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

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

[Out]

ArcSinh[x]

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Rubi [A]  time = 0.0011891, antiderivative size = 2, normalized size of antiderivative = 1., 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 = 0.0233328, size = 42, normalized size = 21. \[ \log \left (1-x^2\right )-\log \left (x^3+\sqrt{1-x^2} \sqrt{1-x^4}-x\right ) \]

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]]

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Maple [B]  time = 0.009, size = 29, normalized size = 14.5 \begin{align*}{{\it Arcsinh} \left ( x \right ) \sqrt{-{x}^{4}+1}{\frac{1}{\sqrt{-{x}^{2}+1}}}{\frac{1}{\sqrt{{x}^{2}+1}}}} \end{align*}

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)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{-x^{2} + 1}}{\sqrt{-x^{4} + 1}}\,{d x} \end{align*}

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)

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Fricas [B]  time = 1.46115, size = 171, normalized size = 85.5 \begin{align*} -\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 ) \end{align*}

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))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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 \end{align*}

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)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{-x^{2} + 1}}{\sqrt{-x^{4} + 1}}\,{d x} \end{align*}

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)