Optimal. Leaf size=19 \[ -\tan ^{-1}\left (\frac {x}{\sqrt {x^4-x^2-1}}\right ) \]
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Rubi [A] time = 0.09, antiderivative size = 19, 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 = {2112, 204} \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 204
Rule 2112
Rubi steps
\begin {align*} \int \frac {1+x^4}{\left (-1+x^4\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 = 4.99, size = 1511, normalized size = 79.53
result too large to display
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [A] time = 0.25, size = 19, 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 = 0.47, size = 30, normalized size = 1.58 \begin {gather*} -\frac {1}{2} \, \arctan \left (\frac {2 \, \sqrt {x^{4} - x^{2} - 1} x}{x^{4} - 2 \, 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^{4} + 1}{\sqrt {x^{4} - x^{2} - 1} {\left (x^{4} - 1\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.41, size = 18, normalized size = 0.95
method | result | size |
elliptic | \(\arctan \left (\frac {\sqrt {x^{4}-x^{2}-1}}{x}\right )\) | \(18\) |
trager | \(\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) x^{4}-2 \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{2}-2 x \sqrt {x^{4}-x^{2}-1}-\RootOf \left (\textit {\_Z}^{2}+1\right )}{\left (-1+x \right ) \left (1+x \right ) \left (x^{2}+1\right )}\right )}{2}\) | \(73\) |
default | \(\frac {2 \sqrt {1-\left (-\frac {1}{2}-\frac {\sqrt {5}}{2}\right ) x^{2}}\, \sqrt {1-\left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right ) x^{2}}\, \EllipticF \left (\frac {x \sqrt {-2-2 \sqrt {5}}}{2}, \frac {i \sqrt {5}}{2}-\frac {i}{2}\right )}{\sqrt {-2-2 \sqrt {5}}\, \sqrt {x^{4}-x^{2}-1}}-\frac {\sqrt {1-\left (-\frac {1}{2}-\frac {\sqrt {5}}{2}\right ) x^{2}}\, \sqrt {1-\left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right ) x^{2}}\, \EllipticPi \left (\sqrt {-\frac {1}{2}-\frac {\sqrt {5}}{2}}\, x , \frac {1}{-\frac {1}{2}-\frac {\sqrt {5}}{2}}, \frac {\sqrt {\frac {\sqrt {5}}{2}-\frac {1}{2}}}{\sqrt {-\frac {1}{2}-\frac {\sqrt {5}}{2}}}\right )}{\sqrt {-\frac {1}{2}-\frac {\sqrt {5}}{2}}\, \sqrt {x^{4}-x^{2}-1}}-\frac {\sqrt {1+\frac {x^{2}}{2}+\frac {\sqrt {5}\, x^{2}}{2}}\, \sqrt {1-\frac {\sqrt {5}\, x^{2}}{2}+\frac {x^{2}}{2}}\, \EllipticPi \left (\sqrt {-\frac {1}{2}-\frac {\sqrt {5}}{2}}\, x , -\frac {1}{-\frac {1}{2}-\frac {\sqrt {5}}{2}}, \frac {\sqrt {\frac {\sqrt {5}}{2}-\frac {1}{2}}}{\sqrt {-\frac {1}{2}-\frac {\sqrt {5}}{2}}}\right )}{\sqrt {-\frac {1}{2}-\frac {\sqrt {5}}{2}}\, \sqrt {x^{4}-x^{2}-1}}\) | \(273\) |
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^{4} + 1}{\sqrt {x^{4} - x^{2} - 1} {\left (x^{4} - 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 = -0.05 \begin {gather*} \int \frac {x^4+1}{\left (x^4-1\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 {x^{4} + 1}{\left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right ) \sqrt {x^{4} - x^{2} - 1}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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