Optimal. Leaf size=72 \[ \frac {1}{3} \sqrt {-3+2 \sqrt {3}} \tanh ^{-1}\left (\frac {\left (1-\sqrt {3}+2 x\right )^2}{2 \sqrt {3 \left (-3+2 \sqrt {3}\right )} \sqrt {-1+4 \sqrt {3} x^2+4 x^4}}\right ) \]
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Rubi [A]
time = 0.09, antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 46, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {1754, 213}
\begin {gather*} \frac {1}{3} \sqrt {2 \sqrt {3}-3} \tanh ^{-1}\left (\frac {\left (2 x-\sqrt {3}+1\right )^2}{2 \sqrt {3 \left (2 \sqrt {3}-3\right )} \sqrt {4 x^4+4 \sqrt {3} x^2-1}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 213
Rule 1754
Rubi steps
\begin {align*} \int \frac {1-\sqrt {3}+2 x}{\left (1+\sqrt {3}+2 x\right ) \sqrt {-1+4 \sqrt {3} x^2+4 x^4}} \, dx &=-\left (\left (4 \left (2-\sqrt {3}\right )\right ) \text {Subst}\left (\int \frac {1}{6 \left (1-\sqrt {3}\right )^4+12 \left (1-\sqrt {3}\right )^3 \left (1+\sqrt {3}\right )+2 x^2} \, dx,x,\frac {\left (1-\sqrt {3}+2 x\right )^2}{\sqrt {-1+4 \sqrt {3} x^2+4 x^4}}\right )\right )\\ &=\frac {1}{3} \sqrt {-3+2 \sqrt {3}} \tanh ^{-1}\left (\frac {\left (1-\sqrt {3}+2 x\right )^2}{2 \sqrt {3 \left (-3+2 \sqrt {3}\right )} \sqrt {-1+4 \sqrt {3} x^2+4 x^4}}\right )\\ \end {align*}
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Mathematica [A]
time = 7.82, size = 81, normalized size = 1.12 \begin {gather*} \frac {1}{3} \sqrt {-3+2 \sqrt {3}} \tanh ^{-1}\left (\frac {\sqrt {9+6 \sqrt {3}} \sqrt {-1+4 \sqrt {3} x^2+4 x^4}}{1+\left (-2-2 \sqrt {3}\right ) x+\left (4+2 \sqrt {3}\right ) x^2}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 4 vs. order
3.
time = 0.35, size = 336, normalized size = 4.67
method | result | size |
default | \(\frac {\sqrt {1-\left (2 \sqrt {3}-4\right ) x^{2}}\, \sqrt {1-\left (4+2 \sqrt {3}\right ) x^{2}}\, \EllipticF \left (x \left (i \sqrt {3}-i\right ), i \sqrt {1+\sqrt {3}\, \left (4+2 \sqrt {3}\right )}\right )}{\left (i \sqrt {3}-i\right ) \sqrt {-1+4 x^{4}+4 x^{2} \sqrt {3}}}-2 \sqrt {3}\, \left (-\frac {\arctanh \left (\frac {4 \sqrt {3}\, \left (-\frac {1}{2}-\frac {\sqrt {3}}{2}\right )^{2}-2+4 x^{2} \sqrt {3}+8 x^{2} \left (-\frac {1}{2}-\frac {\sqrt {3}}{2}\right )^{2}}{2 \sqrt {4 \left (-\frac {1}{2}-\frac {\sqrt {3}}{2}\right )^{4}+4 \sqrt {3}\, \left (-\frac {1}{2}-\frac {\sqrt {3}}{2}\right )^{2}-1}\, \sqrt {-1+4 x^{4}+4 x^{2} \sqrt {3}}}\right )}{4 \sqrt {4 \left (-\frac {1}{2}-\frac {\sqrt {3}}{2}\right )^{4}+4 \sqrt {3}\, \left (-\frac {1}{2}-\frac {\sqrt {3}}{2}\right )^{2}-1}}-\frac {\sqrt {1-\left (2 \sqrt {3}-4\right ) x^{2}}\, \sqrt {1-\left (4+2 \sqrt {3}\right ) x^{2}}\, \EllipticPi \left (\sqrt {2 \sqrt {3}-4}\, x , \frac {1}{\left (2 \sqrt {3}-4\right ) \left (-\frac {1}{2}-\frac {\sqrt {3}}{2}\right )^{2}}, \frac {\sqrt {4+2 \sqrt {3}}}{\sqrt {2 \sqrt {3}-4}}\right )}{2 \sqrt {2 \sqrt {3}-4}\, \left (-\frac {1}{2}-\frac {\sqrt {3}}{2}\right ) \sqrt {-1+4 x^{4}+4 x^{2} \sqrt {3}}}\right )\) | \(336\) |
elliptic | \(\frac {\sqrt {1-\left (2 \sqrt {3}-4\right ) x^{2}}\, \sqrt {1-\left (4+2 \sqrt {3}\right ) x^{2}}\, \EllipticF \left (x \left (i \sqrt {3}-i\right ), i \sqrt {1+\sqrt {3}\, \left (4+2 \sqrt {3}\right )}\right )}{\left (i \sqrt {3}-i\right ) \sqrt {-1+4 x^{4}+4 x^{2} \sqrt {3}}}-\sqrt {3}\, \left (-\frac {\arctanh \left (\frac {4 \sqrt {3}\, \left (-\frac {1}{2}-\frac {\sqrt {3}}{2}\right )^{2}-2+4 x^{2} \sqrt {3}+8 x^{2} \left (-\frac {1}{2}-\frac {\sqrt {3}}{2}\right )^{2}}{2 \sqrt {4 \left (-\frac {1}{2}-\frac {\sqrt {3}}{2}\right )^{4}+4 \sqrt {3}\, \left (-\frac {1}{2}-\frac {\sqrt {3}}{2}\right )^{2}-1}\, \sqrt {-1+4 x^{4}+4 x^{2} \sqrt {3}}}\right )}{2 \sqrt {4 \left (-\frac {1}{2}-\frac {\sqrt {3}}{2}\right )^{4}+4 \sqrt {3}\, \left (-\frac {1}{2}-\frac {\sqrt {3}}{2}\right )^{2}-1}}-\frac {\sqrt {1-\left (2 \sqrt {3}-4\right ) x^{2}}\, \sqrt {1-\left (4+2 \sqrt {3}\right ) x^{2}}\, \EllipticPi \left (\sqrt {2 \sqrt {3}-4}\, x , \frac {1}{\left (2 \sqrt {3}-4\right ) \left (-\frac {1}{2}-\frac {\sqrt {3}}{2}\right )^{2}}, \frac {\sqrt {4+2 \sqrt {3}}}{\sqrt {2 \sqrt {3}-4}}\right )}{\sqrt {2 \sqrt {3}-4}\, \left (-\frac {1}{2}-\frac {\sqrt {3}}{2}\right ) \sqrt {-1+4 x^{4}+4 x^{2} \sqrt {3}}}\right )\) | \(336\) |
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*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 328 vs.
\(2 (52) = 104\).
time = 0.57, size = 328, normalized size = 4.56 \begin {gather*} \frac {1}{12} \, \sqrt {2 \, \sqrt {3} - 3} \log \left (-\frac {2368 \, x^{12} - 6528 \, x^{11} + 12864 \, x^{10} - 19264 \, x^{9} + 14832 \, x^{8} - 10944 \, x^{7} + 6432 \, x^{6} + 5472 \, x^{5} + 3708 \, x^{4} + 2408 \, x^{3} + 804 \, x^{2} + {\left (1728 \, x^{10} - 4800 \, x^{9} + 8208 \, x^{8} - 8928 \, x^{7} + 6048 \, x^{6} - 3024 \, x^{5} - 504 \, x^{4} - 504 \, x^{3} - 324 \, x^{2} + 2 \, \sqrt {3} {\left (496 \, x^{10} - 1408 \, x^{9} + 2304 \, x^{8} - 2640 \, x^{7} + 1848 \, x^{6} - 504 \, x^{5} + 336 \, x^{4} + 204 \, x^{3} + 63 \, x^{2} + 26 \, x + 4\right )} - 72 \, x - 15\right )} \sqrt {4 \, x^{4} + 4 \, \sqrt {3} x^{2} - 1} \sqrt {2 \, \sqrt {3} - 3} + 3 \, \sqrt {3} {\left (448 \, x^{12} - 1280 \, x^{11} + 2560 \, x^{10} - 3200 \, x^{9} + 3696 \, x^{8} - 1920 \, x^{7} - 960 \, x^{5} - 924 \, x^{4} - 400 \, x^{3} - 160 \, x^{2} - 40 \, x - 7\right )} + 204 \, x + 37}{64 \, x^{12} + 384 \, x^{11} + 768 \, x^{10} + 320 \, x^{9} - 720 \, x^{8} - 576 \, x^{7} + 384 \, x^{6} + 288 \, x^{5} - 180 \, x^{4} - 40 \, x^{3} + 48 \, x^{2} - 12 \, x + 1}\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 {2 x - \sqrt {3} + 1}{\left (2 x + 1 + \sqrt {3}\right ) \sqrt {4 x^{4} + 4 \sqrt {3} x^{2} - 1}}\, dx \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*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {2\,x-\sqrt {3}+1}{\sqrt {4\,x^4+4\,\sqrt {3}\,x^2-1}\,\left (2\,x+\sqrt {3}+1\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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