Optimal. Leaf size=89 \[ \frac{2 \sqrt{1-b^2 x^4} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{b} x\right ),-1\right )}{\sqrt{b} \sqrt{b^2 x^4-1}}-\frac{\sqrt{1-b^2 x^4} E\left (\left .\sin ^{-1}\left (\sqrt{b} x\right )\right |-1\right )}{\sqrt{b} \sqrt{b^2 x^4-1}} \]
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Rubi [A] time = 0.0455572, antiderivative size = 89, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {1200, 1199, 423, 424, 248, 221} \[ \frac{2 \sqrt{1-b^2 x^4} F\left (\left .\sin ^{-1}\left (\sqrt{b} x\right )\right |-1\right )}{\sqrt{b} \sqrt{b^2 x^4-1}}-\frac{\sqrt{1-b^2 x^4} E\left (\left .\sin ^{-1}\left (\sqrt{b} x\right )\right |-1\right )}{\sqrt{b} \sqrt{b^2 x^4-1}} \]
Antiderivative was successfully verified.
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Rule 1200
Rule 1199
Rule 423
Rule 424
Rule 248
Rule 221
Rubi steps
\begin{align*} \int \frac{1-b x^2}{\sqrt{-1+b^2 x^4}} \, dx &=\frac{\sqrt{1-b^2 x^4} \int \frac{1-b x^2}{\sqrt{1-b^2 x^4}} \, dx}{\sqrt{-1+b^2 x^4}}\\ &=\frac{\sqrt{1-b^2 x^4} \int \frac{\sqrt{1-b x^2}}{\sqrt{1+b x^2}} \, dx}{\sqrt{-1+b^2 x^4}}\\ &=-\frac{\sqrt{1-b^2 x^4} \int \frac{\sqrt{1+b x^2}}{\sqrt{1-b x^2}} \, dx}{\sqrt{-1+b^2 x^4}}+\frac{\left (2 \sqrt{1-b^2 x^4}\right ) \int \frac{1}{\sqrt{1-b x^2} \sqrt{1+b x^2}} \, dx}{\sqrt{-1+b^2 x^4}}\\ &=-\frac{\sqrt{1-b^2 x^4} E\left (\left .\sin ^{-1}\left (\sqrt{b} x\right )\right |-1\right )}{\sqrt{b} \sqrt{-1+b^2 x^4}}+\frac{\left (2 \sqrt{1-b^2 x^4}\right ) \int \frac{1}{\sqrt{1-b^2 x^4}} \, dx}{\sqrt{-1+b^2 x^4}}\\ &=-\frac{\sqrt{1-b^2 x^4} E\left (\left .\sin ^{-1}\left (\sqrt{b} x\right )\right |-1\right )}{\sqrt{b} \sqrt{-1+b^2 x^4}}+\frac{2 \sqrt{1-b^2 x^4} F\left (\left .\sin ^{-1}\left (\sqrt{b} x\right )\right |-1\right )}{\sqrt{b} \sqrt{-1+b^2 x^4}}\\ \end{align*}
Mathematica [C] time = 0.0209353, size = 74, normalized size = 0.83 \[ -\frac{\sqrt{1-b^2 x^4} \left (b x^3 \, _2F_1\left (\frac{1}{2},\frac{3}{4};\frac{7}{4};b^2 x^4\right )-3 x \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{5}{4};b^2 x^4\right )\right )}{3 \sqrt{b^2 x^4-1}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.046, size = 108, normalized size = 1.2 \begin{align*} -{\sqrt{b{x}^{2}+1}\sqrt{-b{x}^{2}+1} \left ({\it EllipticF} \left ( x\sqrt{-b},i \right ) -{\it EllipticE} \left ( x\sqrt{-b},i \right ) \right ){\frac{1}{\sqrt{-b}}}{\frac{1}{\sqrt{{b}^{2}{x}^{4}-1}}}}+{\sqrt{b{x}^{2}+1}\sqrt{-b{x}^{2}+1}{\it EllipticF} \left ( x\sqrt{-b},i \right ){\frac{1}{\sqrt{-b}}}{\frac{1}{\sqrt{{b}^{2}{x}^{4}-1}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} -\int \frac{b x^{2} - 1}{\sqrt{b^{2} x^{4} - 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\sqrt{b^{2} x^{4} - 1}}{b x^{2} + 1}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.86707, size = 60, normalized size = 0.67 \begin{align*} \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}} \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}} \right )}}{4 \Gamma \left (\frac{5}{4}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int -\frac{b x^{2} - 1}{\sqrt{b^{2} x^{4} - 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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