Integrand size = 24, antiderivative size = 10 \[ \int \frac {1+c^2 x^2}{\sqrt {1-c^4 x^4}} \, dx=\frac {E(\arcsin (c x)|-1)}{c} \]
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Time = 0.01 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {1213, 435} \[ \int \frac {1+c^2 x^2}{\sqrt {1-c^4 x^4}} \, dx=\frac {E(\arcsin (c x)|-1)}{c} \]
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Rule 435
Rule 1213
Rubi steps \begin{align*} \text {integral}& = \int \frac {\sqrt {1+c^2 x^2}}{\sqrt {1-c^2 x^2}} \, dx \\ & = \frac {E\left (\left .\sin ^{-1}(c x)\right |-1\right )}{c} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.02 (sec) , antiderivative size = 47, normalized size of antiderivative = 4.70 \[ \int \frac {1+c^2 x^2}{\sqrt {1-c^4 x^4}} \, dx=x \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},c^4 x^4\right )+\frac {1}{3} c^2 x^3 \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},c^4 x^4\right ) \]
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Result contains higher order function than in optimal. Order 5 vs. order 4.
Time = 1.20 (sec) , antiderivative size = 38, normalized size of antiderivative = 3.80
method | result | size |
meijerg | \(\frac {c^{2} x^{3} {}_{2}^{}{\moversetsp {}{\mundersetsp {}{F_{1}^{}}}}\left (\frac {1}{2},\frac {3}{4};\frac {7}{4};c^{4} x^{4}\right )}{3}+x {}_{2}^{}{\moversetsp {}{\mundersetsp {}{F_{1}^{}}}}\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};c^{4} x^{4}\right )\) | \(38\) |
default | \(\frac {\sqrt {-c^{2} x^{2}+1}\, \sqrt {c^{2} x^{2}+1}\, F\left (x \sqrt {c^{2}}, i\right )}{\sqrt {c^{2}}\, \sqrt {-c^{4} x^{4}+1}}-\frac {\sqrt {-c^{2} x^{2}+1}\, \sqrt {c^{2} x^{2}+1}\, \left (F\left (x \sqrt {c^{2}}, i\right )-E\left (x \sqrt {c^{2}}, i\right )\right )}{\sqrt {c^{2}}\, \sqrt {-c^{4} x^{4}+1}}\) | \(118\) |
elliptic | \(\frac {\sqrt {-c^{2} x^{2}+1}\, \sqrt {c^{2} x^{2}+1}\, F\left (x \sqrt {c^{2}}, i\right )}{\sqrt {c^{2}}\, \sqrt {-c^{4} x^{4}+1}}-\frac {\sqrt {-c^{2} x^{2}+1}\, \sqrt {c^{2} x^{2}+1}\, \left (F\left (x \sqrt {c^{2}}, i\right )-E\left (x \sqrt {c^{2}}, i\right )\right )}{\sqrt {c^{2}}\, \sqrt {-c^{4} x^{4}+1}}\) | \(118\) |
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Leaf count of result is larger than twice the leaf count of optimal. 65 vs. \(2 (9) = 18\).
Time = 0.08 (sec) , antiderivative size = 65, normalized size of antiderivative = 6.50 \[ \int \frac {1+c^2 x^2}{\sqrt {1-c^4 x^4}} \, dx=-\frac {\sqrt {-c^{4} x^{4} + 1} c^{3} - \sqrt {-c^{4}} {\left ({\left (c^{2} + 1\right )} x F(\arcsin \left (\frac {1}{c x}\right )\,|\,-1) - x E(\arcsin \left (\frac {1}{c x}\right )\,|\,-1)\right )}}{c^{5} x} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 71 vs. \(2 (5) = 10\).
Time = 0.90 (sec) , antiderivative size = 71, normalized size of antiderivative = 7.10 \[ \int \frac {1+c^2 x^2}{\sqrt {1-c^4 x^4}} \, dx=\frac {c^{2} x^{3} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {3}{4} \\ \frac {7}{4} \end {matrix}\middle | {c^{4} x^{4} e^{2 i \pi }} \right )}}{4 \Gamma \left (\frac {7}{4}\right )} + \frac {x \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {1}{2} \\ \frac {5}{4} \end {matrix}\middle | {c^{4} x^{4} e^{2 i \pi }} \right )}}{4 \Gamma \left (\frac {5}{4}\right )} \]
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\[ \int \frac {1+c^2 x^2}{\sqrt {1-c^4 x^4}} \, dx=\int { \frac {c^{2} x^{2} + 1}{\sqrt {-c^{4} x^{4} + 1}} \,d x } \]
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\[ \int \frac {1+c^2 x^2}{\sqrt {1-c^4 x^4}} \, dx=\int { \frac {c^{2} x^{2} + 1}{\sqrt {-c^{4} x^{4} + 1}} \,d x } \]
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Timed out. \[ \int \frac {1+c^2 x^2}{\sqrt {1-c^4 x^4}} \, dx=\int \frac {c^2\,x^2+1}{\sqrt {1-c^4\,x^4}} \,d x \]
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