\(\int \frac {1+x}{(-1+x) \sqrt {1+\sqrt {1+\sqrt {1+x}}}} \, dx\) [2329]
Optimal result
Integrand size = 28, antiderivative size = 181 \[
\int \frac {1+x}{(-1+x) \sqrt {1+\sqrt {1+\sqrt {1+x}}}} \, dx=\frac {32}{105} \sqrt {1+\sqrt {1+\sqrt {1+x}}}+\frac {32}{105} \sqrt {1+\sqrt {1+x}} \sqrt {1+\sqrt {1+\sqrt {1+x}}}+\sqrt {1+x} \left (-\frac {48}{35} \sqrt {1+\sqrt {1+\sqrt {1+x}}}+\frac {8}{7} \sqrt {1+\sqrt {1+x}} \sqrt {1+\sqrt {1+\sqrt {1+x}}}\right )+2 \text {RootSum}\left [-2+4 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {\log \left (\sqrt {1+\sqrt {1+\sqrt {1+x}}}-\text {$\#$1}\right )}{\text {$\#$1}}\&\right ]
\]
[Out]
Unintegrable
Rubi [F]
\[
\int \frac {1+x}{(-1+x) \sqrt {1+\sqrt {1+\sqrt {1+x}}}} \, dx=\int \frac {1+x}{(-1+x) \sqrt {1+\sqrt {1+\sqrt {1+x}}}} \, dx
\]
[In]
Int[(1 + x)/((-1 + x)*Sqrt[1 + Sqrt[1 + Sqrt[1 + x]]]),x]
[Out]
(16*(1 + Sqrt[1 + Sqrt[1 + x]])^(3/2))/3 - (24*(1 + Sqrt[1 + Sqrt[1 + x]])^(5/2))/5 + (8*(1 + Sqrt[1 + Sqrt[1
+ x]])^(7/2))/7 + 32*Defer[Subst][Defer[Int][x^2/(-2 + 4*x^4 - 4*x^6 + x^8), x], x, Sqrt[1 + Sqrt[1 + Sqrt[1 +
x]]]] - 48*Defer[Subst][Defer[Int][x^4/(-2 + 4*x^4 - 4*x^6 + x^8), x], x, Sqrt[1 + Sqrt[1 + Sqrt[1 + x]]]] +
16*Defer[Subst][Defer[Int][x^6/(-2 + 4*x^4 - 4*x^6 + x^8), x], x, Sqrt[1 + Sqrt[1 + Sqrt[1 + x]]]]
Rubi steps \begin{align*}
\text {integral}& = 2 \text {Subst}\left (\int \frac {x^3}{\left (-2+x^2\right ) \sqrt {1+\sqrt {1+x}}} \, dx,x,\sqrt {1+x}\right ) \\ & = 4 \text {Subst}\left (\int \frac {x \left (-1+x^2\right )^3}{\sqrt {1+x} \left (-2+\left (-1+x^2\right )^2\right )} \, dx,x,\sqrt {1+\sqrt {1+x}}\right ) \\ & = 4 \text {Subst}\left (\int \frac {(-1+x)^3 x (1+x)^{5/2}}{-2+\left (-1+x^2\right )^2} \, dx,x,\sqrt {1+\sqrt {1+x}}\right ) \\ & = 8 \text {Subst}\left (\int \frac {x^6 \left (-2+x^2\right )^3 \left (-1+x^2\right )}{-2+x^4 \left (-2+x^2\right )^2} \, dx,x,\sqrt {1+\sqrt {1+\sqrt {1+x}}}\right ) \\ & = 8 \text {Subst}\left (\int \left (2 x^2-3 x^4+x^6+\frac {2 x^2 \left (2-3 x^2+x^4\right )}{-2+x^4 \left (-2+x^2\right )^2}\right ) \, dx,x,\sqrt {1+\sqrt {1+\sqrt {1+x}}}\right ) \\ & = \frac {16}{3} \left (1+\sqrt {1+\sqrt {1+x}}\right )^{3/2}-\frac {24}{5} \left (1+\sqrt {1+\sqrt {1+x}}\right )^{5/2}+\frac {8}{7} \left (1+\sqrt {1+\sqrt {1+x}}\right )^{7/2}+16 \text {Subst}\left (\int \frac {x^2 \left (2-3 x^2+x^4\right )}{-2+x^4 \left (-2+x^2\right )^2} \, dx,x,\sqrt {1+\sqrt {1+\sqrt {1+x}}}\right ) \\ & = \frac {16}{3} \left (1+\sqrt {1+\sqrt {1+x}}\right )^{3/2}-\frac {24}{5} \left (1+\sqrt {1+\sqrt {1+x}}\right )^{5/2}+\frac {8}{7} \left (1+\sqrt {1+\sqrt {1+x}}\right )^{7/2}+16 \text {Subst}\left (\int \left (\frac {2 x^2}{-2+4 x^4-4 x^6+x^8}-\frac {3 x^4}{-2+4 x^4-4 x^6+x^8}+\frac {x^6}{-2+4 x^4-4 x^6+x^8}\right ) \, dx,x,\sqrt {1+\sqrt {1+\sqrt {1+x}}}\right ) \\ & = \frac {16}{3} \left (1+\sqrt {1+\sqrt {1+x}}\right )^{3/2}-\frac {24}{5} \left (1+\sqrt {1+\sqrt {1+x}}\right )^{5/2}+\frac {8}{7} \left (1+\sqrt {1+\sqrt {1+x}}\right )^{7/2}+16 \text {Subst}\left (\int \frac {x^6}{-2+4 x^4-4 x^6+x^8} \, dx,x,\sqrt {1+\sqrt {1+\sqrt {1+x}}}\right )+32 \text {Subst}\left (\int \frac {x^2}{-2+4 x^4-4 x^6+x^8} \, dx,x,\sqrt {1+\sqrt {1+\sqrt {1+x}}}\right )-48 \text {Subst}\left (\int \frac {x^4}{-2+4 x^4-4 x^6+x^8} \, dx,x,\sqrt {1+\sqrt {1+\sqrt {1+x}}}\right ) \\
\end{align*}
Mathematica [A] (verified)
Time = 0.18 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.69
\[
\int \frac {1+x}{(-1+x) \sqrt {1+\sqrt {1+\sqrt {1+x}}}} \, dx=\frac {8}{105} \sqrt {1+\sqrt {1+\sqrt {1+x}}} \left (4-18 \sqrt {1+x}+4 \sqrt {1+\sqrt {1+x}}+15 \sqrt {1+x} \sqrt {1+\sqrt {1+x}}\right )+2 \text {RootSum}\left [-2+4 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {\log \left (\sqrt {1+\sqrt {1+\sqrt {1+x}}}-\text {$\#$1}\right )}{\text {$\#$1}}\&\right ]
\]
[In]
Integrate[(1 + x)/((-1 + x)*Sqrt[1 + Sqrt[1 + Sqrt[1 + x]]]),x]
[Out]
(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 + Sqr
t[1 + x]]))/105 + 2*RootSum[-2 + 4*#1^4 - 4*#1^6 + #1^8 & , Log[Sqrt[1 + Sqrt[1 + Sqrt[1 + x]]] - #1]/#1 & ]
Maple [N/A] (verified)
Time = 0.18 (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 |
\(\frac {8 \left (1+\sqrt {1+\sqrt {1+x}}\right )^{\frac {7}{2}}}{7}-\frac {24 \left (1+\sqrt {1+\sqrt {1+x}}\right )^{\frac {5}{2}}}{5}+\frac {16 \left (1+\sqrt {1+\sqrt {1+x}}\right )^{\frac {3}{2}}}{3}+2 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}-4 \textit {\_Z}^{6}+4 \textit {\_Z}^{4}-2\right )}{\sum }\frac {\left (\textit {\_R}^{6}-3 \textit {\_R}^{4}+2 \textit {\_R}^{2}\right ) \ln \left (\sqrt {1+\sqrt {1+\sqrt {1+x}}}-\textit {\_R} \right )}{\textit {\_R}^{7}-3 \textit {\_R}^{5}+2 \textit {\_R}^{3}}\right )\) |
\(117\) |
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[In]
int((1+x)/(-1+x)/(1+(1+(1+x)^(1/2))^(1/2))^(1/2),x,method=_RETURNVERBOSE)
[Out]
8/7*(1+(1+(1+x)^(1/2))^(1/2))^(7/2)-24/5*(1+(1+(1+x)^(1/2))^(1/2))^(5/2)+16/3*(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))
Fricas [C] (verification not implemented)
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 1.09 (sec) , antiderivative size = 2096, normalized size of antiderivative = 11.58
\[
\int \frac {1+x}{(-1+x) \sqrt {1+\sqrt {1+\sqrt {1+x}}}} \, dx=\text {Too large to display}
\]
[In]
integrate((1+x)/(-1+x)/(1+(1+(1+x)^(1/2))^(1/2))^(1/2),x, algorithm="fricas")
[Out]
8/105*((15*sqrt(x + 1) + 4)*sqrt(sqrt(x + 1) + 1) - 18*sqrt(x + 1) + 4)*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)*s
qrt(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) + sqrt(2))^3 - 8*sqrt(2)*sqrt(sqr
t(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)) - 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(sq
rt(2) + 1) + sqrt(2))*(sqrt(2)*sqrt(sqrt(2) + 1) - sqrt(2)) - 12*(sqrt(2)*sqrt(sqrt(2) + 1) - sqrt(2))^2 + 128
))*log(4*(2*(sqrt(2)*sqrt(sqrt(2) + 1) + sqrt(2))^2*(sqrt(2)*sqrt(sqrt(2) + 1) - sqrt(2)) - 2*(sqrt(2)*sqrt(sq
rt(2) + 1) + sqrt(2))*(sqrt(2)*sqrt(sqrt(2) + 1) - 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) - sqrt(2)) - 12*(sqrt(2)*sqrt(sqrt(2)
+ 1) - sqrt(2))^2 + 128)*(sqrt(2)*sqrt(sqrt(2) + 1) + sqrt(2))*(sqrt(2)*sqrt(sqrt(2) + 1) - sqrt(2)))*sqrt(8*s
qrt(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) - sqrt(2)) - 12*(sqrt(2)*sqrt(sqrt(2) + 1) - sqrt(2))^2 + 128)) + 256*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) - sqrt(2)) - 12*(sqrt(2)*sqrt(sqrt(2) + 1) - sqrt(2))^2 + 128))*l
og(-4*(2*(sqrt(2)*sqrt(sqrt(2) + 1) + sqrt(2))^2*(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))^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) - sqrt(2)) - 12*(sqrt(2)*sqrt(sqrt(2) + 1
) - sqrt(2))^2 + 128)*(sqrt(2)*sqrt(sqrt(2) + 1) + sqrt(2))*(sqrt(2)*sqrt(sqrt(2) + 1) - sqrt(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)*sq
rt(sqrt(2) + 1) - sqrt(2)) - 12*(sqrt(2)*sqrt(sqrt(2) + 1) - sqrt(2))^2 + 128)) + 256*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) - sqrt(2)) - 12*(sqrt(2)*sqrt(sqrt(2) + 1) - sqrt(2))^2 + 128))*log(
4*(2*(sqrt(2)*sqrt(sqrt(2) + 1) + sqrt(2))^2*(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))^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) - sqrt(2)) - 12*(sqrt(2)*sqrt(sqrt(2) + 1) -
sqrt(2))^2 + 128)*(sqrt(2)*sqrt(sqrt(2) + 1) + sqrt(2))*(sqrt(2)*sqrt(sqrt(2) + 1) - sqrt(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(s
qrt(2) + 1) - sqrt(2)) - 12*(sqrt(2)*sqrt(sqrt(2) + 1) - sqrt(2))^2 + 128)) + 256*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) - sqrt(2)) - 12*(sqrt(2)*sqrt(sqrt(2) + 1) - sqrt(2))^2 + 128))*log(-4*(
2*(sqrt(2)*sqrt(sqrt(2) + 1) + sqrt(2))^2*(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))^2 + sqrt(-12*(sqrt(2)*sqrt(sqrt(2) + 1) + sqrt(2))^2 + 8*(sq
rt(2)*sqrt(sqrt(2) + 1) + sqrt(2))*(sqrt(2)*sqrt(sqrt(2) + 1) - sqrt(2)) - 12*(sqrt(2)*sqrt(sqrt(2) + 1) - sqr
t(2))^2 + 128)*(sqrt(2)*sqrt(sqrt(2) + 1) + sqrt(2))*(sqrt(2)*sqrt(sqrt(2) + 1) - sqrt(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) - sqrt(2)) - 12*(sqrt(2)*sqrt(sqrt(2) + 1) - sqrt(2))^2 + 128)) + 256*sqrt(sqrt(sqrt(x + 1) + 1) + 1)
)
Sympy [F(-1)]
Timed out. \[
\int \frac {1+x}{(-1+x) \sqrt {1+\sqrt {1+\sqrt {1+x}}}} \, dx=\text {Timed out}
\]
[In]
integrate((1+x)/(-1+x)/(1+(1+(1+x)**(1/2))**(1/2))**(1/2),x)
[Out]
Timed out
Maxima [N/A]
Not integrable
Time = 0.63 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.13
\[
\int \frac {1+x}{(-1+x) \sqrt {1+\sqrt {1+\sqrt {1+x}}}} \, dx=\int { \frac {x + 1}{{\left (x - 1\right )} \sqrt {\sqrt {\sqrt {x + 1} + 1} + 1}} \,d x }
\]
[In]
integrate((1+x)/(-1+x)/(1+(1+(1+x)^(1/2))^(1/2))^(1/2),x, algorithm="maxima")
[Out]
integrate((x + 1)/((x - 1)*sqrt(sqrt(sqrt(x + 1) + 1) + 1)), x)
Giac [N/A]
Not integrable
Time = 0.46 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.13
\[
\int \frac {1+x}{(-1+x) \sqrt {1+\sqrt {1+\sqrt {1+x}}}} \, dx=\int { \frac {x + 1}{{\left (x - 1\right )} \sqrt {\sqrt {\sqrt {x + 1} + 1} + 1}} \,d x }
\]
[In]
integrate((1+x)/(-1+x)/(1+(1+(1+x)^(1/2))^(1/2))^(1/2),x, algorithm="giac")
[Out]
integrate((x + 1)/((x - 1)*sqrt(sqrt(sqrt(x + 1) + 1) + 1)), x)
Mupad [N/A]
Not integrable
Time = 6.42 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.13
\[
\int \frac {1+x}{(-1+x) \sqrt {1+\sqrt {1+\sqrt {1+x}}}} \, dx=\int \frac {x+1}{\sqrt {\sqrt {\sqrt {x+1}+1}+1}\,\left (x-1\right )} \,d x
\]
[In]
int((x + 1)/((((x + 1)^(1/2) + 1)^(1/2) + 1)^(1/2)*(x - 1)),x)
[Out]
int((x + 1)/((((x + 1)^(1/2) + 1)^(1/2) + 1)^(1/2)*(x - 1)), x)