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

Optimal. Leaf size=25 \[ -\frac {1}{2} \tanh ^{-1}\left (\frac {x}{\sqrt {1-x^2}}\right )-\frac {1}{2} \sin ^{-1}(x) \]

[Out]

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

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Rubi [A]  time = 0.01, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {402, 216, 377, 207} \[ -\frac {1}{2} \tanh ^{-1}\left (\frac {x}{\sqrt {1-x^2}}\right )-\frac {1}{2} \sin ^{-1}(x) \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[1 - x^2]/(-1 + 2*x^2),x]

[Out]

-ArcSin[x]/2 - ArcTanh[x/Sqrt[1 - x^2]]/2

Rule 207

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTanh[(Rt[b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])

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 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])

Rubi steps

\begin {align*} \int \frac {\sqrt {1-x^2}}{-1+2 x^2} \, dx &=-\left (\frac {1}{2} \int \frac {1}{\sqrt {1-x^2}} \, dx\right )+\frac {1}{2} \int \frac {1}{\sqrt {1-x^2} \left (-1+2 x^2\right )} \, dx\\ &=-\frac {1}{2} \sin ^{-1}(x)+\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\frac {x}{\sqrt {1-x^2}}\right )\\ &=-\frac {1}{2} \sin ^{-1}(x)-\frac {1}{2} \tanh ^{-1}\left (\frac {x}{\sqrt {1-x^2}}\right )\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 25, normalized size = 1.00 \[ -\frac {1}{2} \tanh ^{-1}\left (\frac {x}{\sqrt {1-x^2}}\right )-\frac {1}{2} \sin ^{-1}(x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*ArcSin[x] - ArcTanh[x/Sqrt[1 - x^2]]/2

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fricas [B]  time = 0.56, size = 74, normalized size = 2.96 \[ \arctan \left (\frac {\sqrt {-x^{2} + 1} - 1}{x}\right ) + \frac {1}{4} \, \log \left (-\frac {x^{2} + \sqrt {-x^{2} + 1} {\left (x + 1\right )} - x - 1}{x^{2}}\right ) - \frac {1}{4} \, \log \left (-\frac {x^{2} - \sqrt {-x^{2} + 1} {\left (x - 1\right )} + x - 1}{x^{2}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [B]  time = 0.61, size = 118, normalized size = 4.72 \[ -\frac {1}{4} \, \pi \mathrm {sgn}\relax (x) - \frac {1}{2} \, \arctan \left (-\frac {x {\left (\frac {{\left (\sqrt {-x^{2} + 1} - 1\right )}^{2}}{x^{2}} - 1\right )}}{2 \, {\left (\sqrt {-x^{2} + 1} - 1\right )}}\right ) - \frac {1}{4} \, \log \left ({\left | -\frac {x}{\sqrt {-x^{2} + 1} - 1} + \frac {\sqrt {-x^{2} + 1} - 1}{x} + 2 \right |}\right ) + \frac {1}{4} \, \log \left ({\left | -\frac {x}{\sqrt {-x^{2} + 1} - 1} + \frac {\sqrt {-x^{2} + 1} - 1}{x} - 2 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*pi*sgn(x) - 1/2*arctan(-1/2*x*((sqrt(-x^2 + 1) - 1)^2/x^2 - 1)/(sqrt(-x^2 + 1) - 1)) - 1/4*log(abs(-x/(sq
rt(-x^2 + 1) - 1) + (sqrt(-x^2 + 1) - 1)/x + 2)) + 1/4*log(abs(-x/(sqrt(-x^2 + 1) - 1) + (sqrt(-x^2 + 1) - 1)/
x - 2))

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maple [B]  time = 0.04, size = 187, normalized size = 7.48 \[ -\frac {\sqrt {2}\, \left (-\frac {\sqrt {2}\, \arctanh \left (\frac {\left (\left (x +\frac {\sqrt {2}}{2}\right ) \sqrt {2}+1\right ) \sqrt {2}}{\sqrt {-4 \left (x +\frac {\sqrt {2}}{2}\right )^{2}+4 \left (x +\frac {\sqrt {2}}{2}\right ) \sqrt {2}+2}}\right )}{4}+\frac {\sqrt {2}\, \arcsin \relax (x )}{4}+\frac {\sqrt {-4 \left (x +\frac {\sqrt {2}}{2}\right )^{2}+4 \left (x +\frac {\sqrt {2}}{2}\right ) \sqrt {2}+2}}{4}\right )}{2}+\frac {\sqrt {2}\, \left (-\frac {\sqrt {2}\, \arctanh \left (\frac {\left (-\left (x -\frac {\sqrt {2}}{2}\right ) \sqrt {2}+1\right ) \sqrt {2}}{\sqrt {-4 \left (x -\frac {\sqrt {2}}{2}\right )^{2}-4 \left (x -\frac {\sqrt {2}}{2}\right ) \sqrt {2}+2}}\right )}{4}-\frac {\sqrt {2}\, \arcsin \relax (x )}{4}+\frac {\sqrt {-4 \left (x -\frac {\sqrt {2}}{2}\right )^{2}-4 \left (x -\frac {\sqrt {2}}{2}\right ) \sqrt {2}+2}}{4}\right )}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 3.07, size = 110, normalized size = 4.40 \[ -\frac {1}{8} \, \sqrt {2} {\left (2 \, \sqrt {2} \arcsin \relax (x) - \sqrt {2} \log \left (\frac {1}{4} \, \sqrt {2} + \frac {\sqrt {2} \sqrt {-x^{2} + 1}}{{\left | 4 \, x + 2 \, \sqrt {2} \right |}} + \frac {1}{{\left | 4 \, x + 2 \, \sqrt {2} \right |}}\right ) + \sqrt {2} \log \left (-\frac {1}{4} \, \sqrt {2} + \frac {\sqrt {2} \sqrt {-x^{2} + 1}}{{\left | 4 \, x - 2 \, \sqrt {2} \right |}} + \frac {1}{{\left | 4 \, x - 2 \, \sqrt {2} \right |}}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*sqrt(2)*(2*sqrt(2)*arcsin(x) - sqrt(2)*log(1/4*sqrt(2) + sqrt(2)*sqrt(-x^2 + 1)/abs(4*x + 2*sqrt(2)) + 1/
abs(4*x + 2*sqrt(2))) + sqrt(2)*log(-1/4*sqrt(2) + sqrt(2)*sqrt(-x^2 + 1)/abs(4*x - 2*sqrt(2)) + 1/abs(4*x - 2
*sqrt(2))))

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mupad [B]  time = 5.35, size = 85, normalized size = 3.40 \[ -\frac {\ln \left (\frac {\sqrt {2}\,\left (\frac {\sqrt {2}\,x}{2}-1\right )\,1{}\mathrm {i}-\sqrt {1-x^2}\,1{}\mathrm {i}}{x-\frac {\sqrt {2}}{2}}\right )}{4}+\frac {\ln \left (\frac {\sqrt {2}\,\left (\frac {\sqrt {2}\,x}{2}+1\right )\,1{}\mathrm {i}+\sqrt {1-x^2}\,1{}\mathrm {i}}{x+\frac {\sqrt {2}}{2}}\right )}{4}-\frac {\mathrm {asin}\relax (x)}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1 - x^2)^(1/2)/(2*x^2 - 1),x)

[Out]

log((2^(1/2)*((2^(1/2)*x)/2 + 1)*1i + (1 - x^2)^(1/2)*1i)/(x + 2^(1/2)/2))/4 - log((2^(1/2)*((2^(1/2)*x)/2 - 1
)*1i - (1 - x^2)^(1/2)*1i)/(x - 2^(1/2)/2))/4 - asin(x)/2

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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