Integrand size = 13, antiderivative size = 124 \[ \int \frac {x^6}{\sqrt {1+x^4}} \, dx=\frac {1}{5} x^3 \sqrt {1+x^4}-\frac {3 x \sqrt {1+x^4}}{5 \left (1+x^2\right )}+\frac {3 \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} E\left (2 \arctan (x)\left |\frac {1}{2}\right .\right )}{5 \sqrt {1+x^4}}-\frac {3 \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan (x),\frac {1}{2}\right )}{10 \sqrt {1+x^4}} \]
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Time = 0.02 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {327, 311, 226, 1210} \[ \int \frac {x^6}{\sqrt {1+x^4}} \, dx=-\frac {3 \left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan (x),\frac {1}{2}\right )}{10 \sqrt {x^4+1}}+\frac {3 \left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} E\left (2 \arctan (x)\left |\frac {1}{2}\right .\right )}{5 \sqrt {x^4+1}}+\frac {1}{5} \sqrt {x^4+1} x^3-\frac {3 \sqrt {x^4+1} x}{5 \left (x^2+1\right )} \]
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Rule 226
Rule 311
Rule 327
Rule 1210
Rubi steps \begin{align*} \text {integral}& = \frac {1}{5} x^3 \sqrt {1+x^4}-\frac {3}{5} \int \frac {x^2}{\sqrt {1+x^4}} \, dx \\ & = \frac {1}{5} x^3 \sqrt {1+x^4}-\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 {1}{5} x^3 \sqrt {1+x^4}-\frac {3 x \sqrt {1+x^4}}{5 \left (1+x^2\right )}+\frac {3 \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 )}{5 \sqrt {1+x^4}}-\frac {3 \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 )}{10 \sqrt {1+x^4}} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.02 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.27 \[ \int \frac {x^6}{\sqrt {1+x^4}} \, dx=\frac {1}{5} x^3 \left (\sqrt {1+x^4}-\operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},-x^4\right )\right ) \]
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Result contains higher order function than in optimal. Order 5 vs. order 4.
Time = 4.44 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.14
method | result | size |
meijerg | \(\frac {x^{7} {}_{2}^{}{\moversetsp {}{\mundersetsp {}{F_{1}^{}}}}\left (\frac {1}{2},\frac {7}{4};\frac {11}{4};-x^{4}\right )}{7}\) | \(17\) |
default | \(\frac {x^{3} \sqrt {x^{4}+1}}{5}-\frac {3 i \sqrt {-i x^{2}+1}\, \sqrt {i x^{2}+1}\, \left (F\left (x \left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ), i\right )-E\left (x \left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ), i\right )\right )}{5 \left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ) \sqrt {x^{4}+1}}\) | \(95\) |
risch | \(\frac {x^{3} \sqrt {x^{4}+1}}{5}-\frac {3 i \sqrt {-i x^{2}+1}\, \sqrt {i x^{2}+1}\, \left (F\left (x \left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ), i\right )-E\left (x \left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ), i\right )\right )}{5 \left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ) \sqrt {x^{4}+1}}\) | \(95\) |
elliptic | \(\frac {x^{3} \sqrt {x^{4}+1}}{5}-\frac {3 i \sqrt {-i x^{2}+1}\, \sqrt {i x^{2}+1}\, \left (F\left (x \left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ), i\right )-E\left (x \left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ), i\right )\right )}{5 \left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ) \sqrt {x^{4}+1}}\) | \(95\) |
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Result contains complex when optimal does not.
Time = 0.10 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.41 \[ \int \frac {x^6}{\sqrt {1+x^4}} \, dx=\frac {-3 i \, \sqrt {i} x E(\arcsin \left (\frac {\sqrt {i}}{x}\right )\,|\,-1) + 3 i \, \sqrt {i} x F(\arcsin \left (\frac {\sqrt {i}}{x}\right )\,|\,-1) + \sqrt {x^{4} + 1} {\left (x^{4} - 3\right )}}{5 \, x} \]
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Result contains complex when optimal does not.
Time = 0.40 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.23 \[ \int \frac {x^6}{\sqrt {1+x^4}} \, dx=\frac {x^{7} \Gamma \left (\frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {7}{4} \\ \frac {11}{4} \end {matrix}\middle | {x^{4} e^{i \pi }} \right )}}{4 \Gamma \left (\frac {11}{4}\right )} \]
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\[ \int \frac {x^6}{\sqrt {1+x^4}} \, dx=\int { \frac {x^{6}}{\sqrt {x^{4} + 1}} \,d x } \]
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\[ \int \frac {x^6}{\sqrt {1+x^4}} \, dx=\int { \frac {x^{6}}{\sqrt {x^{4} + 1}} \,d x } \]
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Timed out. \[ \int \frac {x^6}{\sqrt {1+x^4}} \, dx=\int \frac {x^6}{\sqrt {x^4+1}} \,d x \]
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