3.3032 \(\int \frac{1}{1-4 \left (x^{2 n}\right )^{\frac{1}{n}}} \, dx\)

Optimal. Leaf size=34 \[ \frac{1}{2} x \left (x^{2 n}\right )^{\left .-\frac{1}{2}\right /n} \tanh ^{-1}\left (2 \left (x^{2 n}\right )^{\left .\frac{1}{2}\right /n}\right ) \]

[Out]

(x*ArcTanh[2*(x^(2*n))^(1/(2*n))])/(2*(x^(2*n))^(1/(2*n)))

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Rubi [A]  time = 0.0153499, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ \frac{1}{2} x \left (x^{2 n}\right )^{\left .-\frac{1}{2}\right /n} \tanh ^{-1}\left (2 \left (x^{2 n}\right )^{\left .\frac{1}{2}\right /n}\right ) \]

Antiderivative was successfully verified.

[In]  Int[(1 - 4*(x^(2*n))^n^(-1))^(-1),x]

[Out]

(x*ArcTanh[2*(x^(2*n))^(1/(2*n))])/(2*(x^(2*n))^(1/(2*n)))

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Rubi in Sympy [A]  time = 1.58741, size = 26, normalized size = 0.76 \[ \frac{x \left (x^{2 n}\right )^{- \frac{1}{2 n}} \operatorname{atanh}{\left (2 \left (x^{2 n}\right )^{\frac{1}{2 n}} \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(1-4*(x**(2*n))**(1/n)),x)

[Out]

x*(x**(2*n))**(-1/(2*n))*atanh(2*(x**(2*n))**(1/(2*n)))/2

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Mathematica [A]  time = 2.16826, size = 0, normalized size = 0. \[ \int \frac{1}{1-4 \left (x^{2 n}\right )^{\frac{1}{n}}} \, dx \]

Verification is Not applicable to the result.

[In]  Integrate[(1 - 4*(x^(2*n))^n^(-1))^(-1),x]

[Out]

Integrate[(1 - 4*(x^(2*n))^n^(-1))^(-1), x]

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Maple [A]  time = 0.099, size = 29, normalized size = 0.9 \[{\frac{x}{2} \left ({x}^{2\,n} \right ) ^{-{\frac{1}{2\,n}}}{\it Artanh} \left ( 2\, \left ({x}^{2\,n} \right ) ^{1/2\,{n}^{-1}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(1-4*(x^(2*n))^(1/n)),x)

[Out]

1/2*x*(x^(2*n))^(-1/2/n)*arctanh(2*(x^(2*n))^(1/2/n))

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Maxima [A]  time = 33.7958, size = 31, normalized size = 0.91 \[ \frac{1}{4} \, \log \left (4 \,{\left (x^{n}\right )}^{\left (\frac{1}{n}\right )} + 2\right ) - \frac{1}{4} \, \log \left (4 \, x - 2\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-1/(4*(x^(2*n))^(1/n) - 1),x, algorithm="maxima")

[Out]

1/4*log(4*(x^n)^(1/n) + 2) - 1/4*log(4*x - 2)

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Fricas [A]  time = 0.253853, size = 23, normalized size = 0.68 \[ \frac{1}{4} \, \log \left (2 \, x + 1\right ) - \frac{1}{4} \, \log \left (2 \, x - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-1/(4*(x^(2*n))^(1/n) - 1),x, algorithm="fricas")

[Out]

1/4*log(2*x + 1) - 1/4*log(2*x - 1)

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Sympy [A]  time = 0.180874, 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.

[In]  integrate(1/(1-4*(x**(2*n))**(1/n)),x)

[Out]

-log(x - 1/2)/4 + log(x + 1/2)/4

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GIAC/XCAS [A]  time = 0.216162, size = 20, normalized size = 0.59 \[ \frac{1}{4} \,{\rm ln}\left ({\left | x + \frac{1}{2} \right |}\right ) - \frac{1}{4} \,{\rm ln}\left ({\left | x - \frac{1}{2} \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-1/(4*(x^(2*n))^(1/n) - 1),x, algorithm="giac")

[Out]

1/4*ln(abs(x + 1/2)) - 1/4*ln(abs(x - 1/2))