Optimal. Leaf size=47 \[ \frac{3 \sqrt{x^8+1}}{32 x^8}-\frac{\sqrt{x^8+1}}{16 x^{16}}-\frac{3}{32} \tanh ^{-1}\left (\sqrt{x^8+1}\right ) \]
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Rubi [A] time = 0.0163496, 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, Rules used = {266, 51, 63, 207} \[ \frac{3 \sqrt{x^8+1}}{32 x^8}-\frac{\sqrt{x^8+1}}{16 x^{16}}-\frac{3}{32} \tanh ^{-1}\left (\sqrt{x^8+1}\right ) \]
Antiderivative was successfully verified.
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Rule 266
Rule 51
Rule 63
Rule 207
Rubi steps
\begin{align*} \int \frac{1}{x^{17} \sqrt{1+x^8}} \, dx &=\frac{1}{8} \operatorname{Subst}\left (\int \frac{1}{x^3 \sqrt{1+x}} \, dx,x,x^8\right )\\ &=-\frac{\sqrt{1+x^8}}{16 x^{16}}-\frac{3}{32} \operatorname{Subst}\left (\int \frac{1}{x^2 \sqrt{1+x}} \, dx,x,x^8\right )\\ &=-\frac{\sqrt{1+x^8}}{16 x^{16}}+\frac{3 \sqrt{1+x^8}}{32 x^8}+\frac{3}{64} \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1+x}} \, dx,x,x^8\right )\\ &=-\frac{\sqrt{1+x^8}}{16 x^{16}}+\frac{3 \sqrt{1+x^8}}{32 x^8}+\frac{3}{32} \operatorname{Subst}\left (\int \frac{1}{-1+x^2} \, dx,x,\sqrt{1+x^8}\right )\\ &=-\frac{\sqrt{1+x^8}}{16 x^{16}}+\frac{3 \sqrt{1+x^8}}{32 x^8}-\frac{3}{32} \tanh ^{-1}\left (\sqrt{1+x^8}\right )\\ \end{align*}
Mathematica [C] time = 0.0046953, size = 26, normalized size = 0.55 \[ -\frac{1}{4} \sqrt{x^8+1} \, _2F_1\left (\frac{1}{2},3;\frac{3}{2};x^8+1\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.02, size = 42, normalized size = 0.9 \begin{align*}{\frac{3\,{x}^{16}+{x}^{8}-2}{32\,{x}^{16}}{\frac{1}{\sqrt{{x}^{8}+1}}}}+{\frac{3}{32}\ln \left ({ \left ( \sqrt{{x}^{8}+1}-1 \right ){\frac{1}{\sqrt{{x}^{8}}}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.964054, size = 86, normalized size = 1.83 \begin{align*} -\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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.32907, size = 143, normalized size = 3.04 \begin{align*} -\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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 6.25812, size = 60, normalized size = 1.28 \begin{align*} - \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}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.19435, size = 66, normalized size = 1.4 \begin{align*} \frac{3 \,{\left (x^{8} + 1\right )}^{\frac{3}{2}} - 5 \, \sqrt{x^{8} + 1}}{32 \, x^{16}} - \frac{3}{64} \, \log \left (\sqrt{x^{8} + 1} + 1\right ) + \frac{3}{64} \, \log \left (\sqrt{x^{8} + 1} - 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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