Optimal. Leaf size=35 \[ -\frac {\tan ^{-1}\left (\frac {x}{\sqrt [4]{2}}\right )}{2\ 2^{3/4}}-\frac {\tanh ^{-1}\left (\frac {x}{\sqrt [4]{2}}\right )}{2\ 2^{3/4}} \]
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Rubi [A]
time = 0.01, antiderivative size = 35, 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 {\tan ^{-1}\left (\frac {x}{\sqrt [4]{2}}\right )}{2\ 2^{3/4}}-\frac {\tanh ^{-1}\left (\frac {x}{\sqrt [4]{2}}\right )}{2\ 2^{3/4}} \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 212
Rule 218
Rubi steps
\begin {align*} \int \frac {1}{-2+x^4} \, dx &=-\frac {\int \frac {1}{\sqrt {2}-x^2} \, dx}{2 \sqrt {2}}-\frac {\int \frac {1}{\sqrt {2}+x^2} \, dx}{2 \sqrt {2}}\\ &=-\frac {\tan ^{-1}\left (\frac {x}{\sqrt [4]{2}}\right )}{2\ 2^{3/4}}-\frac {\tanh ^{-1}\left (\frac {x}{\sqrt [4]{2}}\right )}{2\ 2^{3/4}}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 43, normalized size = 1.23 \begin {gather*} -\frac {2 \tan ^{-1}\left (\frac {x}{\sqrt [4]{2}}\right )-\log \left (2-2^{3/4} x\right )+\log \left (2+2^{3/4} x\right )}{4\ 2^{3/4}} \end {gather*}
Antiderivative was successfully verified.
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Mathics [A]
time = 1.93, size = 31, normalized size = 0.89 \begin {gather*} \frac {2^{\frac {1}{4}} \left (\text {Log}\left [-2^{\frac {1}{4}}+x\right ]-2 \text {ArcTan}\left [\frac {2^{\frac {3}{4}} x}{2}\right ]-\text {Log}\left [2^{\frac {1}{4}}+x\right ]\right )}{8} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.03, size = 32, normalized size = 0.91
method | result | size |
risch | \(\frac {\left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{4}-2\right )}{\sum }\frac {\ln \left (-\textit {\_R} +x \right )}{\textit {\_R}^{3}}\right )}{4}\) | \(22\) |
default | \(-\frac {2^{\frac {1}{4}} \left (\ln \left (\frac {x +2^{\frac {1}{4}}}{x -2^{\frac {1}{4}}}\right )+2 \arctan \left (\frac {x 2^{\frac {3}{4}}}{2}\right )\right )}{8}\) | \(32\) |
meijerg | \(\frac {2^{\frac {1}{4}} x \left (\ln \left (1-\frac {2^{\frac {3}{4}} \left (x^{4}\right )^{\frac {1}{4}}}{2}\right )-\ln \left (1+\frac {2^{\frac {3}{4}} \left (x^{4}\right )^{\frac {1}{4}}}{2}\right )-2 \arctan \left (\frac {2^{\frac {3}{4}} \left (x^{4}\right )^{\frac {1}{4}}}{2}\right )\right )}{8 \left (x^{4}\right )^{\frac {1}{4}}}\) | \(54\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.37, size = 34, normalized size = 0.97 \begin {gather*} -\frac {1}{4} \cdot 2^{\frac {1}{4}} \arctan \left (\frac {1}{2} \cdot 2^{\frac {3}{4}} x\right ) + \frac {1}{8} \cdot 2^{\frac {1}{4}} \log \left (\frac {x - 2^{\frac {1}{4}}}{x + 2^{\frac {1}{4}}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 56 vs.
\(2 (25) = 50\).
time = 0.33, size = 56, normalized size = 1.60 \begin {gather*} \frac {1}{8} \cdot 8^{\frac {3}{4}} \arctan \left (-\frac {1}{2} \cdot 8^{\frac {1}{4}} x + \frac {1}{2} \cdot 8^{\frac {1}{4}} \sqrt {x^{2} + \sqrt {2}}\right ) - \frac {1}{32} \cdot 8^{\frac {3}{4}} \log \left (4 \, x + 8^{\frac {3}{4}}\right ) + \frac {1}{32} \cdot 8^{\frac {3}{4}} \log \left (4 \, x - 8^{\frac {3}{4}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.16, size = 46, normalized size = 1.31 \begin {gather*} \frac {\sqrt [4]{2} \log {\left (x - \sqrt [4]{2} \right )}}{8} - \frac {\sqrt [4]{2} \log {\left (x + \sqrt [4]{2} \right )}}{8} - \frac {\sqrt [4]{2} \operatorname {atan}{\left (\frac {2^{\frac {3}{4}} x}{2} \right )}}{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.00, size = 55, normalized size = 1.57 \begin {gather*} \frac {1}{8}\cdot 2^{\frac {1}{4}} \ln \left |x-2^{\frac {1}{4}}\right |-\frac {1}{8}\cdot 2^{\frac {1}{4}} \ln \left |x+2^{\frac {1}{4}}\right |-\frac {1}{4}\cdot 2^{\frac {1}{4}} \arctan \left (\frac {x}{2^{\frac {1}{4}}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.16, size = 20, normalized size = 0.57 \begin {gather*} -\frac {2^{1/4}\,\left (\mathrm {atan}\left (\frac {2^{3/4}\,x}{2}\right )+\mathrm {atanh}\left (\frac {2^{3/4}\,x}{2}\right )\right )}{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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