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

Optimal. Leaf size=265 \[ -\frac{c \left (\frac{2 a+b}{\sqrt{b^2-4 a c}}+1\right ) \tan ^{-1}\left (\frac{x \sqrt{-\sqrt{b^2-4 a c}+b+2 c}}{\sqrt{1-x^2} \sqrt{b-\sqrt{b^2-4 a c}}}\right )}{a \sqrt{b-\sqrt{b^2-4 a c}} \sqrt{-\sqrt{b^2-4 a c}+b+2 c}}-\frac{c \left (1-\frac{2 a+b}{\sqrt{b^2-4 a c}}\right ) \tan ^{-1}\left (\frac{x \sqrt{\sqrt{b^2-4 a c}+b+2 c}}{\sqrt{1-x^2} \sqrt{\sqrt{b^2-4 a c}+b}}\right )}{a \sqrt{\sqrt{b^2-4 a c}+b} \sqrt{\sqrt{b^2-4 a c}+b+2 c}}-\frac{\sqrt{1-x^2}}{a x} \]

[Out]

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

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Rubi [A]  time = 1.84776, antiderivative size = 265, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.172 \[ -\frac{c \left (\frac{2 a+b}{\sqrt{b^2-4 a c}}+1\right ) \tan ^{-1}\left (\frac{x \sqrt{-\sqrt{b^2-4 a c}+b+2 c}}{\sqrt{1-x^2} \sqrt{b-\sqrt{b^2-4 a c}}}\right )}{a \sqrt{b-\sqrt{b^2-4 a c}} \sqrt{-\sqrt{b^2-4 a c}+b+2 c}}-\frac{c \left (1-\frac{2 a+b}{\sqrt{b^2-4 a c}}\right ) \tan ^{-1}\left (\frac{x \sqrt{\sqrt{b^2-4 a c}+b+2 c}}{\sqrt{1-x^2} \sqrt{\sqrt{b^2-4 a c}+b}}\right )}{a \sqrt{\sqrt{b^2-4 a c}+b} \sqrt{\sqrt{b^2-4 a c}+b+2 c}}-\frac{\sqrt{1-x^2}}{a x} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 154.285, size = 248, normalized size = 0.94 \[ \frac{c \left (2 a + b - \sqrt{- 4 a c + b^{2}}\right ) \operatorname{atan}{\left (\frac{x \sqrt{b + 2 c + \sqrt{- 4 a c + b^{2}}}}{\sqrt{b + \sqrt{- 4 a c + b^{2}}} \sqrt{- x^{2} + 1}} \right )}}{a \sqrt{b + \sqrt{- 4 a c + b^{2}}} \sqrt{- 4 a c + b^{2}} \sqrt{b + 2 c + \sqrt{- 4 a c + b^{2}}}} - \frac{c \left (2 a + b + \sqrt{- 4 a c + b^{2}}\right ) \operatorname{atan}{\left (\frac{x \sqrt{b + 2 c - \sqrt{- 4 a c + b^{2}}}}{\sqrt{b - \sqrt{- 4 a c + b^{2}}} \sqrt{- x^{2} + 1}} \right )}}{a \sqrt{b - \sqrt{- 4 a c + b^{2}}} \sqrt{- 4 a c + b^{2}} \sqrt{b + 2 c - \sqrt{- 4 a c + b^{2}}}} - \frac{\sqrt{- x^{2} + 1}}{a x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((-x**2+1)**(1/2)/x**2/(c*x**4+b*x**2+a),x)

[Out]

c*(2*a + b - sqrt(-4*a*c + b**2))*atan(x*sqrt(b + 2*c + sqrt(-4*a*c + b**2))/(sq
rt(b + sqrt(-4*a*c + b**2))*sqrt(-x**2 + 1)))/(a*sqrt(b + sqrt(-4*a*c + b**2))*s
qrt(-4*a*c + b**2)*sqrt(b + 2*c + sqrt(-4*a*c + b**2))) - c*(2*a + b + sqrt(-4*a
*c + b**2))*atan(x*sqrt(b + 2*c - sqrt(-4*a*c + b**2))/(sqrt(b - sqrt(-4*a*c + b
**2))*sqrt(-x**2 + 1)))/(a*sqrt(b - sqrt(-4*a*c + b**2))*sqrt(-4*a*c + b**2)*sqr
t(b + 2*c - sqrt(-4*a*c + b**2))) - sqrt(-x**2 + 1)/(a*x)

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Mathematica [A]  time = 0.493337, size = 0, normalized size = 0. \[ \int \frac{\sqrt{1-x^2}}{x^2 \left (a+b x^2+c x^4\right )} \, dx \]

Verification is Not applicable to the result.

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

[Out]

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

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Maple [C]  time = 0.035, size = 217, normalized size = 0.8 \[ -{\frac{1}{ax} \left ( -{x}^{2}+1 \right ) ^{{\frac{3}{2}}}}-{\frac{x}{a}\sqrt{-{x}^{2}+1}}-{\frac{\arcsin \left ( x \right ) }{a}}+{\frac{1}{4\,a}\sum _{{\it \_R}={\it RootOf} \left ( a{{\it \_Z}}^{8}+ \left ( 4\,a+4\,b \right ){{\it \_Z}}^{6}+ \left ( 6\,a+8\,b+16\,c \right ){{\it \_Z}}^{4}+ \left ( 4\,a+4\,b \right ){{\it \_Z}}^{2}+a \right ) }{\frac{ \left ( a+b \right ){{\it \_R}}^{6}+ \left ( 3\,a+3\,b+4\,c \right ){{\it \_R}}^{4}+ \left ( 3\,a+3\,b+4\,c \right ){{\it \_R}}^{2}+a+b}{{{\it \_R}}^{7}a+3\,{{\it \_R}}^{5}a+3\,{{\it \_R}}^{5}b+3\,{{\it \_R}}^{3}a+4\,{{\it \_R}}^{3}b+8\,{{\it \_R}}^{3}c+{\it \_R}\,a+{\it \_R}\,b}\ln \left ({\frac{1}{x} \left ( \sqrt{-{x}^{2}+1}-1 \right ) }-{\it \_R} \right ) }}-2\,{\frac{1}{a}\arctan \left ({\frac{\sqrt{-{x}^{2}+1}-1}{x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/a/x*(-x^2+1)^(3/2)-1/a*x*(-x^2+1)^(1/2)-1/a*arcsin(x)+1/4/a*sum(((a+b)*_R^6+(
3*a+3*b+4*c)*_R^4+(3*a+3*b+4*c)*_R^2+a+b)/(_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))-2/a*arctan(((-x^2+1)^(1/2)-1)/x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{-x^{2} + 1}}{{\left (c x^{4} + b x^{2} + a\right )} x^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.487556, size = 2804, normalized size = 10.58 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(sqrt(1/2)*(sqrt(-x^2 + 1)*a*x - a*x)*sqrt(-(a*b^2 + b^3 - (2*a^2 + 3*a*b)*c
+ (a^3*b^2 - 4*a^4*c)*sqrt((a^2*b^2 + 2*a*b^3 + b^4 + a^2*c^2 - 2*(a^2*b + a*b^
2)*c)/(a^6*b^2 - 4*a^7*c)))/(a^3*b^2 - 4*a^4*c))*log((2*a*c^2 - 2*(a*c^2 - (a*b
+ b^2)*c)*x^2 - 2*(a*b + b^2)*c + sqrt(1/2)*((a*b^3 + b^4 + 4*a^2*c^2 - (4*a^2*b
+ 5*a*b^2)*c)*sqrt(-x^2 + 1)*x - (a*b^3 + b^4 + 4*a^2*c^2 - (4*a^2*b + 5*a*b^2)
*c)*x - ((a^3*b^3 - 4*a^4*b*c)*sqrt(-x^2 + 1)*x - (a^3*b^3 - 4*a^4*b*c)*x)*sqrt(
(a^2*b^2 + 2*a*b^3 + b^4 + a^2*c^2 - 2*(a^2*b + a*b^2)*c)/(a^6*b^2 - 4*a^7*c)))*
sqrt(-(a*b^2 + b^3 - (2*a^2 + 3*a*b)*c + (a^3*b^2 - 4*a^4*c)*sqrt((a^2*b^2 + 2*a
*b^3 + b^4 + a^2*c^2 - 2*(a^2*b + a*b^2)*c)/(a^6*b^2 - 4*a^7*c)))/(a^3*b^2 - 4*a
^4*c)) - 2*(a*c^2 - (a*b + b^2)*c)*sqrt(-x^2 + 1))/x^2) - sqrt(1/2)*(sqrt(-x^2 +
1)*a*x - a*x)*sqrt(-(a*b^2 + b^3 - (2*a^2 + 3*a*b)*c + (a^3*b^2 - 4*a^4*c)*sqrt
((a^2*b^2 + 2*a*b^3 + b^4 + a^2*c^2 - 2*(a^2*b + a*b^2)*c)/(a^6*b^2 - 4*a^7*c)))
/(a^3*b^2 - 4*a^4*c))*log((2*a*c^2 - 2*(a*c^2 - (a*b + b^2)*c)*x^2 - 2*(a*b + b^
2)*c - sqrt(1/2)*((a*b^3 + b^4 + 4*a^2*c^2 - (4*a^2*b + 5*a*b^2)*c)*sqrt(-x^2 +
1)*x - (a*b^3 + b^4 + 4*a^2*c^2 - (4*a^2*b + 5*a*b^2)*c)*x - ((a^3*b^3 - 4*a^4*b
*c)*sqrt(-x^2 + 1)*x - (a^3*b^3 - 4*a^4*b*c)*x)*sqrt((a^2*b^2 + 2*a*b^3 + b^4 +
a^2*c^2 - 2*(a^2*b + a*b^2)*c)/(a^6*b^2 - 4*a^7*c)))*sqrt(-(a*b^2 + b^3 - (2*a^2
+ 3*a*b)*c + (a^3*b^2 - 4*a^4*c)*sqrt((a^2*b^2 + 2*a*b^3 + b^4 + a^2*c^2 - 2*(a
^2*b + a*b^2)*c)/(a^6*b^2 - 4*a^7*c)))/(a^3*b^2 - 4*a^4*c)) - 2*(a*c^2 - (a*b +
b^2)*c)*sqrt(-x^2 + 1))/x^2) + sqrt(1/2)*(sqrt(-x^2 + 1)*a*x - a*x)*sqrt(-(a*b^2
+ b^3 - (2*a^2 + 3*a*b)*c - (a^3*b^2 - 4*a^4*c)*sqrt((a^2*b^2 + 2*a*b^3 + b^4 +
a^2*c^2 - 2*(a^2*b + a*b^2)*c)/(a^6*b^2 - 4*a^7*c)))/(a^3*b^2 - 4*a^4*c))*log((
2*a*c^2 - 2*(a*c^2 - (a*b + b^2)*c)*x^2 - 2*(a*b + b^2)*c + sqrt(1/2)*((a*b^3 +
b^4 + 4*a^2*c^2 - (4*a^2*b + 5*a*b^2)*c)*sqrt(-x^2 + 1)*x - (a*b^3 + b^4 + 4*a^2
*c^2 - (4*a^2*b + 5*a*b^2)*c)*x + ((a^3*b^3 - 4*a^4*b*c)*sqrt(-x^2 + 1)*x - (a^3
*b^3 - 4*a^4*b*c)*x)*sqrt((a^2*b^2 + 2*a*b^3 + b^4 + a^2*c^2 - 2*(a^2*b + a*b^2)
*c)/(a^6*b^2 - 4*a^7*c)))*sqrt(-(a*b^2 + b^3 - (2*a^2 + 3*a*b)*c - (a^3*b^2 - 4*
a^4*c)*sqrt((a^2*b^2 + 2*a*b^3 + b^4 + a^2*c^2 - 2*(a^2*b + a*b^2)*c)/(a^6*b^2 -
4*a^7*c)))/(a^3*b^2 - 4*a^4*c)) - 2*(a*c^2 - (a*b + b^2)*c)*sqrt(-x^2 + 1))/x^2
) - sqrt(1/2)*(sqrt(-x^2 + 1)*a*x - a*x)*sqrt(-(a*b^2 + b^3 - (2*a^2 + 3*a*b)*c
- (a^3*b^2 - 4*a^4*c)*sqrt((a^2*b^2 + 2*a*b^3 + b^4 + a^2*c^2 - 2*(a^2*b + a*b^2
)*c)/(a^6*b^2 - 4*a^7*c)))/(a^3*b^2 - 4*a^4*c))*log((2*a*c^2 - 2*(a*c^2 - (a*b +
b^2)*c)*x^2 - 2*(a*b + b^2)*c - sqrt(1/2)*((a*b^3 + b^4 + 4*a^2*c^2 - (4*a^2*b
+ 5*a*b^2)*c)*sqrt(-x^2 + 1)*x - (a*b^3 + b^4 + 4*a^2*c^2 - (4*a^2*b + 5*a*b^2)*
c)*x + ((a^3*b^3 - 4*a^4*b*c)*sqrt(-x^2 + 1)*x - (a^3*b^3 - 4*a^4*b*c)*x)*sqrt((
a^2*b^2 + 2*a*b^3 + b^4 + a^2*c^2 - 2*(a^2*b + a*b^2)*c)/(a^6*b^2 - 4*a^7*c)))*s
qrt(-(a*b^2 + b^3 - (2*a^2 + 3*a*b)*c - (a^3*b^2 - 4*a^4*c)*sqrt((a^2*b^2 + 2*a*
b^3 + b^4 + a^2*c^2 - 2*(a^2*b + a*b^2)*c)/(a^6*b^2 - 4*a^7*c)))/(a^3*b^2 - 4*a^
4*c)) - 2*(a*c^2 - (a*b + b^2)*c)*sqrt(-x^2 + 1))/x^2) + 2*x^2 + 2*sqrt(-x^2 + 1
) - 2)/(sqrt(-x^2 + 1)*a*x - a*x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: TypeError