3.3038 \(\int \frac{1}{1+4 (x^{2 n})^{\frac{1}{n}}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.006593, 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, 203} \[ \frac{1}{2} x \left (x^{2 n}\right )^{\left .-\frac{1}{2}\right /n} \tan ^{-1}\left (2 \left (x^{2 n}\right )^{\left .\frac{1}{2}\right /n}\right ) \]

Antiderivative was successfully verified.

[In]

Int[(1 + 4*(x^(2*n))^n^(-1))^(-1),x]

[Out]

(x*ArcTan[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 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[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} \tan ^{-1}\left (2 \left (x^{2 n}\right )^{\left .\frac{1}{2}\right /n}\right )\\ \end{align*}

Mathematica [A]  time = 0.0073461, size = 34, normalized size = 1. \[ \frac{1}{2} x \left (x^{2 n}\right )^{\left .-\frac{1}{2}\right /n} \tan ^{-1}\left (2 \left (x^{2 n}\right )^{\left .\frac{1}{2}\right /n}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.398, size = 29, normalized size = 0.9 \begin{align*}{\frac{x}{2} \left ({x}^{2\,n} \right ) ^{-{\frac{1}{2\,n}}}\arctan \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)*arctan(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 = 1.22548, size = 23, normalized size = 0.68 \begin{align*} \frac{1}{2} \, \arctan \left (2 \, x\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]

1/2*arctan(2*x)

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Sympy [A]  time = 0.095168, size = 5, normalized size = 0.15 \begin{align*} \frac{\operatorname{atan}{\left (2 x \right )}}{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1+4*(x**(2*n))**(1/n)),x)

[Out]

atan(2*x)/2

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Giac [A]  time = 1.11607, size = 8, normalized size = 0.24 \begin{align*} \frac{1}{2} \, \arctan \left (2 \, x\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]

1/2*arctan(2*x)