Integrand size = 39, antiderivative size = 220 \[ \int \frac {\left (\sqrt {x}-\sqrt {-1+x^2}\right )^2}{\left (1+x-x^2\right )^2 \sqrt {-1+x^2}} \, dx=\frac {2-4 x}{5 \left (\sqrt {x}+\sqrt {-1+x^2}\right )}+\frac {1}{25} \sqrt {-110+50 \sqrt {5}} \arctan \left (\frac {1}{2} \sqrt {2+2 \sqrt {5}} \sqrt {x}\right )-\frac {1}{50} \sqrt {-110+50 \sqrt {5}} \arctan \left (\frac {\sqrt {-2+2 \sqrt {5}} \sqrt {-1+x^2}}{2-\left (1-\sqrt {5}\right ) x}\right )-\frac {1}{25} \sqrt {110+50 \sqrt {5}} \text {arctanh}\left (\frac {1}{2} \sqrt {-2+2 \sqrt {5}} \sqrt {x}\right )-\frac {1}{50} \sqrt {110+50 \sqrt {5}} \text {arctanh}\left (\frac {\sqrt {2+2 \sqrt {5}} \sqrt {-1+x^2}}{2-x-\sqrt {5} x}\right ) \]
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)
Time = 6.31 (sec) , antiderivative size = 195, normalized size of antiderivative = 0.89 \[ \int \frac {\left (\sqrt {x}-\sqrt {-1+x^2}\right )^2}{\left (1+x-x^2\right )^2 \sqrt {-1+x^2}} \, dx=\frac {1}{25} \left (-\frac {10 (-1+2 x) \left (-\sqrt {x}+\sqrt {-1+x^2}\right )}{-1-x+x^2}+\sqrt {-110+50 \sqrt {5}} \arctan \left (\sqrt {\frac {1}{2} \left (1+\sqrt {5}\right )} \sqrt {x}\right )-\sqrt {-110+50 \sqrt {5}} \arctan \left (\frac {\sqrt {-2+\sqrt {5}} \sqrt {-1+x^2}}{1+x}\right )-\sqrt {110+50 \sqrt {5}} \text {arctanh}\left (\sqrt {\frac {1}{2} \left (-1+\sqrt {5}\right )} \sqrt {x}\right )+\sqrt {110+50 \sqrt {5}} \text {arctanh}\left (\frac {\sqrt {2+\sqrt {5}} \sqrt {-1+x^2}}{1+x}\right )\right ) \]
((-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
Leaf count is larger than twice the leaf count of optimal. \(541\) vs. \(2(220)=440\).
Time = 0.95 (sec) , antiderivative size = 541, normalized size of antiderivative = 2.46, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.051, 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.
| \(\displaystyle \int \frac {\left (\sqrt {x}-\sqrt {x^2-1}\right )^2}{\left (-x^2+x+1\right )^2 \sqrt {x^2-1}} \, dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {x^2}{\sqrt {x^2-1} \left (x^2-x-1\right )^2}+\frac {x}{\sqrt {x^2-1} \left (x^2-x-1\right )^2}-\frac {2 \sqrt {x}}{\left (x^2-x-1\right )^2}-\frac {1}{\sqrt {x^2-1} \left (x^2-x-1\right )^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{5} \sqrt {\frac {1}{5} \left (2+5 \sqrt {5}\right )} \arctan \left (\frac {2-\left (1-\sqrt {5}\right ) x}{\sqrt {2 \left (\sqrt {5}-1\right )} \sqrt {x^2-1}}\right )-\frac {1}{5} \sqrt {\frac {1}{5} \left (5 \sqrt {5}-2\right )} \arctan \left (\frac {2-\left (1-\sqrt {5}\right ) x}{\sqrt {2 \left (\sqrt {5}-1\right )} \sqrt {x^2-1}}\right )-\frac {1}{5} \sqrt {\frac {1}{10} \left (5 \sqrt {5}-11\right )} \arctan \left (\frac {2-\left (1-\sqrt {5}\right ) x}{\sqrt {2 \left (\sqrt {5}-1\right )} \sqrt {x^2-1}}\right )+\frac {1}{5} \sqrt {\frac {2}{5} \left (5 \sqrt {5}-11\right )} \arctan \left (\sqrt {\frac {2}{\sqrt {5}-1}} \sqrt {x}\right )+\frac {1}{5} \sqrt {\frac {1}{10} \left (11+5 \sqrt {5}\right )} \text {arctanh}\left (\frac {2-\left (1+\sqrt {5}\right ) x}{\sqrt {2 \left (1+\sqrt {5}\right )} \sqrt {x^2-1}}\right )-\frac {1}{5} \sqrt {\frac {1}{5} \left (2+5 \sqrt {5}\right )} \text {arctanh}\left (\frac {2-\left (1+\sqrt {5}\right ) x}{\sqrt {2 \left (1+\sqrt {5}\right )} \sqrt {x^2-1}}\right )-\frac {1}{5} \sqrt {\frac {1}{5} \left (5 \sqrt {5}-2\right )} \text {arctanh}\left (\frac {2-\left (1+\sqrt {5}\right ) x}{\sqrt {2 \left (1+\sqrt {5}\right )} \sqrt {x^2-1}}\right )-\frac {1}{5} \sqrt {\frac {2}{5} \left (11+5 \sqrt {5}\right )} \text {arctanh}\left (\sqrt {\frac {2}{1+\sqrt {5}}} \sqrt {x}\right )-\frac {\sqrt {x^2-1} (1-2 x)}{5 \left (-x^2+x+1\right )}+\frac {2 \sqrt {x} (1-2 x)}{5 \left (-x^2+x+1\right )}-\frac {(3-x) \sqrt {x^2-1}}{5 \left (-x^2+x+1\right )}+\frac {(x+2) \sqrt {x^2-1}}{5 \left (-x^2+x+1\right )}\) |
(2*(1 - 2*x)*Sqrt[x])/(5*(1 + x - x^2)) - ((1 - 2*x)*Sqrt[-1 + x^2])/(5*(1 + x - x^2)) - ((3 - x)*Sqrt[-1 + x^2])/(5*(1 + x - x^2)) + ((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[(-11 + 5*Sqrt[5])/10]*ArcTan[(2 - (1 - Sqrt[5])*x)/(Sqrt[2*(-1 + Sqrt[5])]*Sqrt[-1 + x^2])])/5 - (Sqrt[(-2 + 5 *Sqrt[5])/5]*ArcTan[(2 - (1 - Sqrt[5])*x)/(Sqrt[2*(-1 + Sqrt[5])]*Sqrt[-1 + x^2])])/5 + (Sqrt[(2 + 5*Sqrt[5])/5]*ArcTan[(2 - (1 - Sqrt[5])*x)/(Sqrt[ 2*(-1 + Sqrt[5])]*Sqrt[-1 + x^2])])/5 - (Sqrt[(2*(11 + 5*Sqrt[5]))/5]*ArcT anh[Sqrt[2/(1 + Sqrt[5])]*Sqrt[x]])/5 - (Sqrt[(-2 + 5*Sqrt[5])/5]*ArcTanh[ (2 - (1 + Sqrt[5])*x)/(Sqrt[2*(1 + Sqrt[5])]*Sqrt[-1 + x^2])])/5 - (Sqrt[( 2 + 5*Sqrt[5])/5]*ArcTanh[(2 - (1 + Sqrt[5])*x)/(Sqrt[2*(1 + Sqrt[5])]*Sqr t[-1 + x^2])])/5 + (Sqrt[(11 + 5*Sqrt[5])/10]*ArcTanh[(2 - (1 + Sqrt[5])*x )/(Sqrt[2*(1 + Sqrt[5])]*Sqrt[-1 + x^2])])/5
3.1.10.3.1 Defintions of rubi rules used
Leaf count of result is larger than twice the leaf count of optimal. \(1636\) vs. \(2(158)=316\).
Time = 0.48 (sec) , antiderivative size = 1637, normalized size of antiderivative = 7.44
2/25*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))+2/25*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)))+2/5*x^(1/2)/(x+1/2*5^(1/2)-1/2)-8/ 25*(-5/2+5^(1/2))/(-2+2*5^(1/2))^(1/2)*arctan(2*x^(1/2)/(-2+2*5^(1/2))^(1/ 2))+2/5*x^(1/2)/(x-1/2*5^(1/2)-1/2)-8/25*(5/2+5^(1/2))/(2+2*5^(1/2))^(1/2) *arctanh(2*x^(1/2)/(2+2*5^(1/2))^(1/2))-1/5/(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))^(3/2)+1/10*(5^(1/2)+1)/(1/2+1/2*5^(1/2))*(1/2*(4*(x-1/2*5^(1/2)-1...
Leaf count of result is larger than twice the leaf count of optimal. 450 vs. \(2 (153) = 306\).
Time = 0.25 (sec) , antiderivative size = 450, normalized size of antiderivative = 2.05 \[ \int \frac {\left (\sqrt {x}-\sqrt {-1+x^2}\right )^2}{\left (1+x-x^2\right )^2 \sqrt {-1+x^2}} \, dx=-\frac {\sqrt {5} {\left (x^{2} - x - 1\right )} \sqrt {10 \, \sqrt {5} + 22} \log \left (\sqrt {10 \, \sqrt {5} + 22} {\left (\sqrt {5} - 3\right )} - 4 \, x + 2 \, \sqrt {5} + 4 \, \sqrt {x^{2} - 1} + 2\right ) - \sqrt {5} {\left (x^{2} - x - 1\right )} \sqrt {10 \, \sqrt {5} + 22} \log \left (\sqrt {10 \, \sqrt {5} + 22} {\left (\sqrt {5} - 3\right )} + 4 \, \sqrt {x}\right ) - \sqrt {5} {\left (x^{2} - x - 1\right )} \sqrt {10 \, \sqrt {5} + 22} \log \left (-\sqrt {10 \, \sqrt {5} + 22} {\left (\sqrt {5} - 3\right )} - 4 \, x + 2 \, \sqrt {5} + 4 \, \sqrt {x^{2} - 1} + 2\right ) + \sqrt {5} {\left (x^{2} - x - 1\right )} \sqrt {10 \, \sqrt {5} + 22} \log \left (-\sqrt {10 \, \sqrt {5} + 22} {\left (\sqrt {5} - 3\right )} + 4 \, \sqrt {x}\right ) + \sqrt {5} {\left (x^{2} - x - 1\right )} \sqrt {-10 \, \sqrt {5} + 22} \log \left ({\left (\sqrt {5} + 3\right )} \sqrt {-10 \, \sqrt {5} + 22} - 4 \, x - 2 \, \sqrt {5} + 4 \, \sqrt {x^{2} - 1} + 2\right ) - \sqrt {5} {\left (x^{2} - x - 1\right )} \sqrt {-10 \, \sqrt {5} + 22} \log \left ({\left (\sqrt {5} + 3\right )} \sqrt {-10 \, \sqrt {5} + 22} + 4 \, \sqrt {x}\right ) - \sqrt {5} {\left (x^{2} - x - 1\right )} \sqrt {-10 \, \sqrt {5} + 22} \log \left (-{\left (\sqrt {5} + 3\right )} \sqrt {-10 \, \sqrt {5} + 22} - 4 \, x - 2 \, \sqrt {5} + 4 \, \sqrt {x^{2} - 1} + 2\right ) + \sqrt {5} {\left (x^{2} - x - 1\right )} \sqrt {-10 \, \sqrt {5} + 22} \log \left (-{\left (\sqrt {5} + 3\right )} \sqrt {-10 \, \sqrt {5} + 22} + 4 \, \sqrt {x}\right ) + 40 \, x^{2} + 20 \, \sqrt {x^{2} - 1} {\left (2 \, x - 1\right )} - 20 \, {\left (2 \, x - 1\right )} \sqrt {x} - 40 \, x - 40}{50 \, {\left (x^{2} - x - 1\right )}} \]
-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)
Timed out. \[ \int \frac {\left (\sqrt {x}-\sqrt {-1+x^2}\right )^2}{\left (1+x-x^2\right )^2 \sqrt {-1+x^2}} \, dx=\text {Timed out} \]
\[ \int \frac {\left (\sqrt {x}-\sqrt {-1+x^2}\right )^2}{\left (1+x-x^2\right )^2 \sqrt {-1+x^2}} \, dx=\int { \frac {{\left (\sqrt {x^{2} - 1} - \sqrt {x}\right )}^{2}}{{\left (x^{2} - x - 1\right )}^{2} \sqrt {x^{2} - 1}} \,d x } \]
-2/5*(x^(5/2) - 3*x^(3/2))/(x^2 - x - 1) + integrate(1/5*(x^(3/2) + sqrt(x ))/(x^2 - x - 1), x) + integrate((x^2 + x - 1)/((x^4 - 2*x^3 - x^2 + 2*x + 1)*sqrt(x + 1)*sqrt(x - 1)), x)
Leaf count of result is larger than twice the leaf count of optimal. 358 vs. \(2 (153) = 306\).
Time = 1.65 (sec) , antiderivative size = 358, normalized size of antiderivative = 1.63 \[ \int \frac {\left (\sqrt {x}-\sqrt {-1+x^2}\right )^2}{\left (1+x-x^2\right )^2 \sqrt {-1+x^2}} \, dx=\frac {2}{5} \, \sqrt {\frac {1}{2} \, \sqrt {5} - \frac {11}{10}} \arctan \left (\frac {2 \, x + \sqrt {5} - 2 \, \sqrt {x^{2} - 1} - 1}{\sqrt {2 \, \sqrt {5} - 2}}\right ) + \frac {1}{25} \, \sqrt {50 \, \sqrt {5} - 110} \arctan \left (\frac {\sqrt {x}}{\sqrt {\frac {1}{2} \, \sqrt {5} - \frac {1}{2}}}\right ) - \frac {1}{50} \, \sqrt {50 \, \sqrt {5} + 110} \log \left (\sqrt {x} + \sqrt {\frac {1}{2} \, \sqrt {5} + \frac {1}{2}}\right ) - \frac {1}{5} \, \sqrt {\frac {1}{2} \, \sqrt {5} + \frac {11}{10}} \log \left ({\left | -520 \, x - 78 \, \sqrt {5} \sqrt {50 \, \sqrt {5} + 110} + 260 \, \sqrt {5} + 520 \, \sqrt {x^{2} - 1} + 130 \, \sqrt {50 \, \sqrt {5} + 110} + 260 \right |}\right ) + \frac {1}{5} \, \sqrt {\frac {1}{2} \, \sqrt {5} + \frac {11}{10}} \log \left ({\left | -1040 \, x + 156 \, \sqrt {5} \sqrt {50 \, \sqrt {5} + 110} + 520 \, \sqrt {5} + 1040 \, \sqrt {x^{2} - 1} - 260 \, \sqrt {50 \, \sqrt {5} + 110} + 520 \right |}\right ) + \frac {1}{50} \, \sqrt {50 \, \sqrt {5} + 110} \log \left ({\left | \sqrt {x} - \sqrt {\frac {1}{2} \, \sqrt {5} + \frac {1}{2}} \right |}\right ) + \frac {4 \, {\left ({\left (x - \sqrt {x^{2} - 1}\right )}^{3} + 2 \, {\left (x - \sqrt {x^{2} - 1}\right )}^{2} + 3 \, x - 3 \, \sqrt {x^{2} - 1} - 2\right )}}{5 \, {\left ({\left (x - \sqrt {x^{2} - 1}\right )}^{4} - 2 \, {\left (x - \sqrt {x^{2} - 1}\right )}^{3} - 2 \, {\left (x - \sqrt {x^{2} - 1}\right )}^{2} - 2 \, x + 2 \, \sqrt {x^{2} - 1} + 1\right )}} + \frac {2 \, {\left (2 \, x^{\frac {3}{2}} - \sqrt {x}\right )}}{5 \, {\left (x^{2} - x - 1\right )}} \]
2/5*sqrt(1/2*sqrt(5) - 11/10)*arctan((2*x + sqrt(5) - 2*sqrt(x^2 - 1) - 1) /sqrt(2*sqrt(5) - 2)) + 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*sqr t(5) + 1/2)) - 1/5*sqrt(1/2*sqrt(5) + 11/10)*log(abs(-520*x - 78*sqrt(5)*s qrt(50*sqrt(5) + 110) + 260*sqrt(5) + 520*sqrt(x^2 - 1) + 130*sqrt(50*sqrt (5) + 110) + 260)) + 1/5*sqrt(1/2*sqrt(5) + 11/10)*log(abs(-1040*x + 156*s qrt(5)*sqrt(50*sqrt(5) + 110) + 520*sqrt(5) + 1040*sqrt(x^2 - 1) - 260*sqr t(50*sqrt(5) + 110) + 520)) + 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)
Timed out. \[ \int \frac {\left (\sqrt {x}-\sqrt {-1+x^2}\right )^2}{\left (1+x-x^2\right )^2 \sqrt {-1+x^2}} \, dx=\int \frac {{\left (\sqrt {x^2-1}-\sqrt {x}\right )}^2}{\sqrt {x^2-1}\,{\left (-x^2+x+1\right )}^2} \,d x \]
\[ \int \frac {\left (\sqrt {x}-\sqrt {-1+x^2}\right )^2}{\left (1+x-x^2\right )^2 \sqrt {-1+x^2}} \, dx =\text {Too large to display} \]
int(( - 2*sqrt(x)*sqrt(x**2 - 1) + x**2 + x - 1)/(sqrt(x**2 - 1)*(x**4 - 2 *x**3 - x**2 + 2*x + 1)),x)
(6*sqrt(sqrt(5) - 1)*sqrt(10)*atan((2*sqrt(x))/(sqrt(sqrt(5) - 1)*sqrt(2)) )*x**2 - 6*sqrt(sqrt(5) - 1)*sqrt(10)*atan((2*sqrt(x))/(sqrt(sqrt(5) - 1)* sqrt(2)))*x - 6*sqrt(sqrt(5) - 1)*sqrt(10)*atan((2*sqrt(x))/(sqrt(sqrt(5) - 1)*sqrt(2))) - 10*sqrt(sqrt(5) - 1)*sqrt(2)*atan((2*sqrt(x))/(sqrt(sqrt( 5) - 1)*sqrt(2)))*x**2 + 10*sqrt(sqrt(5) - 1)*sqrt(2)*atan((2*sqrt(x))/(sq rt(sqrt(5) - 1)*sqrt(2)))*x + 10*sqrt(sqrt(5) - 1)*sqrt(2)*atan((2*sqrt(x) )/(sqrt(sqrt(5) - 1)*sqrt(2))) + 3*sqrt(sqrt(5) + 1)*sqrt(10)*log( - sqrt( sqrt(5) + 1) + sqrt(x)*sqrt(2))*x**2 - 3*sqrt(sqrt(5) + 1)*sqrt(10)*log( - sqrt(sqrt(5) + 1) + sqrt(x)*sqrt(2))*x - 3*sqrt(sqrt(5) + 1)*sqrt(10)*log ( - sqrt(sqrt(5) + 1) + sqrt(x)*sqrt(2)) - 3*sqrt(sqrt(5) + 1)*sqrt(10)*lo g(sqrt(sqrt(5) + 1) + sqrt(x)*sqrt(2))*x**2 + 3*sqrt(sqrt(5) + 1)*sqrt(10) *log(sqrt(sqrt(5) + 1) + sqrt(x)*sqrt(2))*x + 3*sqrt(sqrt(5) + 1)*sqrt(10) *log(sqrt(sqrt(5) + 1) + sqrt(x)*sqrt(2)) + 5*sqrt(sqrt(5) + 1)*sqrt(2)*lo g( - sqrt(sqrt(5) + 1) + sqrt(x)*sqrt(2))*x**2 - 5*sqrt(sqrt(5) + 1)*sqrt( 2)*log( - sqrt(sqrt(5) + 1) + sqrt(x)*sqrt(2))*x - 5*sqrt(sqrt(5) + 1)*sqr t(2)*log( - sqrt(sqrt(5) + 1) + sqrt(x)*sqrt(2)) - 5*sqrt(sqrt(5) + 1)*sqr t(2)*log(sqrt(sqrt(5) + 1) + sqrt(x)*sqrt(2))*x**2 + 5*sqrt(sqrt(5) + 1)*s qrt(2)*log(sqrt(sqrt(5) + 1) + sqrt(x)*sqrt(2))*x + 5*sqrt(sqrt(5) + 1)*sq rt(2)*log(sqrt(sqrt(5) + 1) + sqrt(x)*sqrt(2)) + 80*sqrt(x)*x - 40*sqrt(x) + 100*int(x**2/(sqrt(x**2 - 1)*x**4 - 2*sqrt(x**2 - 1)*x**3 - sqrt(x**...