\(\int \frac {1}{x^5 \sqrt {1+x^4} \sqrt {x^2+\sqrt {1+x^4}}} \, dx\) [79]

Optimal result

Integrand size = 30, antiderivative size = 93 \[ \int \frac {1}{x^5 \sqrt {1+x^4} \sqrt {x^2+\sqrt {1+x^4}}} \, dx=\frac {3}{8 x^2 \sqrt {x^2+\sqrt {1+x^4}}}-\frac {\sqrt {x^2+\sqrt {1+x^4}}}{4 x^4}+\frac {3}{8} \arctan \left (\sqrt {x^2+\sqrt {1+x^4}}\right )+\frac {3}{8} \text {arctanh}\left (\sqrt {x^2+\sqrt {1+x^4}}\right ) \] Output:

3/8/x^2/(x^2+(x^4+1)^(1/2))^(1/2)-1/4*(x^2+(x^4+1)^(1/2))^(1/2)/x^4+3/8*ar 
ctan((x^2+(x^4+1)^(1/2))^(1/2))+3/8*arctanh((x^2+(x^4+1)^(1/2))^(1/2))
 

Mathematica [A] (verified)

Time = 0.21 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.94 \[ \int \frac {1}{x^5 \sqrt {1+x^4} \sqrt {x^2+\sqrt {1+x^4}}} \, dx=\frac {1}{8} \left (\frac {-2-x^4-x^2 \sqrt {1+x^4}}{x^4 \left (x^2+\sqrt {1+x^4}\right )^{3/2}}+3 \arctan \left (\sqrt {x^2+\sqrt {1+x^4}}\right )+3 \text {arctanh}\left (\sqrt {x^2+\sqrt {1+x^4}}\right )\right ) \] Input:

Integrate[1/(x^5*Sqrt[1 + x^4]*Sqrt[x^2 + Sqrt[1 + x^4]]),x]
 

Output:

((-2 - x^4 - x^2*Sqrt[1 + x^4])/(x^4*(x^2 + Sqrt[1 + x^4])^(3/2)) + 3*ArcT 
an[Sqrt[x^2 + Sqrt[1 + x^4]]] + 3*ArcTanh[Sqrt[x^2 + Sqrt[1 + x^4]]])/8
 

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.

\(\displaystyle \int \frac {1}{x^5 \sqrt {x^4+1} \sqrt {\sqrt {x^4+1}+x^2}} \, dx\)

\(\Big \downarrow \) 7299

\(\displaystyle \int \frac {1}{x^5 \sqrt {x^4+1} \sqrt {\sqrt {x^4+1}+x^2}}dx\)

Input:

Int[1/(x^5*Sqrt[1 + x^4]*Sqrt[x^2 + Sqrt[1 + x^4]]),x]
 

Output:

$Aborted
 

Defintions of rubi rules used

rule 7299

Int[u_, x_] :> CannotIntegrate[u, x]
 

Maple [F]

\[\int \frac {1}{x^{5} \sqrt {x^{4}+1}\, \sqrt {x^{2}+\sqrt {x^{4}+1}}}d x\]

Input:

int(1/x^5/(x^4+1)^(1/2)/(x^2+(x^4+1)^(1/2))^(1/2),x)
 

Output:

int(1/x^5/(x^4+1)^(1/2)/(x^2+(x^4+1)^(1/2))^(1/2),x)
 

Fricas [A] (verification not implemented)

Time = 0.88 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.37 \[ \int \frac {1}{x^5 \sqrt {1+x^4} \sqrt {x^2+\sqrt {1+x^4}}} \, dx=-\frac {3 \, x^{4} \arctan \left (-\frac {\sqrt {x^{2} + \sqrt {x^{4} + 1}} {\left (x^{2} - \sqrt {x^{4} + 1} - 1\right )}}{x^{2}}\right ) - 3 \, x^{4} \log \left (\frac {\sqrt {x^{2} + \sqrt {x^{4} + 1}} {\left (x^{2} - \sqrt {x^{4} + 1} - 1\right )} - \sqrt {x^{4} + 1} - 1}{x^{2}}\right ) + 2 \, {\left (3 \, x^{4} - 3 \, \sqrt {x^{4} + 1} x^{2} + 2\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}}}{16 \, x^{4}} \] Input:

integrate(1/x^5/(x^4+1)^(1/2)/(x^2+(x^4+1)^(1/2))^(1/2),x, algorithm="fric 
as")
 

Output:

-1/16*(3*x^4*arctan(-sqrt(x^2 + sqrt(x^4 + 1))*(x^2 - sqrt(x^4 + 1) - 1)/x 
^2) - 3*x^4*log((sqrt(x^2 + sqrt(x^4 + 1))*(x^2 - sqrt(x^4 + 1) - 1) - sqr 
t(x^4 + 1) - 1)/x^2) + 2*(3*x^4 - 3*sqrt(x^4 + 1)*x^2 + 2)*sqrt(x^2 + sqrt 
(x^4 + 1)))/x^4
 

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 16.62 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.49 \[ \int \frac {1}{x^5 \sqrt {1+x^4} \sqrt {x^2+\sqrt {1+x^4}}} \, dx=- \frac {\Gamma \left (\frac {3}{4}\right ) \Gamma \left (\frac {5}{4}\right ) \Gamma \left (\frac {7}{4}\right ) {{}_{3}F_{2}\left (\begin {matrix} \frac {3}{4}, \frac {5}{4}, \frac {7}{4} \\ \frac {3}{2}, \frac {11}{4} \end {matrix}\middle | {\frac {e^{i \pi }}{x^{4}}} \right )}}{2 \pi x^{7} \Gamma \left (\frac {11}{4}\right )} \] Input:

integrate(1/x**5/(x**4+1)**(1/2)/(x**2+(x**4+1)**(1/2))**(1/2),x)
 

Output:

-gamma(3/4)*gamma(5/4)*gamma(7/4)*hyper((3/4, 5/4, 7/4), (3/2, 11/4), exp_ 
polar(I*pi)/x**4)/(2*pi*x**7*gamma(11/4))
 

Maxima [F]

\[ \int \frac {1}{x^5 \sqrt {1+x^4} \sqrt {x^2+\sqrt {1+x^4}}} \, dx=\int { \frac {1}{\sqrt {x^{4} + 1} \sqrt {x^{2} + \sqrt {x^{4} + 1}} x^{5}} \,d x } \] Input:

integrate(1/x^5/(x^4+1)^(1/2)/(x^2+(x^4+1)^(1/2))^(1/2),x, algorithm="maxi 
ma")
 

Output:

integrate(1/(sqrt(x^4 + 1)*sqrt(x^2 + sqrt(x^4 + 1))*x^5), x)
 

Giac [F]

\[ \int \frac {1}{x^5 \sqrt {1+x^4} \sqrt {x^2+\sqrt {1+x^4}}} \, dx=\int { \frac {1}{\sqrt {x^{4} + 1} \sqrt {x^{2} + \sqrt {x^{4} + 1}} x^{5}} \,d x } \] Input:

integrate(1/x^5/(x^4+1)^(1/2)/(x^2+(x^4+1)^(1/2))^(1/2),x, algorithm="giac 
")
 

Output:

integrate(1/(sqrt(x^4 + 1)*sqrt(x^2 + sqrt(x^4 + 1))*x^5), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{x^5 \sqrt {1+x^4} \sqrt {x^2+\sqrt {1+x^4}}} \, dx=\int \frac {1}{x^5\,\sqrt {x^4+1}\,\sqrt {\sqrt {x^4+1}+x^2}} \,d x \] Input:

int(1/(x^5*(x^4 + 1)^(1/2)*((x^4 + 1)^(1/2) + x^2)^(1/2)),x)
                                                                                    
                                                                                    
 

Output:

int(1/(x^5*(x^4 + 1)^(1/2)*((x^4 + 1)^(1/2) + x^2)^(1/2)), x)
 

Reduce [F]

\[ \int \frac {1}{x^5 \sqrt {1+x^4} \sqrt {x^2+\sqrt {1+x^4}}} \, dx=-\frac {3 \mathit {atan} \left (\frac {\sqrt {\sqrt {x^{4}+1}+x^{2}}\, \sqrt {x^{4}+1}}{2}-\frac {\sqrt {\sqrt {x^{4}+1}+x^{2}}\, x^{2}}{2}-\frac {\sqrt {\sqrt {x^{4}+1}+x^{2}}}{2}\right )}{16}-\frac {3 \sqrt {\sqrt {x^{4}+1}+x^{2}}}{8}+\int \frac {\sqrt {\sqrt {x^{4}+1}+x^{2}}}{x^{9}+x^{5}}d x +\frac {11 \left (\int \frac {\sqrt {\sqrt {x^{4}+1}+x^{2}}}{x^{5}+x}d x \right )}{8}+\frac {3 \left (\int \frac {\sqrt {\sqrt {x^{4}+1}+x^{2}}\, x^{3}}{x^{4}+1}d x \right )}{8}-\left (\int \frac {\sqrt {\sqrt {x^{4}+1}+x^{2}}\, \sqrt {x^{4}+1}}{x^{7}+x^{3}}d x \right )-\frac {3 \,\mathrm {log}\left (\sqrt {\sqrt {x^{4}+1}+x^{2}}-1\right )}{16}+\frac {3 \,\mathrm {log}\left (\sqrt {\sqrt {x^{4}+1}+x^{2}}+1\right )}{16} \] Input:

int(1/x^5/(x^4+1)^(1/2)/(x^2+(x^4+1)^(1/2))^(1/2),x)
 

Output:

( - 3*atan((sqrt(sqrt(x**4 + 1) + x**2)*sqrt(x**4 + 1) - sqrt(sqrt(x**4 + 
1) + x**2)*x**2 - sqrt(sqrt(x**4 + 1) + x**2))/2) - 6*sqrt(sqrt(x**4 + 1) 
+ x**2) + 16*int(sqrt(sqrt(x**4 + 1) + x**2)/(x**9 + x**5),x) + 22*int(sqr 
t(sqrt(x**4 + 1) + x**2)/(x**5 + x),x) + 6*int((sqrt(sqrt(x**4 + 1) + x**2 
)*x**3)/(x**4 + 1),x) - 16*int((sqrt(sqrt(x**4 + 1) + x**2)*sqrt(x**4 + 1) 
)/(x**7 + x**3),x) - 3*log(sqrt(sqrt(x**4 + 1) + x**2) - 1) + 3*log(sqrt(s 
qrt(x**4 + 1) + x**2) + 1))/16