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

Optimal. Leaf size=14 \[ \frac{1}{2} \tan ^{-1}\left (\sqrt{x^4-1}\right ) \]

[Out]

ArcTan[Sqrt[-1 + x^4]]/2

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Rubi [A]  time = 0.0236733, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ \frac{1}{2} \tan ^{-1}\left (\sqrt{x^4-1}\right ) \]

Antiderivative was successfully verified.

[In]  Int[1/(x*Sqrt[-1 + x^4]),x]

[Out]

ArcTan[Sqrt[-1 + x^4]]/2

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Rubi in Sympy [A]  time = 3.20581, size = 10, normalized size = 0.71 \[ \frac{\operatorname{atan}{\left (\sqrt{x^{4} - 1} \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

atan(sqrt(x**4 - 1))/2

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Mathematica [A]  time = 0.0320399, size = 14, normalized size = 1. \[ \frac{1}{2} \tan ^{-1}\left (\sqrt{x^4-1}\right ) \]

Antiderivative was successfully verified.

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

[Out]

ArcTan[Sqrt[-1 + x^4]]/2

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Maple [A]  time = 0.016, size = 11, normalized size = 0.8 \[ -{\frac{1}{2}\arctan \left ({\frac{1}{\sqrt{{x}^{4}-1}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x/(x^4-1)^(1/2),x)

[Out]

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

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Maxima [A]  time = 1.61973, size = 14, normalized size = 1. \[ \frac{1}{2} \, \arctan \left (\sqrt{x^{4} - 1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*arctan(sqrt(x^4 - 1))

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Fricas [A]  time = 0.275404, size = 14, normalized size = 1. \[ \frac{1}{2} \, \arctan \left (\sqrt{x^{4} - 1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*arctan(sqrt(x^4 - 1))

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Sympy [A]  time = 3.33344, size = 24, normalized size = 1.71 \[ \begin{cases} \frac{i \operatorname{acosh}{\left (\frac{1}{x^{2}} \right )}}{2} & \text{for}\: \left |{\frac{1}{x^{4}}}\right | > 1 \\- \frac{\operatorname{asin}{\left (\frac{1}{x^{2}} \right )}}{2} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((I*acosh(x**(-2))/2, Abs(x**(-4)) > 1), (-asin(x**(-2))/2, True))

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GIAC/XCAS [A]  time = 0.213125, size = 14, normalized size = 1. \[ \frac{1}{2} \, \arctan \left (\sqrt{x^{4} - 1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*arctan(sqrt(x^4 - 1))