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

Optimal. Leaf size=90 \[ \frac{x \sqrt{-b^2 x^4-1}}{b x^2+1}+\frac{\left (b x^2+1\right ) \sqrt{\frac{b^2 x^4+1}{\left (b x^2+1\right )^2}} E\left (2 \tan ^{-1}\left (\sqrt{b} x\right )|\frac{1}{2}\right )}{\sqrt{b} \sqrt{-b^2 x^4-1}} \]

[Out]

(x*Sqrt[-1 - b^2*x^4])/(1 + b*x^2) + ((1 + b*x^2)*Sqrt[(1 + b^2*x^4)/(1 + b*x^2)
^2]*EllipticE[2*ArcTan[Sqrt[b]*x], 1/2])/(Sqrt[b]*Sqrt[-1 - b^2*x^4])

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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]

(x*Sqrt[-1 - b^2*x^4])/(1 + b*x^2) + ((1 + b*x^2)*Sqrt[(1 + b^2*x^4)/(1 + b*x^2)
^2]*EllipticE[2*ArcTan[Sqrt[b]*x], 1/2])/(Sqrt[b]*Sqrt[-1 - b^2*x^4])

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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]

x*sqrt(-b**2*x**4 - 1)/(b*x**2 + 1) + sqrt((b**2*x**4 + 1)/(b*x**2 + 1)**2)*(b*x
**2 + 1)*elliptic_e(2*atan(sqrt(b)*x), 1/2)/(sqrt(b)*sqrt(-b**2*x**4 - 1))

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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]

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

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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]

I/(-I*b)^(1/2)*(1+I*b*x^2)^(1/2)*(1-I*b*x^2)^(1/2)/(-b^2*x^4-1)^(1/2)*(EllipticF
(x*(-I*b)^(1/2),I)-EllipticE(x*(-I*b)^(1/2),I))+1/(-I*b)^(1/2)*(1+I*b*x^2)^(1/2)
*(1-I*b*x^2)^(1/2)/(-b^2*x^4-1)^(1/2)*EllipticF(x*(-I*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. \[ \frac{b x{\rm integral}\left (-\frac{\sqrt{-b^{2} x^{4} - 1}{\left (b x^{2} - 1\right )}}{b^{3} x^{6} + b x^{2}}, x\right ) + \sqrt{-b^{2} x^{4} - 1}}{b x} \]

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]

(b*x*integral(-sqrt(-b^2*x^4 - 1)*(b*x^2 - 1)/(b^3*x^6 + b*x^2), x) + sqrt(-b^2*
x^4 - 1))/(b*x)

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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]

I*b*x**3*gamma(3/4)*hyper((1/2, 3/4), (7/4,), b**2*x**4*exp_polar(I*pi))/(4*gamm
a(7/4)) - I*x*gamma(1/4)*hyper((1/4, 1/2), (5/4,), b**2*x**4*exp_polar(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)