Optimal. Leaf size=15 \[ \tan ^{-1}\left (\frac {x}{\sqrt {1+x^2+x^4}}\right ) \]
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Rubi [A]
time = 0.03, antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {1712, 209}
\begin {gather*} \text {ArcTan}\left (\frac {x}{\sqrt {x^4+x^2+1}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 1712
Rubi steps
\begin {align*} \int \frac {1-x^2}{\left (1+x^2\right ) \sqrt {1+x^2+x^4}} \, dx &=\text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {x}{\sqrt {1+x^2+x^4}}\right )\\ &=\tan ^{-1}\left (\frac {x}{\sqrt {1+x^2+x^4}}\right )\\ \end {align*}
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Mathematica [A]
time = 0.13, size = 15, normalized size = 1.00 \begin {gather*} \tan ^{-1}\left (\frac {x}{\sqrt {1+x^2+x^4}}\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 = 0.22, size = 188, normalized size = 12.53
method | result | size |
elliptic | \(-\arctan \left (\frac {\sqrt {x^{4}+x^{2}+1}}{x}\right )\) | \(18\) |
trager | \(\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) x -\sqrt {x^{4}+x^{2}+1}}{x^{2}+1}\right )\) | \(39\) |
default | \(-\frac {2 \sqrt {1-\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) x^{2}}\, \sqrt {1-\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) x^{2}}\, \EllipticF \left (\frac {x \sqrt {-2+2 i \sqrt {3}}}{2}, \frac {\sqrt {-2+2 i \sqrt {3}}}{2}\right )}{\sqrt {-2+2 i \sqrt {3}}\, \sqrt {x^{4}+x^{2}+1}}+\frac {2 \sqrt {1+\frac {x^{2}}{2}-\frac {i x^{2} \sqrt {3}}{2}}\, \sqrt {1+\frac {x^{2}}{2}+\frac {i x^{2} \sqrt {3}}{2}}\, \EllipticPi \left (\sqrt {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}\, x , -\frac {1}{-\frac {1}{2}+\frac {i \sqrt {3}}{2}}, \frac {\sqrt {-\frac {1}{2}-\frac {i \sqrt {3}}{2}}}{\sqrt {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}\, \sqrt {x^{4}+x^{2}+1}}\) | \(188\) |
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 = 1.31, size = 13, normalized size = 0.87 \begin {gather*} \arctan \left (\frac {x}{\sqrt {x^{4} + x^{2} + 1}}\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^{2}}{x^{2} \sqrt {x^{4} + x^{2} + 1} + \sqrt {x^{4} + x^{2} + 1}}\, dx - \int \left (- \frac {1}{x^{2} \sqrt {x^{4} + x^{2} + 1} + \sqrt {x^{4} + x^{2} + 1}}\right )\, 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 = -0.07 \begin {gather*} -\int \frac {x^2-1}{\left (x^2+1\right )\,\sqrt {x^4+x^2+1}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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