Integrand size = 13, antiderivative size = 58 \[ \int \frac {x^4}{\left (1+x^4\right )^{3/2}} \, dx=-\frac {x}{2 \sqrt {1+x^4}}+\frac {\left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan (x),\frac {1}{2}\right )}{4 \sqrt {1+x^4}} \]
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Time = 0.01 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {294, 226} \[ \int \frac {x^4}{\left (1+x^4\right )^{3/2}} \, dx=\frac {\left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan (x),\frac {1}{2}\right )}{4 \sqrt {x^4+1}}-\frac {x}{2 \sqrt {x^4+1}} \]
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Rule 226
Rule 294
Rubi steps \begin{align*} \text {integral}& = -\frac {x}{2 \sqrt {1+x^4}}+\frac {1}{2} \int \frac {1}{\sqrt {1+x^4}} \, dx \\ & = -\frac {x}{2 \sqrt {1+x^4}}+\frac {\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 )}{4 \sqrt {1+x^4}} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 4.56 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.55 \[ \int \frac {x^4}{\left (1+x^4\right )^{3/2}} \, dx=\frac {1}{2} x \left (-\frac {1}{\sqrt {1+x^4}}+\operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},-x^4\right )\right ) \]
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Result contains higher order function than in optimal. Order 5 vs. order 4.
Time = 4.35 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.29
method | result | size |
meijerg | \(\frac {x^{5} {}_{2}^{}{\moversetsp {}{\mundersetsp {}{F_{1}^{}}}}\left (\frac {5}{4},\frac {3}{2};\frac {9}{4};-x^{4}\right )}{5}\) | \(17\) |
default | \(-\frac {x}{2 \sqrt {x^{4}+1}}+\frac {\sqrt {-i x^{2}+1}\, \sqrt {i x^{2}+1}\, F\left (x \left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ), i\right )}{2 \left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ) \sqrt {x^{4}+1}}\) | \(72\) |
risch | \(-\frac {x}{2 \sqrt {x^{4}+1}}+\frac {\sqrt {-i x^{2}+1}\, \sqrt {i x^{2}+1}\, F\left (x \left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ), i\right )}{2 \left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ) \sqrt {x^{4}+1}}\) | \(72\) |
elliptic | \(-\frac {x}{2 \sqrt {x^{4}+1}}+\frac {\sqrt {-i x^{2}+1}\, \sqrt {i x^{2}+1}\, F\left (x \left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ), i\right )}{2 \left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ) \sqrt {x^{4}+1}}\) | \(72\) |
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Result contains complex when optimal does not.
Time = 0.08 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.67 \[ \int \frac {x^4}{\left (1+x^4\right )^{3/2}} \, dx=\frac {\sqrt {i} {\left (-i \, x^{4} - i\right )} F(\arcsin \left (\sqrt {i} x\right )\,|\,-1) - \sqrt {x^{4} + 1} x}{2 \, {\left (x^{4} + 1\right )}} \]
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Result contains complex when optimal does not.
Time = 0.37 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.50 \[ \int \frac {x^4}{\left (1+x^4\right )^{3/2}} \, dx=\frac {x^{5} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {5}{4}, \frac {3}{2} \\ \frac {9}{4} \end {matrix}\middle | {x^{4} e^{i \pi }} \right )}}{4 \Gamma \left (\frac {9}{4}\right )} \]
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\[ \int \frac {x^4}{\left (1+x^4\right )^{3/2}} \, dx=\int { \frac {x^{4}}{{\left (x^{4} + 1\right )}^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {x^4}{\left (1+x^4\right )^{3/2}} \, dx=\int { \frac {x^{4}}{{\left (x^{4} + 1\right )}^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {x^4}{\left (1+x^4\right )^{3/2}} \, dx=\int \frac {x^4}{{\left (x^4+1\right )}^{3/2}} \,d x \]
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