Optimal. Leaf size=164 \[ \frac {2 \sqrt {\frac {7}{6}-\frac {2}{\sqrt {3}}} (1-x) \sqrt {\frac {x^2+x+1}{\left (-x-\sqrt {3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac {-x+\sqrt {3}+1}{-x-\sqrt {3}+1}\right )|-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {-\frac {1-x}{\left (-x-\sqrt {3}+1\right )^2}} \sqrt {x^3-1}}-\frac {\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {3+2 \sqrt {3}} (1-x)}{\sqrt {x^3-1}}\right )}{3^{3/4}} \]
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Rubi [A] time = 0.22, antiderivative size = 164, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {2141, 219, 2140, 206} \[ \frac {2 \sqrt {\frac {7}{6}-\frac {2}{\sqrt {3}}} (1-x) \sqrt {\frac {x^2+x+1}{\left (-x-\sqrt {3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac {-x+\sqrt {3}+1}{-x-\sqrt {3}+1}\right )|-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {-\frac {1-x}{\left (-x-\sqrt {3}+1\right )^2}} \sqrt {x^3-1}}-\frac {\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {3+2 \sqrt {3}} (1-x)}{\sqrt {x^3-1}}\right )}{3^{3/4}} \]
Antiderivative was successfully verified.
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Rule 206
Rule 219
Rule 2140
Rule 2141
Rubi steps
\begin {align*} \int \frac {x}{\left (1+\sqrt {3}-x\right ) \sqrt {-1+x^3}} \, dx &=-\frac {\left (-1-\sqrt {3}\right ) \int \frac {\left (1+\sqrt {3}\right ) \left (-22+\left (1+\sqrt {3}\right )^3\right )-6 x}{\left (1+\sqrt {3}-x\right ) \sqrt {-1+x^3}} \, dx}{\left (1+\sqrt {3}\right ) \left (-28+\left (1+\sqrt {3}\right )^3\right )}-\frac {\left (-22+\left (1+\sqrt {3}\right )^3\right ) \int \frac {1}{\sqrt {-1+x^3}} \, dx}{-28+\left (1+\sqrt {3}\right )^3}\\ &=\frac {2 \sqrt {\frac {7}{6}-\frac {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 {\left (12 \left (-1-\sqrt {3}\right )\right ) \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 )}{\left (1+\sqrt {3}\right ) \left (-28+\left (1+\sqrt {3}\right )^3\right )}\\ &=-\frac {\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {3+2 \sqrt {3}} (1-x)}{\sqrt {-1+x^3}}\right )}{3^{3/4}}+\frac {2 \sqrt {\frac {7}{6}-\frac {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 = 0.27, size = 230, normalized size = 1.40 \[ \frac {2 i \sqrt {\frac {1-x}{1+\sqrt [3]{-1}}} \left (2 \left (1+\sqrt {3}\right ) \sqrt {x^2+x+1} \Pi \left (\frac {2 i \sqrt {3}}{3+(2+i) \sqrt {3}};\sin ^{-1}\left (\sqrt {\frac {1-(-1)^{2/3} x}{1+\sqrt [3]{-1}}}\right )|\sqrt [3]{-1}\right )+\frac {i \sqrt {\frac {(-1)^{2/3} x+\sqrt [3]{-1}}{1+\sqrt [3]{-1}}} \left (\left (3+(2+i) \sqrt {3}\right ) x+(1+2 i) \sqrt {3}+3 i\right ) F\left (\sin ^{-1}\left (\sqrt {\frac {1-(-1)^{2/3} x}{1+\sqrt [3]{-1}}}\right )|\sqrt [3]{-1}\right )}{\sqrt {\frac {1-(-1)^{2/3} x}{1+\sqrt [3]{-1}}}}\right )}{\left (3+(2+i) \sqrt {3}\right ) \sqrt {x^3-1}} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.48, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {x^{3} - 1} {\left (x^{2} + \sqrt {3} x - x\right )}}{x^{5} - 2 \, x^{4} - 2 \, x^{3} - x^{2} + 2 \, x + 2}, x\right ) \]
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 [A] time = 0.03, size = 255, normalized size = 1.55 \[ -\frac {2 \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {x -1}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \EllipticF \left (\sqrt {\frac {x -1}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {x^{3}-1}}-\frac {2 \left (-1-\sqrt {3}\right ) \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {x -1}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {3}\, \EllipticPi \left (\sqrt {\frac {x -1}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, -\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}{3}, \sqrt {\frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{3 \sqrt {x^{3}-1}} \]
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 \[ -\int \frac {x}{\sqrt {x^{3} - 1} {\left (x - \sqrt {3} - 1\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F(-1)] time = 0.00, size = -1, normalized size = -0.01 \[ \text {Hanged} \]
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 \[ - \int \frac {x}{x \sqrt {x^{3} - 1} - \sqrt {3} \sqrt {x^{3} - 1} - \sqrt {x^{3} - 1}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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