Optimal. Leaf size=90 \[ \frac{x \sqrt{-b^2 x^4-1}}{b x^2+1}+\frac{\left (b x^2+1\right ) \sqrt{\frac{b^2 x^4+1}{\left (b x^2+1\right )^2}} E\left (2 \tan ^{-1}\left (\sqrt{b} x\right )|\frac{1}{2}\right )}{\sqrt{b} \sqrt{-b^2 x^4-1}} \]
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Rubi [A] time = 0.0533591, antiderivative size = 90, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.043 \[ \frac{x \sqrt{-b^2 x^4-1}}{b x^2+1}+\frac{\left (b x^2+1\right ) \sqrt{\frac{b^2 x^4+1}{\left (b x^2+1\right )^2}} E\left (2 \tan ^{-1}\left (\sqrt{b} x\right )|\frac{1}{2}\right )}{\sqrt{b} \sqrt{-b^2 x^4-1}} \]
Antiderivative was successfully verified.
[In] Int[(1 - b*x^2)/Sqrt[-1 - b^2*x^4],x]
[Out]
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Rubi in Sympy [A] time = 8.11608, size = 80, normalized size = 0.89 \[ \frac{x \sqrt{- b^{2} x^{4} - 1}}{b x^{2} + 1} + \frac{\sqrt{\frac{b^{2} x^{4} + 1}{\left (b x^{2} + 1\right )^{2}}} \left (b x^{2} + 1\right ) E\left (2 \operatorname{atan}{\left (\sqrt{b} x \right )}\middle | \frac{1}{2}\right )}{\sqrt{b} \sqrt{- b^{2} x^{4} - 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((-b*x**2+1)/(-b**2*x**4-1)**(1/2),x)
[Out]
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Mathematica [C] time = 0.0698462, size = 79, normalized size = 0.88 \[ -\frac{\sqrt{b^2 x^4+1} \left (E\left (\left .i \sinh ^{-1}\left (\sqrt{i b} x\right )\right |-1\right )-(1-i) F\left (\left .i \sinh ^{-1}\left (\sqrt{i b} x\right )\right |-1\right )\right )}{\sqrt{i b} \sqrt{-b^2 x^4-1}} \]
Antiderivative was successfully verified.
[In] Integrate[(1 - b*x^2)/Sqrt[-1 - b^2*x^4],x]
[Out]
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Maple [C] time = 0.016, size = 122, normalized size = 1.4 \[{i\sqrt{1+ib{x}^{2}}\sqrt{1-ib{x}^{2}} \left ({\it EllipticF} \left ( x\sqrt{-ib},i \right ) -{\it EllipticE} \left ( x\sqrt{-ib},i \right ) \right ){\frac{1}{\sqrt{-ib}}}{\frac{1}{\sqrt{-{b}^{2}{x}^{4}-1}}}}+{1\sqrt{1+ib{x}^{2}}\sqrt{1-ib{x}^{2}}{\it EllipticF} \left ( x\sqrt{-ib},i \right ){\frac{1}{\sqrt{-ib}}}{\frac{1}{\sqrt{-{b}^{2}{x}^{4}-1}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((-b*x^2+1)/(-b^2*x^4-1)^(1/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ -\int \frac{b x^{2} - 1}{\sqrt{-b^{2} x^{4} - 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(b*x^2 - 1)/sqrt(-b^2*x^4 - 1),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[ \frac{b x{\rm integral}\left (-\frac{\sqrt{-b^{2} x^{4} - 1}{\left (b x^{2} - 1\right )}}{b^{3} x^{6} + b x^{2}}, x\right ) + \sqrt{-b^{2} x^{4} - 1}}{b x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(b*x^2 - 1)/sqrt(-b^2*x^4 - 1),x, algorithm="fricas")
[Out]
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Sympy [A] time = 4.01715, size = 70, normalized size = 0.78 \[ \frac{i b x^{3} \Gamma \left (\frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{3}{4} \\ \frac{7}{4} \end{matrix}\middle |{b^{2} x^{4} e^{i \pi }} \right )}}{4 \Gamma \left (\frac{7}{4}\right )} - \frac{i x \Gamma \left (\frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{4}, \frac{1}{2} \\ \frac{5}{4} \end{matrix}\middle |{b^{2} x^{4} e^{i \pi }} \right )}}{4 \Gamma \left (\frac{5}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-b*x**2+1)/(-b**2*x**4-1)**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int -\frac{b x^{2} - 1}{\sqrt{-b^{2} x^{4} - 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(b*x^2 - 1)/sqrt(-b^2*x^4 - 1),x, algorithm="giac")
[Out]