\(\int \frac {\sqrt {1-c^2 x^2}}{\sqrt {1+c^2 x^2}} \, dx\) [24]

Optimal result

Integrand size = 28, antiderivative size = 23 \[ \int \frac {\sqrt {1-c^2 x^2}}{\sqrt {1+c^2 x^2}} \, dx=-\frac {E(\arcsin (c x)|-1)}{c}+\frac {2 \operatorname {EllipticF}(\arcsin (c x),-1)}{c} \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {434, 435, 254, 227} \[ \int \frac {\sqrt {1-c^2 x^2}}{\sqrt {1+c^2 x^2}} \, dx=\frac {2 \operatorname {EllipticF}(\arcsin (c x),-1)}{c}-\frac {E(\arcsin (c x)|-1)}{c} \]

[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*} \text {integral}& = 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] (verified)

Time = 0.01 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.04 \[ \int \frac {\sqrt {1-c^2 x^2}}{\sqrt {1+c^2 x^2}} \, dx=\frac {E\left (\left .\arcsin \left (\sqrt {-c^2} x\right )\right |-1\right )}{\sqrt {-c^2}} \]

[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]

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.48 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.22

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

[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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 73 vs. \(2 (21) = 42\).

Time = 0.09 (sec) , antiderivative size = 73, normalized size of antiderivative = 3.17 \[ \int \frac {\sqrt {1-c^2 x^2}}{\sqrt {1+c^2 x^2}} \, dx=\frac {\sqrt {c^{2} x^{2} + 1} \sqrt {-c^{2} x^{2} + 1} c^{3} + \sqrt {-c^{4}} {\left ({\left (c^{2} - 1\right )} x F(\arcsin \left (\frac {1}{c x}\right )\,|\,-1) + x E(\arcsin \left (\frac {1}{c x}\right )\,|\,-1)\right )}}{c^{5} x} \]

[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^3 + sqrt(-c^4)*((c^2 - 1)*x*elliptic_f(arcsin(1/(c*x)), -1) + x*ellipt
ic_e(arcsin(1/(c*x)), -1)))/(c^5*x)

Sympy [F]

\[ \int \frac {\sqrt {1-c^2 x^2}}{\sqrt {1+c^2 x^2}} \, dx=\int \frac {\sqrt {- \left (c x - 1\right ) \left (c x + 1\right )}}{\sqrt {c^{2} x^{2} + 1}}\, dx \]

[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)

Maxima [F]

\[ \int \frac {\sqrt {1-c^2 x^2}}{\sqrt {1+c^2 x^2}} \, dx=\int { \frac {\sqrt {-c^{2} x^{2} + 1}}{\sqrt {c^{2} x^{2} + 1}} \,d x } \]

[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)

Giac [F]

\[ \int \frac {\sqrt {1-c^2 x^2}}{\sqrt {1+c^2 x^2}} \, dx=\int { \frac {\sqrt {-c^{2} x^{2} + 1}}{\sqrt {c^{2} x^{2} + 1}} \,d x } \]

[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)

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {1-c^2 x^2}}{\sqrt {1+c^2 x^2}} \, dx=\int \frac {\sqrt {1-c^2\,x^2}}{\sqrt {c^2\,x^2+1}} \,d x \]

[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)