Optimal. Leaf size=13 \[ \frac {1}{2} \tan ^{-1}(x)+\frac {1}{2} \tanh ^{-1}(x) \]
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Rubi [A] time = 0.00, antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {28, 21, 212, 206, 203} \begin {gather*} \frac {1}{2} \tan ^{-1}(x)+\frac {1}{2} \tanh ^{-1}(x) \end {gather*}
Antiderivative was successfully verified.
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Rule 21
Rule 28
Rule 203
Rule 206
Rule 212
Rubi steps
\begin {align*} \int \frac {1-x^4}{1-2 x^4+x^8} \, dx &=\int \frac {1-x^4}{\left (-1+x^4\right )^2} \, dx\\ &=-\int \frac {1}{-1+x^4} \, dx\\ &=\frac {1}{2} \int \frac {1}{1-x^2} \, dx+\frac {1}{2} \int \frac {1}{1+x^2} \, dx\\ &=\frac {1}{2} \tan ^{-1}(x)+\frac {1}{2} \tanh ^{-1}(x)\\ \end {align*}
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Mathematica [A] time = 0.01, size = 25, normalized size = 1.92 \begin {gather*} -\frac {1}{4} \log (1-x)+\frac {1}{4} \log (x+1)+\frac {1}{2} \tan ^{-1}(x) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1-x^4}{1-2 x^4+x^8} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 1.42, size = 17, normalized size = 1.31 \begin {gather*} \frac {1}{2} \, \arctan \relax (x) + \frac {1}{4} \, \log \left (x + 1\right ) - \frac {1}{4} \, \log \left (x - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.45, size = 19, normalized size = 1.46 \begin {gather*} \frac {1}{2} \, \arctan \relax (x) + \frac {1}{4} \, \log \left ({\left | x + 1 \right |}\right ) - \frac {1}{4} \, \log \left ({\left | x - 1 \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 10, normalized size = 0.77 \begin {gather*} \frac {\arctanh \relax (x )}{2}+\frac {\arctan \relax (x )}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.60, size = 17, normalized size = 1.31 \begin {gather*} \frac {1}{2} \, \arctan \relax (x) + \frac {1}{4} \, \log \left (x + 1\right ) - \frac {1}{4} \, \log \left (x - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.02, size = 9, normalized size = 0.69 \begin {gather*} \frac {\mathrm {atan}\relax (x)}{2}+\frac {\mathrm {atanh}\relax (x)}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.13, size = 17, normalized size = 1.31 \begin {gather*} - \frac {\log {\left (x - 1 \right )}}{4} + \frac {\log {\left (x + 1 \right )}}{4} + \frac {\operatorname {atan}{\relax (x )}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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