Optimal. Leaf size=31 \[ -\frac {\sqrt {1+x^8}}{8 x^8}+\frac {1}{8} \tanh ^{-1}\left (\sqrt {1+x^8}\right ) \]
[Out]
________________________________________________________________________________________
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 = {272, 44, 65,
213} \begin {gather*} \frac {1}{8} \tanh ^{-1}\left (\sqrt {x^8+1}\right )-\frac {\sqrt {x^8+1}}{8 x^8} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 44
Rule 65
Rule 213
Rule 272
Rubi steps
\begin {align*} \int \frac {1}{x^9 \sqrt {1+x^8}} \, dx &=\frac {1}{8} \text {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} \text {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} \text {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*}
________________________________________________________________________________________
Mathematica [A]
time = 0.03, size = 31, normalized size = 1.00 \begin {gather*} -\frac {\sqrt {1+x^8}}{8 x^8}+\frac {1}{8} \tanh ^{-1}\left (\sqrt {1+x^8}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.30, size = 30, normalized size = 0.97
method | result | size |
trager | \(-\frac {\sqrt {x^{8}+1}}{8 x^{8}}-\frac {\ln \left (\frac {\sqrt {x^{8}+1}-1}{x^{4}}\right )}{8}\) | \(30\) |
risch | \(-\frac {\sqrt {x^{8}+1}}{8 x^{8}}-\frac {\ln \left (\frac {\sqrt {x^{8}+1}-1}{\sqrt {x^{8}}}\right )}{8}\) | \(32\) |
meijerg | \(\frac {\frac {\sqrt {\pi }\, \left (4 x^{8}+8\right )}{8 x^{8}}-\frac {\sqrt {\pi }\, \sqrt {x^{8}+1}}{x^{8}}+\sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {x^{8}+1}}{2}\right )-\frac {\left (1-2 \ln \left (2\right )+8 \ln \left (x \right )\right ) \sqrt {\pi }}{2}-\frac {\sqrt {\pi }}{x^{8}}}{8 \sqrt {\pi }}\) | \(76\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.28, size = 37, normalized size = 1.19 \begin {gather*} -\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 ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.37, size = 44, normalized size = 1.42 \begin {gather*} \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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 1.05, size = 22, normalized size = 0.71 \begin {gather*} \frac {\operatorname {asinh}{\left (\frac {1}{x^{4}} \right )}}{8} - \frac {\sqrt {1 + \frac {1}{x^{8}}}}{8 x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 2.01, size = 37, normalized size = 1.19 \begin {gather*} -\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 ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 1.41, size = 23, normalized size = 0.74 \begin {gather*} \frac {\mathrm {atanh}\left (\sqrt {x^8+1}\right )}{8}-\frac {\sqrt {x^8+1}}{8\,x^8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________