3.22.2 \(\int \frac {1+x^4}{(-1+x^4) \sqrt {1+\sqrt {1+x^2}}} \, dx\) [2102]

3.22.2.1 Optimal result

Integrand size = 28, antiderivative size = 152 \[ \int \frac {1+x^4}{\left (-1+x^4\right ) \sqrt {1+\sqrt {1+x^2}}} \, dx=\frac {2 x}{\sqrt {1+\sqrt {1+x^2}}}-2 \arctan \left (\frac {x}{\sqrt {1+\sqrt {1+x^2}}}\right )+\sqrt {2} \arctan \left (\frac {x}{\sqrt {2} \sqrt {1+\sqrt {1+x^2}}}\right )-\sqrt {1+\sqrt {2}} \arctan \left (\frac {x}{\sqrt {1+\sqrt {2}} \sqrt {1+\sqrt {1+x^2}}}\right )-\sqrt {-1+\sqrt {2}} \text {arctanh}\left (\frac {x}{\sqrt {-1+\sqrt {2}} \sqrt {1+\sqrt {1+x^2}}}\right ) \]

output

2*x/(1+(x^2+1)^(1/2))^(1/2)-2*arctan(x/(1+(x^2+1)^(1/2))^(1/2))+2^(1/2)*ar 
ctan(1/2*x*2^(1/2)/(1+(x^2+1)^(1/2))^(1/2))-(1+2^(1/2))^(1/2)*arctan(x/(1+ 
2^(1/2))^(1/2)/(1+(x^2+1)^(1/2))^(1/2))-(2^(1/2)-1)^(1/2)*arctanh(x/(2^(1/ 
2)-1)^(1/2)/(1+(x^2+1)^(1/2))^(1/2))
 

3.22.2.2 Mathematica [A] (verified)

Time = 0.42 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.00 \[ \int \frac {1+x^4}{\left (-1+x^4\right ) \sqrt {1+\sqrt {1+x^2}}} \, dx=\frac {2 x}{\sqrt {1+\sqrt {1+x^2}}}-2 \arctan \left (\frac {x}{\sqrt {1+\sqrt {1+x^2}}}\right )+\sqrt {2} \arctan \left (\frac {x}{\sqrt {2} \sqrt {1+\sqrt {1+x^2}}}\right )-\sqrt {1+\sqrt {2}} \arctan \left (\frac {\sqrt {-1+\sqrt {2}} x}{\sqrt {1+\sqrt {1+x^2}}}\right )-\sqrt {-1+\sqrt {2}} \text {arctanh}\left (\frac {\sqrt {1+\sqrt {2}} x}{\sqrt {1+\sqrt {1+x^2}}}\right ) \]

input

Integrate[(1 + x^4)/((-1 + x^4)*Sqrt[1 + Sqrt[1 + x^2]]),x]
 

output

(2*x)/Sqrt[1 + Sqrt[1 + x^2]] - 2*ArcTan[x/Sqrt[1 + Sqrt[1 + x^2]]] + Sqrt 
[2]*ArcTan[x/(Sqrt[2]*Sqrt[1 + Sqrt[1 + x^2]])] - Sqrt[1 + Sqrt[2]]*ArcTan 
[(Sqrt[-1 + Sqrt[2]]*x)/Sqrt[1 + Sqrt[1 + x^2]]] - Sqrt[-1 + Sqrt[2]]*ArcT 
anh[(Sqrt[1 + Sqrt[2]]*x)/Sqrt[1 + Sqrt[1 + x^2]]]
 

3.22.2.3 Rubi [F]

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.

\(\displaystyle \int \frac {x^4+1}{\left (x^4-1\right ) \sqrt {\sqrt {x^2+1}+1}} \, dx\)

\(\Big \downarrow \) 7276

\(\displaystyle \int \left (\frac {1}{\sqrt {\sqrt {x^2+1}+1}}+\frac {2}{\left (x^4-1\right ) \sqrt {\sqrt {x^2+1}+1}}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \int \frac {1}{\sqrt {\sqrt {x^2+1}+1}}dx-\frac {1}{2} i \int \frac {1}{(i-x) \sqrt {\sqrt {x^2+1}+1}}dx-\frac {1}{2} \int \frac {1}{(1-x) \sqrt {\sqrt {x^2+1}+1}}dx-\frac {1}{2} i \int \frac {1}{(x+i) \sqrt {\sqrt {x^2+1}+1}}dx-\frac {1}{2} \int \frac {1}{(x+1) \sqrt {\sqrt {x^2+1}+1}}dx\)

input

Int[(1 + x^4)/((-1 + x^4)*Sqrt[1 + Sqrt[1 + x^2]]),x]
 

output

$Aborted
 

3.22.2.3.1 Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 7276

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]
 

3.22.2.4 Maple [F]

\[\int \frac {x^{4}+1}{\left (x^{4}-1\right ) \sqrt {1+\sqrt {x^{2}+1}}}d x\]

input

int((x^4+1)/(x^4-1)/(1+(x^2+1)^(1/2))^(1/2),x)
 

output

int((x^4+1)/(x^4-1)/(1+(x^2+1)^(1/2))^(1/2),x)
 

3.22.2.5 Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 613 vs. \(2 (113) = 226\).

Time = 14.68 (sec) , antiderivative size = 613, normalized size of antiderivative = 4.03 \[ \int \frac {1+x^4}{\left (-1+x^4\right ) \sqrt {1+\sqrt {1+x^2}}} \, dx=-\frac {4 \, \sqrt {2} x \arctan \left (\frac {\sqrt {2} \sqrt {\sqrt {x^{2} + 1} + 1}}{x}\right ) + x \sqrt {-\sqrt {2} - 1} \log \left (-\frac {2 \, \sqrt {x^{2} + 1} {\left (61 \, \sqrt {2} x + 71 \, x\right )} \sqrt {-\sqrt {2} - 1} - {\left (51 \, x^{3} + 2 \, \sqrt {2} {\left (5 \, x^{3} + 66 \, x\right )} + 193 \, x\right )} \sqrt {-\sqrt {2} - 1} + 2 \, {\left (71 \, x^{2} + \sqrt {2} {\left (61 \, x^{2} + 132\right )} - \sqrt {x^{2} + 1} {\left (132 \, \sqrt {2} + 193\right )} + 193\right )} \sqrt {\sqrt {x^{2} + 1} + 1}}{x^{3} - x}\right ) - x \sqrt {-\sqrt {2} - 1} \log \left (\frac {2 \, \sqrt {x^{2} + 1} {\left (61 \, \sqrt {2} x + 71 \, x\right )} \sqrt {-\sqrt {2} - 1} - {\left (51 \, x^{3} + 2 \, \sqrt {2} {\left (5 \, x^{3} + 66 \, x\right )} + 193 \, x\right )} \sqrt {-\sqrt {2} - 1} - 2 \, {\left (71 \, x^{2} + \sqrt {2} {\left (61 \, x^{2} + 132\right )} - \sqrt {x^{2} + 1} {\left (132 \, \sqrt {2} + 193\right )} + 193\right )} \sqrt {\sqrt {x^{2} + 1} + 1}}{x^{3} - x}\right ) - x \sqrt {\sqrt {2} - 1} \log \left (-\frac {{\left (51 \, x^{3} - 2 \, \sqrt {2} {\left (5 \, x^{3} + 66 \, x\right )} + 2 \, \sqrt {x^{2} + 1} {\left (61 \, \sqrt {2} x - 71 \, x\right )} + 193 \, x\right )} \sqrt {\sqrt {2} - 1} + 2 \, {\left (71 \, x^{2} - \sqrt {2} {\left (61 \, x^{2} + 132\right )} + \sqrt {x^{2} + 1} {\left (132 \, \sqrt {2} - 193\right )} + 193\right )} \sqrt {\sqrt {x^{2} + 1} + 1}}{x^{3} - x}\right ) + x \sqrt {\sqrt {2} - 1} \log \left (\frac {{\left (51 \, x^{3} - 2 \, \sqrt {2} {\left (5 \, x^{3} + 66 \, x\right )} + 2 \, \sqrt {x^{2} + 1} {\left (61 \, \sqrt {2} x - 71 \, x\right )} + 193 \, x\right )} \sqrt {\sqrt {2} - 1} - 2 \, {\left (71 \, x^{2} - \sqrt {2} {\left (61 \, x^{2} + 132\right )} + \sqrt {x^{2} + 1} {\left (132 \, \sqrt {2} - 193\right )} + 193\right )} \sqrt {\sqrt {x^{2} + 1} + 1}}{x^{3} - x}\right ) - 2 \, x \arctan \left (\frac {4 \, {\left (x^{4} - 12 \, x^{2} + {\left (5 \, x^{2} - 3\right )} \sqrt {x^{2} + 1} + 3\right )} \sqrt {\sqrt {x^{2} + 1} + 1}}{x^{5} - 46 \, x^{3} + 17 \, x}\right ) - 8 \, \sqrt {\sqrt {x^{2} + 1} + 1} {\left (\sqrt {x^{2} + 1} - 1\right )}}{4 \, x} \]

input

integrate((x^4+1)/(x^4-1)/(1+(x^2+1)^(1/2))^(1/2),x, algorithm="fricas")
 

output

-1/4*(4*sqrt(2)*x*arctan(sqrt(2)*sqrt(sqrt(x^2 + 1) + 1)/x) + x*sqrt(-sqrt 
(2) - 1)*log(-(2*sqrt(x^2 + 1)*(61*sqrt(2)*x + 71*x)*sqrt(-sqrt(2) - 1) - 
(51*x^3 + 2*sqrt(2)*(5*x^3 + 66*x) + 193*x)*sqrt(-sqrt(2) - 1) + 2*(71*x^2 
 + sqrt(2)*(61*x^2 + 132) - sqrt(x^2 + 1)*(132*sqrt(2) + 193) + 193)*sqrt( 
sqrt(x^2 + 1) + 1))/(x^3 - x)) - x*sqrt(-sqrt(2) - 1)*log((2*sqrt(x^2 + 1) 
*(61*sqrt(2)*x + 71*x)*sqrt(-sqrt(2) - 1) - (51*x^3 + 2*sqrt(2)*(5*x^3 + 6 
6*x) + 193*x)*sqrt(-sqrt(2) - 1) - 2*(71*x^2 + sqrt(2)*(61*x^2 + 132) - sq 
rt(x^2 + 1)*(132*sqrt(2) + 193) + 193)*sqrt(sqrt(x^2 + 1) + 1))/(x^3 - x)) 
 - x*sqrt(sqrt(2) - 1)*log(-((51*x^3 - 2*sqrt(2)*(5*x^3 + 66*x) + 2*sqrt(x 
^2 + 1)*(61*sqrt(2)*x - 71*x) + 193*x)*sqrt(sqrt(2) - 1) + 2*(71*x^2 - sqr 
t(2)*(61*x^2 + 132) + sqrt(x^2 + 1)*(132*sqrt(2) - 193) + 193)*sqrt(sqrt(x 
^2 + 1) + 1))/(x^3 - x)) + x*sqrt(sqrt(2) - 1)*log(((51*x^3 - 2*sqrt(2)*(5 
*x^3 + 66*x) + 2*sqrt(x^2 + 1)*(61*sqrt(2)*x - 71*x) + 193*x)*sqrt(sqrt(2) 
 - 1) - 2*(71*x^2 - sqrt(2)*(61*x^2 + 132) + sqrt(x^2 + 1)*(132*sqrt(2) - 
193) + 193)*sqrt(sqrt(x^2 + 1) + 1))/(x^3 - x)) - 2*x*arctan(4*(x^4 - 12*x 
^2 + (5*x^2 - 3)*sqrt(x^2 + 1) + 3)*sqrt(sqrt(x^2 + 1) + 1)/(x^5 - 46*x^3 
+ 17*x)) - 8*sqrt(sqrt(x^2 + 1) + 1)*(sqrt(x^2 + 1) - 1))/x
 

3.22.2.6 Sympy [F]

\[ \int \frac {1+x^4}{\left (-1+x^4\right ) \sqrt {1+\sqrt {1+x^2}}} \, dx=\int \frac {x^{4} + 1}{\left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right ) \sqrt {\sqrt {x^{2} + 1} + 1}}\, dx \]

input

integrate((x**4+1)/(x**4-1)/(1+(x**2+1)**(1/2))**(1/2),x)
 

output

Integral((x**4 + 1)/((x - 1)*(x + 1)*(x**2 + 1)*sqrt(sqrt(x**2 + 1) + 1)), 
 x)
 

3.22.2.7 Maxima [F]

\[ \int \frac {1+x^4}{\left (-1+x^4\right ) \sqrt {1+\sqrt {1+x^2}}} \, dx=\int { \frac {x^{4} + 1}{{\left (x^{4} - 1\right )} \sqrt {\sqrt {x^{2} + 1} + 1}} \,d x } \]

input

integrate((x^4+1)/(x^4-1)/(1+(x^2+1)^(1/2))^(1/2),x, algorithm="maxima")
 

output

integrate((x^4 + 1)/((x^4 - 1)*sqrt(sqrt(x^2 + 1) + 1)), x)
 

3.22.2.8 Giac [F]

\[ \int \frac {1+x^4}{\left (-1+x^4\right ) \sqrt {1+\sqrt {1+x^2}}} \, dx=\int { \frac {x^{4} + 1}{{\left (x^{4} - 1\right )} \sqrt {\sqrt {x^{2} + 1} + 1}} \,d x } \]

input

integrate((x^4+1)/(x^4-1)/(1+(x^2+1)^(1/2))^(1/2),x, algorithm="giac")
 

output

integrate((x^4 + 1)/((x^4 - 1)*sqrt(sqrt(x^2 + 1) + 1)), x)
 

3.22.2.9 Mupad [F(-1)]

Timed out. \[ \int \frac {1+x^4}{\left (-1+x^4\right ) \sqrt {1+\sqrt {1+x^2}}} \, dx=\int \frac {x^4+1}{\left (x^4-1\right )\,\sqrt {\sqrt {x^2+1}+1}} \,d x \]

input

int((x^4 + 1)/((x^4 - 1)*((x^2 + 1)^(1/2) + 1)^(1/2)),x)
 

output

int((x^4 + 1)/((x^4 - 1)*((x^2 + 1)^(1/2) + 1)^(1/2)), x)