3.3.94 \(\int \frac {\sqrt {1-c^2 x^2}}{\sqrt {1+c^2 x^2}} \, dx\) [294]

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

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 23, normalized size of antiderivative = 1.00, 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 = {434, 435, 254, 227} \begin {gather*} \frac {2 F(\text {ArcSin}(c x)|-1)}{c}-\frac {E(\text {ArcSin}(c x)|-1)}{c} \end {gather*}

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 227

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]

Rule 254

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 434

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 435

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[(Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*Ell
ipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d))], 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 &=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*}

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Mathematica [A]
time = 0.49, size = 24, normalized size = 1.04 \begin {gather*} \frac {E\left (\left .\sin ^{-1}\left (\sqrt {-c^2} x\right )\right |-1\right )}{\sqrt {-c^2}} \end {gather*}

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] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.09, size = 28, normalized size = 1.22

method	result	size
default	\(\frac {\left (2 \EllipticF \left (x \,\mathrm {csgn}\left (c \right ) c , i\right )-\EllipticE \left (x \,\mathrm {csgn}\left (c \right ) c , i\right )\right ) \mathrm {csgn}\left (c \right )}{c}\)	\(28\)
elliptic	\(\frac {\sqrt {-c^{4} x^{4}+1}\, \left (\frac {\sqrt {-c^{2} x^{2}+1}\, \sqrt {c^{2} x^{2}+1}\, \EllipticF \left (x \sqrt {c^{2}}, i\right )}{\sqrt {c^{2}}\, \sqrt {-c^{4} x^{4}+1}}+\frac {\sqrt {-c^{2} x^{2}+1}\, \sqrt {c^{2} x^{2}+1}\, \left (\EllipticF \left (x \sqrt {c^{2}}, i\right )-\EllipticE \left (x \sqrt {c^{2}}, i\right )\right )}{\sqrt {c^{2}}\, \sqrt {-c^{4} x^{4}+1}}\right )}{\sqrt {-c^{2} x^{2}+1}\, \sqrt {c^{2} x^{2}+1}}\)	\(153\)

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,method=_RETURNVERBOSE)

[Out]

(2*EllipticF(x*csgn(c)*c,I)-EllipticE(x*csgn(c)*c,I))*csgn(c)/c

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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 [A]
time = 0.22, size = 30, normalized size = 1.30 \begin {gather*} \frac {\sqrt {c^{2} x^{2} + 1} \sqrt {-c^{2} x^{2} + 1}}{c^{2} x} \end {gather*}

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]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {- \left (c x - 1\right ) \left (c x + 1\right )}}{\sqrt {c^{2} x^{2} + 1}}\, dx \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {\sqrt {1-c^2\,x^2}}{\sqrt {c^2\,x^2+1}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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