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

Optimal result

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

Mathematica [A] (verified)

Time = 0.20 (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])]
 

Rubi [A] (verified)

Time = 0.35 (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]])
 

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]
 

Maple [A] (verified)

Time = 0.11 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.87

method	result	size
derivativedivides	\(2 \ln \left (-\frac {1}{2}+\sqrt {x}+\sqrt {-\sqrt {x}+x}\right )-\frac {2 \sqrt {\left (-1+\sqrt {x}\right )^{2}+\sqrt {x}-1}}{-1+\sqrt {x}}+\frac {\sqrt {2}\, \operatorname {arctanh}\left (\frac {\left (1-3 \sqrt {x}\right ) \sqrt {2}}{4 \sqrt {\left (1+\sqrt {x}\right )^{2}-3 \sqrt {x}-1}}\right )}{2}\)	\(76\)
default	\(\frac {\sqrt {-\sqrt {x}+x}\, \left (2 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\left (-1+3 \sqrt {x}\right ) \sqrt {2}}{4 \sqrt {-\sqrt {x}+x}}\right ) \sqrt {x}-\sqrt {2}\, \operatorname {arctanh}\left (\frac {\left (-1+3 \sqrt {x}\right ) \sqrt {2}}{4 \sqrt {-\sqrt {x}+x}}\right ) x -4 \left (-\sqrt {x}+x \right )^{\frac {3}{2}}-\sqrt {2}\, \operatorname {arctanh}\left (\frac {\left (-1+3 \sqrt {x}\right ) \sqrt {2}}{4 \sqrt {-\sqrt {x}+x}}\right )-8 \ln \left (-\frac {1}{2}+\sqrt {x}+\sqrt {-\sqrt {x}+x}\right ) \sqrt {x}+4 \ln \left (-\frac {1}{2}+\sqrt {x}+\sqrt {-\sqrt {x}+x}\right ) x -8 \sqrt {-\sqrt {x}+x}\, \sqrt {x}+4 \sqrt {-\sqrt {x}+x}\, x +4 \ln \left (-\frac {1}{2}+\sqrt {x}+\sqrt {-\sqrt {x}+x}\right )+4 \sqrt {-\sqrt {x}+x}\right )}{2 \sqrt {\sqrt {x}\, \left (-1+\sqrt {x}\right )}\, \left (-1+\sqrt {x}\right )^{2}}\)	\(219\)

Input:

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

Output:

2*ln(-1/2+x^(1/2)+(-x^(1/2)+x)^(1/2))-2/(-1+x^(1/2))*((-1+x^(1/2))^2+x^(1/ 
2)-1)^(1/2)+1/2*2^(1/2)*arctanh(1/4*(1-3*x^(1/2))*2^(1/2)/((1+x^(1/2))^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 = 1.44 (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)
 

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)
 

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)
 

Giac [A] (verification not implemented)

Time = 1.00 (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))
 

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)
 

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 186, normalized size of antiderivative = 2.14 \[ \int \frac {\sqrt {x}}{(-1+x) \sqrt {-\sqrt {x}+x}} \, dx=\frac {-4 x^{\frac {1}{4}} \sqrt {\sqrt {x}-1}-\sqrt {x}\, \sqrt {2}\, \mathrm {log}\left (\sqrt {\sqrt {x}-1}+x^{\frac {1}{4}}-\sqrt {2}\, i +i \right )-\sqrt {x}\, \sqrt {2}\, \mathrm {log}\left (\sqrt {\sqrt {x}-1}+x^{\frac {1}{4}}+\sqrt {2}\, i -i \right )+\sqrt {x}\, \sqrt {2}\, \mathrm {log}\left (2 x^{\frac {1}{4}} \sqrt {\sqrt {x}-1}+2 \sqrt {x}+2 \sqrt {2}+2\right )+8 \sqrt {x}\, \mathrm {log}\left (\sqrt {\sqrt {x}-1}+x^{\frac {1}{4}}\right )-4 \sqrt {x}+\sqrt {2}\, \mathrm {log}\left (\sqrt {\sqrt {x}-1}+x^{\frac {1}{4}}-\sqrt {2}\, i +i \right )+\sqrt {2}\, \mathrm {log}\left (\sqrt {\sqrt {x}-1}+x^{\frac {1}{4}}+\sqrt {2}\, i -i \right )-\sqrt {2}\, \mathrm {log}\left (2 x^{\frac {1}{4}} \sqrt {\sqrt {x}-1}+2 \sqrt {x}+2 \sqrt {2}+2\right )-8 \,\mathrm {log}\left (\sqrt {\sqrt {x}-1}+x^{\frac {1}{4}}\right )+4}{2 \sqrt {x}-2} \] Input:

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

Output:

( - 4*x**(1/4)*sqrt(sqrt(x) - 1) - sqrt(x)*sqrt(2)*log(sqrt(sqrt(x) - 1) + 
 x**(1/4) - sqrt(2)*i + i) - sqrt(x)*sqrt(2)*log(sqrt(sqrt(x) - 1) + x**(1 
/4) + sqrt(2)*i - i) + sqrt(x)*sqrt(2)*log(2*x**(1/4)*sqrt(sqrt(x) - 1) + 
2*sqrt(x) + 2*sqrt(2) + 2) + 8*sqrt(x)*log(sqrt(sqrt(x) - 1) + x**(1/4)) - 
 4*sqrt(x) + sqrt(2)*log(sqrt(sqrt(x) - 1) + x**(1/4) - sqrt(2)*i + i) + s 
qrt(2)*log(sqrt(sqrt(x) - 1) + x**(1/4) + sqrt(2)*i - i) - sqrt(2)*log(2*x 
**(1/4)*sqrt(sqrt(x) - 1) + 2*sqrt(x) + 2*sqrt(2) + 2) - 8*log(sqrt(sqrt(x 
) - 1) + x**(1/4)) + 4)/(2*(sqrt(x) - 1))