3.727 \(\int \frac{\sqrt{-1-\sqrt{x}+x}}{(-1+x) \sqrt{x}} \, dx\)

Optimal. Leaf size=89 \[ \tan ^{-1}\left (\frac{3-\sqrt{x}}{2 \sqrt{x-\sqrt{x}-1}}\right )-2 \tanh ^{-1}\left (\frac{1-2 \sqrt{x}}{2 \sqrt{x-\sqrt{x}-1}}\right )-\tanh ^{-1}\left (\frac{3 \sqrt{x}+1}{2 \sqrt{x-\sqrt{x}-1}}\right ) \]

[Out]

ArcTan[(3 - Sqrt[x])/(2*Sqrt[-1 - Sqrt[x] + x])] - 2*ArcTanh[(1 - 2*Sqrt[x])/(2*Sqrt[-1 - Sqrt[x] + x])] - Arc
Tanh[(1 + 3*Sqrt[x])/(2*Sqrt[-1 - Sqrt[x] + x])]

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Rubi [A]  time = 0.263023, antiderivative size = 89, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.24, Rules used = {990, 621, 206, 1033, 724, 204} \[ \tan ^{-1}\left (\frac{3-\sqrt{x}}{2 \sqrt{x-\sqrt{x}-1}}\right )-2 \tanh ^{-1}\left (\frac{1-2 \sqrt{x}}{2 \sqrt{x-\sqrt{x}-1}}\right )-\tanh ^{-1}\left (\frac{3 \sqrt{x}+1}{2 \sqrt{x-\sqrt{x}-1}}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

ArcTan[(3 - Sqrt[x])/(2*Sqrt[-1 - Sqrt[x] + x])] - 2*ArcTanh[(1 - 2*Sqrt[x])/(2*Sqrt[-1 - Sqrt[x] + x])] - Arc
Tanh[(1 + 3*Sqrt[x])/(2*Sqrt[-1 - Sqrt[x] + x])]

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

Mathematica [A]  time = 0.0621889, size = 89, normalized size = 1. \[ \tan ^{-1}\left (\frac{3-\sqrt{x}}{2 \sqrt{x-\sqrt{x}-1}}\right )-2 \tanh ^{-1}\left (\frac{1-2 \sqrt{x}}{2 \sqrt{x-\sqrt{x}-1}}\right )-\tanh ^{-1}\left (\frac{3 \sqrt{x}+1}{2 \sqrt{x-\sqrt{x}-1}}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

ArcTan[(3 - Sqrt[x])/(2*Sqrt[-1 - Sqrt[x] + x])] - 2*ArcTanh[(1 - 2*Sqrt[x])/(2*Sqrt[-1 - Sqrt[x] + x])] - Arc
Tanh[(1 + 3*Sqrt[x])/(2*Sqrt[-1 - Sqrt[x] + x])]

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Maple [A]  time = 0.013, size = 130, normalized size = 1.5 \begin{align*} \sqrt{ \left ( -1+\sqrt{x} \right ) ^{2}+\sqrt{x}-2}+{\frac{1}{2}\ln \left ( \sqrt{x}-{\frac{1}{2}}+\sqrt{ \left ( -1+\sqrt{x} \right ) ^{2}+\sqrt{x}-2} \right ) }-\arctan \left ({\frac{1}{2} \left ( \sqrt{x}-3 \right ){\frac{1}{\sqrt{ \left ( -1+\sqrt{x} \right ) ^{2}+\sqrt{x}-2}}}} \right ) -\sqrt{ \left ( 1+\sqrt{x} \right ) ^{2}-3\,\sqrt{x}-2}+{\frac{3}{2}\ln \left ( \sqrt{x}-{\frac{1}{2}}+\sqrt{ \left ( 1+\sqrt{x} \right ) ^{2}-3\,\sqrt{x}-2} \right ) }+{\it Artanh} \left ({\frac{1}{2} \left ( -1-3\,\sqrt{x} \right ){\frac{1}{\sqrt{ \left ( 1+\sqrt{x} \right ) ^{2}-3\,\sqrt{x}-2}}}} \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 23.5296, size = 248, normalized size = 2.79 \begin{align*} -\arctan \left (\frac{{\left ({\left (x - 4\right )} \sqrt{x} - 2 \, x + 3\right )} \sqrt{x - \sqrt{x} - 1}}{2 \,{\left (x^{2} - 3 \, x + 1\right )}}\right ) + \log \left (-\frac{8 \, x^{2} + 2 \,{\left ({\left (4 \, x - 5\right )} \sqrt{x} + 2 \, x - 1\right )} \sqrt{x - \sqrt{x} - 1} - 17 \, x - 2 \, \sqrt{x} + 11}{x - 1}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-arctan(1/2*((x - 4)*sqrt(x) - 2*x + 3)*sqrt(x - sqrt(x) - 1)/(x^2 - 3*x + 1)) + log(-(8*x^2 + 2*((4*x - 5)*sq
rt(x) + 2*x - 1)*sqrt(x - sqrt(x) - 1) - 17*x - 2*sqrt(x) + 11)/(x - 1))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: NotImplementedError