Optimal. Leaf size=147 \[ \frac {2 \sqrt {\frac {7}{6}+\frac {2}{\sqrt {3}}} (x+1) \sqrt {\frac {x^2-x+1}{\left (x+\sqrt {3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac {x-\sqrt {3}+1}{x+\sqrt {3}+1}\right )|-7-4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {\frac {x+1}{\left (x+\sqrt {3}+1\right )^2}} \sqrt {x^3+1}}-\frac {\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2 \sqrt {3}-3} (x+1)}{\sqrt {x^3+1}}\right )}{3^{3/4}} \]
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Rubi [A] time = 0.22, antiderivative size = 147, 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, 218, 2140, 206} \[ \frac {2 \sqrt {\frac {7}{6}+\frac {2}{\sqrt {3}}} (x+1) \sqrt {\frac {x^2-x+1}{\left (x+\sqrt {3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac {x-\sqrt {3}+1}{x+\sqrt {3}+1}\right )|-7-4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {\frac {x+1}{\left (x+\sqrt {3}+1\right )^2}} \sqrt {x^3+1}}-\frac {\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2 \sqrt {3}-3} (x+1)}{\sqrt {x^3+1}}\right )}{3^{3/4}} \]
Antiderivative was successfully verified.
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Rule 206
Rule 218
Rule 2140
Rule 2141
Rubi steps
\begin {align*} \int \frac {x}{\left (1-\sqrt {3}+x\right ) \sqrt {1+x^3}} \, dx &=\frac {\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}{6 \left (3+\sqrt {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 {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 )}{3+\sqrt {3}}\\ &=-\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 = 0.57, size = 225, normalized size = 1.53 \[ \frac {2 \sqrt {\frac {x+1}{1+\sqrt [3]{-1}}} \left (\frac {\sqrt {\frac {\sqrt [3]{-1}-(-1)^{2/3} x}{1+\sqrt [3]{-1}}} \left (\left ((1+2 i) \sqrt {3}-3 i\right ) x-(2+i) \sqrt {3}+3\right ) F\left (\sin ^{-1}\left (\sqrt {\frac {(-1)^{2/3} x+1}{1+\sqrt [3]{-1}}}\right )|\sqrt [3]{-1}\right )}{\sqrt {\frac {(-1)^{2/3} x+1}{1+\sqrt [3]{-1}}}}-2 \left (\sqrt {3}-1\right ) \sqrt {x^2-x+1} \Pi \left (\frac {2 \sqrt {3}}{-3 i+(1+2 i) \sqrt {3}};\sin ^{-1}\left (\sqrt {\frac {(-1)^{2/3} x+1}{1+\sqrt [3]{-1}}}\right )|\sqrt [3]{-1}\right )\right )}{\left ((1+2 i) \sqrt {3}-3 i\right ) \sqrt {x^3+1}} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.50, 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] 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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maple [B] time = 0.04, size = 255, normalized size = 1.73 \[ \frac {2 \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {x +1}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \EllipticF \left (\sqrt {\frac {x +1}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {x^{3}+1}}+\frac {2 \left (1-\sqrt {3}\right ) \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {x +1}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {3}\, \EllipticPi \left (\sqrt {\frac {x +1}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, -\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}{3}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{3 \sqrt {x^{3}+1}} \]
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 \[ \int \frac {x}{\sqrt {\left (x + 1\right ) \left (x^{2} - x + 1\right )} \left (x - \sqrt {3} + 1\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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