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 = 67, normalized size of antiderivative = 1.40, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {281, 327, 223, 212} \[ \int \frac {x^{20}}{\sqrt {-1+x^6}} \, dx=\frac {5}{48} \text {arctanh}\left (\frac {x^3}{\sqrt {x^6-1}}\right )+\frac {1}{18} \sqrt {x^6-1} x^{15}+\frac {5}{72} \sqrt {x^6-1} x^9+\frac {5}{48} \sqrt {x^6-1} x^3 \]
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Rule 212
Rule 223
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}{48} \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {x^3}{\sqrt {-1+x^6}}\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 {arctanh}\left (\frac {x^3}{\sqrt {-1+x^6}}\right ) \\ \end{align*}
Time = 0.15 (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 = 1.07 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.83
method | result | size |
trager | \(\frac {x^{3} \left (8 x^{12}+10 x^{6}+15\right ) \sqrt {x^{6}-1}}{144}+\frac {5 \ln \left (x^{3}+\sqrt {x^{6}-1}\right )}{48}\) | \(40\) |
pseudoelliptic | \(\frac {\sqrt {x^{6}-1}\, \left (8 x^{15}+10 x^{9}+15 x^{3}\right )}{144}+\frac {5 \ln \left (x^{3}+\sqrt {x^{6}-1}\right )}{48}\) | \(41\) |
risch | \(\frac {x^{3} \left (8 x^{12}+10 x^{6}+15\right ) \sqrt {x^{6}-1}}{144}+\frac {5 \sqrt {-\operatorname {signum}\left (x^{6}-1\right )}\, \arcsin \left (x^{3}\right )}{48 \sqrt {\operatorname {signum}\left (x^{6}-1\right )}}\) | \(50\) |
meijerg | \(\frac {i \sqrt {-\operatorname {signum}\left (x^{6}-1\right )}\, \left (\frac {i \sqrt {\pi }\, x^{3} \left (56 x^{12}+70 x^{6}+105\right ) \sqrt {-x^{6}+1}}{168}-\frac {5 i \sqrt {\pi }\, \arcsin \left (x^{3}\right )}{8}\right )}{6 \sqrt {\pi }\, \sqrt {\operatorname {signum}\left (x^{6}-1\right )}}\) | \(66\) |
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none
Time = 0.24 (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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Result contains complex when optimal does not.
Time = 7.49 (sec) , antiderivative size = 143, normalized size of antiderivative = 2.98 \[ \int \frac {x^{20}}{\sqrt {-1+x^6}} \, dx=\begin {cases} \frac {x^{21}}{18 \sqrt {x^{6} - 1}} + \frac {x^{15}}{72 \sqrt {x^{6} - 1}} + \frac {5 x^{9}}{144 \sqrt {x^{6} - 1}} - \frac {5 x^{3}}{48 \sqrt {x^{6} - 1}} + \frac {5 \operatorname {acosh}{\left (x^{3} \right )}}{48} & \text {for}\: \left |{x^{6}}\right | > 1 \\- \frac {i x^{21}}{18 \sqrt {1 - x^{6}}} - \frac {i x^{15}}{72 \sqrt {1 - x^{6}}} - \frac {5 i x^{9}}{144 \sqrt {1 - x^{6}}} + \frac {5 i x^{3}}{48 \sqrt {1 - x^{6}}} - \frac {5 i \operatorname {asin}{\left (x^{3} \right )}}{48} & \text {otherwise} \end {cases} \]
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Leaf count of result is larger than twice the leaf count of optimal. 109 vs. \(2 (40) = 80\).
Time = 0.19 (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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none
Time = 0.31 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.29 \[ \int \frac {x^{20}}{\sqrt {-1+x^6}} \, dx=\frac {1}{144} \, {\left (2 \, {\left (4 \, x^{6} + 5\right )} x^{6} + 15\right )} \sqrt {x^{6} - 1} x^{3} + \frac {5 \, {\left (\log \left (\sqrt {-\frac {1}{x^{6}} + 1} + 1\right ) - \log \left (-\sqrt {-\frac {1}{x^{6}} + 1} + 1\right )\right )}}{96 \, \mathrm {sgn}\left (x\right )} \]
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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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