Integrand size = 17, antiderivative size = 27 \[ \int \frac {1}{x^4 \sqrt [4]{-x^2+x^4}} \, dx=\frac {2 \left (3+4 x^2\right ) \left (-x^2+x^4\right )^{3/4}}{21 x^5} \]
[Out]
Time = 0.03 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.52, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {2041, 2039} \[ \int \frac {1}{x^4 \sqrt [4]{-x^2+x^4}} \, dx=\frac {2 \left (x^4-x^2\right )^{3/4}}{7 x^5}+\frac {8 \left (x^4-x^2\right )^{3/4}}{21 x^3} \]
[In]
[Out]
Rule 2039
Rule 2041
Rubi steps \begin{align*} \text {integral}& = \frac {2 \left (-x^2+x^4\right )^{3/4}}{7 x^5}+\frac {4}{7} \int \frac {1}{x^2 \sqrt [4]{-x^2+x^4}} \, dx \\ & = \frac {2 \left (-x^2+x^4\right )^{3/4}}{7 x^5}+\frac {8 \left (-x^2+x^4\right )^{3/4}}{21 x^3} \\ \end{align*}
Time = 0.15 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x^4 \sqrt [4]{-x^2+x^4}} \, dx=\frac {2 \left (x^2 \left (-1+x^2\right )\right )^{3/4} \left (3+4 x^2\right )}{21 x^5} \]
[In]
[Out]
Time = 0.82 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.89
method | result | size |
trager | \(\frac {2 \left (4 x^{2}+3\right ) \left (x^{4}-x^{2}\right )^{\frac {3}{4}}}{21 x^{5}}\) | \(24\) |
pseudoelliptic | \(\frac {2 \left (4 x^{2}+3\right ) \left (x^{4}-x^{2}\right )^{\frac {3}{4}}}{21 x^{5}}\) | \(24\) |
risch | \(\frac {-\frac {2}{21} x^{2}-\frac {2}{7}+\frac {8}{21} x^{4}}{x^{3} \left (x^{2} \left (x^{2}-1\right )\right )^{\frac {1}{4}}}\) | \(29\) |
gosper | \(\frac {2 \left (1+x \right ) \left (x -1\right ) \left (4 x^{2}+3\right )}{21 x^{3} \left (x^{4}-x^{2}\right )^{\frac {1}{4}}}\) | \(30\) |
meijerg | \(-\frac {2 {\left (-\operatorname {signum}\left (x^{2}-1\right )\right )}^{\frac {1}{4}} \left (1+\frac {4 x^{2}}{3}\right ) \left (-x^{2}+1\right )^{\frac {3}{4}}}{7 \operatorname {signum}\left (x^{2}-1\right )^{\frac {1}{4}} x^{\frac {7}{2}}}\) | \(40\) |
[In]
[Out]
none
Time = 0.24 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.85 \[ \int \frac {1}{x^4 \sqrt [4]{-x^2+x^4}} \, dx=\frac {2 \, {\left (x^{4} - x^{2}\right )}^{\frac {3}{4}} {\left (4 \, x^{2} + 3\right )}}{21 \, x^{5}} \]
[In]
[Out]
\[ \int \frac {1}{x^4 \sqrt [4]{-x^2+x^4}} \, dx=\int \frac {1}{x^{4} \sqrt [4]{x^{2} \left (x - 1\right ) \left (x + 1\right )}}\, dx \]
[In]
[Out]
none
Time = 0.19 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.07 \[ \int \frac {1}{x^4 \sqrt [4]{-x^2+x^4}} \, dx=\frac {2 \, {\left (4 \, x^{5} - x^{3} - 3 \, x\right )}}{21 \, {\left (x + 1\right )}^{\frac {1}{4}} {\left (x - 1\right )}^{\frac {1}{4}} x^{\frac {9}{2}}} \]
[In]
[Out]
none
Time = 0.27 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.85 \[ \int \frac {1}{x^4 \sqrt [4]{-x^2+x^4}} \, dx=-\frac {2}{7} \, {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {7}{4}} + \frac {2}{3} \, {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {3}{4}} \]
[In]
[Out]
Time = 5.29 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.30 \[ \int \frac {1}{x^4 \sqrt [4]{-x^2+x^4}} \, dx=\frac {8\,x^2\,{\left (x^4-x^2\right )}^{3/4}+6\,{\left (x^4-x^2\right )}^{3/4}}{21\,x^5} \]
[In]
[Out]