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 = 0.03, 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, 203} \[ \frac {1}{2} \tan ^{-1}\left (\sqrt {\frac {1}{2} \left (3-\sqrt {5}\right )} x\right ) \]
Antiderivative was successfully verified.
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Rule 6
Rule 203
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] time = 0.02, size = 39, normalized size = 1.62 \[ \frac {1}{4} i \log \left (-2 i x+\sqrt {5}+1\right )-\frac {1}{4} i \log \left (2 i x+\sqrt {5}+1\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 11, normalized size = 0.46 \[ \frac {1}{2} \, \arctan \left (\frac {1}{2} \, x {\left (\sqrt {5} - 1\right )}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.36, size = 13, normalized size = 0.54 \[ \frac {1}{2} \, \arctan \left (\frac {2 \, x}{\sqrt {5} + 1}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.03, size = 32, normalized size = 1.33 \[ \frac {4 \arctan \left (\frac {4 x}{2+2 \sqrt {5}}\right )}{\left (\sqrt {5}-1\right ) \left (2+2 \sqrt {5}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.98, size = 11, normalized size = 0.46 \[ \frac {1}{2} \, \arctan \left (\frac {1}{2} \, x {\left (\sqrt {5} - 1\right )}\right ) \]
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 \[ \frac {\sqrt {2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,x\,\left (\sqrt {5}+1\right )}{4\,\left (\frac {\sqrt {5}}{4}+\frac {1}{4}\right )\,\sqrt {\sqrt {5}+3}}\right )\,\left (\sqrt {5}+1\right )}{4\,\sqrt {\sqrt {5}+3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.58, size = 15, normalized size = 0.62 \[ - \frac {\operatorname {atan}{\left (x \left (\frac {1}{2} - \frac {\sqrt {5}}{2}\right ) \right )}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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