Integrand size = 13, antiderivative size = 156 \[ \int \frac {1}{x^6 \left (1+x^4\right )^{3/2}} \, dx=\frac {1}{2 x^5 \sqrt {1+x^4}}-\frac {7 \sqrt {1+x^4}}{10 x^5}+\frac {21 \sqrt {1+x^4}}{10 x}-\frac {21 x \sqrt {1+x^4}}{10 \left (1+x^2\right )}+\frac {21 \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} E\left (2 \arctan (x)\left |\frac {1}{2}\right .\right )}{10 \sqrt {1+x^4}}-\frac {21 \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan (x),\frac {1}{2}\right )}{20 \sqrt {1+x^4}} \]
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Time = 0.03 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {296, 331, 311, 226, 1210} \[ \int \frac {1}{x^6 \left (1+x^4\right )^{3/2}} \, dx=-\frac {21 \left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan (x),\frac {1}{2}\right )}{20 \sqrt {x^4+1}}+\frac {21 \left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} E\left (2 \arctan (x)\left |\frac {1}{2}\right .\right )}{10 \sqrt {x^4+1}}+\frac {21 \sqrt {x^4+1}}{10 x}-\frac {7 \sqrt {x^4+1}}{10 x^5}+\frac {1}{2 \sqrt {x^4+1} x^5}-\frac {21 \sqrt {x^4+1} x}{10 \left (x^2+1\right )} \]
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Rule 226
Rule 296
Rule 311
Rule 331
Rule 1210
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2 x^5 \sqrt {1+x^4}}+\frac {7}{2} \int \frac {1}{x^6 \sqrt {1+x^4}} \, dx \\ & = \frac {1}{2 x^5 \sqrt {1+x^4}}-\frac {7 \sqrt {1+x^4}}{10 x^5}-\frac {21}{10} \int \frac {1}{x^2 \sqrt {1+x^4}} \, dx \\ & = \frac {1}{2 x^5 \sqrt {1+x^4}}-\frac {7 \sqrt {1+x^4}}{10 x^5}+\frac {21 \sqrt {1+x^4}}{10 x}-\frac {21}{10} \int \frac {x^2}{\sqrt {1+x^4}} \, dx \\ & = \frac {1}{2 x^5 \sqrt {1+x^4}}-\frac {7 \sqrt {1+x^4}}{10 x^5}+\frac {21 \sqrt {1+x^4}}{10 x}-\frac {21}{10} \int \frac {1}{\sqrt {1+x^4}} \, dx+\frac {21}{10} \int \frac {1-x^2}{\sqrt {1+x^4}} \, dx \\ & = \frac {1}{2 x^5 \sqrt {1+x^4}}-\frac {7 \sqrt {1+x^4}}{10 x^5}+\frac {21 \sqrt {1+x^4}}{10 x}-\frac {21 x \sqrt {1+x^4}}{10 \left (1+x^2\right )}+\frac {21 \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} E\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{10 \sqrt {1+x^4}}-\frac {21 \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{20 \sqrt {1+x^4}} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.01 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.14 \[ \int \frac {1}{x^6 \left (1+x^4\right )^{3/2}} \, dx=-\frac {\operatorname {Hypergeometric2F1}\left (-\frac {5}{4},\frac {3}{2},-\frac {1}{4},-x^4\right )}{5 x^5} \]
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Result contains higher order function than in optimal. Order 5 vs. order 4.
Time = 4.32 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.11
method | result | size |
meijerg | \(-\frac {{}_{2}^{}{\moversetsp {}{\mundersetsp {}{F_{1}^{}}}}\left (-\frac {5}{4},\frac {3}{2};-\frac {1}{4};-x^{4}\right )}{5 x^{5}}\) | \(17\) |
risch | \(\frac {21 x^{8}+14 x^{4}-2}{10 x^{5} \sqrt {x^{4}+1}}-\frac {21 i \sqrt {-i x^{2}+1}\, \sqrt {i x^{2}+1}\, \left (F\left (x \left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ), i\right )-E\left (x \left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ), i\right )\right )}{10 \left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ) \sqrt {x^{4}+1}}\) | \(107\) |
default | \(\frac {x^{3}}{2 \sqrt {x^{4}+1}}-\frac {\sqrt {x^{4}+1}}{5 x^{5}}+\frac {8 \sqrt {x^{4}+1}}{5 x}-\frac {21 i \sqrt {-i x^{2}+1}\, \sqrt {i x^{2}+1}\, \left (F\left (x \left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ), i\right )-E\left (x \left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ), i\right )\right )}{10 \left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ) \sqrt {x^{4}+1}}\) | \(119\) |
elliptic | \(\frac {x^{3}}{2 \sqrt {x^{4}+1}}-\frac {\sqrt {x^{4}+1}}{5 x^{5}}+\frac {8 \sqrt {x^{4}+1}}{5 x}-\frac {21 i \sqrt {-i x^{2}+1}\, \sqrt {i x^{2}+1}\, \left (F\left (x \left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ), i\right )-E\left (x \left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ), i\right )\right )}{10 \left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ) \sqrt {x^{4}+1}}\) | \(119\) |
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Result contains complex when optimal does not.
Time = 0.10 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.52 \[ \int \frac {1}{x^6 \left (1+x^4\right )^{3/2}} \, dx=-\frac {21 \, \sqrt {i} {\left (-i \, x^{9} - i \, x^{5}\right )} E(\arcsin \left (\sqrt {i} x\right )\,|\,-1) + 21 \, \sqrt {i} {\left (i \, x^{9} + i \, x^{5}\right )} F(\arcsin \left (\sqrt {i} x\right )\,|\,-1) - {\left (21 \, x^{8} + 14 \, x^{4} - 2\right )} \sqrt {x^{4} + 1}}{10 \, {\left (x^{9} + x^{5}\right )}} \]
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Result contains complex when optimal does not.
Time = 0.54 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.23 \[ \int \frac {1}{x^6 \left (1+x^4\right )^{3/2}} \, dx=\frac {\Gamma \left (- \frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {5}{4}, \frac {3}{2} \\ - \frac {1}{4} \end {matrix}\middle | {x^{4} e^{i \pi }} \right )}}{4 x^{5} \Gamma \left (- \frac {1}{4}\right )} \]
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\[ \int \frac {1}{x^6 \left (1+x^4\right )^{3/2}} \, dx=\int { \frac {1}{{\left (x^{4} + 1\right )}^{\frac {3}{2}} x^{6}} \,d x } \]
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\[ \int \frac {1}{x^6 \left (1+x^4\right )^{3/2}} \, dx=\int { \frac {1}{{\left (x^{4} + 1\right )}^{\frac {3}{2}} x^{6}} \,d x } \]
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Timed out. \[ \int \frac {1}{x^6 \left (1+x^4\right )^{3/2}} \, dx=\int \frac {1}{x^6\,{\left (x^4+1\right )}^{3/2}} \,d x \]
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