Optimal. Leaf size=44 \[ -\frac {2 \tan ^{-1}\left (\frac {\sqrt {2 \sqrt {3}-3} (x+1)}{\sqrt {-x^3-1}}\right )}{\sqrt {2 \sqrt {3}-3}} \]
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Rubi [A] time = 0.09, antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {2140, 203} \[ -\frac {2 \tan ^{-1}\left (\frac {\sqrt {2 \sqrt {3}-3} (x+1)}{\sqrt {-x^3-1}}\right )}{\sqrt {2 \sqrt {3}-3}} \]
Antiderivative was successfully verified.
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Rule 203
Rule 2140
Rubi steps
\begin {align*} \int \frac {1+\sqrt {3}+x}{\left (1-\sqrt {3}+x\right ) \sqrt {-1-x^3}} \, dx &=-\left (2 \operatorname {Subst}\left (\int \frac {1}{1-\left (3-2 \sqrt {3}\right ) x^2} \, dx,x,\frac {1+x}{\sqrt {-1-x^3}}\right )\right )\\ &=-\frac {2 \tan ^{-1}\left (\frac {\sqrt {-3+2 \sqrt {3}} (1+x)}{\sqrt {-1-x^3}}\right )}{\sqrt {-3+2 \sqrt {3}}}\\ \end {align*}
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Mathematica [C] time = 0.38, size = 269, normalized size = 6.11 \[ -\frac {2 \sqrt {6} \sqrt {\frac {i (x+1)}{\sqrt {3}+3 i}} \left (4 i \sqrt {-2 i x+\sqrt {3}+i} \sqrt {x^2-x+1} \Pi \left (\frac {2 i \sqrt {3}}{-3+(2+i) \sqrt {3}};\sin ^{-1}\left (\frac {\sqrt {-2 i x+\sqrt {3}+i}}{\sqrt {2} \sqrt [4]{3}}\right )|\frac {2 \sqrt {3}}{3 i+\sqrt {3}}\right )+\sqrt {2 i x+\sqrt {3}-i} \left (\left (\sqrt {3}+(-2-i)\right ) x-i \sqrt {3}+(1+2 i)\right ) F\left (\sin ^{-1}\left (\frac {\sqrt {-2 i x+\sqrt {3}+i}}{\sqrt {2} \sqrt [4]{3}}\right )|\frac {2 \sqrt {3}}{3 i+\sqrt {3}}\right )\right )}{\left (-3+(2+i) \sqrt {3}\right ) \sqrt {-2 i x+\sqrt {3}+i} \sqrt {-x^3-1}} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.49, size = 59, normalized size = 1.34 \[ \frac {1}{3} \, \sqrt {3} \sqrt {2 \, \sqrt {3} + 3} \arctan \left (\frac {\sqrt {-x^{3} - 1} {\left (\sqrt {3} {\left (x^{2} - 4 \, x - 2\right )} + 6 \, x + 6\right )} \sqrt {2 \, \sqrt {3} + 3}}{6 \, {\left (x^{3} + 1\right )}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.07, size = 247, normalized size = 5.61 \[ -\frac {2 i \sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {x +1}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {-i \left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \EllipticF \left (\frac {\sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \sqrt {\frac {i \sqrt {3}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )}{3 \sqrt {-x^{3}-1}}-\frac {4 i \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {x +1}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {-i \left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \EllipticPi \left (\frac {\sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \frac {i \sqrt {3}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}-\sqrt {3}}, \sqrt {\frac {i \sqrt {3}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {-x^{3}-1}\, \left (\frac {3}{2}+\frac {i \sqrt {3}}{2}-\sqrt {3}\right )} \]
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 \[ \int \frac {x + \sqrt {3} + 1}{\sqrt {-x^{3} - 1} {\left (x - \sqrt {3} + 1\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F(-1)] time = 0.00, size = -1, normalized size = -0.02 \[ \text {Hanged} \]
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 \[ \int \frac {x + 1 + \sqrt {3}}{\sqrt {- \left (x + 1\right ) \left (x^{2} - x + 1\right )} \left (x - \sqrt {3} + 1\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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