3.3027 \(\int \frac{1}{1+4 \sqrt{x^4}} \, dx\)

Optimal. Leaf size=22 \[ \frac{x \tan ^{-1}\left (2 \sqrt [4]{x^4}\right )}{2 \sqrt [4]{x^4}} \]

[Out]

(x*ArcTan[2*(x^4)^(1/4)])/(2*(x^4)^(1/4))

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Rubi [A]  time = 0.0133759, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{x \tan ^{-1}\left (2 \sqrt [4]{x^4}\right )}{2 \sqrt [4]{x^4}} \]

Antiderivative was successfully verified.

[In]  Int[(1 + 4*Sqrt[x^4])^(-1),x]

[Out]

(x*ArcTan[2*(x^4)^(1/4)])/(2*(x^4)^(1/4))

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Rubi in Sympy [A]  time = 1.25802, size = 19, normalized size = 0.86 \[ \frac{x \operatorname{atan}{\left (2 \sqrt [4]{x^{4}} \right )}}{2 \sqrt [4]{x^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*atan(2*(x**4)**(1/4))/(2*(x**4)**(1/4))

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Mathematica [A]  time = 0.0406807, size = 0, normalized size = 0. \[ \int \frac{1}{1+4 \sqrt{x^4}} \, dx \]

Verification is Not applicable to the result.

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

[Out]

Integrate[(1 + 4*Sqrt[x^4])^(-1), x]

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Maple [A]  time = 0.013, size = 29, normalized size = 1.3 \[{\frac{1}{2}\arctan \left ( 2\,\sqrt{{\frac{\sqrt{{x}^{4}}}{{x}^{2}}}}x \right ){\frac{1}{\sqrt{{\frac{1}{{x}^{2}}\sqrt{{x}^{4}}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(1+4*(x^4)^(1/2)),x)

[Out]

1/2/((x^4)^(1/2)/x^2)^(1/2)*arctan(2*((x^4)^(1/2)/x^2)^(1/2)*x)

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Maxima [A]  time = 1.59791, size = 8, normalized size = 0.36 \[ \frac{1}{2} \, \arctan \left (2 \, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*arctan(2*x)

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Fricas [A]  time = 0.23709, size = 8, normalized size = 0.36 \[ \frac{1}{2} \, \arctan \left (2 \, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*arctan(2*x)

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Sympy [A]  time = 0.18316, size = 5, normalized size = 0.23 \[ \frac{\operatorname{atan}{\left (2 x \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

atan(2*x)/2

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GIAC/XCAS [A]  time = 0.210987, size = 8, normalized size = 0.36 \[ \frac{1}{2} \, \arctan \left (2 \, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*arctan(2*x)