Optimal. Leaf size=14 \[ -\frac{1}{4} \tanh ^{-1}\left (\sqrt{x^8+1}\right ) \]
[Out]
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Rubi [A] time = 0.0262751, 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 \[ -\frac{1}{4} \tanh ^{-1}\left (\sqrt{x^8+1}\right ) \]
Antiderivative was successfully verified.
[In] Int[1/(x*Sqrt[1 + x^8]),x]
[Out]
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Rubi in Sympy [A] time = 3.31774, size = 12, normalized size = 0.86 \[ - \frac{\operatorname{atanh}{\left (\sqrt{x^{8} + 1} \right )}}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x/(x**8+1)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0163348, size = 14, normalized size = 1. \[ -\frac{1}{4} \tanh ^{-1}\left (\sqrt{x^8+1}\right ) \]
Antiderivative was successfully verified.
[In] Integrate[1/(x*Sqrt[1 + x^8]),x]
[Out]
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Maple [A] time = 0.009, size = 19, normalized size = 1.4 \[{\frac{1}{4}\ln \left ({1 \left ( \sqrt{{x}^{8}+1}-1 \right ){\frac{1}{\sqrt{{x}^{8}}}}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x/(x^8+1)^(1/2),x)
[Out]
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Maxima [A] time = 1.43722, size = 34, normalized size = 2.43 \[ -\frac{1}{8} \, \log \left (\sqrt{x^{8} + 1} + 1\right ) + \frac{1}{8} \, \log \left (\sqrt{x^{8} + 1} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(x^8 + 1)*x),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.240708, size = 34, normalized size = 2.43 \[ -\frac{1}{8} \, \log \left (\sqrt{x^{8} + 1} + 1\right ) + \frac{1}{8} \, \log \left (\sqrt{x^{8} + 1} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(x^8 + 1)*x),x, algorithm="fricas")
[Out]
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Sympy [A] time = 3.34482, size = 8, normalized size = 0.57 \[ - \frac{\operatorname{asinh}{\left (\frac{1}{x^{4}} \right )}}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/x/(x**8+1)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.229492, size = 34, normalized size = 2.43 \[ -\frac{1}{8} \,{\rm ln}\left (\sqrt{x^{8} + 1} + 1\right ) + \frac{1}{8} \,{\rm ln}\left (\sqrt{x^{8} + 1} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(x^8 + 1)*x),x, algorithm="giac")
[Out]