Integrand size = 13, antiderivative size = 35 \[ \int \frac {x^8}{\sqrt {1+x^6}} \, dx=\frac {1}{6} x^3 \sqrt {1+x^6}-\frac {1}{6} \log \left (x^3+\sqrt {1+x^6}\right ) \]
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Time = 0.01 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.71, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {281, 327, 221} \[ \int \frac {x^8}{\sqrt {1+x^6}} \, dx=\frac {1}{6} x^3 \sqrt {x^6+1}-\frac {\text {arcsinh}\left (x^3\right )}{6} \]
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Rule 221
Rule 281
Rule 327
Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} \text {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} \text {Subst}\left (\int \frac {1}{\sqrt {1+x^2}} \, dx,x,x^3\right ) \\ & = \frac {1}{6} x^3 \sqrt {1+x^6}-\frac {\text {arcsinh}\left (x^3\right )}{6} \\ \end{align*}
Time = 0.11 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.00 \[ \int \frac {x^8}{\sqrt {1+x^6}} \, dx=\frac {1}{6} x^3 \sqrt {1+x^6}-\frac {1}{6} \log \left (x^3+\sqrt {1+x^6}\right ) \]
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Time = 1.11 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.57
method | result | size |
risch | \(\frac {x^{3} \sqrt {x^{6}+1}}{6}-\frac {\operatorname {arcsinh}\left (x^{3}\right )}{6}\) | \(20\) |
pseudoelliptic | \(\frac {x^{3} \sqrt {x^{6}+1}}{6}-\frac {\operatorname {arcsinh}\left (x^{3}\right )}{6}\) | \(20\) |
trager | \(\frac {x^{3} \sqrt {x^{6}+1}}{6}-\frac {\ln \left (x^{3}+\sqrt {x^{6}+1}\right )}{6}\) | \(28\) |
meijerg | \(\frac {\sqrt {\pi }\, x^{3} \sqrt {x^{6}+1}-\sqrt {\pi }\, \operatorname {arcsinh}\left (x^{3}\right )}{6 \sqrt {\pi }}\) | \(30\) |
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none
Time = 0.25 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.83 \[ \int \frac {x^8}{\sqrt {1+x^6}} \, dx=\frac {1}{6} \, \sqrt {x^{6} + 1} x^{3} + \frac {1}{6} \, \log \left (-x^{3} + \sqrt {x^{6} + 1}\right ) \]
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Time = 0.97 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.54 \[ \int \frac {x^8}{\sqrt {1+x^6}} \, dx=\frac {x^{3} \sqrt {x^{6} + 1}}{6} - \frac {\operatorname {asinh}{\left (x^{3} \right )}}{6} \]
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Leaf count of result is larger than twice the leaf count of optimal. 58 vs. \(2 (27) = 54\).
Time = 0.22 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.66 \[ \int \frac {x^8}{\sqrt {1+x^6}} \, dx=\frac {\sqrt {x^{6} + 1}}{6 \, x^{3} {\left (\frac {x^{6} + 1}{x^{6}} - 1\right )}} - \frac {1}{12} \, \log \left (\frac {\sqrt {x^{6} + 1}}{x^{3}} + 1\right ) + \frac {1}{12} \, \log \left (\frac {\sqrt {x^{6} + 1}}{x^{3}} - 1\right ) \]
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Time = 0.30 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.20 \[ \int \frac {x^8}{\sqrt {1+x^6}} \, dx=\frac {1}{6} \, \sqrt {x^{6} + 1} x^{3} - \frac {\log \left (\sqrt {\frac {1}{x^{6}} + 1} + 1\right ) - \log \left (\sqrt {\frac {1}{x^{6}} + 1} - 1\right )}{12 \, \mathrm {sgn}\left (x\right )} \]
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Timed out. \[ \int \frac {x^8}{\sqrt {1+x^6}} \, dx=\int \frac {x^8}{\sqrt {x^6+1}} \,d x \]
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