3.8.20 \(\int \frac {1}{\sqrt {-1+x} \sqrt {-\sqrt {-1+x}+x}} \, dx\) [720]

Optimal. Leaf size=20 \[ -2 \sinh ^{-1}\left (\frac {1-2 \sqrt {-1+x}}{\sqrt {3}}\right ) \]

[Out]

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

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Rubi [A]
time = 0.05, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {633, 221} \begin {gather*} -2 \sinh ^{-1}\left (\frac {1-2 \sqrt {x-1}}{\sqrt {3}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[-1 + x]*Sqrt[-Sqrt[-1 + x] + x]),x]

[Out]

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

Rule 221

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

Rule 633

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

Rubi steps

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

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Mathematica [A]
time = 0.00, size = 31, normalized size = 1.55 \begin {gather*} -2 \log \left (1-2 \sqrt {-1+x}+2 \sqrt {-\sqrt {-1+x}+x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-2*Log[1 - 2*Sqrt[-1 + x] + 2*Sqrt[-Sqrt[-1 + x] + x]]

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Maple [A]
time = 0.37, size = 16, normalized size = 0.80

method	result	size
derivativedivides	\(2 \arcsinh \left (\frac {2 \sqrt {3}\, \left (\sqrt {-1+x}-\frac {1}{2}\right )}{3}\right )\)	\(16\)
default	\(2 \arcsinh \left (\frac {2 \sqrt {3}\, \left (\sqrt {-1+x}-\frac {1}{2}\right )}{3}\right )\)	\(16\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(-1+x)^(1/2)/(x-(-1+x)^(1/2))^(1/2),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 35 vs. \(2 (17) = 34\).
time = 0.57, size = 35, normalized size = 1.75 \begin {gather*} \log \left (4 \, \sqrt {x - \sqrt {x - 1}} {\left (2 \, \sqrt {x - 1} - 1\right )} + 8 \, x - 8 \, \sqrt {x - 1} - 3\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(4*sqrt(x - sqrt(x - 1))*(2*sqrt(x - 1) - 1) + 8*x - 8*sqrt(x - 1) - 3)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 1.42, size = 25, normalized size = 1.25 \begin {gather*} -2 \, \log \left (2 \, \sqrt {x - \sqrt {x - 1}} - 2 \, \sqrt {x - 1} + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*log(2*sqrt(x - sqrt(x - 1)) - 2*sqrt(x - 1) + 1)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.05 \begin {gather*} \int \frac {1}{\sqrt {x-\sqrt {x-1}}\,\sqrt {x-1}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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