Optimal. Leaf size=63 \[ -\frac {1}{3} \sqrt {3+2 \sqrt {3}} \tan ^{-1}\left (\frac {\left (1+\sqrt {3}+x\right )^2}{\sqrt {3 \left (3+2 \sqrt {3}\right )} \sqrt {-4-4 \sqrt {3} x^2+x^4}}\right ) \]
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Rubi [A]
time = 0.08, antiderivative size = 63, 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 = {1754, 209}
\begin {gather*} -\frac {1}{3} \sqrt {3+2 \sqrt {3}} \text {ArcTan}\left (\frac {\left (x+\sqrt {3}+1\right )^2}{\sqrt {3 \left (3+2 \sqrt {3}\right )} \sqrt {x^4-4 \sqrt {3} x^2-4}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 1754
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 ) \text {Subst}\left (\int \frac {1}{6 \left (1-\sqrt {3}\right ) \left (1+\sqrt {3}\right )^3+3 \left (1+\sqrt {3}\right )^4+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}} \tan ^{-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 [A]
time = 8.20, size = 77, normalized size = 1.22 \begin {gather*} -\frac {1}{3} \sqrt {3+2 \sqrt {3}} \tan ^{-1}\left (\frac {\sqrt {-9+6 \sqrt {3}} \sqrt {-4-4 \sqrt {3} x^2+x^4}}{-2+\left (2-2 \sqrt {3}\right ) x+\left (-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 = 0.43, size = 311, normalized size = 4.94
method | result | size |
default | \(\frac {\sqrt {1-\left (-1-\frac {\sqrt {3}}{2}\right ) x^{2}}\, \sqrt {1-\left (1-\frac {\sqrt {3}}{2}\right ) x^{2}}\, \EllipticF \left (x \left (\frac {i}{2}+\frac {i \sqrt {3}}{2}\right ), i \sqrt {1-4 \sqrt {3}\, \left (1-\frac {\sqrt {3}}{2}\right )}\right )}{\left (\frac {i}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {-4+x^{4}-4 \sqrt {3}\, x^{2}}}+2 \sqrt {3}\, \left (-\frac {\arctanh \left (\frac {-4 \left (\sqrt {3}-1\right )^{2} \sqrt {3}-8-4 \sqrt {3}\, x^{2}+2 x^{2} \left (\sqrt {3}-1\right )^{2}}{2 \sqrt {\left (\sqrt {3}-1\right )^{4}-4 \left (\sqrt {3}-1\right )^{2} \sqrt {3}-4}\, \sqrt {-4+x^{4}-4 \sqrt {3}\, x^{2}}}\right )}{2 \sqrt {\left (\sqrt {3}-1\right )^{4}-4 \left (\sqrt {3}-1\right )^{2} \sqrt {3}-4}}-\frac {\sqrt {1-\left (-1-\frac {\sqrt {3}}{2}\right ) x^{2}}\, \sqrt {1-\left (1-\frac {\sqrt {3}}{2}\right ) x^{2}}\, \EllipticPi \left (\sqrt {-1-\frac {\sqrt {3}}{2}}\, x , \frac {1}{\left (-1-\frac {\sqrt {3}}{2}\right ) \left (\sqrt {3}-1\right )^{2}}, \frac {\sqrt {1-\frac {\sqrt {3}}{2}}}{\sqrt {-1-\frac {\sqrt {3}}{2}}}\right )}{\sqrt {-1-\frac {\sqrt {3}}{2}}\, \left (\sqrt {3}-1\right ) \sqrt {-4+x^{4}-4 \sqrt {3}\, x^{2}}}\right )\) | \(311\) |
elliptic | \(\frac {\sqrt {1-\left (-1-\frac {\sqrt {3}}{2}\right ) x^{2}}\, \sqrt {1-\left (1-\frac {\sqrt {3}}{2}\right ) x^{2}}\, \EllipticF \left (x \left (\frac {i}{2}+\frac {i \sqrt {3}}{2}\right ), i \sqrt {1-4 \sqrt {3}\, \left (1-\frac {\sqrt {3}}{2}\right )}\right )}{\left (\frac {i}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {-4+x^{4}-4 \sqrt {3}\, x^{2}}}+2 \sqrt {3}\, \left (-\frac {\arctanh \left (\frac {-4 \left (\sqrt {3}-1\right )^{2} \sqrt {3}-8-4 \sqrt {3}\, x^{2}+2 x^{2} \left (\sqrt {3}-1\right )^{2}}{2 \sqrt {\left (\sqrt {3}-1\right )^{4}-4 \left (\sqrt {3}-1\right )^{2} \sqrt {3}-4}\, \sqrt {-4+x^{4}-4 \sqrt {3}\, x^{2}}}\right )}{2 \sqrt {\left (\sqrt {3}-1\right )^{4}-4 \left (\sqrt {3}-1\right )^{2} \sqrt {3}-4}}-\frac {\sqrt {1-\left (-1-\frac {\sqrt {3}}{2}\right ) x^{2}}\, \sqrt {1-\left (1-\frac {\sqrt {3}}{2}\right ) x^{2}}\, \EllipticPi \left (\sqrt {-1-\frac {\sqrt {3}}{2}}\, x , \frac {1}{\left (-1-\frac {\sqrt {3}}{2}\right ) \left (\sqrt {3}-1\right )^{2}}, \frac {\sqrt {1-\frac {\sqrt {3}}{2}}}{\sqrt {-1-\frac {\sqrt {3}}{2}}}\right )}{\sqrt {-1-\frac {\sqrt {3}}{2}}\, \left (\sqrt {3}-1\right ) \sqrt {-4+x^{4}-4 \sqrt {3}\, x^{2}}}\right )\) | \(311\) |
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. 112 vs.
\(2 (45) = 90\).
time = 1.25, size = 112, normalized size = 1.78 \begin {gather*} \frac {1}{6} \, \sqrt {2 \, \sqrt {3} + 3} \arctan \left (-\frac {{\left (9 \, x^{4} - 30 \, x^{3} + 18 \, x^{2} - 2 \, \sqrt {3} {\left (2 \, x^{4} - 10 \, x^{3} + 3 \, x^{2} - 10 \, x + 2\right )} + 24\right )} \sqrt {x^{4} - 4 \, \sqrt {3} x^{2} - 4} \sqrt {2 \, \sqrt {3} + 3}}{11 \, x^{6} - 42 \, x^{5} + 66 \, x^{4} - 176 \, x^{3} - 132 \, x^{2} - 168 \, x - 88}\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 {x + 1 + \sqrt {3}}{\left (x - \sqrt {3} + 1\right ) \sqrt {x^{4} - 4 \sqrt {3} x^{2} - 4}}\, 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 = -0.02 \begin {gather*} \int \frac {x+\sqrt {3}+1}{\sqrt {x^4-4\,\sqrt {3}\,x^2-4}\,\left (x-\sqrt {3}+1\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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