Optimal. Leaf size=17 \[ -\tan ^{-1}\left (\frac {x}{\sqrt {x^4+x^2+1}}\right ) \]
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Rubi [A] time = 0.04, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {1698, 203} \begin {gather*} -\tan ^{-1}\left (\frac {x}{\sqrt {x^4+x^2+1}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 203
Rule 1698
Rubi steps
\begin {align*} \int \frac {-1+x^2}{\left (1+x^2\right ) \sqrt {1+x^2+x^4}} \, dx &=-\operatorname {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 [C] time = 0.13, size = 93, normalized size = 5.47 \begin {gather*} \frac {(-1)^{2/3} \sqrt {\sqrt [3]{-1} x^2+1} \sqrt {1-(-1)^{2/3} x^2} \left (F\left (i \sinh ^{-1}\left ((-1)^{5/6} x\right )|(-1)^{2/3}\right )-2 \Pi \left (\sqrt [3]{-1};i \sinh ^{-1}\left ((-1)^{5/6} x\right )|(-1)^{2/3}\right )\right )}{\sqrt {x^4+x^2+1}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.17, size = 17, 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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fricas [A] time = 0.50, size = 15, normalized size = 0.88 \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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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{2} - 1}{\sqrt {x^{4} + x^{2} + 1} {\left (x^{2} + 1\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.42, size = 16, normalized size = 0.94
method | result | size |
elliptic | \(\arctan \left (\frac {\sqrt {x^{4}+x^{2}+1}}{x}\right )\) | \(16\) |
trager | \(\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) x +\sqrt {x^{4}+x^{2}+1}}{x^{2}+1}\right )\) | \(36\) |
default | \(\frac {2 \sqrt {1-\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) x^{2}}\, \sqrt {1-\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) x^{2}}\, \EllipticF \left (\frac {x \sqrt {-2+2 i \sqrt {3}}}{2}, \frac {\sqrt {-2+2 i \sqrt {3}}}{2}\right )}{\sqrt {-2+2 i \sqrt {3}}\, \sqrt {x^{4}+x^{2}+1}}-\frac {2 \sqrt {1+\frac {x^{2}}{2}-\frac {i \sqrt {3}\, x^{2}}{2}}\, \sqrt {1+\frac {x^{2}}{2}+\frac {i \sqrt {3}\, x^{2}}{2}}\, \EllipticPi \left (\sqrt {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}\, x , -\frac {1}{-\frac {1}{2}+\frac {i \sqrt {3}}{2}}, \frac {\sqrt {-\frac {1}{2}-\frac {i \sqrt {3}}{2}}}{\sqrt {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}\, \sqrt {x^{4}+x^{2}+1}}\) | \(188\) |
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*} \int \frac {x^{2} - 1}{\sqrt {x^{4} + x^{2} + 1} {\left (x^{2} + 1\right )}}\,{d x} \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 = -0.06 \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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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (x - 1\right ) \left (x + 1\right )}{\sqrt {\left (x^{2} - x + 1\right ) \left (x^{2} + x + 1\right )} \left (x^{2} + 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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