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

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 = 0.10464, 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 \[ \tan ^{-1}\left (\frac{x}{\sqrt{\left (x^{2 n}+1\right )^{\frac{1}{n}}-x^2}}\right ) \]

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)]]

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Rubi in Sympy [A]  time = 3.96785, size = 19, normalized size = 0.79 \[ \operatorname{atan}{\left (\frac{x}{\sqrt{- x^{2} + \left (x^{2 n} + 1\right )^{\frac{1}{n}}}} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

atan(x/sqrt(-x**2 + (x**(2*n) + 1)**(1/n)))

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Mathematica [A]  time = 0.0517336, 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 = 0.106, size = 0, normalized size = 0. \[ \int{\frac{1}{1+{x}^{2\,n}}{\frac{1}{\sqrt{-{x}^{2}+\sqrt [n]{1+{x}^{2\,n}}}}}}\, dx \]

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. \[ \int \frac{1}{\sqrt{-x^{2} +{\left (x^{2 \, n} + 1\right )}^{\left (\frac{1}{n}\right )}}{\left (x^{2 \, n} + 1\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(-x^2 + (x^(2*n) + 1)^(1/n))*(x^(2*n) + 1)),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. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: TypeError

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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