Integrand size = 16, antiderivative size = 70 \[ \int \sqrt {1-\sqrt {x}-x} \, dx=-\frac {1}{4} \left (1+2 \sqrt {x}\right ) \sqrt {1-\sqrt {x}-x}-\frac {2}{3} \left (1-\sqrt {x}-x\right )^{3/2}-\frac {5}{8} \arcsin \left (\frac {1+2 \sqrt {x}}{\sqrt {5}}\right ) \] Output:
-1/4*(1+2*x^(1/2))*(1-x^(1/2)-x)^(1/2)-2/3*(1-x^(1/2)-x)^(3/2)-5/8*arcsin( 1/5*(1+2*x^(1/2))*5^(1/2))
Time = 0.16 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.91 \[ \int \sqrt {1-\sqrt {x}-x} \, dx=\frac {1}{12} \sqrt {1-\sqrt {x}-x} \left (-11+2 \sqrt {x}+8 x\right )-\frac {5}{4} \arctan \left (\frac {\sqrt {x}}{-1+\sqrt {1-\sqrt {x}-x}}\right ) \] Input:
Integrate[Sqrt[1 - Sqrt[x] - x],x]
Output:
(Sqrt[1 - Sqrt[x] - x]*(-11 + 2*Sqrt[x] + 8*x))/12 - (5*ArcTan[Sqrt[x]/(-1 + Sqrt[1 - Sqrt[x] - x])])/4
Time = 0.35 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.10, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {1680, 1160, 1087, 1090, 223}
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.
\(\displaystyle \int \sqrt {-x-\sqrt {x}+1} \, dx\) |
\(\Big \downarrow \) 1680 |
\(\displaystyle 2 \int \sqrt {-x-\sqrt {x}+1} \sqrt {x}d\sqrt {x}\) |
\(\Big \downarrow \) 1160 |
\(\displaystyle 2 \left (-\frac {1}{2} \int \sqrt {-x-\sqrt {x}+1}d\sqrt {x}-\frac {1}{3} \left (-x-\sqrt {x}+1\right )^{3/2}\right )\) |
\(\Big \downarrow \) 1087 |
\(\displaystyle 2 \left (\frac {1}{2} \left (-\frac {5}{8} \int \frac {1}{\sqrt {-x-\sqrt {x}+1}}d\sqrt {x}-\frac {1}{4} \sqrt {-x-\sqrt {x}+1} \left (2 \sqrt {x}+1\right )\right )-\frac {1}{3} \left (-x-\sqrt {x}+1\right )^{3/2}\right )\) |
\(\Big \downarrow \) 1090 |
\(\displaystyle 2 \left (\frac {1}{2} \left (\frac {1}{8} \sqrt {5} \int \frac {1}{\sqrt {1-\frac {x}{5}}}d\left (-2 \sqrt {x}-1\right )-\frac {1}{4} \left (2 \sqrt {x}+1\right ) \sqrt {-x-\sqrt {x}+1}\right )-\frac {1}{3} \left (-x-\sqrt {x}+1\right )^{3/2}\right )\) |
\(\Big \downarrow \) 223 |
\(\displaystyle 2 \left (\frac {1}{2} \left (\frac {5}{8} \arcsin \left (\frac {-2 \sqrt {x}-1}{\sqrt {5}}\right )-\frac {1}{4} \left (2 \sqrt {x}+1\right ) \sqrt {-x-\sqrt {x}+1}\right )-\frac {1}{3} \left (-x-\sqrt {x}+1\right )^{3/2}\right )\) |
Input:
Int[Sqrt[1 - Sqrt[x] - x],x]
Output:
2*(-1/3*(1 - Sqrt[x] - x)^(3/2) + (-1/4*((1 + 2*Sqrt[x])*Sqrt[1 - Sqrt[x] - x]) + (5*ArcSin[(-1 - 2*Sqrt[x])/Sqrt[5]])/8)/2)
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[Rt[-b, 2]*(x/Sqrt [a])]/Rt[-b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && NegQ[b]
Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(b + 2*c*x) *((a + b*x + c*x^2)^p/(2*c*(2*p + 1))), x] - Simp[p*((b^2 - 4*a*c)/(2*c*(2* p + 1))) Int[(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] && GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[3*p])
Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/(2*c*(-4* (c/(b^2 - 4*a*c)))^p) Subst[Int[Simp[1 - x^2/(b^2 - 4*a*c), x]^p, x], x, b + 2*c*x], x] /; FreeQ[{a, b, c, p}, x] && GtQ[4*a - b^2/c, 0]
Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol ] :> Simp[e*((a + b*x + c*x^2)^(p + 1)/(2*c*(p + 1))), x] + Simp[(2*c*d - b *e)/(2*c) Int[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[p, -1]
Int[((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[n]}, Simp[k Subst[Int[x^(k - 1)*(a + b*x^(k*n) + c*x^(2*k* n))^p, x], x, x^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && EqQ[n2, 2*n] && Fr actionQ[n]
Time = 0.02 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.71
method | result | size |
derivativedivides | \(-\frac {2 \left (1-\sqrt {x}-x \right )^{\frac {3}{2}}}{3}+\frac {\left (-2 \sqrt {x}-1\right ) \sqrt {1-\sqrt {x}-x}}{4}-\frac {5 \arcsin \left (\frac {2 \sqrt {5}\, \left (\sqrt {x}+\frac {1}{2}\right )}{5}\right )}{8}\) | \(50\) |
default | \(-\frac {2 \left (1-\sqrt {x}-x \right )^{\frac {3}{2}}}{3}+\frac {\left (-2 \sqrt {x}-1\right ) \sqrt {1-\sqrt {x}-x}}{4}-\frac {5 \arcsin \left (\frac {2 \sqrt {5}\, \left (\sqrt {x}+\frac {1}{2}\right )}{5}\right )}{8}\) | \(50\) |
Input:
int((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))
Time = 0.60 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.99 \[ \int \sqrt {1-\sqrt {x}-x} \, dx=\frac {1}{12} \, {\left (8 \, x + 2 \, \sqrt {x} - 11\right )} \sqrt {-x - \sqrt {x} + 1} + \frac {5}{16} \, \arctan \left (\frac {4 \, {\left (2 \, {\left (8 \, x - 7\right )} \sqrt {x} - 8 \, x - 3\right )} \sqrt {-x - \sqrt {x} + 1}}{64 \, x^{2} - 112 \, x + 9}\right ) \] Input:
integrate((1-x^(1/2)-x)^(1/2),x, algorithm="fricas")
Output:
1/12*(8*x + 2*sqrt(x) - 11)*sqrt(-x - 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 = 0.33 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.69 \[ \int \sqrt {1-\sqrt {x}-x} \, dx=2 \sqrt {- \sqrt {x} - x + 1} \left (\frac {\sqrt {x}}{12} + \frac {x}{3} - \frac {11}{24}\right ) - \frac {5 \operatorname {asin}{\left (\frac {2 \sqrt {5} \left (\sqrt {x} + \frac {1}{2}\right )}{5} \right )}}{8} \] Input:
integrate((1-x**(1/2)-x)**(1/2),x)
Output:
2*sqrt(-sqrt(x) - x + 1)*(sqrt(x)/12 + x/3 - 11/24) - 5*asin(2*sqrt(5)*(sq rt(x) + 1/2)/5)/8
\[ \int \sqrt {1-\sqrt {x}-x} \, dx=\int { \sqrt {-x - \sqrt {x} + 1} \,d x } \] Input:
integrate((1-x^(1/2)-x)^(1/2),x, algorithm="maxima")
Output:
integrate(sqrt(-x - sqrt(x) + 1), x)
Time = 0.12 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.63 \[ \int \sqrt {1-\sqrt {x}-x} \, dx=\frac {1}{12} \, {\left (2 \, \sqrt {x} {\left (4 \, \sqrt {x} + 1\right )} - 11\right )} \sqrt {-x - \sqrt {x} + 1} - \frac {5}{8} \, \arcsin \left (\frac {1}{5} \, \sqrt {5} {\left (2 \, \sqrt {x} + 1\right )}\right ) \] Input:
integrate((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) - 5/8*arcsin( 1/5*sqrt(5)*(2*sqrt(x) + 1))
Timed out. \[ \int \sqrt {1-\sqrt {x}-x} \, dx=\int \sqrt {1-\sqrt {x}-x} \,d x \] Input:
int((1 - x^(1/2) - x)^(1/2),x)
Output:
int((1 - x^(1/2) - x)^(1/2), x)
Time = 0.17 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.83 \[ \int \sqrt {1-\sqrt {x}-x} \, dx=-\frac {5 \mathit {asin} \left (\frac {2 \sqrt {x}+1}{\sqrt {5}}\right )}{8}+\frac {\sqrt {x}\, \sqrt {-\sqrt {x}-x +1}}{6}+\frac {2 \sqrt {-\sqrt {x}-x +1}\, x}{3}-\frac {11 \sqrt {-\sqrt {x}-x +1}}{12}+\frac {5 \sqrt {5}}{12} \] Input:
int((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) + 16*sqrt( - sqrt(x) - x + 1)*x - 22*sqrt( - sqrt(x) - x + 1) + 10*sqrt(5) )/24