Integrand size = 36, antiderivative size = 590 \[ \int \frac {\left (1+x^4\right ) \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{1-x^4} \, dx=\frac {\left (6+16 x-48 x^2\right ) \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}+(15-8 x) \sqrt {x+\sqrt {1+x^2}} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}+\sqrt {1+x^2} \left ((16-48 x) \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-8 \sqrt {x+\sqrt {1+x^2}} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}\right )}{60 x+60 \sqrt {1+x^2}}-\frac {1}{4} \text {arctanh}\left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}\right )-\frac {1}{2} \text {RootSum}\left [-2+4 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {2 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right )-2 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^4+\log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^6}{2 \text {$\#$1}^3-3 \text {$\#$1}^5+\text {$\#$1}^7}\&\right ]+\text {RootSum}\left [2-4 \text {$\#$1}^2+6 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {\log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}}{1-2 \text {$\#$1}^2+\text {$\#$1}^4}\&\right ]+\frac {1}{2} \text {RootSum}\left [2-8 \text {$\#$1}^2+8 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {-2 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right )+6 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^2-4 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^4+\log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^6}{-2 \text {$\#$1}+4 \text {$\#$1}^3-3 \text {$\#$1}^5+\text {$\#$1}^7}\&\right ] \]
Time = 0.77 (sec) , antiderivative size = 551, normalized size of antiderivative = 0.93 \[ \int \frac {\left (1+x^4\right ) \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{1-x^4} \, dx=\frac {1}{4} \left (-\frac {\sqrt {1+\sqrt {x+\sqrt {1+x^2}}} \left (-6+48 x^2-16 \sqrt {1+x^2}-15 \sqrt {x+\sqrt {1+x^2}}+8 \sqrt {1+x^2} \sqrt {x+\sqrt {1+x^2}}+8 x \left (-2+6 \sqrt {1+x^2}+\sqrt {x+\sqrt {1+x^2}}\right )\right )}{15 \left (x+\sqrt {1+x^2}\right )}-\text {arctanh}\left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}\right )-2 \text {RootSum}\left [-2+4 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {2 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right )-2 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^4+\log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^6}{2 \text {$\#$1}^3-3 \text {$\#$1}^5+\text {$\#$1}^7}\&\right ]+4 \text {RootSum}\left [2-4 \text {$\#$1}^2+6 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {\log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}}{1-2 \text {$\#$1}^2+\text {$\#$1}^4}\&\right ]+2 \text {RootSum}\left [2-8 \text {$\#$1}^2+8 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {-2 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right )+6 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^2-4 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^4+\log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^6}{-2 \text {$\#$1}+4 \text {$\#$1}^3-3 \text {$\#$1}^5+\text {$\#$1}^7}\&\right ]\right ) \]
(-1/15*(Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]]*(-6 + 48*x^2 - 16*Sqrt[1 + x^2] - 15*Sqrt[x + Sqrt[1 + x^2]] + 8*Sqrt[1 + x^2]*Sqrt[x + Sqrt[1 + x^2]] + 8 *x*(-2 + 6*Sqrt[1 + x^2] + Sqrt[x + Sqrt[1 + x^2]])))/(x + Sqrt[1 + x^2]) - ArcTanh[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]]] - 2*RootSum[-2 + 4*#1^4 - 4*# 1^6 + #1^8 & , (2*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1] - 2*Log[Sqrt [1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^4 + Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^6)/(2*#1^3 - 3*#1^5 + #1^7) & ] + 4*RootSum[2 - 4*#1^2 + 6 *#1^4 - 4*#1^6 + #1^8 & , (Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1) /(1 - 2*#1^2 + #1^4) & ] + 2*RootSum[2 - 8*#1^2 + 8*#1^4 - 4*#1^6 + #1^8 & , (-2*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1] + 6*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^2 - 4*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^4 + Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^6)/(-2*#1 + 4*#1 ^3 - 3*#1^5 + #1^7) & ])/4
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.
\(\displaystyle \int \frac {\left (x^4+1\right ) \sqrt {\sqrt {\sqrt {x^2+1}+x}+1}}{1-x^4} \, dx\) |
\(\Big \downarrow \) 7276 |
\(\displaystyle \int \left (\frac {2 \sqrt {\sqrt {\sqrt {x^2+1}+x}+1}}{1-x^4}-\sqrt {\sqrt {\sqrt {x^2+1}+x}+1}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\int \sqrt {\sqrt {x+\sqrt {x^2+1}}+1}dx+\frac {1}{2} i \int \frac {\sqrt {\sqrt {x+\sqrt {x^2+1}}+1}}{i-x}dx+\frac {1}{2} \int \frac {\sqrt {\sqrt {x+\sqrt {x^2+1}}+1}}{1-x}dx+\frac {1}{2} i \int \frac {\sqrt {\sqrt {x+\sqrt {x^2+1}}+1}}{x+i}dx+\frac {1}{2} \int \frac {\sqrt {\sqrt {x+\sqrt {x^2+1}}+1}}{x+1}dx\) |
3.32.4.3.1 Defintions of rubi rules used
Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionE xpand[u/(a + b*x^n), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ [n, 0]
Not integrable
Time = 0.02 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.05
\[\int \frac {\left (x^{4}+1\right ) \sqrt {1+\sqrt {x +\sqrt {x^{2}+1}}}}{-x^{4}+1}d x\]
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 0.36 (sec) , antiderivative size = 1162, normalized size of antiderivative = 1.97 \[ \int \frac {\left (1+x^4\right ) \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{1-x^4} \, dx=\text {Too large to display} \]
-1/60*((15*x - 15*sqrt(x^2 + 1) + 8)*sqrt(x + sqrt(x^2 + 1)) + 54*x - 6*sq rt(x^2 + 1) - 16)*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1) - 1/2*sqrt(2*sqrt(4*I) - 4)*log(I*sqrt(2*sqrt(4*I) - 4) + 2*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1)) + 1/2*sqrt(2*sqrt(4*I) - 4)*log(-I*sqrt(2*sqrt(4*I) - 4) + 2*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1)) - 1/2*sqrt(-2*sqrt(4*I) - 4)*log(I*sqrt(-2*sqrt(4*I) - 4) + 2*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1)) + 1/2*sqrt(-2*sqrt(4*I) - 4)*l og(-I*sqrt(-2*sqrt(4*I) - 4) + 2*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1)) + 1/2* sqrt(2*sqrt(-4*I) - 4)*log(I*sqrt(2*sqrt(-4*I) - 4) + 2*sqrt(sqrt(x + sqrt (x^2 + 1)) + 1)) - 1/2*sqrt(2*sqrt(-4*I) - 4)*log(-I*sqrt(2*sqrt(-4*I) - 4 ) + 2*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1)) + 1/2*sqrt(-2*sqrt(-4*I) - 4)*log (I*sqrt(-2*sqrt(-4*I) - 4) + 2*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1)) - 1/2*sq rt(-2*sqrt(-4*I) - 4)*log(-I*sqrt(-2*sqrt(-4*I) - 4) + 2*sqrt(sqrt(x + sqr t(x^2 + 1)) + 1)) + 1/2*sqrt(sqrt(sqrt(2) + 1) + 1)*log(sqrt(sqrt(x + sqrt (x^2 + 1)) + 1) + sqrt(sqrt(sqrt(2) + 1) + 1)) - 1/2*sqrt(sqrt(sqrt(2) + 1 ) + 1)*log(sqrt(sqrt(x + sqrt(x^2 + 1)) + 1) - sqrt(sqrt(sqrt(2) + 1) + 1) ) + 1/2*sqrt(-sqrt(sqrt(2) + 1) + 1)*log(sqrt(sqrt(x + sqrt(x^2 + 1)) + 1) + sqrt(-sqrt(sqrt(2) + 1) + 1)) - 1/2*sqrt(-sqrt(sqrt(2) + 1) + 1)*log(sq rt(sqrt(x + sqrt(x^2 + 1)) + 1) - sqrt(-sqrt(sqrt(2) + 1) + 1)) - 1/2*sqrt (sqrt(sqrt(2) - 1) + 1)*log(sqrt(sqrt(x + sqrt(x^2 + 1)) + 1) + sqrt(sqrt( sqrt(2) - 1) + 1)) + 1/2*sqrt(sqrt(sqrt(2) - 1) + 1)*log(sqrt(sqrt(x + ...
Not integrable
Time = 19.27 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.09 \[ \int \frac {\left (1+x^4\right ) \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{1-x^4} \, dx=- \int \frac {\sqrt {\sqrt {x + \sqrt {x^{2} + 1}} + 1}}{x^{4} - 1}\, dx - \int \frac {x^{4} \sqrt {\sqrt {x + \sqrt {x^{2} + 1}} + 1}}{x^{4} - 1}\, dx \]
-Integral(sqrt(sqrt(x + sqrt(x**2 + 1)) + 1)/(x**4 - 1), x) - Integral(x** 4*sqrt(sqrt(x + sqrt(x**2 + 1)) + 1)/(x**4 - 1), x)
Not integrable
Time = 0.89 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.05 \[ \int \frac {\left (1+x^4\right ) \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{1-x^4} \, dx=\int { -\frac {{\left (x^{4} + 1\right )} \sqrt {\sqrt {x + \sqrt {x^{2} + 1}} + 1}}{x^{4} - 1} \,d x } \]
Not integrable
Time = 1.59 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.05 \[ \int \frac {\left (1+x^4\right ) \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{1-x^4} \, dx=\int { -\frac {{\left (x^{4} + 1\right )} \sqrt {\sqrt {x + \sqrt {x^{2} + 1}} + 1}}{x^{4} - 1} \,d x } \]
Not integrable
Time = 8.74 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.05 \[ \int \frac {\left (1+x^4\right ) \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{1-x^4} \, dx=\int -\frac {\sqrt {\sqrt {x+\sqrt {x^2+1}}+1}\,\left (x^4+1\right )}{x^4-1} \,d x \]