Optimal. Leaf size=74 \[ -\frac {1}{2} \tan ^{-1}\left (\sqrt {3}-2 x\right )+\frac {1}{2} \tan ^{-1}\left (\sqrt {3}+2 x\right )-\frac {\log \left (1-\sqrt {3} x+x^2\right )}{4 \sqrt {3}}+\frac {\log \left (1+\sqrt {3} x+x^2\right )}{4 \sqrt {3}} \]
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Rubi [A]
time = 0.03, antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 5, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {1108, 648, 632,
210, 642} \begin {gather*} -\frac {1}{2} \text {ArcTan}\left (\sqrt {3}-2 x\right )+\frac {1}{2} \text {ArcTan}\left (2 x+\sqrt {3}\right )-\frac {\log \left (x^2-\sqrt {3} x+1\right )}{4 \sqrt {3}}+\frac {\log \left (x^2+\sqrt {3} x+1\right )}{4 \sqrt {3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 632
Rule 642
Rule 648
Rule 1108
Rubi steps
\begin {align*} \int \frac {1}{1-x^2+x^4} \, dx &=\frac {\int \frac {\sqrt {3}-x}{1-\sqrt {3} x+x^2} \, dx}{2 \sqrt {3}}+\frac {\int \frac {\sqrt {3}+x}{1+\sqrt {3} x+x^2} \, dx}{2 \sqrt {3}}\\ &=\frac {1}{4} \int \frac {1}{1-\sqrt {3} x+x^2} \, dx+\frac {1}{4} \int \frac {1}{1+\sqrt {3} x+x^2} \, dx-\frac {\int \frac {-\sqrt {3}+2 x}{1-\sqrt {3} x+x^2} \, dx}{4 \sqrt {3}}+\frac {\int \frac {\sqrt {3}+2 x}{1+\sqrt {3} x+x^2} \, dx}{4 \sqrt {3}}\\ &=-\frac {\log \left (1-\sqrt {3} x+x^2\right )}{4 \sqrt {3}}+\frac {\log \left (1+\sqrt {3} x+x^2\right )}{4 \sqrt {3}}-\frac {1}{2} \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,-\sqrt {3}+2 x\right )-\frac {1}{2} \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,\sqrt {3}+2 x\right )\\ &=-\frac {1}{2} \tan ^{-1}\left (\sqrt {3}-2 x\right )+\frac {1}{2} \tan ^{-1}\left (\sqrt {3}+2 x\right )-\frac {\log \left (1-\sqrt {3} x+x^2\right )}{4 \sqrt {3}}+\frac {\log \left (1+\sqrt {3} x+x^2\right )}{4 \sqrt {3}}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 0.05, size = 77, normalized size = 1.04 \begin {gather*} \frac {i \left (\sqrt {-1-i \sqrt {3}} \tan ^{-1}\left (\frac {1}{2} \left (1-i \sqrt {3}\right ) x\right )-\sqrt {-1+i \sqrt {3}} \tan ^{-1}\left (\frac {1}{2} \left (1+i \sqrt {3}\right ) x\right )\right )}{\sqrt {6}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.04, size = 57, normalized size = 0.77
method | result | size |
risch | \(\frac {\left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{4}-\textit {\_Z}^{2}+1\right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{2 \textit {\_R}^{3}-\textit {\_R}}\right )}{2}\) | \(35\) |
default | \(\frac {\arctan \left (2 x -\sqrt {3}\right )}{2}+\frac {\arctan \left (2 x +\sqrt {3}\right )}{2}-\frac {\ln \left (1+x^{2}-x \sqrt {3}\right ) \sqrt {3}}{12}+\frac {\ln \left (1+x^{2}+x \sqrt {3}\right ) \sqrt {3}}{12}\) | \(57\) |
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. 163 vs.
\(2 (56) = 112\).
time = 0.34, size = 163, normalized size = 2.20 \begin {gather*} -\frac {1}{6} \, \sqrt {6} \sqrt {3} \sqrt {2} \arctan \left (-\frac {1}{3} \, \sqrt {6} \sqrt {3} \sqrt {2} x + \frac {1}{36} \, \sqrt {6} \sqrt {3} \sqrt {2} \sqrt {-72 \, \sqrt {6} \sqrt {2} x + 144 \, x^{2} + 144} + \sqrt {3}\right ) - \frac {1}{6} \, \sqrt {6} \sqrt {3} \sqrt {2} \arctan \left (-\frac {1}{3} \, \sqrt {6} \sqrt {3} \sqrt {2} x + \frac {1}{3} \, \sqrt {6} \sqrt {3} \sqrt {\sqrt {6} \sqrt {2} x + 2 \, x^{2} + 2} - \sqrt {3}\right ) + \frac {1}{24} \, \sqrt {6} \sqrt {2} \log \left (72 \, \sqrt {6} \sqrt {2} x + 144 \, x^{2} + 144\right ) - \frac {1}{24} \, \sqrt {6} \sqrt {2} \log \left (-72 \, \sqrt {6} \sqrt {2} x + 144 \, x^{2} + 144\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.09, size = 63, normalized size = 0.85 \begin {gather*} - \frac {\sqrt {3} \log {\left (x^{2} - \sqrt {3} x + 1 \right )}}{12} + \frac {\sqrt {3} \log {\left (x^{2} + \sqrt {3} x + 1 \right )}}{12} + \frac {\operatorname {atan}{\left (2 x - \sqrt {3} \right )}}{2} + \frac {\operatorname {atan}{\left (2 x + \sqrt {3} \right )}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 3.58, size = 56, normalized size = 0.76 \begin {gather*} \frac {1}{12} \, \sqrt {3} \log \left (x^{2} + \sqrt {3} x + 1\right ) - \frac {1}{12} \, \sqrt {3} \log \left (x^{2} - \sqrt {3} x + 1\right ) + \frac {1}{2} \, \arctan \left (2 \, x + \sqrt {3}\right ) + \frac {1}{2} \, \arctan \left (2 \, x - \sqrt {3}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.19, size = 47, normalized size = 0.64 \begin {gather*} \mathrm {atan}\left (\frac {2\,x}{-1+\sqrt {3}\,1{}\mathrm {i}}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )+\mathrm {atan}\left (\frac {2\,x}{1+\sqrt {3}\,1{}\mathrm {i}}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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