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

Optimal. Leaf size=122 \[ -\frac{\tan ^{-1}\left (\frac{1-2 x^2}{\sqrt{3}}\right )}{\sqrt{3}}-\frac{\tan ^{-1}\left (\frac{x}{\sqrt{-\frac{-\sqrt{3}+i}{\sqrt{3}+i}} \sqrt{1-x^2}}\right )}{\sqrt{3}}-\frac{\tan ^{-1}\left (\frac{\sqrt{-\frac{-\sqrt{3}+i}{\sqrt{3}+i}} x}{\sqrt{1-x^2}}\right )}{\sqrt{3}}+\sin ^{-1}(x) \]

[Out]

ArcSin[x] - ArcTan[(1 - 2*x^2)/Sqrt[3]]/Sqrt[3] - ArcTan[x/(Sqrt[-((I - Sqrt[3])
/(I + Sqrt[3]))]*Sqrt[1 - x^2])]/Sqrt[3] - ArcTan[(Sqrt[-((I - Sqrt[3])/(I + Sqr
t[3]))]*x)/Sqrt[1 - x^2]]/Sqrt[3]

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Rubi [A]  time = 0.335202, antiderivative size = 122, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 9, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.529 \[ -\frac{\tan ^{-1}\left (\frac{1-2 x^2}{\sqrt{3}}\right )}{\sqrt{3}}-\frac{\tan ^{-1}\left (\frac{x}{\sqrt{-\frac{-\sqrt{3}+i}{\sqrt{3}+i}} \sqrt{1-x^2}}\right )}{\sqrt{3}}-\frac{\tan ^{-1}\left (\frac{\sqrt{-\frac{-\sqrt{3}+i}{\sqrt{3}+i}} x}{\sqrt{1-x^2}}\right )}{\sqrt{3}}+\sin ^{-1}(x) \]

Antiderivative was successfully verified.

[In]  Int[(x^(-1) + Sqrt[1 - x^2])^(-1),x]

[Out]

ArcSin[x] - ArcTan[(1 - 2*x^2)/Sqrt[3]]/Sqrt[3] - ArcTan[x/(Sqrt[-((I - Sqrt[3])
/(I + Sqrt[3]))]*Sqrt[1 - x^2])]/Sqrt[3] - ArcTan[(Sqrt[-((I - Sqrt[3])/(I + Sqr
t[3]))]*x)/Sqrt[1 - x^2]]/Sqrt[3]

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Rubi in Sympy [A]  time = 52.7528, size = 112, normalized size = 0.92 \[ \frac{2 \sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (\frac{1}{3} + \frac{\sqrt{- x^{2} + 1} - 1}{3 x}\right ) \right )}}{3} - \frac{2 \sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (\frac{1}{3} - \frac{\sqrt{- x^{2} + 1} - 1}{x} - \frac{2 \left (\sqrt{- x^{2} + 1} - 1\right )^{2}}{3 x^{2}} - \frac{\left (\sqrt{- x^{2} + 1} - 1\right )^{3}}{3 x^{3}}\right ) \right )}}{3} - 2 \operatorname{atan}{\left (\frac{\sqrt{- x^{2} + 1} - 1}{x} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(1/x+(-x**2+1)**(1/2)),x)

[Out]

2*sqrt(3)*atan(sqrt(3)*(1/3 + (sqrt(-x**2 + 1) - 1)/(3*x)))/3 - 2*sqrt(3)*atan(s
qrt(3)*(1/3 - (sqrt(-x**2 + 1) - 1)/x - 2*(sqrt(-x**2 + 1) - 1)**2/(3*x**2) - (s
qrt(-x**2 + 1) - 1)**3/(3*x**3)))/3 - 2*atan((sqrt(-x**2 + 1) - 1)/x)

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Mathematica [B]  time = 6.67332, size = 2681, normalized size = 21.98 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[(x^(-1) + Sqrt[1 - x^2])^(-1),x]

[Out]

((1 + x*Sqrt[1 - x^2])*ArcSin[x])/(x*(x^(-1) + Sqrt[1 - x^2])) + ((-I + Sqrt[3])
*(1 + x*Sqrt[1 - x^2])*ArcTan[(x*(7*I - Sqrt[3] + (8*I)*Sqrt[3]*x + (7*I)*x^2 +
Sqrt[3]*x^2))/(-6*I + 2*Sqrt[3] + 3*x - (11*I)*Sqrt[3]*x - (18*I)*x^2 - 2*Sqrt[3
]*x^2 - 3*x^3 - (3*I)*Sqrt[3]*x^3 - (2*I)*Sqrt[2*(1 - I*Sqrt[3])]*Sqrt[1 - x^2]
- (2*I)*Sqrt[6*(1 - I*Sqrt[3])]*x*Sqrt[1 - x^2] - (2*I)*Sqrt[2*(1 - I*Sqrt[3])]*
x^2*Sqrt[1 - x^2])])/(2*Sqrt[6*(1 - I*Sqrt[3])]*x*(x^(-1) + Sqrt[1 - x^2])) - ((
-I + Sqrt[3])*(1 + x*Sqrt[1 - x^2])*ArcTan[(x*(7*I - Sqrt[3] - (8*I)*Sqrt[3]*x +
(7*I)*x^2 + Sqrt[3]*x^2))/(6*I - 2*Sqrt[3] + 3*x - (11*I)*Sqrt[3]*x + (18*I)*x^
2 + 2*Sqrt[3]*x^2 - 3*x^3 - (3*I)*Sqrt[3]*x^3 + (2*I)*Sqrt[2*(1 - I*Sqrt[3])]*Sq
rt[1 - x^2] - (2*I)*Sqrt[6*(1 - I*Sqrt[3])]*x*Sqrt[1 - x^2] + (2*I)*Sqrt[2*(1 -
I*Sqrt[3])]*x^2*Sqrt[1 - x^2])])/(2*Sqrt[6*(1 - I*Sqrt[3])]*x*(x^(-1) + Sqrt[1 -
x^2])) - ((I + Sqrt[3])*(1 + x*Sqrt[1 - x^2])*ArcTan[(x*(-7*I - Sqrt[3] - (8*I)
*Sqrt[3]*x - (7*I)*x^2 + Sqrt[3]*x^2))/(-6*I - 2*Sqrt[3] - 3*x - (11*I)*Sqrt[3]*
x - (18*I)*x^2 + 2*Sqrt[3]*x^2 + 3*x^3 - (3*I)*Sqrt[3]*x^3 - (2*I)*Sqrt[2*(1 + I
*Sqrt[3])]*Sqrt[1 - x^2] - (2*I)*Sqrt[6*(1 + I*Sqrt[3])]*x*Sqrt[1 - x^2] - (2*I)
*Sqrt[2*(1 + I*Sqrt[3])]*x^2*Sqrt[1 - x^2])])/(2*Sqrt[6*(1 + I*Sqrt[3])]*x*(x^(-
1) + Sqrt[1 - x^2])) + ((I + Sqrt[3])*(1 + x*Sqrt[1 - x^2])*ArcTan[(x*(-7*I - Sq
rt[3] + (8*I)*Sqrt[3]*x - (7*I)*x^2 + Sqrt[3]*x^2))/(6*I + 2*Sqrt[3] - 3*x - (11
*I)*Sqrt[3]*x + (18*I)*x^2 - 2*Sqrt[3]*x^2 + 3*x^3 - (3*I)*Sqrt[3]*x^3 + (2*I)*S
qrt[2*(1 + I*Sqrt[3])]*Sqrt[1 - x^2] - (2*I)*Sqrt[6*(1 + I*Sqrt[3])]*x*Sqrt[1 -
x^2] + (2*I)*Sqrt[2*(1 + I*Sqrt[3])]*x^2*Sqrt[1 - x^2])])/(2*Sqrt[6*(1 + I*Sqrt[
3])]*x*(x^(-1) + Sqrt[1 - x^2])) + ((I/4)*(-I + Sqrt[3])*(1 + x*Sqrt[1 - x^2])*L
og[(-I + Sqrt[3] - 2*x)^2*(I + Sqrt[3] - 2*x)^2])/(Sqrt[6*(1 - I*Sqrt[3])]*x*(x^
(-1) + Sqrt[1 - x^2])) - ((I/4)*(I + Sqrt[3])*(1 + x*Sqrt[1 - x^2])*Log[(-I + Sq
rt[3] - 2*x)^2*(I + Sqrt[3] - 2*x)^2])/(Sqrt[6*(1 + I*Sqrt[3])]*x*(x^(-1) + Sqrt
[1 - x^2])) - ((I/4)*(-I + Sqrt[3])*(1 + x*Sqrt[1 - x^2])*Log[(-I + Sqrt[3] + 2*
x)^2*(I + Sqrt[3] + 2*x)^2])/(Sqrt[6*(1 - I*Sqrt[3])]*x*(x^(-1) + Sqrt[1 - x^2])
) + ((I/4)*(I + Sqrt[3])*(1 + x*Sqrt[1 - x^2])*Log[(-I + Sqrt[3] + 2*x)^2*(I + S
qrt[3] + 2*x)^2])/(Sqrt[6*(1 + I*Sqrt[3])]*x*(x^(-1) + Sqrt[1 - x^2])) - ((I/2)*
(1 + x*Sqrt[1 - x^2])*Log[-1/2 - (I/2)*Sqrt[3] + x^2])/(Sqrt[3]*x*(x^(-1) + Sqrt
[1 - x^2])) + ((I/2)*(1 + x*Sqrt[1 - x^2])*Log[-1/2 + (I/2)*Sqrt[3] + x^2])/(Sqr
t[3]*x*(x^(-1) + Sqrt[1 - x^2])) - ((I/4)*(-I + Sqrt[3])*(1 + x*Sqrt[1 - x^2])*L
og[3*I + Sqrt[3] - 3*x - (5*I)*Sqrt[3]*x + (10*I)*x^2 + 3*x^3 - (3*I)*Sqrt[3]*x^
3 + I*x^4 - Sqrt[3]*x^4 + (2*I)*Sqrt[2*(1 - I*Sqrt[3])]*Sqrt[1 - x^2] - (3*I)*Sq
rt[6*(1 - I*Sqrt[3])]*x*Sqrt[1 - x^2] + (5*I)*Sqrt[2*(1 - I*Sqrt[3])]*x^2*Sqrt[1
- x^2] - I*Sqrt[6*(1 - I*Sqrt[3])]*x^3*Sqrt[1 - x^2]])/(Sqrt[6*(1 - I*Sqrt[3])]
*x*(x^(-1) + Sqrt[1 - x^2])) + ((I/4)*(-I + Sqrt[3])*(1 + x*Sqrt[1 - x^2])*Log[3
*I + Sqrt[3] + 3*x + (5*I)*Sqrt[3]*x + (10*I)*x^2 - 3*x^3 + (3*I)*Sqrt[3]*x^3 +
I*x^4 - Sqrt[3]*x^4 + (2*I)*Sqrt[2*(1 - I*Sqrt[3])]*Sqrt[1 - x^2] + (3*I)*Sqrt[6
*(1 - I*Sqrt[3])]*x*Sqrt[1 - x^2] + (5*I)*Sqrt[2*(1 - I*Sqrt[3])]*x^2*Sqrt[1 - x
^2] + I*Sqrt[6*(1 - I*Sqrt[3])]*x^3*Sqrt[1 - x^2]])/(Sqrt[6*(1 - I*Sqrt[3])]*x*(
x^(-1) + Sqrt[1 - x^2])) - ((I/4)*(I + Sqrt[3])*(1 + x*Sqrt[1 - x^2])*Log[-3*I +
Sqrt[3] + 3*x - (5*I)*Sqrt[3]*x - (10*I)*x^2 - 3*x^3 - (3*I)*Sqrt[3]*x^3 - I*x^
4 - Sqrt[3]*x^4 - (2*I)*Sqrt[2*(1 + I*Sqrt[3])]*Sqrt[1 - x^2] - (3*I)*Sqrt[6*(1
+ I*Sqrt[3])]*x*Sqrt[1 - x^2] - (5*I)*Sqrt[2*(1 + I*Sqrt[3])]*x^2*Sqrt[1 - x^2]
- I*Sqrt[6*(1 + I*Sqrt[3])]*x^3*Sqrt[1 - x^2]])/(Sqrt[6*(1 + I*Sqrt[3])]*x*(x^(-
1) + Sqrt[1 - x^2])) + ((I/4)*(I + Sqrt[3])*(1 + x*Sqrt[1 - x^2])*Log[-3*I + Sqr
t[3] - 3*x + (5*I)*Sqrt[3]*x - (10*I)*x^2 + 3*x^3 + (3*I)*Sqrt[3]*x^3 - I*x^4 -
Sqrt[3]*x^4 - (2*I)*Sqrt[2*(1 + I*Sqrt[3])]*Sqrt[1 - x^2] + (3*I)*Sqrt[6*(1 + I*
Sqrt[3])]*x*Sqrt[1 - x^2] - (5*I)*Sqrt[2*(1 + I*Sqrt[3])]*x^2*Sqrt[1 - x^2] + I*
Sqrt[6*(1 + I*Sqrt[3])]*x^3*Sqrt[1 - x^2]])/(Sqrt[6*(1 + I*Sqrt[3])]*x*(x^(-1) +
Sqrt[1 - x^2]))

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Maple [B]  time = 0.047, size = 234, normalized size = 1.9 \[{\frac{i}{6}}\sqrt{3}\ln \left ({\frac{1}{{x}^{2}} \left ( \sqrt{-{x}^{2}+1}-1 \right ) ^{2}}+{\frac{i\sqrt{3}+1}{x} \left ( \sqrt{-{x}^{2}+1}-1 \right ) }-1 \right ) -{\frac{i}{6}}\sqrt{3}\ln \left ({\frac{1}{{x}^{2}} \left ( \sqrt{-{x}^{2}+1}-1 \right ) ^{2}}+{\frac{1-i\sqrt{3}}{x} \left ( \sqrt{-{x}^{2}+1}-1 \right ) }-1 \right ) -2\,\arctan \left ({\frac{\sqrt{-{x}^{2}+1}-1}{x}} \right ) +{\frac{i}{6}}\sqrt{3}\ln \left ({\frac{1}{{x}^{2}} \left ( \sqrt{-{x}^{2}+1}-1 \right ) ^{2}}+{\frac{-1+i\sqrt{3}}{x} \left ( \sqrt{-{x}^{2}+1}-1 \right ) }-1 \right ) -{\frac{i}{6}}\sqrt{3}\ln \left ({\frac{1}{{x}^{2}} \left ( \sqrt{-{x}^{2}+1}-1 \right ) ^{2}}+{\frac{-i\sqrt{3}-1}{x} \left ( \sqrt{-{x}^{2}+1}-1 \right ) }-1 \right ) +{\frac{\sqrt{3}}{3}\arctan \left ({\frac{ \left ( 2\,{x}^{2}-1 \right ) \sqrt{3}}{3}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(1/x+(-x^2+1)^(1/2)),x)

[Out]

1/6*I*3^(1/2)*ln(((-x^2+1)^(1/2)-1)^2/x^2+(I*3^(1/2)+1)*((-x^2+1)^(1/2)-1)/x-1)-
1/6*I*3^(1/2)*ln(((-x^2+1)^(1/2)-1)^2/x^2+(1-I*3^(1/2))*((-x^2+1)^(1/2)-1)/x-1)-
2*arctan(((-x^2+1)^(1/2)-1)/x)+1/6*I*3^(1/2)*ln(((-x^2+1)^(1/2)-1)^2/x^2+(-1+I*3
^(1/2))*((-x^2+1)^(1/2)-1)/x-1)-1/6*I*3^(1/2)*ln(((-x^2+1)^(1/2)-1)^2/x^2+(-I*3^
(1/2)-1)*((-x^2+1)^(1/2)-1)/x-1)+1/3*3^(1/2)*arctan(1/3*(2*x^2-1)*3^(1/2))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(-x^2 + 1) + 1/x),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.260602, size = 151, normalized size = 1.24 \[ -\frac{1}{3} \, \sqrt{3}{\left (2 \, \sqrt{3} \arctan \left (\frac{\sqrt{-x^{2} + 1} - 1}{x}\right ) - \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x^{2} - 1\right )}\right ) + \arctan \left (-\frac{2 \, \sqrt{3}{\left (2 \, x^{2} - 1\right )} \sqrt{-x^{2} + 1} + \sqrt{3}{\left (2 \, x^{4} - 5 \, x^{2} + 2\right )}}{3 \,{\left (2 \, x^{3} -{\left (x^{3} - 2 \, x\right )} \sqrt{-x^{2} + 1} - 2 \, x\right )}}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(-x^2 + 1) + 1/x),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x}{x \sqrt{- x^{2} + 1} + 1}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(1/x+(-x**2+1)**(1/2)),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.274593, size = 261, normalized size = 2.14 \[ \frac{1}{2} \, \pi{\rm sign}\left (x\right ) - \frac{1}{6} \, \sqrt{3}{\left (\pi{\rm sign}\left (x\right ) + 2 \, \arctan \left (-\frac{\sqrt{3} x{\left (\frac{\sqrt{-x^{2} + 1} - 1}{x} + \frac{{\left (\sqrt{-x^{2} + 1} - 1\right )}^{2}}{x^{2}} - 1\right )}}{3 \,{\left (\sqrt{-x^{2} + 1} - 1\right )}}\right )\right )} - \frac{1}{6} \, \sqrt{3}{\left (\pi{\rm sign}\left (x\right ) + 2 \, \arctan \left (\frac{\sqrt{3} x{\left (\frac{\sqrt{-x^{2} + 1} - 1}{x} - \frac{{\left (\sqrt{-x^{2} + 1} - 1\right )}^{2}}{x^{2}} + 1\right )}}{3 \,{\left (\sqrt{-x^{2} + 1} - 1\right )}}\right )\right )} + \frac{1}{3} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x^{2} - 1\right )}\right ) + \arctan \left (-\frac{x{\left (\frac{{\left (\sqrt{-x^{2} + 1} - 1\right )}^{2}}{x^{2}} - 1\right )}}{2 \,{\left (\sqrt{-x^{2} + 1} - 1\right )}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(-x^2 + 1) + 1/x),x, algorithm="giac")

[Out]

1/2*pi*sign(x) - 1/6*sqrt(3)*(pi*sign(x) + 2*arctan(-1/3*sqrt(3)*x*((sqrt(-x^2 +
1) - 1)/x + (sqrt(-x^2 + 1) - 1)^2/x^2 - 1)/(sqrt(-x^2 + 1) - 1))) - 1/6*sqrt(3
)*(pi*sign(x) + 2*arctan(1/3*sqrt(3)*x*((sqrt(-x^2 + 1) - 1)/x - (sqrt(-x^2 + 1)
- 1)^2/x^2 + 1)/(sqrt(-x^2 + 1) - 1))) + 1/3*sqrt(3)*arctan(1/3*sqrt(3)*(2*x^2
- 1)) + arctan(-1/2*x*((sqrt(-x^2 + 1) - 1)^2/x^2 - 1)/(sqrt(-x^2 + 1) - 1))