3.3031 \(\int \frac{1}{1+4 \left (x^{2 n}\right )^{\frac{1}{n}}} \, dx\)

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 = 0.0159857, 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 \[ \frac{1}{2} x \left (x^{2 n}\right )^{\left .-\frac{1}{2}\right /n} \tan ^{-1}\left (2 \left (x^{2 n}\right )^{\left .\frac{1}{2}\right /n}\right ) \]

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)))

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Rubi in Sympy [A]  time = 1.56992, size = 26, normalized size = 0.76 \[ \frac{x \left (x^{2 n}\right )^{- \frac{1}{2 n}} \operatorname{atan}{\left (2 \left (x^{2 n}\right )^{\frac{1}{2 n}} \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(1+4*(x**(2*n))**(1/n)),x)

[Out]

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

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Mathematica [A]  time = 2.17893, size = 0, normalized size = 0. \[ \int \frac{1}{1+4 \left (x^{2 n}\right )^{\frac{1}{n}}} \, dx \]

Verification is Not applicable to the result.

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

[Out]

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

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Maple [A]  time = 0.107, size = 29, normalized size = 0.9 \[{\frac{x}{2} \left ({x}^{2\,n} \right ) ^{-{\frac{1}{2\,n}}}\arctan \left ( 2\, \left ({x}^{2\,n} \right ) ^{1/2\,{n}^{-1}} \right ) } \]

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 [A]  time = 22.419, size = 8, normalized size = 0.24 \[ \frac{1}{2} \, \arctan \left (2 \, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(4*(x^(2*n))^(1/n) + 1),x, algorithm="maxima")

[Out]

1/2*arctan(2*x)

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Fricas [A]  time = 0.249771, size = 8, normalized size = 0.24 \[ \frac{1}{2} \, \arctan \left (2 \, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(4*(x^(2*n))^(1/n) + 1),x, algorithm="fricas")

[Out]

1/2*arctan(2*x)

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Sympy [A]  time = 0.174254, size = 5, normalized size = 0.15 \[ \frac{\operatorname{atan}{\left (2 x \right )}}{2} \]

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/XCAS [A]  time = 0.216322, size = 8, normalized size = 0.24 \[ \frac{1}{2} \, \arctan \left (2 \, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(4*(x^(2*n))^(1/n) + 1),x, algorithm="giac")

[Out]

1/2*arctan(2*x)