Integrand size = 15, antiderivative size = 53 \[ \int \frac {1}{x^6 \sqrt {1-x^4}} \, dx=-\frac {\sqrt {1-x^4}}{5 x^5}-\frac {3 \sqrt {1-x^4}}{5 x}-\frac {3}{5} E(\arcsin (x)|-1)+\frac {3}{5} \operatorname {EllipticF}(\arcsin (x),-1) \]
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Time = 0.01 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {331, 313, 227, 1195, 435} \[ \int \frac {1}{x^6 \sqrt {1-x^4}} \, dx=\frac {3}{5} \operatorname {EllipticF}(\arcsin (x),-1)-\frac {3}{5} E(\arcsin (x)|-1)-\frac {3 \sqrt {1-x^4}}{5 x}-\frac {\sqrt {1-x^4}}{5 x^5} \]
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Rule 227
Rule 313
Rule 331
Rule 435
Rule 1195
Rubi steps \begin{align*} \text {integral}& = -\frac {\sqrt {1-x^4}}{5 x^5}+\frac {3}{5} \int \frac {1}{x^2 \sqrt {1-x^4}} \, dx \\ & = -\frac {\sqrt {1-x^4}}{5 x^5}-\frac {3 \sqrt {1-x^4}}{5 x}-\frac {3}{5} \int \frac {x^2}{\sqrt {1-x^4}} \, dx \\ & = -\frac {\sqrt {1-x^4}}{5 x^5}-\frac {3 \sqrt {1-x^4}}{5 x}+\frac {3}{5} \int \frac {1}{\sqrt {1-x^4}} \, dx-\frac {3}{5} \int \frac {1+x^2}{\sqrt {1-x^4}} \, dx \\ & = -\frac {\sqrt {1-x^4}}{5 x^5}-\frac {3 \sqrt {1-x^4}}{5 x}+\frac {3}{5} F\left (\left .\sin ^{-1}(x)\right |-1\right )-\frac {3}{5} \int \frac {\sqrt {1+x^2}}{\sqrt {1-x^2}} \, dx \\ & = -\frac {\sqrt {1-x^4}}{5 x^5}-\frac {3 \sqrt {1-x^4}}{5 x}-\frac {3}{5} E\left (\left .\sin ^{-1}(x)\right |-1\right )+\frac {3}{5} F\left (\left .\sin ^{-1}(x)\right |-1\right ) \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.01 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.38 \[ \int \frac {1}{x^6 \sqrt {1-x^4}} \, dx=-\frac {\operatorname {Hypergeometric2F1}\left (-\frac {5}{4},\frac {1}{2},-\frac {1}{4},x^4\right )}{5 x^5} \]
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Result contains higher order function than in optimal. Order 5 vs. order 4.
Time = 4.47 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.28
method | result | size |
meijerg | \(-\frac {{}_{2}^{}{\moversetsp {}{\mundersetsp {}{F_{1}^{}}}}\left (-\frac {5}{4},\frac {1}{2};-\frac {1}{4};x^{4}\right )}{5 x^{5}}\) | \(15\) |
risch | \(\frac {3 x^{8}-2 x^{4}-1}{5 x^{5} \sqrt {-x^{4}+1}}+\frac {3 \sqrt {-x^{2}+1}\, \sqrt {x^{2}+1}\, \left (F\left (x , i\right )-E\left (x , i\right )\right )}{5 \sqrt {-x^{4}+1}}\) | \(66\) |
default | \(-\frac {\sqrt {-x^{4}+1}}{5 x^{5}}-\frac {3 \sqrt {-x^{4}+1}}{5 x}+\frac {3 \sqrt {-x^{2}+1}\, \sqrt {x^{2}+1}\, \left (F\left (x , i\right )-E\left (x , i\right )\right )}{5 \sqrt {-x^{4}+1}}\) | \(68\) |
elliptic | \(-\frac {\sqrt {-x^{4}+1}}{5 x^{5}}-\frac {3 \sqrt {-x^{4}+1}}{5 x}+\frac {3 \sqrt {-x^{2}+1}\, \sqrt {x^{2}+1}\, \left (F\left (x , i\right )-E\left (x , i\right )\right )}{5 \sqrt {-x^{4}+1}}\) | \(68\) |
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none
Time = 0.08 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.77 \[ \int \frac {1}{x^6 \sqrt {1-x^4}} \, dx=-\frac {3 \, x^{5} E(\arcsin \left (x\right )\,|\,-1) - 3 \, x^{5} F(\arcsin \left (x\right )\,|\,-1) + {\left (3 \, x^{4} + 1\right )} \sqrt {-x^{4} + 1}}{5 \, x^{5}} \]
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Time = 0.49 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.70 \[ \int \frac {1}{x^6 \sqrt {1-x^4}} \, dx=\frac {\Gamma \left (- \frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {5}{4}, \frac {1}{2} \\ - \frac {1}{4} \end {matrix}\middle | {x^{4} e^{2 i \pi }} \right )}}{4 x^{5} \Gamma \left (- \frac {1}{4}\right )} \]
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\[ \int \frac {1}{x^6 \sqrt {1-x^4}} \, dx=\int { \frac {1}{\sqrt {-x^{4} + 1} x^{6}} \,d x } \]
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\[ \int \frac {1}{x^6 \sqrt {1-x^4}} \, dx=\int { \frac {1}{\sqrt {-x^{4} + 1} x^{6}} \,d x } \]
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Timed out. \[ \int \frac {1}{x^6 \sqrt {1-x^4}} \, dx=\int \frac {1}{x^6\,\sqrt {1-x^4}} \,d x \]
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