Integrand size = 32, antiderivative size = 171 \[ \int \left (\frac {1}{\sqrt {1-\sqrt {x}}}-\sqrt {1-\sqrt {x}-x}\right ) \, dx=-\frac {8}{3} \sqrt {1-\sqrt {x}}+\left (\frac {2}{3}+\frac {1}{4} \sqrt {\left (1+2 \sqrt {x}\right )^2}\right ) \sqrt {1-\sqrt {x}-x}-\frac {4}{3} \sqrt {1-\sqrt {x}} \sqrt {x}-\frac {2}{3} \sqrt {1-\sqrt {x}-x} \sqrt {x}-\frac {2}{3} \sqrt {1-\sqrt {x}-x} x+\frac {5}{8} i \log \left (i \sqrt {\left (1+2 \sqrt {x}\right )^2}-2 \sqrt {1-\sqrt {x}-x}\right ) \] Output:
-8/3*(1-x^(1/2))^(1/2)+(2/3+1/4*((1+2*x^(1/2))^2)^(1/2))*(1-x^(1/2)-x)^(1/ 2)-4/3*(1-x^(1/2))^(1/2)*x^(1/2)-2/3*(1-x^(1/2)-x)^(1/2)*x^(1/2)-2/3*(1-x^ (1/2)-x)^(1/2)*x+5/8*I*ln(I*((1+2*x^(1/2))^2)^(1/2)-2*(1-x^(1/2)-x)^(1/2))
Time = 0.12 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.51 \[ \int \left (\frac {1}{\sqrt {1-\sqrt {x}}}-\sqrt {1-\sqrt {x}-x}\right ) \, dx=-\frac {4}{3} \sqrt {1-\sqrt {x}} \left (2+\sqrt {x}\right )-\frac {1}{12} \sqrt {1-\sqrt {x}-x} \left (-11+2 \sqrt {x}+8 x\right )+\frac {5}{4} \arctan \left (\frac {\sqrt {x}}{-1+\sqrt {1-\sqrt {x}-x}}\right ) \] Input:
Integrate[1/Sqrt[1 - Sqrt[x]] - Sqrt[1 - Sqrt[x] - x],x]
Output:
(-4*Sqrt[1 - Sqrt[x]]*(2 + Sqrt[x]))/3 - (Sqrt[1 - Sqrt[x] - x]*(-11 + 2*S qrt[x] + 8*x))/12 + (5*ArcTan[Sqrt[x]/(-1 + Sqrt[1 - Sqrt[x] - x])])/4
Time = 0.24 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.60, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.031, Rules used = {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.
\(\displaystyle \int \left (\frac {1}{\sqrt {1-\sqrt {x}}}-\sqrt {-x-\sqrt {x}+1}\right ) \, dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {5}{8} \arcsin \left (\frac {2 \sqrt {x}+1}{\sqrt {5}}\right )+\frac {4}{3} \left (1-\sqrt {x}\right )^{3/2}-4 \sqrt {1-\sqrt {x}}+\frac {2}{3} \left (-x-\sqrt {x}+1\right )^{3/2}+\frac {1}{4} \left (2 \sqrt {x}+1\right ) \sqrt {-x-\sqrt {x}+1}\) |
Input:
Int[1/Sqrt[1 - Sqrt[x]] - Sqrt[1 - Sqrt[x] - x],x]
Output:
-4*Sqrt[1 - Sqrt[x]] + (4*(1 - Sqrt[x])^(3/2))/3 + ((1 + 2*Sqrt[x])*Sqrt[1 - Sqrt[x] - x])/4 + (2*(1 - Sqrt[x] - x)^(3/2))/3 + (5*ArcSin[(1 + 2*Sqrt [x])/Sqrt[5]])/8
Time = 0.09 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.42
method | result | size |
derivativedivides | \(\frac {2 \left (1-\sqrt {x}-x \right )^{\frac {3}{2}}}{3}-\frac {\left (-2 \sqrt {x}-1\right ) \sqrt {1-\sqrt {x}-x}}{4}+\frac {5 \arcsin \left (\frac {2 \sqrt {5}\, \left (\sqrt {x}+\frac {1}{2}\right )}{5}\right )}{8}+\frac {4 \left (1-\sqrt {x}\right )^{\frac {3}{2}}}{3}-4 \sqrt {1-\sqrt {x}}\) | \(72\) |
default | \(\frac {2 \left (1-\sqrt {x}-x \right )^{\frac {3}{2}}}{3}-\frac {\left (-2 \sqrt {x}-1\right ) \sqrt {1-\sqrt {x}-x}}{4}+\frac {5 \arcsin \left (\frac {2 \sqrt {5}\, \left (\sqrt {x}+\frac {1}{2}\right )}{5}\right )}{8}+\frac {4 \left (1-\sqrt {x}\right )^{\frac {3}{2}}}{3}-4 \sqrt {1-\sqrt {x}}\) | \(72\) |
Input:
int(1/(1-x^(1/2))^(1/2)-(1-x^(1/2)-x)^(1/2),x,method=_RETURNVERBOSE)
Output:
2/3*(1-x^(1/2)-x)^(3/2)-1/4*(-2*x^(1/2)-1)*(1-x^(1/2)-x)^(1/2)+5/8*arcsin( 2/5*5^(1/2)*(x^(1/2)+1/2))+4/3*(1-x^(1/2))^(3/2)-4*(1-x^(1/2))^(1/2)
Time = 0.64 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.50 \[ \int \left (\frac {1}{\sqrt {1-\sqrt {x}}}-\sqrt {1-\sqrt {x}-x}\right ) \, dx=-\frac {1}{12} \, {\left (8 \, x + 2 \, \sqrt {x} - 11\right )} \sqrt {-x - \sqrt {x} + 1} - \frac {4}{3} \, {\left (\sqrt {x} + 2\right )} \sqrt {-\sqrt {x} + 1} - \frac {5}{16} \, \arctan \left (\frac {4 \, {\left (2 \, {\left (8 \, x - 7\right )} \sqrt {x} - 8 \, x - 3\right )} \sqrt {-x - \sqrt {x} + 1}}{64 \, x^{2} - 112 \, x + 9}\right ) \] Input:
integrate(1/(1-x^(1/2))^(1/2)-(1-x^(1/2)-x)^(1/2),x, algorithm="fricas")
Output:
-1/12*(8*x + 2*sqrt(x) - 11)*sqrt(-x - sqrt(x) + 1) - 4/3*(sqrt(x) + 2)*sq rt(-sqrt(x) + 1) - 5/16*arctan(4*(2*(8*x - 7)*sqrt(x) - 8*x - 3)*sqrt(-x - sqrt(x) + 1)/(64*x^2 - 112*x + 9))
Time = 0.75 (sec) , antiderivative size = 296, normalized size of antiderivative = 1.73 \[ \int \left (\frac {1}{\sqrt {1-\sqrt {x}}}-\sqrt {1-\sqrt {x}-x}\right ) \, dx=- 2 \sqrt {- \sqrt {x} - x + 1} \left (\frac {\sqrt {x}}{12} + \frac {x}{3} - \frac {11}{24}\right ) + \begin {cases} \frac {4 i x^{\frac {5}{2}} \sqrt {\sqrt {x} - 1}}{- 3 x^{\frac {5}{2}} + 3 x^{2}} - \frac {8 x^{\frac {5}{2}}}{- 3 x^{\frac {5}{2}} + 3 x^{2}} + \frac {4 i x^{3} \sqrt {\sqrt {x} - 1}}{- 3 x^{\frac {5}{2}} + 3 x^{2}} - \frac {8 i x^{2} \sqrt {\sqrt {x} - 1}}{- 3 x^{\frac {5}{2}} + 3 x^{2}} + \frac {8 x^{2}}{- 3 x^{\frac {5}{2}} + 3 x^{2}} & \text {for}\: \left |{\sqrt {x}}\right | > 1 \\\frac {4 x^{\frac {5}{2}} \sqrt {1 - \sqrt {x}}}{- 3 x^{\frac {5}{2}} + 3 x^{2}} - \frac {8 x^{\frac {5}{2}}}{- 3 x^{\frac {5}{2}} + 3 x^{2}} + \frac {4 x^{3} \sqrt {1 - \sqrt {x}}}{- 3 x^{\frac {5}{2}} + 3 x^{2}} - \frac {8 x^{2} \sqrt {1 - \sqrt {x}}}{- 3 x^{\frac {5}{2}} + 3 x^{2}} + \frac {8 x^{2}}{- 3 x^{\frac {5}{2}} + 3 x^{2}} & \text {otherwise} \end {cases} + \frac {5 \operatorname {asin}{\left (\frac {2 \sqrt {5} \left (\sqrt {x} + \frac {1}{2}\right )}{5} \right )}}{8} \] Input:
integrate(1/(1-x**(1/2))**(1/2)-(1-x**(1/2)-x)**(1/2),x)
Output:
-2*sqrt(-sqrt(x) - x + 1)*(sqrt(x)/12 + x/3 - 11/24) + Piecewise((4*I*x**( 5/2)*sqrt(sqrt(x) - 1)/(-3*x**(5/2) + 3*x**2) - 8*x**(5/2)/(-3*x**(5/2) + 3*x**2) + 4*I*x**3*sqrt(sqrt(x) - 1)/(-3*x**(5/2) + 3*x**2) - 8*I*x**2*sqr t(sqrt(x) - 1)/(-3*x**(5/2) + 3*x**2) + 8*x**2/(-3*x**(5/2) + 3*x**2), Abs (sqrt(x)) > 1), (4*x**(5/2)*sqrt(1 - sqrt(x))/(-3*x**(5/2) + 3*x**2) - 8*x **(5/2)/(-3*x**(5/2) + 3*x**2) + 4*x**3*sqrt(1 - sqrt(x))/(-3*x**(5/2) + 3 *x**2) - 8*x**2*sqrt(1 - sqrt(x))/(-3*x**(5/2) + 3*x**2) + 8*x**2/(-3*x**( 5/2) + 3*x**2), True)) + 5*asin(2*sqrt(5)*(sqrt(x) + 1/2)/5)/8
\[ \int \left (\frac {1}{\sqrt {1-\sqrt {x}}}-\sqrt {1-\sqrt {x}-x}\right ) \, dx=\int { -\sqrt {-x - \sqrt {x} + 1} + \frac {1}{\sqrt {-\sqrt {x} + 1}} \,d x } \] Input:
integrate(1/(1-x^(1/2))^(1/2)-(1-x^(1/2)-x)^(1/2),x, algorithm="maxima")
Output:
4/3*(-sqrt(x) + 1)^(3/2) - 4*sqrt(-sqrt(x) + 1) - integrate(sqrt(-x - sqrt (x) + 1), x)
Time = 0.22 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.39 \[ \int \left (\frac {1}{\sqrt {1-\sqrt {x}}}-\sqrt {1-\sqrt {x}-x}\right ) \, dx=-\frac {1}{12} \, {\left (2 \, \sqrt {x} {\left (4 \, \sqrt {x} + 1\right )} - 11\right )} \sqrt {-x - \sqrt {x} + 1} + \frac {4}{3} \, {\left (-\sqrt {x} + 1\right )}^{\frac {3}{2}} - 4 \, \sqrt {-\sqrt {x} + 1} + \frac {5}{8} \, \arcsin \left (\frac {1}{5} \, \sqrt {5} {\left (2 \, \sqrt {x} + 1\right )}\right ) \] Input:
integrate(1/(1-x^(1/2))^(1/2)-(1-x^(1/2)-x)^(1/2),x, algorithm="giac")
Output:
-1/12*(2*sqrt(x)*(4*sqrt(x) + 1) - 11)*sqrt(-x - sqrt(x) + 1) + 4/3*(-sqrt (x) + 1)^(3/2) - 4*sqrt(-sqrt(x) + 1) + 5/8*arcsin(1/5*sqrt(5)*(2*sqrt(x) + 1))
Timed out. \[ \int \left (\frac {1}{\sqrt {1-\sqrt {x}}}-\sqrt {1-\sqrt {x}-x}\right ) \, dx=-\int \sqrt {1-\sqrt {x}-x}-\frac {1}{\sqrt {1-\sqrt {x}}} \,d x \] Input:
int(1/(1 - x^(1/2))^(1/2) - (1 - x^(1/2) - x)^(1/2),x)
Output:
-int((1 - x^(1/2) - x)^(1/2) - 1/(1 - x^(1/2))^(1/2), x)
Time = 0.19 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.46 \[ \int \left (\frac {1}{\sqrt {1-\sqrt {x}}}-\sqrt {1-\sqrt {x}-x}\right ) \, dx=\frac {5 \mathit {asin} \left (\frac {2 \sqrt {x}+1}{\sqrt {5}}\right )}{8}-\frac {\sqrt {x}\, \sqrt {-\sqrt {x}-x +1}}{6}-\frac {2 \sqrt {-\sqrt {x}-x +1}\, x}{3}+\frac {11 \sqrt {-\sqrt {x}-x +1}}{12}-\frac {4 \sqrt {x}\, \sqrt {-\sqrt {x}+1}}{3}-\frac {8 \sqrt {-\sqrt {x}+1}}{3}-\frac {5 \sqrt {5}}{12} \] Input:
int(1/(1-x^(1/2))^(1/2)-(1-x^(1/2)-x)^(1/2),x)
Output:
(15*asin((2*sqrt(x) + 1)/sqrt(5)) - 4*sqrt(x)*sqrt( - sqrt(x) - x + 1) - 1 6*sqrt( - sqrt(x) - x + 1)*x + 22*sqrt( - sqrt(x) - x + 1) - 32*sqrt(x)*sq rt( - sqrt(x) + 1) - 64*sqrt( - sqrt(x) + 1) - 10*sqrt(5))/24