3.9.69 \(\int \frac {\sqrt {1+x^2}}{\sqrt {1-x^4}} \, dx\) [869]

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

[Out]

arcsin(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 = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {26, 222} \begin {gather*} \text {ArcSin}(x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[1 + x^2]/Sqrt[1 - x^4],x]

[Out]

ArcSin[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 222

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

________________________________________________________________________________________

Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(32\) vs. \(2(2)=4\).
time = 0.54, size = 32, normalized size = 16.00 \begin {gather*} -\tan ^{-1}\left (\frac {x \sqrt {1+x^2} \sqrt {1-x^4}}{-1+x^4}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-ArcTan[(x*Sqrt[1 + x^2]*Sqrt[1 - x^4])/(-1 + x^4)]

________________________________________________________________________________________

Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(28\) vs. \(2(2)=4\).
time = 0.57, size = 29, normalized size = 14.50

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

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

________________________________________________________________________________________

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)

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 27 vs. \(2 (2) = 4\).
time = 0.36, size = 27, normalized size = 13.50 \begin {gather*} -\arctan \left (\frac {\sqrt {-x^{4} + 1} \sqrt {x^{2} + 1}}{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]

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

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {x^{2} + 1}}{\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**2 + 1)/sqrt(-(x - 1)*(x + 1)*(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^4+1)^(1/2),x, algorithm="giac")

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________