Optimal. Leaf size=70 \[ -\frac {1}{3} \sqrt {3+2 \sqrt {3}} \tan ^{-1}\left (\frac {\left (1+\sqrt {3}+2 x\right )^2}{2 \sqrt {3 \left (3+2 \sqrt {3}\right )} \sqrt {-1-4 \sqrt {3} x^2+4 x^4}}\right ) \]
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Rubi [A]
time = 0.08, antiderivative size = 70, 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, 209}
\begin {gather*} -\frac {1}{3} \sqrt {3+2 \sqrt {3}} \text {ArcTan}\left (\frac {\left (2 x+\sqrt {3}+1\right )^2}{2 \sqrt {3 \left (3+2 \sqrt {3}\right )} \sqrt {4 x^4-4 \sqrt {3} x^2-1}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 209
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}{12 \left (1-\sqrt {3}\right ) \left (1+\sqrt {3}\right )^3+6 \left (1+\sqrt {3}\right )^4+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}} \tan ^{-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 = 7.83, size = 81, normalized size = 1.16 \begin {gather*} -\frac {1}{3} \sqrt {3+2 \sqrt {3}} \tan ^{-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 = 0.39, size = 337, normalized size = 4.81
method | result | size |
elliptic | \(\frac {\sqrt {1-\left (-4-2 \sqrt {3}\right ) x^{2}}\, \sqrt {1-\left (-2 \sqrt {3}+4\right ) x^{2}}\, \EllipticF \left (x \left (i+i \sqrt {3}\right ), i \sqrt {1-\sqrt {3}\, \left (-2 \sqrt {3}+4\right )}\right )}{\left (i+i \sqrt {3}\right ) \sqrt {-1+4 x^{4}-4 x^{2} \sqrt {3}}}+\sqrt {3}\, \left (-\frac {\arctanh \left (\frac {-4 \sqrt {3}\, \left (\frac {\sqrt {3}}{2}-\frac {1}{2}\right )^{2}-2-4 x^{2} \sqrt {3}+8 x^{2} \left (\frac {\sqrt {3}}{2}-\frac {1}{2}\right )^{2}}{2 \sqrt {4 \left (\frac {\sqrt {3}}{2}-\frac {1}{2}\right )^{4}-4 \sqrt {3}\, \left (\frac {\sqrt {3}}{2}-\frac {1}{2}\right )^{2}-1}\, \sqrt {-1+4 x^{4}-4 x^{2} \sqrt {3}}}\right )}{2 \sqrt {4 \left (\frac {\sqrt {3}}{2}-\frac {1}{2}\right )^{4}-4 \sqrt {3}\, \left (\frac {\sqrt {3}}{2}-\frac {1}{2}\right )^{2}-1}}-\frac {\sqrt {1-\left (-4-2 \sqrt {3}\right ) x^{2}}\, \sqrt {1-\left (-2 \sqrt {3}+4\right ) x^{2}}\, \EllipticPi \left (\sqrt {-4-2 \sqrt {3}}\, x , \frac {1}{\left (-4-2 \sqrt {3}\right ) \left (\frac {\sqrt {3}}{2}-\frac {1}{2}\right )^{2}}, \frac {\sqrt {-2 \sqrt {3}+4}}{\sqrt {-4-2 \sqrt {3}}}\right )}{\sqrt {-4-2 \sqrt {3}}\, \left (\frac {\sqrt {3}}{2}-\frac {1}{2}\right ) \sqrt {-1+4 x^{4}-4 x^{2} \sqrt {3}}}\right )\) | \(336\) |
default | \(\frac {\sqrt {1-\left (-4-2 \sqrt {3}\right ) x^{2}}\, \sqrt {1-\left (-2 \sqrt {3}+4\right ) x^{2}}\, \EllipticF \left (x \left (i+i \sqrt {3}\right ), i \sqrt {1-\sqrt {3}\, \left (-2 \sqrt {3}+4\right )}\right )}{\left (i+i \sqrt {3}\right ) \sqrt {-1+4 x^{4}-4 x^{2} \sqrt {3}}}+2 \sqrt {3}\, \left (-\frac {\arctanh \left (\frac {-4 \sqrt {3}\, \left (\frac {\sqrt {3}}{2}-\frac {1}{2}\right )^{2}-2-4 x^{2} \sqrt {3}+8 x^{2} \left (\frac {\sqrt {3}}{2}-\frac {1}{2}\right )^{2}}{2 \sqrt {4 \left (\frac {\sqrt {3}}{2}-\frac {1}{2}\right )^{4}-4 \sqrt {3}\, \left (\frac {\sqrt {3}}{2}-\frac {1}{2}\right )^{2}-1}\, \sqrt {-1+4 x^{4}-4 x^{2} \sqrt {3}}}\right )}{4 \sqrt {4 \left (\frac {\sqrt {3}}{2}-\frac {1}{2}\right )^{4}-4 \sqrt {3}\, \left (\frac {\sqrt {3}}{2}-\frac {1}{2}\right )^{2}-1}}-\frac {\sqrt {1-\left (-4-2 \sqrt {3}\right ) x^{2}}\, \sqrt {1-\left (-2 \sqrt {3}+4\right ) x^{2}}\, \EllipticPi \left (\sqrt {-4-2 \sqrt {3}}\, x , \frac {1}{\left (-4-2 \sqrt {3}\right ) \left (\frac {\sqrt {3}}{2}-\frac {1}{2}\right )^{2}}, \frac {\sqrt {-2 \sqrt {3}+4}}{\sqrt {-4-2 \sqrt {3}}}\right )}{2 \sqrt {-4-2 \sqrt {3}}\, \left (\frac {\sqrt {3}}{2}-\frac {1}{2}\right ) \sqrt {-1+4 x^{4}-4 x^{2} \sqrt {3}}}\right )\) | \(337\) |
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. 114 vs.
\(2 (50) = 100\).
time = 0.52, size = 114, normalized size = 1.63 \begin {gather*} \frac {1}{6} \, \sqrt {2 \, \sqrt {3} + 3} \arctan \left (-\frac {{\left (36 \, x^{4} - 60 \, x^{3} + 18 \, x^{2} - \sqrt {3} {\left (16 \, x^{4} - 40 \, x^{3} + 6 \, x^{2} - 10 \, x + 1\right )} + 6\right )} \sqrt {4 \, x^{4} - 4 \, \sqrt {3} x^{2} - 1} \sqrt {2 \, \sqrt {3} + 3}}{88 \, x^{6} - 168 \, x^{5} + 132 \, x^{4} - 176 \, x^{3} - 66 \, x^{2} - 42 \, x - 11}\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 + 1 + \sqrt {3}}{\left (2 x - \sqrt {3} + 1\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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