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 = 0.33 (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 = 0.17 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.84
method | result | size |
gosper | \(\frac {\left (x^{8}+x^{4}-1\right )^{\frac {3}{4}}}{3 x^{3}}\) | \(16\) |
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 | \(\frac {\left (x^{8}+x^{4}-1\right )^{\frac {3}{4}}}{3 x^{3}}\) | \(16\) |
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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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\[ \int \frac {1+x^8}{x^4 \sqrt [4]{-1+x^4+x^8}} \, dx=\int \frac {x^{8} + 1}{x^{4} \sqrt [4]{x^{8} + x^{4} - 1}}\, dx \]
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none
Time = 0.21 (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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\[ \int \frac {1+x^8}{x^4 \sqrt [4]{-1+x^4+x^8}} \, dx=\int { \frac {x^{8} + 1}{{\left (x^{8} + x^{4} - 1\right )}^{\frac {1}{4}} x^{4}} \,d x } \]
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Time = 5.05 (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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