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

Optimal. Leaf size=31 \[ \sqrt{2} \tan ^{-1}\left (\frac{\sqrt{1-x^2}}{\sqrt{2} \sqrt{x-1}}\right ) \]

[Out]

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

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Rubi [A]  time = 0.0103923, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {661, 203} \[ \sqrt{2} \tan ^{-1}\left (\frac{\sqrt{1-x^2}}{\sqrt{2} \sqrt{x-1}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[-1 + x]*Sqrt[1 - x^2]),x]

[Out]

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

Rule 661

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

Rule 203

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

Rubi steps

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

Mathematica [A]  time = 0.027545, size = 46, normalized size = 1.48 \[ -\frac{\sqrt{2} \sqrt{x-1} \sqrt{x+1} \tanh ^{-1}\left (\frac{\sqrt{x+1}}{\sqrt{2}}\right )}{\sqrt{1-x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.085, size = 39, normalized size = 1.3 \begin{align*}{\sqrt{2}\sqrt{-{x}^{2}+1}\arctan \left ({\frac{\sqrt{2}}{2}\sqrt{-1-x}} \right ){\frac{1}{\sqrt{-1+x}}}{\frac{1}{\sqrt{-1-x}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/(-1+x)^(1/2)*(-x^2+1)^(1/2)/(-1-x)^(1/2)*2^(1/2)*arctan(1/2*(-1-x)^(1/2)*2^(1/2))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 2.1309, size = 84, normalized size = 2.71 \begin{align*} \sqrt{2} \arctan \left (\frac{\sqrt{2} \sqrt{-x^{2} + 1} \sqrt{x - 1}}{x^{2} - 1}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{- \left (x - 1\right ) \left (x + 1\right )} \sqrt{x - 1}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [C]  time = 1.24628, size = 49, normalized size = 1.58 \begin{align*} \frac{1}{2} i \,{\left (\sqrt{2} \log \left (\sqrt{2} + \sqrt{x + 1}\right ) - \sqrt{2} \log \left (-\sqrt{2} + \sqrt{x + 1}\right )\right )} \mathrm{sgn}\left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*I*(sqrt(2)*log(sqrt(2) + sqrt(x + 1)) - sqrt(2)*log(-sqrt(2) + sqrt(x + 1)))*sgn(x)