Optimal. Leaf size=24 \[ \frac {1}{2} \tan ^{-1}\left (\sqrt {\frac {1}{2} \left (3-\sqrt {5}\right )} x\right ) \]
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Rubi [A]
time = 0.02, antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {6, 209}
\begin {gather*} \frac {1}{2} \text {ArcTan}\left (\sqrt {\frac {1}{2} \left (3-\sqrt {5}\right )} x\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 6
Rule 209
Rubi steps
\begin {align*} \int \frac {1}{1+\sqrt {5}-x^2+\sqrt {5} x^2} \, dx &=\int \frac {1}{1+\sqrt {5}+\left (-1+\sqrt {5}\right ) x^2} \, dx\\ &=\frac {1}{2} \tan ^{-1}\left (\sqrt {\frac {1}{2} \left (3-\sqrt {5}\right )} x\right )\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 0.02, size = 39, normalized size = 1.62 \begin {gather*} \frac {1}{4} i \log \left (1+\sqrt {5}-2 i x\right )-\frac {1}{4} i \log \left (1+\sqrt {5}+2 i x\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(31\) vs.
\(2(12)=24\).
time = 0.24, size = 32, normalized size = 1.33
method | result | size |
risch | \(\frac {\arctan \left (\frac {2 x}{\sqrt {5}+1}\right )}{4}-\frac {\arctan \left (\frac {2 x}{-\sqrt {5}-1}\right )}{4}\) | \(30\) |
default | \(\frac {4 \arctan \left (\frac {4 x}{2 \sqrt {5}+2}\right )}{\left (\sqrt {5}-1\right ) \left (2 \sqrt {5}+2\right )}\) | \(32\) |
meijerg | \(\frac {\arctan \left (\frac {x \sqrt {\sqrt {5}-1}}{\sqrt {\sqrt {5}+1}}\right )}{\sqrt {\sqrt {5}+1}\, \sqrt {\sqrt {5}-1}}\) | \(33\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.81, size = 11, normalized size = 0.46 \begin {gather*} \frac {1}{2} \, \arctan \left (\frac {1}{2} \, x {\left (\sqrt {5} - 1\right )}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 13, normalized size = 0.54 \begin {gather*} \frac {1}{2} \, \arctan \left (\frac {1}{2} \, \sqrt {5} x - \frac {1}{2} \, x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.32, size = 15, normalized size = 0.62 \begin {gather*} - \frac {\operatorname {atan}{\left (x \left (\frac {1}{2} - \frac {\sqrt {5}}{2}\right ) \right )}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 2.96, size = 13, normalized size = 0.54 \begin {gather*} \frac {1}{2} \, \arctan \left (\frac {2 \, x}{\sqrt {5} + 1}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.10, size = 45, normalized size = 1.88 \begin {gather*} \frac {\sqrt {2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,x\,\left (\sqrt {5}+1\right )}{4\,\left (\frac {\sqrt {5}}{4}+\frac {1}{4}\right )\,\sqrt {\sqrt {5}+3}}\right )\,\left (\sqrt {5}+1\right )}{4\,\sqrt {\sqrt {5}+3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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