Integrand size = 45, antiderivative size = 428 \[ \int \frac {\left (1+x^4\right ) \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\sqrt {1+x^2} \left (1-x^4\right )} \, dx=\frac {\left (-5-4 x^2\right ) \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-4 x \sqrt {1+x^2} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{1+x^2+x \sqrt {1+x^2}}+4 \text {arctanh}\left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}\right )-\frac {1}{2} \text {RootSum}\left [-1-8 \text {$\#$1}^2-8 \text {$\#$1}^3+14 \text {$\#$1}^4+32 \text {$\#$1}^5+24 \text {$\#$1}^6+8 \text {$\#$1}^7+\text {$\#$1}^8\&,\frac {\log \left (-1+\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right )}{-1+\text {$\#$1}+3 \text {$\#$1}^2+\text {$\#$1}^3}\&\right ]+\frac {1}{4} \text {RootSum}\left [1+16 \text {$\#$1}^4+32 \text {$\#$1}^5+24 \text {$\#$1}^6+8 \text {$\#$1}^7+\text {$\#$1}^8\&,\frac {\log \left (-1+\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right )}{8 \text {$\#$1}^3+20 \text {$\#$1}^4+18 \text {$\#$1}^5+7 \text {$\#$1}^6+\text {$\#$1}^7}\&\right ]+\frac {1}{2} \text {RootSum}\left [-1+8 \text {$\#$1}^2+8 \text {$\#$1}^3+18 \text {$\#$1}^4+32 \text {$\#$1}^5+24 \text {$\#$1}^6+8 \text {$\#$1}^7+\text {$\#$1}^8\&,\frac {\log \left (-1+\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right )+\log \left (-1+\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}}{1+4 \text {$\#$1}^2+4 \text {$\#$1}^3+\text {$\#$1}^4}\&\right ] \]
Time = 0.00 (sec) , antiderivative size = 306, normalized size of antiderivative = 0.71 \[ \int \frac {\left (1+x^4\right ) \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\sqrt {1+x^2} \left (1-x^4\right )} \, dx=-\frac {\left (5+4 x^2+4 x \sqrt {1+x^2}\right ) \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{1+x^2+x \sqrt {1+x^2}}+4 \text {arctanh}\left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}\right )-\frac {1}{2} \text {RootSum}\left [-2+4 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {\log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right )}{-2 \text {$\#$1}+\text {$\#$1}^3}\&\right ]+\frac {1}{4} \text {RootSum}\left [2-4 \text {$\#$1}^2+6 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {\log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right )}{-\text {$\#$1}+3 \text {$\#$1}^3-3 \text {$\#$1}^5+\text {$\#$1}^7}\&\right ]+\frac {1}{2} \text {RootSum}\left [2-8 \text {$\#$1}^2+8 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {\log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}}{2-2 \text {$\#$1}^2+\text {$\#$1}^4}\&\right ] \]
-(((5 + 4*x^2 + 4*x*Sqrt[1 + x^2])*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]])/(1 + x^2 + x*Sqrt[1 + x^2])) + 4*ArcTanh[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]]] - RootSum[-2 + 4*#1^4 - 4*#1^6 + #1^8 & , Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2 ]]] - #1]/(-2*#1 + #1^3) & ]/2 + RootSum[2 - 4*#1^2 + 6*#1^4 - 4*#1^6 + #1 ^8 & , Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]/(-#1 + 3*#1^3 - 3*#1^5 + #1^7) & ]/4 + RootSum[2 - 8*#1^2 + 8*#1^4 - 4*#1^6 + #1^8 & , (Log[Sqrt[ 1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1)/(2 - 2*#1^2 + #1^4) & ]/2
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.
| \(\displaystyle \int \frac {\left (x^4+1\right ) \sqrt {\sqrt {\sqrt {x^2+1}+x}+1}}{\sqrt {x^2+1} \left (1-x^4\right )} \, dx\) |
| \(\Big \downarrow \) 1388 |
| \(\displaystyle \int \frac {\left (x^4+1\right ) \sqrt {\sqrt {\sqrt {x^2+1}+x}+1}}{\left (1-x^2\right ) \left (x^2+1\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \int \left (-\frac {\sqrt {\sqrt {\sqrt {x^2+1}+x}+1} x^2}{\left (x^2+1\right )^{3/2}}+\frac {2 \sqrt {\sqrt {\sqrt {x^2+1}+x}+1}}{\left (1-x^2\right ) \left (x^2+1\right )^{3/2}}-\frac {\sqrt {\sqrt {\sqrt {x^2+1}+x}+1}}{\left (x^2+1\right )^{3/2}}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\int \frac {\sqrt {\sqrt {x+\sqrt {x^2+1}}+1}}{\left (x^2+1\right )^{3/2}}dx+\int \frac {\sqrt {\sqrt {x+\sqrt {x^2+1}}+1}}{(1-x) \left (x^2+1\right )^{3/2}}dx-\int \frac {x^2 \sqrt {\sqrt {x+\sqrt {x^2+1}}+1}}{\left (x^2+1\right )^{3/2}}dx+\int \frac {\sqrt {\sqrt {x+\sqrt {x^2+1}}+1}}{(x+1) \left (x^2+1\right )^{3/2}}dx\) |
3.31.24.3.1 Defintions of rubi rules used
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_)/((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 = 0.00 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.09
\[\int \frac {\left (x^{4}+1\right ) \sqrt {1+\sqrt {x +\sqrt {x^{2}+1}}}}{\sqrt {x^{2}+1}\, \left (-x^{4}+1\right )}d x\]
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 0.35 (sec) , antiderivative size = 1598, normalized size of antiderivative = 3.73 \[ \int \frac {\left (1+x^4\right ) \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\sqrt {1+x^2} \left (1-x^4\right )} \, dx=\text {Too large to display} \]
integrate((x^4+1)*(1+(x+(x^2+1)^(1/2))^(1/2))^(1/2)/(x^2+1)^(1/2)/(-x^4+1) ,x, algorithm="fricas")
1/8*(sqrt(2)*(x^2 + 1)*sqrt(sqrt(2*I*sqrt(2) + 1) - 1)*log(((sqrt(2) - I)* sqrt(2*I*sqrt(2) + 1) + 3*sqrt(2) - 3*I)*sqrt(sqrt(2*I*sqrt(2) + 1) - 1) + 6*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1)) - sqrt(2)*(x^2 + 1)*sqrt(sqrt(2*I*sq rt(2) + 1) - 1)*log(-((sqrt(2) - I)*sqrt(2*I*sqrt(2) + 1) + 3*sqrt(2) - 3* I)*sqrt(sqrt(2*I*sqrt(2) + 1) - 1) + 6*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1)) - sqrt(2)*(x^2 + 1)*sqrt(-sqrt(2*I*sqrt(2) + 1) - 1)*log(((sqrt(2) - I)*sq rt(2*I*sqrt(2) + 1) - 3*sqrt(2) + 3*I)*sqrt(-sqrt(2*I*sqrt(2) + 1) - 1) + 6*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1)) + sqrt(2)*(x^2 + 1)*sqrt(-sqrt(2*I*sq rt(2) + 1) - 1)*log(-((sqrt(2) - I)*sqrt(2*I*sqrt(2) + 1) - 3*sqrt(2) + 3* I)*sqrt(-sqrt(2*I*sqrt(2) + 1) - 1) + 6*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1)) + sqrt(2)*(x^2 + 1)*sqrt(sqrt(-2*I*sqrt(2) + 1) - 1)*log(((sqrt(2) + I)*s qrt(-2*I*sqrt(2) + 1) + 3*sqrt(2) + 3*I)*sqrt(sqrt(-2*I*sqrt(2) + 1) - 1) + 6*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1)) - sqrt(2)*(x^2 + 1)*sqrt(sqrt(-2*I* sqrt(2) + 1) - 1)*log(-((sqrt(2) + I)*sqrt(-2*I*sqrt(2) + 1) + 3*sqrt(2) + 3*I)*sqrt(sqrt(-2*I*sqrt(2) + 1) - 1) + 6*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1)) - sqrt(2)*(x^2 + 1)*sqrt(-sqrt(-2*I*sqrt(2) + 1) - 1)*log(((sqrt(2) + I)*sqrt(-2*I*sqrt(2) + 1) - 3*sqrt(2) - 3*I)*sqrt(-sqrt(-2*I*sqrt(2) + 1) - 1) + 6*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1)) + sqrt(2)*(x^2 + 1)*sqrt(-sqrt (-2*I*sqrt(2) + 1) - 1)*log(-((sqrt(2) + I)*sqrt(-2*I*sqrt(2) + 1) - 3*sqr t(2) - 3*I)*sqrt(-sqrt(-2*I*sqrt(2) + 1) - 1) + 6*sqrt(sqrt(x + sqrt(x^...
Not integrable
Time = 32.35 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.20 \[ \int \frac {\left (1+x^4\right ) \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\sqrt {1+x^2} \left (1-x^4\right )} \, dx=- \int \frac {\sqrt {\sqrt {x + \sqrt {x^{2} + 1}} + 1}}{x^{4} \sqrt {x^{2} + 1} - \sqrt {x^{2} + 1}}\, dx - \int \frac {x^{4} \sqrt {\sqrt {x + \sqrt {x^{2} + 1}} + 1}}{x^{4} \sqrt {x^{2} + 1} - \sqrt {x^{2} + 1}}\, dx \]
-Integral(sqrt(sqrt(x + sqrt(x**2 + 1)) + 1)/(x**4*sqrt(x**2 + 1) - sqrt(x **2 + 1)), x) - Integral(x**4*sqrt(sqrt(x + sqrt(x**2 + 1)) + 1)/(x**4*sqr t(x**2 + 1) - sqrt(x**2 + 1)), x)
Not integrable
Time = 0.96 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.09 \[ \int \frac {\left (1+x^4\right ) \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\sqrt {1+x^2} \left (1-x^4\right )} \, dx=\int { -\frac {{\left (x^{4} + 1\right )} \sqrt {\sqrt {x + \sqrt {x^{2} + 1}} + 1}}{{\left (x^{4} - 1\right )} \sqrt {x^{2} + 1}} \,d x } \]
integrate((x^4+1)*(1+(x+(x^2+1)^(1/2))^(1/2))^(1/2)/(x^2+1)^(1/2)/(-x^4+1) ,x, algorithm="maxima")
Exception generated. \[ \int \frac {\left (1+x^4\right ) \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\sqrt {1+x^2} \left (1-x^4\right )} \, dx=\text {Exception raised: RuntimeError} \]
integrate((x^4+1)*(1+(x+(x^2+1)^(1/2))^(1/2))^(1/2)/(x^2+1)^(1/2)/(-x^4+1) ,x, algorithm="giac")
Exception raised: RuntimeError >> an error occurred running a Giac command :INPUT:sage2OUTPUT:Unable to divide, perhaps due to rounding error%%%{%%{p oly1[-16232886178711429450219333683647001563848436365999727949750274764264 752867482521
Not integrable
Time = 0.00 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.09 \[ \int \frac {\left (1+x^4\right ) \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\sqrt {1+x^2} \left (1-x^4\right )} \, dx=\int -\frac {\sqrt {\sqrt {x+\sqrt {x^2+1}}+1}\,\left (x^4+1\right )}{\sqrt {x^2+1}\,\left (x^4-1\right )} \,d x \]