Integrand size = 27, antiderivative size = 35 \[ \int \left (\left (1-x^6\right )^{2/3}+\frac {\left (1-x^6\right )^{2/3}}{x^6}\right ) \, dx=-\frac {\left (1-x^6\right )^{2/3}}{5 x^5}+\frac {1}{5} x \left (1-x^6\right )^{2/3} \]
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Result contains higher order function than in optimal. Order 5 vs. order 2 in optimal.
Time = 0.01 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.03, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {251, 371} \[ \int \left (\left (1-x^6\right )^{2/3}+\frac {\left (1-x^6\right )^{2/3}}{x^6}\right ) \, dx=x \operatorname {Hypergeometric2F1}\left (-\frac {2}{3},\frac {1}{6},\frac {7}{6},x^6\right )-\frac {\operatorname {Hypergeometric2F1}\left (-\frac {5}{6},-\frac {2}{3},\frac {1}{6},x^6\right )}{5 x^5} \]
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Rule 251
Rule 371
Rubi steps \begin{align*} \text {integral}& = \int \left (1-x^6\right )^{2/3} \, dx+\int \frac {\left (1-x^6\right )^{2/3}}{x^6} \, dx \\ & = -\frac {\, _2F_1\left (-\frac {5}{6},-\frac {2}{3};\frac {1}{6};x^6\right )}{5 x^5}+x \, _2F_1\left (-\frac {2}{3},\frac {1}{6};\frac {7}{6};x^6\right ) \\ \end{align*}
Time = 0.16 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.51 \[ \int \left (\left (1-x^6\right )^{2/3}+\frac {\left (1-x^6\right )^{2/3}}{x^6}\right ) \, dx=-\frac {\left (1-x^6\right )^{5/3}}{5 x^5} \]
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Time = 1.13 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.57
method | result | size |
trager | \(\frac {\left (x^{6}-1\right ) \left (-x^{6}+1\right )^{\frac {2}{3}}}{5 x^{5}}\) | \(20\) |
risch | \(-\frac {x^{12}-2 x^{6}+1}{5 x^{5} \left (-x^{6}+1\right )^{\frac {1}{3}}}\) | \(25\) |
meijerg | \(x {}_{2}^{}{\moversetsp {}{\mundersetsp {}{F_{1}^{}}}}\left (-\frac {2}{3},\frac {1}{6};\frac {7}{6};x^{6}\right )-\frac {{}_{2}^{}{\moversetsp {}{\mundersetsp {}{F_{1}^{}}}}\left (-\frac {5}{6},-\frac {2}{3};\frac {1}{6};x^{6}\right )}{5 x^{5}}\) | \(27\) |
gosper | \(\frac {\left (-x^{6}+1\right )^{\frac {2}{3}} \left (x^{2}-x +1\right ) \left (x^{2}+x +1\right ) \left (x +1\right ) \left (x -1\right )}{5 x^{5}}\) | \(35\) |
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none
Time = 0.28 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.54 \[ \int \left (\left (1-x^6\right )^{2/3}+\frac {\left (1-x^6\right )^{2/3}}{x^6}\right ) \, dx=\frac {{\left (x^{6} - 1\right )} {\left (-x^{6} + 1\right )}^{\frac {2}{3}}}{5 \, x^{5}} \]
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Result contains complex when optimal does not.
Time = 0.66 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.94 \[ \int \left (\left (1-x^6\right )^{2/3}+\frac {\left (1-x^6\right )^{2/3}}{x^6}\right ) \, dx=\frac {x \Gamma \left (\frac {1}{6}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{3}, \frac {1}{6} \\ \frac {7}{6} \end {matrix}\middle | {x^{6} e^{2 i \pi }} \right )}}{6 \Gamma \left (\frac {7}{6}\right )} + \frac {\Gamma \left (- \frac {5}{6}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {5}{6}, - \frac {2}{3} \\ \frac {1}{6} \end {matrix}\middle | {x^{6} e^{2 i \pi }} \right )}}{6 x^{5} \Gamma \left (\frac {1}{6}\right )} \]
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none
Time = 0.30 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.09 \[ \int \left (\left (1-x^6\right )^{2/3}+\frac {\left (1-x^6\right )^{2/3}}{x^6}\right ) \, dx=\frac {{\left (x^{6} - 1\right )} {\left (x^{2} + x + 1\right )}^{\frac {2}{3}} {\left (-x^{2} + x - 1\right )}^{\frac {2}{3}} {\left (x + 1\right )}^{\frac {2}{3}} {\left (x - 1\right )}^{\frac {2}{3}}}{5 \, x^{5}} \]
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\[ \int \left (\left (1-x^6\right )^{2/3}+\frac {\left (1-x^6\right )^{2/3}}{x^6}\right ) \, dx=\int { {\left (-x^{6} + 1\right )}^{\frac {2}{3}} + \frac {{\left (-x^{6} + 1\right )}^{\frac {2}{3}}}{x^{6}} \,d x } \]
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Time = 19.88 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.40 \[ \int \left (\left (1-x^6\right )^{2/3}+\frac {\left (1-x^6\right )^{2/3}}{x^6}\right ) \, dx=-\frac {{\left (1-x^6\right )}^{5/3}}{5\,x^5} \]
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