Optimal. Leaf size=67 \[ \frac {1}{4} \tan ^{-1}\left (\frac {\frac {x^4}{2}-x^2-\frac {1}{2}}{x \sqrt {x^4-1}}\right )-\frac {1}{4} \tanh ^{-1}\left (\frac {\frac {x^4}{2}+x^2-\frac {1}{2}}{x \sqrt {x^4-1}}\right ) \]
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Rubi [A] time = 0.01, antiderivative size = 45, normalized size of antiderivative = 0.67, number of steps used = 1, number of rules used = 1, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {405} \begin {gather*} -\frac {1}{2} \tan ^{-1}\left (\frac {x \left (1-x^2\right )}{\sqrt {x^4-1}}\right )-\frac {1}{2} \tanh ^{-1}\left (\frac {x \left (x^2+1\right )}{\sqrt {x^4-1}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 405
Rubi steps
\begin {align*} \int \frac {\sqrt {-1+x^4}}{1+x^4} \, dx &=-\frac {1}{2} \tan ^{-1}\left (\frac {x \left (1-x^2\right )}{\sqrt {-1+x^4}}\right )-\frac {1}{2} \tanh ^{-1}\left (\frac {x \left (1+x^2\right )}{\sqrt {-1+x^4}}\right )\\ \end {align*}
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Mathematica [C] time = 0.11, size = 108, normalized size = 1.61 \begin {gather*} -\frac {5 x \sqrt {x^4-1} F_1\left (\frac {1}{4};-\frac {1}{2},1;\frac {5}{4};x^4,-x^4\right )}{\left (x^4+1\right ) \left (2 x^4 \left (2 F_1\left (\frac {5}{4};-\frac {1}{2},2;\frac {9}{4};x^4,-x^4\right )+F_1\left (\frac {5}{4};\frac {1}{2},1;\frac {9}{4};x^4,-x^4\right )\right )-5 F_1\left (\frac {1}{4};-\frac {1}{2},1;\frac {5}{4};x^4,-x^4\right )\right )} \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [C] time = 0.19, size = 53, normalized size = 0.79 \begin {gather*} \left (-\frac {1}{4}+\frac {i}{4}\right ) \tan ^{-1}\left (\frac {(1+i) x}{\sqrt {-1+x^4}}\right )+\left (\frac {1}{4}+\frac {i}{4}\right ) \tan ^{-1}\left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {-1+x^4}}{x}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.56, size = 51, normalized size = 0.76 \begin {gather*} \frac {1}{2} \, \arctan \left (\frac {\sqrt {x^{4} - 1} x}{x^{2} + 1}\right ) + \frac {1}{4} \, \log \left (\frac {x^{4} + 2 \, x^{2} - 2 \, \sqrt {x^{4} - 1} x - 1}{x^{4} + 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 {\sqrt {x^{4} - 1}}{x^{4} + 1}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.60, size = 101, normalized size = 1.51
method | result | size |
default | \(\frac {\left (\frac {\sqrt {2}\, \arctan \left (\frac {\sqrt {x^{4}-1}}{x}+1\right )}{4}+\frac {\sqrt {2}\, \arctan \left (\frac {\sqrt {x^{4}-1}}{x}-1\right )}{4}+\frac {\sqrt {2}\, \ln \left (\frac {\frac {x^{4}-1}{2 x^{2}}-\frac {\sqrt {x^{4}-1}}{x}+1}{\frac {x^{4}-1}{2 x^{2}}+\frac {\sqrt {x^{4}-1}}{x}+1}\right )}{8}\right ) \sqrt {2}}{2}\) | \(101\) |
elliptic | \(\frac {\left (\frac {\sqrt {2}\, \arctan \left (\frac {\sqrt {x^{4}-1}}{x}+1\right )}{4}+\frac {\sqrt {2}\, \arctan \left (\frac {\sqrt {x^{4}-1}}{x}-1\right )}{4}+\frac {\sqrt {2}\, \ln \left (\frac {\frac {x^{4}-1}{2 x^{2}}-\frac {\sqrt {x^{4}-1}}{x}+1}{\frac {x^{4}-1}{2 x^{2}}+\frac {\sqrt {x^{4}-1}}{x}+1}\right )}{8}\right ) \sqrt {2}}{2}\) | \(101\) |
trager | \(-\frac {\ln \left (-\frac {-4 \RootOf \left (8 \textit {\_Z}^{2}+4 \textit {\_Z} +1\right ) x +\sqrt {x^{4}-1}}{4 x^{2} \RootOf \left (8 \textit {\_Z}^{2}+4 \textit {\_Z} +1\right )+x^{2}-1}\right )}{2}-\ln \left (-\frac {-4 \RootOf \left (8 \textit {\_Z}^{2}+4 \textit {\_Z} +1\right ) x +\sqrt {x^{4}-1}}{4 x^{2} \RootOf \left (8 \textit {\_Z}^{2}+4 \textit {\_Z} +1\right )+x^{2}-1}\right ) \RootOf \left (8 \textit {\_Z}^{2}+4 \textit {\_Z} +1\right )+\RootOf \left (8 \textit {\_Z}^{2}+4 \textit {\_Z} +1\right ) \ln \left (-\frac {4 \RootOf \left (8 \textit {\_Z}^{2}+4 \textit {\_Z} +1\right ) x +\sqrt {x^{4}-1}+2 x}{4 x^{2} \RootOf \left (8 \textit {\_Z}^{2}+4 \textit {\_Z} +1\right )+x^{2}+1}\right )\) | \(176\) |
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 {\sqrt {x^{4} - 1}}{x^{4} + 1}\,{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 = -0.01 \begin {gather*} \int \frac {\sqrt {x^4-1}}{x^4+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 {\sqrt {\left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right )}}{x^{4} + 1}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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