\(\int \sqrt {1+\sqrt {1+\sqrt {1+x}}} \, dx\) [1922]

Optimal result

Integrand size = 19, antiderivative size = 133 \[ \int \sqrt {1+\sqrt {1+\sqrt {1+x}}} \, dx=\frac {8}{315} (27+35 x) \sqrt {1+\sqrt {1+\sqrt {1+x}}}-\frac {64}{315} \sqrt {1+\sqrt {1+x}} \sqrt {1+\sqrt {1+\sqrt {1+x}}}+\sqrt {1+x} \left (\frac {8}{315} \sqrt {1+\sqrt {1+\sqrt {1+x}}}+\frac {8}{63} \sqrt {1+\sqrt {1+x}} \sqrt {1+\sqrt {1+\sqrt {1+x}}}\right ) \]

[Out]

8/315*(27+35*x)*(1+(1+(1+x)^(1/2))^(1/2))^(1/2)-64/315*(1+(1+x)^(1/2))^(1/2)*(1+(1+(1+x)^(1/2))^(1/2))^(1/2)+(
1+x)^(1/2)*(8/315*(1+(1+(1+x)^(1/2))^(1/2))^(1/2)+8/63*(1+(1+x)^(1/2))^(1/2)*(1+(1+(1+x)^(1/2))^(1/2))^(1/2))

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.53, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {378, 1412, 786} \[ \int \sqrt {1+\sqrt {1+\sqrt {1+x}}} \, dx=\frac {8}{9} \left (\sqrt {\sqrt {x+1}+1}+1\right )^{9/2}-\frac {24}{7} \left (\sqrt {\sqrt {x+1}+1}+1\right )^{7/2}+\frac {16}{5} \left (\sqrt {\sqrt {x+1}+1}+1\right )^{5/2} \]

[In]

Int[Sqrt[1 + Sqrt[1 + Sqrt[1 + x]]],x]

[Out]

(16*(1 + Sqrt[1 + Sqrt[1 + x]])^(5/2))/5 - (24*(1 + Sqrt[1 + Sqrt[1 + x]])^(7/2))/7 + (8*(1 + Sqrt[1 + Sqrt[1
+ x]])^(9/2))/9

Rule 378

Int[((a_) + (b_.)*(v_)^(n_))^(p_.)*(x_)^(m_.), x_Symbol] :> With[{c = Coefficient[v, x, 0], d = Coefficient[v,
x, 1]}, Dist[1/d^(m + 1), Subst[Int[SimplifyIntegrand[(x - c)^m*(a + b*x^n)^p, x], x], x, v], x] /; NeQ[c, 0]
] /; FreeQ[{a, b, n, p}, x] && LinearQ[v, x] && IntegerQ[m]

Rule 786

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegr
and[(d + e*x)^m*(f + g*x)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && IGtQ[p, 0]

Rule 1412

Int[((a_) + (c_.)*(x_)^(n2_.))^(p_.)*((d_) + (e_.)*(x_)^(n_))^(q_.), x_Symbol] :> With[{g = Denominator[n]}, D
ist[g, Subst[Int[x^(g - 1)*(d + e*x^(g*n))^q*(a + c*x^(2*g*n))^p, x], x, x^(1/g)], x]] /; FreeQ[{a, c, d, e, p
, q}, x] && EqQ[n2, 2*n] && FractionQ[n]

Rubi steps \begin{align*} \text {integral}& = 2 \text {Subst}\left (\int x \sqrt {1+\sqrt {1+x}} \, dx,x,\sqrt {1+x}\right ) \\ & = 2 \text {Subst}\left (\int \sqrt {1+\sqrt {x}} (-1+x) \, dx,x,1+\sqrt {1+x}\right ) \\ & = 4 \text {Subst}\left (\int x \sqrt {1+x} \left (-1+x^2\right ) \, dx,x,\sqrt {1+\sqrt {1+x}}\right ) \\ & = 4 \text {Subst}\left (\int \left (2 (1+x)^{3/2}-3 (1+x)^{5/2}+(1+x)^{7/2}\right ) \, dx,x,\sqrt {1+\sqrt {1+x}}\right ) \\ & = \frac {16}{5} \left (1+\sqrt {1+\sqrt {1+x}}\right )^{5/2}-\frac {24}{7} \left (1+\sqrt {1+\sqrt {1+x}}\right )^{7/2}+\frac {8}{9} \left (1+\sqrt {1+\sqrt {1+x}}\right )^{9/2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.54 \[ \int \sqrt {1+\sqrt {1+\sqrt {1+x}}} \, dx=\frac {8}{315} \sqrt {1+\sqrt {1+\sqrt {1+x}}} \left (27+35 x+\sqrt {1+x}-8 \sqrt {1+\sqrt {1+x}}+5 \sqrt {1+x} \sqrt {1+\sqrt {1+x}}\right ) \]

[In]

Integrate[Sqrt[1 + Sqrt[1 + Sqrt[1 + x]]],x]

[Out]

(8*Sqrt[1 + Sqrt[1 + Sqrt[1 + x]]]*(27 + 35*x + Sqrt[1 + x] - 8*Sqrt[1 + Sqrt[1 + x]] + 5*Sqrt[1 + x]*Sqrt[1 +
Sqrt[1 + x]]))/315

Maple [A] (verified)

Time = 0.14 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.35

method	result	size
derivativedivides	\(\frac {8 \left (1+\sqrt {1+\sqrt {1+x}}\right )^{\frac {9}{2}}}{9}-\frac {24 \left (1+\sqrt {1+\sqrt {1+x}}\right )^{\frac {7}{2}}}{7}+\frac {16 \left (1+\sqrt {1+\sqrt {1+x}}\right )^{\frac {5}{2}}}{5}\)	\(47\)
default	\(\frac {8 \left (1+\sqrt {1+\sqrt {1+x}}\right )^{\frac {9}{2}}}{9}-\frac {24 \left (1+\sqrt {1+\sqrt {1+x}}\right )^{\frac {7}{2}}}{7}+\frac {16 \left (1+\sqrt {1+\sqrt {1+x}}\right )^{\frac {5}{2}}}{5}\)	\(47\)

[In]

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

[Out]

8/9*(1+(1+(1+x)^(1/2))^(1/2))^(9/2)-24/7*(1+(1+(1+x)^(1/2))^(1/2))^(7/2)+16/5*(1+(1+(1+x)^(1/2))^(1/2))^(5/2)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.33 \[ \int \sqrt {1+\sqrt {1+\sqrt {1+x}}} \, dx=\frac {8}{315} \, {\left ({\left (5 \, \sqrt {x + 1} - 8\right )} \sqrt {\sqrt {x + 1} + 1} + 35 \, x + \sqrt {x + 1} + 27\right )} \sqrt {\sqrt {\sqrt {x + 1} + 1} + 1} \]

[In]

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

[Out]

8/315*((5*sqrt(x + 1) - 8)*sqrt(sqrt(x + 1) + 1) + 35*x + sqrt(x + 1) + 27)*sqrt(sqrt(sqrt(x + 1) + 1) + 1)

Sympy [A] (verification not implemented)

Time = 1.31 (sec) , antiderivative size = 230, normalized size of antiderivative = 1.73 \[ \int \sqrt {1+\sqrt {1+\sqrt {1+x}}} \, dx=- \frac {\sqrt {2} \sqrt {x + 1} \sqrt {\sqrt {x + 1} + 1} \sqrt {\sqrt {\sqrt {x + 1} + 1} + 1} \Gamma \left (- \frac {1}{4}\right ) \Gamma \left (\frac {1}{4}\right )}{63 \pi } - \frac {\sqrt {2} \sqrt {x + 1} \sqrt {\sqrt {\sqrt {x + 1} + 1} + 1} \Gamma \left (- \frac {1}{4}\right ) \Gamma \left (\frac {1}{4}\right )}{315 \pi } - \frac {\sqrt {2} \left (x + 1\right ) \sqrt {\sqrt {\sqrt {x + 1} + 1} + 1} \Gamma \left (- \frac {1}{4}\right ) \Gamma \left (\frac {1}{4}\right )}{9 \pi } + \frac {8 \sqrt {2} \sqrt {\sqrt {x + 1} + 1} \sqrt {\sqrt {\sqrt {x + 1} + 1} + 1} \Gamma \left (- \frac {1}{4}\right ) \Gamma \left (\frac {1}{4}\right )}{315 \pi } + \frac {8 \sqrt {2} \sqrt {\sqrt {\sqrt {x + 1} + 1} + 1} \Gamma \left (- \frac {1}{4}\right ) \Gamma \left (\frac {1}{4}\right )}{315 \pi } \]

[In]

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

[Out]

-sqrt(2)*sqrt(x + 1)*sqrt(sqrt(x + 1) + 1)*sqrt(sqrt(sqrt(x + 1) + 1) + 1)*gamma(-1/4)*gamma(1/4)/(63*pi) - sq
rt(2)*sqrt(x + 1)*sqrt(sqrt(sqrt(x + 1) + 1) + 1)*gamma(-1/4)*gamma(1/4)/(315*pi) - sqrt(2)*(x + 1)*sqrt(sqrt(
sqrt(x + 1) + 1) + 1)*gamma(-1/4)*gamma(1/4)/(9*pi) + 8*sqrt(2)*sqrt(sqrt(x + 1) + 1)*sqrt(sqrt(sqrt(x + 1) +
1) + 1)*gamma(-1/4)*gamma(1/4)/(315*pi) + 8*sqrt(2)*sqrt(sqrt(sqrt(x + 1) + 1) + 1)*gamma(-1/4)*gamma(1/4)/(31
5*pi)

Maxima [C] (verification not implemented)

Result contains higher order function than in optimal. Order 3 vs. order 2.

Time = 0.19 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.35 \[ \int \sqrt {1+\sqrt {1+\sqrt {1+x}}} \, dx=\frac {8}{9} \, {\left (\sqrt {\sqrt {x + 1} + 1} + 1\right )}^{\frac {9}{2}} - \frac {24}{7} \, {\left (\sqrt {\sqrt {x + 1} + 1} + 1\right )}^{\frac {7}{2}} + \frac {16}{5} \, {\left (\sqrt {\sqrt {x + 1} + 1} + 1\right )}^{\frac {5}{2}} \]

[In]

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

[Out]

8/9*(sqrt(sqrt(x + 1) + 1) + 1)^(9/2) - 24/7*(sqrt(sqrt(x + 1) + 1) + 1)^(7/2) + 16/5*(sqrt(sqrt(x + 1) + 1) +
1)^(5/2)

Giac [C] (verification not implemented)

Result contains higher order function than in optimal. Order 3 vs. order 2.

Time = 0.35 (sec) , antiderivative size = 312, normalized size of antiderivative = 2.35 \[ \int \sqrt {1+\sqrt {1+\sqrt {1+x}}} \, dx=\frac {8}{315} \, {\left ({\left (35 \, {\left (\sqrt {\sqrt {x + 1} + 1} + 1\right )}^{\frac {9}{2}} - 180 \, {\left (\sqrt {\sqrt {x + 1} + 1} + 1\right )}^{\frac {7}{2}} + 378 \, {\left (\sqrt {\sqrt {x + 1} + 1} + 1\right )}^{\frac {5}{2}} - 420 \, {\left (\sqrt {\sqrt {x + 1} + 1} + 1\right )}^{\frac {3}{2}} + 315 \, \sqrt {\sqrt {\sqrt {x + 1} + 1} + 1}\right )} \mathrm {sgn}\left (4 \, {\left (\sqrt {x + 1} + 1\right )}^{2} - 8 \, \sqrt {x + 1} - 7\right ) + 9 \, {\left (5 \, {\left (\sqrt {\sqrt {x + 1} + 1} + 1\right )}^{\frac {7}{2}} - 21 \, {\left (\sqrt {\sqrt {x + 1} + 1} + 1\right )}^{\frac {5}{2}} + 35 \, {\left (\sqrt {\sqrt {x + 1} + 1} + 1\right )}^{\frac {3}{2}} - 35 \, \sqrt {\sqrt {\sqrt {x + 1} + 1} + 1}\right )} \mathrm {sgn}\left (4 \, {\left (\sqrt {x + 1} + 1\right )}^{2} - 8 \, \sqrt {x + 1} - 7\right ) - 21 \, {\left (3 \, {\left (\sqrt {\sqrt {x + 1} + 1} + 1\right )}^{\frac {5}{2}} - 10 \, {\left (\sqrt {\sqrt {x + 1} + 1} + 1\right )}^{\frac {3}{2}} + 15 \, \sqrt {\sqrt {\sqrt {x + 1} + 1} + 1}\right )} \mathrm {sgn}\left (4 \, {\left (\sqrt {x + 1} + 1\right )}^{2} - 8 \, \sqrt {x + 1} - 7\right ) - 105 \, {\left ({\left (\sqrt {\sqrt {x + 1} + 1} + 1\right )}^{\frac {3}{2}} - 3 \, \sqrt {\sqrt {\sqrt {x + 1} + 1} + 1}\right )} \mathrm {sgn}\left (4 \, {\left (\sqrt {x + 1} + 1\right )}^{2} - 8 \, \sqrt {x + 1} - 7\right )\right )} \mathrm {sgn}\left (4 \, x + 1\right ) \]

[In]

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

[Out]

8/315*((35*(sqrt(sqrt(x + 1) + 1) + 1)^(9/2) - 180*(sqrt(sqrt(x + 1) + 1) + 1)^(7/2) + 378*(sqrt(sqrt(x + 1) +
1) + 1)^(5/2) - 420*(sqrt(sqrt(x + 1) + 1) + 1)^(3/2) + 315*sqrt(sqrt(sqrt(x + 1) + 1) + 1))*sgn(4*(sqrt(x +
1) + 1)^2 - 8*sqrt(x + 1) - 7) + 9*(5*(sqrt(sqrt(x + 1) + 1) + 1)^(7/2) - 21*(sqrt(sqrt(x + 1) + 1) + 1)^(5/2)
+ 35*(sqrt(sqrt(x + 1) + 1) + 1)^(3/2) - 35*sqrt(sqrt(sqrt(x + 1) + 1) + 1))*sgn(4*(sqrt(x + 1) + 1)^2 - 8*sq
rt(x + 1) - 7) - 21*(3*(sqrt(sqrt(x + 1) + 1) + 1)^(5/2) - 10*(sqrt(sqrt(x + 1) + 1) + 1)^(3/2) + 15*sqrt(sqrt
(sqrt(x + 1) + 1) + 1))*sgn(4*(sqrt(x + 1) + 1)^2 - 8*sqrt(x + 1) - 7) - 105*((sqrt(sqrt(x + 1) + 1) + 1)^(3/2
) - 3*sqrt(sqrt(sqrt(x + 1) + 1) + 1))*sgn(4*(sqrt(x + 1) + 1)^2 - 8*sqrt(x + 1) - 7))*sgn(4*x + 1)

Mupad [F(-1)]

Timed out. \[ \int \sqrt {1+\sqrt {1+\sqrt {1+x}}} \, dx=\int \sqrt {\sqrt {\sqrt {x+1}+1}+1} \,d x \]

[In]

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

[Out]

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