3.442 \(\int \frac{1}{x \sqrt{1+x^3}} \, dx\)

Optimal. Leaf size=14 \[ -\frac{2}{3} \tanh ^{-1}\left (\sqrt{x^3+1}\right ) \]

[Out]

(-2*ArcTanh[Sqrt[1 + x^3]])/3

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Rubi [A]  time = 0.0240662, 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 \[ -\frac{2}{3} \tanh ^{-1}\left (\sqrt{x^3+1}\right ) \]

Antiderivative was successfully verified.

[In]  Int[1/(x*Sqrt[1 + x^3]),x]

[Out]

(-2*ArcTanh[Sqrt[1 + x^3]])/3

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Rubi in Sympy [A]  time = 3.27789, size = 14, normalized size = 1. \[ - \frac{2 \operatorname{atanh}{\left (\sqrt{x^{3} + 1} \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*atanh(sqrt(x**3 + 1))/3

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Mathematica [A]  time = 0.0167783, size = 14, normalized size = 1. \[ -\frac{2}{3} \tanh ^{-1}\left (\sqrt{x^3+1}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-2*ArcTanh[Sqrt[1 + x^3]])/3

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Maple [A]  time = 0.17, size = 11, normalized size = 0.8 \[ -{\frac{2}{3}{\it Artanh} \left ( \sqrt{{x}^{3}+1} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x/(x^3+1)^(1/2),x)

[Out]

-2/3*arctanh((x^3+1)^(1/2))

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Maxima [A]  time = 1.43723, size = 34, normalized size = 2.43 \[ -\frac{1}{3} \, \log \left (\sqrt{x^{3} + 1} + 1\right ) + \frac{1}{3} \, \log \left (\sqrt{x^{3} + 1} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*log(sqrt(x^3 + 1) + 1) + 1/3*log(sqrt(x^3 + 1) - 1)

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Fricas [A]  time = 0.227784, size = 34, normalized size = 2.43 \[ -\frac{1}{3} \, \log \left (\sqrt{x^{3} + 1} + 1\right ) + \frac{1}{3} \, \log \left (\sqrt{x^{3} + 1} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*log(sqrt(x^3 + 1) + 1) + 1/3*log(sqrt(x^3 + 1) - 1)

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Sympy [A]  time = 3.3641, size = 12, normalized size = 0.86 \[ - \frac{2 \operatorname{asinh}{\left (\frac{1}{x^{\frac{3}{2}}} \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*asinh(x**(-3/2))/3

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GIAC/XCAS [A]  time = 0.214755, size = 35, normalized size = 2.5 \[ -\frac{1}{3} \,{\rm ln}\left (\sqrt{x^{3} + 1} + 1\right ) + \frac{1}{3} \,{\rm ln}\left ({\left | \sqrt{x^{3} + 1} - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*ln(sqrt(x^3 + 1) + 1) + 1/3*ln(abs(sqrt(x^3 + 1) - 1))