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

Optimal. Leaf size=23 \[ \frac{2 F\left (\left .\sin ^{-1}(c x)\right |-1\right )}{c}-\frac{E\left (\left .\sin ^{-1}(c x)\right |-1\right )}{c} \]

[Out]

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

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Rubi [A]  time = 0.112811, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{2 F\left (\left .\sin ^{-1}(c x)\right |-1\right )}{c}-\frac{E\left (\left .\sin ^{-1}(c x)\right |-1\right )}{c} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 27.2071, size = 20, normalized size = 0.87 \[ - \frac{E\left (\operatorname{asin}{\left (c x \right )}\middle | -1\right )}{c} + \frac{2 F\left (\operatorname{asin}{\left (c x \right )}\middle | -1\right )}{c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((-c**2*x**2+1)/(-c**4*x**4+1)**(1/2),x)

[Out]

-elliptic_e(asin(c*x), -1)/c + 2*elliptic_f(asin(c*x), -1)/c

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Mathematica [C]  time = 0.055079, size = 52, normalized size = 2.26 \[ \frac{i \left (E\left (\left .i \sinh ^{-1}\left (\sqrt{-c^2} x\right )\right |-1\right )-2 F\left (\left .i \sinh ^{-1}\left (\sqrt{-c^2} x\right )\right |-1\right )\right )}{\sqrt{-c^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.01, size = 117, normalized size = 5.1 \[{1\sqrt{-{c}^{2}{x}^{2}+1}\sqrt{{c}^{2}{x}^{2}+1}{\it EllipticF} \left ( x\sqrt{{c}^{2}},i \right ){\frac{1}{\sqrt{{c}^{2}}}}{\frac{1}{\sqrt{-{c}^{4}{x}^{4}+1}}}}+{1\sqrt{-{c}^{2}{x}^{2}+1}\sqrt{{c}^{2}{x}^{2}+1} \left ({\it EllipticF} \left ( x\sqrt{{c}^{2}},i \right ) -{\it EllipticE} \left ( x\sqrt{{c}^{2}},i \right ) \right ){\frac{1}{\sqrt{{c}^{2}}}}{\frac{1}{\sqrt{-{c}^{4}{x}^{4}+1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-c^2*x^2+1)/(-c^4*x^4+1)^(1/2),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ -\int \frac{c^{2} x^{2} - 1}{\sqrt{-c^{4} x^{4} + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(c^2*x^2 - 1)/sqrt(-c^4*x^4 + 1),x, algorithm="maxima")

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (-\frac{c^{2} x^{2} - 1}{\sqrt{-c^{4} x^{4} + 1}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(c^2*x^2 - 1)/sqrt(-c^4*x^4 + 1),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 4.14197, size = 71, normalized size = 3.09 \[ - \frac{c^{2} x^{3} \Gamma \left (\frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{3}{4} \\ \frac{7}{4} \end{matrix}\middle |{c^{4} x^{4} e^{2 i \pi }} \right )}}{4 \Gamma \left (\frac{7}{4}\right )} + \frac{x \Gamma \left (\frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{4}, \frac{1}{2} \\ \frac{5}{4} \end{matrix}\middle |{c^{4} x^{4} e^{2 i \pi }} \right )}}{4 \Gamma \left (\frac{5}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-c**2*x**2+1)/(-c**4*x**4+1)**(1/2),x)

[Out]

-c**2*x**3*gamma(3/4)*hyper((1/2, 3/4), (7/4,), c**4*x**4*exp_polar(2*I*pi))/(4*
gamma(7/4)) + x*gamma(1/4)*hyper((1/4, 1/2), (5/4,), c**4*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. \[ \int -\frac{c^{2} x^{2} - 1}{\sqrt{-c^{4} x^{4} + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(c^2*x^2 - 1)/sqrt(-c^4*x^4 + 1),x, algorithm="giac")

[Out]

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