3.1.9 \(\int \frac {1}{\sqrt {-1+x^2} (\sqrt {x}+\sqrt {-1+x^2})^2} \, dx\) [9]

3.1.9.1 Optimal result

Integrand size = 27, antiderivative size = 220 \[ \int \frac {1}{\sqrt {-1+x^2} \left (\sqrt {x}+\sqrt {-1+x^2}\right )^2} \, dx=\frac {2-4 x}{5 \left (\sqrt {x}+\sqrt {-1+x^2}\right )}+\frac {1}{25} \sqrt {-110+50 \sqrt {5}} \arctan \left (\frac {1}{2} \sqrt {2+2 \sqrt {5}} \sqrt {x}\right )-\frac {1}{50} \sqrt {-110+50 \sqrt {5}} \arctan \left (\frac {\sqrt {-2+2 \sqrt {5}} \sqrt {-1+x^2}}{2-\left (1-\sqrt {5}\right ) x}\right )-\frac {1}{25} \sqrt {110+50 \sqrt {5}} \text {arctanh}\left (\frac {1}{2} \sqrt {-2+2 \sqrt {5}} \sqrt {x}\right )-\frac {1}{50} \sqrt {110+50 \sqrt {5}} \text {arctanh}\left (\frac {\sqrt {2+2 \sqrt {5}} \sqrt {-1+x^2}}{2-x-\sqrt {5} x}\right ) \]

output

1/5*(2-4*x)/(x^(1/2)+(x^2-1)^(1/2))-1/50*arctan((x^2-1)^(1/2)*(-2+2*5^(1/2 
))^(1/2)/(2-x*(-5^(1/2)+1)))*(-110+50*5^(1/2))^(1/2)+1/25*arctan(1/2*x^(1/ 
2)*(2+2*5^(1/2))^(1/2))*(-110+50*5^(1/2))^(1/2)-1/25*arctanh(1/2*x^(1/2)*( 
-2+2*5^(1/2))^(1/2))*(110+50*5^(1/2))^(1/2)-1/50*arctanh((x^2-1)^(1/2)*(2+ 
2*5^(1/2))^(1/2)/(2-x-x*5^(1/2)))*(110+50*5^(1/2))^(1/2)
 

3.1.9.2 Mathematica [A] (verified)

Time = 6.10 (sec) , antiderivative size = 195, normalized size of antiderivative = 0.89 \[ \int \frac {1}{\sqrt {-1+x^2} \left (\sqrt {x}+\sqrt {-1+x^2}\right )^2} \, dx=\frac {1}{25} \left (-\frac {10 (-1+2 x) \left (-\sqrt {x}+\sqrt {-1+x^2}\right )}{-1-x+x^2}+\sqrt {-110+50 \sqrt {5}} \arctan \left (\sqrt {\frac {1}{2} \left (1+\sqrt {5}\right )} \sqrt {x}\right )-\sqrt {-110+50 \sqrt {5}} \arctan \left (\frac {\sqrt {-2+\sqrt {5}} \sqrt {-1+x^2}}{1+x}\right )-\sqrt {110+50 \sqrt {5}} \text {arctanh}\left (\sqrt {\frac {1}{2} \left (-1+\sqrt {5}\right )} \sqrt {x}\right )+\sqrt {110+50 \sqrt {5}} \text {arctanh}\left (\frac {\sqrt {2+\sqrt {5}} \sqrt {-1+x^2}}{1+x}\right )\right ) \]

input

Integrate[1/(Sqrt[-1 + x^2]*(Sqrt[x] + Sqrt[-1 + x^2])^2),x]
 

output

((-10*(-1 + 2*x)*(-Sqrt[x] + Sqrt[-1 + x^2]))/(-1 - x + x^2) + Sqrt[-110 + 
 50*Sqrt[5]]*ArcTan[Sqrt[(1 + Sqrt[5])/2]*Sqrt[x]] - Sqrt[-110 + 50*Sqrt[5 
]]*ArcTan[(Sqrt[-2 + Sqrt[5]]*Sqrt[-1 + x^2])/(1 + x)] - Sqrt[110 + 50*Sqr 
t[5]]*ArcTanh[Sqrt[(-1 + Sqrt[5])/2]*Sqrt[x]] + Sqrt[110 + 50*Sqrt[5]]*Arc 
Tanh[(Sqrt[2 + Sqrt[5]]*Sqrt[-1 + x^2])/(1 + x)])/25
 

3.1.9.3 Rubi [A] (verified)

Time = 0.69 (sec) , antiderivative size = 365, normalized size of antiderivative = 1.66, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {7293, 2009}

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.

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {2}{5} \sqrt {\frac {1}{5} \left (5 \sqrt {5}-2\right )} \arctan \left (\frac {2-\left (1-\sqrt {5}\right ) x}{\sqrt {2 \left (\sqrt {5}-1\right )} \sqrt {x^2-1}}\right )+\sqrt {\frac {2}{5 \left (\sqrt {5}-1\right )}} \arctan \left (\frac {2-\left (1-\sqrt {5}\right ) x}{\sqrt {2 \left (\sqrt {5}-1\right )} \sqrt {x^2-1}}\right )+\frac {1}{5} \sqrt {\frac {2}{5} \left (5 \sqrt {5}-11\right )} \arctan \left (\sqrt {\frac {2}{\sqrt {5}-1}} \sqrt {x}\right )-\frac {2}{5} \sqrt {\frac {1}{5} \left (2+5 \sqrt {5}\right )} \text {arctanh}\left (\frac {2-\left (1+\sqrt {5}\right ) x}{\sqrt {2 \left (1+\sqrt {5}\right )} \sqrt {x^2-1}}\right )+\sqrt {\frac {2}{5 \left (1+\sqrt {5}\right )}} \text {arctanh}\left (\frac {2-\left (1+\sqrt {5}\right ) x}{\sqrt {2 \left (1+\sqrt {5}\right )} \sqrt {x^2-1}}\right )-\frac {1}{5} \sqrt {\frac {2}{5} \left (11+5 \sqrt {5}\right )} \text {arctanh}\left (\sqrt {\frac {2}{1+\sqrt {5}}} \sqrt {x}\right )-\frac {2 \sqrt {x^2-1} (1-2 x)}{5 \left (-x^2+x+1\right )}+\frac {2 \sqrt {x} (1-2 x)}{5 \left (-x^2+x+1\right )}\)

input

Int[1/(Sqrt[-1 + x^2]*(Sqrt[x] + Sqrt[-1 + x^2])^2),x]
 

output

(2*(1 - 2*x)*Sqrt[x])/(5*(1 + x - x^2)) - (2*(1 - 2*x)*Sqrt[-1 + x^2])/(5* 
(1 + x - x^2)) + (Sqrt[(2*(-11 + 5*Sqrt[5]))/5]*ArcTan[Sqrt[2/(-1 + Sqrt[5 
])]*Sqrt[x]])/5 + Sqrt[2/(5*(-1 + Sqrt[5]))]*ArcTan[(2 - (1 - Sqrt[5])*x)/ 
(Sqrt[2*(-1 + Sqrt[5])]*Sqrt[-1 + x^2])] - (2*Sqrt[(-2 + 5*Sqrt[5])/5]*Arc 
Tan[(2 - (1 - Sqrt[5])*x)/(Sqrt[2*(-1 + Sqrt[5])]*Sqrt[-1 + x^2])])/5 - (S 
qrt[(2*(11 + 5*Sqrt[5]))/5]*ArcTanh[Sqrt[2/(1 + Sqrt[5])]*Sqrt[x]])/5 + Sq 
rt[2/(5*(1 + Sqrt[5]))]*ArcTanh[(2 - (1 + Sqrt[5])*x)/(Sqrt[2*(1 + Sqrt[5] 
)]*Sqrt[-1 + x^2])] - (2*Sqrt[(2 + 5*Sqrt[5])/5]*ArcTanh[(2 - (1 + Sqrt[5] 
)*x)/(Sqrt[2*(1 + Sqrt[5])]*Sqrt[-1 + x^2])])/5
 

3.1.9.3.1 Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 7293

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v] 
]
 

3.1.9.4 Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1205\) vs. \(2(158)=316\).

Time = 0.32 (sec) , antiderivative size = 1206, normalized size of antiderivative = 5.48

method	result	size
default	\(\text {Expression too large to display}\)	\(1206\)

input

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

output

-6/25*5^(1/2)/(-2+2*5^(1/2))^(1/2)*arctan(2*(1-5^(1/2)+(-5^(1/2)+1)*(x+1/2 
*5^(1/2)-1/2))/(-2+2*5^(1/2))^(1/2)/(4*(x+1/2*5^(1/2)-1/2)^2+4*(-5^(1/2)+1 
)*(x+1/2*5^(1/2)-1/2)+2-2*5^(1/2))^(1/2))-6/25*5^(1/2)/(2+2*5^(1/2))^(1/2) 
*arctanh(2*(1+5^(1/2)+(5^(1/2)+1)*(x-1/2*5^(1/2)-1/2))/(2+2*5^(1/2))^(1/2) 
/(4*(x-1/2*5^(1/2)-1/2)^2+4*(5^(1/2)+1)*(x-1/2*5^(1/2)-1/2)+2+2*5^(1/2))^( 
1/2))+(2/5+2/5*5^(1/2))*(-1/4/(1/2+1/2*5^(1/2))/(x-1/2*5^(1/2)-1/2)*((x-1/ 
2*5^(1/2)-1/2)^2+(5^(1/2)+1)*(x-1/2*5^(1/2)-1/2)+1/2+1/2*5^(1/2))^(1/2)+1/ 
4*(5^(1/2)+1)/(1/2+1/2*5^(1/2))/(2+2*5^(1/2))^(1/2)*arctanh(2*(1+5^(1/2)+( 
5^(1/2)+1)*(x-1/2*5^(1/2)-1/2))/(2+2*5^(1/2))^(1/2)/(4*(x-1/2*5^(1/2)-1/2) 
^2+4*(5^(1/2)+1)*(x-1/2*5^(1/2)-1/2)+2+2*5^(1/2))^(1/2)))+(2/5-2/5*5^(1/2) 
)*(-1/4/(1/2-1/2*5^(1/2))/(x+1/2*5^(1/2)-1/2)*((x+1/2*5^(1/2)-1/2)^2+(-5^( 
1/2)+1)*(x+1/2*5^(1/2)-1/2)+1/2-1/2*5^(1/2))^(1/2)-1/4*(-5^(1/2)+1)/(1/2-1 
/2*5^(1/2))/(-2+2*5^(1/2))^(1/2)*arctan(2*(1-5^(1/2)+(-5^(1/2)+1)*(x+1/2*5 
^(1/2)-1/2))/(-2+2*5^(1/2))^(1/2)/(4*(x+1/2*5^(1/2)-1/2)^2+4*(-5^(1/2)+1)* 
(x+1/2*5^(1/2)-1/2)+2-2*5^(1/2))^(1/2)))+1/5*(5^(1/2)+1)^2*(-1/4/(1/2+1/2* 
5^(1/2))/(x-1/2*5^(1/2)-1/2)*((x-1/2*5^(1/2)-1/2)^2+(5^(1/2)+1)*(x-1/2*5^( 
1/2)-1/2)+1/2+1/2*5^(1/2))^(1/2)+1/4*(5^(1/2)+1)/(1/2+1/2*5^(1/2))/(2+2*5^ 
(1/2))^(1/2)*arctanh(2*(1+5^(1/2)+(5^(1/2)+1)*(x-1/2*5^(1/2)-1/2))/(2+2*5^ 
(1/2))^(1/2)/(4*(x-1/2*5^(1/2)-1/2)^2+4*(5^(1/2)+1)*(x-1/2*5^(1/2)-1/2)+2+ 
2*5^(1/2))^(1/2)))+1/5*(5^(1/2)-1)^2*(-1/4/(1/2-1/2*5^(1/2))/(x+1/2*5^(...
 

3.1.9.5 Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 450 vs. \(2 (153) = 306\).

Time = 0.24 (sec) , antiderivative size = 450, normalized size of antiderivative = 2.05 \[ \int \frac {1}{\sqrt {-1+x^2} \left (\sqrt {x}+\sqrt {-1+x^2}\right )^2} \, dx=-\frac {\sqrt {5} {\left (x^{2} - x - 1\right )} \sqrt {10 \, \sqrt {5} + 22} \log \left (\sqrt {10 \, \sqrt {5} + 22} {\left (\sqrt {5} - 3\right )} - 4 \, x + 2 \, \sqrt {5} + 4 \, \sqrt {x^{2} - 1} + 2\right ) - \sqrt {5} {\left (x^{2} - x - 1\right )} \sqrt {10 \, \sqrt {5} + 22} \log \left (\sqrt {10 \, \sqrt {5} + 22} {\left (\sqrt {5} - 3\right )} + 4 \, \sqrt {x}\right ) - \sqrt {5} {\left (x^{2} - x - 1\right )} \sqrt {10 \, \sqrt {5} + 22} \log \left (-\sqrt {10 \, \sqrt {5} + 22} {\left (\sqrt {5} - 3\right )} - 4 \, x + 2 \, \sqrt {5} + 4 \, \sqrt {x^{2} - 1} + 2\right ) + \sqrt {5} {\left (x^{2} - x - 1\right )} \sqrt {10 \, \sqrt {5} + 22} \log \left (-\sqrt {10 \, \sqrt {5} + 22} {\left (\sqrt {5} - 3\right )} + 4 \, \sqrt {x}\right ) + \sqrt {5} {\left (x^{2} - x - 1\right )} \sqrt {-10 \, \sqrt {5} + 22} \log \left ({\left (\sqrt {5} + 3\right )} \sqrt {-10 \, \sqrt {5} + 22} - 4 \, x - 2 \, \sqrt {5} + 4 \, \sqrt {x^{2} - 1} + 2\right ) - \sqrt {5} {\left (x^{2} - x - 1\right )} \sqrt {-10 \, \sqrt {5} + 22} \log \left ({\left (\sqrt {5} + 3\right )} \sqrt {-10 \, \sqrt {5} + 22} + 4 \, \sqrt {x}\right ) - \sqrt {5} {\left (x^{2} - x - 1\right )} \sqrt {-10 \, \sqrt {5} + 22} \log \left (-{\left (\sqrt {5} + 3\right )} \sqrt {-10 \, \sqrt {5} + 22} - 4 \, x - 2 \, \sqrt {5} + 4 \, \sqrt {x^{2} - 1} + 2\right ) + \sqrt {5} {\left (x^{2} - x - 1\right )} \sqrt {-10 \, \sqrt {5} + 22} \log \left (-{\left (\sqrt {5} + 3\right )} \sqrt {-10 \, \sqrt {5} + 22} + 4 \, \sqrt {x}\right ) + 40 \, x^{2} + 20 \, \sqrt {x^{2} - 1} {\left (2 \, x - 1\right )} - 20 \, {\left (2 \, x - 1\right )} \sqrt {x} - 40 \, x - 40}{50 \, {\left (x^{2} - x - 1\right )}} \]

input

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

output

-1/50*(sqrt(5)*(x^2 - x - 1)*sqrt(10*sqrt(5) + 22)*log(sqrt(10*sqrt(5) + 2 
2)*(sqrt(5) - 3) - 4*x + 2*sqrt(5) + 4*sqrt(x^2 - 1) + 2) - sqrt(5)*(x^2 - 
 x - 1)*sqrt(10*sqrt(5) + 22)*log(sqrt(10*sqrt(5) + 22)*(sqrt(5) - 3) + 4* 
sqrt(x)) - sqrt(5)*(x^2 - x - 1)*sqrt(10*sqrt(5) + 22)*log(-sqrt(10*sqrt(5 
) + 22)*(sqrt(5) - 3) - 4*x + 2*sqrt(5) + 4*sqrt(x^2 - 1) + 2) + sqrt(5)*( 
x^2 - x - 1)*sqrt(10*sqrt(5) + 22)*log(-sqrt(10*sqrt(5) + 22)*(sqrt(5) - 3 
) + 4*sqrt(x)) + sqrt(5)*(x^2 - x - 1)*sqrt(-10*sqrt(5) + 22)*log((sqrt(5) 
 + 3)*sqrt(-10*sqrt(5) + 22) - 4*x - 2*sqrt(5) + 4*sqrt(x^2 - 1) + 2) - sq 
rt(5)*(x^2 - x - 1)*sqrt(-10*sqrt(5) + 22)*log((sqrt(5) + 3)*sqrt(-10*sqrt 
(5) + 22) + 4*sqrt(x)) - sqrt(5)*(x^2 - x - 1)*sqrt(-10*sqrt(5) + 22)*log( 
-(sqrt(5) + 3)*sqrt(-10*sqrt(5) + 22) - 4*x - 2*sqrt(5) + 4*sqrt(x^2 - 1) 
+ 2) + sqrt(5)*(x^2 - x - 1)*sqrt(-10*sqrt(5) + 22)*log(-(sqrt(5) + 3)*sqr 
t(-10*sqrt(5) + 22) + 4*sqrt(x)) + 40*x^2 + 20*sqrt(x^2 - 1)*(2*x - 1) - 2 
0*(2*x - 1)*sqrt(x) - 40*x - 40)/(x^2 - x - 1)
 

3.1.9.6 Sympy [F]

\[ \int \frac {1}{\sqrt {-1+x^2} \left (\sqrt {x}+\sqrt {-1+x^2}\right )^2} \, dx=\int \frac {1}{\sqrt {\left (x - 1\right ) \left (x + 1\right )} \left (\sqrt {x} + \sqrt {x^{2} - 1}\right )^{2}}\, dx \]

input

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

output

Integral(1/(sqrt((x - 1)*(x + 1))*(sqrt(x) + sqrt(x**2 - 1))**2), x)
 

3.1.9.7 Maxima [F]

\[ \int \frac {1}{\sqrt {-1+x^2} \left (\sqrt {x}+\sqrt {-1+x^2}\right )^2} \, dx=\int { \frac {1}{\sqrt {x^{2} - 1} {\left (\sqrt {x^{2} - 1} + \sqrt {x}\right )}^{2}} \,d x } \]

input

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

output

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

3.1.9.8 Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 367 vs. \(2 (153) = 306\).

Time = 1.49 (sec) , antiderivative size = 367, normalized size of antiderivative = 1.67 \[ \int \frac {1}{\sqrt {-1+x^2} \left (\sqrt {x}+\sqrt {-1+x^2}\right )^2} \, dx=\frac {2}{5} \, \sqrt {\frac {1}{10}} \sqrt {5 \, \sqrt {5} - 11} \arctan \left (\frac {2 \, x + \sqrt {5} - 2 \, \sqrt {x^{2} - 1} - 1}{\sqrt {2 \, \sqrt {5} - 2}}\right ) + \frac {1}{5} \, \sqrt {\frac {1}{10}} \sqrt {5 \, \sqrt {5} + 11} \log \left ({\left | -153040 \, x + 22956 \, \sqrt {5} \sqrt {50 \, \sqrt {5} + 110} + 76520 \, \sqrt {5} + 153040 \, \sqrt {x^{2} - 1} - 38260 \, \sqrt {50 \, \sqrt {5} + 110} + 76520 \right |}\right ) - \frac {1}{5} \, \sqrt {\frac {1}{10}} \sqrt {5 \, \sqrt {5} + 11} \log \left ({\left | -153040 \, x - 22956 \, \sqrt {5} \sqrt {50 \, \sqrt {5} + 110} + 76520 \, \sqrt {5} + 153040 \, \sqrt {x^{2} - 1} + 38260 \, \sqrt {50 \, \sqrt {5} + 110} + 76520 \right |}\right ) + \frac {1}{25} \, \sqrt {50 \, \sqrt {5} - 110} \arctan \left (\frac {\sqrt {x}}{\sqrt {\frac {1}{2} \, \sqrt {5} - \frac {1}{2}}}\right ) - \frac {1}{50} \, \sqrt {50 \, \sqrt {5} + 110} \log \left (\sqrt {x} + \sqrt {\frac {1}{2} \, \sqrt {5} + \frac {1}{2}}\right ) + \frac {1}{50} \, \sqrt {50 \, \sqrt {5} + 110} \log \left ({\left | \sqrt {x} - \sqrt {\frac {1}{2} \, \sqrt {5} + \frac {1}{2}} \right |}\right ) + \frac {4 \, {\left ({\left (x - \sqrt {x^{2} - 1}\right )}^{3} + 2 \, {\left (x - \sqrt {x^{2} - 1}\right )}^{2} + 3 \, x - 3 \, \sqrt {x^{2} - 1} - 2\right )}}{5 \, {\left ({\left (x - \sqrt {x^{2} - 1}\right )}^{4} - 2 \, {\left (x - \sqrt {x^{2} - 1}\right )}^{3} - 2 \, {\left (x - \sqrt {x^{2} - 1}\right )}^{2} - 2 \, x + 2 \, \sqrt {x^{2} - 1} + 1\right )}} + \frac {2 \, {\left (2 \, x^{\frac {3}{2}} - \sqrt {x}\right )}}{5 \, {\left (x^{2} - x - 1\right )}} \]

input

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

output

2/5*sqrt(1/10)*sqrt(5*sqrt(5) - 11)*arctan((2*x + sqrt(5) - 2*sqrt(x^2 - 1 
) - 1)/sqrt(2*sqrt(5) - 2)) + 1/5*sqrt(1/10)*sqrt(5*sqrt(5) + 11)*log(abs( 
-153040*x + 22956*sqrt(5)*sqrt(50*sqrt(5) + 110) + 76520*sqrt(5) + 153040* 
sqrt(x^2 - 1) - 38260*sqrt(50*sqrt(5) + 110) + 76520)) - 1/5*sqrt(1/10)*sq 
rt(5*sqrt(5) + 11)*log(abs(-153040*x - 22956*sqrt(5)*sqrt(50*sqrt(5) + 110 
) + 76520*sqrt(5) + 153040*sqrt(x^2 - 1) + 38260*sqrt(50*sqrt(5) + 110) + 
76520)) + 1/25*sqrt(50*sqrt(5) - 110)*arctan(sqrt(x)/sqrt(1/2*sqrt(5) - 1/ 
2)) - 1/50*sqrt(50*sqrt(5) + 110)*log(sqrt(x) + sqrt(1/2*sqrt(5) + 1/2)) + 
 1/50*sqrt(50*sqrt(5) + 110)*log(abs(sqrt(x) - sqrt(1/2*sqrt(5) + 1/2))) + 
 4/5*((x - sqrt(x^2 - 1))^3 + 2*(x - sqrt(x^2 - 1))^2 + 3*x - 3*sqrt(x^2 - 
 1) - 2)/((x - sqrt(x^2 - 1))^4 - 2*(x - sqrt(x^2 - 1))^3 - 2*(x - sqrt(x^ 
2 - 1))^2 - 2*x + 2*sqrt(x^2 - 1) + 1) + 2/5*(2*x^(3/2) - sqrt(x))/(x^2 - 
x - 1)
 

3.1.9.9 Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt {-1+x^2} \left (\sqrt {x}+\sqrt {-1+x^2}\right )^2} \, dx=\int \frac {1}{\sqrt {x^2-1}\,{\left (\sqrt {x^2-1}+\sqrt {x}\right )}^2} \,d x \]

input

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

output

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