3.9.71 \(\int \frac {\sqrt {1-x^2}}{\sqrt {1-x^4}} \, dx\) [871]

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

[Out]

arcsinh(x)

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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, 221} \begin {gather*} \sinh ^{-1}(x) \end {gather*}

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 221

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*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(42\) vs. \(2(2)=4\).
time = 0.52, size = 42, normalized size = 21.00 \begin {gather*} \log \left (1-x^2\right )-\log \left (-x+x^3+\sqrt {1-x^2} \sqrt {1-x^4}\right ) \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. \(28\) vs. \(2(2)=4\).
time = 0.50, size = 29, normalized size = 14.50

method	result	size
default	\(\frac {\sqrt {-x^{4}+1}\, \arcsinh \left (x \right )}{\sqrt {-x^{2}+1}\, \sqrt {x^{2}+1}}\)	\(29\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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 = 0.00, 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^4+1)^(1/2),x, algorithm="maxima")

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 81 vs. \(2 (2) = 4\).
time = 0.36, size = 81, normalized size = 40.50 \begin {gather*} -\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 {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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 {gather*}

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 = 0.00, 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^4+1)^(1/2),x, algorithm="giac")

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.50 \begin {gather*} \int \frac {\sqrt {1-x^2}}{\sqrt {1-x^4}} \,d x \end {gather*}

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)

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