Optimal. Leaf size=31 \[ \frac {1}{8} \tanh ^{-1}\left (\sqrt {x^8+1}\right )-\frac {\sqrt {x^8+1}}{8 x^8} \]
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Rubi [A] time = 0.01, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {266, 51, 63, 207} \[ \frac {1}{8} \tanh ^{-1}\left (\sqrt {x^8+1}\right )-\frac {\sqrt {x^8+1}}{8 x^8} \]
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 207
Rule 266
Rubi steps
\begin {align*} \int \frac {1}{x^9 \sqrt {1+x^8}} \, dx &=\frac {1}{8} \operatorname {Subst}\left (\int \frac {1}{x^2 \sqrt {1+x}} \, dx,x,x^8\right )\\ &=-\frac {\sqrt {1+x^8}}{8 x^8}-\frac {1}{16} \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1+x}} \, dx,x,x^8\right )\\ &=-\frac {\sqrt {1+x^8}}{8 x^8}-\frac {1}{8} \operatorname {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\sqrt {1+x^8}\right )\\ &=-\frac {\sqrt {1+x^8}}{8 x^8}+\frac {1}{8} \tanh ^{-1}\left (\sqrt {1+x^8}\right )\\ \end {align*}
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Mathematica [A] time = 0.02, size = 31, normalized size = 1.00 \[ \frac {1}{8} \tanh ^{-1}\left (\sqrt {x^8+1}\right )-\frac {\sqrt {x^8+1}}{8 x^8} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.72, size = 44, normalized size = 1.42 \[ \frac {x^{8} \log \left (\sqrt {x^{8} + 1} + 1\right ) - x^{8} \log \left (\sqrt {x^{8} + 1} - 1\right ) - 2 \, \sqrt {x^{8} + 1}}{16 \, x^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 37, normalized size = 1.19 \[ -\frac {\sqrt {x^{8} + 1}}{8 \, x^{8}} + \frac {1}{16} \, \log \left (\sqrt {x^{8} + 1} + 1\right ) - \frac {1}{16} \, \log \left (\sqrt {x^{8} + 1} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 32, normalized size = 1.03 \[ -\frac {\ln \left (\frac {\sqrt {x^{8}+1}-1}{\sqrt {x^{8}}}\right )}{8}-\frac {\sqrt {x^{8}+1}}{8 x^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.05, size = 37, normalized size = 1.19 \[ -\frac {\sqrt {x^{8} + 1}}{8 \, x^{8}} + \frac {1}{16} \, \log \left (\sqrt {x^{8} + 1} + 1\right ) - \frac {1}{16} \, \log \left (\sqrt {x^{8} + 1} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.41, size = 23, normalized size = 0.74 \[ \frac {\mathrm {atanh}\left (\sqrt {x^8+1}\right )}{8}-\frac {\sqrt {x^8+1}}{8\,x^8} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.29, size = 22, normalized size = 0.71 \[ \frac {\operatorname {asinh}{\left (\frac {1}{x^{4}} \right )}}{8} - \frac {\sqrt {1 + \frac {1}{x^{8}}}}{8 x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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