Integrand size = 13, antiderivative size = 140 \[ \int \frac {x^{10}}{\sqrt {1+x^4}} \, dx=-\frac {7}{45} x^3 \sqrt {1+x^4}+\frac {1}{9} x^7 \sqrt {1+x^4}+\frac {7 x \sqrt {1+x^4}}{15 \left (1+x^2\right )}-\frac {7 \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} E\left (2 \arctan (x)\left |\frac {1}{2}\right .\right )}{15 \sqrt {1+x^4}}+\frac {7 \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan (x),\frac {1}{2}\right )}{30 \sqrt {1+x^4}} \]
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Time = 0.02 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {327, 311, 226, 1210} \[ \int \frac {x^{10}}{\sqrt {1+x^4}} \, dx=\frac {7 \left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan (x),\frac {1}{2}\right )}{30 \sqrt {x^4+1}}-\frac {7 \left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} E\left (2 \arctan (x)\left |\frac {1}{2}\right .\right )}{15 \sqrt {x^4+1}}+\frac {1}{9} \sqrt {x^4+1} x^7-\frac {7}{45} \sqrt {x^4+1} x^3+\frac {7 \sqrt {x^4+1} x}{15 \left (x^2+1\right )} \]
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Rule 226
Rule 311
Rule 327
Rule 1210
Rubi steps \begin{align*} \text {integral}& = \frac {1}{9} x^7 \sqrt {1+x^4}-\frac {7}{9} \int \frac {x^6}{\sqrt {1+x^4}} \, dx \\ & = -\frac {7}{45} x^3 \sqrt {1+x^4}+\frac {1}{9} x^7 \sqrt {1+x^4}+\frac {7}{15} \int \frac {x^2}{\sqrt {1+x^4}} \, dx \\ & = -\frac {7}{45} x^3 \sqrt {1+x^4}+\frac {1}{9} x^7 \sqrt {1+x^4}+\frac {7}{15} \int \frac {1}{\sqrt {1+x^4}} \, dx-\frac {7}{15} \int \frac {1-x^2}{\sqrt {1+x^4}} \, dx \\ & = -\frac {7}{45} x^3 \sqrt {1+x^4}+\frac {1}{9} x^7 \sqrt {1+x^4}+\frac {7 x \sqrt {1+x^4}}{15 \left (1+x^2\right )}-\frac {7 \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 )}{15 \sqrt {1+x^4}}+\frac {7 \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 )}{30 \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 = 42, normalized size of antiderivative = 0.30 \[ \int \frac {x^{10}}{\sqrt {1+x^4}} \, dx=\frac {1}{45} x^3 \left (\sqrt {1+x^4} \left (-7+5 x^4\right )+7 \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},-x^4\right )\right ) \]
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Result contains higher order function than in optimal. Order 5 vs. order 4.
Time = 4.42 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.12
method | result | size |
meijerg | \(\frac {x^{11} {}_{2}^{}{\moversetsp {}{\mundersetsp {}{F_{1}^{}}}}\left (\frac {1}{2},\frac {11}{4};\frac {15}{4};-x^{4}\right )}{11}\) | \(17\) |
risch | \(\frac {x^{3} \left (5 x^{4}-7\right ) \sqrt {x^{4}+1}}{45}+\frac {7 i \sqrt {-i x^{2}+1}\, \sqrt {i x^{2}+1}\, \left (F\left (x \left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ), i\right )-E\left (x \left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ), i\right )\right )}{15 \left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ) \sqrt {x^{4}+1}}\) | \(102\) |
default | \(\frac {x^{7} \sqrt {x^{4}+1}}{9}-\frac {7 x^{3} \sqrt {x^{4}+1}}{45}+\frac {7 i \sqrt {-i x^{2}+1}\, \sqrt {i x^{2}+1}\, \left (F\left (x \left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ), i\right )-E\left (x \left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ), i\right )\right )}{15 \left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ) \sqrt {x^{4}+1}}\) | \(107\) |
elliptic | \(\frac {x^{7} \sqrt {x^{4}+1}}{9}-\frac {7 x^{3} \sqrt {x^{4}+1}}{45}+\frac {7 i \sqrt {-i x^{2}+1}\, \sqrt {i x^{2}+1}\, \left (F\left (x \left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ), i\right )-E\left (x \left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ), i\right )\right )}{15 \left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ) \sqrt {x^{4}+1}}\) | \(107\) |
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Result contains complex when optimal does not.
Time = 0.10 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.41 \[ \int \frac {x^{10}}{\sqrt {1+x^4}} \, dx=\frac {21 i \, \sqrt {i} x E(\arcsin \left (\frac {\sqrt {i}}{x}\right )\,|\,-1) - 21 i \, \sqrt {i} x F(\arcsin \left (\frac {\sqrt {i}}{x}\right )\,|\,-1) + {\left (5 \, x^{8} - 7 \, x^{4} + 21\right )} \sqrt {x^{4} + 1}}{45 \, x} \]
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Result contains complex when optimal does not.
Time = 0.51 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.21 \[ \int \frac {x^{10}}{\sqrt {1+x^4}} \, dx=\frac {x^{11} \Gamma \left (\frac {11}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {11}{4} \\ \frac {15}{4} \end {matrix}\middle | {x^{4} e^{i \pi }} \right )}}{4 \Gamma \left (\frac {15}{4}\right )} \]
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\[ \int \frac {x^{10}}{\sqrt {1+x^4}} \, dx=\int { \frac {x^{10}}{\sqrt {x^{4} + 1}} \,d x } \]
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\[ \int \frac {x^{10}}{\sqrt {1+x^4}} \, dx=\int { \frac {x^{10}}{\sqrt {x^{4} + 1}} \,d x } \]
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Timed out. \[ \int \frac {x^{10}}{\sqrt {1+x^4}} \, dx=\int \frac {x^{10}}{\sqrt {x^4+1}} \,d x \]
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