Integrand size = 18, antiderivative size = 16 \[ \int \frac {-2+x^6}{x^4 \sqrt [4]{1+x^6}} \, dx=\frac {2 \left (1+x^6\right )^{3/4}}{3 x^3} \]
[Out]
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} \[ \int \frac {-2+x^6}{x^4 \sqrt [4]{1+x^6}} \, dx=\frac {2 \left (x^6+1\right )^{3/4}}{3 x^3} \]
[In]
[Out]
Rule 460
Rubi steps \begin{align*} \text {integral}& = \frac {2 \left (1+x^6\right )^{3/4}}{3 x^3} \\ \end{align*}
Time = 0.16 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \frac {-2+x^6}{x^4 \sqrt [4]{1+x^6}} \, dx=\frac {2 \left (1+x^6\right )^{3/4}}{3 x^3} \]
[In]
[Out]
Time = 0.81 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.81
method | result | size |
trager | \(\frac {2 \left (x^{6}+1\right )^{\frac {3}{4}}}{3 x^{3}}\) | \(13\) |
risch | \(\frac {2 \left (x^{6}+1\right )^{\frac {3}{4}}}{3 x^{3}}\) | \(13\) |
pseudoelliptic | \(\frac {2 \left (x^{6}+1\right )^{\frac {3}{4}}}{3 x^{3}}\) | \(13\) |
gosper | \(\frac {2 \left (x^{2}+1\right ) \left (x^{4}-x^{2}+1\right )}{3 x^{3} \left (x^{6}+1\right )^{\frac {1}{4}}}\) | \(28\) |
meijerg | \(\frac {2 \operatorname {hypergeom}\left (\left [-\frac {1}{2}, \frac {1}{4}\right ], \left [\frac {1}{2}\right ], -x^{6}\right )}{3 x^{3}}+\frac {x^{3} \operatorname {hypergeom}\left (\left [\frac {1}{4}, \frac {1}{2}\right ], \left [\frac {3}{2}\right ], -x^{6}\right )}{3}\) | \(34\) |
[In]
[Out]
none
Time = 0.24 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.75 \[ \int \frac {-2+x^6}{x^4 \sqrt [4]{1+x^6}} \, dx=\frac {2 \, {\left (x^{6} + 1\right )}^{\frac {3}{4}}}{3 \, x^{3}} \]
[In]
[Out]
Result contains complex when optimal does not.
Time = 1.32 (sec) , antiderivative size = 42, normalized size of antiderivative = 2.62 \[ \int \frac {-2+x^6}{x^4 \sqrt [4]{1+x^6}} \, dx=\frac {x^{3} {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {1}{2} \\ \frac {3}{2} \end {matrix}\middle | {x^{6} e^{i \pi }} \right )}}{3} + \frac {2 {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {1}{4} \\ \frac {1}{2} \end {matrix}\middle | {x^{6} e^{i \pi }} \right )}}{3 x^{3}} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 29 vs. \(2 (12) = 24\).
Time = 0.30 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.81 \[ \int \frac {-2+x^6}{x^4 \sqrt [4]{1+x^6}} \, dx=\frac {2 \, {\left (x^{6} + 1\right )}}{3 \, {\left (x^{4} - x^{2} + 1\right )}^{\frac {1}{4}} {\left (x^{2} + 1\right )}^{\frac {1}{4}} x^{3}} \]
[In]
[Out]
\[ \int \frac {-2+x^6}{x^4 \sqrt [4]{1+x^6}} \, dx=\int { \frac {x^{6} - 2}{{\left (x^{6} + 1\right )}^{\frac {1}{4}} x^{4}} \,d x } \]
[In]
[Out]
Time = 0.04 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.75 \[ \int \frac {-2+x^6}{x^4 \sqrt [4]{1+x^6}} \, dx=\frac {2\,{\left (x^6+1\right )}^{3/4}}{3\,x^3} \]
[In]
[Out]