Optimal. Leaf size=31 \[ -\frac {\sqrt {1+x^4}}{4 x^4}+\frac {1}{4} \tanh ^{-1}\left (\sqrt {1+x^4}\right ) \]
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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 = {272, 44, 65,
213} \begin {gather*} \frac {1}{4} \tanh ^{-1}\left (\sqrt {x^4+1}\right )-\frac {\sqrt {x^4+1}}{4 x^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 44
Rule 65
Rule 213
Rule 272
Rubi steps
\begin {align*} \int \frac {1}{x^5 \sqrt {1+x^4}} \, dx &=\frac {1}{4} \text {Subst}\left (\int \frac {1}{x^2 \sqrt {1+x}} \, dx,x,x^4\right )\\ &=-\frac {\sqrt {1+x^4}}{4 x^4}-\frac {1}{8} \text {Subst}\left (\int \frac {1}{x \sqrt {1+x}} \, dx,x,x^4\right )\\ &=-\frac {\sqrt {1+x^4}}{4 x^4}-\frac {1}{4} \text {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\sqrt {1+x^4}\right )\\ &=-\frac {\sqrt {1+x^4}}{4 x^4}+\frac {1}{4} \tanh ^{-1}\left (\sqrt {1+x^4}\right )\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 31, normalized size = 1.00 \begin {gather*} -\frac {\sqrt {1+x^4}}{4 x^4}+\frac {1}{4} \tanh ^{-1}\left (\sqrt {1+x^4}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.19, size = 24, normalized size = 0.77
method | result | size |
default | \(-\frac {\sqrt {x^{4}+1}}{4 x^{4}}+\frac {\arctanh \left (\frac {1}{\sqrt {x^{4}+1}}\right )}{4}\) | \(24\) |
risch | \(-\frac {\sqrt {x^{4}+1}}{4 x^{4}}+\frac {\arctanh \left (\frac {1}{\sqrt {x^{4}+1}}\right )}{4}\) | \(24\) |
elliptic | \(-\frac {\sqrt {x^{4}+1}}{4 x^{4}}+\frac {\arctanh \left (\frac {1}{\sqrt {x^{4}+1}}\right )}{4}\) | \(24\) |
trager | \(-\frac {\sqrt {x^{4}+1}}{4 x^{4}}+\frac {\ln \left (\frac {1+\sqrt {x^{4}+1}}{x^{2}}\right )}{4}\) | \(30\) |
meijerg | \(\frac {\frac {\sqrt {\pi }\, \left (4 x^{4}+8\right )}{8 x^{4}}-\frac {\sqrt {\pi }\, \sqrt {x^{4}+1}}{x^{4}}+\sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {x^{4}+1}}{2}\right )-\frac {\left (1-2 \ln \left (2\right )+4 \ln \left (x \right )\right ) \sqrt {\pi }}{2}-\frac {\sqrt {\pi }}{x^{4}}}{4 \sqrt {\pi }}\) | \(76\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 37, normalized size = 1.19 \begin {gather*} -\frac {\sqrt {x^{4} + 1}}{4 \, x^{4}} + \frac {1}{8} \, \log \left (\sqrt {x^{4} + 1} + 1\right ) - \frac {1}{8} \, \log \left (\sqrt {x^{4} + 1} - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 44, normalized size = 1.42 \begin {gather*} \frac {x^{4} \log \left (\sqrt {x^{4} + 1} + 1\right ) - x^{4} \log \left (\sqrt {x^{4} + 1} - 1\right ) - 2 \, \sqrt {x^{4} + 1}}{8 \, x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.91, size = 22, normalized size = 0.71 \begin {gather*} \frac {\operatorname {asinh}{\left (\frac {1}{x^{2}} \right )}}{4} - \frac {\sqrt {1 + \frac {1}{x^{4}}}}{4 x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.30, size = 37, normalized size = 1.19 \begin {gather*} -\frac {\sqrt {x^{4} + 1}}{4 \, x^{4}} + \frac {1}{8} \, \log \left (\sqrt {x^{4} + 1} + 1\right ) - \frac {1}{8} \, \log \left (\sqrt {x^{4} + 1} - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.21, size = 23, normalized size = 0.74 \begin {gather*} \frac {\mathrm {atanh}\left (\sqrt {x^4+1}\right )}{4}-\frac {\sqrt {x^4+1}}{4\,x^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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