Optimal. Leaf size=47 \[ \frac{3 \sqrt{x^8+1}}{32 x^8}-\frac{3}{32} \tanh ^{-1}\left (\sqrt{x^8+1}\right )-\frac{\sqrt{x^8+1}}{16 x^{16}} \]
[Out]
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Rubi [A] time = 0.0528254, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308 \[ \frac{3 \sqrt{x^8+1}}{32 x^8}-\frac{3}{32} \tanh ^{-1}\left (\sqrt{x^8+1}\right )-\frac{\sqrt{x^8+1}}{16 x^{16}} \]
Antiderivative was successfully verified.
[In] Int[1/(x^17*Sqrt[1 + x^8]),x]
[Out]
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Rubi in Sympy [A] time = 5.06273, size = 41, normalized size = 0.87 \[ - \frac{3 \operatorname{atanh}{\left (\sqrt{x^{8} + 1} \right )}}{32} + \frac{3 \sqrt{x^{8} + 1}}{32 x^{8}} - \frac{\sqrt{x^{8} + 1}}{16 x^{16}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x**17/(x**8+1)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0433811, size = 37, normalized size = 0.79 \[ \frac{1}{32} \left (\frac{\sqrt{x^8+1} \left (3 x^8-2\right )}{x^{16}}-3 \tanh ^{-1}\left (\sqrt{x^8+1}\right )\right ) \]
Antiderivative was successfully verified.
[In] Integrate[1/(x^17*Sqrt[1 + x^8]),x]
[Out]
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Maple [A] time = 0.037, size = 42, normalized size = 0.9 \[{\frac{3\,{x}^{16}+{x}^{8}-2}{32\,{x}^{16}}{\frac{1}{\sqrt{{x}^{8}+1}}}}+{\frac{3}{32}\ln \left ({1 \left ( \sqrt{{x}^{8}+1}-1 \right ){\frac{1}{\sqrt{{x}^{8}}}}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x^17/(x^8+1)^(1/2),x)
[Out]
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Maxima [A] time = 1.43816, size = 86, normalized size = 1.83 \[ -\frac{3 \,{\left (x^{8} + 1\right )}^{\frac{3}{2}} - 5 \, \sqrt{x^{8} + 1}}{32 \,{\left (2 \, x^{8} -{\left (x^{8} + 1\right )}^{2} + 1\right )}} - \frac{3}{64} \, \log \left (\sqrt{x^{8} + 1} + 1\right ) + \frac{3}{64} \, \log \left (\sqrt{x^{8} + 1} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(x^8 + 1)*x^17),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.237902, size = 70, normalized size = 1.49 \[ -\frac{3 \, x^{16} \log \left (\sqrt{x^{8} + 1} + 1\right ) - 3 \, x^{16} \log \left (\sqrt{x^{8} + 1} - 1\right ) - 2 \,{\left (3 \, x^{8} - 2\right )} \sqrt{x^{8} + 1}}{64 \, x^{16}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(x^8 + 1)*x^17),x, algorithm="fricas")
[Out]
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Sympy [A] time = 27.263, size = 60, normalized size = 1.28 \[ - \frac{3 \operatorname{asinh}{\left (\frac{1}{x^{4}} \right )}}{32} + \frac{3}{32 x^{4} \sqrt{1 + \frac{1}{x^{8}}}} + \frac{1}{32 x^{12} \sqrt{1 + \frac{1}{x^{8}}}} - \frac{1}{16 x^{20} \sqrt{1 + \frac{1}{x^{8}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/x**17/(x**8+1)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.228642, size = 66, normalized size = 1.4 \[ \frac{3 \,{\left (x^{8} + 1\right )}^{\frac{3}{2}} - 5 \, \sqrt{x^{8} + 1}}{32 \, x^{16}} - \frac{3}{64} \,{\rm ln}\left (\sqrt{x^{8} + 1} + 1\right ) + \frac{3}{64} \,{\rm ln}\left (\sqrt{x^{8} + 1} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(x^8 + 1)*x^17),x, algorithm="giac")
[Out]