Integrand size = 23, antiderivative size = 90 \[ \int \frac {1-b x^2}{\sqrt {-1-b^2 x^4}} \, dx=\frac {x \sqrt {-1-b^2 x^4}}{1+b x^2}+\frac {\left (1+b x^2\right ) \sqrt {\frac {1+b^2 x^4}{\left (1+b x^2\right )^2}} E\left (2 \arctan \left (\sqrt {b} x\right )|\frac {1}{2}\right )}{\sqrt {b} \sqrt {-1-b^2 x^4}} \]
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Time = 0.01 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {1210} \[ \int \frac {1-b x^2}{\sqrt {-1-b^2 x^4}} \, dx=\frac {\left (b x^2+1\right ) \sqrt {\frac {b^2 x^4+1}{\left (b x^2+1\right )^2}} E\left (2 \arctan \left (\sqrt {b} x\right )|\frac {1}{2}\right )}{\sqrt {b} \sqrt {-b^2 x^4-1}}+\frac {x \sqrt {-b^2 x^4-1}}{b x^2+1} \]
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Rule 1210
Rubi steps \begin{align*} \text {integral}& = \frac {x \sqrt {-1-b^2 x^4}}{1+b x^2}+\frac {\left (1+b x^2\right ) \sqrt {\frac {1+b^2 x^4}{\left (1+b x^2\right )^2}} E\left (2 \tan ^{-1}\left (\sqrt {b} x\right )|\frac {1}{2}\right )}{\sqrt {b} \sqrt {-1-b^2 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 = 76, normalized size of antiderivative = 0.84 \[ \int \frac {1-b x^2}{\sqrt {-1-b^2 x^4}} \, dx=-\frac {\sqrt {1+b^2 x^4} \left (-3 x \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},-b^2 x^4\right )+b x^3 \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},-b^2 x^4\right )\right )}{3 \sqrt {-1-b^2 x^4}} \]
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 2.49 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.00
method | result | size |
meijerg | \(-\frac {b \sqrt {\operatorname {signum}\left (b^{2} x^{4}+1\right )}\, x^{3} {}_{2}^{}{\moversetsp {}{\mundersetsp {}{F_{1}^{}}}}\left (\frac {1}{2},\frac {3}{4};\frac {7}{4};-b^{2} x^{4}\right )}{3 \sqrt {-\operatorname {signum}\left (b^{2} x^{4}+1\right )}}+\frac {\sqrt {\operatorname {signum}\left (b^{2} x^{4}+1\right )}\, x {}_{2}^{}{\moversetsp {}{\mundersetsp {}{F_{1}^{}}}}\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};-b^{2} x^{4}\right )}{\sqrt {-\operatorname {signum}\left (b^{2} x^{4}+1\right )}}\) | \(90\) |
default | \(\frac {\sqrt {i b \,x^{2}+1}\, \sqrt {-i b \,x^{2}+1}\, F\left (x \sqrt {-i b}, i\right )}{\sqrt {-i b}\, \sqrt {-b^{2} x^{4}-1}}+\frac {i \sqrt {i b \,x^{2}+1}\, \sqrt {-i b \,x^{2}+1}\, \left (F\left (x \sqrt {-i b}, i\right )-E\left (x \sqrt {-i b}, i\right )\right )}{\sqrt {-i b}\, \sqrt {-b^{2} x^{4}-1}}\) | \(122\) |
elliptic | \(\frac {\sqrt {i b \,x^{2}+1}\, \sqrt {-i b \,x^{2}+1}\, F\left (x \sqrt {-i b}, i\right )}{\sqrt {-i b}\, \sqrt {-b^{2} x^{4}-1}}+\frac {i \sqrt {i b \,x^{2}+1}\, \sqrt {-i b \,x^{2}+1}\, \left (F\left (x \sqrt {-i b}, i\right )-E\left (x \sqrt {-i b}, i\right )\right )}{\sqrt {-i b}\, \sqrt {-b^{2} x^{4}-1}}\) | \(122\) |
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none
Time = 0.08 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.97 \[ \int \frac {1-b x^2}{\sqrt {-1-b^2 x^4}} \, dx=-\frac {\sqrt {-b^{2}} {\left (b + 1\right )} x \left (-\frac {1}{b^{2}}\right )^{\frac {3}{4}} F(\arcsin \left (\frac {\left (-\frac {1}{b^{2}}\right )^{\frac {1}{4}}}{x}\right )\,|\,-1) - \sqrt {-b^{2}} x \left (-\frac {1}{b^{2}}\right )^{\frac {3}{4}} E(\arcsin \left (\frac {\left (-\frac {1}{b^{2}}\right )^{\frac {1}{4}}}{x}\right )\,|\,-1) - \sqrt {-b^{2} x^{4} - 1}}{b x} \]
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Result contains complex when optimal does not.
Time = 0.84 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.78 \[ \int \frac {1-b x^2}{\sqrt {-1-b^2 x^4}} \, dx=\frac {i b x^{3} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {3}{4} \\ \frac {7}{4} \end {matrix}\middle | {b^{2} x^{4} e^{i \pi }} \right )}}{4 \Gamma \left (\frac {7}{4}\right )} - \frac {i x \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {1}{2} \\ \frac {5}{4} \end {matrix}\middle | {b^{2} x^{4} e^{i \pi }} \right )}}{4 \Gamma \left (\frac {5}{4}\right )} \]
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\[ \int \frac {1-b x^2}{\sqrt {-1-b^2 x^4}} \, dx=\int { -\frac {b x^{2} - 1}{\sqrt {-b^{2} x^{4} - 1}} \,d x } \]
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\[ \int \frac {1-b x^2}{\sqrt {-1-b^2 x^4}} \, dx=\int { -\frac {b x^{2} - 1}{\sqrt {-b^{2} x^{4} - 1}} \,d x } \]
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Timed out. \[ \int \frac {1-b x^2}{\sqrt {-1-b^2 x^4}} \, dx=-\int \frac {b\,x^2-1}{\sqrt {-b^2\,x^4-1}} \,d x \]
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