Optimal. Leaf size=89 \[ \frac{\left (b x^2+1\right ) \sqrt{\frac{b^2 x^4+1}{\left (b x^2+1\right )^2}} E\left (2 \tan ^{-1}\left (\sqrt{b} x\right )|\frac{1}{2}\right )}{\sqrt{b} \sqrt{b^2 x^4+1}}-\frac{x \sqrt{b^2 x^4+1}}{b x^2+1} \]
[Out]
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Rubi [A] time = 0.0517291, antiderivative size = 89, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045 \[ \frac{\left (b x^2+1\right ) \sqrt{\frac{b^2 x^4+1}{\left (b x^2+1\right )^2}} E\left (2 \tan ^{-1}\left (\sqrt{b} x\right )|\frac{1}{2}\right )}{\sqrt{b} \sqrt{b^2 x^4+1}}-\frac{x \sqrt{b^2 x^4+1}}{b x^2+1} \]
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 = 7.22608, size = 76, normalized size = 0.85 \[ - \frac{x \sqrt{b^{2} x^{4} + 1}}{b x^{2} + 1} + \frac{\sqrt{\frac{b^{2} x^{4} + 1}{\left (b x^{2} + 1\right )^{2}}} \left (b x^{2} + 1\right ) E\left (2 \operatorname{atan}{\left (\sqrt{b} x \right )}\middle | \frac{1}{2}\right )}{\sqrt{b} \sqrt{b^{2} x^{4} + 1}} \]
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 = 0.0644465, size = 52, normalized size = 0.58 \[ -\frac{E\left (\left .i \sinh ^{-1}\left (\sqrt{i b} x\right )\right |-1\right )-(1-i) F\left (\left .i \sinh ^{-1}\left (\sqrt{i b} x\right )\right |-1\right )}{\sqrt{i b}} \]
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 = 0.019, size = 120, 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. \[{\rm integral}\left (-\frac{b x^{2} - 1}{\sqrt{b^{2} x^{4} + 1}}, x\right ) \]
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 = 3.86314, size = 66, normalized size = 0.74 \[ - \frac{b x^{3} \Gamma \left (\frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{3}{4} \\ \frac{7}{4} \end{matrix}\middle |{b^{2} x^{4} e^{i \pi }} \right )}}{4 \Gamma \left (\frac{7}{4}\right )} + \frac{x \Gamma \left (\frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{4}, \frac{1}{2} \\ \frac{5}{4} \end{matrix}\middle |{b^{2} x^{4} e^{i \pi }} \right )}}{4 \Gamma \left (\frac{5}{4}\right )} \]
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]