Integrand size = 13, antiderivative size = 150 \[ \int \frac {x^6}{\sqrt {-1+x^4}} \, dx=\frac {3 x \left (1+x^2\right )}{5 \sqrt {-1+x^4}}+\frac {1}{5} x^3 \sqrt {-1+x^4}-\frac {3 \sqrt {2} \sqrt {-1+x^2} \sqrt {1+x^2} E\left (\arcsin \left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right )|\frac {1}{2}\right )}{5 \sqrt {-1+x^4}}+\frac {3 \sqrt {-1+x^2} \sqrt {1+x^2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right ),\frac {1}{2}\right )}{5 \sqrt {2} \sqrt {-1+x^4}} \]
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Time = 0.02 (sec) , antiderivative size = 150, 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, 312, 228, 1199} \[ \int \frac {x^6}{\sqrt {-1+x^4}} \, dx=\frac {3 \sqrt {x^2-1} \sqrt {x^2+1} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} x}{\sqrt {x^2-1}}\right ),\frac {1}{2}\right )}{5 \sqrt {2} \sqrt {x^4-1}}-\frac {3 \sqrt {2} \sqrt {x^2-1} \sqrt {x^2+1} E\left (\arcsin \left (\frac {\sqrt {2} x}{\sqrt {x^2-1}}\right )|\frac {1}{2}\right )}{5 \sqrt {x^4-1}}+\frac {1}{5} \sqrt {x^4-1} x^3+\frac {3 \left (x^2+1\right ) x}{5 \sqrt {x^4-1}} \]
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Rule 228
Rule 312
Rule 327
Rule 1199
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 {3 x \left (1+x^2\right )}{5 \sqrt {-1+x^4}}+\frac {1}{5} x^3 \sqrt {-1+x^4}-\frac {3 \sqrt {2} \sqrt {-1+x^2} \sqrt {1+x^2} E\left (\sin ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right )|\frac {1}{2}\right )}{5 \sqrt {-1+x^4}}+\frac {3 \sqrt {-1+x^2} \sqrt {1+x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right )|\frac {1}{2}\right )}{5 \sqrt {2} \sqrt {-1+x^4}} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.03 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.31 \[ \int \frac {x^6}{\sqrt {-1+x^4}} \, dx=\frac {x^3 \left (-1+x^4+\sqrt {1-x^4} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},x^4\right )\right )}{5 \sqrt {-1+x^4}} \]
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 4.61 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.22
method | result | size |
meijerg | \(\frac {\sqrt {-\operatorname {signum}\left (x^{4}-1\right )}\, x^{7} {}_{2}^{}{\moversetsp {}{\mundersetsp {}{F_{1}^{}}}}\left (\frac {1}{2},\frac {7}{4};\frac {11}{4};x^{4}\right )}{7 \sqrt {\operatorname {signum}\left (x^{4}-1\right )}}\) | \(33\) |
default | \(\frac {x^{3} \sqrt {x^{4}-1}}{5}-\frac {3 i \sqrt {x^{2}+1}\, \sqrt {-x^{2}+1}\, \left (F\left (i x , i\right )-E\left (i x , i\right )\right )}{5 \sqrt {x^{4}-1}}\) | \(57\) |
risch | \(\frac {x^{3} \sqrt {x^{4}-1}}{5}-\frac {3 i \sqrt {x^{2}+1}\, \sqrt {-x^{2}+1}\, \left (F\left (i x , i\right )-E\left (i x , i\right )\right )}{5 \sqrt {x^{4}-1}}\) | \(57\) |
elliptic | \(\frac {x^{3} \sqrt {x^{4}-1}}{5}-\frac {3 i \sqrt {x^{2}+1}\, \sqrt {-x^{2}+1}\, \left (F\left (i x , i\right )-E\left (i x , i\right )\right )}{5 \sqrt {x^{4}-1}}\) | \(57\) |
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none
Time = 0.09 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.25 \[ \int \frac {x^6}{\sqrt {-1+x^4}} \, dx=\frac {3 \, x E(\arcsin \left (\frac {1}{x}\right )\,|\,-1) - 3 \, x F(\arcsin \left (\frac {1}{x}\right )\,|\,-1) + {\left (x^{4} + 3\right )} \sqrt {x^{4} - 1}}{5 \, x} \]
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Result contains complex when optimal does not.
Time = 0.42 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.18 \[ \int \frac {x^6}{\sqrt {-1+x^4}} \, dx=- \frac {i x^{7} \Gamma \left (\frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {7}{4} \\ \frac {11}{4} \end {matrix}\middle | {x^{4}} \right )}}{4 \Gamma \left (\frac {11}{4}\right )} \]
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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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