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

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

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Rubi [A]  time = 0.01, antiderivative size = 31, normalized size of antiderivative = 1.00, 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} \begin {gather*} \sqrt {2} \tan ^{-1}\left (\frac {\sqrt {1-x^2}}{\sqrt {2} \sqrt {x-1}}\right ) \end {gather*}

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 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])

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]

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*}

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Mathematica [A]  time = 0.03, size = 46, normalized size = 1.48 \begin {gather*} -\frac {\sqrt {2} \sqrt {x-1} \sqrt {x+1} \tanh ^{-1}\left (\frac {\sqrt {x+1}}{\sqrt {2}}\right )}{\sqrt {1-x^2}} \end {gather*}

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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IntegrateAlgebraic [A]  time = 0.07, size = 38, normalized size = 1.23 \begin {gather*} -\sqrt {2} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {x-1}}{\sqrt {-(x-1)^2-2 (x-1)}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.39, size = 30, normalized size = 0.97 \begin {gather*} \sqrt {2} \arctan \left (\frac {\sqrt {2} \sqrt {-x^{2} + 1} \sqrt {x - 1}}{x^{2} - 1}\right ) \end {gather*}

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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giac [C]  time = 0.18, size = 36, normalized size = 1.16 \begin {gather*} \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}\relax (x) \end {gather*}

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)

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maple [A]  time = 0.06, size = 39, normalized size = 1.26 \begin {gather*} \frac {\sqrt {-x^{2}+1}\, \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {-x -1}}{2}\right )}{\sqrt {x -1}\, \sqrt {-x -1}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {-x^{2} + 1} \sqrt {x - 1}}\,{d x} \end {gather*}

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {1}{\sqrt {1-x^2}\,\sqrt {x-1}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

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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