Integrand size = 28, antiderivative size = 181 \[ \int \frac {1+x}{(-1+x) \sqrt {1+\sqrt {1+\sqrt {1+x}}}} \, dx=\frac {32}{105} \sqrt {1+\sqrt {1+\sqrt {1+x}}}+\frac {32}{105} \sqrt {1+\sqrt {1+x}} \sqrt {1+\sqrt {1+\sqrt {1+x}}}+\sqrt {1+x} \left (-\frac {48}{35} \sqrt {1+\sqrt {1+\sqrt {1+x}}}+\frac {8}{7} \sqrt {1+\sqrt {1+x}} \sqrt {1+\sqrt {1+\sqrt {1+x}}}\right )+2 \text {RootSum}\left [-2+4 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {\log \left (\sqrt {1+\sqrt {1+\sqrt {1+x}}}-\text {$\#$1}\right )}{\text {$\#$1}}\&\right ] \]
Time = 0.00 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.69 \[ \int \frac {1+x}{(-1+x) \sqrt {1+\sqrt {1+\sqrt {1+x}}}} \, dx=\frac {8}{105} \sqrt {1+\sqrt {1+\sqrt {1+x}}} \left (4-18 \sqrt {1+x}+4 \sqrt {1+\sqrt {1+x}}+15 \sqrt {1+x} \sqrt {1+\sqrt {1+x}}\right )+2 \text {RootSum}\left [-2+4 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {\log \left (\sqrt {1+\sqrt {1+\sqrt {1+x}}}-\text {$\#$1}\right )}{\text {$\#$1}}\&\right ] \]
(8*Sqrt[1 + Sqrt[1 + Sqrt[1 + x]]]*(4 - 18*Sqrt[1 + x] + 4*Sqrt[1 + Sqrt[1 + x]] + 15*Sqrt[1 + x]*Sqrt[1 + Sqrt[1 + x]]))/105 + 2*RootSum[-2 + 4*#1^ 4 - 4*#1^6 + #1^8 & , Log[Sqrt[1 + Sqrt[1 + Sqrt[1 + x]]] - #1]/#1 & ]
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.
\(\displaystyle \int \frac {x+1}{(x-1) \sqrt {\sqrt {\sqrt {x+1}+1}+1}} \, dx\) |
\(\Big \downarrow \) 7267 |
\(\displaystyle 2 \int -\frac {(x+1)^{3/2}}{(1-x) \sqrt {\sqrt {\sqrt {x+1}+1}+1}}d\sqrt {x+1}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -2 \int \frac {(x+1)^{3/2}}{(1-x) \sqrt {\sqrt {\sqrt {x+1}+1}+1}}d\sqrt {x+1}\) |
\(\Big \downarrow \) 7267 |
\(\displaystyle -4 \int \frac {x^3 \sqrt {\sqrt {x+1}+1}}{\left (-(x+1)^2+2 (x+1)+1\right ) \sqrt {\sqrt {\sqrt {x+1}+1}+1}}d\sqrt {\sqrt {x+1}+1}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle 4 \int -\frac {x^3 \sqrt {\sqrt {x+1}+1}}{\left (-(x+1)^2+2 (x+1)+1\right ) \sqrt {\sqrt {\sqrt {x+1}+1}+1}}d\sqrt {\sqrt {x+1}+1}\) |
\(\Big \downarrow \) 2003 |
\(\displaystyle 4 \int \frac {\sqrt {\sqrt {x+1}+1} \left (1-\sqrt {\sqrt {x+1}+1}\right )^3 \left (\sqrt {\sqrt {x+1}+1}+1\right )^{5/2}}{-(x+1)^2+2 (x+1)+1}d\sqrt {\sqrt {x+1}+1}\) |
\(\Big \downarrow \) 7267 |
\(\displaystyle 8 \int \frac {(1-x)^3 x (x+1)^3}{-(x+1)^4+4 (x+1)^3-4 (x+1)^2+2}d\sqrt {\sqrt {\sqrt {x+1}+1}+1}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -8 \int -\frac {(1-x)^3 x (x+1)^3}{-(x+1)^4+4 (x+1)^3-4 (x+1)^2+2}d\sqrt {\sqrt {\sqrt {x+1}+1}+1}\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle -8 \int \left (-(x+1)^3+3 (x+1)^2+\frac {2 \left ((x+1)^2-3 (x+1)+2\right ) (x+1)}{-(x+1)^4+4 (x+1)^3-4 (x+1)^2+2}-2 (x+1)\right )d\sqrt {\sqrt {\sqrt {x+1}+1}+1}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 8 \left (4 \int \frac {x+1}{(x+1)^4-4 (x+1)^3+4 (x+1)^2-2}d\sqrt {\sqrt {\sqrt {x+1}+1}+1}-6 \int \frac {(x+1)^2}{(x+1)^4-4 (x+1)^3+4 (x+1)^2-2}d\sqrt {\sqrt {\sqrt {x+1}+1}+1}+2 \int \frac {(x+1)^3}{(x+1)^4-4 (x+1)^3+4 (x+1)^2-2}d\sqrt {\sqrt {\sqrt {x+1}+1}+1}+\frac {1}{7} (x+1)^{7/2}-\frac {3}{5} (x+1)^{5/2}+\frac {2}{3} (x+1)^{3/2}\right )\) |
3.24.30.3.1 Defintions of rubi rules used
Int[(u_)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] : > Int[u*(c + d*x)^(n + p)*(a/c + (b/d)*x)^p, x] /; FreeQ[{a, b, c, d, n, p} , x] && EqQ[b*c^2 + a*d^2, 0] && (IntegerQ[p] || (GtQ[a, 0] && GtQ[c, 0] && !IntegerQ[n]))
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 = 0.17 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.65
method | result | size |
derivativedivides | \(\frac {8 \left (1+\sqrt {1+\sqrt {1+x}}\right )^{\frac {7}{2}}}{7}-\frac {24 \left (1+\sqrt {1+\sqrt {1+x}}\right )^{\frac {5}{2}}}{5}+\frac {16 \left (1+\sqrt {1+\sqrt {1+x}}\right )^{\frac {3}{2}}}{3}+2 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}-4 \textit {\_Z}^{6}+4 \textit {\_Z}^{4}-2\right )}{\sum }\frac {\left (\textit {\_R}^{6}-3 \textit {\_R}^{4}+2 \textit {\_R}^{2}\right ) \ln \left (\sqrt {1+\sqrt {1+\sqrt {1+x}}}-\textit {\_R} \right )}{\textit {\_R}^{7}-3 \textit {\_R}^{5}+2 \textit {\_R}^{3}}\right )\) | \(117\) |
default | \(\frac {8 \left (1+\sqrt {1+\sqrt {1+x}}\right )^{\frac {7}{2}}}{7}-\frac {24 \left (1+\sqrt {1+\sqrt {1+x}}\right )^{\frac {5}{2}}}{5}+\frac {16 \left (1+\sqrt {1+\sqrt {1+x}}\right )^{\frac {3}{2}}}{3}+2 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}-4 \textit {\_Z}^{6}+4 \textit {\_Z}^{4}-2\right )}{\sum }\frac {\left (\textit {\_R}^{6}-3 \textit {\_R}^{4}+2 \textit {\_R}^{2}\right ) \ln \left (\sqrt {1+\sqrt {1+\sqrt {1+x}}}-\textit {\_R} \right )}{\textit {\_R}^{7}-3 \textit {\_R}^{5}+2 \textit {\_R}^{3}}\right )\) | \(117\) |
8/7*(1+(1+(1+x)^(1/2))^(1/2))^(7/2)-24/5*(1+(1+(1+x)^(1/2))^(1/2))^(5/2)+1 6/3*(1+(1+(1+x)^(1/2))^(1/2))^(3/2)+2*sum((_R^6-3*_R^4+2*_R^2)/(_R^7-3*_R^ 5+2*_R^3)*ln((1+(1+(1+x)^(1/2))^(1/2))^(1/2)-_R),_R=RootOf(_Z^8-4*_Z^6+4*_ Z^4-2))
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 1.01 (sec) , antiderivative size = 2096, normalized size of antiderivative = 11.58 \[ \int \frac {1+x}{(-1+x) \sqrt {1+\sqrt {1+\sqrt {1+x}}}} \, dx=\text {Too large to display} \]
8/105*((15*sqrt(x + 1) + 4)*sqrt(sqrt(x + 1) + 1) - 18*sqrt(x + 1) + 4)*sq rt(sqrt(sqrt(x + 1) + 1) + 1) - sqrt(2*sqrt(2)*sqrt(sqrt(2) + 1) - 2*sqrt( 2))*log(16*((sqrt(2)*sqrt(sqrt(2) + 1) + sqrt(2))^3 - (sqrt(2)*sqrt(sqrt(2 ) + 1) + sqrt(2))^2*(sqrt(2)*sqrt(sqrt(2) + 1) - sqrt(2)) + (sqrt(2)*sqrt( sqrt(2) + 1) + sqrt(2))*(sqrt(2)*sqrt(sqrt(2) + 1) - sqrt(2))^2 - 8*sqrt(2 )*sqrt(sqrt(2) + 1) - 8*sqrt(2))*sqrt(2*sqrt(2)*sqrt(sqrt(2) + 1) - 2*sqrt (2)) + 128*sqrt(sqrt(sqrt(x + 1) + 1) + 1)) + sqrt(2*sqrt(2)*sqrt(sqrt(2) + 1) - 2*sqrt(2))*log(-16*((sqrt(2)*sqrt(sqrt(2) + 1) + sqrt(2))^3 - (sqrt (2)*sqrt(sqrt(2) + 1) + sqrt(2))^2*(sqrt(2)*sqrt(sqrt(2) + 1) - sqrt(2)) + (sqrt(2)*sqrt(sqrt(2) + 1) + sqrt(2))*(sqrt(2)*sqrt(sqrt(2) + 1) - sqrt(2 ))^2 - 8*sqrt(2)*sqrt(sqrt(2) + 1) - 8*sqrt(2))*sqrt(2*sqrt(2)*sqrt(sqrt(2 ) + 1) - 2*sqrt(2)) + 128*sqrt(sqrt(sqrt(x + 1) + 1) + 1)) + sqrt(-2*sqrt( 2)*sqrt(sqrt(2) + 1) - 2*sqrt(2))*log(16*((sqrt(2)*sqrt(sqrt(2) + 1) + sqr t(2))^3 - 8*sqrt(2)*sqrt(sqrt(2) + 1) - 8*sqrt(2) - 16)*sqrt(-2*sqrt(2)*sq rt(sqrt(2) + 1) - 2*sqrt(2)) + 128*sqrt(sqrt(sqrt(x + 1) + 1) + 1)) - sqrt (-2*sqrt(2)*sqrt(sqrt(2) + 1) - 2*sqrt(2))*log(-16*((sqrt(2)*sqrt(sqrt(2) + 1) + sqrt(2))^3 - 8*sqrt(2)*sqrt(sqrt(2) + 1) - 8*sqrt(2) - 16)*sqrt(-2* sqrt(2)*sqrt(sqrt(2) + 1) - 2*sqrt(2)) + 128*sqrt(sqrt(sqrt(x + 1) + 1) + 1)) - 1/2*sqrt(8*sqrt(2) + 2*sqrt(-12*(sqrt(2)*sqrt(sqrt(2) + 1) + sqrt(2) )^2 + 8*(sqrt(2)*sqrt(sqrt(2) + 1) + sqrt(2))*(sqrt(2)*sqrt(sqrt(2) + 1...
Timed out. \[ \int \frac {1+x}{(-1+x) \sqrt {1+\sqrt {1+\sqrt {1+x}}}} \, dx=\text {Timed out} \]
Not integrable
Time = 0.58 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.13 \[ \int \frac {1+x}{(-1+x) \sqrt {1+\sqrt {1+\sqrt {1+x}}}} \, dx=\int { \frac {x + 1}{{\left (x - 1\right )} \sqrt {\sqrt {\sqrt {x + 1} + 1} + 1}} \,d x } \]
Not integrable
Time = 0.46 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.13 \[ \int \frac {1+x}{(-1+x) \sqrt {1+\sqrt {1+\sqrt {1+x}}}} \, dx=\int { \frac {x + 1}{{\left (x - 1\right )} \sqrt {\sqrt {\sqrt {x + 1} + 1} + 1}} \,d x } \]
Not integrable
Time = 0.00 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.13 \[ \int \frac {1+x}{(-1+x) \sqrt {1+\sqrt {1+\sqrt {1+x}}}} \, dx=\int \frac {x+1}{\sqrt {\sqrt {\sqrt {x+1}+1}+1}\,\left (x-1\right )} \,d x \]