Integrand size = 15, antiderivative size = 58 \[ \int x^8 \sqrt {-1+4 x^6} \, dx=-\frac {1}{96} x^3 \sqrt {-1+4 x^6}+\frac {1}{12} x^9 \sqrt {-1+4 x^6}-\frac {1}{192} \text {arctanh}\left (\frac {2 x^3}{\sqrt {-1+4 x^6}}\right ) \]
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Time = 0.02 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {281, 285, 327, 223, 212} \[ \int x^8 \sqrt {-1+4 x^6} \, dx=-\frac {1}{192} \text {arctanh}\left (\frac {2 x^3}{\sqrt {4 x^6-1}}\right )+\frac {1}{12} \sqrt {4 x^6-1} x^9-\frac {1}{96} \sqrt {4 x^6-1} x^3 \]
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Rule 212
Rule 223
Rule 281
Rule 285
Rule 327
Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} \text {Subst}\left (\int x^2 \sqrt {-1+4 x^2} \, dx,x,x^3\right ) \\ & = \frac {1}{12} x^9 \sqrt {-1+4 x^6}-\frac {1}{12} \text {Subst}\left (\int \frac {x^2}{\sqrt {-1+4 x^2}} \, dx,x,x^3\right ) \\ & = -\frac {1}{96} x^3 \sqrt {-1+4 x^6}+\frac {1}{12} x^9 \sqrt {-1+4 x^6}-\frac {1}{96} \text {Subst}\left (\int \frac {1}{\sqrt {-1+4 x^2}} \, dx,x,x^3\right ) \\ & = -\frac {1}{96} x^3 \sqrt {-1+4 x^6}+\frac {1}{12} x^9 \sqrt {-1+4 x^6}-\frac {1}{96} \text {Subst}\left (\int \frac {1}{1-4 x^2} \, dx,x,\frac {x^3}{\sqrt {-1+4 x^6}}\right ) \\ & = -\frac {1}{96} x^3 \sqrt {-1+4 x^6}+\frac {1}{12} x^9 \sqrt {-1+4 x^6}-\frac {1}{192} \tanh ^{-1}\left (\frac {2 x^3}{\sqrt {-1+4 x^6}}\right ) \\ \end{align*}
Time = 0.30 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.83 \[ \int x^8 \sqrt {-1+4 x^6} \, dx=\frac {1}{96} x^3 \sqrt {-1+4 x^6} \left (-1+8 x^6\right )-\frac {1}{192} \log \left (2 x^3+\sqrt {-1+4 x^6}\right ) \]
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Time = 4.84 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.71
method | result | size |
trager | \(\frac {x^{3} \left (8 x^{6}-1\right ) \sqrt {4 x^{6}-1}}{96}-\frac {\ln \left (2 x^{3}+\sqrt {4 x^{6}-1}\right )}{192}\) | \(41\) |
pseudoelliptic | \(-\frac {\ln \left (2 x^{3}+\sqrt {4 x^{6}-1}\right )}{192}+\frac {\left (8 x^{9}-x^{3}\right ) \sqrt {4 x^{6}-1}}{96}\) | \(42\) |
risch | \(\frac {x^{3} \left (8 x^{6}-1\right ) \sqrt {4 x^{6}-1}}{96}-\frac {\sqrt {-\operatorname {signum}\left (4 x^{6}-1\right )}\, \arcsin \left (2 x^{3}\right )}{192 \sqrt {\operatorname {signum}\left (4 x^{6}-1\right )}}\) | \(53\) |
meijerg | \(-\frac {i \sqrt {\operatorname {signum}\left (4 x^{6}-1\right )}\, \left (-\frac {i \sqrt {\pi }\, x^{3} \left (-24 x^{6}+3\right ) \sqrt {-4 x^{6}+1}}{3}+\frac {i \sqrt {\pi }\, \arcsin \left (2 x^{3}\right )}{2}\right )}{96 \sqrt {\pi }\, \sqrt {-\operatorname {signum}\left (4 x^{6}-1\right )}}\) | \(67\) |
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Time = 0.28 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.71 \[ \int x^8 \sqrt {-1+4 x^6} \, dx=\frac {1}{96} \, {\left (8 \, x^{9} - x^{3}\right )} \sqrt {4 \, x^{6} - 1} + \frac {1}{192} \, \log \left (-2 \, x^{3} + \sqrt {4 \, x^{6} - 1}\right ) \]
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Result contains complex when optimal does not.
Time = 1.91 (sec) , antiderivative size = 119, normalized size of antiderivative = 2.05 \[ \int x^8 \sqrt {-1+4 x^6} \, dx=\begin {cases} \frac {x^{15}}{3 \sqrt {4 x^{6} - 1}} - \frac {x^{9}}{8 \sqrt {4 x^{6} - 1}} + \frac {x^{3}}{96 \sqrt {4 x^{6} - 1}} - \frac {\operatorname {acosh}{\left (2 x^{3} \right )}}{192} & \text {for}\: \left |{x^{6}}\right | > \frac {1}{4} \\- \frac {i x^{15}}{3 \sqrt {1 - 4 x^{6}}} + \frac {i x^{9}}{8 \sqrt {1 - 4 x^{6}}} - \frac {i x^{3}}{96 \sqrt {1 - 4 x^{6}}} + \frac {i \operatorname {asin}{\left (2 x^{3} \right )}}{192} & \text {otherwise} \end {cases} \]
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Leaf count of result is larger than twice the leaf count of optimal. 97 vs. \(2 (46) = 92\).
Time = 0.19 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.67 \[ \int x^8 \sqrt {-1+4 x^6} \, dx=-\frac {\frac {4 \, \sqrt {4 \, x^{6} - 1}}{x^{3}} + \frac {{\left (4 \, x^{6} - 1\right )}^{\frac {3}{2}}}{x^{9}}}{96 \, {\left (\frac {8 \, {\left (4 \, x^{6} - 1\right )}}{x^{6}} - \frac {{\left (4 \, x^{6} - 1\right )}^{2}}{x^{12}} - 16\right )}} - \frac {1}{384} \, \log \left (\frac {\sqrt {4 \, x^{6} - 1}}{x^{3}} + 2\right ) + \frac {1}{384} \, \log \left (\frac {\sqrt {4 \, x^{6} - 1}}{x^{3}} - 2\right ) \]
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Time = 0.31 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.98 \[ \int x^8 \sqrt {-1+4 x^6} \, dx=\frac {1}{96} \, {\left (8 \, x^{6} - 1\right )} \sqrt {4 \, x^{6} - 1} x^{3} - \frac {\log \left (\sqrt {-\frac {1}{x^{6}} + 4} + 2\right ) - \log \left (-\sqrt {-\frac {1}{x^{6}} + 4} + 2\right )}{384 \, \mathrm {sgn}\left (x\right )} \]
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Timed out. \[ \int x^8 \sqrt {-1+4 x^6} \, dx=\int x^8\,\sqrt {4\,x^6-1} \,d x \]
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