\(\int \frac {\sqrt {-1-\sqrt {x}+x}}{(-1+x) \sqrt {x}} \, dx\) [130]

Optimal result

Integrand size = 25, antiderivative size = 89 \[ \int \frac {\sqrt {-1-\sqrt {x}+x}}{(-1+x) \sqrt {x}} \, dx=\arctan \left (\frac {3-\sqrt {x}}{2 \sqrt {-1-\sqrt {x}+x}}\right )-2 \text {arctanh}\left (\frac {1-2 \sqrt {x}}{2 \sqrt {-1-\sqrt {x}+x}}\right )-\text {arctanh}\left (\frac {1+3 \sqrt {x}}{2 \sqrt {-1-\sqrt {x}+x}}\right ) \] Output:

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

Mathematica [A] (verified)

Time = 0.18 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.91 \[ \int \frac {\sqrt {-1-\sqrt {x}+x}}{(-1+x) \sqrt {x}} \, dx=-2 \arctan \left (1-\sqrt {x}+\sqrt {-1-\sqrt {x}+x}\right )-2 \text {arctanh}\left (1+\sqrt {x}-\sqrt {-1-\sqrt {x}+x}\right )-2 \log \left (1-2 \sqrt {x}+2 \sqrt {-1-\sqrt {x}+x}\right ) \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.51 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.09, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.440, Rules used = {2035, 25, 1321, 25, 1092, 219, 1366, 25, 1154, 217, 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.

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

\(\Big \downarrow \) 2035

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 1321

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 1092

\(\displaystyle 2 \left (2 \int \frac {1}{4-x}d\left (-\frac {1-2 \sqrt {x}}{\sqrt {x-\sqrt {x}-1}}\right )+\int \frac {\sqrt {x}}{(1-x) \sqrt {x-\sqrt {x}-1}}d\sqrt {x}\right )\)

\(\Big \downarrow \) 219

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

\(\Big \downarrow \) 1366

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 1154

\(\displaystyle 2 \left (-\int \frac {1}{-x-4}d\frac {3-\sqrt {x}}{\sqrt {x-\sqrt {x}-1}}+\int \frac {1}{4-x}d\left (-\frac {3 \sqrt {x}+1}{\sqrt {x-\sqrt {x}-1}}\right )-\text {arctanh}\left (\frac {1-2 \sqrt {x}}{2 \sqrt {x-\sqrt {x}-1}}\right )\right )\)

\(\Big \downarrow \) 217

\(\displaystyle 2 \left (\int \frac {1}{4-x}d\left (-\frac {3 \sqrt {x}+1}{\sqrt {x-\sqrt {x}-1}}\right )+\frac {1}{2} \arctan \left (\frac {3-\sqrt {x}}{2 \sqrt {x-\sqrt {x}-1}}\right )-\text {arctanh}\left (\frac {1-2 \sqrt {x}}{2 \sqrt {x-\sqrt {x}-1}}\right )\right )\)

\(\Big \downarrow \) 219

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 25

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

rule 217

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

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 1092

Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[2   Subst[I 
nt[1/(4*c - x^2), x], x, (b + 2*c*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a 
, b, c}, x]
 

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 1321

Int[Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2]/((d_) + (f_.)*(x_)^2), x_Symbol] 
 :> Simp[c/f   Int[1/Sqrt[a + b*x + c*x^2], x], x] - Simp[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 1366

Int[((g_.) + (h_.)*(x_))/(((a_) + (c_.)*(x_)^2)*Sqrt[(d_.) + (e_.)*(x_) + ( 
f_.)*(x_)^2]), x_Symbol] :> With[{q = Rt[(-a)*c, 2]}, Simp[(h/2 + c*(g/(2*q 
)))   Int[1/((-q + c*x)*Sqrt[d + e*x + f*x^2]), x], x] + Simp[(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 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]
 

Maple [A] (verified)

Time = 0.04 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.46

method	result	size
derivativedivides	\(\sqrt {\left (-1+\sqrt {x}\right )^{2}-2+\sqrt {x}}+\frac {\ln \left (-\frac {1}{2}+\sqrt {x}+\sqrt {\left (-1+\sqrt {x}\right )^{2}-2+\sqrt {x}}\right )}{2}-\arctan \left (\frac {-3+\sqrt {x}}{2 \sqrt {\left (-1+\sqrt {x}\right )^{2}-2+\sqrt {x}}}\right )-\sqrt {\left (1+\sqrt {x}\right )^{2}-2-3 \sqrt {x}}+\frac {3 \ln \left (-\frac {1}{2}+\sqrt {x}+\sqrt {\left (1+\sqrt {x}\right )^{2}-2-3 \sqrt {x}}\right )}{2}+\operatorname {arctanh}\left (\frac {-1-3 \sqrt {x}}{2 \sqrt {\left (1+\sqrt {x}\right )^{2}-2-3 \sqrt {x}}}\right )\)	\(130\)
default	\(\sqrt {\left (-1+\sqrt {x}\right )^{2}-2+\sqrt {x}}+\frac {\ln \left (-\frac {1}{2}+\sqrt {x}+\sqrt {\left (-1+\sqrt {x}\right )^{2}-2+\sqrt {x}}\right )}{2}-\arctan \left (\frac {-3+\sqrt {x}}{2 \sqrt {\left (-1+\sqrt {x}\right )^{2}-2+\sqrt {x}}}\right )-\sqrt {\left (1+\sqrt {x}\right )^{2}-2-3 \sqrt {x}}+\frac {3 \ln \left (-\frac {1}{2}+\sqrt {x}+\sqrt {\left (1+\sqrt {x}\right )^{2}-2-3 \sqrt {x}}\right )}{2}+\operatorname {arctanh}\left (\frac {-1-3 \sqrt {x}}{2 \sqrt {\left (1+\sqrt {x}\right )^{2}-2-3 \sqrt {x}}}\right )\)	\(130\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 2.27 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.98 \[ \int \frac {\sqrt {-1-\sqrt {x}+x}}{(-1+x) \sqrt {x}} \, dx=-\arctan \left (\frac {{\left ({\left (x - 4\right )} \sqrt {x} - 2 \, x + 3\right )} \sqrt {x - \sqrt {x} - 1}}{2 \, {\left (x^{2} - 3 \, x + 1\right )}}\right ) + \log \left (-\frac {8 \, x^{2} + 2 \, {\left ({\left (4 \, x - 5\right )} \sqrt {x} + 2 \, x - 1\right )} \sqrt {x - \sqrt {x} - 1} - 17 \, x - 2 \, \sqrt {x} + 11}{x - 1}\right ) \] Input:

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

Output:

-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)*sqrt(x) + 2*x - 1)*sqrt(x - sqrt(x) - 1) 
 - 17*x - 2*sqrt(x) + 11)/(x - 1))
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [A] (verification not implemented)

Time = 0.29 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.91 \[ \int \frac {\sqrt {-1-\sqrt {x}+x}}{(-1+x) \sqrt {x}} \, dx=-2 \, \arctan \left (\sqrt {x - \sqrt {x} - 1} - \sqrt {x} + 1\right ) - \log \left (-\sqrt {x - \sqrt {x} - 1} + \sqrt {x} + 2\right ) + \log \left (-\sqrt {x - \sqrt {x} - 1} + \sqrt {x}\right ) - 2 \, \log \left ({\left | 2 \, \sqrt {x - \sqrt {x} - 1} - 2 \, \sqrt {x} + 1 \right |}\right ) \] Input:

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

Output:

-2*arctan(sqrt(x - sqrt(x) - 1) - sqrt(x) + 1) - log(-sqrt(x - sqrt(x) - 1 
) + sqrt(x) + 2) + log(-sqrt(x - sqrt(x) - 1) + sqrt(x)) - 2*log(abs(2*sqr 
t(x - sqrt(x) - 1) - 2*sqrt(x) + 1))
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.96 \[ \int \frac {\sqrt {-1-\sqrt {x}+x}}{(-1+x) \sqrt {x}} \, dx=-2 \mathit {atan} \left (\sqrt {-\sqrt {x}+x -1}+\sqrt {x}-1\right )-\mathrm {log}\left (\frac {10 \sqrt {-\sqrt {x}+x -1}+10 \sqrt {x}}{\sqrt {5}}\right )+2 \,\mathrm {log}\left (\frac {2 \sqrt {-\sqrt {x}+x -1}+2 \sqrt {x}-1}{\sqrt {5}}\right )+\mathrm {log}\left (\frac {2 \sqrt {-\sqrt {x}+x -1}+2 \sqrt {x}+4}{\sqrt {5}}\right ) \] Input:

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

Output:

 - 2*atan(sqrt( - sqrt(x) + x - 1) + sqrt(x) - 1) - log((10*sqrt( - sqrt(x 
) + x - 1) + 10*sqrt(x))/sqrt(5)) + 2*log((2*sqrt( - sqrt(x) + x - 1) + 2* 
sqrt(x) - 1)/sqrt(5)) + log((2*sqrt( - sqrt(x) + x - 1) + 2*sqrt(x) + 4)/s 
qrt(5))