3.3039 \(\int \frac{1}{1-4 (x^{2 n})^{\frac{1}{n}}} \, dx\)

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 = 0.0056382, 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, 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.

[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)))

Rule 254

Int[((a_) + (b_.)*((c_.)*(x_)^(q_.))^(n_))^(p_.), x_Symbol] :> Dist[x/(c*x^q)^(1/q), Subst[Int[(a + b*x^(n*q))
^p, x], x, (c*x^q)^(1/q)], x] /; FreeQ[{a, b, c, n, p, q}, x] && IntegerQ[n*q] && NeQ[x, (c*x^q)^(1/q)]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

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*}

Mathematica [A]  time = 0.004507, size = 34, normalized size = 1. \[ \frac{1}{2} x \left (x^{2 n}\right )^{\left .-\frac{1}{2}\right /n} \tanh ^{-1}\left (2 \left (x^{2 n}\right )^{\left .\frac{1}{2}\right /n}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(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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Maple [A]  time = 0.34, size = 29, normalized size = 0.9 \begin{align*}{\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 ) } \end{align*}

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 [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\int \frac{1}{4 \,{\left (x^{2 \, n}\right )}^{\left (\frac{1}{n}\right )} - 1}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-integrate(1/(4*(x^(2*n))^(1/n) - 1), x)

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Fricas [A]  time = 1.28178, size = 50, normalized size = 1.47 \begin{align*} \frac{1}{4} \, \log \left (2 \, x + 1\right ) - \frac{1}{4} \, \log \left (2 \, x - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*log(2*x + 1) - 1/4*log(2*x - 1)

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Sympy [A]  time = 0.096625, size = 15, normalized size = 0.44 \begin{align*} - \frac{\log{\left (x - \frac{1}{2} \right )}}{4} + \frac{\log{\left (x + \frac{1}{2} \right )}}{4} \end{align*}

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 [A]  time = 1.1428, size = 20, normalized size = 0.59 \begin{align*} \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{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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