Optimal. Leaf size=31 \[ \frac{1}{3} \tanh ^{-1}\left (\sqrt{x^3+1}\right )-\frac{\sqrt{x^3+1}}{3 x^3} \]
[Out]
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Rubi [A] time = 0.0376508, antiderivative size = 31, 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 \[ \frac{1}{3} \tanh ^{-1}\left (\sqrt{x^3+1}\right )-\frac{\sqrt{x^3+1}}{3 x^3} \]
Antiderivative was successfully verified.
[In] Int[1/(x^4*Sqrt[1 + x^3]),x]
[Out]
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Rubi in Sympy [A] time = 4.13414, size = 24, normalized size = 0.77 \[ \frac{\operatorname{atanh}{\left (\sqrt{x^{3} + 1} \right )}}{3} - \frac{\sqrt{x^{3} + 1}}{3 x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x**4/(x**3+1)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0282817, size = 31, normalized size = 1. \[ \frac{1}{3} \tanh ^{-1}\left (\sqrt{x^3+1}\right )-\frac{\sqrt{x^3+1}}{3 x^3} \]
Antiderivative was successfully verified.
[In] Integrate[1/(x^4*Sqrt[1 + x^3]),x]
[Out]
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Maple [A] time = 0.029, size = 24, normalized size = 0.8 \[{\frac{1}{3}{\it Artanh} \left ( \sqrt{{x}^{3}+1} \right ) }-{\frac{1}{3\,{x}^{3}}\sqrt{{x}^{3}+1}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x^4/(x^3+1)^(1/2),x)
[Out]
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Maxima [A] time = 1.44629, size = 50, normalized size = 1.61 \[ -\frac{\sqrt{x^{3} + 1}}{3 \, x^{3}} + \frac{1}{6} \, \log \left (\sqrt{x^{3} + 1} + 1\right ) - \frac{1}{6} \, \log \left (\sqrt{x^{3} + 1} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(x^3 + 1)*x^4),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.228784, size = 59, normalized size = 1.9 \[ \frac{x^{3} \log \left (\sqrt{x^{3} + 1} + 1\right ) - x^{3} \log \left (\sqrt{x^{3} + 1} - 1\right ) - 2 \, \sqrt{x^{3} + 1}}{6 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(x^3 + 1)*x^4),x, algorithm="fricas")
[Out]
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Sympy [A] time = 6.32317, size = 26, normalized size = 0.84 \[ \frac{\operatorname{asinh}{\left (\frac{1}{x^{\frac{3}{2}}} \right )}}{3} - \frac{\sqrt{1 + \frac{1}{x^{3}}}}{3 x^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/x**4/(x**3+1)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.217323, size = 51, normalized size = 1.65 \[ -\frac{\sqrt{x^{3} + 1}}{3 \, x^{3}} + \frac{1}{6} \,{\rm ln}\left (\sqrt{x^{3} + 1} + 1\right ) - \frac{1}{6} \,{\rm ln}\left ({\left | \sqrt{x^{3} + 1} - 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(x^3 + 1)*x^4),x, algorithm="giac")
[Out]