3.31.39 \(\int \frac {1}{1-4 (x^{2 n})^{\frac {1}{n}}} \, dx\) [3039]

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]

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

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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.

[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 212

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

Rule 260

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)]

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.

[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 = 0.28, 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.

[In]

int(1/(1-4*(x^(2*n))^(1/n)),x,method=_RETURNVERBOSE)

[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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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 = 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.

[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 = 0.03, 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.

[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 = 1.29, 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.

[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))

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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.

[In]

int(-1/(4*(x^(2*n))^(1/n) - 1),x)

[Out]

-int(1/(4*(x^(2*n))^(1/n) - 1), x)

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