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 = 0.01, 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 = {254, 206} \[ \frac {1}{2} x \left (x^{2 n}\right )^{\left .-\frac {1}{2}\right /n} \tanh ^{-1}\left (2 \left (x^{2 n}\right )^{\left .\frac {1}{2}\right /n}\right ) \]
Antiderivative was successfully verified.
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Rule 206
Rule 254
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 ) \operatorname {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 = 0.00, size = 34, normalized size = 1.00 \[ \frac {1}{2} x \left (x^{2 n}\right )^{\left .-\frac {1}{2}\right /n} \tanh ^{-1}\left (2 \left (x^{2 n}\right )^{\left .\frac {1}{2}\right /n}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.88, size = 17, normalized size = 0.50 \[ \frac {1}{4} \, \log \left (2 \, x + 1\right ) - \frac {1}{4} \, \log \left (2 \, x - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 15, normalized size = 0.44 \[ \frac {1}{4} \, \log \left ({\left | x + \frac {1}{2} \right |}\right ) - \frac {1}{4} \, \log \left ({\left | x - \frac {1}{2} \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.13, size = 42, normalized size = 1.24 \[ \frac {x \left (x^{2 n}\right )^{-\frac {1}{2 n}} \arctanh \left (2 \left (x^{2 n}\right )^{\frac {1}{n}} \left (x^{2 n}\right )^{-\frac {1}{2 n}}\right )}{2} \]
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 \[ -\int \frac {1}{4 \, {\left (x^{2 \, n}\right )}^{\left (\frac {1}{n}\right )} - 1}\,{d x} \]
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 \[ -\int \frac {1}{4\,{\left (x^{2\,n}\right )}^{1/n}-1} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.14, size = 15, normalized size = 0.44 \[ - \frac {\log {\left (x - \frac {1}{2} \right )}}{4} + \frac {\log {\left (x + \frac {1}{2} \right )}}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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