Integrand size = 23, antiderivative size = 29 \[ \int \frac {\left (-1+x^4\right ) \left (1+x^4\right )}{\left (1+x^2+x^4\right )^{5/2}} \, dx=-\frac {x \left (3+2 x^2+3 x^4\right )}{3 \left (1+x^2+x^4\right )^{3/2}} \]
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Time = 0.03 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.21, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {1692, 1602} \[ \int \frac {\left (-1+x^4\right ) \left (1+x^4\right )}{\left (1+x^2+x^4\right )^{5/2}} \, dx=\frac {x^3}{3 \left (x^4+x^2+1\right )^{3/2}}-\frac {x}{\sqrt {x^4+x^2+1}} \]
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Rule 1602
Rule 1692
Rubi steps \begin{align*} \text {integral}& = \frac {x^3}{3 \left (1+x^2+x^4\right )^{3/2}}+\frac {1}{9} \int \frac {-9+9 x^4}{\left (1+x^2+x^4\right )^{3/2}} \, dx \\ & = \frac {x^3}{3 \left (1+x^2+x^4\right )^{3/2}}-\frac {x}{\sqrt {1+x^2+x^4}} \\ \end{align*}
Time = 0.31 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00 \[ \int \frac {\left (-1+x^4\right ) \left (1+x^4\right )}{\left (1+x^2+x^4\right )^{5/2}} \, dx=-\frac {x \left (3+2 x^2+3 x^4\right )}{3 \left (1+x^2+x^4\right )^{3/2}} \]
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Time = 0.55 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.83
method | result | size |
pseudoelliptic | \(-\frac {\left (x^{4}+\frac {2}{3} x^{2}+1\right ) x}{\left (x^{4}+x^{2}+1\right )^{\frac {3}{2}}}\) | \(24\) |
trager | \(-\frac {x \left (3 x^{4}+2 x^{2}+3\right )}{3 \left (x^{4}+x^{2}+1\right )^{\frac {3}{2}}}\) | \(26\) |
risch | \(-\frac {x \left (3 x^{4}+2 x^{2}+3\right )}{3 \left (x^{4}+x^{2}+1\right )^{\frac {3}{2}}}\) | \(26\) |
gosper | \(-\frac {\left (3 x^{4}+2 x^{2}+3\right ) x \left (x^{2}+x +1\right ) \left (x^{2}-x +1\right )}{3 \left (x^{4}+x^{2}+1\right )^{\frac {5}{2}}}\) | \(40\) |
elliptic | \(\frac {\left (-\frac {\sqrt {2}\, x}{\sqrt {x^{4}+x^{2}+1}}+\frac {\sqrt {2}\, x^{3}}{3 \left (x^{4}+x^{2}+1\right )^{\frac {3}{2}}}\right ) \sqrt {2}}{2}\) | \(41\) |
default | \(-\frac {\frac {1}{9} x -\frac {1}{9} x^{3}}{\left (x^{4}+x^{2}+1\right )^{\frac {3}{2}}}+\frac {\frac {14}{27} x^{3}-\frac {5}{27} x}{\sqrt {x^{4}+x^{2}+1}}+\frac {\frac {2}{9} x^{3}+\frac {1}{9} x}{\left (x^{4}+x^{2}+1\right )^{\frac {3}{2}}}-\frac {2 \left (\frac {7}{27} x^{3}+\frac {11}{27} x \right )}{\sqrt {x^{4}+x^{2}+1}}\) | \(85\) |
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Time = 0.26 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.66 \[ \int \frac {\left (-1+x^4\right ) \left (1+x^4\right )}{\left (1+x^2+x^4\right )^{5/2}} \, dx=-\frac {{\left (3 \, x^{5} + 2 \, x^{3} + 3 \, x\right )} \sqrt {x^{4} + x^{2} + 1}}{3 \, {\left (x^{8} + 2 \, x^{6} + 3 \, x^{4} + 2 \, x^{2} + 1\right )}} \]
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\[ \int \frac {\left (-1+x^4\right ) \left (1+x^4\right )}{\left (1+x^2+x^4\right )^{5/2}} \, dx=\int \frac {\left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right ) \left (x^{4} + 1\right )}{\left (\left (x^{2} - x + 1\right ) \left (x^{2} + x + 1\right )\right )^{\frac {5}{2}}}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 56 vs. \(2 (25) = 50\).
Time = 0.25 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.93 \[ \int \frac {\left (-1+x^4\right ) \left (1+x^4\right )}{\left (1+x^2+x^4\right )^{5/2}} \, dx=-\frac {{\left (3 \, x^{5} + 2 \, x^{3} + 3 \, x\right )} \sqrt {x^{2} + x + 1} \sqrt {x^{2} - x + 1}}{3 \, {\left (x^{8} + 2 \, x^{6} + 3 \, x^{4} + 2 \, x^{2} + 1\right )}} \]
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Time = 0.27 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.90 \[ \int \frac {\left (-1+x^4\right ) \left (1+x^4\right )}{\left (1+x^2+x^4\right )^{5/2}} \, dx=-\frac {{\left ({\left (3 \, x^{2} + 2\right )} x^{2} + 3\right )} x}{3 \, {\left (x^{4} + x^{2} + 1\right )}^{\frac {3}{2}}} \]
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Time = 0.14 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.86 \[ \int \frac {\left (-1+x^4\right ) \left (1+x^4\right )}{\left (1+x^2+x^4\right )^{5/2}} \, dx=-\frac {x\,\left (3\,x^4+2\,x^2+3\right )}{3\,{\left (x^4+x^2+1\right )}^{3/2}} \]
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