3.30.9 \(\int \frac {1+x^4}{(-1+x^4) \sqrt {x+\sqrt {1+x^2}}} \, dx\) [2909]

3.30.9.1 Optimal result

Integrand size = 28, antiderivative size = 326 \[ \int \frac {1+x^4}{\left (-1+x^4\right ) \sqrt {x+\sqrt {1+x^2}}} \, dx=\frac {4}{3} x \sqrt {-x+\sqrt {1+x^2}}+\frac {2}{3} \sqrt {1+x^2} \sqrt {-x+\sqrt {1+x^2}}+\sqrt {-1+\sqrt {2}} \arctan \left (\frac {\sqrt {-x+\sqrt {1+x^2}}}{\sqrt {-1+\sqrt {2}}}\right )-\sqrt {1+\sqrt {2}} \arctan \left (\frac {\sqrt {-x+\sqrt {1+x^2}}}{\sqrt {1+\sqrt {2}}}\right )+\sqrt {2} \arctan \left (\frac {-\frac {1}{\sqrt {2}}-\frac {x}{\sqrt {2}}+\frac {\sqrt {1+x^2}}{\sqrt {2}}}{\sqrt {-x+\sqrt {1+x^2}}}\right )-\sqrt {-1+\sqrt {2}} \text {arctanh}\left (\frac {\sqrt {-x+\sqrt {1+x^2}}}{\sqrt {-1+\sqrt {2}}}\right )+\sqrt {1+\sqrt {2}} \text {arctanh}\left (\frac {\sqrt {-x+\sqrt {1+x^2}}}{\sqrt {1+\sqrt {2}}}\right )-\sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {-x+\sqrt {1+x^2}}}{1-x+\sqrt {1+x^2}}\right ) \]

output

4/3*x*(-x+(x^2+1)^(1/2))^(1/2)+2/3*(x^2+1)^(1/2)*(-x+(x^2+1)^(1/2))^(1/2)+ 
(2^(1/2)-1)^(1/2)*arctan((-x+(x^2+1)^(1/2))^(1/2)/(2^(1/2)-1)^(1/2))-(1+2^ 
(1/2))^(1/2)*arctan((-x+(x^2+1)^(1/2))^(1/2)/(1+2^(1/2))^(1/2))+2^(1/2)*ar 
ctan((-1/2*2^(1/2)-1/2*x*2^(1/2)+1/2*(x^2+1)^(1/2)*2^(1/2))/(-x+(x^2+1)^(1 
/2))^(1/2))-(2^(1/2)-1)^(1/2)*arctanh((-x+(x^2+1)^(1/2))^(1/2)/(2^(1/2)-1) 
^(1/2))+(1+2^(1/2))^(1/2)*arctanh((-x+(x^2+1)^(1/2))^(1/2)/(1+2^(1/2))^(1/ 
2))-2^(1/2)*arctanh(2^(1/2)*(-x+(x^2+1)^(1/2))^(1/2)/(1-x+(x^2+1)^(1/2)))
 

3.30.9.2 Mathematica [A] (verified)

Time = 0.38 (sec) , antiderivative size = 279, normalized size of antiderivative = 0.86 \[ \int \frac {1+x^4}{\left (-1+x^4\right ) \sqrt {x+\sqrt {1+x^2}}} \, dx=-\frac {1}{3 \left (x+\sqrt {1+x^2}\right )^{3/2}}+\sqrt {x+\sqrt {1+x^2}}-\sqrt {2} \arctan \left (\frac {-1+x+\sqrt {1+x^2}}{\sqrt {2} \sqrt {x+\sqrt {1+x^2}}}\right )-\sqrt {-1+\sqrt {2}} \arctan \left (\sqrt {-1+\sqrt {2}} \sqrt {x+\sqrt {1+x^2}}\right )+\sqrt {1+\sqrt {2}} \arctan \left (\sqrt {1+\sqrt {2}} \sqrt {x+\sqrt {1+x^2}}\right )-\sqrt {-1+\sqrt {2}} \text {arctanh}\left (\sqrt {-1+\sqrt {2}} \sqrt {x+\sqrt {1+x^2}}\right )+\sqrt {1+\sqrt {2}} \text {arctanh}\left (\sqrt {1+\sqrt {2}} \sqrt {x+\sqrt {1+x^2}}\right )-\sqrt {2} \text {arctanh}\left (\frac {1+x+\sqrt {1+x^2}}{\sqrt {2} \sqrt {x+\sqrt {1+x^2}}}\right ) \]

input

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

output

-1/3*1/(x + Sqrt[1 + x^2])^(3/2) + Sqrt[x + Sqrt[1 + x^2]] - Sqrt[2]*ArcTa 
n[(-1 + x + Sqrt[1 + x^2])/(Sqrt[2]*Sqrt[x + Sqrt[1 + x^2]])] - Sqrt[-1 + 
Sqrt[2]]*ArcTan[Sqrt[-1 + Sqrt[2]]*Sqrt[x + Sqrt[1 + x^2]]] + Sqrt[1 + Sqr 
t[2]]*ArcTan[Sqrt[1 + Sqrt[2]]*Sqrt[x + Sqrt[1 + x^2]]] - Sqrt[-1 + Sqrt[2 
]]*ArcTanh[Sqrt[-1 + Sqrt[2]]*Sqrt[x + Sqrt[1 + x^2]]] + Sqrt[1 + Sqrt[2]] 
*ArcTanh[Sqrt[1 + Sqrt[2]]*Sqrt[x + Sqrt[1 + x^2]]] - Sqrt[2]*ArcTanh[(1 + 
 x + Sqrt[1 + x^2])/(Sqrt[2]*Sqrt[x + Sqrt[1 + x^2]])]
 

3.30.9.3 Rubi [A] (verified)

Time = 1.03 (sec) , antiderivative size = 341, normalized size of antiderivative = 1.05, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {7276, 2009}

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.

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

\(\Big \downarrow \) 7276

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

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {\arctan \left (\sqrt {\sqrt {2}-1} \sqrt {\sqrt {x^2+1}+x}\right )}{\sqrt {1+\sqrt {2}}}+\frac {\arctan \left (\sqrt {1+\sqrt {2}} \sqrt {\sqrt {x^2+1}+x}\right )}{\sqrt {\sqrt {2}-1}}+\sqrt {2} \arctan \left (1-\sqrt {2} \sqrt {\sqrt {x^2+1}+x}\right )-\sqrt {2} \arctan \left (\sqrt {2} \sqrt {\sqrt {x^2+1}+x}+1\right )-\frac {\text {arctanh}\left (\sqrt {\sqrt {2}-1} \sqrt {\sqrt {x^2+1}+x}\right )}{\sqrt {1+\sqrt {2}}}+\frac {\text {arctanh}\left (\sqrt {1+\sqrt {2}} \sqrt {\sqrt {x^2+1}+x}\right )}{\sqrt {\sqrt {2}-1}}+\sqrt {\sqrt {x^2+1}+x}-\frac {1}{3 \left (\sqrt {x^2+1}+x\right )^{3/2}}+\frac {\log \left (\sqrt {x^2+1}-\sqrt {2} \sqrt {\sqrt {x^2+1}+x}+x+1\right )}{\sqrt {2}}-\frac {\log \left (\sqrt {x^2+1}+\sqrt {2} \sqrt {\sqrt {x^2+1}+x}+x+1\right )}{\sqrt {2}}\)

input

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

output

-1/3*1/(x + Sqrt[1 + x^2])^(3/2) + Sqrt[x + Sqrt[1 + x^2]] - ArcTan[Sqrt[- 
1 + Sqrt[2]]*Sqrt[x + Sqrt[1 + x^2]]]/Sqrt[1 + Sqrt[2]] + ArcTan[Sqrt[1 + 
Sqrt[2]]*Sqrt[x + Sqrt[1 + x^2]]]/Sqrt[-1 + Sqrt[2]] + Sqrt[2]*ArcTan[1 - 
Sqrt[2]*Sqrt[x + Sqrt[1 + x^2]]] - Sqrt[2]*ArcTan[1 + Sqrt[2]*Sqrt[x + Sqr 
t[1 + x^2]]] - ArcTanh[Sqrt[-1 + Sqrt[2]]*Sqrt[x + Sqrt[1 + x^2]]]/Sqrt[1 
+ Sqrt[2]] + ArcTanh[Sqrt[1 + Sqrt[2]]*Sqrt[x + Sqrt[1 + x^2]]]/Sqrt[-1 + 
Sqrt[2]] + Log[1 + x + Sqrt[1 + x^2] - Sqrt[2]*Sqrt[x + Sqrt[1 + x^2]]]/Sq 
rt[2] - Log[1 + x + Sqrt[1 + x^2] + Sqrt[2]*Sqrt[x + Sqrt[1 + x^2]]]/Sqrt[ 
2]
 

3.30.9.3.1 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]
 

3.30.9.4 Maple [F]

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

input

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

output

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

3.30.9.5 Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.27 (sec) , antiderivative size = 429, normalized size of antiderivative = 1.32 \[ \int \frac {1+x^4}{\left (-1+x^4\right ) \sqrt {x+\sqrt {1+x^2}}} \, dx=-\frac {2}{3} \, {\left (x^{2} - \sqrt {x^{2} + 1} x - 1\right )} \sqrt {x + \sqrt {x^{2} + 1}} - \frac {1}{2} \, \sqrt {\sqrt {2} - 1} \log \left ({\left (\sqrt {2} + 1\right )} \sqrt {\sqrt {2} - 1} + \sqrt {x + \sqrt {x^{2} + 1}}\right ) + \frac {1}{2} \, \sqrt {\sqrt {2} - 1} \log \left (-{\left (\sqrt {2} + 1\right )} \sqrt {\sqrt {2} - 1} + \sqrt {x + \sqrt {x^{2} + 1}}\right ) + \frac {1}{2} \, \sqrt {\sqrt {2} + 1} \log \left (\sqrt {\sqrt {2} + 1} {\left (\sqrt {2} - 1\right )} + \sqrt {x + \sqrt {x^{2} + 1}}\right ) - \frac {1}{2} \, \sqrt {\sqrt {2} + 1} \log \left (-\sqrt {\sqrt {2} + 1} {\left (\sqrt {2} - 1\right )} + \sqrt {x + \sqrt {x^{2} + 1}}\right ) - \frac {1}{2} \, \sqrt {-\sqrt {2} + 1} \log \left ({\left (\sqrt {2} + 1\right )} \sqrt {-\sqrt {2} + 1} + \sqrt {x + \sqrt {x^{2} + 1}}\right ) + \frac {1}{2} \, \sqrt {-\sqrt {2} + 1} \log \left (-{\left (\sqrt {2} + 1\right )} \sqrt {-\sqrt {2} + 1} + \sqrt {x + \sqrt {x^{2} + 1}}\right ) + \frac {1}{2} \, \sqrt {-\sqrt {2} - 1} \log \left ({\left (\sqrt {2} - 1\right )} \sqrt {-\sqrt {2} - 1} + \sqrt {x + \sqrt {x^{2} + 1}}\right ) - \frac {1}{2} \, \sqrt {-\sqrt {2} - 1} \log \left (-{\left (\sqrt {2} - 1\right )} \sqrt {-\sqrt {2} - 1} + \sqrt {x + \sqrt {x^{2} + 1}}\right ) - \left (\frac {1}{2} i + \frac {1}{2}\right ) \, \sqrt {2} \log \left (\left (i + 1\right ) \, \sqrt {2} + 2 \, \sqrt {x + \sqrt {x^{2} + 1}}\right ) + \left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {2} \log \left (-\left (i - 1\right ) \, \sqrt {2} + 2 \, \sqrt {x + \sqrt {x^{2} + 1}}\right ) - \left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {2} \log \left (\left (i - 1\right ) \, \sqrt {2} + 2 \, \sqrt {x + \sqrt {x^{2} + 1}}\right ) + \left (\frac {1}{2} i + \frac {1}{2}\right ) \, \sqrt {2} \log \left (-\left (i + 1\right ) \, \sqrt {2} + 2 \, \sqrt {x + \sqrt {x^{2} + 1}}\right ) \]

input

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

output

-2/3*(x^2 - sqrt(x^2 + 1)*x - 1)*sqrt(x + sqrt(x^2 + 1)) - 1/2*sqrt(sqrt(2 
) - 1)*log((sqrt(2) + 1)*sqrt(sqrt(2) - 1) + sqrt(x + sqrt(x^2 + 1))) + 1/ 
2*sqrt(sqrt(2) - 1)*log(-(sqrt(2) + 1)*sqrt(sqrt(2) - 1) + sqrt(x + sqrt(x 
^2 + 1))) + 1/2*sqrt(sqrt(2) + 1)*log(sqrt(sqrt(2) + 1)*(sqrt(2) - 1) + sq 
rt(x + sqrt(x^2 + 1))) - 1/2*sqrt(sqrt(2) + 1)*log(-sqrt(sqrt(2) + 1)*(sqr 
t(2) - 1) + sqrt(x + sqrt(x^2 + 1))) - 1/2*sqrt(-sqrt(2) + 1)*log((sqrt(2) 
 + 1)*sqrt(-sqrt(2) + 1) + sqrt(x + sqrt(x^2 + 1))) + 1/2*sqrt(-sqrt(2) + 
1)*log(-(sqrt(2) + 1)*sqrt(-sqrt(2) + 1) + sqrt(x + sqrt(x^2 + 1))) + 1/2* 
sqrt(-sqrt(2) - 1)*log((sqrt(2) - 1)*sqrt(-sqrt(2) - 1) + sqrt(x + sqrt(x^ 
2 + 1))) - 1/2*sqrt(-sqrt(2) - 1)*log(-(sqrt(2) - 1)*sqrt(-sqrt(2) - 1) + 
sqrt(x + sqrt(x^2 + 1))) - (1/2*I + 1/2)*sqrt(2)*log((I + 1)*sqrt(2) + 2*s 
qrt(x + sqrt(x^2 + 1))) + (1/2*I - 1/2)*sqrt(2)*log(-(I - 1)*sqrt(2) + 2*s 
qrt(x + sqrt(x^2 + 1))) - (1/2*I - 1/2)*sqrt(2)*log((I - 1)*sqrt(2) + 2*sq 
rt(x + sqrt(x^2 + 1))) + (1/2*I + 1/2)*sqrt(2)*log(-(I + 1)*sqrt(2) + 2*sq 
rt(x + sqrt(x^2 + 1)))
 

3.30.9.6 Sympy [F]

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

input

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

output

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

3.30.9.7 Maxima [F]

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

input

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

output

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

3.30.9.8 Giac [F]

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

input

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

output

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

3.30.9.9 Mupad [F(-1)]

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

input

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

output

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