Optimal. Leaf size=65 \[ \frac {1}{3} \sqrt {2 \sqrt {3}-3} \tanh ^{-1}\left (\frac {\left (x-\sqrt {3}+1\right )^2}{\sqrt {3 \left (2 \sqrt {3}-3\right )} \sqrt {x^4+4 \sqrt {3} x^2-4}}\right ) \]
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Rubi [A] time = 0.13, antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {1740, 207} \[ \frac {1}{3} \sqrt {2 \sqrt {3}-3} \tanh ^{-1}\left (\frac {\left (x-\sqrt {3}+1\right )^2}{\sqrt {3 \left (2 \sqrt {3}-3\right )} \sqrt {x^4+4 \sqrt {3} x^2-4}}\right ) \]
Antiderivative was successfully verified.
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Rule 207
Rule 1740
Rubi steps
\begin {align*} \int \frac {1-\sqrt {3}+x}{\left (1+\sqrt {3}+x\right ) \sqrt {-4+4 \sqrt {3} x^2+x^4}} \, dx &=-\left (\left (4 \left (2-\sqrt {3}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{3 \left (1-\sqrt {3}\right )^4+6 \left (1-\sqrt {3}\right )^3 \left (1+\sqrt {3}\right )+4 x^2} \, dx,x,\frac {\left (1-\sqrt {3}+x\right )^2}{\sqrt {-4+4 \sqrt {3} x^2+x^4}}\right )\right )\\ &=\frac {1}{3} \sqrt {-3+2 \sqrt {3}} \tanh ^{-1}\left (\frac {\left (1-\sqrt {3}+x\right )^2}{\sqrt {3 \left (-3+2 \sqrt {3}\right )} \sqrt {-4+4 \sqrt {3} x^2+x^4}}\right )\\ \end {align*}
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Mathematica [C] time = 3.21, size = 685, normalized size = 10.54 \[ \frac {\left (x+\sqrt {3}-1\right )^2 \sqrt {-x^3+\left (\sqrt {3}-1\right ) x^2-2 \left (2+\sqrt {3}\right ) x+2 \left (1+\sqrt {3}\right )} \sqrt {\frac {-\frac {4}{x+\sqrt {3}-1}+\sqrt {3}+1}{3+\sqrt {3}+i \sqrt {2 \left (2+\sqrt {3}\right )}}} \left (\left (\frac {2 \left (2 i \sqrt {3}-\sqrt {2 \left (2+\sqrt {3}\right )}+\sqrt {6 \left (2+\sqrt {3}\right )}\right )}{x+\sqrt {3}-1}+i \left (-1+\sqrt {3}+i \sqrt {2 \left (2+\sqrt {3}\right )}\right )\right ) \sqrt {\sqrt {2 \left (2+\sqrt {3}\right )}+i \left (\frac {8}{x+\sqrt {3}-1}-\sqrt {3}+1\right )} \operatorname {EllipticF}\left (\sin ^{-1}\left (\frac {\sqrt {\sqrt {2 \left (2+\sqrt {3}\right )}-i \left (\frac {8}{x+\sqrt {3}-1}-\sqrt {3}+1\right )}}{2^{3/4} \sqrt [4]{2+\sqrt {3}}}\right ),\frac {2 i \sqrt {2 \left (2+\sqrt {3}\right )}}{3+\sqrt {3}+i \sqrt {2 \left (2+\sqrt {3}\right )}}\right )+2 \sqrt {6} \sqrt {\frac {x^2+2 \sqrt {3}+4}{\left (x+\sqrt {3}-1\right )^2}} \sqrt {\sqrt {2 \left (2+\sqrt {3}\right )}-i \left (\frac {8}{x+\sqrt {3}-1}-\sqrt {3}+1\right )} \Pi \left (\frac {2 \sqrt {2 \left (2+\sqrt {3}\right )}}{\sqrt {2 \left (2+\sqrt {3}\right )}+i \left (3+\sqrt {3}\right )};\sin ^{-1}\left (\frac {\sqrt {\sqrt {2 \left (2+\sqrt {3}\right )}-i \left (-\sqrt {3}+1+\frac {8}{x+\sqrt {3}-1}\right )}}{2^{3/4} \sqrt [4]{2+\sqrt {3}}}\right )|\frac {2 i \sqrt {2 \left (2+\sqrt {3}\right )}}{3+\sqrt {3}+i \sqrt {2 \left (2+\sqrt {3}\right )}}\right )\right )}{\left (\sqrt {2 \left (2+\sqrt {3}\right )}+i \left (3+\sqrt {3}\right )\right ) \sqrt {-\frac {x^3}{2}+\frac {1}{2} \left (\sqrt {3}-1\right ) x^2-\left (2+\sqrt {3}\right ) x+\sqrt {3}+1} \sqrt {x^4+4 \sqrt {3} x^2-4} \sqrt {\sqrt {2 \left (2+\sqrt {3}\right )}-i \left (\frac {8}{x+\sqrt {3}-1}-\sqrt {3}+1\right )}} \]
Warning: Unable to verify antiderivative.
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fricas [B] time = 0.71, size = 323, normalized size = 4.97 \[ \frac {1}{12} \, \sqrt {2 \, \sqrt {3} - 3} \log \left (-\frac {37 \, x^{12} - 204 \, x^{11} + 804 \, x^{10} - 2408 \, x^{9} + 3708 \, x^{8} - 5472 \, x^{7} + 6432 \, x^{6} + 10944 \, x^{5} + 14832 \, x^{4} + 19264 \, x^{3} + 12864 \, x^{2} + {\left (54 \, x^{10} - 300 \, x^{9} + 1026 \, x^{8} - 2232 \, x^{7} + 3024 \, x^{6} - 3024 \, x^{5} - 1008 \, x^{4} - 2016 \, x^{3} - 2592 \, x^{2} + \sqrt {3} {\left (31 \, x^{10} - 176 \, x^{9} + 576 \, x^{8} - 1320 \, x^{7} + 1848 \, x^{6} - 1008 \, x^{5} + 1344 \, x^{4} + 1632 \, x^{3} + 1008 \, x^{2} + 832 \, x + 256\right )} - 1152 \, x - 480\right )} \sqrt {x^{4} + 4 \, \sqrt {3} x^{2} - 4} \sqrt {2 \, \sqrt {3} - 3} + 3 \, \sqrt {3} {\left (7 \, x^{12} - 40 \, x^{11} + 160 \, x^{10} - 400 \, x^{9} + 924 \, x^{8} - 960 \, x^{7} - 1920 \, x^{5} - 3696 \, x^{4} - 3200 \, x^{3} - 2560 \, x^{2} - 1280 \, x - 448\right )} + 6528 \, x + 2368}{x^{12} + 12 \, x^{11} + 48 \, x^{10} + 40 \, x^{9} - 180 \, x^{8} - 288 \, x^{7} + 384 \, x^{6} + 576 \, x^{5} - 720 \, x^{4} - 320 \, x^{3} + 768 \, x^{2} - 384 \, x + 64}\right ) \]
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 \[ \int \frac {x - \sqrt {3} + 1}{\sqrt {x^{4} + 4 \, \sqrt {3} x^{2} - 4} {\left (x + \sqrt {3} + 1\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.16, size = 327, normalized size = 5.03 \[ \frac {\sqrt {-\left (-1+\frac {\sqrt {3}}{2}\right ) x^{2}+1}\, \sqrt {-\left (1+\frac {\sqrt {3}}{2}\right ) x^{2}+1}\, \EllipticF \left (\left (\frac {i \sqrt {3}}{2}-\frac {i}{2}\right ) x , i \sqrt {1+4 \sqrt {3}\, \left (1+\frac {\sqrt {3}}{2}\right )}\right )}{\left (\frac {i \sqrt {3}}{2}-\frac {i}{2}\right ) \sqrt {x^{4}+4 \sqrt {3}\, x^{2}-4}}-2 \sqrt {3}\, \left (-\frac {\sqrt {-\left (-1+\frac {\sqrt {3}}{2}\right ) x^{2}+1}\, \sqrt {-\left (1+\frac {\sqrt {3}}{2}\right ) x^{2}+1}\, \EllipticPi \left (\sqrt {-1+\frac {\sqrt {3}}{2}}\, x , \frac {1}{\left (-1+\frac {\sqrt {3}}{2}\right ) \left (-1-\sqrt {3}\right )^{2}}, \frac {\sqrt {1+\frac {\sqrt {3}}{2}}}{\sqrt {-1+\frac {\sqrt {3}}{2}}}\right )}{\sqrt {-1+\frac {\sqrt {3}}{2}}\, \left (-1-\sqrt {3}\right ) \sqrt {x^{4}+4 \sqrt {3}\, x^{2}-4}}-\frac {\arctanh \left (\frac {4 \sqrt {3}\, x^{2}+2 \left (-1-\sqrt {3}\right )^{2} x^{2}+4 \sqrt {3}\, \left (-1-\sqrt {3}\right )^{2}-8}{2 \sqrt {\left (-1-\sqrt {3}\right )^{4}+4 \sqrt {3}\, \left (-1-\sqrt {3}\right )^{2}-4}\, \sqrt {x^{4}+4 \sqrt {3}\, x^{2}-4}}\right )}{2 \sqrt {\left (-1-\sqrt {3}\right )^{4}+4 \sqrt {3}\, \left (-1-\sqrt {3}\right )^{2}-4}}\right ) \]
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 \[ \int \frac {x - \sqrt {3} + 1}{\sqrt {x^{4} + 4 \, \sqrt {3} x^{2} - 4} {\left (x + \sqrt {3} + 1\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {x-\sqrt {3}+1}{\left (x+\sqrt {3}+1\right )\,\sqrt {x^4+4\,\sqrt {3}\,x^2-4}} \,d x \]
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 \[ \int \frac {x - \sqrt {3} + 1}{\left (x + 1 + \sqrt {3}\right ) \sqrt {x^{4} + 4 \sqrt {3} x^{2} - 4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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