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

Optimal. Leaf size=23 \[ \frac {1}{2} \sqrt {1-x^2} x+\frac {1}{2} \sin ^{-1}(x) \]

[Out]

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

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Rubi [A]  time = 0.00, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {26, 195, 216} \[ \frac {1}{2} \sqrt {1-x^2} x+\frac {1}{2} \sin ^{-1}(x) \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[1 - x^4]/Sqrt[1 + x^2],x]

[Out]

(x*Sqrt[1 - x^2])/2 + ArcSin[x]/2

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 195

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^p)/(n*p + 1), x] + Dist[(a*n*p)/(n*p + 1),
Int[(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || (EqQ[n, 2
] && IntegerQ[4*p]) || (EqQ[n, 2] && IntegerQ[3*p]) || LtQ[Denominator[p + 1/n], Denominator[p]])

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

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Mathematica [B]  time = 0.04, size = 50, normalized size = 2.17 \[ \frac {1}{2} \left (\frac {\sqrt {1-x^4} x}{\sqrt {x^2+1}}+\tan ^{-1}\left (\frac {x \sqrt {x^2+1}}{\sqrt {1-x^4}}\right )\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 0.45, size = 60, normalized size = 2.61 \[ \frac {\sqrt {-x^{4} + 1} \sqrt {x^{2} + 1} x - {\left (x^{2} + 1\right )} \arctan \left (\frac {\sqrt {-x^{4} + 1} \sqrt {x^{2} + 1}}{x^{3} + x}\right )}{2 \, {\left (x^{2} + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 0.01, size = 42, normalized size = 1.83 \[ \frac {\sqrt {-x^{4}+1}\, \left (\sqrt {-x^{2}+1}\, x +\arcsin \relax (x )\right )}{2 \sqrt {x^{2}+1}\, \sqrt {-x^{2}+1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \[ \int \frac {\sqrt {1-x^4}}{\sqrt {x^2+1}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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