Optimal. Leaf size=17 \[ -\tanh ^{-1}\left (\frac {x}{\sqrt {x^4+x^2+1}}\right ) \]
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Rubi [A] time = 0.08, 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 = {2112, 206} \begin {gather*} -\tanh ^{-1}\left (\frac {x}{\sqrt {x^4+x^2+1}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 206
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 )\\ &=-\tanh ^{-1}\left (\frac {x}{\sqrt {1+x^2+x^4}}\right )\\ \end {align*}
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Mathematica [C] time = 0.23, size = 120, normalized size = 7.06 \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 )-\Pi \left (-(-1)^{5/6};i \sinh ^{-1}\left ((-1)^{5/6} x\right )|(-1)^{2/3}\right )-\Pi \left ((-1)^{5/6};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 = 0.24, size = 17, normalized size = 1.00 \begin {gather*} -\tanh ^{-1}\left (\frac {x}{\sqrt {1+x^2+x^4}}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.48, size = 34, normalized size = 2.00 \begin {gather*} \frac {1}{2} \, \log \left (\frac {x^{4} + 2 \, x^{2} - 2 \, \sqrt {x^{4} + x^{2} + 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 {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 = 0.29, size = 18, normalized size = 1.06
method | result | size |
elliptic | \(-\arctanh \left (\frac {\sqrt {x^{4}+x^{2}+1}}{x}\right )\) | \(18\) |
trager | \(\frac {\ln \left (-\frac {-x^{4}+2 \sqrt {x^{4}+x^{2}+1}\, x -2 x^{2}-1}{x^{4}+1}\right )}{2}\) | \(38\) |
default | \(\frac {2 \sqrt {1-\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) x^{2}}\, \sqrt {1-\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) x^{2}}\, \EllipticF \left (\frac {x \sqrt {-2+2 i \sqrt {3}}}{2}, \frac {\sqrt {-2+2 i \sqrt {3}}}{2}\right )}{\sqrt {-2+2 i \sqrt {3}}\, \sqrt {x^{4}+x^{2}+1}}+\frac {\left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (\textit {\_Z}^{4}+1\right )}{\sum }\underline {\hspace {1.25 ex}}\alpha \left (-\frac {\arctanh \left (\frac {\left (2 \underline {\hspace {1.25 ex}}\alpha ^{2}+1\right ) \left (-3 \underline {\hspace {1.25 ex}}\alpha ^{2}+5 x^{2}+4\right )}{10 \sqrt {\underline {\hspace {1.25 ex}}\alpha ^{2}}\, \sqrt {x^{4}+x^{2}+1}}\right )}{\sqrt {\underline {\hspace {1.25 ex}}\alpha ^{2}}}+\frac {\sqrt {2}\, \underline {\hspace {1.25 ex}}\alpha ^{3} \sqrt {x^{2}+2-i \sqrt {3}\, x^{2}}\, \sqrt {x^{2}+2+i \sqrt {3}\, x^{2}}\, \EllipticPi \left (\sqrt {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}\, x , \frac {i \underline {\hspace {1.25 ex}}\alpha ^{2} \sqrt {3}}{2}+\frac {\underline {\hspace {1.25 ex}}\alpha ^{2}}{2}, \frac {\sqrt {-\frac {1}{2}-\frac {i \sqrt {3}}{2}}}{\sqrt {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {i \sqrt {3}-1}\, \sqrt {x^{4}+x^{2}+1}}\right )\right )}{4}\) | \(250\) |
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 = -0.06 \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 {\left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right )}{\sqrt {\left (x^{2} - x + 1\right ) \left (x^{2} + x + 1\right )} \left (x^{4} + 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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