Optimal. Leaf size=35 \[ -\frac {\tan ^{-1}\left (\sqrt [4]{5} x\right )}{2 \sqrt [4]{5}}-\frac {\tanh ^{-1}\left (\sqrt [4]{5} x\right )}{2 \sqrt [4]{5}} \]
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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 = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {218, 212, 209}
\begin {gather*} -\frac {\text {ArcTan}\left (\sqrt [4]{5} x\right )}{2 \sqrt [4]{5}}-\frac {\tanh ^{-1}\left (\sqrt [4]{5} x\right )}{2 \sqrt [4]{5}} \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 212
Rule 218
Rubi steps
\begin {align*} \int \frac {1}{-1+5 x^4} \, dx &=-\left (\frac {1}{2} \int \frac {1}{1-\sqrt {5} x^2} \, dx\right )-\frac {1}{2} \int \frac {1}{1+\sqrt {5} x^2} \, dx\\ &=-\frac {\tan ^{-1}\left (\sqrt [4]{5} x\right )}{2 \sqrt [4]{5}}-\frac {\tanh ^{-1}\left (\sqrt [4]{5} x\right )}{2 \sqrt [4]{5}}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 43, normalized size = 1.23 \begin {gather*} -\frac {2 \tan ^{-1}\left (\sqrt [4]{5} x\right )-\log \left (1-\sqrt [4]{5} x\right )+\log \left (1+\sqrt [4]{5} x\right )}{4 \sqrt [4]{5}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.03, size = 33, normalized size = 0.94
method | result | size |
risch | \(\frac {\left (\munderset {\textit {\_R} =\RootOf \left (5 \textit {\_Z}^{4}-1\right )}{\sum }\frac {\ln \left (-\textit {\_R} +x \right )}{\textit {\_R}^{3}}\right )}{20}\) | \(24\) |
default | \(-\frac {5^{\frac {3}{4}} \left (\ln \left (\frac {x +\frac {5^{\frac {3}{4}}}{5}}{x -\frac {5^{\frac {3}{4}}}{5}}\right )+2 \arctan \left (5^{\frac {1}{4}} x \right )\right )}{20}\) | \(33\) |
meijerg | \(\frac {5^{\frac {3}{4}} x \left (\ln \left (1-5^{\frac {1}{4}} \left (x^{4}\right )^{\frac {1}{4}}\right )-\ln \left (1+5^{\frac {1}{4}} \left (x^{4}\right )^{\frac {1}{4}}\right )-2 \arctan \left (5^{\frac {1}{4}} \left (x^{4}\right )^{\frac {1}{4}}\right )\right )}{20 \left (x^{4}\right )^{\frac {1}{4}}}\) | \(52\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 4.55, size = 41, normalized size = 1.17 \begin {gather*} -\frac {1}{10} \cdot 5^{\frac {3}{4}} \arctan \left (5^{\frac {1}{4}} x\right ) + \frac {1}{20} \cdot 5^{\frac {3}{4}} \log \left (\frac {\sqrt {5} x - 5^{\frac {1}{4}}}{\sqrt {5} x + 5^{\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. 60 vs.
\(2 (23) = 46\).
time = 0.95, size = 60, normalized size = 1.71 \begin {gather*} \frac {1}{5} \cdot 5^{\frac {3}{4}} \arctan \left (-5^{\frac {1}{4}} x + \frac {1}{5} \cdot 5^{\frac {1}{4}} \sqrt {25 \, x^{2} + 5 \, \sqrt {5}}\right ) - \frac {1}{20} \cdot 5^{\frac {3}{4}} \log \left (5 \, x + 5^{\frac {3}{4}}\right ) + \frac {1}{20} \cdot 5^{\frac {3}{4}} \log \left (5 \, x - 5^{\frac {3}{4}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.15, size = 48, normalized size = 1.37 \begin {gather*} \frac {5^{\frac {3}{4}} \log {\left (x - \frac {5^{\frac {3}{4}}}{5} \right )}}{20} - \frac {5^{\frac {3}{4}} \log {\left (x + \frac {5^{\frac {3}{4}}}{5} \right )}}{20} - \frac {5^{\frac {3}{4}} \operatorname {atan}{\left (\sqrt [4]{5} x \right )}}{10} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.45, size = 39, normalized size = 1.11 \begin {gather*} -\frac {1}{10} \cdot 5^{\frac {3}{4}} \arctan \left (5 \, \left (\frac {1}{5}\right )^{\frac {3}{4}} x\right ) - \frac {1}{20} \cdot 5^{\frac {3}{4}} \log \left ({\left | x + \left (\frac {1}{5}\right )^{\frac {1}{4}} \right |}\right ) + \frac {1}{20} \cdot 5^{\frac {3}{4}} \log \left ({\left | x - \left (\frac {1}{5}\right )^{\frac {1}{4}} \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.18, size = 18, normalized size = 0.51 \begin {gather*} -\frac {5^{3/4}\,\left (\mathrm {atan}\left (5^{1/4}\,x\right )+\mathrm {atanh}\left (5^{1/4}\,x\right )\right )}{10} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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