Optimal. Leaf size=34 \[ \frac {1}{2} x \left (x^{2 n}\right )^{\left .-\frac {1}{2}\right /n} \tanh ^{-1}\left (2 \left (x^{2 n}\right )^{\left .\frac {1}{2}\right /n}\right ) \]
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Rubi [A]
time = 0.00, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {260, 212}
\begin {gather*} \frac {1}{2} x \left (x^{2 n}\right )^{\left .-\frac {1}{2}\right /n} \tanh ^{-1}\left (2 \left (x^{2 n}\right )^{\left .\frac {1}{2}\right /n}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 260
Rubi steps
\begin {align*} \int \frac {1}{1-4 \left (x^{2 n}\right )^{\frac {1}{n}}} \, dx &=\left (x \left (x^{2 n}\right )^{\left .-\frac {1}{2}\right /n}\right ) \text {Subst}\left (\int \frac {1}{1-4 x^2} \, dx,x,\left (x^{2 n}\right )^{\left .\frac {1}{2}\right /n}\right )\\ &=\frac {1}{2} x \left (x^{2 n}\right )^{\left .-\frac {1}{2}\right /n} \tanh ^{-1}\left (2 \left (x^{2 n}\right )^{\left .\frac {1}{2}\right /n}\right )\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 34, normalized size = 1.00 \begin {gather*} \frac {1}{2} x \left (x^{2 n}\right )^{\left .-\frac {1}{2}\right /n} \tanh ^{-1}\left (2 \left (x^{2 n}\right )^{\left .\frac {1}{2}\right /n}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.28, size = 29, normalized size = 0.85
method | result | size |
meijerg | \(\frac {x \left (x^{2 n}\right )^{-\frac {1}{2 n}} \arctanh \left (2 \left (x^{2 n}\right )^{\frac {1}{2 n}}\right )}{2}\) | \(29\) |
risch | \(\frac {x \left (x^{2 n}\right )^{-\frac {1}{2 n}} \arctanh \left (2 \left (x^{2 n}\right )^{\frac {1}{n}} \left (x^{2 n}\right )^{-\frac {1}{2 n}}\right )}{2}\) | \(42\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 17, normalized size = 0.50 \begin {gather*} \frac {1}{4} \, \log \left (2 \, x + 1\right ) - \frac {1}{4} \, \log \left (2 \, x - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.03, size = 15, normalized size = 0.44 \begin {gather*} - \frac {\log {\left (x - \frac {1}{2} \right )}}{4} + \frac {\log {\left (x + \frac {1}{2} \right )}}{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.29, size = 15, normalized size = 0.44 \begin {gather*} \frac {1}{4} \, \log \left ({\left | x + \frac {1}{2} \right |}\right ) - \frac {1}{4} \, \log \left ({\left | x - \frac {1}{2} \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} -\int \frac {1}{4\,{\left (x^{2\,n}\right )}^{1/n}-1} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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