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

Optimal. Leaf size=27 \[ \sqrt{\frac{1}{1-x^2}} \sqrt{1-x^2} \sin ^{-1}(x) \]

[Out]

Sqrt[(1 - x^2)^(-1)]*Sqrt[1 - x^2]*ArcSin[x]

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Rubi [A]  time = 0.0206499, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {6688, 6720, 216} \[ \sqrt{\frac{1}{1-x^2}} \sqrt{1-x^2} \sin ^{-1}(x) \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[(1 + x^2)/(1 - x^4)],x]

[Out]

Sqrt[(1 - x^2)^(-1)]*Sqrt[1 - x^2]*ArcSin[x]

Rule 6688

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

Rule 6720

Int[(u_.)*((a_.)*(v_)^(m_.))^(p_), x_Symbol] :> Dist[(a^IntPart[p]*(a*v^m)^FracPart[p])/v^(m*FracPart[p]), Int
[u*v^(m*p), x], x] /; FreeQ[{a, m, p}, x] &&  !IntegerQ[p] &&  !FreeQ[v, x] &&  !(EqQ[a, 1] && EqQ[m, 1]) &&
!(EqQ[v, x] && EqQ[m, 1])

Rule 216

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

Rubi steps

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

Mathematica [A]  time = 0.0049208, size = 27, normalized size = 1. \[ \sqrt{\frac{1}{1-x^2}} \sqrt{1-x^2} \sin ^{-1}(x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

Sqrt[(1 - x^2)^(-1)]*Sqrt[1 - x^2]*ArcSin[x]

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Maple [A]  time = 0.003, size = 30, normalized size = 1.1 \begin{align*} \sqrt{- \left ({x}^{2}-1 \right ) ^{-1}}\sqrt{{x}^{2}-1}\ln \left ( x+\sqrt{{x}^{2}-1} \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-1/(x^2-1))^(1/2)*(x^2-1)^(1/2)*ln(x+(x^2-1)^(1/2))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.47535, size = 65, normalized size = 2.41 \begin{align*} 2 \, \arctan \left (\frac{{\left (x^{2} - 1\right )} \sqrt{-\frac{1}{x^{2} - 1}} + 1}{x}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*arctan(((x^2 - 1)*sqrt(-1/(x^2 - 1)) + 1)/x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt((x**2 + 1)/(1 - x**4)), x)

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Giac [A]  time = 1.13012, size = 14, normalized size = 0.52 \begin{align*} -\arcsin \left (x\right ) \mathrm{sgn}\left (x^{2} - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-arcsin(x)*sgn(x^2 - 1)