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

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

[Out]

-ArcTanh[Sqrt[1 + x^4]]/2

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Rubi [A]  time = 0.0232999, 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 \[ -\frac{1}{2} \tanh ^{-1}\left (\sqrt{x^4+1}\right ) \]

Antiderivative was successfully verified.

[In]  Int[1/(x*Sqrt[1 + x^4]),x]

[Out]

-ArcTanh[Sqrt[1 + x^4]]/2

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Rubi in Sympy [A]  time = 3.21453, size = 12, normalized size = 0.86 \[ - \frac{\operatorname{atanh}{\left (\sqrt{x^{4} + 1} \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-atanh(sqrt(x**4 + 1))/2

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Mathematica [A]  time = 0.0303389, size = 14, normalized size = 1. \[ -\frac{1}{2} \tanh ^{-1}\left (\sqrt{x^4+1}\right ) \]

Antiderivative was successfully verified.

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

[Out]

-ArcTanh[Sqrt[1 + x^4]]/2

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Maple [A]  time = 0.012, size = 11, normalized size = 0.8 \[ -{\frac{1}{2}{\it Artanh} \left ({\frac{1}{\sqrt{{x}^{4}+1}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x/(x^4+1)^(1/2),x)

[Out]

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

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Maxima [A]  time = 1.43781, size = 34, normalized size = 2.43 \[ -\frac{1}{4} \, \log \left (\sqrt{x^{4} + 1} + 1\right ) + \frac{1}{4} \, \log \left (\sqrt{x^{4} + 1} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/4*log(sqrt(x^4 + 1) + 1) + 1/4*log(sqrt(x^4 + 1) - 1)

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Fricas [A]  time = 0.271323, size = 34, normalized size = 2.43 \[ -\frac{1}{4} \, \log \left (\sqrt{x^{4} + 1} + 1\right ) + \frac{1}{4} \, \log \left (\sqrt{x^{4} + 1} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/4*log(sqrt(x^4 + 1) + 1) + 1/4*log(sqrt(x^4 + 1) - 1)

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Sympy [A]  time = 3.25413, size = 8, normalized size = 0.57 \[ - \frac{\operatorname{asinh}{\left (\frac{1}{x^{2}} \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-asinh(x**(-2))/2

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GIAC/XCAS [A]  time = 0.215251, size = 34, normalized size = 2.43 \[ -\frac{1}{4} \,{\rm ln}\left (\sqrt{x^{4} + 1} + 1\right ) + \frac{1}{4} \,{\rm ln}\left (\sqrt{x^{4} + 1} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/4*ln(sqrt(x^4 + 1) + 1) + 1/4*ln(sqrt(x^4 + 1) - 1)