Optimal. Leaf size=35 \[ \frac {1}{6} x^3 \sqrt {x^6+1}-\frac {1}{6} \log \left (\sqrt {x^6+1}+x^3\right ) \]
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Rubi [A] time = 0.01, antiderivative size = 25, normalized size of antiderivative = 0.71, number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {275, 321, 215} \begin {gather*} \frac {1}{6} x^3 \sqrt {x^6+1}-\frac {1}{6} \sinh ^{-1}\left (x^3\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 215
Rule 275
Rule 321
Rubi steps
\begin {align*} \int \frac {x^8}{\sqrt {1+x^6}} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {1+x^2}} \, dx,x,x^3\right )\\ &=\frac {1}{6} x^3 \sqrt {1+x^6}-\frac {1}{6} \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+x^2}} \, dx,x,x^3\right )\\ &=\frac {1}{6} x^3 \sqrt {1+x^6}-\frac {1}{6} \sinh ^{-1}\left (x^3\right )\\ \end {align*}
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Mathematica [A] time = 0.01, size = 25, normalized size = 0.71 \begin {gather*} \frac {1}{6} x^3 \sqrt {x^6+1}-\frac {1}{6} \sinh ^{-1}\left (x^3\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.18, size = 35, normalized size = 1.00 \begin {gather*} \frac {1}{6} x^3 \sqrt {x^6+1}-\frac {1}{6} \log \left (\sqrt {x^6+1}+x^3\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.39, size = 29, normalized size = 0.83 \begin {gather*} \frac {1}{6} \, \sqrt {x^{6} + 1} x^{3} + \frac {1}{6} \, \log \left (-x^{3} + \sqrt {x^{6} + 1}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{8}}{\sqrt {x^{6} + 1}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.21, size = 20, normalized size = 0.57 \begin {gather*} \frac {x^{3} \sqrt {x^{6}+1}}{6}-\frac {\arcsinh \left (x^{3}\right )}{6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.34, size = 58, normalized size = 1.66 \begin {gather*} \frac {\sqrt {x^{6} + 1}}{6 \, x^{3} {\left (\frac {x^{6} + 1}{x^{6}} - 1\right )}} - \frac {1}{12} \, \log \left (\frac {\sqrt {x^{6} + 1}}{x^{3}} + 1\right ) + \frac {1}{12} \, \log \left (\frac {\sqrt {x^{6} + 1}}{x^{3}} - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {x^8}{\sqrt {x^6+1}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.52, size = 19, normalized size = 0.54 \begin {gather*} \frac {x^{3} \sqrt {x^{6} + 1}}{6} - \frac {\operatorname {asinh}{\left (x^{3} \right )}}{6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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