Integrand size = 13, antiderivative size = 48 \[ \int \frac {x^{20}}{\sqrt {1+x^6}} \, dx=\frac {1}{144} \sqrt {1+x^6} \left (15 x^3-10 x^9+8 x^{15}\right )-\frac {5}{48} \log \left (x^3+\sqrt {1+x^6}\right ) \]
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Time = 0.02 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.19, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {281, 327, 221} \[ \int \frac {x^{20}}{\sqrt {1+x^6}} \, dx=-\frac {5 \text {arcsinh}\left (x^3\right )}{48}+\frac {1}{18} \sqrt {x^6+1} x^{15}-\frac {5}{72} \sqrt {x^6+1} x^9+\frac {5}{48} \sqrt {x^6+1} x^3 \]
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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^6}{\sqrt {1+x^2}} \, dx,x,x^3\right ) \\ & = \frac {1}{18} x^{15} \sqrt {1+x^6}-\frac {5}{18} \text {Subst}\left (\int \frac {x^4}{\sqrt {1+x^2}} \, dx,x,x^3\right ) \\ & = -\frac {5}{72} x^9 \sqrt {1+x^6}+\frac {1}{18} x^{15} \sqrt {1+x^6}+\frac {5}{24} \text {Subst}\left (\int \frac {x^2}{\sqrt {1+x^2}} \, dx,x,x^3\right ) \\ & = \frac {5}{48} x^3 \sqrt {1+x^6}-\frac {5}{72} x^9 \sqrt {1+x^6}+\frac {1}{18} x^{15} \sqrt {1+x^6}-\frac {5}{48} \text {Subst}\left (\int \frac {1}{\sqrt {1+x^2}} \, dx,x,x^3\right ) \\ & = \frac {5}{48} x^3 \sqrt {1+x^6}-\frac {5}{72} x^9 \sqrt {1+x^6}+\frac {1}{18} x^{15} \sqrt {1+x^6}-\frac {5 \text {arcsinh}\left (x^3\right )}{48} \\ \end{align*}
Time = 0.14 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.98 \[ \int \frac {x^{20}}{\sqrt {1+x^6}} \, dx=\frac {1}{144} x^3 \sqrt {1+x^6} \left (15-10 x^6+8 x^{12}\right )-\frac {5}{48} \log \left (x^3+\sqrt {1+x^6}\right ) \]
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Time = 1.06 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.67
method | result | size |
risch | \(\frac {x^{3} \left (8 x^{12}-10 x^{6}+15\right ) \sqrt {x^{6}+1}}{144}-\frac {5 \,\operatorname {arcsinh}\left (x^{3}\right )}{48}\) | \(32\) |
pseudoelliptic | \(\frac {\sqrt {x^{6}+1}\, \left (8 x^{15}-10 x^{9}+15 x^{3}\right )}{144}-\frac {5 \,\operatorname {arcsinh}\left (x^{3}\right )}{48}\) | \(33\) |
trager | \(\frac {x^{3} \left (8 x^{12}-10 x^{6}+15\right ) \sqrt {x^{6}+1}}{144}-\frac {5 \ln \left (x^{3}+\sqrt {x^{6}+1}\right )}{48}\) | \(40\) |
meijerg | \(\frac {\frac {\sqrt {\pi }\, x^{3} \left (56 x^{12}-70 x^{6}+105\right ) \sqrt {x^{6}+1}}{168}-\frac {5 \sqrt {\pi }\, \operatorname {arcsinh}\left (x^{3}\right )}{8}}{6 \sqrt {\pi }}\) | \(43\) |
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Time = 0.25 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.88 \[ \int \frac {x^{20}}{\sqrt {1+x^6}} \, dx=\frac {1}{144} \, {\left (8 \, x^{15} - 10 \, x^{9} + 15 \, x^{3}\right )} \sqrt {x^{6} + 1} + \frac {5}{48} \, \log \left (-x^{3} + \sqrt {x^{6} + 1}\right ) \]
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Time = 6.77 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.35 \[ \int \frac {x^{20}}{\sqrt {1+x^6}} \, dx=\frac {x^{21}}{18 \sqrt {x^{6} + 1}} - \frac {x^{15}}{72 \sqrt {x^{6} + 1}} + \frac {5 x^{9}}{144 \sqrt {x^{6} + 1}} + \frac {5 x^{3}}{48 \sqrt {x^{6} + 1}} - \frac {5 \operatorname {asinh}{\left (x^{3} \right )}}{48} \]
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Leaf count of result is larger than twice the leaf count of optimal. 109 vs. \(2 (40) = 80\).
Time = 0.17 (sec) , antiderivative size = 109, normalized size of antiderivative = 2.27 \[ \int \frac {x^{20}}{\sqrt {1+x^6}} \, dx=\frac {\frac {33 \, \sqrt {x^{6} + 1}}{x^{3}} - \frac {40 \, {\left (x^{6} + 1\right )}^{\frac {3}{2}}}{x^{9}} + \frac {15 \, {\left (x^{6} + 1\right )}^{\frac {5}{2}}}{x^{15}}}{144 \, {\left (\frac {3 \, {\left (x^{6} + 1\right )}}{x^{6}} - \frac {3 \, {\left (x^{6} + 1\right )}^{2}}{x^{12}} + \frac {{\left (x^{6} + 1\right )}^{3}}{x^{18}} - 1\right )}} - \frac {5}{96} \, \log \left (\frac {\sqrt {x^{6} + 1}}{x^{3}} + 1\right ) + \frac {5}{96} \, \log \left (\frac {\sqrt {x^{6} + 1}}{x^{3}} - 1\right ) \]
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Time = 0.29 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.17 \[ \int \frac {x^{20}}{\sqrt {1+x^6}} \, dx=\frac {1}{144} \, {\left (2 \, {\left (4 \, x^{6} - 5\right )} x^{6} + 15\right )} \sqrt {x^{6} + 1} x^{3} - \frac {5 \, {\left (\log \left (\sqrt {\frac {1}{x^{6}} + 1} + 1\right ) - \log \left (\sqrt {\frac {1}{x^{6}} + 1} - 1\right )\right )}}{96 \, \mathrm {sgn}\left (x\right )} \]
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Timed out. \[ \int \frac {x^{20}}{\sqrt {1+x^6}} \, dx=\int \frac {x^{20}}{\sqrt {x^6+1}} \,d x \]
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