3.938 \(\int \frac{1}{x \left (1+x^4\right )^{3/2}} \, dx\)

Optimal. Leaf size=28 \[ \frac{1}{2 \sqrt{x^4+1}}-\frac{1}{2} \tanh ^{-1}\left (\sqrt{x^4+1}\right ) \]

[Out]

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

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Rubi [A]  time = 0.033269, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308 \[ \frac{1}{2 \sqrt{x^4+1}}-\frac{1}{2} \tanh ^{-1}\left (\sqrt{x^4+1}\right ) \]

Antiderivative was successfully verified.

[In]  Int[1/(x*(1 + x^4)^(3/2)),x]

[Out]

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

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Rubi in Sympy [A]  time = 3.94977, size = 22, normalized size = 0.79 \[ - \frac{\operatorname{atanh}{\left (\sqrt{x^{4} + 1} \right )}}{2} + \frac{1}{2 \sqrt{x^{4} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x/(x**4+1)**(3/2),x)

[Out]

-atanh(sqrt(x**4 + 1))/2 + 1/(2*sqrt(x**4 + 1))

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Mathematica [A]  time = 0.0361846, size = 28, normalized size = 1. \[ \frac{1}{2 \sqrt{x^4+1}}-\frac{1}{2} \tanh ^{-1}\left (\sqrt{x^4+1}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x*(1 + x^4)^(3/2)),x]

[Out]

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

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Maple [A]  time = 0.016, size = 21, normalized size = 0.8 \[{\frac{1}{2}{\frac{1}{\sqrt{{x}^{4}+1}}}}-{\frac{1}{2}{\it Artanh} \left ({\frac{1}{\sqrt{{x}^{4}+1}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x/(x^4+1)^(3/2),x)

[Out]

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

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Maxima [A]  time = 1.41593, size = 46, normalized size = 1.64 \[ \frac{1}{2 \, \sqrt{x^{4} + 1}} - \frac{1}{4} \, \log \left (\sqrt{x^{4} + 1} + 1\right ) + \frac{1}{4} \, \log \left (\sqrt{x^{4} + 1} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^4 + 1)^(3/2)*x),x, algorithm="maxima")

[Out]

1/2/sqrt(x^4 + 1) - 1/4*log(sqrt(x^4 + 1) + 1) + 1/4*log(sqrt(x^4 + 1) - 1)

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Fricas [A]  time = 0.267229, size = 65, normalized size = 2.32 \[ -\frac{\sqrt{x^{4} + 1} \log \left (\sqrt{x^{4} + 1} + 1\right ) - \sqrt{x^{4} + 1} \log \left (\sqrt{x^{4} + 1} - 1\right ) - 2}{4 \, \sqrt{x^{4} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^4 + 1)^(3/2)*x),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 4.70878, size = 87, normalized size = 3.11 \[ \frac{x^{4} \log{\left (x^{4} \right )}}{4 x^{4} + 4} - \frac{2 x^{4} \log{\left (\sqrt{x^{4} + 1} + 1 \right )}}{4 x^{4} + 4} + \frac{2 \sqrt{x^{4} + 1}}{4 x^{4} + 4} + \frac{\log{\left (x^{4} \right )}}{4 x^{4} + 4} - \frac{2 \log{\left (\sqrt{x^{4} + 1} + 1 \right )}}{4 x^{4} + 4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x/(x**4+1)**(3/2),x)

[Out]

x**4*log(x**4)/(4*x**4 + 4) - 2*x**4*log(sqrt(x**4 + 1) + 1)/(4*x**4 + 4) + 2*sq
rt(x**4 + 1)/(4*x**4 + 4) + log(x**4)/(4*x**4 + 4) - 2*log(sqrt(x**4 + 1) + 1)/(
4*x**4 + 4)

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GIAC/XCAS [A]  time = 0.221631, size = 46, normalized size = 1.64 \[ \frac{1}{2 \, \sqrt{x^{4} + 1}} - \frac{1}{4} \,{\rm ln}\left (\sqrt{x^{4} + 1} + 1\right ) + \frac{1}{4} \,{\rm ln}\left (\sqrt{x^{4} + 1} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^4 + 1)^(3/2)*x),x, algorithm="giac")

[Out]

1/2/sqrt(x^4 + 1) - 1/4*ln(sqrt(x^4 + 1) + 1) + 1/4*ln(sqrt(x^4 + 1) - 1)