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

Optimal. Leaf size=39 \[ \sqrt {x-\sqrt {x^2-1}}+\frac {1}{3 \left (x-\sqrt {x^2-1}\right )^{3/2}} \]

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Rubi [A]  time = 0.02, antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {2117, 14} \begin {gather*} \sqrt {x-\sqrt {x^2-1}}+\frac {1}{3 \left (x-\sqrt {x^2-1}\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[x - Sqrt[-1 + x^2]],x]

[Out]

1/(3*(x - Sqrt[-1 + x^2])^(3/2)) + Sqrt[x - Sqrt[-1 + x^2]]

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 2117

Int[((g_.) + (h_.)*((d_.) + (e_.)*(x_) + (f_.)*Sqrt[(a_) + (c_.)*(x_)^2])^(n_))^(p_.), x_Symbol] :> Dist[1/(2*
e), Subst[Int[((g + h*x^n)^p*(d^2 + a*f^2 - 2*d*x + x^2))/(d - x)^2, x], x, d + e*x + f*Sqrt[a + c*x^2]], x] /
; FreeQ[{a, c, d, e, f, g, h, n}, x] && EqQ[e^2 - c*f^2, 0] && IntegerQ[p]

Rubi steps

\begin {align*} \int \frac {1}{\sqrt {x-\sqrt {-1+x^2}}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {-1+x^2}{x^{5/2}} \, dx,x,x-\sqrt {-1+x^2}\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (-\frac {1}{x^{5/2}}+\frac {1}{\sqrt {x}}\right ) \, dx,x,x-\sqrt {-1+x^2}\right )\\ &=\frac {1}{3 \left (x-\sqrt {-1+x^2}\right )^{3/2}}+\sqrt {x-\sqrt {-1+x^2}}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 39, normalized size = 1.00 \begin {gather*} \sqrt {x-\sqrt {x^2-1}}+\frac {1}{3 \left (x-\sqrt {x^2-1}\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

1/(3*(x - Sqrt[-1 + x^2])^(3/2)) + Sqrt[x - Sqrt[-1 + x^2]]

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IntegrateAlgebraic [A]  time = 0.06, size = 39, normalized size = 1.00 \begin {gather*} \frac {1}{3 \left (x-\sqrt {-1+x^2}\right )^{3/2}}+\sqrt {x-\sqrt {-1+x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

1/(3*(x - Sqrt[-1 + x^2])^(3/2)) + Sqrt[x - Sqrt[-1 + x^2]]

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fricas [A]  time = 0.48, size = 29, normalized size = 0.74 \begin {gather*} \frac {2}{3} \, {\left (x^{2} + \sqrt {x^{2} - 1} x + 1\right )} \sqrt {x - \sqrt {x^{2} - 1}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {1}{\sqrt {x -\sqrt {x^{2}-1}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x - (x^2 - 1)^(1/2))^(1/2),x)

[Out]

int(1/(x - (x^2 - 1)^(1/2))^(1/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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