Integrand size = 18, antiderivative size = 16 \[ \int \frac {\left (-2+x^6\right ) \left (1+x^6\right )^{3/4}}{x^8} \, dx=\frac {2 \left (1+x^6\right )^{7/4}}{7 x^7} \]
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Time = 0.00 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {460} \[ \int \frac {\left (-2+x^6\right ) \left (1+x^6\right )^{3/4}}{x^8} \, dx=\frac {2 \left (x^6+1\right )^{7/4}}{7 x^7} \]
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Rule 460
Rubi steps \begin{align*} \text {integral}& = \frac {2 \left (1+x^6\right )^{7/4}}{7 x^7} \\ \end{align*}
Time = 0.27 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \frac {\left (-2+x^6\right ) \left (1+x^6\right )^{3/4}}{x^8} \, dx=\frac {2 \left (1+x^6\right )^{7/4}}{7 x^7} \]
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Time = 0.80 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.81
method | result | size |
trager | \(\frac {2 \left (x^{6}+1\right )^{\frac {7}{4}}}{7 x^{7}}\) | \(13\) |
pseudoelliptic | \(\frac {2 \left (x^{6}+1\right )^{\frac {7}{4}}}{7 x^{7}}\) | \(13\) |
risch | \(\frac {\frac {2}{7} x^{12}+\frac {4}{7} x^{6}+\frac {2}{7}}{x^{7} \left (x^{6}+1\right )^{\frac {1}{4}}}\) | \(23\) |
gosper | \(\frac {2 \left (x^{2}+1\right ) \left (x^{4}-x^{2}+1\right ) \left (x^{6}+1\right )^{\frac {3}{4}}}{7 x^{7}}\) | \(28\) |
meijerg | \(\frac {2 \operatorname {hypergeom}\left (\left [-\frac {7}{6}, -\frac {3}{4}\right ], \left [-\frac {1}{6}\right ], -x^{6}\right )}{7 x^{7}}-\frac {\operatorname {hypergeom}\left (\left [-\frac {3}{4}, -\frac {1}{6}\right ], \left [\frac {5}{6}\right ], -x^{6}\right )}{x}\) | \(34\) |
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none
Time = 0.26 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.75 \[ \int \frac {\left (-2+x^6\right ) \left (1+x^6\right )^{3/4}}{x^8} \, dx=\frac {2 \, {\left (x^{6} + 1\right )}^{\frac {7}{4}}}{7 \, x^{7}} \]
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Result contains complex when optimal does not.
Time = 1.47 (sec) , antiderivative size = 71, normalized size of antiderivative = 4.44 \[ \int \frac {\left (-2+x^6\right ) \left (1+x^6\right )^{3/4}}{x^8} \, dx=\frac {\Gamma \left (- \frac {1}{6}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{4}, - \frac {1}{6} \\ \frac {5}{6} \end {matrix}\middle | {x^{6} e^{i \pi }} \right )}}{6 x \Gamma \left (\frac {5}{6}\right )} - \frac {\Gamma \left (- \frac {7}{6}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {7}{6}, - \frac {3}{4} \\ - \frac {1}{6} \end {matrix}\middle | {x^{6} e^{i \pi }} \right )}}{3 x^{7} \Gamma \left (- \frac {1}{6}\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 29 vs. \(2 (12) = 24\).
Time = 0.32 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.81 \[ \int \frac {\left (-2+x^6\right ) \left (1+x^6\right )^{3/4}}{x^8} \, dx=\frac {2 \, {\left (x^{6} + 1\right )} {\left (x^{4} - x^{2} + 1\right )}^{\frac {3}{4}} {\left (x^{2} + 1\right )}^{\frac {3}{4}}}{7 \, x^{7}} \]
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\[ \int \frac {\left (-2+x^6\right ) \left (1+x^6\right )^{3/4}}{x^8} \, dx=\int { \frac {{\left (x^{6} + 1\right )}^{\frac {3}{4}} {\left (x^{6} - 2\right )}}{x^{8}} \,d x } \]
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Time = 5.41 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.69 \[ \int \frac {\left (-2+x^6\right ) \left (1+x^6\right )^{3/4}}{x^8} \, dx=\frac {2\,{\left (x^6+1\right )}^{3/4}+2\,x^6\,{\left (x^6+1\right )}^{3/4}}{7\,x^7} \]
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