\(\int \frac {\sqrt {1+c^2 x^2}}{\sqrt {1-c^2 x^2}} \, dx\) [22]

Optimal result

Integrand size = 28, antiderivative size = 10 \[ \int \frac {\sqrt {1+c^2 x^2}}{\sqrt {1-c^2 x^2}} \, dx=\frac {E(\arcsin (c x)|-1)}{c} \]

[Out]

EllipticE(c*x,I)/c

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.036, Rules used = {435} \[ \int \frac {\sqrt {1+c^2 x^2}}{\sqrt {1-c^2 x^2}} \, dx=\frac {E(\arcsin (c x)|-1)}{c} \]

[In]

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

[Out]

EllipticE[ArcSin[c*x], -1]/c

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}& = \frac {E\left (\left .\sin ^{-1}(c x)\right |-1\right )}{c} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.41 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {1+c^2 x^2}}{\sqrt {1-c^2 x^2}} \, dx=\frac {E(\arcsin (c x)|-1)}{c} \]

[In]

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

[Out]

EllipticE[ArcSin[c*x], -1]/c

Maple [C] (verified)

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

Time = 1.98 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.50

method	result	size
default	\(\frac {E\left (x \,\operatorname {csgn}\left (c \right ) c , i\right ) \operatorname {csgn}\left (c \right )}{c}\)	\(15\)
elliptic	\(\frac {\sqrt {-c^{4} x^{4}+1}\, \left (\frac {\sqrt {-c^{2} x^{2}+1}\, \sqrt {c^{2} x^{2}+1}\, F\left (x \sqrt {c^{2}}, i\right )}{\sqrt {c^{2}}\, \sqrt {-c^{4} x^{4}+1}}-\frac {\sqrt {-c^{2} x^{2}+1}\, \sqrt {c^{2} x^{2}+1}\, \left (F\left (x \sqrt {c^{2}}, i\right )-E\left (x \sqrt {c^{2}}, i\right )\right )}{\sqrt {c^{2}}\, \sqrt {-c^{4} x^{4}+1}}\right )}{\sqrt {c^{2} x^{2}+1}\, \sqrt {-c^{2} x^{2}+1}}\)	\(154\)

[In]

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

[Out]

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. 76 vs. \(2 (9) = 18\).

Time = 0.08 (sec) , antiderivative size = 76, normalized size of antiderivative = 7.60 \[ \int \frac {\sqrt {1+c^2 x^2}}{\sqrt {1-c^2 x^2}} \, dx=-\frac {\sqrt {c^{2} x^{2} + 1} \sqrt {-c^{2} x^{2} + 1} c^{3} - \sqrt {-c^{4}} {\left ({\left (c^{2} + 1\right )} x F(\arcsin \left (\frac {1}{c x}\right )\,|\,-1) - x E(\arcsin \left (\frac {1}{c x}\right )\,|\,-1)\right )}}{c^{5} x} \]

[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*ellip
tic_e(arcsin(1/(c*x)), -1)))/(c^5*x)

Sympy [F]

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

[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)

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

[In]

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

[Out]

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