3.383 \(\int \frac{\sqrt{1-x^2}}{a+b x^2+c x^4} \, dx\)

Optimal. Leaf size=220 \[ \frac{\sqrt{-\sqrt{b^2-4 a c}+b+2 c} \tan ^{-1}\left (\frac{x \sqrt{-\sqrt{b^2-4 a c}+b+2 c}}{\sqrt{1-x^2} \sqrt{b-\sqrt{b^2-4 a c}}}\right )}{\sqrt{b^2-4 a c} \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{\sqrt{\sqrt{b^2-4 a c}+b+2 c} \tan ^{-1}\left (\frac{x \sqrt{\sqrt{b^2-4 a c}+b+2 c}}{\sqrt{1-x^2} \sqrt{\sqrt{b^2-4 a c}+b}}\right )}{\sqrt{b^2-4 a c} \sqrt{\sqrt{b^2-4 a c}+b}} \]

[Out]

(Sqrt[b + 2*c - Sqrt[b^2 - 4*a*c]]*ArcTan[(Sqrt[b + 2*c - Sqrt[b^2 - 4*a*c]]*x)/(Sqrt[b - Sqrt[b^2 - 4*a*c]]*S
qrt[1 - x^2])])/(Sqrt[b^2 - 4*a*c]*Sqrt[b - Sqrt[b^2 - 4*a*c]]) - (Sqrt[b + 2*c + Sqrt[b^2 - 4*a*c]]*ArcTan[(S
qrt[b + 2*c + Sqrt[b^2 - 4*a*c]]*x)/(Sqrt[b + Sqrt[b^2 - 4*a*c]]*Sqrt[1 - x^2])])/(Sqrt[b^2 - 4*a*c]*Sqrt[b +
Sqrt[b^2 - 4*a*c]])

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Rubi [A]  time = 0.294096, antiderivative size = 220, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192, Rules used = {1174, 402, 216, 377, 205} \[ \frac{\sqrt{-\sqrt{b^2-4 a c}+b+2 c} \tan ^{-1}\left (\frac{x \sqrt{-\sqrt{b^2-4 a c}+b+2 c}}{\sqrt{1-x^2} \sqrt{b-\sqrt{b^2-4 a c}}}\right )}{\sqrt{b^2-4 a c} \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{\sqrt{\sqrt{b^2-4 a c}+b+2 c} \tan ^{-1}\left (\frac{x \sqrt{\sqrt{b^2-4 a c}+b+2 c}}{\sqrt{1-x^2} \sqrt{\sqrt{b^2-4 a c}+b}}\right )}{\sqrt{b^2-4 a c} \sqrt{\sqrt{b^2-4 a c}+b}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[1 - x^2]/(a + b*x^2 + c*x^4),x]

[Out]

(Sqrt[b + 2*c - Sqrt[b^2 - 4*a*c]]*ArcTan[(Sqrt[b + 2*c - Sqrt[b^2 - 4*a*c]]*x)/(Sqrt[b - Sqrt[b^2 - 4*a*c]]*S
qrt[1 - x^2])])/(Sqrt[b^2 - 4*a*c]*Sqrt[b - Sqrt[b^2 - 4*a*c]]) - (Sqrt[b + 2*c + Sqrt[b^2 - 4*a*c]]*ArcTan[(S
qrt[b + 2*c + Sqrt[b^2 - 4*a*c]]*x)/(Sqrt[b + Sqrt[b^2 - 4*a*c]]*Sqrt[1 - x^2])])/(Sqrt[b^2 - 4*a*c]*Sqrt[b +
Sqrt[b^2 - 4*a*c]])

Rule 1174

Int[((d_) + (e_.)*(x_)^2)^(q_)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{r = Rt[b^2 - 4*a*c, 2]
}, Dist[(2*c)/r, Int[(d + e*x^2)^q/(b - r + 2*c*x^2), x], x] - Dist[(2*c)/r, Int[(d + e*x^2)^q/(b + r + 2*c*x^
2), x], x]] /; FreeQ[{a, b, c, d, e, q}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] &&  !Integ
erQ[q]

Rule 402

Int[((a_) + (b_.)*(x_)^2)^(p_.)/((c_) + (d_.)*(x_)^2), x_Symbol] :> Dist[b/d, Int[(a + b*x^2)^(p - 1), x], x]
- Dist[(b*c - a*d)/d, Int[(a + b*x^2)^(p - 1)/(c + d*x^2), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d,
0] && GtQ[p, 0] && (EqQ[p, 1/2] || EqQ[Denominator[p], 4])

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]

Rule 377

Int[((a_) + (b_.)*(x_)^(n_))^(p_)/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Subst[Int[1/(c - (b*c - a*d)*x^n), x]
, x, x/(a + b*x^n)^(1/n)] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[n*p + 1, 0] && IntegerQ[n]

Rule 205

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

Rubi steps

\begin{align*} \int \frac{\sqrt{1-x^2}}{a+b x^2+c x^4} \, dx &=\frac{(2 c) \int \frac{\sqrt{1-x^2}}{b-\sqrt{b^2-4 a c}+2 c x^2} \, dx}{\sqrt{b^2-4 a c}}-\frac{(2 c) \int \frac{\sqrt{1-x^2}}{b+\sqrt{b^2-4 a c}+2 c x^2} \, dx}{\sqrt{b^2-4 a c}}\\ &=\frac{\left (b+2 c-\sqrt{b^2-4 a c}\right ) \int \frac{1}{\sqrt{1-x^2} \left (b-\sqrt{b^2-4 a c}+2 c x^2\right )} \, dx}{\sqrt{b^2-4 a c}}-\frac{\left (b+2 c+\sqrt{b^2-4 a c}\right ) \int \frac{1}{\sqrt{1-x^2} \left (b+\sqrt{b^2-4 a c}+2 c x^2\right )} \, dx}{\sqrt{b^2-4 a c}}\\ &=\frac{\left (b+2 c-\sqrt{b^2-4 a c}\right ) \operatorname{Subst}\left (\int \frac{1}{b-\sqrt{b^2-4 a c}-\left (-b-2 c+\sqrt{b^2-4 a c}\right ) x^2} \, dx,x,\frac{x}{\sqrt{1-x^2}}\right )}{\sqrt{b^2-4 a c}}-\frac{\left (b+2 c+\sqrt{b^2-4 a c}\right ) \operatorname{Subst}\left (\int \frac{1}{b+\sqrt{b^2-4 a c}-\left (-b-2 c-\sqrt{b^2-4 a c}\right ) x^2} \, dx,x,\frac{x}{\sqrt{1-x^2}}\right )}{\sqrt{b^2-4 a c}}\\ &=\frac{\sqrt{b+2 c-\sqrt{b^2-4 a c}} \tan ^{-1}\left (\frac{\sqrt{b+2 c-\sqrt{b^2-4 a c}} x}{\sqrt{b-\sqrt{b^2-4 a c}} \sqrt{1-x^2}}\right )}{\sqrt{b^2-4 a c} \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{\sqrt{b+2 c+\sqrt{b^2-4 a c}} \tan ^{-1}\left (\frac{\sqrt{b+2 c+\sqrt{b^2-4 a c}} x}{\sqrt{b+\sqrt{b^2-4 a c}} \sqrt{1-x^2}}\right )}{\sqrt{b^2-4 a c} \sqrt{b+\sqrt{b^2-4 a c}}}\\ \end{align*}

Mathematica [B]  time = 6.06606, size = 2959, normalized size = 13.45 \[ \text{Result too large to show} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[1 - x^2]/(a + b*x^2 + c*x^4),x]

[Out]

(Log[-(Sqrt[-(b/c) + Sqrt[b^2 - 4*a*c]/c]/Sqrt[2]) + x]/Sqrt[1 + (b/c - Sqrt[b^2 - 4*a*c]/c)/2] - Log[2 - Sqrt
[2]*Sqrt[-(b/c) + Sqrt[b^2 - 4*a*c]/c]*x + 2*Sqrt[1 + (b/c - Sqrt[b^2 - 4*a*c]/c)/2]*Sqrt[1 - x^2]]/Sqrt[1 + (
b/c - Sqrt[b^2 - 4*a*c]/c)/2])/(Sqrt[2]*c*Sqrt[-(b/c) + Sqrt[b^2 - 4*a*c]/c]*(-(Sqrt[-(b/c) - Sqrt[b^2 - 4*a*c
]/c]/Sqrt[2]) + Sqrt[-(b/c) + Sqrt[b^2 - 4*a*c]/c]/Sqrt[2])*(Sqrt[-(b/c) - Sqrt[b^2 - 4*a*c]/c]/Sqrt[2] + Sqrt
[-(b/c) + Sqrt[b^2 - 4*a*c]/c]/Sqrt[2])) - (Sqrt[-(b/c) + Sqrt[b^2 - 4*a*c]/c]*(Log[-(Sqrt[-(b/c) + Sqrt[b^2 -
4*a*c]/c]/Sqrt[2]) + x]/Sqrt[1 + (b/c - Sqrt[b^2 - 4*a*c]/c)/2] - Log[2 - Sqrt[2]*Sqrt[-(b/c) + Sqrt[b^2 - 4*
a*c]/c]*x + 2*Sqrt[1 + (b/c - Sqrt[b^2 - 4*a*c]/c)/2]*Sqrt[1 - x^2]]/Sqrt[1 + (b/c - Sqrt[b^2 - 4*a*c]/c)/2]))
/(2*Sqrt[2]*c*(-(Sqrt[-(b/c) - Sqrt[b^2 - 4*a*c]/c]/Sqrt[2]) + Sqrt[-(b/c) + Sqrt[b^2 - 4*a*c]/c]/Sqrt[2])*(Sq
rt[-(b/c) - Sqrt[b^2 - 4*a*c]/c]/Sqrt[2] + Sqrt[-(b/c) + Sqrt[b^2 - 4*a*c]/c]/Sqrt[2])) - (Log[Sqrt[-(b/c) + S
qrt[b^2 - 4*a*c]/c]/Sqrt[2] + x]/Sqrt[1 + (b/c - Sqrt[b^2 - 4*a*c]/c)/2] - Log[2 + Sqrt[2]*Sqrt[-(b/c) + Sqrt[
b^2 - 4*a*c]/c]*x + 2*Sqrt[1 + (b/c - Sqrt[b^2 - 4*a*c]/c)/2]*Sqrt[1 - x^2]]/Sqrt[1 + (b/c - Sqrt[b^2 - 4*a*c]
/c)/2])/(Sqrt[2]*c*Sqrt[-(b/c) + Sqrt[b^2 - 4*a*c]/c]*(-(Sqrt[-(b/c) - Sqrt[b^2 - 4*a*c]/c]/Sqrt[2]) - Sqrt[-(
b/c) + Sqrt[b^2 - 4*a*c]/c]/Sqrt[2])*(Sqrt[-(b/c) - Sqrt[b^2 - 4*a*c]/c]/Sqrt[2] - Sqrt[-(b/c) + Sqrt[b^2 - 4*
a*c]/c]/Sqrt[2])) + (Sqrt[-(b/c) + Sqrt[b^2 - 4*a*c]/c]*(Log[Sqrt[-(b/c) + Sqrt[b^2 - 4*a*c]/c]/Sqrt[2] + x]/S
qrt[1 + (b/c - Sqrt[b^2 - 4*a*c]/c)/2] - Log[2 + Sqrt[2]*Sqrt[-(b/c) + Sqrt[b^2 - 4*a*c]/c]*x + 2*Sqrt[1 + (b/
c - Sqrt[b^2 - 4*a*c]/c)/2]*Sqrt[1 - x^2]]/Sqrt[1 + (b/c - Sqrt[b^2 - 4*a*c]/c)/2]))/(2*Sqrt[2]*c*(-(Sqrt[-(b/
c) - Sqrt[b^2 - 4*a*c]/c]/Sqrt[2]) - Sqrt[-(b/c) + Sqrt[b^2 - 4*a*c]/c]/Sqrt[2])*(Sqrt[-(b/c) - Sqrt[b^2 - 4*a
*c]/c]/Sqrt[2] - Sqrt[-(b/c) + Sqrt[b^2 - 4*a*c]/c]/Sqrt[2])) + (Log[-(Sqrt[-(b/c) - Sqrt[b^2 - 4*a*c]/c]/Sqrt
[2]) + x]/Sqrt[1 + (b/c + Sqrt[b^2 - 4*a*c]/c)/2] - Log[2 - Sqrt[2]*Sqrt[-(b/c) - Sqrt[b^2 - 4*a*c]/c]*x + 2*S
qrt[1 + (b/c + Sqrt[b^2 - 4*a*c]/c)/2]*Sqrt[1 - x^2]]/Sqrt[1 + (b/c + Sqrt[b^2 - 4*a*c]/c)/2])/(Sqrt[2]*c*Sqrt
[-(b/c) - Sqrt[b^2 - 4*a*c]/c]*(Sqrt[-(b/c) - Sqrt[b^2 - 4*a*c]/c]/Sqrt[2] - Sqrt[-(b/c) + Sqrt[b^2 - 4*a*c]/c
]/Sqrt[2])*(Sqrt[-(b/c) - Sqrt[b^2 - 4*a*c]/c]/Sqrt[2] + Sqrt[-(b/c) + Sqrt[b^2 - 4*a*c]/c]/Sqrt[2])) - (Sqrt[
-(b/c) - Sqrt[b^2 - 4*a*c]/c]*(Log[-(Sqrt[-(b/c) - Sqrt[b^2 - 4*a*c]/c]/Sqrt[2]) + x]/Sqrt[1 + (b/c + Sqrt[b^2
- 4*a*c]/c)/2] - Log[2 - Sqrt[2]*Sqrt[-(b/c) - Sqrt[b^2 - 4*a*c]/c]*x + 2*Sqrt[1 + (b/c + Sqrt[b^2 - 4*a*c]/c
)/2]*Sqrt[1 - x^2]]/Sqrt[1 + (b/c + Sqrt[b^2 - 4*a*c]/c)/2]))/(2*Sqrt[2]*c*(Sqrt[-(b/c) - Sqrt[b^2 - 4*a*c]/c]
/Sqrt[2] - Sqrt[-(b/c) + Sqrt[b^2 - 4*a*c]/c]/Sqrt[2])*(Sqrt[-(b/c) - Sqrt[b^2 - 4*a*c]/c]/Sqrt[2] + Sqrt[-(b/
c) + Sqrt[b^2 - 4*a*c]/c]/Sqrt[2])) - (Log[Sqrt[-(b/c) - Sqrt[b^2 - 4*a*c]/c]/Sqrt[2] + x]/Sqrt[1 + (b/c + Sqr
t[b^2 - 4*a*c]/c)/2] - Log[2 + Sqrt[2]*Sqrt[-(b/c) - Sqrt[b^2 - 4*a*c]/c]*x + 2*Sqrt[1 + (b/c + Sqrt[b^2 - 4*a
*c]/c)/2]*Sqrt[1 - x^2]]/Sqrt[1 + (b/c + Sqrt[b^2 - 4*a*c]/c)/2])/(Sqrt[2]*c*Sqrt[-(b/c) - Sqrt[b^2 - 4*a*c]/c
]*(-(Sqrt[-(b/c) - Sqrt[b^2 - 4*a*c]/c]/Sqrt[2]) - Sqrt[-(b/c) + Sqrt[b^2 - 4*a*c]/c]/Sqrt[2])*(-(Sqrt[-(b/c)
- Sqrt[b^2 - 4*a*c]/c]/Sqrt[2]) + Sqrt[-(b/c) + Sqrt[b^2 - 4*a*c]/c]/Sqrt[2])) + (Sqrt[-(b/c) - Sqrt[b^2 - 4*a
*c]/c]*(Log[Sqrt[-(b/c) - Sqrt[b^2 - 4*a*c]/c]/Sqrt[2] + x]/Sqrt[1 + (b/c + Sqrt[b^2 - 4*a*c]/c)/2] - Log[2 +
Sqrt[2]*Sqrt[-(b/c) - Sqrt[b^2 - 4*a*c]/c]*x + 2*Sqrt[1 + (b/c + Sqrt[b^2 - 4*a*c]/c)/2]*Sqrt[1 - x^2]]/Sqrt[1
+ (b/c + Sqrt[b^2 - 4*a*c]/c)/2]))/(2*Sqrt[2]*c*(-(Sqrt[-(b/c) - Sqrt[b^2 - 4*a*c]/c]/Sqrt[2]) - Sqrt[-(b/c)
+ Sqrt[b^2 - 4*a*c]/c]/Sqrt[2])*(-(Sqrt[-(b/c) - Sqrt[b^2 - 4*a*c]/c]/Sqrt[2]) + Sqrt[-(b/c) + Sqrt[b^2 - 4*a*
c]/c]/Sqrt[2]))

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Maple [C]  time = 0.011, size = 130, normalized size = 0.6 \begin{align*} -{\frac{1}{4}\sum _{{\it \_R}={\it RootOf} \left ( a{{\it \_Z}}^{8}+ \left ( 4\,a+4\,b \right ){{\it \_Z}}^{6}+ \left ( 6\,a+8\,b+16\,c \right ){{\it \_Z}}^{4}+ \left ( 4\,a+4\,b \right ){{\it \_Z}}^{2}+a \right ) }{\frac{{{\it \_R}}^{6}-{{\it \_R}}^{4}-{{\it \_R}}^{2}+1}{{{\it \_R}}^{7}a+3\,{{\it \_R}}^{5}a+3\,{{\it \_R}}^{5}b+3\,{{\it \_R}}^{3}a+4\,{{\it \_R}}^{3}b+8\,{{\it \_R}}^{3}c+{\it \_R}\,a+{\it \_R}\,b}\ln \left ({\frac{1}{x} \left ( \sqrt{-{x}^{2}+1}-1 \right ) }-{\it \_R} \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-x^2+1)^(1/2)/(c*x^4+b*x^2+a),x)

[Out]

-1/4*sum((_R^6-_R^4-_R^2+1)/(_R^7*a+3*_R^5*a+3*_R^5*b+3*_R^3*a+4*_R^3*b+8*_R^3*c+_R*a+_R*b)*ln(((-x^2+1)^(1/2)
-1)/x-_R),_R=RootOf(a*_Z^8+(4*a+4*b)*_Z^6+(6*a+8*b+16*c)*_Z^4+(4*a+4*b)*_Z^2+a))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(-x^2 + 1)/(c*x^4 + b*x^2 + a), x)

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Fricas [B]  time = 2.47431, size = 1651, normalized size = 7.5 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*sqrt(1/2)*sqrt(-(2*a + b + (a*b^2 - 4*a^2*c)/sqrt(a^2*b^2 - 4*a^3*c))/(a*b^2 - 4*a^2*c))*log(-(x^2 + sqrt(
1/2)*((a*b^2 - 4*a^2*c)*sqrt(-x^2 + 1)*x - (a*b^2 - 4*a^2*c)*x)*sqrt(-(2*a + b + (a*b^2 - 4*a^2*c)/sqrt(a^2*b^
2 - 4*a^3*c))/(a*b^2 - 4*a^2*c))/sqrt(a^2*b^2 - 4*a^3*c) + sqrt(-x^2 + 1) - 1)/x^2) - 1/2*sqrt(1/2)*sqrt(-(2*a
+ b + (a*b^2 - 4*a^2*c)/sqrt(a^2*b^2 - 4*a^3*c))/(a*b^2 - 4*a^2*c))*log(-(x^2 - sqrt(1/2)*((a*b^2 - 4*a^2*c)*
sqrt(-x^2 + 1)*x - (a*b^2 - 4*a^2*c)*x)*sqrt(-(2*a + b + (a*b^2 - 4*a^2*c)/sqrt(a^2*b^2 - 4*a^3*c))/(a*b^2 - 4
*a^2*c))/sqrt(a^2*b^2 - 4*a^3*c) + sqrt(-x^2 + 1) - 1)/x^2) - 1/2*sqrt(1/2)*sqrt(-(2*a + b - (a*b^2 - 4*a^2*c)
/sqrt(a^2*b^2 - 4*a^3*c))/(a*b^2 - 4*a^2*c))*log(-(x^2 + sqrt(1/2)*((a*b^2 - 4*a^2*c)*sqrt(-x^2 + 1)*x - (a*b^
2 - 4*a^2*c)*x)*sqrt(-(2*a + b - (a*b^2 - 4*a^2*c)/sqrt(a^2*b^2 - 4*a^3*c))/(a*b^2 - 4*a^2*c))/sqrt(a^2*b^2 -
4*a^3*c) + sqrt(-x^2 + 1) - 1)/x^2) + 1/2*sqrt(1/2)*sqrt(-(2*a + b - (a*b^2 - 4*a^2*c)/sqrt(a^2*b^2 - 4*a^3*c)
)/(a*b^2 - 4*a^2*c))*log(-(x^2 - sqrt(1/2)*((a*b^2 - 4*a^2*c)*sqrt(-x^2 + 1)*x - (a*b^2 - 4*a^2*c)*x)*sqrt(-(2
*a + b - (a*b^2 - 4*a^2*c)/sqrt(a^2*b^2 - 4*a^3*c))/(a*b^2 - 4*a^2*c))/sqrt(a^2*b^2 - 4*a^3*c) + sqrt(-x^2 + 1
) - 1)/x^2)

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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 )}}{a + b x^{2} + c x^{4}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out