Integrand size = 21, antiderivative size = 19 \[ \int \frac {-2+x^6}{x^4 \sqrt [4]{1+x^4+x^6}} \, dx=\frac {2 \left (1+x^4+x^6\right )^{3/4}}{3 x^3} \]
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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} \[ \int \frac {-2+x^6}{x^4 \sqrt [4]{1+x^4+x^6}} \, dx=\frac {2 \left (x^6+x^4+1\right )^{3/4}}{3 x^3} \]
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Rule 1604
Rubi steps \begin{align*} \text {integral}& = \frac {2 \left (1+x^4+x^6\right )^{3/4}}{3 x^3} \\ \end{align*}
Time = 0.42 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00 \[ \int \frac {-2+x^6}{x^4 \sqrt [4]{1+x^4+x^6}} \, dx=\frac {2 \left (1+x^4+x^6\right )^{3/4}}{3 x^3} \]
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Time = 0.14 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.84
method | result | size |
gosper | \(\frac {2 \left (x^{6}+x^{4}+1\right )^{\frac {3}{4}}}{3 x^{3}}\) | \(16\) |
trager | \(\frac {2 \left (x^{6}+x^{4}+1\right )^{\frac {3}{4}}}{3 x^{3}}\) | \(16\) |
risch | \(\frac {2 \left (x^{6}+x^{4}+1\right )^{\frac {3}{4}}}{3 x^{3}}\) | \(16\) |
pseudoelliptic | \(\frac {2 \left (x^{6}+x^{4}+1\right )^{\frac {3}{4}}}{3 x^{3}}\) | \(16\) |
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none
Time = 0.24 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.79 \[ \int \frac {-2+x^6}{x^4 \sqrt [4]{1+x^4+x^6}} \, dx=\frac {2 \, {\left (x^{6} + x^{4} + 1\right )}^{\frac {3}{4}}}{3 \, x^{3}} \]
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\[ \int \frac {-2+x^6}{x^4 \sqrt [4]{1+x^4+x^6}} \, dx=\int \frac {x^{6} - 2}{x^{4} \sqrt [4]{x^{6} + x^{4} + 1}}\, dx \]
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none
Time = 0.22 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.79 \[ \int \frac {-2+x^6}{x^4 \sqrt [4]{1+x^4+x^6}} \, dx=\frac {2 \, {\left (x^{6} + x^{4} + 1\right )}^{\frac {3}{4}}}{3 \, x^{3}} \]
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\[ \int \frac {-2+x^6}{x^4 \sqrt [4]{1+x^4+x^6}} \, dx=\int { \frac {x^{6} - 2}{{\left (x^{6} + x^{4} + 1\right )}^{\frac {1}{4}} x^{4}} \,d x } \]
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Time = 0.07 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.79 \[ \int \frac {-2+x^6}{x^4 \sqrt [4]{1+x^4+x^6}} \, dx=\frac {2\,{\left (x^6+x^4+1\right )}^{3/4}}{3\,x^3} \]
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