Integrand size = 21, antiderivative size = 19 \[ \int \frac {-1+x^8}{x^4 \sqrt [4]{1+x^4+x^8}} \, dx=\frac {\left (1+x^4+x^8\right )^{3/4}}{3 x^3} \]
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Time = 0.01 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {1604} \[ \int \frac {-1+x^8}{x^4 \sqrt [4]{1+x^4+x^8}} \, dx=\frac {\left (x^8+x^4+1\right )^{3/4}}{3 x^3} \]
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Rule 1604
Rubi steps \begin{align*} \text {integral}& = \frac {\left (1+x^4+x^8\right )^{3/4}}{3 x^3} \\ \end{align*}
Time = 0.34 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00 \[ \int \frac {-1+x^8}{x^4 \sqrt [4]{1+x^4+x^8}} \, dx=\frac {\left (1+x^4+x^8\right )^{3/4}}{3 x^3} \]
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Time = 0.18 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.84
method | result | size |
trager | \(\frac {\left (x^{8}+x^{4}+1\right )^{\frac {3}{4}}}{3 x^{3}}\) | \(16\) |
risch | \(\frac {\left (x^{8}+x^{4}+1\right )^{\frac {3}{4}}}{3 x^{3}}\) | \(16\) |
pseudoelliptic | \(\frac {\left (x^{8}+x^{4}+1\right )^{\frac {3}{4}}}{3 x^{3}}\) | \(16\) |
gosper | \(\frac {\left (x^{2}+x +1\right ) \left (x^{2}-x +1\right ) \left (x^{4}-x^{2}+1\right )}{3 \left (x^{8}+x^{4}+1\right )^{\frac {1}{4}} x^{3}}\) | \(40\) |
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none
Time = 0.24 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.79 \[ \int \frac {-1+x^8}{x^4 \sqrt [4]{1+x^4+x^8}} \, dx=\frac {{\left (x^{8} + x^{4} + 1\right )}^{\frac {3}{4}}}{3 \, x^{3}} \]
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\[ \int \frac {-1+x^8}{x^4 \sqrt [4]{1+x^4+x^8}} \, dx=\int \frac {\left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right ) \left (x^{4} + 1\right )}{x^{4} \sqrt [4]{\left (x^{2} - x + 1\right ) \left (x^{2} + x + 1\right ) \left (x^{4} - x^{2} + 1\right )}}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 43 vs. \(2 (15) = 30\).
Time = 0.23 (sec) , antiderivative size = 43, normalized size of antiderivative = 2.26 \[ \int \frac {-1+x^8}{x^4 \sqrt [4]{1+x^4+x^8}} \, dx=\frac {x^{8} + x^{4} + 1}{3 \, {\left (x^{4} - x^{2} + 1\right )}^{\frac {1}{4}} {\left (x^{2} + x + 1\right )}^{\frac {1}{4}} {\left (x^{2} - x + 1\right )}^{\frac {1}{4}} x^{3}} \]
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\[ \int \frac {-1+x^8}{x^4 \sqrt [4]{1+x^4+x^8}} \, dx=\int { \frac {x^{8} - 1}{{\left (x^{8} + x^{4} + 1\right )}^{\frac {1}{4}} x^{4}} \,d x } \]
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Time = 4.99 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.79 \[ \int \frac {-1+x^8}{x^4 \sqrt [4]{1+x^4+x^8}} \, dx=\frac {{\left (x^8+x^4+1\right )}^{3/4}}{3\,x^3} \]
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