3.871 \(\int \frac {\sqrt {1-x^2}}{\sqrt {1-x^4}} \, dx\)

Optimal. Leaf size=2 \[ \sinh ^{-1}(x) \]

[Out]

arcsinh(x)

________________________________________________________________________________________

Rubi [A]  time = 0.00, antiderivative size = 2, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {26, 215} \[ \sinh ^{-1}(x) \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[1 - x^2]/Sqrt[1 - x^4],x]

[Out]

ArcSinh[x]

Rule 26

Int[(u_.)*((a_) + (b_.)*(x_)^(n_.))^(m_.)*((c_) + (d_.)*(x_)^(j_))^(p_.), x_Symbol] :> Dist[(-(b^2/d))^m, Int[
u/(a - b*x^n)^m, x], x] /; FreeQ[{a, b, c, d, m, n, p}, x] && EqQ[j, 2*n] && EqQ[p, -m] && EqQ[b^2*c + a^2*d,
0] && GtQ[a, 0] && LtQ[d, 0]

Rule 215

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[(Rt[b, 2]*x)/Sqrt[a]]/Rt[b, 2], x] /; FreeQ[{a, b},
x] && GtQ[a, 0] && PosQ[b]

Rubi steps

\begin {align*} \int \frac {\sqrt {1-x^2}}{\sqrt {1-x^4}} \, dx &=\int \frac {1}{\sqrt {1+x^2}} \, dx\\ &=\sinh ^{-1}(x)\\ \end {align*}

________________________________________________________________________________________

Mathematica [B]  time = 0.02, size = 42, normalized size = 21.00 \[ \log \left (1-x^2\right )-\log \left (x^3+\sqrt {1-x^2} \sqrt {1-x^4}-x\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[1 - x^2]/Sqrt[1 - x^4],x]

[Out]

Log[1 - x^2] - Log[-x + x^3 + Sqrt[1 - x^2]*Sqrt[1 - x^4]]

________________________________________________________________________________________

fricas [B]  time = 0.44, size = 81, normalized size = 40.50 \[ -\frac {1}{2} \, \log \left (\frac {x^{3} + \sqrt {-x^{4} + 1} \sqrt {-x^{2} + 1} - x}{x^{3} - x}\right ) + \frac {1}{2} \, \log \left (-\frac {x^{3} - \sqrt {-x^{4} + 1} \sqrt {-x^{2} + 1} - x}{x^{3} - x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x^2+1)^(1/2)/(-x^4+1)^(1/2),x, algorithm="fricas")

[Out]

-1/2*log((x^3 + sqrt(-x^4 + 1)*sqrt(-x^2 + 1) - x)/(x^3 - x)) + 1/2*log(-(x^3 - sqrt(-x^4 + 1)*sqrt(-x^2 + 1)
- x)/(x^3 - x))

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {-x^{2} + 1}}{\sqrt {-x^{4} + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x^2+1)^(1/2)/(-x^4+1)^(1/2),x, algorithm="giac")

[Out]

integrate(sqrt(-x^2 + 1)/sqrt(-x^4 + 1), x)

________________________________________________________________________________________

maple [B]  time = 0.01, size = 29, normalized size = 14.50 \[ \frac {\sqrt {-x^{4}+1}\, \arcsinh \relax (x )}{\sqrt {-x^{2}+1}\, \sqrt {x^{2}+1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-x^2+1)^(1/2)/(-x^4+1)^(1/2),x)

[Out]

1/(-x^2+1)^(1/2)/(x^2+1)^(1/2)*(-x^4+1)^(1/2)*arcsinh(x)

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {-x^{2} + 1}}{\sqrt {-x^{4} + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x^2+1)^(1/2)/(-x^4+1)^(1/2),x, algorithm="maxima")

[Out]

integrate(sqrt(-x^2 + 1)/sqrt(-x^4 + 1), x)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.50 \[ \int \frac {\sqrt {1-x^2}}{\sqrt {1-x^4}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1 - x^2)^(1/2)/(1 - x^4)^(1/2),x)

[Out]

int((1 - x^2)^(1/2)/(1 - x^4)^(1/2), x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x**2+1)**(1/2)/(-x**4+1)**(1/2),x)

[Out]

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

________________________________________________________________________________________