Integrand size = 15, antiderivative size = 25 \[ \int \frac {1}{x^4 \sqrt [4]{x^2+x^4}} \, dx=\frac {2 \left (-3+4 x^2\right ) \left (x^2+x^4\right )^{3/4}}{21 x^5} \]
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Time = 0.03 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.48, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {2041, 2039} \[ \int \frac {1}{x^4 \sqrt [4]{x^2+x^4}} \, dx=\frac {8 \left (x^4+x^2\right )^{3/4}}{21 x^3}-\frac {2 \left (x^4+x^2\right )^{3/4}}{7 x^5} \]
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Rule 2039
Rule 2041
Rubi steps \begin{align*} \text {integral}& = -\frac {2 \left (x^2+x^4\right )^{3/4}}{7 x^5}-\frac {4}{7} \int \frac {1}{x^2 \sqrt [4]{x^2+x^4}} \, dx \\ & = -\frac {2 \left (x^2+x^4\right )^{3/4}}{7 x^5}+\frac {8 \left (x^2+x^4\right )^{3/4}}{21 x^3} \\ \end{align*}
Time = 0.15 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x^4 \sqrt [4]{x^2+x^4}} \, dx=\frac {2 \left (-3+4 x^2\right ) \left (x^2+x^4\right )^{3/4}}{21 x^5} \]
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Time = 0.79 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.80
method | result | size |
meijerg | \(-\frac {2 \left (1-\frac {4 x^{2}}{3}\right ) \left (x^{2}+1\right )^{\frac {3}{4}}}{7 x^{\frac {7}{2}}}\) | \(20\) |
trager | \(\frac {2 \left (4 x^{2}-3\right ) \left (x^{4}+x^{2}\right )^{\frac {3}{4}}}{21 x^{5}}\) | \(22\) |
pseudoelliptic | \(\frac {2 \left (4 x^{2}-3\right ) \left (x^{2} \left (x^{2}+1\right )\right )^{\frac {3}{4}}}{21 x^{5}}\) | \(24\) |
gosper | \(\frac {2 \left (x^{2}+1\right ) \left (4 x^{2}-3\right )}{21 x^{3} \left (x^{4}+x^{2}\right )^{\frac {1}{4}}}\) | \(27\) |
risch | \(\frac {\frac {2}{21} x^{2}-\frac {2}{7}+\frac {8}{21} x^{4}}{x^{3} \left (x^{2} \left (x^{2}+1\right )\right )^{\frac {1}{4}}}\) | \(27\) |
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Time = 0.26 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.84 \[ \int \frac {1}{x^4 \sqrt [4]{x^2+x^4}} \, dx=\frac {2 \, {\left (x^{4} + x^{2}\right )}^{\frac {3}{4}} {\left (4 \, x^{2} - 3\right )}}{21 \, x^{5}} \]
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\[ \int \frac {1}{x^4 \sqrt [4]{x^2+x^4}} \, dx=\int \frac {1}{x^{4} \sqrt [4]{x^{2} \left (x^{2} + 1\right )}}\, dx \]
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Time = 0.28 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.96 \[ \int \frac {1}{x^4 \sqrt [4]{x^2+x^4}} \, dx=\frac {2 \, {\left (4 \, x^{5} + x^{3} - 3 \, x\right )}}{21 \, {\left (x^{2} + 1\right )}^{\frac {1}{4}} x^{\frac {9}{2}}} \]
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Time = 0.27 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.76 \[ \int \frac {1}{x^4 \sqrt [4]{x^2+x^4}} \, dx=-\frac {2}{7} \, {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {7}{4}} + \frac {2}{3} \, {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {3}{4}} \]
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Time = 5.08 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.24 \[ \int \frac {1}{x^4 \sqrt [4]{x^2+x^4}} \, dx=-\frac {6\,{\left (x^4+x^2\right )}^{3/4}-8\,x^2\,{\left (x^4+x^2\right )}^{3/4}}{21\,x^5} \]
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