Integrand size = 19, antiderivative size = 133 \[ \int \sqrt {1+\sqrt {1+\sqrt {1+x}}} \, dx=\frac {8}{315} (27+35 x) \sqrt {1+\sqrt {1+\sqrt {1+x}}}-\frac {64}{315} \sqrt {1+\sqrt {1+x}} \sqrt {1+\sqrt {1+\sqrt {1+x}}}+\sqrt {1+x} \left (\frac {8}{315} \sqrt {1+\sqrt {1+\sqrt {1+x}}}+\frac {8}{63} \sqrt {1+\sqrt {1+x}} \sqrt {1+\sqrt {1+\sqrt {1+x}}}\right ) \] Output:
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))
Time = 0.06 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.54 \[ \int \sqrt {1+\sqrt {1+\sqrt {1+x}}} \, dx=\frac {8}{315} \sqrt {1+\sqrt {1+\sqrt {1+x}}} \left (27+35 x+\sqrt {1+x}-8 \sqrt {1+\sqrt {1+x}}+5 \sqrt {1+x} \sqrt {1+\sqrt {1+x}}\right ) \] Input:
Integrate[Sqrt[1 + Sqrt[1 + Sqrt[1 + x]]],x]
Output:
(8*Sqrt[1 + Sqrt[1 + Sqrt[1 + x]]]*(27 + 35*x + Sqrt[1 + x] - 8*Sqrt[1 + S qrt[1 + x]] + 5*Sqrt[1 + x]*Sqrt[1 + Sqrt[1 + x]]))/315
Time = 0.27 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.41, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.368, Rules used = {7267, 896, 25, 1388, 900, 86, 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.
\(\displaystyle \int \sqrt {\sqrt {\sqrt {x+1}+1}+1} \, dx\) |
\(\Big \downarrow \) 7267 |
\(\displaystyle 2 \int \sqrt {x+1} \sqrt {\sqrt {\sqrt {x+1}+1}+1}d\sqrt {x+1}\) |
\(\Big \downarrow \) 896 |
\(\displaystyle 2 \int \sqrt {x+1} \sqrt {\sqrt [4]{x+1}+1}d\left (\sqrt {x+1}+1\right )\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -2 \int -\sqrt {x+1} \sqrt {\sqrt [4]{x+1}+1}d\left (\sqrt {x+1}+1\right )\) |
\(\Big \downarrow \) 1388 |
\(\displaystyle -2 \int \left (1-\sqrt [4]{x+1}\right ) \left (\sqrt [4]{x+1}+1\right )^{3/2}d\left (\sqrt {x+1}+1\right )\) |
\(\Big \downarrow \) 900 |
\(\displaystyle -4 \int -(x+1)^{3/4} \left (\sqrt {x+1}+2\right )^{3/2}d\sqrt [4]{x+1}\) |
\(\Big \downarrow \) 86 |
\(\displaystyle -4 \int \left (-\left (\sqrt {x+1}+2\right )^{7/2}+3 \left (\sqrt {x+1}+2\right )^{5/2}-2 \left (\sqrt {x+1}+2\right )^{3/2}\right )d\sqrt [4]{x+1}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -4 \left (-\frac {2}{9} \left (\sqrt {x+1}+2\right )^{9/2}+\frac {6}{7} \left (\sqrt {x+1}+2\right )^{7/2}-\frac {4}{5} \left (\sqrt {x+1}+2\right )^{5/2}\right )\) |
Input:
Int[Sqrt[1 + Sqrt[1 + Sqrt[1 + x]]],x]
Output:
-4*((-4*(2 + Sqrt[1 + x])^(5/2))/5 + (6*(2 + Sqrt[1 + x])^(7/2))/7 - (2*(2 + Sqrt[1 + x])^(9/2))/9)
Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_ .), x_] :> Int[ExpandIntegrand[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && ((ILtQ[n, 0] && ILtQ[p, 0]) || EqQ[p, 1 ] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))
Int[((a_) + (b_.)*(v_)^(n_))^(p_.)*(x_)^(m_.), x_Symbol] :> With[{c = Coeff icient[v, x, 0], d = Coefficient[v, x, 1]}, Simp[1/d^(m + 1) Subst[Int[Si mplifyIntegrand[(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]
Int[((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol ] :> With[{g = Denominator[n]}, Simp[g Subst[Int[x^(g - 1)*(a + b*x^(g*n) )^p*(c + d*x^(g*n))^q, x], x, x^(1/g)], x]] /; FreeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && FractionQ[n]
Int[(u_.)*((a_) + (c_.)*(x_)^(n2_.))^(p_.)*((d_) + (e_.)*(x_)^(n_))^(q_.), x_Symbol] :> Int[u*(d + e*x^n)^(p + q)*(a/d + (c/e)*x^n)^p, x] /; FreeQ[{a, c, d, e, n, p, q}, x] && EqQ[n2, 2*n] && EqQ[c*d^2 + a*e^2, 0] && (Integer Q[p] || (GtQ[a, 0] && GtQ[d, 0]))
Int[u_, x_Symbol] :> With[{lst = SubstForFractionalPowerOfLinear[u, x]}, Si mp[lst[[2]]*lst[[4]] Subst[Int[lst[[1]], x], x, lst[[3]]^(1/lst[[2]])], x ] /; !FalseQ[lst] && SubstForFractionalPowerQ[u, lst[[3]], x]]
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 | \(\frac {8 \left (1+\sqrt {1+\sqrt {1+x}}\right )^{\frac {9}{2}}}{9}-\frac {24 \left (1+\sqrt {1+\sqrt {1+x}}\right )^{\frac {7}{2}}}{7}+\frac {16 \left (1+\sqrt {1+\sqrt {1+x}}\right )^{\frac {5}{2}}}{5}\) | \(47\) |
Input:
int((1+(1+(1+x)^(1/2))^(1/2))^(1/2),x,method=_RETURNVERBOSE)
Output:
8/9*(1+(1+(1+x)^(1/2))^(1/2))^(9/2)-24/7*(1+(1+(1+x)^(1/2))^(1/2))^(7/2)+1 6/5*(1+(1+(1+x)^(1/2))^(1/2))^(5/2)
Time = 0.09 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.33 \[ \int \sqrt {1+\sqrt {1+\sqrt {1+x}}} \, dx=\frac {8}{315} \, {\left ({\left (5 \, \sqrt {x + 1} - 8\right )} \sqrt {\sqrt {x + 1} + 1} + 35 \, x + \sqrt {x + 1} + 27\right )} \sqrt {\sqrt {\sqrt {x + 1} + 1} + 1} \] Input:
integrate((1+(1+(1+x)^(1/2))^(1/2))^(1/2),x, algorithm="fricas")
Output:
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)
Time = 1.50 (sec) , antiderivative size = 230, normalized size of antiderivative = 1.73 \[ \int \sqrt {1+\sqrt {1+\sqrt {1+x}}} \, dx=- \frac {\sqrt {2} \sqrt {x + 1} \sqrt {\sqrt {x + 1} + 1} \sqrt {\sqrt {\sqrt {x + 1} + 1} + 1} \Gamma \left (- \frac {1}{4}\right ) \Gamma \left (\frac {1}{4}\right )}{63 \pi } - \frac {\sqrt {2} \sqrt {x + 1} \sqrt {\sqrt {\sqrt {x + 1} + 1} + 1} \Gamma \left (- \frac {1}{4}\right ) \Gamma \left (\frac {1}{4}\right )}{315 \pi } - \frac {\sqrt {2} \left (x + 1\right ) \sqrt {\sqrt {\sqrt {x + 1} + 1} + 1} \Gamma \left (- \frac {1}{4}\right ) \Gamma \left (\frac {1}{4}\right )}{9 \pi } + \frac {8 \sqrt {2} \sqrt {\sqrt {x + 1} + 1} \sqrt {\sqrt {\sqrt {x + 1} + 1} + 1} \Gamma \left (- \frac {1}{4}\right ) \Gamma \left (\frac {1}{4}\right )}{315 \pi } + \frac {8 \sqrt {2} \sqrt {\sqrt {\sqrt {x + 1} + 1} + 1} \Gamma \left (- \frac {1}{4}\right ) \Gamma \left (\frac {1}{4}\right )}{315 \pi } \] Input:
integrate((1+(1+(1+x)**(1/2))**(1/2))**(1/2),x)
Output:
-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) - sqrt(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(sq rt(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)/(315*pi )
Result contains higher order function than in optimal. Order 3 vs. order 2.
Time = 0.04 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.35 \[ \int \sqrt {1+\sqrt {1+\sqrt {1+x}}} \, dx=\frac {8}{9} \, {\left (\sqrt {\sqrt {x + 1} + 1} + 1\right )}^{\frac {9}{2}} - \frac {24}{7} \, {\left (\sqrt {\sqrt {x + 1} + 1} + 1\right )}^{\frac {7}{2}} + \frac {16}{5} \, {\left (\sqrt {\sqrt {x + 1} + 1} + 1\right )}^{\frac {5}{2}} \] Input:
integrate((1+(1+(1+x)^(1/2))^(1/2))^(1/2),x, algorithm="maxima")
Output:
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)
Result contains higher order function than in optimal. Order 3 vs. order 2.
Time = 0.38 (sec) , antiderivative size = 312, normalized size of antiderivative = 2.35 \[ \int \sqrt {1+\sqrt {1+\sqrt {1+x}}} \, dx=\frac {8}{315} \, {\left ({\left (35 \, {\left (\sqrt {\sqrt {x + 1} + 1} + 1\right )}^{\frac {9}{2}} - 180 \, {\left (\sqrt {\sqrt {x + 1} + 1} + 1\right )}^{\frac {7}{2}} + 378 \, {\left (\sqrt {\sqrt {x + 1} + 1} + 1\right )}^{\frac {5}{2}} - 420 \, {\left (\sqrt {\sqrt {x + 1} + 1} + 1\right )}^{\frac {3}{2}} + 315 \, \sqrt {\sqrt {\sqrt {x + 1} + 1} + 1}\right )} \mathrm {sgn}\left (4 \, {\left (\sqrt {x + 1} + 1\right )}^{2} - 8 \, \sqrt {x + 1} - 7\right ) + 9 \, {\left (5 \, {\left (\sqrt {\sqrt {x + 1} + 1} + 1\right )}^{\frac {7}{2}} - 21 \, {\left (\sqrt {\sqrt {x + 1} + 1} + 1\right )}^{\frac {5}{2}} + 35 \, {\left (\sqrt {\sqrt {x + 1} + 1} + 1\right )}^{\frac {3}{2}} - 35 \, \sqrt {\sqrt {\sqrt {x + 1} + 1} + 1}\right )} \mathrm {sgn}\left (4 \, {\left (\sqrt {x + 1} + 1\right )}^{2} - 8 \, \sqrt {x + 1} - 7\right ) - 21 \, {\left (3 \, {\left (\sqrt {\sqrt {x + 1} + 1} + 1\right )}^{\frac {5}{2}} - 10 \, {\left (\sqrt {\sqrt {x + 1} + 1} + 1\right )}^{\frac {3}{2}} + 15 \, \sqrt {\sqrt {\sqrt {x + 1} + 1} + 1}\right )} \mathrm {sgn}\left (4 \, {\left (\sqrt {x + 1} + 1\right )}^{2} - 8 \, \sqrt {x + 1} - 7\right ) - 105 \, {\left ({\left (\sqrt {\sqrt {x + 1} + 1} + 1\right )}^{\frac {3}{2}} - 3 \, \sqrt {\sqrt {\sqrt {x + 1} + 1} + 1}\right )} \mathrm {sgn}\left (4 \, {\left (\sqrt {x + 1} + 1\right )}^{2} - 8 \, \sqrt {x + 1} - 7\right )\right )} \mathrm {sgn}\left (4 \, x + 1\right ) \] Input:
integrate((1+(1+(1+x)^(1/2))^(1/2))^(1/2),x, algorithm="giac")
Output:
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*sqrt(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)
Timed out. \[ \int \sqrt {1+\sqrt {1+\sqrt {1+x}}} \, dx=\int \sqrt {\sqrt {\sqrt {x+1}+1}+1} \,d x \] Input:
int((((x + 1)^(1/2) + 1)^(1/2) + 1)^(1/2),x)
Output:
int((((x + 1)^(1/2) + 1)^(1/2) + 1)^(1/2), x)
Time = 0.21 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.32 \[ \int \sqrt {1+\sqrt {1+\sqrt {1+x}}} \, dx=\frac {8 \sqrt {\sqrt {\sqrt {x +1}+1}+1}\, \left (5 \sqrt {x +1}\, \sqrt {\sqrt {x +1}+1}-8 \sqrt {\sqrt {x +1}+1}+\sqrt {x +1}+35 x +27\right )}{315} \] Input:
int((1+(1+(1+x)^(1/2))^(1/2))^(1/2),x)
Output:
(8*sqrt(sqrt(sqrt(x + 1) + 1) + 1)*(5*sqrt(x + 1)*sqrt(sqrt(x + 1) + 1) - 8*sqrt(sqrt(x + 1) + 1) + sqrt(x + 1) + 35*x + 27))/315