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, 218,
212, 209} \begin {gather*} \frac {\text {ArcTan}(x)}{2}+\frac {1}{2} \tanh ^{-1}(x) \end {gather*}
Antiderivative was successfully verified.
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Rule 21
Rule 28
Rule 209
Rule 212
Rule 218
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 = 0.00, size = 25, normalized size = 1.92 \begin {gather*} \frac {1}{2} \tan ^{-1}(x)-\frac {1}{4} \log (1-x)+\frac {1}{4} \log (1+x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.02, size = 10, normalized size = 0.77
method | result | size |
default | \(\frac {\arctan \left (x \right )}{2}+\frac {\arctanh \left (x \right )}{2}\) | \(10\) |
risch | \(\frac {\arctan \left (x \right )}{2}-\frac {\ln \left (-1+x \right )}{4}+\frac {\ln \left (1+x \right )}{4}\) | \(18\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.48, size = 17, normalized size = 1.31 \begin {gather*} \frac {1}{2} \, \arctan \left (x\right ) + \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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Fricas [A]
time = 0.33, size = 17, normalized size = 1.31 \begin {gather*} \frac {1}{2} \, \arctan \left (x\right ) + \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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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 17 vs.
\(2 (8) = 16\).
time = 0.05, 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}{\left (x \right )}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 19 vs.
\(2 (9) = 18\).
time = 5.84, size = 19, normalized size = 1.46 \begin {gather*} \frac {1}{2} \, \arctan \left (x\right ) + \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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Mupad [B]
time = 0.02, size = 9, normalized size = 0.69 \begin {gather*} \frac {\mathrm {atan}\left (x\right )}{2}+\frac {\mathrm {atanh}\left (x\right )}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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