3.22.77 \(\int \frac {-4-2 x+2 x^2+x^4}{x (-2+x^2) \sqrt [4]{\frac {2+x^2}{-2+x^2}} (8-10 x+4 x^2+4 x^3-4 x^4+x^5)} \, dx\) [2177]

3.22.77.1 Optimal result

Integrand size = 66, antiderivative size = 161 \[ \int \frac {-4-2 x+2 x^2+x^4}{x \left (-2+x^2\right ) \sqrt [4]{\frac {2+x^2}{-2+x^2}} \left (8-10 x+4 x^2+4 x^3-4 x^4+x^5\right )} \, dx=-\frac {\arctan \left (\frac {-\frac {1}{\sqrt {2}}+\sqrt {2} x-\frac {x^2}{\sqrt {2}}+\frac {\sqrt {\frac {2+x^2}{-2+x^2}}}{\sqrt {2}}}{(-1+x) \sqrt [4]{\frac {2+x^2}{-2+x^2}}}\right )}{2 \sqrt {2}}+\frac {\text {arctanh}\left (\frac {\frac {1}{\sqrt {2}}-\sqrt {2} x+\frac {x^2}{\sqrt {2}}+\frac {\sqrt {\frac {2+x^2}{-2+x^2}}}{\sqrt {2}}}{(-1+x) \sqrt [4]{\frac {2+x^2}{-2+x^2}}}\right )}{2 \sqrt {2}} \]

output

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

3.22.77.2 Mathematica [F]

\[ \int \frac {-4-2 x+2 x^2+x^4}{x \left (-2+x^2\right ) \sqrt [4]{\frac {2+x^2}{-2+x^2}} \left (8-10 x+4 x^2+4 x^3-4 x^4+x^5\right )} \, dx=\int \frac {-4-2 x+2 x^2+x^4}{x \left (-2+x^2\right ) \sqrt [4]{\frac {2+x^2}{-2+x^2}} \left (8-10 x+4 x^2+4 x^3-4 x^4+x^5\right )} \, dx \]

input

Integrate[(-4 - 2*x + 2*x^2 + x^4)/(x*(-2 + x^2)*((2 + x^2)/(-2 + x^2))^(1 
/4)*(8 - 10*x + 4*x^2 + 4*x^3 - 4*x^4 + x^5)),x]
 

output

Integrate[(-4 - 2*x + 2*x^2 + x^4)/(x*(-2 + x^2)*((2 + x^2)/(-2 + x^2))^(1 
/4)*(8 - 10*x + 4*x^2 + 4*x^3 - 4*x^4 + x^5)), x]
 

3.22.77.3 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.

\(\displaystyle \int \frac {x^4+2 x^2-2 x-4}{x \left (x^2-2\right ) \sqrt [4]{\frac {x^2+2}{x^2-2}} \left (x^5-4 x^4+4 x^3+4 x^2-10 x+8\right )} \, dx\)

\(\Big \downarrow \) 2058

\(\displaystyle \frac {\sqrt [4]{x^2+2} \int -\frac {-x^4-2 x^2+2 x+4}{x \left (x^2-2\right )^{3/4} \sqrt [4]{x^2+2} \left (x^5-4 x^4+4 x^3+4 x^2-10 x+8\right )}dx}{\sqrt [4]{x^2-2} \sqrt [4]{-\frac {x^2+2}{2-x^2}}}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {\sqrt [4]{x^2+2} \int \frac {-x^4-2 x^2+2 x+4}{x \left (x^2-2\right )^{3/4} \sqrt [4]{x^2+2} \left (x^5-4 x^4+4 x^3+4 x^2-10 x+8\right )}dx}{\sqrt [4]{x^2-2} \sqrt [4]{-\frac {x^2+2}{2-x^2}}}\)

\(\Big \downarrow \) 7293

\(\displaystyle -\frac {\sqrt [4]{x^2+2} \int \left (\frac {-x^4+2 x^3-4 x^2-8 x+14}{2 \left (x^2-2\right )^{3/4} \sqrt [4]{x^2+2} \left (x^5-4 x^4+4 x^3+4 x^2-10 x+8\right )}+\frac {1}{2 x \left (x^2-2\right )^{3/4} \sqrt [4]{x^2+2}}\right )dx}{\sqrt [4]{x^2-2} \sqrt [4]{-\frac {x^2+2}{2-x^2}}}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {\sqrt [4]{x^2+2} \left (7 \int \frac {1}{\left (x^2-2\right )^{3/4} \sqrt [4]{x^2+2} \left (x^5-4 x^4+4 x^3+4 x^2-10 x+8\right )}dx-4 \int \frac {x}{\left (x^2-2\right )^{3/4} \sqrt [4]{x^2+2} \left (x^5-4 x^4+4 x^3+4 x^2-10 x+8\right )}dx-2 \int \frac {x^2}{\left (x^2-2\right )^{3/4} \sqrt [4]{x^2+2} \left (x^5-4 x^4+4 x^3+4 x^2-10 x+8\right )}dx+\int \frac {x^3}{\left (x^2-2\right )^{3/4} \sqrt [4]{x^2+2} \left (x^5-4 x^4+4 x^3+4 x^2-10 x+8\right )}dx-\frac {1}{2} \int \frac {x^4}{\left (x^2-2\right )^{3/4} \sqrt [4]{x^2+2} \left (x^5-4 x^4+4 x^3+4 x^2-10 x+8\right )}dx+\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{x^2+2}}{\sqrt [4]{x^2-2}}\right )}{4 \sqrt {2}}-\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{x^2+2}}{\sqrt [4]{x^2-2}}+1\right )}{4 \sqrt {2}}-\frac {\log \left (\frac {\sqrt {x^2+2}}{\sqrt {x^2-2}}-\frac {\sqrt {2} \sqrt [4]{x^2+2}}{\sqrt [4]{x^2-2}}+1\right )}{8 \sqrt {2}}+\frac {\log \left (\frac {\sqrt {x^2+2}}{\sqrt {x^2-2}}+\frac {\sqrt {2} \sqrt [4]{x^2+2}}{\sqrt [4]{x^2-2}}+1\right )}{8 \sqrt {2}}\right )}{\sqrt [4]{x^2-2} \sqrt [4]{-\frac {x^2+2}{2-x^2}}}\)

input

Int[(-4 - 2*x + 2*x^2 + x^4)/(x*(-2 + x^2)*((2 + x^2)/(-2 + x^2))^(1/4)*(8 
 - 10*x + 4*x^2 + 4*x^3 - 4*x^4 + x^5)),x]
 

output

$Aborted
 

3.22.77.3.1 Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 2009

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

rule 2058

Int[(u_.)*((e_.)*((a_.) + (b_.)*(x_)^(n_.))^(q_.)*((c_) + (d_.)*(x_)^(n_))^ 
(r_.))^(p_), x_Symbol] :> Simp[Simp[(e*(a + b*x^n)^q*(c + d*x^n)^r)^p/((a + 
 b*x^n)^(p*q)*(c + d*x^n)^(p*r))]   Int[u*(a + b*x^n)^(p*q)*(c + d*x^n)^(p* 
r), x], x] /; FreeQ[{a, b, c, d, e, n, p, q, r}, x]
 

rule 7293

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v] 
]
 

3.22.77.4 Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 4.12 (sec) , antiderivative size = 1066, normalized size of antiderivative = 6.62

method	result	size
trager	\(\text {Expression too large to display}\)	\(1066\)

input

int((x^4+2*x^2-2*x-4)/x/(x^2-2)/((x^2+2)/(x^2-2))^(1/4)/(x^5-4*x^4+4*x^3+4 
*x^2-10*x+8),x,method=_RETURNVERBOSE)
 

output

1/4*RootOf(_Z^4+1)^3*ln((-RootOf(_Z^4+1)^3*x^6+2*(-(-x^2-2)/(x^2-2))^(1/4) 
*x^5*RootOf(_Z^4+1)^2+4*RootOf(_Z^4+1)^3*x^5-2*x^4*(-(-x^2-2)/(x^2-2))^(1/ 
2)*RootOf(_Z^4+1)-6*(-(-x^2-2)/(x^2-2))^(1/4)*x^4*RootOf(_Z^4+1)^2-4*RootO 
f(_Z^4+1)^3*x^4+2*(-(-x^2-2)/(x^2-2))^(3/4)*x^3+4*x^3*(-(-x^2-2)/(x^2-2))^ 
(1/2)*RootOf(_Z^4+1)+2*(-(-x^2-2)/(x^2-2))^(1/4)*x^3*RootOf(_Z^4+1)^2-4*Ro 
otOf(_Z^4+1)^3*x^3-2*(-(-x^2-2)/(x^2-2))^(3/4)*x^2+2*x^2*(-(-x^2-2)/(x^2-2 
))^(1/2)*RootOf(_Z^4+1)+10*(-(-x^2-2)/(x^2-2))^(1/4)*x^2*RootOf(_Z^4+1)^2+ 
12*RootOf(_Z^4+1)^3*x^2-4*(-(-x^2-2)/(x^2-2))^(3/4)*x-8*(-(-x^2-2)/(x^2-2) 
)^(1/2)*RootOf(_Z^4+1)*x-12*(-(-x^2-2)/(x^2-2))^(1/4)*x*RootOf(_Z^4+1)^2-8 
*RootOf(_Z^4+1)^3*x+4*(-(-x^2-2)/(x^2-2))^(3/4)+4*(-(-x^2-2)/(x^2-2))^(1/2 
)*RootOf(_Z^4+1)+4*(-(-x^2-2)/(x^2-2))^(1/4)*RootOf(_Z^4+1)^2+4*RootOf(_Z^ 
4+1)^3)/x/(x^5-4*x^4+4*x^3+4*x^2-10*x+8))+1/4*RootOf(_Z^4+1)*ln((-2*(-(-x^ 
2-2)/(x^2-2))^(1/2)*x^4*RootOf(_Z^4+1)^3+4*(-(-x^2-2)/(x^2-2))^(1/2)*x^3*R 
ootOf(_Z^4+1)^3-2*(-(-x^2-2)/(x^2-2))^(1/4)*x^5*RootOf(_Z^4+1)^2+2*(-(-x^2 
-2)/(x^2-2))^(1/2)*x^2*RootOf(_Z^4+1)^3+6*(-(-x^2-2)/(x^2-2))^(1/4)*x^4*Ro 
otOf(_Z^4+1)^2-RootOf(_Z^4+1)*x^6+2*(-(-x^2-2)/(x^2-2))^(3/4)*x^3-8*(-(-x^ 
2-2)/(x^2-2))^(1/2)*RootOf(_Z^4+1)^3*x-2*(-(-x^2-2)/(x^2-2))^(1/4)*x^3*Roo 
tOf(_Z^4+1)^2+4*RootOf(_Z^4+1)*x^5-2*(-(-x^2-2)/(x^2-2))^(3/4)*x^2+4*(-(-x 
^2-2)/(x^2-2))^(1/2)*RootOf(_Z^4+1)^3-10*(-(-x^2-2)/(x^2-2))^(1/4)*x^2*Roo 
tOf(_Z^4+1)^2-4*RootOf(_Z^4+1)*x^4-4*(-(-x^2-2)/(x^2-2))^(3/4)*x+12*(-(...
 

3.22.77.5 Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 25.38 (sec) , antiderivative size = 741, normalized size of antiderivative = 4.60 \[ \int \frac {-4-2 x+2 x^2+x^4}{x \left (-2+x^2\right ) \sqrt [4]{\frac {2+x^2}{-2+x^2}} \left (8-10 x+4 x^2+4 x^3-4 x^4+x^5\right )} \, dx=\text {Too large to display} \]

input

integrate((x^4+2*x^2-2*x-4)/x/(x^2-2)/((x^2+2)/(x^2-2))^(1/4)/(x^5-4*x^4+4 
*x^3+4*x^2-10*x+8),x, algorithm="fricas")
 

output

(1/16*I - 1/16)*sqrt(2)*log(-(2*sqrt(2)*((I + 1)*x^4 - (2*I + 2)*x^3 - (I 
+ 1)*x^2 + (4*I + 4)*x - 2*I - 2)*sqrt((x^2 + 2)/(x^2 - 2)) - 4*(x^3 - x^2 
 - 2*x + 2)*((x^2 + 2)/(x^2 - 2))^(3/4) - sqrt(2)*(-(I - 1)*x^6 + (4*I - 4 
)*x^5 - (4*I - 4)*x^4 - (4*I - 4)*x^3 + (12*I - 12)*x^2 - (8*I - 8)*x + 4* 
I - 4) + 4*(-I*x^5 + 3*I*x^4 - I*x^3 - 5*I*x^2 + 6*I*x - 2*I)*((x^2 + 2)/( 
x^2 - 2))^(1/4))/(x^6 - 4*x^5 + 4*x^4 + 4*x^3 - 10*x^2 + 8*x)) - (1/16*I + 
 1/16)*sqrt(2)*log(-(2*sqrt(2)*(-(I - 1)*x^4 + (2*I - 2)*x^3 + (I - 1)*x^2 
 - (4*I - 4)*x + 2*I - 2)*sqrt((x^2 + 2)/(x^2 - 2)) - 4*(x^3 - x^2 - 2*x + 
 2)*((x^2 + 2)/(x^2 - 2))^(3/4) - sqrt(2)*((I + 1)*x^6 - (4*I + 4)*x^5 + ( 
4*I + 4)*x^4 + (4*I + 4)*x^3 - (12*I + 12)*x^2 + (8*I + 8)*x - 4*I - 4) + 
4*(I*x^5 - 3*I*x^4 + I*x^3 + 5*I*x^2 - 6*I*x + 2*I)*((x^2 + 2)/(x^2 - 2))^ 
(1/4))/(x^6 - 4*x^5 + 4*x^4 + 4*x^3 - 10*x^2 + 8*x)) + (1/16*I + 1/16)*sqr 
t(2)*log(-(2*sqrt(2)*((I - 1)*x^4 - (2*I - 2)*x^3 - (I - 1)*x^2 + (4*I - 4 
)*x - 2*I + 2)*sqrt((x^2 + 2)/(x^2 - 2)) - 4*(x^3 - x^2 - 2*x + 2)*((x^2 + 
 2)/(x^2 - 2))^(3/4) - sqrt(2)*(-(I + 1)*x^6 + (4*I + 4)*x^5 - (4*I + 4)*x 
^4 - (4*I + 4)*x^3 + (12*I + 12)*x^2 - (8*I + 8)*x + 4*I + 4) + 4*(I*x^5 - 
 3*I*x^4 + I*x^3 + 5*I*x^2 - 6*I*x + 2*I)*((x^2 + 2)/(x^2 - 2))^(1/4))/(x^ 
6 - 4*x^5 + 4*x^4 + 4*x^3 - 10*x^2 + 8*x)) - (1/16*I - 1/16)*sqrt(2)*log(- 
(2*sqrt(2)*(-(I + 1)*x^4 + (2*I + 2)*x^3 + (I + 1)*x^2 - (4*I + 4)*x + 2*I 
 + 2)*sqrt((x^2 + 2)/(x^2 - 2)) - 4*(x^3 - x^2 - 2*x + 2)*((x^2 + 2)/(x...
 

3.22.77.6 Sympy [F(-1)]

Timed out. \[ \int \frac {-4-2 x+2 x^2+x^4}{x \left (-2+x^2\right ) \sqrt [4]{\frac {2+x^2}{-2+x^2}} \left (8-10 x+4 x^2+4 x^3-4 x^4+x^5\right )} \, dx=\text {Timed out} \]

input

integrate((x**4+2*x**2-2*x-4)/x/(x**2-2)/((x**2+2)/(x**2-2))**(1/4)/(x**5- 
4*x**4+4*x**3+4*x**2-10*x+8),x)
 

output

Timed out
 

3.22.77.7 Maxima [F]

\[ \int \frac {-4-2 x+2 x^2+x^4}{x \left (-2+x^2\right ) \sqrt [4]{\frac {2+x^2}{-2+x^2}} \left (8-10 x+4 x^2+4 x^3-4 x^4+x^5\right )} \, dx=\int { \frac {x^{4} + 2 \, x^{2} - 2 \, x - 4}{{\left (x^{5} - 4 \, x^{4} + 4 \, x^{3} + 4 \, x^{2} - 10 \, x + 8\right )} {\left (x^{2} - 2\right )} x \left (\frac {x^{2} + 2}{x^{2} - 2}\right )^{\frac {1}{4}}} \,d x } \]

input

integrate((x^4+2*x^2-2*x-4)/x/(x^2-2)/((x^2+2)/(x^2-2))^(1/4)/(x^5-4*x^4+4 
*x^3+4*x^2-10*x+8),x, algorithm="maxima")
 

output

integrate((x^4 + 2*x^2 - 2*x - 4)/((x^5 - 4*x^4 + 4*x^3 + 4*x^2 - 10*x + 8 
)*(x^2 - 2)*x*((x^2 + 2)/(x^2 - 2))^(1/4)), x)
 

3.22.77.8 Giac [F]

\[ \int \frac {-4-2 x+2 x^2+x^4}{x \left (-2+x^2\right ) \sqrt [4]{\frac {2+x^2}{-2+x^2}} \left (8-10 x+4 x^2+4 x^3-4 x^4+x^5\right )} \, dx=\int { \frac {x^{4} + 2 \, x^{2} - 2 \, x - 4}{{\left (x^{5} - 4 \, x^{4} + 4 \, x^{3} + 4 \, x^{2} - 10 \, x + 8\right )} {\left (x^{2} - 2\right )} x \left (\frac {x^{2} + 2}{x^{2} - 2}\right )^{\frac {1}{4}}} \,d x } \]

input

integrate((x^4+2*x^2-2*x-4)/x/(x^2-2)/((x^2+2)/(x^2-2))^(1/4)/(x^5-4*x^4+4 
*x^3+4*x^2-10*x+8),x, algorithm="giac")
 

output

integrate((x^4 + 2*x^2 - 2*x - 4)/((x^5 - 4*x^4 + 4*x^3 + 4*x^2 - 10*x + 8 
)*(x^2 - 2)*x*((x^2 + 2)/(x^2 - 2))^(1/4)), x)
 

3.22.77.9 Mupad [F(-1)]

Timed out. \[ \int \frac {-4-2 x+2 x^2+x^4}{x \left (-2+x^2\right ) \sqrt [4]{\frac {2+x^2}{-2+x^2}} \left (8-10 x+4 x^2+4 x^3-4 x^4+x^5\right )} \, dx=\int -\frac {-x^4-2\,x^2+2\,x+4}{x\,{\left (\frac {x^2+2}{x^2-2}\right )}^{1/4}\,\left (x^2-2\right )\,\left (x^5-4\,x^4+4\,x^3+4\,x^2-10\,x+8\right )} \,d x \]

input

int(-(2*x - 2*x^2 - x^4 + 4)/(x*((x^2 + 2)/(x^2 - 2))^(1/4)*(x^2 - 2)*(4*x 
^2 - 10*x + 4*x^3 - 4*x^4 + x^5 + 8)),x)
 

output

int(-(2*x - 2*x^2 - x^4 + 4)/(x*((x^2 + 2)/(x^2 - 2))^(1/4)*(x^2 - 2)*(4*x 
^2 - 10*x + 4*x^3 - 4*x^4 + x^5 + 8)), x)