Optimal. Leaf size=28 \[ \frac {\sqrt {x^6+1}}{3}+\frac {1}{3} \tanh ^{-1}\left (\sqrt {x^6+1}\right ) \]
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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 = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {446, 80, 63, 207} \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 63
Rule 80
Rule 207
Rule 446
Rubi steps
\begin {align*} \int \frac {-1+x^6}{x \sqrt {1+x^6}} \, dx &=\frac {1}{6} \operatorname {Subst}\left (\int \frac {-1+x}{x \sqrt {1+x}} \, dx,x,x^6\right )\\ &=\frac {\sqrt {1+x^6}}{3}-\frac {1}{6} \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1+x}} \, dx,x,x^6\right )\\ &=\frac {\sqrt {1+x^6}}{3}-\frac {1}{3} \operatorname {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 = 0.01, size = 24, normalized size = 0.86 \begin {gather*} \frac {1}{3} \left (\sqrt {x^6+1}+\tanh ^{-1}\left (\sqrt {x^6+1}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.04, 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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fricas [A] time = 0.48, 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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giac [A] time = 0.29, 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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maple [A] time = 0.22, size = 27, normalized size = 0.96
method | result | size |
trager | \(\frac {\sqrt {x^{6}+1}}{3}+\frac {\ln \left (\frac {\sqrt {x^{6}+1}+1}{x^{3}}\right )}{3}\) | \(27\) |
default | \(\frac {\sqrt {x^{6}+1}}{3}-\frac {\ln \left (\frac {\sqrt {x^{6}+1}-1}{\sqrt {x^{6}}}\right )}{3}\) | \(29\) |
meijerg | \(\frac {-2 \sqrt {\pi }+2 \sqrt {\pi }\, \sqrt {x^{6}+1}}{6 \sqrt {\pi }}-\frac {\left (-2 \ln \relax (2)+6 \ln \relax (x )\right ) \sqrt {\pi }-2 \ln \left (\frac {1}{2}+\frac {\sqrt {x^{6}+1}}{2}\right ) \sqrt {\pi }}{6 \sqrt {\pi }}\) | \(61\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.55, 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 = 0.30, size = 20, normalized size = 0.71 \begin {gather*} \frac {\mathrm {atanh}\left (\sqrt {x^6+1}\right )}{3}+\frac {\sqrt {x^6+1}}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 28.44, size = 39, normalized size = 1.39 \begin {gather*} \frac {\sqrt {x^{6} + 1}}{3} - \frac {\log {\left (-1 + \frac {1}{\sqrt {x^{6} + 1}} \right )}}{6} + \frac {\log {\left (1 + \frac {1}{\sqrt {x^{6} + 1}} \right )}}{6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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