Integrand size = 13, antiderivative size = 16 \[ \int \frac {\left (1+x^4\right )^{3/4}}{x^8} \, dx=-\frac {\left (1+x^4\right )^{7/4}}{7 x^7} \]
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Time = 0.00 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {270} \[ \int \frac {\left (1+x^4\right )^{3/4}}{x^8} \, dx=-\frac {\left (x^4+1\right )^{7/4}}{7 x^7} \]
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Rule 270
Rubi steps \begin{align*} \text {integral}& = -\frac {\left (1+x^4\right )^{7/4}}{7 x^7} \\ \end{align*}
Time = 0.12 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \frac {\left (1+x^4\right )^{3/4}}{x^8} \, dx=-\frac {\left (1+x^4\right )^{7/4}}{7 x^7} \]
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Time = 0.83 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.81
method | result | size |
gosper | \(-\frac {\left (x^{4}+1\right )^{\frac {7}{4}}}{7 x^{7}}\) | \(13\) |
trager | \(-\frac {\left (x^{4}+1\right )^{\frac {7}{4}}}{7 x^{7}}\) | \(13\) |
meijerg | \(-\frac {\left (x^{4}+1\right )^{\frac {7}{4}}}{7 x^{7}}\) | \(13\) |
pseudoelliptic | \(-\frac {\left (x^{4}+1\right )^{\frac {7}{4}}}{7 x^{7}}\) | \(13\) |
risch | \(-\frac {x^{8}+2 x^{4}+1}{7 x^{7} \left (x^{4}+1\right )^{\frac {1}{4}}}\) | \(23\) |
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Time = 0.24 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.75 \[ \int \frac {\left (1+x^4\right )^{3/4}}{x^8} \, dx=-\frac {{\left (x^{4} + 1\right )}^{\frac {7}{4}}}{7 \, x^{7}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 53 vs. \(2 (14) = 28\).
Time = 0.51 (sec) , antiderivative size = 53, normalized size of antiderivative = 3.31 \[ \int \frac {\left (1+x^4\right )^{3/4}}{x^8} \, dx=\frac {\left (1 + \frac {1}{x^{4}}\right )^{\frac {3}{4}} \Gamma \left (- \frac {7}{4}\right )}{4 \Gamma \left (- \frac {3}{4}\right )} + \frac {\left (1 + \frac {1}{x^{4}}\right )^{\frac {3}{4}} \Gamma \left (- \frac {7}{4}\right )}{4 x^{4} \Gamma \left (- \frac {3}{4}\right )} \]
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Time = 0.20 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.75 \[ \int \frac {\left (1+x^4\right )^{3/4}}{x^8} \, dx=-\frac {{\left (x^{4} + 1\right )}^{\frac {7}{4}}}{7 \, x^{7}} \]
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\[ \int \frac {\left (1+x^4\right )^{3/4}}{x^8} \, dx=\int { \frac {{\left (x^{4} + 1\right )}^{\frac {3}{4}}}{x^{8}} \,d x } \]
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Time = 5.15 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.50 \[ \int \frac {\left (1+x^4\right )^{3/4}}{x^8} \, dx=-\frac {{\left (x^4+1\right )}^{3/4}+x^4\,{\left (x^4+1\right )}^{3/4}}{7\,x^7} \]
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