\(\int (\frac {1}{\sqrt {1-\sqrt {x}}}-\sqrt {1-\sqrt {x}-x}) \, dx\) [2258]

Optimal result

Integrand size = 32, antiderivative size = 171 \[ \int \left (\frac {1}{\sqrt {1-\sqrt {x}}}-\sqrt {1-\sqrt {x}-x}\right ) \, dx=-\frac {8}{3} \sqrt {1-\sqrt {x}}+\left (\frac {2}{3}+\frac {1}{4} \sqrt {\left (1+2 \sqrt {x}\right )^2}\right ) \sqrt {1-\sqrt {x}-x}-\frac {4}{3} \sqrt {1-\sqrt {x}} \sqrt {x}-\frac {2}{3} \sqrt {1-\sqrt {x}-x} \sqrt {x}-\frac {2}{3} \sqrt {1-\sqrt {x}-x} x+\frac {5}{8} i \log \left (i \sqrt {\left (1+2 \sqrt {x}\right )^2}-2 \sqrt {1-\sqrt {x}-x}\right ) \]

[Out]

-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
))

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.60, number of steps used = 9, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.219, Rules used = {196, 45, 1355, 654, 626, 633, 222} \[ \int \left (\frac {1}{\sqrt {1-\sqrt {x}}}-\sqrt {1-\sqrt {x}-x}\right ) \, dx=\frac {5}{8} \arcsin \left (\frac {2 \sqrt {x}+1}{\sqrt {5}}\right )+\frac {4}{3} \left (1-\sqrt {x}\right )^{3/2}-4 \sqrt {1-\sqrt {x}}+\frac {2}{3} \left (-x-\sqrt {x}+1\right )^{3/2}+\frac {1}{4} \left (2 \sqrt {x}+1\right ) \sqrt {-x-\sqrt {x}+1} \]

[In]

Int[1/Sqrt[1 - Sqrt[x]] - Sqrt[1 - Sqrt[x] - x],x]

[Out]

-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

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 196

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(1/n - 1)*(a + b*x)^p, x], x, x^n], x] /
; FreeQ[{a, b, p}, x] && FractionQ[n] && IntegerQ[1/n]

Rule 222

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]

Rule 626

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] - Dist[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]
&& NeQ[b^2 - 4*a*c, 0] && GtQ[p, 0] && IntegerQ[4*p]

Rule 633

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[1/(2*c*(-4*(c/(b^2 - 4*a*c)))^p), Subst[Int[Si
mp[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]

Rule 654

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] + Dist[(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[2*c*d - b*e, 0] && NeQ[p, -1]

Rule 1355

Int[((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[n]}, Dist[k, Subst[I
nt[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] &
& FractionQ[n]

Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{\sqrt {1-\sqrt {x}}} \, dx-\int \sqrt {1-\sqrt {x}-x} \, dx \\ & = 2 \text {Subst}\left (\int \frac {x}{\sqrt {1-x}} \, dx,x,\sqrt {x}\right )-2 \text {Subst}\left (\int x \sqrt {1-x-x^2} \, dx,x,\sqrt {x}\right ) \\ & = \frac {2}{3} \left (1-\sqrt {x}-x\right )^{3/2}+2 \text {Subst}\left (\int \left (\frac {1}{\sqrt {1-x}}-\sqrt {1-x}\right ) \, dx,x,\sqrt {x}\right )+\text {Subst}\left (\int \sqrt {1-x-x^2} \, dx,x,\sqrt {x}\right ) \\ & = -4 \sqrt {1-\sqrt {x}}+\frac {4}{3} \left (1-\sqrt {x}\right )^{3/2}+\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} \text {Subst}\left (\int \frac {1}{\sqrt {1-x-x^2}} \, dx,x,\sqrt {x}\right ) \\ & = -4 \sqrt {1-\sqrt {x}}+\frac {4}{3} \left (1-\sqrt {x}\right )^{3/2}+\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 {1}{8} \sqrt {5} \text {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^2}{5}}} \, dx,x,-1-2 \sqrt {x}\right ) \\ & = -4 \sqrt {1-\sqrt {x}}+\frac {4}{3} \left (1-\sqrt {x}\right )^{3/2}+\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 ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.51 \[ \int \left (\frac {1}{\sqrt {1-\sqrt {x}}}-\sqrt {1-\sqrt {x}-x}\right ) \, dx=-\frac {4}{3} \sqrt {1-\sqrt {x}} \left (2+\sqrt {x}\right )-\frac {1}{12} \sqrt {1-\sqrt {x}-x} \left (-11+2 \sqrt {x}+8 x\right )+\frac {5}{4} \arctan \left (\frac {\sqrt {x}}{-1+\sqrt {1-\sqrt {x}-x}}\right ) \]

[In]

Integrate[1/Sqrt[1 - Sqrt[x]] - Sqrt[1 - Sqrt[x] - x],x]

[Out]

(-4*Sqrt[1 - Sqrt[x]]*(2 + Sqrt[x]))/3 - (Sqrt[1 - Sqrt[x] - x]*(-11 + 2*Sqrt[x] + 8*x))/12 + (5*ArcTan[Sqrt[x
]/(-1 + Sqrt[1 - Sqrt[x] - x])])/4

Maple [A] (verified)

Time = 0.10 (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	\(\frac {2 \left (1-\sqrt {x}-x \right )^{\frac {3}{2}}}{3}-\frac {\left (-2 \sqrt {x}-1\right ) \sqrt {1-\sqrt {x}-x}}{4}+\frac {5 \arcsin \left (\frac {2 \sqrt {5}\, \left (\sqrt {x}+\frac {1}{2}\right )}{5}\right )}{8}+\frac {4 \left (1-\sqrt {x}\right )^{\frac {3}{2}}}{3}-4 \sqrt {1-\sqrt {x}}\)	\(72\)

[In]

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

[Out]

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)

Fricas [A] (verification not implemented)

none

Time = 0.77 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.58 \[ \int \left (\frac {1}{\sqrt {1-\sqrt {x}}}-\sqrt {1-\sqrt {x}-x}\right ) \, dx=-\frac {1}{12} \, {\left (8 \, x + 2 \, \sqrt {x} - 11\right )} \sqrt {-x - \sqrt {x} + 1} - \frac {4}{3} \, {\left (\sqrt {x} + 2\right )} \sqrt {-\sqrt {x} + 1} - \frac {5}{16} \, \arctan \left (-\frac {{\left (8 \, x^{2} - {\left (16 \, x^{2} - 38 \, x + 11\right )} \sqrt {x} - 9 \, x + 3\right )} \sqrt {-x - \sqrt {x} + 1}}{4 \, {\left (4 \, x^{3} - 13 \, x^{2} + 7 \, x - 1\right )}}\right ) \]

[In]

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

[Out]

-1/12*(8*x + 2*sqrt(x) - 11)*sqrt(-x - sqrt(x) + 1) - 4/3*(sqrt(x) + 2)*sqrt(-sqrt(x) + 1) - 5/16*arctan(-1/4*
(8*x^2 - (16*x^2 - 38*x + 11)*sqrt(x) - 9*x + 3)*sqrt(-x - sqrt(x) + 1)/(4*x^3 - 13*x^2 + 7*x - 1))

Sympy [A] (verification not implemented)

Time = 0.99 (sec) , antiderivative size = 296, normalized size of antiderivative = 1.73 \[ \int \left (\frac {1}{\sqrt {1-\sqrt {x}}}-\sqrt {1-\sqrt {x}-x}\right ) \, dx=- 2 \sqrt {- \sqrt {x} - x + 1} \left (\frac {\sqrt {x}}{12} + \frac {x}{3} - \frac {11}{24}\right ) + \begin {cases} \frac {4 i x^{\frac {5}{2}} \sqrt {\sqrt {x} - 1}}{- 3 x^{\frac {5}{2}} + 3 x^{2}} - \frac {8 x^{\frac {5}{2}}}{- 3 x^{\frac {5}{2}} + 3 x^{2}} + \frac {4 i x^{3} \sqrt {\sqrt {x} - 1}}{- 3 x^{\frac {5}{2}} + 3 x^{2}} - \frac {8 i x^{2} \sqrt {\sqrt {x} - 1}}{- 3 x^{\frac {5}{2}} + 3 x^{2}} + \frac {8 x^{2}}{- 3 x^{\frac {5}{2}} + 3 x^{2}} & \text {for}\: \left |{\sqrt {x}}\right | > 1 \\\frac {4 x^{\frac {5}{2}} \sqrt {1 - \sqrt {x}}}{- 3 x^{\frac {5}{2}} + 3 x^{2}} - \frac {8 x^{\frac {5}{2}}}{- 3 x^{\frac {5}{2}} + 3 x^{2}} + \frac {4 x^{3} \sqrt {1 - \sqrt {x}}}{- 3 x^{\frac {5}{2}} + 3 x^{2}} - \frac {8 x^{2} \sqrt {1 - \sqrt {x}}}{- 3 x^{\frac {5}{2}} + 3 x^{2}} + \frac {8 x^{2}}{- 3 x^{\frac {5}{2}} + 3 x^{2}} & \text {otherwise} \end {cases} + \frac {5 \operatorname {asin}{\left (\frac {2 \sqrt {5} \left (\sqrt {x} + \frac {1}{2}\right )}{5} \right )}}{8} \]

[In]

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

[Out]

-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*s
qrt(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

Maxima [F]

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

[In]

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

[Out]

4/3*(-sqrt(x) + 1)^(3/2) - 4*sqrt(-sqrt(x) + 1) - integrate(sqrt(-x - sqrt(x) + 1), x)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.39 \[ \int \left (\frac {1}{\sqrt {1-\sqrt {x}}}-\sqrt {1-\sqrt {x}-x}\right ) \, dx=-\frac {1}{12} \, {\left (2 \, \sqrt {x} {\left (4 \, \sqrt {x} + 1\right )} - 11\right )} \sqrt {-x - \sqrt {x} + 1} + \frac {4}{3} \, {\left (-\sqrt {x} + 1\right )}^{\frac {3}{2}} - 4 \, \sqrt {-\sqrt {x} + 1} + \frac {5}{8} \, \arcsin \left (\frac {1}{5} \, \sqrt {5} {\left (2 \, \sqrt {x} + 1\right )}\right ) \]

[In]

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

[Out]

-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))

Mupad [F(-1)]

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

[In]

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

[Out]

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