Integrand size = 9, antiderivative size = 72 \[ \int \frac {1}{\left (1+x^4\right )^{5/2}} \, dx=\frac {x}{6 \left (1+x^4\right )^{3/2}}+\frac {5 x}{12 \sqrt {1+x^4}}+\frac {5 \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan (x),\frac {1}{2}\right )}{24 \sqrt {1+x^4}} \]
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Time = 0.01 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {205, 226} \[ \int \frac {1}{\left (1+x^4\right )^{5/2}} \, dx=\frac {5 \left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan (x),\frac {1}{2}\right )}{24 \sqrt {x^4+1}}+\frac {5 x}{12 \sqrt {x^4+1}}+\frac {x}{6 \left (x^4+1\right )^{3/2}} \]
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Rule 205
Rule 226
Rubi steps \begin{align*} \text {integral}& = \frac {x}{6 \left (1+x^4\right )^{3/2}}+\frac {5}{6} \int \frac {1}{\left (1+x^4\right )^{3/2}} \, dx \\ & = \frac {x}{6 \left (1+x^4\right )^{3/2}}+\frac {5 x}{12 \sqrt {1+x^4}}+\frac {5}{12} \int \frac {1}{\sqrt {1+x^4}} \, dx \\ & = \frac {x}{6 \left (1+x^4\right )^{3/2}}+\frac {5 x}{12 \sqrt {1+x^4}}+\frac {5 \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 )}{24 \sqrt {1+x^4}} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 5.19 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.68 \[ \int \frac {1}{\left (1+x^4\right )^{5/2}} \, dx=\frac {x}{6 \left (1+x^4\right )^{3/2}}+\frac {5 x}{12 \sqrt {1+x^4}}+\frac {5}{12} x \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},-x^4\right ) \]
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Result contains higher order function than in optimal. Order 5 vs. order 4.
Time = 4.21 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.19
method | result | size |
meijerg | \(x {}_{2}^{}{\moversetsp {}{\mundersetsp {}{F_{1}^{}}}}\left (\frac {1}{4},\frac {5}{2};\frac {5}{4};-x^{4}\right )\) | \(14\) |
risch | \(\frac {x \left (5 x^{4}+7\right )}{12 \left (x^{4}+1\right )^{\frac {3}{2}}}+\frac {5 \sqrt {-i x^{2}+1}\, \sqrt {i x^{2}+1}\, F\left (x \left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ), i\right )}{12 \left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ) \sqrt {x^{4}+1}}\) | \(79\) |
default | \(\frac {x}{6 \left (x^{4}+1\right )^{\frac {3}{2}}}+\frac {5 x}{12 \sqrt {x^{4}+1}}+\frac {5 \sqrt {-i x^{2}+1}\, \sqrt {i x^{2}+1}\, F\left (x \left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ), i\right )}{12 \left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ) \sqrt {x^{4}+1}}\) | \(82\) |
elliptic | \(\frac {x}{6 \left (x^{4}+1\right )^{\frac {3}{2}}}+\frac {5 x}{12 \sqrt {x^{4}+1}}+\frac {5 \sqrt {-i x^{2}+1}\, \sqrt {i x^{2}+1}\, F\left (x \left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ), i\right )}{12 \left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ) \sqrt {x^{4}+1}}\) | \(82\) |
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Result contains complex when optimal does not.
Time = 0.10 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.81 \[ \int \frac {1}{\left (1+x^4\right )^{5/2}} \, dx=-\frac {5 \, \sqrt {i} {\left (i \, x^{8} + 2 i \, x^{4} + i\right )} F(\arcsin \left (\sqrt {i} x\right )\,|\,-1) - {\left (5 \, x^{5} + 7 \, x\right )} \sqrt {x^{4} + 1}}{12 \, {\left (x^{8} + 2 \, x^{4} + 1\right )}} \]
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Result contains complex when optimal does not.
Time = 0.44 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.38 \[ \int \frac {1}{\left (1+x^4\right )^{5/2}} \, dx=\frac {x \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {5}{2} \\ \frac {5}{4} \end {matrix}\middle | {x^{4} e^{i \pi }} \right )}}{4 \Gamma \left (\frac {5}{4}\right )} \]
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\[ \int \frac {1}{\left (1+x^4\right )^{5/2}} \, dx=\int { \frac {1}{{\left (x^{4} + 1\right )}^{\frac {5}{2}}} \,d x } \]
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\[ \int \frac {1}{\left (1+x^4\right )^{5/2}} \, dx=\int { \frac {1}{{\left (x^{4} + 1\right )}^{\frac {5}{2}}} \,d x } \]
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Time = 0.08 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.17 \[ \int \frac {1}{\left (1+x^4\right )^{5/2}} \, dx=x\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{4},\frac {5}{2};\ \frac {5}{4};\ -x^4\right ) \]
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