^{-1}\left (\sqrt {\frac {1-(-1)^{2/3} x}{1+\sqrt [3]{-1}}}\right )|\sqrt [3]{-1}\right )\right )+\frac {3 \sqrt {\frac {(-1)^{2/3} x+\sqrt [3]{-1}}{1+\sqrt [3]{-1}}} \left (\left (\sqrt {3}+i\right ) x+2 i\right ) 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 )}{3 \left (\sqrt {3}+i\right ) \sqrt {x^3-1}} \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [A] time = 1.54, size = 89, normalized size = 1.00 \begin {gather*} -\frac {\sqrt {2} \tan ^{-1}\left (\frac {\sqrt {2 \sqrt {3}-3} \sqrt {x^3-1}}{x^2+x+1}\right )}{3^{3/4}}-\frac {\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {3+2 \sqrt {3}} \sqrt {x^3-1}}{x^2+x+1}\right )}{3^{3/4}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.48, size = 230, normalized size = 2.58 \begin {gather*} \frac {1}{27} \cdot 27^{\frac {3}{4}} \sqrt {2} \arctan \left (\frac {\sqrt {x^{3} - 1} {\left (27^{\frac {3}{4}} \sqrt {2} - 9 \cdot 27^{\frac {1}{4}} \sqrt {2}\right )}}{18 \, {\left (x^{2} + x + 1\right )}}\right ) - \frac {1}{108} \cdot 27^{\frac {3}{4}} \sqrt {2} \log \left (\frac {2 \, {\left (9 \, x^{4} + 18 \, x^{3} + 54 \, x^{2} + 36 \, \sqrt {3} {\left (x^{3} - 1\right )} + \sqrt {x^{3} - 1} {\left (27^{\frac {3}{4}} \sqrt {2} {\left (x^{2} + 4 \, x - 2\right )} + 9 \cdot 27^{\frac {1}{4}} \sqrt {2} {\left (x^{2} + 2\right )}\right )} - 36 \, x + 36\right )}}{x^{4} - 4 \, x^{3} + 8 \, x + 4}\right ) + \frac {1}{108} \cdot 27^{\frac {3}{4}} \sqrt {2} \log \left (\frac {2 \, {\left (9 \, x^{4} + 18 \, x^{3} + 54 \, x^{2} + 36 \, \sqrt {3} {\left (x^{3} - 1\right )} - \sqrt {x^{3} - 1} {\left (27^{\frac {3}{4}} \sqrt {2} {\left (x^{2} + 4 \, x - 2\right )} + 9 \cdot 27^{\frac {1}{4}} \sqrt {2} {\left (x^{2} + 2\right )}\right )} - 36 \, x + 36\right )}}{x^{4} - 4 \, x^{3} + 8 \, 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}{\sqrt {x^{3} - 1} {\left (x^{2} - 2 \, x - 2\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.32, size = 1516, normalized size = 17.03
result too large to display
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}{\sqrt {x^{3} - 1} {\left (x^{2} - 2 \, x - 2\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.20, size = 509, normalized size = 5.72 \begin {gather*} -\frac {2\,\left (\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\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}}}\,\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 )}{\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 )}}-\frac {\left (2\,\sqrt {3}-6\right )\,\left (\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\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}}}\,\Pi \left (\frac {\sqrt {3}\,\left (\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{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 )}{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 )}}+\frac {\left (2\,\sqrt {3}+6\right )\,\left (\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\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}}}\,\Pi \left (-\frac {\sqrt {3}\,\left (\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{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 )}{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}{\sqrt {\left (x - 1\right ) \left (x^{2} + x + 1\right )} \left (x^{2} - 2 x - 2\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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