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 = {212, 206, 203} \[ -\frac {\tan ^{-1}\left (\sqrt [4]{5} x\right )}{2 \sqrt [4]{5}}-\frac {\tanh ^{-1}\left (\sqrt [4]{5} x\right )}{2 \sqrt [4]{5}} \]
Antiderivative was successfully verified.
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Rule 203
Rule 206
Rule 212
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 = 0.02, size = 43, normalized size = 1.23 \[ -\frac {-\log \left (1-\sqrt [4]{5} x\right )+\log \left (\sqrt [4]{5} x+1\right )+2 \tan ^{-1}\left (\sqrt [4]{5} x\right )}{4 \sqrt [4]{5}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.44, size = 58, normalized size = 1.66 \[ \frac {1}{5} \cdot 5^{\frac {3}{4}} \arctan \left (\frac {1}{5} \cdot 5^{\frac {3}{4}} \sqrt {5 \, x^{2} + \sqrt {5}} - 5^{\frac {1}{4}} x\right ) - \frac {1}{20} \cdot 5^{\frac {3}{4}} \log \left (5 \, x + 5^{\frac {3}{4}}\right ) + \frac {1}{20} \cdot 5^{\frac {3}{4}} \log \left (5 \, x - 5^{\frac {3}{4}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.08, size = 39, normalized size = 1.11 \[ -\frac {1}{10} \cdot 5^{\frac {3}{4}} \arctan \left (5 \, \left (\frac {1}{5}\right )^{\frac {3}{4}} x\right ) - \frac {1}{20} \cdot 5^{\frac {3}{4}} \log \left ({\left | x + \left (\frac {1}{5}\right )^{\frac {1}{4}} \right |}\right ) + \frac {1}{20} \cdot 5^{\frac {3}{4}} \log \left ({\left | x - \left (\frac {1}{5}\right )^{\frac {1}{4}} \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 36, normalized size = 1.03 \[ -\frac {5^{\frac {3}{4}} \arctan \left (5^{\frac {1}{4}} x \right )}{10}-\frac {5^{\frac {3}{4}} \ln \left (\frac {x +\frac {5^{\frac {3}{4}}}{5}}{x -\frac {5^{\frac {3}{4}}}{5}}\right )}{20} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.98, size = 41, normalized size = 1.17 \[ -\frac {1}{10} \cdot 5^{\frac {3}{4}} \arctan \left (5^{\frac {1}{4}} x\right ) + \frac {1}{20} \cdot 5^{\frac {3}{4}} \log \left (\frac {\sqrt {5} x - 5^{\frac {1}{4}}}{\sqrt {5} x + 5^{\frac {1}{4}}}\right ) \]
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 \[ -\frac {5^{3/4}\,\left (\mathrm {atan}\left (5^{1/4}\,x\right )+\mathrm {atanh}\left (5^{1/4}\,x\right )\right )}{10} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.32, size = 48, normalized size = 1.37 \[ \frac {5^{\frac {3}{4}} \log {\left (x - \frac {5^{\frac {3}{4}}}{5} \right )}}{20} - \frac {5^{\frac {3}{4}} \log {\left (x + \frac {5^{\frac {3}{4}}}{5} \right )}}{20} - \frac {5^{\frac {3}{4}} \operatorname {atan}{\left (\sqrt [4]{5} x \right )}}{10} \]
Verification of antiderivative is not currently implemented for this CAS.
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