\(\int \frac {-b^{12}+a^{12} x^{12}}{\sqrt {b^4+a^4 x^4} (b^{12}+a^{12} x^{12})} \, dx\) [2920]

Optimal result

Integrand size = 42, antiderivative size = 333 \[ \int \frac {-b^{12}+a^{12} x^{12}}{\sqrt {b^4+a^4 x^4} \left (b^{12}+a^{12} x^{12}\right )} \, dx=-\frac {x}{3 \sqrt {b^4+a^4 x^4}}+\frac {1}{6} \text {RootSum}\left [16 a^8 b^8-32 a^6 b^6 \text {$\#$1}^2-24 a^4 b^4 \text {$\#$1}^4-8 a^2 b^2 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {8 a^6 b^6 \log (x)-8 a^6 b^6 \log \left (b^2+a^2 x^2+\sqrt {b^4+a^4 x^4}-x \text {$\#$1}\right )-4 a^4 b^4 \log (x) \text {$\#$1}^2+4 a^4 b^4 \log \left (b^2+a^2 x^2+\sqrt {b^4+a^4 x^4}-x \text {$\#$1}\right ) \text {$\#$1}^2-2 a^2 b^2 \log (x) \text {$\#$1}^4+2 a^2 b^2 \log \left (b^2+a^2 x^2+\sqrt {b^4+a^4 x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4+\log (x) \text {$\#$1}^6-\log \left (b^2+a^2 x^2+\sqrt {b^4+a^4 x^4}-x \text {$\#$1}\right ) \text {$\#$1}^6}{8 a^6 b^6 \text {$\#$1}+12 a^4 b^4 \text {$\#$1}^3+6 a^2 b^2 \text {$\#$1}^5-\text {$\#$1}^7}\&\right ] \] Output:

Unintegrable
 

Mathematica [A] (verified)

Time = 1.05 (sec) , antiderivative size = 331, normalized size of antiderivative = 0.99 \[ \int \frac {-b^{12}+a^{12} x^{12}}{\sqrt {b^4+a^4 x^4} \left (b^{12}+a^{12} x^{12}\right )} \, dx=-\frac {x}{3 \sqrt {b^4+a^4 x^4}}+\frac {1}{6} \text {RootSum}\left [16 a^8 b^8-32 a^6 b^6 \text {$\#$1}^2-24 a^4 b^4 \text {$\#$1}^4-8 a^2 b^2 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {-8 a^6 b^6 \log (x)+8 a^6 b^6 \log \left (b^2+a^2 x^2+\sqrt {b^4+a^4 x^4}-x \text {$\#$1}\right )+4 a^4 b^4 \log (x) \text {$\#$1}^2-4 a^4 b^4 \log \left (b^2+a^2 x^2+\sqrt {b^4+a^4 x^4}-x \text {$\#$1}\right ) \text {$\#$1}^2+2 a^2 b^2 \log (x) \text {$\#$1}^4-2 a^2 b^2 \log \left (b^2+a^2 x^2+\sqrt {b^4+a^4 x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4-\log (x) \text {$\#$1}^6+\log \left (b^2+a^2 x^2+\sqrt {b^4+a^4 x^4}-x \text {$\#$1}\right ) \text {$\#$1}^6}{-8 a^6 b^6 \text {$\#$1}-12 a^4 b^4 \text {$\#$1}^3-6 a^2 b^2 \text {$\#$1}^5+\text {$\#$1}^7}\&\right ] \] Input:

Integrate[(-b^12 + a^12*x^12)/(Sqrt[b^4 + a^4*x^4]*(b^12 + a^12*x^12)),x]
 

Output:

-1/3*x/Sqrt[b^4 + a^4*x^4] + RootSum[16*a^8*b^8 - 32*a^6*b^6*#1^2 - 24*a^4 
*b^4*#1^4 - 8*a^2*b^2*#1^6 + #1^8 & , (-8*a^6*b^6*Log[x] + 8*a^6*b^6*Log[b 
^2 + a^2*x^2 + Sqrt[b^4 + a^4*x^4] - x*#1] + 4*a^4*b^4*Log[x]*#1^2 - 4*a^4 
*b^4*Log[b^2 + a^2*x^2 + Sqrt[b^4 + a^4*x^4] - x*#1]*#1^2 + 2*a^2*b^2*Log[ 
x]*#1^4 - 2*a^2*b^2*Log[b^2 + a^2*x^2 + Sqrt[b^4 + a^4*x^4] - x*#1]*#1^4 - 
 Log[x]*#1^6 + Log[b^2 + a^2*x^2 + Sqrt[b^4 + a^4*x^4] - x*#1]*#1^6)/(-8*a 
^6*b^6*#1 - 12*a^4*b^4*#1^3 - 6*a^2*b^2*#1^5 + #1^7) & ]/6
 

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.

\(\displaystyle \int \frac {a^{12} x^{12}-b^{12}}{\sqrt {a^4 x^4+b^4} \left (a^{12} x^{12}+b^{12}\right )} \, dx\)

\(\Big \downarrow \) 7276

\(\displaystyle \int \left (\frac {1}{\sqrt {a^4 x^4+b^4}}-\frac {2 b^{12}}{\sqrt {a^4 x^4+b^4} \left (a^{12} x^{12}+b^{12}\right )}\right )dx\)

\(\Big \downarrow \) 7299

\(\displaystyle \int \left (\frac {1}{\sqrt {a^4 x^4+b^4}}-\frac {2 b^{12}}{\sqrt {a^4 x^4+b^4} \left (a^{12} x^{12}+b^{12}\right )}\right )dx\)

Input:

Int[(-b^12 + a^12*x^12)/(Sqrt[b^4 + a^4*x^4]*(b^12 + a^12*x^12)),x]
 

Output:

$Aborted
 

Maple [N/A] (verified)

Time = 4.72 (sec) , antiderivative size = 172, normalized size of antiderivative = 0.52

method	result	size
elliptic	\(\frac {\left (-\frac {\sqrt {2}\, x}{3 \sqrt {a^{4} x^{4}+b^{4}}}+\frac {\sqrt {2}\, \left (2 \arctan \left (\frac {\sqrt {a^{4} x^{4}+b^{4}}}{x \sqrt {\sqrt {3}\, \sqrt {a^{4} b^{4}}}}\right )-\ln \left (\frac {\frac {\sqrt {a^{4} x^{4}+b^{4}}\, \sqrt {2}}{2 x}+\frac {\sqrt {2}\, \sqrt {\sqrt {3}\, \sqrt {a^{4} b^{4}}}}{2}}{\frac {\sqrt {a^{4} x^{4}+b^{4}}\, \sqrt {2}}{2 x}-\frac {\sqrt {2}\, \sqrt {\sqrt {3}\, \sqrt {a^{4} b^{4}}}}{2}}\right )\right )}{6 \sqrt {\sqrt {3}\, \sqrt {a^{4} b^{4}}}}\right ) \sqrt {2}}{2}\)	\(172\)
default	\(-\frac {6 \sqrt {a^{4} x^{4}+b^{4}}\, \left (a^{4} b^{4}\right )^{\frac {1}{4}} x -3^{\frac {3}{4}} \left (a^{4} x^{4}+b^{4}\right ) \left (2 \arctan \left (\frac {\sqrt {a^{4} x^{4}+b^{4}}\, 3^{\frac {3}{4}}}{3 x \left (a^{4} b^{4}\right )^{\frac {1}{4}}}\right )-\ln \left (\frac {x 3^{\frac {1}{4}} \left (a^{4} b^{4}\right )^{\frac {1}{4}}+\sqrt {a^{4} x^{4}+b^{4}}}{-x 3^{\frac {1}{4}} \left (a^{4} b^{4}\right )^{\frac {1}{4}}+\sqrt {a^{4} x^{4}+b^{4}}}\right )\right )}{18 \left (a^{4} b^{4}\right )^{\frac {1}{4}} \left (a^{2} x^{2}-\sqrt {2}\, \sqrt {a^{2} b^{2}}\, x +b^{2}\right ) \left (a^{2} x^{2}+\sqrt {2}\, \sqrt {a^{2} b^{2}}\, x +b^{2}\right )}\)	\(206\)
pseudoelliptic	\(-\frac {6 \sqrt {a^{4} x^{4}+b^{4}}\, \left (a^{4} b^{4}\right )^{\frac {1}{4}} x -3^{\frac {3}{4}} \left (a^{4} x^{4}+b^{4}\right ) \left (2 \arctan \left (\frac {\sqrt {a^{4} x^{4}+b^{4}}\, 3^{\frac {3}{4}}}{3 x \left (a^{4} b^{4}\right )^{\frac {1}{4}}}\right )-\ln \left (\frac {x 3^{\frac {1}{4}} \left (a^{4} b^{4}\right )^{\frac {1}{4}}+\sqrt {a^{4} x^{4}+b^{4}}}{-x 3^{\frac {1}{4}} \left (a^{4} b^{4}\right )^{\frac {1}{4}}+\sqrt {a^{4} x^{4}+b^{4}}}\right )\right )}{18 \left (a^{4} b^{4}\right )^{\frac {1}{4}} \left (a^{2} x^{2}-\sqrt {2}\, \sqrt {a^{2} b^{2}}\, x +b^{2}\right ) \left (a^{2} x^{2}+\sqrt {2}\, \sqrt {a^{2} b^{2}}\, x +b^{2}\right )}\)	\(206\)

Input:

int((a^12*x^12-b^12)/(a^4*x^4+b^4)^(1/2)/(a^12*x^12+b^12),x,method=_RETURN 
VERBOSE)
 

Output:

1/2*(-1/3/(a^4*x^4+b^4)^(1/2)*2^(1/2)*x+1/6*2^(1/2)/(3^(1/2)*(a^4*b^4)^(1/ 
2))^(1/2)*(2*arctan((a^4*x^4+b^4)^(1/2)/x/(3^(1/2)*(a^4*b^4)^(1/2))^(1/2)) 
-ln((1/2*(a^4*x^4+b^4)^(1/2)*2^(1/2)/x+1/2*2^(1/2)*(3^(1/2)*(a^4*b^4)^(1/2 
))^(1/2))/(1/2*(a^4*x^4+b^4)^(1/2)*2^(1/2)/x-1/2*2^(1/2)*(3^(1/2)*(a^4*b^4 
)^(1/2))^(1/2)))))*2^(1/2)
 

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 3 vs. order 1.

Time = 1.62 (sec) , antiderivative size = 402, normalized size of antiderivative = 1.21 \[ \int \frac {-b^{12}+a^{12} x^{12}}{\sqrt {b^4+a^4 x^4} \left (b^{12}+a^{12} x^{12}\right )} \, dx=-\frac {12 \, \sqrt {a^{4} x^{4} + b^{4}} a b x + 2 \cdot 3^{\frac {3}{4}} {\left (a^{4} x^{4} + b^{4}\right )} \arctan \left (\frac {2 \, {\left (3^{\frac {3}{4}} a^{3} b^{3} x^{3} + 3^{\frac {1}{4}} {\left (a^{5} b x^{5} + a b^{5} x\right )}\right )} \sqrt {a^{4} x^{4} + b^{4}}}{a^{8} x^{8} - a^{4} b^{4} x^{4} + b^{8}}\right ) + 3^{\frac {3}{4}} {\left (a^{4} x^{4} + b^{4}\right )} \log \left (-\frac {3^{\frac {3}{4}} {\left (a^{8} x^{8} + 5 \, a^{4} b^{4} x^{4} + b^{8}\right )} + 6 \, {\left (a^{5} b x^{5} + \sqrt {3} a^{3} b^{3} x^{3} + a b^{5} x\right )} \sqrt {a^{4} x^{4} + b^{4}} + 6 \cdot 3^{\frac {1}{4}} {\left (a^{6} b^{2} x^{6} + a^{2} b^{6} x^{2}\right )}}{a^{8} x^{8} - a^{4} b^{4} x^{4} + b^{8}}\right ) - 3^{\frac {3}{4}} {\left (a^{4} x^{4} + b^{4}\right )} \log \left (\frac {3^{\frac {3}{4}} {\left (a^{8} x^{8} + 5 \, a^{4} b^{4} x^{4} + b^{8}\right )} - 6 \, {\left (a^{5} b x^{5} + \sqrt {3} a^{3} b^{3} x^{3} + a b^{5} x\right )} \sqrt {a^{4} x^{4} + b^{4}} + 6 \cdot 3^{\frac {1}{4}} {\left (a^{6} b^{2} x^{6} + a^{2} b^{6} x^{2}\right )}}{a^{8} x^{8} - a^{4} b^{4} x^{4} + b^{8}}\right )}{36 \, {\left (a^{5} b x^{4} + a b^{5}\right )}} \] Input:

integrate((a^12*x^12-b^12)/(a^4*x^4+b^4)^(1/2)/(a^12*x^12+b^12),x, algorit 
hm="fricas")
 

Output:

-1/36*(12*sqrt(a^4*x^4 + b^4)*a*b*x + 2*3^(3/4)*(a^4*x^4 + b^4)*arctan(2*( 
3^(3/4)*a^3*b^3*x^3 + 3^(1/4)*(a^5*b*x^5 + a*b^5*x))*sqrt(a^4*x^4 + b^4)/( 
a^8*x^8 - a^4*b^4*x^4 + b^8)) + 3^(3/4)*(a^4*x^4 + b^4)*log(-(3^(3/4)*(a^8 
*x^8 + 5*a^4*b^4*x^4 + b^8) + 6*(a^5*b*x^5 + sqrt(3)*a^3*b^3*x^3 + a*b^5*x 
)*sqrt(a^4*x^4 + b^4) + 6*3^(1/4)*(a^6*b^2*x^6 + a^2*b^6*x^2))/(a^8*x^8 - 
a^4*b^4*x^4 + b^8)) - 3^(3/4)*(a^4*x^4 + b^4)*log((3^(3/4)*(a^8*x^8 + 5*a^ 
4*b^4*x^4 + b^8) - 6*(a^5*b*x^5 + sqrt(3)*a^3*b^3*x^3 + a*b^5*x)*sqrt(a^4* 
x^4 + b^4) + 6*3^(1/4)*(a^6*b^2*x^6 + a^2*b^6*x^2))/(a^8*x^8 - a^4*b^4*x^4 
 + b^8)))/(a^5*b*x^4 + a*b^5)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {-b^{12}+a^{12} x^{12}}{\sqrt {b^4+a^4 x^4} \left (b^{12}+a^{12} x^{12}\right )} \, dx=\text {Timed out} \] Input:

integrate((a**12*x**12-b**12)/(a**4*x**4+b**4)**(1/2)/(a**12*x**12+b**12), 
x)
 

Output:

Timed out
 

Maxima [N/A]

Not integrable

Time = 0.12 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.13 \[ \int \frac {-b^{12}+a^{12} x^{12}}{\sqrt {b^4+a^4 x^4} \left (b^{12}+a^{12} x^{12}\right )} \, dx=\int { \frac {a^{12} x^{12} - b^{12}}{{\left (a^{12} x^{12} + b^{12}\right )} \sqrt {a^{4} x^{4} + b^{4}}} \,d x } \] Input:

integrate((a^12*x^12-b^12)/(a^4*x^4+b^4)^(1/2)/(a^12*x^12+b^12),x, algorit 
hm="maxima")
 

Output:

integrate((a^12*x^12 - b^12)/((a^12*x^12 + b^12)*sqrt(a^4*x^4 + b^4)), x)
 

Giac [N/A]

Not integrable

Time = 0.59 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.13 \[ \int \frac {-b^{12}+a^{12} x^{12}}{\sqrt {b^4+a^4 x^4} \left (b^{12}+a^{12} x^{12}\right )} \, dx=\int { \frac {a^{12} x^{12} - b^{12}}{{\left (a^{12} x^{12} + b^{12}\right )} \sqrt {a^{4} x^{4} + b^{4}}} \,d x } \] Input:

integrate((a^12*x^12-b^12)/(a^4*x^4+b^4)^(1/2)/(a^12*x^12+b^12),x, algorit 
hm="giac")
 

Output:

integrate((a^12*x^12 - b^12)/((a^12*x^12 + b^12)*sqrt(a^4*x^4 + b^4)), x)
 

Mupad [N/A]

Not integrable

Time = 10.07 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.13 \[ \int \frac {-b^{12}+a^{12} x^{12}}{\sqrt {b^4+a^4 x^4} \left (b^{12}+a^{12} x^{12}\right )} \, dx=\int -\frac {b^{12}-a^{12}\,x^{12}}{\sqrt {a^4\,x^4+b^4}\,\left (a^{12}\,x^{12}+b^{12}\right )} \,d x \] Input:

int(-(b^12 - a^12*x^12)/((b^4 + a^4*x^4)^(1/2)*(b^12 + a^12*x^12)),x)
 

Output:

int(-(b^12 - a^12*x^12)/((b^4 + a^4*x^4)^(1/2)*(b^12 + a^12*x^12)), x)
 

Reduce [N/A]

Not integrable

Time = 1.37 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.33 \[ \int \frac {-b^{12}+a^{12} x^{12}}{\sqrt {b^4+a^4 x^4} \left (b^{12}+a^{12} x^{12}\right )} \, dx=-\left (\int \frac {\sqrt {a^{4} x^{4}+b^{4}}}{a^{16} x^{16}+a^{12} b^{4} x^{12}+a^{4} b^{12} x^{4}+b^{16}}d x \right ) b^{12}+\left (\int \frac {\sqrt {a^{4} x^{4}+b^{4}}\, x^{12}}{a^{16} x^{16}+a^{12} b^{4} x^{12}+a^{4} b^{12} x^{4}+b^{16}}d x \right ) a^{12} \] Input:

int((a^12*x^12-b^12)/(a^4*x^4+b^4)^(1/2)/(a^12*x^12+b^12),x)
 

Output:

 - int(sqrt(a**4*x**4 + b**4)/(a**16*x**16 + a**12*b**4*x**12 + a**4*b**12 
*x**4 + b**16),x)*b**12 + int((sqrt(a**4*x**4 + b**4)*x**12)/(a**16*x**16 
+ a**12*b**4*x**12 + a**4*b**12*x**4 + b**16),x)*a**12