Integrand size = 32, antiderivative size = 156 \[ \int \frac {1+x^2}{\left (-1+x^2\right ) \sqrt {1+\sqrt {1+\sqrt {1+x}}}} \, dx=\frac {-4 (-151-30 x)+64 \sqrt {1+\sqrt {1+x}}+\sqrt {1+x} \left (8-24 \sqrt {1+\sqrt {1+x}}\right )}{105 \sqrt {1+\sqrt {1+\sqrt {1+x}}}}+2 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {1+\sqrt {1+\sqrt {1+x}}}}{\sqrt {2}}\right )+\text {RootSum}\left [-2+4 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {\log \left (\sqrt {1+\sqrt {1+\sqrt {1+x}}}-\text {$\#$1}\right )}{\text {$\#$1}}\&\right ] \] Output:
Unintegrable
Time = 0.00 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.96 \[ \int \frac {1+x^2}{\left (-1+x^2\right ) \sqrt {1+\sqrt {1+\sqrt {1+x}}}} \, dx=\frac {484+8 \sqrt {1+x}+120 (1+x)-8 \sqrt {1+\sqrt {1+x}} \left (-8+3 \sqrt {1+x}\right )}{105 \sqrt {1+\sqrt {1+\sqrt {1+x}}}}+2 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {1+\sqrt {1+\sqrt {1+x}}}}{\sqrt {2}}\right )+\text {RootSum}\left [-2+4 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {\log \left (\sqrt {1+\sqrt {1+\sqrt {1+x}}}-\text {$\#$1}\right )}{\text {$\#$1}}\&\right ] \] Input:
Integrate[(1 + x^2)/((-1 + x^2)*Sqrt[1 + Sqrt[1 + Sqrt[1 + x]]]),x]
Output:
(484 + 8*Sqrt[1 + x] + 120*(1 + x) - 8*Sqrt[1 + Sqrt[1 + x]]*(-8 + 3*Sqrt[ 1 + x]))/(105*Sqrt[1 + Sqrt[1 + Sqrt[1 + x]]]) + 2*Sqrt[2]*ArcTanh[Sqrt[1 + Sqrt[1 + Sqrt[1 + x]]]/Sqrt[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.
\(\displaystyle \int \frac {x^2+1}{\left (x^2-1\right ) \sqrt {\sqrt {\sqrt {x+1}+1}+1}} \, dx\) |
\(\Big \downarrow \) 7267 |
\(\displaystyle 2 \int -\frac {(x+1)^2-2 (x+1)+2}{(1-x) \sqrt {x+1} \sqrt {\sqrt {\sqrt {x+1}+1}+1}}d\sqrt {x+1}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -2 \int \frac {(x+1)^2-2 (x+1)+2}{(1-x) \sqrt {x+1} \sqrt {\sqrt {\sqrt {x+1}+1}+1}}d\sqrt {x+1}\) |
\(\Big \downarrow \) 7267 |
\(\displaystyle 4 \int \frac {\sqrt {\sqrt {x+1}+1} \left ((x+1)^4-4 (x+1)^3+4 (x+1)^2+1\right )}{\left ((x+1)^3-3 (x+1)^2+x+2\right ) \sqrt {\sqrt {\sqrt {x+1}+1}+1}}d\sqrt {\sqrt {x+1}+1}\) |
\(\Big \downarrow \) 2461 |
\(\displaystyle 4 \int \left (\frac {x \sqrt {\sqrt {x+1}+1} \left ((x+1)^4-4 (x+1)^3+4 (x+1)^2+1\right )}{2 \left ((x+1)^2-2 (x+1)-1\right ) \sqrt {\sqrt {\sqrt {x+1}+1}+1}}-\frac {\sqrt {\sqrt {x+1}+1} \left ((x+1)^4-4 (x+1)^3+4 (x+1)^2+1\right )}{2 x \sqrt {\sqrt {\sqrt {x+1}+1}+1}}\right )d\sqrt {\sqrt {x+1}+1}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 4 \left (4 \text {Subst}\left (\int \frac {x+1}{(x+1)^4-4 (x+1)^3+4 (x+1)^2-2}d\sqrt {\sqrt {x+1}+1},\sqrt {\sqrt {x+1}+1},\sqrt {\sqrt {\sqrt {x+1}+1}+1}\right )-6 \text {Subst}\left (\int \frac {(x+1)^2}{(x+1)^4-4 (x+1)^3+4 (x+1)^2-2}d\sqrt {\sqrt {x+1}+1},\sqrt {\sqrt {x+1}+1},\sqrt {\sqrt {\sqrt {x+1}+1}+1}\right )+2 \text {Subst}\left (\int \frac {(x+1)^3}{(x+1)^4-4 (x+1)^3+4 (x+1)^2-2}d\sqrt {\sqrt {x+1}+1},\sqrt {\sqrt {x+1}+1},\sqrt {\sqrt {\sqrt {x+1}+1}+1}\right )+\frac {\text {arctanh}\left (\frac {\sqrt {\sqrt {\sqrt {x+1}+1}+1}}{\sqrt {2}}\right )}{\sqrt {2}}+\frac {2}{7} \left (\sqrt {\sqrt {x+1}+1}+1\right )^{7/2}-\frac {6}{5} \left (\sqrt {\sqrt {x+1}+1}+1\right )^{5/2}+\frac {4}{3} \left (\sqrt {\sqrt {x+1}+1}+1\right )^{3/2}+\frac {1}{\sqrt {\sqrt {\sqrt {x+1}+1}+1}}\right )\) |
Input:
Int[(1 + x^2)/((-1 + x^2)*Sqrt[1 + Sqrt[1 + Sqrt[1 + x]]]),x]
Output:
$Aborted
Time = 0.19 (sec) , antiderivative size = 154, normalized size of antiderivative = 0.99
method | result | size |
derivativedivides | \(\frac {8 \left (1+\sqrt {1+\sqrt {1+x}}\right )^{\frac {7}{2}}}{7}-\frac {24 \left (1+\sqrt {1+\sqrt {1+x}}\right )^{\frac {5}{2}}}{5}+\frac {16 \left (1+\sqrt {1+\sqrt {1+x}}\right )^{\frac {3}{2}}}{3}+2 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {1+\sqrt {1+\sqrt {1+x}}}\, \sqrt {2}}{2}\right )+\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}-4 \textit {\_Z}^{6}+4 \textit {\_Z}^{4}-2\right )}{\sum }\frac {\left (\textit {\_R}^{6}-3 \textit {\_R}^{4}+2 \textit {\_R}^{2}\right ) \ln \left (\sqrt {1+\sqrt {1+\sqrt {1+x}}}-\textit {\_R} \right )}{\textit {\_R}^{7}-3 \textit {\_R}^{5}+2 \textit {\_R}^{3}}\right )+\frac {4}{\sqrt {1+\sqrt {1+\sqrt {1+x}}}}\) | \(154\) |
default | \(\frac {8 \left (1+\sqrt {1+\sqrt {1+x}}\right )^{\frac {7}{2}}}{7}-\frac {24 \left (1+\sqrt {1+\sqrt {1+x}}\right )^{\frac {5}{2}}}{5}+\frac {16 \left (1+\sqrt {1+\sqrt {1+x}}\right )^{\frac {3}{2}}}{3}+2 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {1+\sqrt {1+\sqrt {1+x}}}\, \sqrt {2}}{2}\right )+\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}-4 \textit {\_Z}^{6}+4 \textit {\_Z}^{4}-2\right )}{\sum }\frac {\left (\textit {\_R}^{6}-3 \textit {\_R}^{4}+2 \textit {\_R}^{2}\right ) \ln \left (\sqrt {1+\sqrt {1+\sqrt {1+x}}}-\textit {\_R} \right )}{\textit {\_R}^{7}-3 \textit {\_R}^{5}+2 \textit {\_R}^{3}}\right )+\frac {4}{\sqrt {1+\sqrt {1+\sqrt {1+x}}}}\) | \(154\) |
Input:
int((x^2+1)/(x^2-1)/(1+(1+(1+x)^(1/2))^(1/2))^(1/2),x,method=_RETURNVERBOS E)
Output:
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*2^(1/2)*arctanh(1/2*(1+(1+(1+x)^(1/2 ))^(1/2))^(1/2)*2^(1/2))+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))+4/(1+ (1+(1+x)^(1/2))^(1/2))^(1/2)
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 0.81 (sec) , antiderivative size = 2034, normalized size of antiderivative = 13.04 \[ \int \frac {1+x^2}{\left (-1+x^2\right ) \sqrt {1+\sqrt {1+\sqrt {1+x}}}} \, dx=\text {Too large to display} \] Input:
integrate((x^2+1)/(x^2-1)/(1+(1+(1+x)^(1/2))^(1/2))^(1/2),x, algorithm="fr icas")
Output:
1/210*(105*(x + 1)*sqrt(2*sqrt(2) + 2*sqrt(-3/4*(sqrt(2) + sqrt(2*sqrt(2) + 2))^2 - 1/2*(sqrt(2) + sqrt(2*sqrt(2) + 2))*(sqrt(2) - sqrt(2*sqrt(2) + 2)) - 3/4*(sqrt(2) - sqrt(2*sqrt(2) + 2))^2 + 8))*log(1/8*((sqrt(2) + sqrt (2*sqrt(2) + 2))^2*(sqrt(2) - sqrt(2*sqrt(2) + 2)) + (sqrt(2) + sqrt(2*sqr t(2) + 2))*(sqrt(2) - sqrt(2*sqrt(2) + 2))^2 - 2*sqrt(-3/4*(sqrt(2) + sqrt (2*sqrt(2) + 2))^2 - 1/2*(sqrt(2) + sqrt(2*sqrt(2) + 2))*(sqrt(2) - sqrt(2 *sqrt(2) + 2)) - 3/4*(sqrt(2) - sqrt(2*sqrt(2) + 2))^2 + 8)*(sqrt(2) + sqr t(2*sqrt(2) + 2))*(sqrt(2) - sqrt(2*sqrt(2) + 2)))*sqrt(2*sqrt(2) + 2*sqrt (-3/4*(sqrt(2) + sqrt(2*sqrt(2) + 2))^2 - 1/2*(sqrt(2) + sqrt(2*sqrt(2) + 2))*(sqrt(2) - sqrt(2*sqrt(2) + 2)) - 3/4*(sqrt(2) - sqrt(2*sqrt(2) + 2))^ 2 + 8)) + 2*sqrt(sqrt(sqrt(x + 1) + 1) + 1)) - 105*(x + 1)*sqrt(2*sqrt(2) + 2*sqrt(-3/4*(sqrt(2) + sqrt(2*sqrt(2) + 2))^2 - 1/2*(sqrt(2) + sqrt(2*sq rt(2) + 2))*(sqrt(2) - sqrt(2*sqrt(2) + 2)) - 3/4*(sqrt(2) - sqrt(2*sqrt(2 ) + 2))^2 + 8))*log(-1/8*((sqrt(2) + sqrt(2*sqrt(2) + 2))^2*(sqrt(2) - sqr t(2*sqrt(2) + 2)) + (sqrt(2) + sqrt(2*sqrt(2) + 2))*(sqrt(2) - sqrt(2*sqrt (2) + 2))^2 - 2*sqrt(-3/4*(sqrt(2) + sqrt(2*sqrt(2) + 2))^2 - 1/2*(sqrt(2) + sqrt(2*sqrt(2) + 2))*(sqrt(2) - sqrt(2*sqrt(2) + 2)) - 3/4*(sqrt(2) - s qrt(2*sqrt(2) + 2))^2 + 8)*(sqrt(2) + sqrt(2*sqrt(2) + 2))*(sqrt(2) - sqrt (2*sqrt(2) + 2)))*sqrt(2*sqrt(2) + 2*sqrt(-3/4*(sqrt(2) + sqrt(2*sqrt(2) + 2))^2 - 1/2*(sqrt(2) + sqrt(2*sqrt(2) + 2))*(sqrt(2) - sqrt(2*sqrt(2) ...
Timed out. \[ \int \frac {1+x^2}{\left (-1+x^2\right ) \sqrt {1+\sqrt {1+\sqrt {1+x}}}} \, dx=\text {Timed out} \] Input:
integrate((x**2+1)/(x**2-1)/(1+(1+(1+x)**(1/2))**(1/2))**(1/2),x)
Output:
Timed out
Not integrable
Time = 0.43 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.18 \[ \int \frac {1+x^2}{\left (-1+x^2\right ) \sqrt {1+\sqrt {1+\sqrt {1+x}}}} \, dx=\int { \frac {x^{2} + 1}{{\left (x^{2} - 1\right )} \sqrt {\sqrt {\sqrt {x + 1} + 1} + 1}} \,d x } \] Input:
integrate((x^2+1)/(x^2-1)/(1+(1+(1+x)^(1/2))^(1/2))^(1/2),x, algorithm="ma xima")
Output:
integrate((x^2 + 1)/((x^2 - 1)*sqrt(sqrt(sqrt(x + 1) + 1) + 1)), x)
Not integrable
Time = 6.10 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.18 \[ \int \frac {1+x^2}{\left (-1+x^2\right ) \sqrt {1+\sqrt {1+\sqrt {1+x}}}} \, dx=\int { \frac {x^{2} + 1}{{\left (x^{2} - 1\right )} \sqrt {\sqrt {\sqrt {x + 1} + 1} + 1}} \,d x } \] Input:
integrate((x^2+1)/(x^2-1)/(1+(1+(1+x)^(1/2))^(1/2))^(1/2),x, algorithm="gi ac")
Output:
integrate((x^2 + 1)/((x^2 - 1)*sqrt(sqrt(sqrt(x + 1) + 1) + 1)), x)
Not integrable
Time = 0.00 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.18 \[ \int \frac {1+x^2}{\left (-1+x^2\right ) \sqrt {1+\sqrt {1+\sqrt {1+x}}}} \, dx=\int \frac {x^2+1}{\left (x^2-1\right )\,\sqrt {\sqrt {\sqrt {x+1}+1}+1}} \,d x \] Input:
int((x^2 + 1)/((x^2 - 1)*(((x + 1)^(1/2) + 1)^(1/2) + 1)^(1/2)),x)
Output:
int((x^2 + 1)/((x^2 - 1)*(((x + 1)^(1/2) + 1)^(1/2) + 1)^(1/2)), x)
Not integrable
Time = 200.04 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.18 \[ \int \frac {1+x^2}{\left (-1+x^2\right ) \sqrt {1+\sqrt {1+\sqrt {1+x}}}} \, dx=\int \frac {x^{2}+1}{\left (x^{2}-1\right ) \sqrt {1+\sqrt {1+\sqrt {x +1}}}}d x \] Input:
int((x^2+1)/(x^2-1)/(1+(1+(1+x)^(1/2))^(1/2))^(1/2),x)
Output:
int((x^2+1)/(x^2-1)/(1+(1+(1+x)^(1/2))^(1/2))^(1/2),x)