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

Optimal. Leaf size=75 \[ -\frac{1}{2} \tan ^{-1}\left (\frac{3-x}{2 \sqrt{x^2-x-1}}\right )+\tanh ^{-1}\left (\frac{1-2 x}{2 \sqrt{x^2-x-1}}\right )+\frac{1}{2} \tanh ^{-1}\left (\frac{3 x+1}{2 \sqrt{x^2-x-1}}\right ) \]

[Out]

-ArcTan[(3 - x)/(2*Sqrt[-1 - x + x^2])]/2 + ArcTanh[(1 - 2*x)/(2*Sqrt[-1 - x + x^2])] + ArcTanh[(1 + 3*x)/(2*S
qrt[-1 - x + x^2])]/2

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Rubi [A]  time = 0.0582723, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {990, 621, 206, 1033, 724, 204} \[ -\frac{1}{2} \tan ^{-1}\left (\frac{3-x}{2 \sqrt{x^2-x-1}}\right )+\tanh ^{-1}\left (\frac{1-2 x}{2 \sqrt{x^2-x-1}}\right )+\frac{1}{2} \tanh ^{-1}\left (\frac{3 x+1}{2 \sqrt{x^2-x-1}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[-1 - x + x^2]/(1 - x^2),x]

[Out]

-ArcTan[(3 - x)/(2*Sqrt[-1 - x + x^2])]/2 + ArcTanh[(1 - 2*x)/(2*Sqrt[-1 - x + x^2])] + ArcTanh[(1 + 3*x)/(2*S
qrt[-1 - x + x^2])]/2

Rule 990

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

Rule 621

Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[2, Subst[Int[1/(4*c - x^2), x], x, (b + 2*c*x)
/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 206

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

Rule 1033

Int[((g_.) + (h_.)*(x_))/(((a_) + (c_.)*(x_)^2)*Sqrt[(d_.) + (e_.)*(x_) + (f_.)*(x_)^2]), x_Symbol] :> With[{q
= Rt[-(a*c), 2]}, Dist[h/2 + (c*g)/(2*q), Int[1/((-q + c*x)*Sqrt[d + e*x + f*x^2]), x], x] + Dist[h/2 - (c*g)
/(2*q), Int[1/((q + c*x)*Sqrt[d + e*x + f*x^2]), x], x]] /; FreeQ[{a, c, d, e, f, g, h}, x] && NeQ[e^2 - 4*d*f
, 0] && PosQ[-(a*c)]

Rule 724

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[-2, Subst[Int[1/(4*c*d
^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, (2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a,
b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[2*c*d - b*e, 0]

Rule 204

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

Rubi steps

\begin{align*} \int \frac{\sqrt{-1-x+x^2}}{1-x^2} \, dx &=-\int \frac{1}{\sqrt{-1-x+x^2}} \, dx-\int \frac{x}{\left (1-x^2\right ) \sqrt{-1-x+x^2}} \, dx\\ &=-\left (\frac{1}{2} \int \frac{1}{(-1-x) \sqrt{-1-x+x^2}} \, dx\right )-\frac{1}{2} \int \frac{1}{(1-x) \sqrt{-1-x+x^2}} \, dx-2 \operatorname{Subst}\left (\int \frac{1}{4-x^2} \, dx,x,\frac{-1+2 x}{\sqrt{-1-x+x^2}}\right )\\ &=\tanh ^{-1}\left (\frac{1-2 x}{2 \sqrt{-1-x+x^2}}\right )+\operatorname{Subst}\left (\int \frac{1}{-4-x^2} \, dx,x,\frac{3-x}{\sqrt{-1-x+x^2}}\right )+\operatorname{Subst}\left (\int \frac{1}{4-x^2} \, dx,x,\frac{1+3 x}{\sqrt{-1-x+x^2}}\right )\\ &=-\frac{1}{2} \tan ^{-1}\left (\frac{3-x}{2 \sqrt{-1-x+x^2}}\right )+\tanh ^{-1}\left (\frac{1-2 x}{2 \sqrt{-1-x+x^2}}\right )+\frac{1}{2} \tanh ^{-1}\left (\frac{1+3 x}{2 \sqrt{-1-x+x^2}}\right )\\ \end{align*}

Mathematica [A]  time = 0.0389001, size = 75, normalized size = 1. \[ -\frac{1}{2} \tan ^{-1}\left (\frac{3-x}{2 \sqrt{x^2-x-1}}\right )+\tanh ^{-1}\left (\frac{1-2 x}{2 \sqrt{x^2-x-1}}\right )+\frac{1}{2} \tanh ^{-1}\left (\frac{3 x+1}{2 \sqrt{x^2-x-1}}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-ArcTan[(3 - x)/(2*Sqrt[-1 - x + x^2])]/2 + ArcTanh[(1 - 2*x)/(2*Sqrt[-1 - x + x^2])] + ArcTanh[(1 + 3*x)/(2*S
qrt[-1 - x + x^2])]/2

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Maple [A]  time = 0.055, size = 102, normalized size = 1.4 \begin{align*}{\frac{1}{2}\sqrt{ \left ( 1+x \right ) ^{2}-2-3\,x}}-{\frac{3}{4}\ln \left ( -{\frac{1}{2}}+x+\sqrt{ \left ( 1+x \right ) ^{2}-2-3\,x} \right ) }-{\frac{1}{2}{\it Artanh} \left ({\frac{-1-3\,x}{2}{\frac{1}{\sqrt{ \left ( 1+x \right ) ^{2}-2-3\,x}}}} \right ) }-{\frac{1}{2}\sqrt{ \left ( -1+x \right ) ^{2}-2+x}}-{\frac{1}{4}\ln \left ( -{\frac{1}{2}}+x+\sqrt{ \left ( -1+x \right ) ^{2}-2+x} \right ) }+{\frac{1}{2}\arctan \left ({\frac{-3+x}{2}{\frac{1}{\sqrt{ \left ( -1+x \right ) ^{2}-2+x}}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.71787, size = 112, normalized size = 1.49 \begin{align*} \frac{1}{2} \, \arcsin \left (\frac{2 \, \sqrt{5} x}{5 \,{\left | 2 \, x - 2 \right |}} - \frac{6 \, \sqrt{5}}{5 \,{\left | 2 \, x - 2 \right |}}\right ) - \log \left (x + \sqrt{x^{2} - x - 1} - \frac{1}{2}\right ) - \frac{1}{2} \, \log \left (\frac{2 \, \sqrt{x^{2} - x - 1}}{{\left | 2 \, x + 2 \right |}} + \frac{2}{{\left | 2 \, x + 2 \right |}} - \frac{3}{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*arcsin(2/5*sqrt(5)*x/abs(2*x - 2) - 6/5*sqrt(5)/abs(2*x - 2)) - log(x + sqrt(x^2 - x - 1) - 1/2) - 1/2*log
(2*sqrt(x^2 - x - 1)/abs(2*x + 2) + 2/abs(2*x + 2) - 3/2)

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Fricas [A]  time = 1.80182, size = 197, normalized size = 2.63 \begin{align*} \arctan \left (-x + \sqrt{x^{2} - x - 1} + 1\right ) - \frac{1}{2} \, \log \left (-x + \sqrt{x^{2} - x - 1}\right ) + \frac{1}{2} \, \log \left (-x + \sqrt{x^{2} - x - 1} - 2\right ) + \log \left (-2 \, x + 2 \, \sqrt{x^{2} - x - 1} + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.27606, size = 99, normalized size = 1.32 \begin{align*} \arctan \left (-x + \sqrt{x^{2} - x - 1} + 1\right ) - \frac{1}{2} \, \log \left ({\left | -x + \sqrt{x^{2} - x - 1} \right |}\right ) + \frac{1}{2} \, \log \left ({\left | -x + \sqrt{x^{2} - x - 1} - 2 \right |}\right ) + \log \left ({\left | -2 \, x + 2 \, \sqrt{x^{2} - x - 1} + 1 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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