Integrand size = 15, antiderivative size = 61 \[ \int \frac {x^{12}}{\left (1-x^4\right )^{3/2}} \, dx=\frac {x^9}{2 \sqrt {1-x^4}}+\frac {15}{14} x \sqrt {1-x^4}+\frac {9}{14} x^5 \sqrt {1-x^4}-\frac {15}{14} \operatorname {EllipticF}(\arcsin (x),-1) \]
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Time = 0.02 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {294, 327, 227} \[ \int \frac {x^{12}}{\left (1-x^4\right )^{3/2}} \, dx=-\frac {15}{14} \operatorname {EllipticF}(\arcsin (x),-1)+\frac {15}{14} \sqrt {1-x^4} x+\frac {x^9}{2 \sqrt {1-x^4}}+\frac {9}{14} \sqrt {1-x^4} x^5 \]
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Rule 227
Rule 294
Rule 327
Rubi steps \begin{align*} \text {integral}& = \frac {x^9}{2 \sqrt {1-x^4}}-\frac {9}{2} \int \frac {x^8}{\sqrt {1-x^4}} \, dx \\ & = \frac {x^9}{2 \sqrt {1-x^4}}+\frac {9}{14} x^5 \sqrt {1-x^4}-\frac {45}{14} \int \frac {x^4}{\sqrt {1-x^4}} \, dx \\ & = \frac {x^9}{2 \sqrt {1-x^4}}+\frac {15}{14} x \sqrt {1-x^4}+\frac {9}{14} x^5 \sqrt {1-x^4}-\frac {15}{14} \int \frac {1}{\sqrt {1-x^4}} \, dx \\ & = \frac {x^9}{2 \sqrt {1-x^4}}+\frac {15}{14} x \sqrt {1-x^4}+\frac {9}{14} x^5 \sqrt {1-x^4}-\frac {15}{14} F\left (\left .\sin ^{-1}(x)\right |-1\right ) \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 6.45 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.89 \[ \int \frac {x^{12}}{\left (1-x^4\right )^{3/2}} \, dx=-\frac {x \left (-15+6 x^4+2 x^8+15 \sqrt {1-x^4} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},x^4\right )\right )}{14 \sqrt {1-x^4}} \]
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Result contains higher order function than in optimal. Order 5 vs. order 4.
Time = 5.00 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.25
method | result | size |
meijerg | \(\frac {x^{13} {}_{2}^{}{\moversetsp {}{\mundersetsp {}{F_{1}^{}}}}\left (\frac {3}{2},\frac {13}{4};\frac {17}{4};x^{4}\right )}{13}\) | \(15\) |
risch | \(-\frac {x \left (2 x^{8}+6 x^{4}-15\right )}{14 \sqrt {-x^{4}+1}}-\frac {15 \sqrt {-x^{2}+1}\, \sqrt {x^{2}+1}\, F\left (x , i\right )}{14 \sqrt {-x^{4}+1}}\) | \(57\) |
default | \(\frac {x}{2 \sqrt {-x^{4}+1}}+\frac {x^{5} \sqrt {-x^{4}+1}}{7}+\frac {4 x \sqrt {-x^{4}+1}}{7}-\frac {15 \sqrt {-x^{2}+1}\, \sqrt {x^{2}+1}\, F\left (x , i\right )}{14 \sqrt {-x^{4}+1}}\) | \(71\) |
elliptic | \(\frac {x}{2 \sqrt {-x^{4}+1}}+\frac {x^{5} \sqrt {-x^{4}+1}}{7}+\frac {4 x \sqrt {-x^{4}+1}}{7}-\frac {15 \sqrt {-x^{2}+1}\, \sqrt {x^{2}+1}\, F\left (x , i\right )}{14 \sqrt {-x^{4}+1}}\) | \(71\) |
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none
Time = 0.10 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.82 \[ \int \frac {x^{12}}{\left (1-x^4\right )^{3/2}} \, dx=-\frac {15 \, {\left (i \, x^{4} - i\right )} F(\arcsin \left (\frac {1}{x}\right )\,|\,-1) - {\left (2 \, x^{9} + 6 \, x^{5} - 15 \, x\right )} \sqrt {-x^{4} + 1}}{14 \, {\left (x^{4} - 1\right )}} \]
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Time = 0.60 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.51 \[ \int \frac {x^{12}}{\left (1-x^4\right )^{3/2}} \, dx=\frac {x^{13} \Gamma \left (\frac {13}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{2}, \frac {13}{4} \\ \frac {17}{4} \end {matrix}\middle | {x^{4} e^{2 i \pi }} \right )}}{4 \Gamma \left (\frac {17}{4}\right )} \]
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\[ \int \frac {x^{12}}{\left (1-x^4\right )^{3/2}} \, dx=\int { \frac {x^{12}}{{\left (-x^{4} + 1\right )}^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {x^{12}}{\left (1-x^4\right )^{3/2}} \, dx=\int { \frac {x^{12}}{{\left (-x^{4} + 1\right )}^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {x^{12}}{\left (1-x^4\right )^{3/2}} \, dx=\int \frac {x^{12}}{{\left (1-x^4\right )}^{3/2}} \,d x \]
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