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

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

[Out]

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

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Rubi [A]  time = 0.0260898, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {423, 424, 248, 221} \[ \frac{2 F\left (\left .\sin ^{-1}(c x)\right |-1\right )}{c}-\frac{E\left (\left .\sin ^{-1}(c x)\right |-1\right )}{c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 423

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Dist[b/d, Int[Sqrt[c + d*x^2]/Sqrt[a + b
*x^2], x], x] - Dist[(b*c - a*d)/d, Int[1/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]), x], x] /; FreeQ[{a, b, c, d}, x]
&& PosQ[d/c] && NegQ[b/a]

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]

Rule 248

Int[((a1_.) + (b1_.)*(x_)^(n_))^(p_.)*((a2_.) + (b2_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[(a1*a2 + b1*b2*x^(2*
n))^p, x] /; FreeQ[{a1, b1, a2, b2, n, p}, x] && EqQ[a2*b1 + a1*b2, 0] && (IntegerQ[p] || (GtQ[a1, 0] && GtQ[a
2, 0]))

Rule 221

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Simp[EllipticF[ArcSin[(Rt[-b, 4]*x)/Rt[a, 4]], -1]/(Rt[a, 4]*Rt[
-b, 4]), x] /; FreeQ[{a, b}, x] && NegQ[b/a] && GtQ[a, 0]

Rubi steps

\begin{align*} \int \frac{\sqrt{1-c^2 x^2}}{\sqrt{1+c^2 x^2}} \, dx &=2 \int \frac{1}{\sqrt{1-c^2 x^2} \sqrt{1+c^2 x^2}} \, dx-\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}+2 \int \frac{1}{\sqrt{1-c^4 x^4}} \, dx\\ &=-\frac{E\left (\left .\sin ^{-1}(c x)\right |-1\right )}{c}+\frac{2 F\left (\left .\sin ^{-1}(c x)\right |-1\right )}{c}\\ \end{align*}

Mathematica [A]  time = 0.0075945, size = 24, normalized size = 1.04 \[ \frac{E\left (\left .\sin ^{-1}\left (\sqrt{-c^2} x\right )\right |-1\right )}{\sqrt{-c^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

EllipticE[ArcSin[Sqrt[-c^2]*x], -1]/Sqrt[-c^2]

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Maple [C]  time = 0.049, size = 28, normalized size = 1.2 \begin{align*}{\frac{ \left ( 2\,{\it EllipticF} \left ( x{\it csgn} \left ( c \right ) c,i \right ) -{\it EllipticE} \left ( x{\it csgn} \left ( c \right ) c,i \right ) \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]

(2*EllipticF(x*csgn(c)*c,I)-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}}, 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), x)

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