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

3.12.76.1 Optimal result

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

output

-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)))
 

3.12.76.2 Mathematica [A] (verified)

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

input

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

output

(-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])]
 

3.12.76.3 Rubi [A] (verified)

Time = 0.37 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.05, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {2035, 25, 2144, 25, 1091, 219, 1316, 1123, 1154, 219}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 2035

\(\displaystyle 2 \int -\frac {x}{(1-x) \sqrt {x-\sqrt {x}}}d\sqrt {x}\)

\(\Big \downarrow \) 25

\(\displaystyle -2 \int \frac {x}{(1-x) \sqrt {x-\sqrt {x}}}d\sqrt {x}\)

\(\Big \downarrow \) 2144

\(\displaystyle 2 \left (\int \frac {1}{\sqrt {x-\sqrt {x}}}d\sqrt {x}+\int -\frac {1}{(1-x) \sqrt {x-\sqrt {x}}}d\sqrt {x}\right )\)

\(\Big \downarrow \) 25

\(\displaystyle 2 \left (\int \frac {1}{\sqrt {x-\sqrt {x}}}d\sqrt {x}-\int \frac {1}{(1-x) \sqrt {x-\sqrt {x}}}d\sqrt {x}\right )\)

\(\Big \downarrow \) 1091

\(\displaystyle 2 \left (2 \int \frac {1}{1-x}d\frac {\sqrt {x}}{\sqrt {x-\sqrt {x}}}-\int \frac {1}{(1-x) \sqrt {x-\sqrt {x}}}d\sqrt {x}\right )\)

\(\Big \downarrow \) 219

\(\displaystyle 2 \left (2 \text {arctanh}\left (\frac {\sqrt {x}}{\sqrt {x-\sqrt {x}}}\right )-\int \frac {1}{(1-x) \sqrt {x-\sqrt {x}}}d\sqrt {x}\right )\)

\(\Big \downarrow \) 1316

\(\displaystyle 2 \left (-\frac {1}{2} \int \frac {1}{\left (1-\sqrt {x}\right ) \sqrt {x-\sqrt {x}}}d\sqrt {x}-\frac {1}{2} \int \frac {1}{\left (\sqrt {x}+1\right ) \sqrt {x-\sqrt {x}}}d\sqrt {x}+2 \text {arctanh}\left (\frac {\sqrt {x}}{\sqrt {x-\sqrt {x}}}\right )\right )\)

\(\Big \downarrow \) 1123

\(\displaystyle 2 \left (-\frac {1}{2} \int \frac {1}{\left (\sqrt {x}+1\right ) \sqrt {x-\sqrt {x}}}d\sqrt {x}+2 \text {arctanh}\left (\frac {\sqrt {x}}{\sqrt {x-\sqrt {x}}}\right )+\frac {\sqrt {x-\sqrt {x}}}{1-\sqrt {x}}\right )\)

\(\Big \downarrow \) 1154

\(\displaystyle 2 \left (\int \frac {1}{8-x}d\frac {1-3 \sqrt {x}}{\sqrt {x-\sqrt {x}}}+2 \text {arctanh}\left (\frac {\sqrt {x}}{\sqrt {x-\sqrt {x}}}\right )+\frac {\sqrt {x-\sqrt {x}}}{1-\sqrt {x}}\right )\)

\(\Big \downarrow \) 219

\(\displaystyle 2 \left (\frac {\text {arctanh}\left (\frac {1-3 \sqrt {x}}{2 \sqrt {2} \sqrt {x-\sqrt {x}}}\right )}{2 \sqrt {2}}+2 \text {arctanh}\left (\frac {\sqrt {x}}{\sqrt {x-\sqrt {x}}}\right )+\frac {\sqrt {x-\sqrt {x}}}{1-\sqrt {x}}\right )\)

input

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

output

2*(Sqrt[-Sqrt[x] + x]/(1 - Sqrt[x]) + ArcTanh[(1 - 3*Sqrt[x])/(2*Sqrt[2]*S 
qrt[-Sqrt[x] + x])]/(2*Sqrt[2]) + 2*ArcTanh[Sqrt[x]/Sqrt[-Sqrt[x] + x]])
 

3.12.76.3.1 Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 219

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] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 1091

Int[1/Sqrt[(b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[2   Subst[Int[1/(1 
 - c*x^2), x], x, x/Sqrt[b*x + c*x^2]], x] /; FreeQ[{b, c}, x]
 

rule 1123

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S 
ymbol] :> Simp[e*(d + e*x)^m*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(2*c*d - b 
*e))), x] /; FreeQ[{a, b, c, d, e, m, p}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 
0] && EqQ[m + 2*p + 2, 0]
 

rule 1154

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Sym 
bol] :> Simp[-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]
 

rule 1316

Int[1/(((a_) + (c_.)*(x_)^2)*Sqrt[(d_.) + (e_.)*(x_) + (f_.)*(x_)^2]), x_Sy 
mbol] :> Simp[1/2   Int[1/((a - Rt[(-a)*c, 2]*x)*Sqrt[d + e*x + f*x^2]), x] 
, x] + Simp[1/2   Int[1/((a + Rt[(-a)*c, 2]*x)*Sqrt[d + e*x + f*x^2]), x], 
x] /; FreeQ[{a, c, d, e, f}, x] && NeQ[e^2 - 4*d*f, 0] && PosQ[(-a)*c]
 

rule 2035

Int[(Fx_)*(x_)^(m_), x_Symbol] :> With[{k = Denominator[m]}, Simp[k   Subst 
[Int[x^(k*(m + 1) - 1)*SubstPower[Fx, x, k], x], x, x^(1/k)], x]] /; Fracti 
onQ[m] && AlgebraicFunctionQ[Fx, x]
 

rule 2144

Int[(Px_)/(((a_) + (c_.)*(x_)^2)*Sqrt[(d_.) + (e_.)*(x_) + (f_.)*(x_)^2]), 
x_Symbol] :> With[{A = Coeff[Px, x, 0], B = Coeff[Px, x, 1], C = Coeff[Px, 
x, 2]}, Simp[C/c   Int[1/Sqrt[d + e*x + f*x^2], x], x] + Simp[1/c   Int[(A* 
c - a*C + B*c*x)/((a + c*x^2)*Sqrt[d + e*x + f*x^2]), x], x]] /; FreeQ[{a, 
c, d, e, f}, x] && PolyQ[Px, x, 2]
 

3.12.76.4 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	\(-\frac {\sqrt {-\sqrt {x}+x}\, \left (-2 \sqrt {x}\, \operatorname {arctanh}\left (\frac {\left (-1+3 \sqrt {x}\right ) \sqrt {2}}{4 \sqrt {-\sqrt {x}+x}}\right ) \sqrt {2}+4 \left (-\sqrt {x}+x \right )^{\frac {3}{2}}+x \,\operatorname {arctanh}\left (\frac {\left (-1+3 \sqrt {x}\right ) \sqrt {2}}{4 \sqrt {-\sqrt {x}+x}}\right ) \sqrt {2}+8 \sqrt {x}\, \sqrt {-\sqrt {x}+x}+8 \sqrt {x}\, \ln \left (-\frac {1}{2}+\sqrt {x}+\sqrt {-\sqrt {x}+x}\right )-4 x \sqrt {-\sqrt {x}+x}-4 x \ln \left (-\frac {1}{2}+\sqrt {x}+\sqrt {-\sqrt {x}+x}\right )+\sqrt {2}\, \operatorname {arctanh}\left (\frac {\left (-1+3 \sqrt {x}\right ) \sqrt {2}}{4 \sqrt {-\sqrt {x}+x}}\right )-4 \sqrt {-\sqrt {x}+x}-4 \ln \left (-\frac {1}{2}+\sqrt {x}+\sqrt {-\sqrt {x}+x}\right )\right )}{2 \sqrt {\sqrt {x}\, \left (\sqrt {x}-1\right )}\, \left (\sqrt {x}-1\right )^{2}}\)	\(217\)

input

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

output

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))
 

3.12.76.5 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 \[ \int \frac {\sqrt {x}}{(-1+x) \sqrt {-\sqrt {x}+x}} \, dx=\frac {\sqrt {2} {\left (x - 1\right )} \log \left (-\frac {17 \, x^{2} - 4 \, {\left (\sqrt {2} {\left (3 \, x + 5\right )} \sqrt {x} - \sqrt {2} {\left (7 \, x + 1\right )}\right )} \sqrt {x - \sqrt {x}} - 16 \, {\left (3 \, x + 1\right )} \sqrt {x} + 46 \, x + 1}{x^{2} - 2 \, x + 1}\right ) + 4 \, {\left (x - 1\right )} \log \left (-4 \, \sqrt {x - \sqrt {x}} {\left (2 \, \sqrt {x} - 1\right )} - 8 \, x + 8 \, \sqrt {x} - 1\right ) - 8 \, \sqrt {x - \sqrt {x}} {\left (\sqrt {x} + 1\right )}}{4 \, {\left (x - 1\right )}} \]

input

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

output

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*sqr 
t(x) - 1) - 8*sqrt(x - sqrt(x))*(sqrt(x) + 1))/(x - 1)
 

3.12.76.6 Sympy [F]

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

input

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

output

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

3.12.76.7 Maxima [F]

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

input

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

output

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

3.12.76.8 Giac [A] (verification not implemented)

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

input

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

output

-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))
 

3.12.76.9 Mupad [F(-1)]

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

input

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

output

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