3.2.91 \(\int \frac {\sqrt {1-x^2}}{\sqrt {1+x^2}} \, dx\) [191]

Optimal. Leaf size=13 \[ -E\left (\left .\sin ^{-1}(x)\right |-1\right )+2 F\left (\left .\sin ^{-1}(x)\right |-1\right ) \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.01, antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {434, 435, 254, 227} \begin {gather*} 2 F(\text {ArcSin}(x)|-1)-E(\text {ArcSin}(x)|-1) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[1 - x^2]/Sqrt[1 + x^2],x]

[Out]

-EllipticE[ArcSin[x], -1] + 2*EllipticF[ArcSin[x], -1]

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

________________________________________________________________________________________

Mathematica [C] Result contains complex when optimal does not.
time = 0.28, size = 12, normalized size = 0.92 \begin {gather*} -i E\left (\left .i \sinh ^{-1}(x)\right |-1\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-I)*EllipticE[I*ArcSinh[x], -1]

________________________________________________________________________________________

Maple [A]
time = 0.08, size = 14, normalized size = 1.08

method	result	size
default	\(-\EllipticE \left (x , i\right )+2 \EllipticF \left (x , i\right )\)	\(14\)
elliptic	\(\frac {\sqrt {-x^{4}+1}\, \left (\frac {\sqrt {-x^{2}+1}\, \sqrt {x^{2}+1}\, \EllipticF \left (x , i\right )}{\sqrt {-x^{4}+1}}+\frac {\sqrt {-x^{2}+1}\, \sqrt {x^{2}+1}\, \left (\EllipticF \left (x , i\right )-\EllipticE \left (x , i\right )\right )}{\sqrt {-x^{4}+1}}\right )}{\sqrt {x^{2}+1}\, \sqrt {-x^{2}+1}}\)	\(95\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

Fricas [A]
time = 0.31, size = 20, normalized size = 1.54 \begin {gather*} \frac {\sqrt {x^{2} + 1} \sqrt {-x^{2} + 1}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(x^2 + 1)*sqrt(-x^2 + 1)/x

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.08 \begin {gather*} \int \frac {\sqrt {1-x^2}}{\sqrt {x^2+1}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________