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

Optimal. Leaf size=28 \[ \frac{2 \sqrt{x^3+1}}{3}-\frac{2}{3} \tanh ^{-1}\left (\sqrt{x^3+1}\right ) \]

[Out]

(2*Sqrt[1 + x^3])/3 - (2*ArcTanh[Sqrt[1 + x^3]])/3

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Rubi [A]  time = 0.0285713, 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 \[ \frac{2 \sqrt{x^3+1}}{3}-\frac{2}{3} \tanh ^{-1}\left (\sqrt{x^3+1}\right ) \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[1 + x^3]/x,x]

[Out]

(2*Sqrt[1 + x^3])/3 - (2*ArcTanh[Sqrt[1 + x^3]])/3

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Rubi in Sympy [A]  time = 1.9181, size = 24, normalized size = 0.86 \[ \frac{2 \sqrt{x^{3} + 1}}{3} - \frac{2 \operatorname{atanh}{\left (\sqrt{x^{3} + 1} \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*sqrt(x**3 + 1)/3 - 2*atanh(sqrt(x**3 + 1))/3

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Mathematica [A]  time = 0.0162647, size = 28, normalized size = 1. \[ \frac{2 \sqrt{x^3+1}}{3}-\frac{2}{3} \tanh ^{-1}\left (\sqrt{x^3+1}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[1 + x^3]/x,x]

[Out]

(2*Sqrt[1 + x^3])/3 - (2*ArcTanh[Sqrt[1 + x^3]])/3

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Maple [A]  time = 0.024, size = 21, normalized size = 0.8 \[ -{\frac{2}{3}{\it Artanh} \left ( \sqrt{{x}^{3}+1} \right ) }+{\frac{2}{3}\sqrt{{x}^{3}+1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((x^3+1)^(1/2)/x,x)

[Out]

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

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Maxima [A]  time = 1.43411, size = 46, normalized size = 1.64 \[ \frac{2}{3} \, \sqrt{x^{3} + 1} - \frac{1}{3} \, \log \left (\sqrt{x^{3} + 1} + 1\right ) + \frac{1}{3} \, \log \left (\sqrt{x^{3} + 1} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3*sqrt(x^3 + 1) - 1/3*log(sqrt(x^3 + 1) + 1) + 1/3*log(sqrt(x^3 + 1) - 1)

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Fricas [A]  time = 0.212808, size = 46, normalized size = 1.64 \[ \frac{2}{3} \, \sqrt{x^{3} + 1} - \frac{1}{3} \, \log \left (\sqrt{x^{3} + 1} + 1\right ) + \frac{1}{3} \, \log \left (\sqrt{x^{3} + 1} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3*sqrt(x^3 + 1) - 1/3*log(sqrt(x^3 + 1) + 1) + 1/3*log(sqrt(x^3 + 1) - 1)

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Sympy [A]  time = 1.81773, size = 48, normalized size = 1.71 \[ \frac{2 x^{\frac{3}{2}}}{3 \sqrt{1 + \frac{1}{x^{3}}}} - \frac{2 \operatorname{asinh}{\left (\frac{1}{x^{\frac{3}{2}}} \right )}}{3} + \frac{2}{3 x^{\frac{3}{2}} \sqrt{1 + \frac{1}{x^{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x**3+1)**(1/2)/x,x)

[Out]

2*x**(3/2)/(3*sqrt(1 + x**(-3))) - 2*asinh(x**(-3/2))/3 + 2/(3*x**(3/2)*sqrt(1 +
x**(-3)))

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GIAC/XCAS [A]  time = 0.197238, size = 47, normalized size = 1.68 \[ \frac{2}{3} \, \sqrt{x^{3} + 1} - \frac{1}{3} \,{\rm ln}\left (\sqrt{x^{3} + 1} + 1\right ) + \frac{1}{3} \,{\rm ln}\left ({\left | \sqrt{x^{3} + 1} - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3*sqrt(x^3 + 1) - 1/3*ln(sqrt(x^3 + 1) + 1) + 1/3*ln(abs(sqrt(x^3 + 1) - 1))