3.329 \(\int \frac{1}{(1+x^{2 n}) \sqrt{-x^2+(1+x^{2 n})^{\frac{1}{n}}}} \, dx\)

Optimal. Leaf size=24 \[ \tan ^{-1}\left (\frac{x}{\sqrt{\left (x^{2 n}+1\right )^{\frac{1}{n}}-x^2}}\right ) \]

[Out]

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

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Rubi [A]  time = 0.0678271, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065, Rules used = {2128, 203} \[ \tan ^{-1}\left (\frac{x}{\sqrt{\left (x^{2 n}+1\right )^{\frac{1}{n}}-x^2}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[1/((1 + x^(2*n))*Sqrt[-x^2 + (1 + x^(2*n))^n^(-1)]),x]

[Out]

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

Rule 2128

Int[1/(((a_) + (b_.)*(x_)^(n_.))*Sqrt[(c_.)*(x_)^2 + (d_.)*((a_) + (b_.)*(x_)^(n_.))^(p_.)]), x_Symbol] :> Dis
t[1/a, Subst[Int[1/(1 - c*x^2), x], x, x/Sqrt[c*x^2 + d*(a + b*x^n)^(2/n)]], x] /; FreeQ[{a, b, c, d, n}, x] &
& EqQ[p, 2/n]

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}{\left (1+x^{2 n}\right ) \sqrt{-x^2+\left (1+x^{2 n}\right )^{\frac{1}{n}}}} \, dx &=\operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\frac{x}{\sqrt{-x^2+\left (1+x^{2 n}\right )^{\frac{1}{n}}}}\right )\\ &=\tan ^{-1}\left (\frac{x}{\sqrt{-x^2+\left (1+x^{2 n}\right )^{\frac{1}{n}}}}\right )\\ \end{align*}

Mathematica [A]  time = 0.0813074, size = 26, normalized size = 1.08 \[ \cot ^{-1}\left (\frac{\sqrt{\left (x^{2 n}+1\right )^{\frac{1}{n}}-x^2}}{x}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

ArcCot[Sqrt[-x^2 + (1 + x^(2*n))^n^(-1)]/x]

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Maple [F]  time = 0.079, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{1+{x}^{2\,n}}{\frac{1}{\sqrt{-{x}^{2}+\sqrt [n]{1+{x}^{2\,n}}}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(1+x^(2*n))/(-x^2+(1+x^(2*n))^(1/n))^(1/2),x)

[Out]

int(1/(1+x^(2*n))/(-x^2+(1+x^(2*n))^(1/n))^(1/2),x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: UnboundLocalError

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/(sqrt(-x**2 + (x**(2*n) + 1)**(1/n))*(x**(2*n) + 1)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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