Optimal. Leaf size=67 \[ \frac {\tan ^{-1}\left (\frac {x}{\sqrt {2-\sqrt {3}}}\right )}{2 \sqrt {3 \left (2-\sqrt {3}\right )}}-\frac {\tan ^{-1}\left (\frac {x}{\sqrt {2+\sqrt {3}}}\right )}{2 \sqrt {3 \left (2+\sqrt {3}\right )}} \]
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Rubi [A]
time = 0.01, antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {1107, 209}
\begin {gather*} \frac {\text {ArcTan}\left (\frac {x}{\sqrt {2-\sqrt {3}}}\right )}{2 \sqrt {3 \left (2-\sqrt {3}\right )}}-\frac {\text {ArcTan}\left (\frac {x}{\sqrt {2+\sqrt {3}}}\right )}{2 \sqrt {3 \left (2+\sqrt {3}\right )}} \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 1107
Rubi steps
\begin {align*} \int \frac {1}{1+4 x^2+x^4} \, dx &=\frac {\int \frac {1}{2-\sqrt {3}+x^2} \, dx}{2 \sqrt {3}}-\frac {\int \frac {1}{2+\sqrt {3}+x^2} \, dx}{2 \sqrt {3}}\\ &=\frac {\tan ^{-1}\left (\frac {x}{\sqrt {2-\sqrt {3}}}\right )}{2 \sqrt {3 \left (2-\sqrt {3}\right )}}-\frac {\tan ^{-1}\left (\frac {x}{\sqrt {2+\sqrt {3}}}\right )}{2 \sqrt {3 \left (2+\sqrt {3}\right )}}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 67, normalized size = 1.00 \begin {gather*} \frac {\tan ^{-1}\left (\frac {x}{\sqrt {2-\sqrt {3}}}\right )}{2 \sqrt {3 \left (2-\sqrt {3}\right )}}-\frac {\tan ^{-1}\left (\frac {x}{\sqrt {2+\sqrt {3}}}\right )}{2 \sqrt {3 \left (2+\sqrt {3}\right )}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.03, size = 60, normalized size = 0.90
method | result | size |
risch | \(\frac {\left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{4}+4 \textit {\_Z}^{2}+1\right )}{\sum }\frac {\ln \left (-\textit {\_R} +x \right )}{\textit {\_R}^{3}+2 \textit {\_R}}\right )}{4}\) | \(33\) |
default | \(-\frac {\sqrt {3}\, \arctan \left (\frac {2 x}{\sqrt {6}+\sqrt {2}}\right )}{3 \left (\sqrt {6}+\sqrt {2}\right )}+\frac {\sqrt {3}\, \arctan \left (\frac {2 x}{\sqrt {6}-\sqrt {2}}\right )}{3 \sqrt {6}-3 \sqrt {2}}\) | \(60\) |
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 [A]
time = 0.61, size = 87, normalized size = 1.30 \begin {gather*} -\frac {1}{3} \, \sqrt {3} \sqrt {\sqrt {3} + 2} \arctan \left (-{\left (x - \sqrt {x^{2} - \sqrt {3} + 2}\right )} \sqrt {\sqrt {3} + 2}\right ) + \frac {1}{3} \, \sqrt {3} \sqrt {-\sqrt {3} + 2} \arctan \left (-x \sqrt {-\sqrt {3} + 2} + \sqrt {x^{2} + \sqrt {3} + 2} \sqrt {-\sqrt {3} + 2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.10, size = 92, normalized size = 1.37 \begin {gather*} - 2 \sqrt {\frac {1}{24} - \frac {\sqrt {3}}{48}} \operatorname {atan}{\left (\frac {x}{\sqrt {3} \sqrt {2 - \sqrt {3}} + 2 \sqrt {2 - \sqrt {3}}} \right )} - 2 \sqrt {\frac {\sqrt {3}}{48} + \frac {1}{24}} \operatorname {atan}{\left (\frac {x}{- 2 \sqrt {\sqrt {3} + 2} + \sqrt {3} \sqrt {\sqrt {3} + 2}} \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.49, size = 51, normalized size = 0.76 \begin {gather*} \frac {1}{12} \, {\left (\sqrt {6} - 3 \, \sqrt {2}\right )} \arctan \left (\frac {2 \, x}{\sqrt {6} + \sqrt {2}}\right ) + \frac {1}{12} \, {\left (\sqrt {6} + 3 \, \sqrt {2}\right )} \arctan \left (\frac {2 \, x}{\sqrt {6} - \sqrt {2}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.20, size = 117, normalized size = 1.75 \begin {gather*} 2\,\mathrm {atanh}\left (\frac {24\,x\,\sqrt {\frac {\sqrt {3}}{48}-\frac {1}{24}}}{2\,\sqrt {3}-4}-\frac {16\,\sqrt {3}\,x\,\sqrt {\frac {\sqrt {3}}{48}-\frac {1}{24}}}{2\,\sqrt {3}-4}\right )\,\sqrt {\frac {\sqrt {3}}{48}-\frac {1}{24}}-2\,\mathrm {atanh}\left (\frac {24\,x\,\sqrt {-\frac {\sqrt {3}}{48}-\frac {1}{24}}}{2\,\sqrt {3}+4}+\frac {16\,\sqrt {3}\,x\,\sqrt {-\frac {\sqrt {3}}{48}-\frac {1}{24}}}{2\,\sqrt {3}+4}\right )\,\sqrt {-\frac {\sqrt {3}}{48}-\frac {1}{24}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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