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 = 3, number of rules used = 3, integrand size = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {218, 212, 209}
\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 209
Rule 212
Rule 218
Rubi steps
\begin {align*} \int \frac {1}{-1+x^4} \, dx &=-\left (\frac {1}{2} \int \frac {1}{1-x^2} \, dx\right )-\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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Mathics [A]
time = 1.67, size = 17, normalized size = 1.31 \begin {gather*} -\frac {\text {ArcTan}\left [x\right ]}{2}-\frac {\text {Log}\left [1+x\right ]}{4}+\frac {\text {Log}\left [-1+x\right ]}{4} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.07, size = 10, normalized size = 0.77
method | result | size |
default | \(-\frac {\arctan \left (x \right )}{2}-\frac {\arctanh \left (x \right )}{2}\) | \(10\) |
risch | \(\frac {\ln \left (-1+x \right )}{4}-\frac {\arctan \left (x \right )}{2}-\frac {\ln \left (1+x \right )}{4}\) | \(18\) |
meijerg | \(\frac {x \left (\ln \left (1-\left (x^{4}\right )^{\frac {1}{4}}\right )-\ln \left (1+\left (x^{4}\right )^{\frac {1}{4}}\right )-2 \arctan \left (\left (x^{4}\right )^{\frac {1}{4}}\right )\right )}{4 \left (x^{4}\right )^{\frac {1}{4}}}\) | \(38\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.35, 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 = 0.34, 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 [A]
time = 0.06, 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 = 0.00, size = 24, normalized size = 1.85 \begin {gather*} \frac {\ln \left |x-1\right |}{4}-\frac {\ln \left |x+1\right |}{4}-\frac {\arctan x}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.03, 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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