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

Optimal. Leaf size=10 \[ \frac{E\left (\left .\sin ^{-1}(c x)\right |-1\right )}{c} \]

[Out]

EllipticE[ArcSin[c*x], -1]/c

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Rubi [A]  time = 0.0093696, antiderivative size = 10, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.036, Rules used = {424} \[ \frac{E\left (\left .\sin ^{-1}(c x)\right |-1\right )}{c} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[1 + c^2*x^2]/Sqrt[1 - c^2*x^2],x]

[Out]

EllipticE[ArcSin[c*x], -1]/c

Rule 424

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[(Sqrt[a]*EllipticE[ArcSin[Rt[-(d/c)
, 2]*x], (b*c)/(a*d)])/(Sqrt[c]*Rt[-(d/c), 2]), x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[
a, 0]

Rubi steps

\begin{align*} \int \frac{\sqrt{1+c^2 x^2}}{\sqrt{1-c^2 x^2}} \, dx &=\frac{E\left (\left .\sin ^{-1}(c x)\right |-1\right )}{c}\\ \end{align*}

Mathematica [A]  time = 0.0071131, size = 10, normalized size = 1. \[ \frac{E\left (\left .\sin ^{-1}(c x)\right |-1\right )}{c} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[1 + c^2*x^2]/Sqrt[1 - c^2*x^2],x]

[Out]

EllipticE[ArcSin[c*x], -1]/c

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Maple [C]  time = 0.063, size = 15, normalized size = 1.5 \begin{align*}{\frac{{\it EllipticE} \left ( x{\it csgn} \left ( c \right ) c,i \right ){\it csgn} \left ( c \right ) }{c}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c^2*x^2+1)^(1/2)/(-c^2*x^2+1)^(1/2),x)

[Out]

EllipticE(x*csgn(c)*c,I)*csgn(c)/c

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c^2*x^2+1)^(1/2)/(-c^2*x^2+1)^(1/2),x, algorithm="maxima")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c^2*x^2+1)^(1/2)/(-c^2*x^2+1)^(1/2),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{c^{2} x^{2} + 1}}{\sqrt{- \left (c x - 1\right ) \left (c x + 1\right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(c**2*x**2 + 1)/sqrt(-(c*x - 1)*(c*x + 1)), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{c^{2} x^{2} + 1}}{\sqrt{-c^{2} x^{2} + 1}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c^2*x^2+1)^(1/2)/(-c^2*x^2+1)^(1/2),x, algorithm="giac")

[Out]

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