\(\int \frac {(1+x^2)^2}{(1-x^2) (1-6 x^2+x^4)^{3/4}} \, dx\) [1317]

Optimal result

Integrand size = 31, antiderivative size = 95 \[ \int \frac {\left (1+x^2\right )^2}{\left (1-x^2\right ) \left (1-6 x^2+x^4\right )^{3/4}} \, dx=\arctan \left (\frac {i+x}{\sqrt [4]{1-6 x^2+x^4}}\right )-\arctan \left (\frac {\sqrt [4]{1-6 x^2+x^4}}{-i+x}\right )-\text {arctanh}\left (\frac {i+x}{\sqrt [4]{1-6 x^2+x^4}}\right )-\text {arctanh}\left (\frac {\sqrt [4]{1-6 x^2+x^4}}{-i+x}\right ) \] Output:

arctan((I+x)/(x^4-6*x^2+1)^(1/4))-arctan((x^4-6*x^2+1)^(1/4)/(-I+x))-arcta 
nh((I+x)/(x^4-6*x^2+1)^(1/4))-arctanh((x^4-6*x^2+1)^(1/4)/(-I+x))
 

Mathematica [A] (verified)

Time = 6.71 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.00 \[ \int \frac {\left (1+x^2\right )^2}{\left (1-x^2\right ) \left (1-6 x^2+x^4\right )^{3/4}} \, dx=\arctan \left (\frac {-i+x}{\sqrt [4]{1-6 x^2+x^4}}\right )-\arctan \left (\frac {\sqrt [4]{1-6 x^2+x^4}}{i+x}\right )-\text {arctanh}\left (\frac {-i+x}{\sqrt [4]{1-6 x^2+x^4}}\right )-\text {arctanh}\left (\frac {\sqrt [4]{1-6 x^2+x^4}}{i+x}\right ) \] Input:

Integrate[(1 + x^2)^2/((1 - x^2)*(1 - 6*x^2 + x^4)^(3/4)),x]
 

Output:

ArcTan[(-I + x)/(1 - 6*x^2 + x^4)^(1/4)] - ArcTan[(1 - 6*x^2 + x^4)^(1/4)/ 
(I + x)] - ArcTanh[(-I + x)/(1 - 6*x^2 + x^4)^(1/4)] - ArcTanh[(1 - 6*x^2 
+ x^4)^(1/4)/(I + x)]
 

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.

\(\displaystyle \int \frac {\left (x^2+1\right )^2}{\left (1-x^2\right ) \left (x^4-6 x^2+1\right )^{3/4}} \, dx\)

\(\Big \downarrow \) 2260

\(\displaystyle \int \frac {\left (x^2+1\right )^2}{\left (1-x^2\right ) \left (x^4-6 x^2+1\right )^{3/4}}dx\)

Input:

Int[(1 + x^2)^2/((1 - x^2)*(1 - 6*x^2 + x^4)^(3/4)),x]
 

Output:

$Aborted
 

Maple [C] (verified)

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

Time = 2.41 (sec) , antiderivative size = 233, normalized size of antiderivative = 2.45

method	result	size
trager	\(\frac {\ln \left (-\frac {\left (x^{4}-6 x^{2}+1\right )^{\frac {3}{4}} x -\sqrt {x^{4}-6 x^{2}+1}\, x^{2}+\left (x^{4}-6 x^{2}+1\right )^{\frac {1}{4}} x^{3}-x^{4}+\sqrt {x^{4}-6 x^{2}+1}-3 \left (x^{4}-6 x^{2}+1\right )^{\frac {1}{4}} x +5 x^{2}}{\left (-1+x \right ) \left (1+x \right )}\right )}{2}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \sqrt {x^{4}-6 x^{2}+1}\, x^{2}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{4}-\left (x^{4}-6 x^{2}+1\right )^{\frac {3}{4}} x +\left (x^{4}-6 x^{2}+1\right )^{\frac {1}{4}} x^{3}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \sqrt {x^{4}-6 x^{2}+1}+5 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{2}-3 \left (x^{4}-6 x^{2}+1\right )^{\frac {1}{4}} x}{\left (-1+x \right ) \left (1+x \right )}\right )}{2}\)	\(233\)

Input:

int((x^2+1)^2/(-x^2+1)/(x^4-6*x^2+1)^(3/4),x,method=_RETURNVERBOSE)
 

Output:

1/2*ln(-((x^4-6*x^2+1)^(3/4)*x-(x^4-6*x^2+1)^(1/2)*x^2+(x^4-6*x^2+1)^(1/4) 
*x^3-x^4+(x^4-6*x^2+1)^(1/2)-3*(x^4-6*x^2+1)^(1/4)*x+5*x^2)/(-1+x)/(1+x))+ 
1/2*RootOf(_Z^2+1)*ln((RootOf(_Z^2+1)*(x^4-6*x^2+1)^(1/2)*x^2-RootOf(_Z^2+ 
1)*x^4-(x^4-6*x^2+1)^(3/4)*x+(x^4-6*x^2+1)^(1/4)*x^3-RootOf(_Z^2+1)*(x^4-6 
*x^2+1)^(1/2)+5*RootOf(_Z^2+1)*x^2-3*(x^4-6*x^2+1)^(1/4)*x)/(-1+x)/(1+x))
 

Fricas [F(-2)]

Exception generated. \[ \int \frac {\left (1+x^2\right )^2}{\left (1-x^2\right ) \left (1-6 x^2+x^4\right )^{3/4}} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((x^2+1)^2/(-x^2+1)/(x^4-6*x^2+1)^(3/4),x, algorithm="fricas")
 

Output:

Exception raised: TypeError >>  Error detected within library code:   inte 
grate: implementation incomplete (residue poly has multiple non-linear fac 
tors)
 

Sympy [F]

\[ \int \frac {\left (1+x^2\right )^2}{\left (1-x^2\right ) \left (1-6 x^2+x^4\right )^{3/4}} \, dx=- \int \frac {2 x^{2}}{x^{2} \left (x^{4} - 6 x^{2} + 1\right )^{\frac {3}{4}} - \left (x^{4} - 6 x^{2} + 1\right )^{\frac {3}{4}}}\, dx - \int \frac {x^{4}}{x^{2} \left (x^{4} - 6 x^{2} + 1\right )^{\frac {3}{4}} - \left (x^{4} - 6 x^{2} + 1\right )^{\frac {3}{4}}}\, dx - \int \frac {1}{x^{2} \left (x^{4} - 6 x^{2} + 1\right )^{\frac {3}{4}} - \left (x^{4} - 6 x^{2} + 1\right )^{\frac {3}{4}}}\, dx \] Input:

integrate((x**2+1)**2/(-x**2+1)/(x**4-6*x**2+1)**(3/4),x)
 

Output:

-Integral(2*x**2/(x**2*(x**4 - 6*x**2 + 1)**(3/4) - (x**4 - 6*x**2 + 1)**( 
3/4)), x) - Integral(x**4/(x**2*(x**4 - 6*x**2 + 1)**(3/4) - (x**4 - 6*x** 
2 + 1)**(3/4)), x) - Integral(1/(x**2*(x**4 - 6*x**2 + 1)**(3/4) - (x**4 - 
 6*x**2 + 1)**(3/4)), x)
 

Maxima [F]

\[ \int \frac {\left (1+x^2\right )^2}{\left (1-x^2\right ) \left (1-6 x^2+x^4\right )^{3/4}} \, dx=\int { -\frac {{\left (x^{2} + 1\right )}^{2}}{{\left (x^{4} - 6 \, x^{2} + 1\right )}^{\frac {3}{4}} {\left (x^{2} - 1\right )}} \,d x } \] Input:

integrate((x^2+1)^2/(-x^2+1)/(x^4-6*x^2+1)^(3/4),x, algorithm="maxima")
 

Output:

-integrate((x^2 + 1)^2/((x^4 - 6*x^2 + 1)^(3/4)*(x^2 - 1)), x)
 

Giac [F]

\[ \int \frac {\left (1+x^2\right )^2}{\left (1-x^2\right ) \left (1-6 x^2+x^4\right )^{3/4}} \, dx=\int { -\frac {{\left (x^{2} + 1\right )}^{2}}{{\left (x^{4} - 6 \, x^{2} + 1\right )}^{\frac {3}{4}} {\left (x^{2} - 1\right )}} \,d x } \] Input:

integrate((x^2+1)^2/(-x^2+1)/(x^4-6*x^2+1)^(3/4),x, algorithm="giac")
 

Output:

integrate(-(x^2 + 1)^2/((x^4 - 6*x^2 + 1)^(3/4)*(x^2 - 1)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (1+x^2\right )^2}{\left (1-x^2\right ) \left (1-6 x^2+x^4\right )^{3/4}} \, dx=\int -\frac {{\left (x^2+1\right )}^2}{\left (x^2-1\right )\,{\left (x^4-6\,x^2+1\right )}^{3/4}} \,d x \] Input:

int(-(x^2 + 1)^2/((x^2 - 1)*(x^4 - 6*x^2 + 1)^(3/4)),x)
 

Output:

int(-(x^2 + 1)^2/((x^2 - 1)*(x^4 - 6*x^2 + 1)^(3/4)), x)
 

Reduce [F]

\[ \int \frac {\left (1+x^2\right )^2}{\left (1-x^2\right ) \left (1-6 x^2+x^4\right )^{3/4}} \, dx=-\left (\int \frac {x^{4}}{\left (x^{4}-6 x^{2}+1\right )^{\frac {3}{4}} x^{2}-\left (x^{4}-6 x^{2}+1\right )^{\frac {3}{4}}}d x \right )-2 \left (\int \frac {x^{2}}{\left (x^{4}-6 x^{2}+1\right )^{\frac {3}{4}} x^{2}-\left (x^{4}-6 x^{2}+1\right )^{\frac {3}{4}}}d x \right )-\left (\int \frac {1}{\left (x^{4}-6 x^{2}+1\right )^{\frac {3}{4}} x^{2}-\left (x^{4}-6 x^{2}+1\right )^{\frac {3}{4}}}d x \right ) \] Input:

int((x^2+1)^2/(-x^2+1)/(x^4-6*x^2+1)^(3/4),x)
 

Output:

 - int(x**4/((x**4 - 6*x**2 + 1)**(3/4)*x**2 - (x**4 - 6*x**2 + 1)**(3/4)) 
,x) - 2*int(x**2/((x**4 - 6*x**2 + 1)**(3/4)*x**2 - (x**4 - 6*x**2 + 1)**( 
3/4)),x) - int(1/((x**4 - 6*x**2 + 1)**(3/4)*x**2 - (x**4 - 6*x**2 + 1)**( 
3/4)),x)