3.23 \(\int \frac{1-b x^2}{\sqrt{1-b^2 x^4}} \, dx\)

Optimal. Leaf size=35 \[ \frac{2 F\left (\left .\sin ^{-1}\left (\sqrt{b} x\right )\right |-1\right )}{\sqrt{b}}-\frac{E\left (\left .\sin ^{-1}\left (\sqrt{b} x\right )\right |-1\right )}{\sqrt{b}} \]

[Out]

-(EllipticE[ArcSin[Sqrt[b]*x], -1]/Sqrt[b]) + (2*EllipticF[ArcSin[Sqrt[b]*x], -1
])/Sqrt[b]

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Rubi [A]  time = 0.112191, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217 \[ \frac{2 F\left (\left .\sin ^{-1}\left (\sqrt{b} x\right )\right |-1\right )}{\sqrt{b}}-\frac{E\left (\left .\sin ^{-1}\left (\sqrt{b} x\right )\right |-1\right )}{\sqrt{b}} \]

Antiderivative was successfully verified.

[In]  Int[(1 - b*x^2)/Sqrt[1 - b^2*x^4],x]

[Out]

-(EllipticE[ArcSin[Sqrt[b]*x], -1]/Sqrt[b]) + (2*EllipticF[ArcSin[Sqrt[b]*x], -1
])/Sqrt[b]

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Rubi in Sympy [A]  time = 26.392, size = 34, normalized size = 0.97 \[ - \frac{E\left (\operatorname{asin}{\left (\sqrt{b} x \right )}\middle | -1\right )}{\sqrt{b}} + \frac{2 F\left (\operatorname{asin}{\left (\sqrt{b} x \right )}\middle | -1\right )}{\sqrt{b}} \]

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]

-elliptic_e(asin(sqrt(b)*x), -1)/sqrt(b) + 2*elliptic_f(asin(sqrt(b)*x), -1)/sqr
t(b)

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Mathematica [C]  time = 0.0503164, size = 46, normalized size = 1.31 \[ \frac{i \left (E\left (\left .i \sinh ^{-1}\left (\sqrt{-b} x\right )\right |-1\right )-2 F\left (\left .i \sinh ^{-1}\left (\sqrt{-b} x\right )\right |-1\right )\right )}{\sqrt{-b}} \]

Antiderivative was successfully verified.

[In]  Integrate[(1 - b*x^2)/Sqrt[1 - b^2*x^4],x]

[Out]

(I*(EllipticE[I*ArcSinh[Sqrt[-b]*x], -1] - 2*EllipticF[I*ArcSinh[Sqrt[-b]*x], -1
]))/Sqrt[-b]

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Maple [B]  time = 0.008, size = 99, normalized size = 2.8 \[{1\sqrt{-b{x}^{2}+1}\sqrt{b{x}^{2}+1} \left ({\it EllipticF} \left ( x\sqrt{b},i \right ) -{\it EllipticE} \left ( x\sqrt{b},i \right ) \right ){\frac{1}{\sqrt{b}}}{\frac{1}{\sqrt{-{b}^{2}{x}^{4}+1}}}}+{1\sqrt{-b{x}^{2}+1}\sqrt{b{x}^{2}+1}{\it EllipticF} \left ( x\sqrt{b},i \right ){\frac{1}{\sqrt{b}}}{\frac{1}{\sqrt{-{b}^{2}{x}^{4}+1}}}} \]

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]

1/b^(1/2)*(-b*x^2+1)^(1/2)*(b*x^2+1)^(1/2)/(-b^2*x^4+1)^(1/2)*(EllipticF(x*b^(1/
2),I)-EllipticE(x*b^(1/2),I))+1/b^(1/2)*(-b*x^2+1)^(1/2)*(b*x^2+1)^(1/2)/(-b^2*x
^4+1)^(1/2)*EllipticF(x*b^(1/2),I)

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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]

-integrate((b*x^2 - 1)/sqrt(-b^2*x^4 + 1), x)

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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]

integral(-(b*x^2 - 1)/sqrt(-b^2*x^4 + 1), x)

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Sympy [A]  time = 4.03753, size = 70, normalized size = 2. \[ - \frac{b x^{3} \Gamma \left (\frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{3}{4} \\ \frac{7}{4} \end{matrix}\middle |{b^{2} x^{4} e^{2 i \pi }} \right )}}{4 \Gamma \left (\frac{7}{4}\right )} + \frac{x \Gamma \left (\frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{4}, \frac{1}{2} \\ \frac{5}{4} \end{matrix}\middle |{b^{2} x^{4} e^{2 i \pi }} \right )}}{4 \Gamma \left (\frac{5}{4}\right )} \]

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]

-b*x**3*gamma(3/4)*hyper((1/2, 3/4), (7/4,), b**2*x**4*exp_polar(2*I*pi))/(4*gam
ma(7/4)) + x*gamma(1/4)*hyper((1/4, 1/2), (5/4,), b**2*x**4*exp_polar(2*I*pi))/(
4*gamma(5/4))

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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]

integrate(-(b*x^2 - 1)/sqrt(-b^2*x^4 + 1), x)