\(\int \frac {\sqrt {x}}{(-1+x) \sqrt {-\sqrt {x}+x}} \, dx\) [1176]

Optimal result

Integrand size = 24, antiderivative size = 87 \[ \int \frac {\sqrt {x}}{(-1+x) \sqrt {-\sqrt {x}+x}} \, dx=-\frac {2 \sqrt {-\sqrt {x}+x}}{-1+\sqrt {x}}+4 \text {arctanh}\left (\frac {\sqrt {-\sqrt {x}+x}}{-1+\sqrt {x}}\right )-\sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {-\sqrt {x}+x}}{-1+\sqrt {x}}\right ) \]

[Out]

-2*(-x^(1/2)+x)^(1/2)/(-1+x^(1/2))+4*arctanh((-x^(1/2)+x)^(1/2)/(-1+x^(1/2)))-2^(1/2)*arctanh(2^(1/2)*(-x^(1/2
)+x)^(1/2)/(-1+x^(1/2)))

Rubi [A] (verified)

Time = 0.16 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.41, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {2081, 1563, 862, 100, 163, 56, 221, 95, 212} \[ \int \frac {\sqrt {x}}{(-1+x) \sqrt {-\sqrt {x}+x}} \, dx=\frac {4 \sqrt {\sqrt {x}-1} \sqrt [4]{x} \text {arcsinh}\left (\sqrt {\sqrt {x}-1}\right )}{\sqrt {x-\sqrt {x}}}-\frac {\sqrt {2} \sqrt {\sqrt {x}-1} \sqrt [4]{x} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{x}}{\sqrt {\sqrt {x}-1}}\right )}{\sqrt {x-\sqrt {x}}}-\frac {2 \sqrt {x}}{\sqrt {x-\sqrt {x}}} \]

[In]

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

[Out]

(-2*Sqrt[x])/Sqrt[-Sqrt[x] + x] + (4*Sqrt[-1 + Sqrt[x]]*x^(1/4)*ArcSinh[Sqrt[-1 + Sqrt[x]]])/Sqrt[-Sqrt[x] + x
] - (Sqrt[2]*Sqrt[-1 + Sqrt[x]]*x^(1/4)*ArcTanh[(Sqrt[2]*x^(1/4))/Sqrt[-1 + Sqrt[x]]])/Sqrt[-Sqrt[x] + x]

Rule 56

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

Rule 95

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> With[{q = Denomin
ator[m]}, Dist[q, Subst[Int[x^(q*(m + 1) - 1)/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^
(1/q)], x]] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && LtQ[-1, m, 0] && SimplerQ[
a + b*x, c + d*x]

Rule 100

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*c -
a*d)*(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*((e + f*x)^(p + 1)/(b*(b*e - a*f)*(m + 1))), x] + Dist[1/(b*(b*e - a*
f)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 2)*(e + f*x)^p*Simp[a*d*(d*e*(n - 1) + c*f*(p + 1)) + b*c*(d
*e*(m - n + 2) - c*f*(m + p + 2)) + d*(a*d*f*(n + p) + b*(d*e*(m + 1) - c*f*(m + n + p + 1)))*x, x], x], x] /;
FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && GtQ[n, 1] && (IntegersQ[2*m, 2*n, 2*p] || IntegersQ[m, n + p
] || IntegersQ[p, m + n])

Rule 163

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

Rule 212

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

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 862

Int[((d_) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[(d + e*x)
^(m + p)*(f + g*x)^n*(a/d + (c/e)*x)^p, x] /; FreeQ[{a, c, d, e, f, g, m, n}, x] && NeQ[e*f - d*g, 0] && EqQ[c
*d^2 + a*e^2, 0] && (IntegerQ[p] || (GtQ[a, 0] && GtQ[d, 0] && EqQ[m + p, 0]))

Rule 1563

Int[(x_)^(m_.)*((a_) + (c_.)*(x_)^(n2_.))^(p_)*((d_) + (e_.)*(x_)^(n_))^(q_.), x_Symbol] :> With[{g = Denomina
tor[n]}, Dist[g, Subst[Int[x^(g*(m + 1) - 1)*(d + e*x^(g*n))^q*(a + c*x^(2*g*n))^p, x], x, x^(1/g)], x]] /; Fr
eeQ[{a, c, d, e, m, p, q}, x] && EqQ[n2, 2*n] && FractionQ[n]

Rule 2081

Int[(u_.)*(P_)^(p_.), x_Symbol] :> With[{m = MinimumMonomialExponent[P, x]}, Dist[P^FracPart[p]/(x^(m*FracPart
[p])*Distrib[1/x^m, P]^FracPart[p]), Int[u*x^(m*p)*Distrib[1/x^m, P]^p, x], x]] /; FreeQ[p, x] &&  !IntegerQ[p
] && SumQ[P] && EveryQ[BinomialQ[#1, x] & , P] &&  !PolyQ[P, x, 2]

Rubi steps \begin{align*} \text {integral}& = \frac {\left (\sqrt {-1+\sqrt {x}} \sqrt [4]{x}\right ) \int \frac {\sqrt [4]{x}}{\sqrt {-1+\sqrt {x}} (-1+x)} \, dx}{\sqrt {-\sqrt {x}+x}} \\ & = \frac {\left (2 \sqrt {-1+\sqrt {x}} \sqrt [4]{x}\right ) \text {Subst}\left (\int \frac {x^{3/2}}{\sqrt {-1+x} \left (-1+x^2\right )} \, dx,x,\sqrt {x}\right )}{\sqrt {-\sqrt {x}+x}} \\ & = \frac {\left (2 \sqrt {-1+\sqrt {x}} \sqrt [4]{x}\right ) \text {Subst}\left (\int \frac {x^{3/2}}{(-1+x)^{3/2} (1+x)} \, dx,x,\sqrt {x}\right )}{\sqrt {-\sqrt {x}+x}} \\ & = -\frac {2 \sqrt {x}}{\sqrt {-\sqrt {x}+x}}-\frac {\left (2 \sqrt {-1+\sqrt {x}} \sqrt [4]{x}\right ) \text {Subst}\left (\int \frac {-\frac {1}{2}-x}{\sqrt {-1+x} \sqrt {x} (1+x)} \, dx,x,\sqrt {x}\right )}{\sqrt {-\sqrt {x}+x}} \\ & = -\frac {2 \sqrt {x}}{\sqrt {-\sqrt {x}+x}}-\frac {\left (\sqrt {-1+\sqrt {x}} \sqrt [4]{x}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-1+x} \sqrt {x} (1+x)} \, dx,x,\sqrt {x}\right )}{\sqrt {-\sqrt {x}+x}}+\frac {\left (2 \sqrt {-1+\sqrt {x}} \sqrt [4]{x}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-1+x} \sqrt {x}} \, dx,x,\sqrt {x}\right )}{\sqrt {-\sqrt {x}+x}} \\ & = -\frac {2 \sqrt {x}}{\sqrt {-\sqrt {x}+x}}-\frac {\left (2 \sqrt {-1+\sqrt {x}} \sqrt [4]{x}\right ) \text {Subst}\left (\int \frac {1}{1-2 x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt {-1+\sqrt {x}}}\right )}{\sqrt {-\sqrt {x}+x}}+\frac {\left (4 \sqrt {-1+\sqrt {x}} \sqrt [4]{x}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1+x^2}} \, dx,x,\sqrt {-1+\sqrt {x}}\right )}{\sqrt {-\sqrt {x}+x}} \\ & = -\frac {2 \sqrt {x}}{\sqrt {-\sqrt {x}+x}}+\frac {4 \sqrt {-1+\sqrt {x}} \sqrt [4]{x} \text {arcsinh}\left (\sqrt {-1+\sqrt {x}}\right )}{\sqrt {-\sqrt {x}+x}}-\frac {\sqrt {2} \sqrt {-1+\sqrt {x}} \sqrt [4]{x} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{x}}{\sqrt {-1+\sqrt {x}}}\right )}{\sqrt {-\sqrt {x}+x}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.19 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {x}}{(-1+x) \sqrt {-\sqrt {x}+x}} \, dx=-\frac {2 \sqrt {-\sqrt {x}+x}}{-1+\sqrt {x}}+4 \text {arctanh}\left (\frac {\sqrt {-\sqrt {x}+x}}{-1+\sqrt {x}}\right )-\sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {-\sqrt {x}+x}}{-1+\sqrt {x}}\right ) \]

[In]

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

[Out]

(-2*Sqrt[-Sqrt[x] + x])/(-1 + Sqrt[x]) + 4*ArcTanh[Sqrt[-Sqrt[x] + x]/(-1 + Sqrt[x])] - Sqrt[2]*ArcTanh[(Sqrt[
2]*Sqrt[-Sqrt[x] + x])/(-1 + Sqrt[x])]

Maple [A] (verified)

Time = 1.13 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.87

method	result	size
derivativedivides	\(2 \ln \left (-\frac {1}{2}+\sqrt {x}+\sqrt {-\sqrt {x}+x}\right )-\frac {2 \sqrt {\left (\sqrt {x}-1\right )^{2}+\sqrt {x}-1}}{\sqrt {x}-1}+\frac {\sqrt {2}\, \operatorname {arctanh}\left (\frac {\left (1-3 \sqrt {x}\right ) \sqrt {2}}{4 \sqrt {\left (\sqrt {x}+1\right )^{2}-3 \sqrt {x}-1}}\right )}{2}\)	\(76\)
default	\(-\frac {\sqrt {-\sqrt {x}+x}\, \left (-2 \sqrt {x}\, \operatorname {arctanh}\left (\frac {\left (-1+3 \sqrt {x}\right ) \sqrt {2}}{4 \sqrt {-\sqrt {x}+x}}\right ) \sqrt {2}+4 \left (-\sqrt {x}+x \right )^{\frac {3}{2}}+x \,\operatorname {arctanh}\left (\frac {\left (-1+3 \sqrt {x}\right ) \sqrt {2}}{4 \sqrt {-\sqrt {x}+x}}\right ) \sqrt {2}+8 \sqrt {x}\, \sqrt {-\sqrt {x}+x}+8 \sqrt {x}\, \ln \left (-\frac {1}{2}+\sqrt {x}+\sqrt {-\sqrt {x}+x}\right )-4 x \sqrt {-\sqrt {x}+x}-4 x \ln \left (-\frac {1}{2}+\sqrt {x}+\sqrt {-\sqrt {x}+x}\right )+\sqrt {2}\, \operatorname {arctanh}\left (\frac {\left (-1+3 \sqrt {x}\right ) \sqrt {2}}{4 \sqrt {-\sqrt {x}+x}}\right )-4 \sqrt {-\sqrt {x}+x}-4 \ln \left (-\frac {1}{2}+\sqrt {x}+\sqrt {-\sqrt {x}+x}\right )\right )}{2 \sqrt {\sqrt {x}\, \left (\sqrt {x}-1\right )}\, \left (\sqrt {x}-1\right )^{2}}\)	\(217\)

[In]

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

[Out]

2*ln(-1/2+x^(1/2)+(-x^(1/2)+x)^(1/2))-2/(x^(1/2)-1)*((x^(1/2)-1)^2+x^(1/2)-1)^(1/2)+1/2*2^(1/2)*arctanh(1/4*(1
-3*x^(1/2))*2^(1/2)/((x^(1/2)+1)^2-3*x^(1/2)-1)^(1/2))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 132 vs. \(2 (65) = 130\).

Time = 1.19 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.52 \[ \int \frac {\sqrt {x}}{(-1+x) \sqrt {-\sqrt {x}+x}} \, dx=\frac {\sqrt {2} {\left (x - 1\right )} \log \left (-\frac {17 \, x^{2} - 4 \, {\left (\sqrt {2} {\left (3 \, x + 5\right )} \sqrt {x} - \sqrt {2} {\left (7 \, x + 1\right )}\right )} \sqrt {x - \sqrt {x}} - 16 \, {\left (3 \, x + 1\right )} \sqrt {x} + 46 \, x + 1}{x^{2} - 2 \, x + 1}\right ) + 4 \, {\left (x - 1\right )} \log \left (-4 \, \sqrt {x - \sqrt {x}} {\left (2 \, \sqrt {x} - 1\right )} - 8 \, x + 8 \, \sqrt {x} - 1\right ) - 8 \, \sqrt {x - \sqrt {x}} {\left (\sqrt {x} + 1\right )}}{4 \, {\left (x - 1\right )}} \]

[In]

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

[Out]

1/4*(sqrt(2)*(x - 1)*log(-(17*x^2 - 4*(sqrt(2)*(3*x + 5)*sqrt(x) - sqrt(2)*(7*x + 1))*sqrt(x - sqrt(x)) - 16*(
3*x + 1)*sqrt(x) + 46*x + 1)/(x^2 - 2*x + 1)) + 4*(x - 1)*log(-4*sqrt(x - sqrt(x))*(2*sqrt(x) - 1) - 8*x + 8*s
qrt(x) - 1) - 8*sqrt(x - sqrt(x))*(sqrt(x) + 1))/(x - 1)

Sympy [F]

\[ \int \frac {\sqrt {x}}{(-1+x) \sqrt {-\sqrt {x}+x}} \, dx=\int \frac {\sqrt {x}}{\sqrt {- \sqrt {x} + x} \left (x - 1\right )}\, dx \]

[In]

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

[Out]

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

Maxima [F]

\[ \int \frac {\sqrt {x}}{(-1+x) \sqrt {-\sqrt {x}+x}} \, dx=\int { \frac {\sqrt {x}}{\sqrt {x - \sqrt {x}} {\left (x - 1\right )}} \,d x } \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.58 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.10 \[ \int \frac {\sqrt {x}}{(-1+x) \sqrt {-\sqrt {x}+x}} \, dx=-\frac {1}{2} \, \sqrt {2} \log \left (\frac {2 \, {\left (\sqrt {2} - \sqrt {x - \sqrt {x}} + \sqrt {x} + 1\right )}}{{\left | 2 \, \sqrt {2} + 2 \, \sqrt {x - \sqrt {x}} - 2 \, \sqrt {x} - 2 \right |}}\right ) - \frac {2}{\sqrt {x - \sqrt {x}} - \sqrt {x} + 1} - 2 \, \log \left ({\left | 2 \, \sqrt {x - \sqrt {x}} - 2 \, \sqrt {x} + 1 \right |}\right ) \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {x}}{(-1+x) \sqrt {-\sqrt {x}+x}} \, dx=\int \frac {\sqrt {x}}{\sqrt {x-\sqrt {x}}\,\left (x-1\right )} \,d x \]

[In]

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

[Out]

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