Optimal. Leaf size=28 \[ \frac {\sqrt {1+x^6}}{3}-\frac {1}{3} \tanh ^{-1}\left (\sqrt {1+x^6}\right ) \]
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Rubi [A]
time = 0.01, antiderivative size = 28, 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, 52, 65,
213} \begin {gather*} \frac {\sqrt {x^6+1}}{3}-\frac {1}{3} \tanh ^{-1}\left (\sqrt {x^6+1}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 52
Rule 65
Rule 213
Rule 272
Rubi steps
\begin {align*} \int \frac {\sqrt {1+x^6}}{x} \, dx &=\frac {1}{6} \text {Subst}\left (\int \frac {\sqrt {1+x}}{x} \, dx,x,x^6\right )\\ &=\frac {\sqrt {1+x^6}}{3}+\frac {1}{6} \text {Subst}\left (\int \frac {1}{x \sqrt {1+x}} \, dx,x,x^6\right )\\ &=\frac {\sqrt {1+x^6}}{3}+\frac {1}{3} \text {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\sqrt {1+x^6}\right )\\ &=\frac {\sqrt {1+x^6}}{3}-\frac {1}{3} \tanh ^{-1}\left (\sqrt {1+x^6}\right )\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 28, normalized size = 1.00 \begin {gather*} \frac {\sqrt {1+x^6}}{3}-\frac {1}{3} \tanh ^{-1}\left (\sqrt {1+x^6}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.34, size = 27, normalized size = 0.96
method | result | size |
trager | \(\frac {\sqrt {x^{6}+1}}{3}-\frac {\ln \left (\frac {\sqrt {x^{6}+1}+1}{x^{3}}\right )}{3}\) | \(27\) |
meijerg | \(-\frac {4 \sqrt {\pi }-4 \sqrt {\pi }\, \sqrt {x^{6}+1}+4 \sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {x^{6}+1}}{2}\right )-2 \left (2-2 \ln \left (2\right )+6 \ln \left (x \right )\right ) \sqrt {\pi }}{12 \sqrt {\pi }}\) | \(56\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.25, size = 34, normalized size = 1.21 \begin {gather*} \frac {1}{3} \, \sqrt {x^{6} + 1} - \frac {1}{6} \, \log \left (\sqrt {x^{6} + 1} + 1\right ) + \frac {1}{6} \, \log \left (\sqrt {x^{6} + 1} - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 34, normalized size = 1.21 \begin {gather*} \frac {1}{3} \, \sqrt {x^{6} + 1} - \frac {1}{6} \, \log \left (\sqrt {x^{6} + 1} + 1\right ) + \frac {1}{6} \, \log \left (\sqrt {x^{6} + 1} - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.63, size = 39, normalized size = 1.39 \begin {gather*} \frac {x^{3}}{3 \sqrt {1 + \frac {1}{x^{6}}}} - \frac {\operatorname {asinh}{\left (\frac {1}{x^{3}} \right )}}{3} + \frac {1}{3 x^{3} \sqrt {1 + \frac {1}{x^{6}}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.39, size = 34, normalized size = 1.21 \begin {gather*} \frac {1}{3} \, \sqrt {x^{6} + 1} - \frac {1}{6} \, \log \left (\sqrt {x^{6} + 1} + 1\right ) + \frac {1}{6} \, \log \left (\sqrt {x^{6} + 1} - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.27, size = 20, normalized size = 0.71 \begin {gather*} \frac {\sqrt {x^6+1}}{3}-\frac {\mathrm {atanh}\left (\sqrt {x^6+1}\right )}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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