Optimal. Leaf size=28 \[ \frac{\sqrt{x^8+1}}{4}-\frac{1}{4} \tanh ^{-1}\left (\sqrt{x^8+1}\right ) \]
[Out]
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Rubi [A] time = 0.0367779, 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 \[ \frac{\sqrt{x^8+1}}{4}-\frac{1}{4} \tanh ^{-1}\left (\sqrt{x^8+1}\right ) \]
Antiderivative was successfully verified.
[In] Int[Sqrt[1 + x^8]/x,x]
[Out]
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Rubi in Sympy [A] time = 4.10398, size = 20, normalized size = 0.71 \[ \frac{\sqrt{x^{8} + 1}}{4} - \frac{\operatorname{atanh}{\left (\sqrt{x^{8} + 1} \right )}}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((x**8+1)**(1/2)/x,x)
[Out]
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Mathematica [A] time = 0.0214165, size = 28, normalized size = 1. \[ \frac{\sqrt{x^8+1}}{4}-\frac{1}{4} \tanh ^{-1}\left (\sqrt{x^8+1}\right ) \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[1 + x^8]/x,x]
[Out]
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Maple [B] time = 0.055, size = 56, normalized size = 2. \[ -{\frac{1}{16\,\sqrt{\pi }} \left ( -2\, \left ( 2-2\,\ln \left ( 2 \right ) +8\,\ln \left ( x \right ) \right ) \sqrt{\pi }+4\,\sqrt{\pi }-4\,\sqrt{\pi }\sqrt{{x}^{8}+1}+4\,\sqrt{\pi }\ln \left ( 1/2+1/2\,\sqrt{{x}^{8}+1} \right ) \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((x^8+1)^(1/2)/x,x)
[Out]
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Maxima [A] time = 1.45816, size = 46, normalized size = 1.64 \[ \frac{1}{4} \, \sqrt{x^{8} + 1} - \frac{1}{8} \, \log \left (\sqrt{x^{8} + 1} + 1\right ) + \frac{1}{8} \, \log \left (\sqrt{x^{8} + 1} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(x^8 + 1)/x,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.225928, size = 46, normalized size = 1.64 \[ \frac{1}{4} \, \sqrt{x^{8} + 1} - \frac{1}{8} \, \log \left (\sqrt{x^{8} + 1} + 1\right ) + \frac{1}{8} \, \log \left (\sqrt{x^{8} + 1} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(x^8 + 1)/x,x, algorithm="fricas")
[Out]
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Sympy [A] time = 4.08762, size = 39, normalized size = 1.39 \[ \frac{x^{4}}{4 \sqrt{1 + \frac{1}{x^{8}}}} - \frac{\operatorname{asinh}{\left (\frac{1}{x^{4}} \right )}}{4} + \frac{1}{4 x^{4} \sqrt{1 + \frac{1}{x^{8}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((x**8+1)**(1/2)/x,x)
[Out]
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GIAC/XCAS [A] time = 0.225921, size = 46, normalized size = 1.64 \[ \frac{1}{4} \, \sqrt{x^{8} + 1} - \frac{1}{8} \,{\rm ln}\left (\sqrt{x^{8} + 1} + 1\right ) + \frac{1}{8} \,{\rm ln}\left (\sqrt{x^{8} + 1} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(x^8 + 1)/x,x, algorithm="giac")
[Out]