Optimal. Leaf size=156 \[ \frac{\left (b x^2+1\right ) \sqrt{\frac{b^2 x^4+1}{\left (b x^2+1\right )^2}} \text{EllipticF}\left (2 \tan ^{-1}\left (\sqrt{b} x\right ),\frac{1}{2}\right )}{\sqrt{b} \sqrt{-b^2 x^4-1}}-\frac{x \sqrt{-b^2 x^4-1}}{b x^2+1}-\frac{\left (b x^2+1\right ) \sqrt{\frac{b^2 x^4+1}{\left (b x^2+1\right )^2}} E\left (2 \tan ^{-1}\left (\sqrt{b} x\right )|\frac{1}{2}\right )}{\sqrt{b} \sqrt{-b^2 x^4-1}} \]
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Rubi [A] time = 0.03229, antiderivative size = 156, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {1198, 220, 1196} \[ -\frac{x \sqrt{-b^2 x^4-1}}{b x^2+1}+\frac{\left (b x^2+1\right ) \sqrt{\frac{b^2 x^4+1}{\left (b x^2+1\right )^2}} F\left (2 \tan ^{-1}\left (\sqrt{b} x\right )|\frac{1}{2}\right )}{\sqrt{b} \sqrt{-b^2 x^4-1}}-\frac{\left (b x^2+1\right ) \sqrt{\frac{b^2 x^4+1}{\left (b x^2+1\right )^2}} E\left (2 \tan ^{-1}\left (\sqrt{b} x\right )|\frac{1}{2}\right )}{\sqrt{b} \sqrt{-b^2 x^4-1}} \]
Antiderivative was successfully verified.
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Rule 1198
Rule 220
Rule 1196
Rubi steps
\begin{align*} \int \frac{1+b x^2}{\sqrt{-1-b^2 x^4}} \, dx &=2 \int \frac{1}{\sqrt{-1-b^2 x^4}} \, dx-\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 \tan ^{-1}\left (\sqrt{b} x\right )|\frac{1}{2}\right )}{\sqrt{b} \sqrt{-1-b^2 x^4}}+\frac{\left (1+b x^2\right ) \sqrt{\frac{1+b^2 x^4}{\left (1+b x^2\right )^2}} F\left (2 \tan ^{-1}\left (\sqrt{b} x\right )|\frac{1}{2}\right )}{\sqrt{b} \sqrt{-1-b^2 x^4}}\\ \end{align*}
Mathematica [C] time = 0.0186291, size = 76, normalized size = 0.49 \[ \frac{\sqrt{b^2 x^4+1} \left (b x^3 \, _2F_1\left (\frac{1}{2},\frac{3}{4};\frac{7}{4};-b^2 x^4\right )+3 x \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{5}{4};-b^2 x^4\right )\right )}{3 \sqrt{-b^2 x^4-1}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.046, size = 122, normalized size = 0.8 \begin{align*}{-i\sqrt{1+ib{x}^{2}}\sqrt{1-ib{x}^{2}} \left ({\it EllipticF} \left ( x\sqrt{-ib},i \right ) -{\it EllipticE} \left ( x\sqrt{-ib},i \right ) \right ){\frac{1}{\sqrt{-ib}}}{\frac{1}{\sqrt{-{b}^{2}{x}^{4}-1}}}}+{\sqrt{1+ib{x}^{2}}\sqrt{1-ib{x}^{2}}{\it EllipticF} \left ( x\sqrt{-ib},i \right ){\frac{1}{\sqrt{-ib}}}{\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*} \frac{b x{\rm integral}\left (-\frac{\sqrt{-b^{2} x^{4} - 1}{\left (b x^{2} + 1\right )}}{b^{3} x^{6} + b x^{2}}, x\right ) - \sqrt{-b^{2} x^{4} - 1}}{b x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 1.51971, size = 71, normalized size = 0.46 \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} 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 )} \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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