3.3028 \(\int \frac{1}{1-4 \sqrt{x^4}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0115792, 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 \[ \frac{x \tanh ^{-1}\left (2 \sqrt [4]{x^4}\right )}{2 \sqrt [4]{x^4}} \]

Antiderivative was successfully verified.

[In]  Int[(1 - 4*Sqrt[x^4])^(-1),x]

[Out]

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

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Rubi in Sympy [A]  time = 1.28666, size = 19, normalized size = 0.86 \[ \frac{x \operatorname{atanh}{\left (2 \sqrt [4]{x^{4}} \right )}}{2 \sqrt [4]{x^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0336869, 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 = 0.007, size = 29, normalized size = 1.3 \[{\frac{1}{2}{\it Artanh} \left ( 2\,\sqrt{{\frac{\sqrt{{x}^{4}}}{{x}^{2}}}}x \right ){\frac{1}{\sqrt{{\frac{1}{{x}^{2}}\sqrt{{x}^{4}}}}}}} \]

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

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Maxima [A]  time = 1.36199, size = 23, normalized size = 1.05 \[ \frac{1}{4} \, \log \left (2 \, x + 1\right ) - \frac{1}{4} \, \log \left (2 \, x - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*log(2*x + 1) - 1/4*log(2*x - 1)

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Fricas [A]  time = 0.245985, size = 23, normalized size = 1.05 \[ \frac{1}{4} \, \log \left (2 \, x + 1\right ) - \frac{1}{4} \, \log \left (2 \, x - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*log(2*x + 1) - 1/4*log(2*x - 1)

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Sympy [A]  time = 0.188317, size = 15, normalized size = 0.68 \[ - \frac{\log{\left (x - \frac{1}{2} \right )}}{4} + \frac{\log{\left (x + \frac{1}{2} \right )}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-log(x - 1/2)/4 + log(x + 1/2)/4

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GIAC/XCAS [A]  time = 0.215656, size = 20, normalized size = 0.91 \[ \frac{1}{4} \,{\rm ln}\left ({\left | x + \frac{1}{2} \right |}\right ) - \frac{1}{4} \,{\rm ln}\left ({\left | x - \frac{1}{2} \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*ln(abs(x + 1/2)) - 1/4*ln(abs(x - 1/2))