\(\int \frac {b^{12}+a^{12} x^{12}}{\sqrt {-b^4+a^4 x^4} (-b^{12}+a^{12} x^{12})} \, dx\) [2973]

Optimal result

Integrand size = 44, antiderivative size = 376 \[ \int \frac {b^{12}+a^{12} x^{12}}{\sqrt {-b^4+a^4 x^4} \left (-b^{12}+a^{12} x^{12}\right )} \, dx=-\frac {x}{3 \sqrt {-b^4+a^4 x^4}}+\frac {1}{6} \text {RootSum}\left [16 a^8 b^8+32 i a^6 b^6 \text {$\#$1}^2+24 a^4 b^4 \text {$\#$1}^4-8 i a^2 b^2 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {8 a^6 b^6 \log (x)-8 a^6 b^6 \log \left (i b^2+a^2 x^2+\sqrt {-b^4+a^4 x^4}-x \text {$\#$1}\right )+4 i a^4 b^4 \log (x) \text {$\#$1}^2-4 i a^4 b^4 \log \left (i b^2+a^2 x^2+\sqrt {-b^4+a^4 x^4}-x \text {$\#$1}\right ) \text {$\#$1}^2+2 a^2 b^2 \log (x) \text {$\#$1}^4-2 a^2 b^2 \log \left (i b^2+a^2 x^2+\sqrt {-b^4+a^4 x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4+i \log (x) \text {$\#$1}^6-i \log \left (i b^2+a^2 x^2+\sqrt {-b^4+a^4 x^4}-x \text {$\#$1}\right ) \text {$\#$1}^6}{8 a^6 b^6 \text {$\#$1}-12 i a^4 b^4 \text {$\#$1}^3-6 a^2 b^2 \text {$\#$1}^5-i \text {$\#$1}^7}\&\right ] \] Output:

Unintegrable
 

Mathematica [A] (verified)

Time = 1.65 (sec) , antiderivative size = 517, normalized size of antiderivative = 1.38 \[ \int \frac {b^{12}+a^{12} x^{12}}{\sqrt {-b^4+a^4 x^4} \left (-b^{12}+a^{12} x^{12}\right )} \, dx=-\frac {x}{3 \sqrt {-b^4+a^4 x^4}}+\left (\frac {1}{6}-\frac {i}{6}\right ) \text {RootSum}\left [4 a^4 b^4+(4-4 i) a^3 b^3 \text {$\#$1}-(2+2 i) a b \text {$\#$1}^3-\text {$\#$1}^4\&,\frac {-2 a^2 b^2 \log (x)+2 a^2 b^2 \log \left (i b^2+a^2 x^2+\sqrt {-b^4+a^4 x^4}-x \text {$\#$1}\right )-(1-i) a b \log (x) \text {$\#$1}+(1-i) a b \log \left (i b^2+a^2 x^2+\sqrt {-b^4+a^4 x^4}-x \text {$\#$1}\right ) \text {$\#$1}+i \log (x) \text {$\#$1}^2-i \log \left (i b^2+a^2 x^2+\sqrt {-b^4+a^4 x^4}-x \text {$\#$1}\right ) \text {$\#$1}^2}{2 a^3 b^3-3 i a b \text {$\#$1}^2-(1+i) \text {$\#$1}^3}\&\right ]-\left (\frac {1}{6}-\frac {i}{6}\right ) \text {RootSum}\left [4 a^4 b^4-(4-4 i) a^3 b^3 \text {$\#$1}+(2+2 i) a b \text {$\#$1}^3-\text {$\#$1}^4\&,\frac {-2 a^2 b^2 \log (x)+2 a^2 b^2 \log \left (i b^2+a^2 x^2+\sqrt {-b^4+a^4 x^4}-x \text {$\#$1}\right )+(1-i) a b \log (x) \text {$\#$1}-(1-i) a b \log \left (i b^2+a^2 x^2+\sqrt {-b^4+a^4 x^4}-x \text {$\#$1}\right ) \text {$\#$1}+i \log (x) \text {$\#$1}^2-i \log \left (i b^2+a^2 x^2+\sqrt {-b^4+a^4 x^4}-x \text {$\#$1}\right ) \text {$\#$1}^2}{2 a^3 b^3-3 i a b \text {$\#$1}^2+(1+i) \text {$\#$1}^3}\&\right ] \] 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] + (1/6 - I/6)*RootSum[4*a^4*b^4 + (4 - 4*I)*a^ 
3*b^3*#1 - (2 + 2*I)*a*b*#1^3 - #1^4 & , (-2*a^2*b^2*Log[x] + 2*a^2*b^2*Lo 
g[I*b^2 + a^2*x^2 + Sqrt[-b^4 + a^4*x^4] - x*#1] - (1 - I)*a*b*Log[x]*#1 + 
 (1 - I)*a*b*Log[I*b^2 + a^2*x^2 + Sqrt[-b^4 + a^4*x^4] - x*#1]*#1 + I*Log 
[x]*#1^2 - I*Log[I*b^2 + a^2*x^2 + Sqrt[-b^4 + a^4*x^4] - x*#1]*#1^2)/(2*a 
^3*b^3 - (3*I)*a*b*#1^2 - (1 + I)*#1^3) & ] - (1/6 - I/6)*RootSum[4*a^4*b^ 
4 - (4 - 4*I)*a^3*b^3*#1 + (2 + 2*I)*a*b*#1^3 - #1^4 & , (-2*a^2*b^2*Log[x 
] + 2*a^2*b^2*Log[I*b^2 + a^2*x^2 + Sqrt[-b^4 + a^4*x^4] - x*#1] + (1 - I) 
*a*b*Log[x]*#1 - (1 - I)*a*b*Log[I*b^2 + a^2*x^2 + Sqrt[-b^4 + a^4*x^4] - 
x*#1]*#1 + I*Log[x]*#1^2 - I*Log[I*b^2 + a^2*x^2 + Sqrt[-b^4 + a^4*x^4] - 
x*#1]*#1^2)/(2*a^3*b^3 - (3*I)*a*b*#1^2 + (1 + I)*#1^3) & ]
 

Rubi [A] (warning: unable to verify)

Time = 4.96 (sec) , antiderivative size = 455, normalized size of antiderivative = 1.21, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, 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.

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

\(\Big \downarrow \) 7276

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {2 b \sqrt {1-\frac {a^4 x^4}{b^4}} \operatorname {EllipticF}\left (\arcsin \left (\frac {a x}{b}\right ),-1\right )}{3 a \sqrt {a^4 x^4-b^4}}+\frac {b \sqrt {1-\frac {a^4 x^4}{b^4}} E\left (\left .\arcsin \left (\frac {a x}{b}\right )\right |-1\right )}{6 a \sqrt {a^4 x^4-b^4}}-\frac {b \sqrt {1-\frac {a^4 x^4}{b^4}} \operatorname {EllipticPi}\left (-\sqrt [3]{-1},\arcsin \left (\frac {a x}{b}\right ),-1\right )}{3 a \sqrt {a^4 x^4-b^4}}-\frac {b \sqrt {1-\frac {a^4 x^4}{b^4}} \operatorname {EllipticPi}\left ((-1)^{2/3},\arcsin \left (\frac {a x}{b}\right ),-1\right )}{3 a \sqrt {a^4 x^4-b^4}}-\frac {b \sqrt {1-\frac {a^4 x^4}{b^4}} \operatorname {EllipticPi}\left (\frac {\sqrt [3]{-1} a^4}{\left (-a^6\right )^{2/3}},\arcsin \left (\frac {a x}{b}\right ),-1\right )}{3 a \sqrt {a^4 x^4-b^4}}-\frac {x \left (a^2 x^2+b^2\right )}{6 b^2 \sqrt {a^4 x^4-b^4}}-\frac {b \sqrt {1-\frac {a^4 x^4}{b^4}} \operatorname {EllipticPi}\left (\frac {\sqrt [3]{-a^6}}{a^2},\arcsin \left (\frac {a x}{b}\right ),-1\right )}{3 a \sqrt {a^4 x^4-b^4}}-\frac {b \sqrt {1-\frac {a^4 x^4}{b^4}} \operatorname {EllipticPi}\left (\frac {(-1)^{2/3} \sqrt [3]{-a^6}}{a^2},\arcsin \left (\frac {a x}{b}\right ),-1\right )}{3 a \sqrt {a^4 x^4-b^4}}\)

Input:

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

Output:

-1/6*(x*(b^2 + a^2*x^2))/(b^2*Sqrt[-b^4 + a^4*x^4]) + (b*Sqrt[1 - (a^4*x^4 
)/b^4]*EllipticE[ArcSin[(a*x)/b], -1])/(6*a*Sqrt[-b^4 + a^4*x^4]) + (2*b*S 
qrt[1 - (a^4*x^4)/b^4]*EllipticF[ArcSin[(a*x)/b], -1])/(3*a*Sqrt[-b^4 + a^ 
4*x^4]) - (b*Sqrt[1 - (a^4*x^4)/b^4]*EllipticPi[-(-1)^(1/3), ArcSin[(a*x)/ 
b], -1])/(3*a*Sqrt[-b^4 + a^4*x^4]) - (b*Sqrt[1 - (a^4*x^4)/b^4]*EllipticP 
i[(-1)^(2/3), ArcSin[(a*x)/b], -1])/(3*a*Sqrt[-b^4 + a^4*x^4]) - (b*Sqrt[1 
 - (a^4*x^4)/b^4]*EllipticPi[((-1)^(1/3)*a^4)/(-a^6)^(2/3), ArcSin[(a*x)/b 
], -1])/(3*a*Sqrt[-b^4 + a^4*x^4]) - (b*Sqrt[1 - (a^4*x^4)/b^4]*EllipticPi 
[(-a^6)^(1/3)/a^2, ArcSin[(a*x)/b], -1])/(3*a*Sqrt[-b^4 + a^4*x^4]) - (b*S 
qrt[1 - (a^4*x^4)/b^4]*EllipticPi[((-1)^(2/3)*(-a^6)^(1/3))/a^2, ArcSin[(a 
*x)/b], -1])/(3*a*Sqrt[-b^4 + a^4*x^4])
 

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]
 

Maple [N/A] (verified)

Time = 15.18 (sec) , antiderivative size = 276, normalized size of antiderivative = 0.73

method	result	size
elliptic	\(\frac {\left (-\frac {\sqrt {2}\, x}{3 \sqrt {a^{4} x^{4}-b^{4}}}+\frac {\ln \left (\frac {\frac {a^{4} x^{4}-b^{4}}{2 x^{2}}-\frac {\sqrt {\sqrt {3}\, \sqrt {a^{4} b^{4}}}\, \sqrt {a^{4} x^{4}-b^{4}}\, \sqrt {2}}{2 x}+\frac {\sqrt {3}\, \sqrt {a^{4} b^{4}}}{2}}{\frac {a^{4} x^{4}-b^{4}}{2 x^{2}}+\frac {\sqrt {\sqrt {3}\, \sqrt {a^{4} b^{4}}}\, \sqrt {a^{4} x^{4}-b^{4}}\, \sqrt {2}}{2 x}+\frac {\sqrt {3}\, \sqrt {a^{4} b^{4}}}{2}}\right )+2 \arctan \left (\frac {\sqrt {a^{4} x^{4}-b^{4}}\, \sqrt {2}}{\sqrt {\sqrt {3}\, \sqrt {a^{4} b^{4}}}\, x}+1\right )+2 \arctan \left (\frac {\sqrt {a^{4} x^{4}-b^{4}}\, \sqrt {2}}{\sqrt {\sqrt {3}\, \sqrt {a^{4} b^{4}}}\, x}-1\right )}{6 \sqrt {\sqrt {3}\, \sqrt {a^{4} b^{4}}}}\right ) \sqrt {2}}{2}\)	\(276\)
default	\(\frac {4 a^{4} \left (2 \sqrt {6}\, \sqrt {\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {i a^{2} b^{2}}\, a b}\, \sqrt {\left (-\frac {1}{2}-\frac {i}{2}\right ) \sqrt {i a^{2} b^{2}}\, a b}\, \sqrt {a^{4} x^{4}-b^{4}}\, x +\left (a x +b \right ) \left (a x -b \right ) \left (\left (\ln \left (\frac {\left (2 \,6^{\frac {1}{4}} \sqrt {\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {i a^{2} b^{2}}\, a b}\, \sqrt {a^{4} x^{4}-b^{4}}-\sqrt {6}\, \left (a^{2} x^{2}+i b^{2}\right ) \sqrt {i a^{2} b^{2}}+\left (\left (-1-i\right ) x^{2} a^{2}-4 i a b x +\left (1-i\right ) b^{2}\right ) a b \right ) a^{2}}{2 a^{2} x^{2}+\left (1+i\right ) x b a +\sqrt {6}\, \sqrt {i a^{2} b^{2}}\, x +2 i b^{2}}\right )+\ln \left (-\frac {a^{2} \left (2 \,6^{\frac {1}{4}} \sqrt {\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {i a^{2} b^{2}}\, a b}\, \sqrt {a^{4} x^{4}-b^{4}}+\sqrt {6}\, \left (a^{2} x^{2}+i b^{2}\right ) \sqrt {i a^{2} b^{2}}+a \left (\left (1+i\right ) x^{2} a^{2}-4 i a b x +\left (-1+i\right ) b^{2}\right ) b \right )}{-2 a^{2} x^{2}+\left (1+i\right ) x b a +\sqrt {6}\, \sqrt {i a^{2} b^{2}}\, x -2 i b^{2}}\right )+2 \ln \left (2\right )\right ) \sqrt {\left (-\frac {1}{2}-\frac {i}{2}\right ) \sqrt {i a^{2} b^{2}}\, a b}+\sqrt {\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {i a^{2} b^{2}}\, a b}\, \left (\ln \left (\frac {a^{2} \left (2 \,6^{\frac {1}{4}} \sqrt {\left (-\frac {1}{2}-\frac {i}{2}\right ) \sqrt {i a^{2} b^{2}}\, a b}\, \sqrt {a^{4} x^{4}-b^{4}}+\sqrt {6}\, \left (a^{2} x^{2}+i b^{2}\right ) \sqrt {i a^{2} b^{2}}+\left (\left (-1-i\right ) x^{2} a^{2}-4 i a b x +\left (1-i\right ) b^{2}\right ) a b \right )}{2 a^{2} x^{2}+\left (1+i\right ) x b a +2 i b^{2}-\sqrt {6}\, \sqrt {i a^{2} b^{2}}\, x}\right )+\ln \left (\frac {a^{2} \left (2 \,6^{\frac {1}{4}} \sqrt {\left (-\frac {1}{2}-\frac {i}{2}\right ) \sqrt {i a^{2} b^{2}}\, a b}\, \sqrt {a^{4} x^{4}-b^{4}}-\sqrt {6}\, \left (a^{2} x^{2}+i b^{2}\right ) \sqrt {i a^{2} b^{2}}+a \left (\left (1+i\right ) x^{2} a^{2}-4 i a b x +\left (-1+i\right ) b^{2}\right ) b \right )}{2 a^{2} x^{2}+\left (-1-i\right ) x b a +\sqrt {6}\, \sqrt {i a^{2} b^{2}}\, x +2 i b^{2}}\right )+2 \ln \left (2\right )\right )\right ) \left (a^{2} x^{2}+b^{2}\right ) 6^{\frac {1}{4}}\right ) \sqrt {6}\, b^{4}}{27 \sqrt {\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {i a^{2} b^{2}}\, a b}\, \sqrt {\left (-\frac {1}{2}-\frac {i}{2}\right ) \sqrt {i a^{2} b^{2}}\, a b}\, \left (a x +b \right ) \left (\sqrt {6}\, \sqrt {i a^{2} b^{2}}+\left (1+i\right ) a b \right ) \left (\frac {\sqrt {6}\, \sqrt {i a^{2} b^{2}}}{3}+\left (1+i\right ) a b \right ) \left (a x -b \right ) \left (-i \sqrt {6}\, \sqrt {i a^{2} b^{2}}+\left (-1+i\right ) a b \right ) \left (a x +i b \right ) \left (i a x +b \right ) \left (-\frac {\sqrt {6}\, \sqrt {i a^{2} b^{2}}}{3}+\left (1+i\right ) a b \right )}\)	\(891\)
pseudoelliptic	\(\frac {4 a^{4} \left (2 \sqrt {6}\, \sqrt {\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {i a^{2} b^{2}}\, a b}\, \sqrt {\left (-\frac {1}{2}-\frac {i}{2}\right ) \sqrt {i a^{2} b^{2}}\, a b}\, \sqrt {a^{4} x^{4}-b^{4}}\, x +\left (a x +b \right ) \left (a x -b \right ) \left (\left (\ln \left (\frac {\left (2 \,6^{\frac {1}{4}} \sqrt {\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {i a^{2} b^{2}}\, a b}\, \sqrt {a^{4} x^{4}-b^{4}}-\sqrt {6}\, \left (a^{2} x^{2}+i b^{2}\right ) \sqrt {i a^{2} b^{2}}+\left (\left (-1-i\right ) x^{2} a^{2}-4 i a b x +\left (1-i\right ) b^{2}\right ) a b \right ) a^{2}}{2 a^{2} x^{2}+\left (1+i\right ) x b a +\sqrt {6}\, \sqrt {i a^{2} b^{2}}\, x +2 i b^{2}}\right )+\ln \left (-\frac {a^{2} \left (2 \,6^{\frac {1}{4}} \sqrt {\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {i a^{2} b^{2}}\, a b}\, \sqrt {a^{4} x^{4}-b^{4}}+\sqrt {6}\, \left (a^{2} x^{2}+i b^{2}\right ) \sqrt {i a^{2} b^{2}}+a \left (\left (1+i\right ) x^{2} a^{2}-4 i a b x +\left (-1+i\right ) b^{2}\right ) b \right )}{-2 a^{2} x^{2}+\left (1+i\right ) x b a +\sqrt {6}\, \sqrt {i a^{2} b^{2}}\, x -2 i b^{2}}\right )+2 \ln \left (2\right )\right ) \sqrt {\left (-\frac {1}{2}-\frac {i}{2}\right ) \sqrt {i a^{2} b^{2}}\, a b}+\sqrt {\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {i a^{2} b^{2}}\, a b}\, \left (\ln \left (\frac {a^{2} \left (2 \,6^{\frac {1}{4}} \sqrt {\left (-\frac {1}{2}-\frac {i}{2}\right ) \sqrt {i a^{2} b^{2}}\, a b}\, \sqrt {a^{4} x^{4}-b^{4}}+\sqrt {6}\, \left (a^{2} x^{2}+i b^{2}\right ) \sqrt {i a^{2} b^{2}}+\left (\left (-1-i\right ) x^{2} a^{2}-4 i a b x +\left (1-i\right ) b^{2}\right ) a b \right )}{2 a^{2} x^{2}+\left (1+i\right ) x b a +2 i b^{2}-\sqrt {6}\, \sqrt {i a^{2} b^{2}}\, x}\right )+\ln \left (\frac {a^{2} \left (2 \,6^{\frac {1}{4}} \sqrt {\left (-\frac {1}{2}-\frac {i}{2}\right ) \sqrt {i a^{2} b^{2}}\, a b}\, \sqrt {a^{4} x^{4}-b^{4}}-\sqrt {6}\, \left (a^{2} x^{2}+i b^{2}\right ) \sqrt {i a^{2} b^{2}}+a \left (\left (1+i\right ) x^{2} a^{2}-4 i a b x +\left (-1+i\right ) b^{2}\right ) b \right )}{2 a^{2} x^{2}+\left (-1-i\right ) x b a +\sqrt {6}\, \sqrt {i a^{2} b^{2}}\, x +2 i b^{2}}\right )+2 \ln \left (2\right )\right )\right ) \left (a^{2} x^{2}+b^{2}\right ) 6^{\frac {1}{4}}\right ) \sqrt {6}\, b^{4}}{27 \sqrt {\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {i a^{2} b^{2}}\, a b}\, \sqrt {\left (-\frac {1}{2}-\frac {i}{2}\right ) \sqrt {i a^{2} b^{2}}\, a b}\, \left (a x +b \right ) \left (\sqrt {6}\, \sqrt {i a^{2} b^{2}}+\left (1+i\right ) a b \right ) \left (\frac {\sqrt {6}\, \sqrt {i a^{2} b^{2}}}{3}+\left (1+i\right ) a b \right ) \left (a x -b \right ) \left (-i \sqrt {6}\, \sqrt {i a^{2} b^{2}}+\left (-1+i\right ) a b \right ) \left (a x +i b \right ) \left (i a x +b \right ) \left (-\frac {\sqrt {6}\, \sqrt {i a^{2} b^{2}}}{3}+\left (1+i\right ) a b \right )}\)	\(891\)

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

Fricas [C] (verification not implemented)

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

Time = 8.03 (sec) , antiderivative size = 796, normalized size of antiderivative = 2.12 \[ \int \frac {b^{12}+a^{12} x^{12}}{\sqrt {-b^4+a^4 x^4} \left (-b^{12}+a^{12} x^{12}\right )} \, dx =\text {Too large to display} \] 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:

-1/144*(48*sqrt(a^4*x^4 - b^4)*a*b*x + 2*12^(3/4)*(a^4*x^4 - b^4)*arctan(( 
a^16*x^16 + 2*a^12*b^4*x^12 + 3*a^8*b^8*x^8 + 2*a^4*b^12*x^4 + b^16 + 4*sq 
rt(3)*(a^14*b^2*x^14 - a^2*b^14*x^2) + sqrt(a^4*x^4 - b^4)*(3*12^(3/4)*(a^ 
11*b^3*x^11 - 3*a^7*b^7*x^7 + a^3*b^11*x^3) + 2*12^(1/4)*(a^13*b*x^13 - 12 
*a^9*b^5*x^9 + 12*a^5*b^9*x^5 - a*b^13*x)))/(a^16*x^16 - 46*a^12*b^4*x^12 
+ 99*a^8*b^8*x^8 - 46*a^4*b^12*x^4 + b^16)) + 2*12^(3/4)*(a^4*x^4 - b^4)*a 
rctan(-(a^16*x^16 + 2*a^12*b^4*x^12 + 3*a^8*b^8*x^8 + 2*a^4*b^12*x^4 + b^1 
6 + 4*sqrt(3)*(a^14*b^2*x^14 - a^2*b^14*x^2) - sqrt(a^4*x^4 - b^4)*(3*12^( 
3/4)*(a^11*b^3*x^11 - 3*a^7*b^7*x^7 + a^3*b^11*x^3) + 2*12^(1/4)*(a^13*b*x 
^13 - 12*a^9*b^5*x^9 + 12*a^5*b^9*x^5 - a*b^13*x)))/(a^16*x^16 - 46*a^12*b 
^4*x^12 + 99*a^8*b^8*x^8 - 46*a^4*b^12*x^4 + b^16)) + 12^(3/4)*(a^4*x^4 - 
b^4)*log(2*(12*a^6*b^2*x^6 - 12*a^2*b^6*x^2 + sqrt(3)*(a^8*x^8 + a^4*b^4*x 
^4 + b^8) + (6*12^(1/4)*a^3*b^3*x^3 + 12^(3/4)*(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)) - 12^(3/4)*(a^4*x^4 - b^4) 
*log(2*(12*a^6*b^2*x^6 - 12*a^2*b^6*x^2 + sqrt(3)*(a^8*x^8 + a^4*b^4*x^4 + 
 b^8) - (6*12^(1/4)*a^3*b^3*x^3 + 12^(3/4)*(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)))/(a^5*b*x^4 - a*b^5)
 

Sympy [N/A]

Not integrable

Time = 34.56 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.34 \[ \int \frac {b^{12}+a^{12} x^{12}}{\sqrt {-b^4+a^4 x^4} \left (-b^{12}+a^{12} x^{12}\right )} \, dx=\int \frac {\left (a^{4} x^{4} + b^{4}\right ) \left (a^{8} x^{8} - a^{4} b^{4} x^{4} + b^{8}\right )}{\sqrt {\left (a x - b\right ) \left (a x + b\right ) \left (a^{2} x^{2} + b^{2}\right )} \left (a x - b\right ) \left (a x + b\right ) \left (a^{2} x^{2} + b^{2}\right ) \left (a^{2} x^{2} - a b x + b^{2}\right ) \left (a^{2} x^{2} + a b x + b^{2}\right ) \left (a^{4} x^{4} - a^{2} b^{2} x^{2} + b^{4}\right )}\, dx \] Input:

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

Output:

Integral((a**4*x**4 + b**4)*(a**8*x**8 - a**4*b**4*x**4 + b**8)/(sqrt((a*x 
 - b)*(a*x + b)*(a**2*x**2 + b**2))*(a*x - b)*(a*x + b)*(a**2*x**2 + b**2) 
*(a**2*x**2 - a*b*x + b**2)*(a**2*x**2 + a*b*x + b**2)*(a**4*x**4 - a**2*b 
**2*x**2 + b**4)), x)
 

Maxima [N/A]

Not integrable

Time = 0.28 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.12 \[ \int \frac {b^{12}+a^{12} x^{12}}{\sqrt {-b^4+a^4 x^4} \left (-b^{12}+a^{12} x^{12}\right )} \, dx=\int { \frac {a^{12} x^{12} + b^{12}}{{\left (a^{12} x^{12} - b^{12}\right )} \sqrt {a^{4} x^{4} - b^{4}}} \,d x } \] 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 = 0.54 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.12 \[ \int \frac {b^{12}+a^{12} x^{12}}{\sqrt {-b^4+a^4 x^4} \left (-b^{12}+a^{12} x^{12}\right )} \, dx=\int { \frac {a^{12} x^{12} + b^{12}}{{\left (a^{12} x^{12} - b^{12}\right )} \sqrt {a^{4} x^{4} - b^{4}}} \,d x } \] 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 = 10.81 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.12 \[ \int \frac {b^{12}+a^{12} x^{12}}{\sqrt {-b^4+a^4 x^4} \left (-b^{12}+a^{12} x^{12}\right )} \, dx=\int -\frac {a^{12}\,x^{12}+b^{12}}{\sqrt {a^4\,x^4-b^4}\,\left (b^{12}-a^{12}\,x^{12}\right )} \,d x \] Input:

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

Output:

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

Reduce [N/A]

Not integrable

Time = 0.57 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.31 \[ \int \frac {b^{12}+a^{12} x^{12}}{\sqrt {-b^4+a^4 x^4} \left (-b^{12}+a^{12} x^{12}\right )} \, dx=\left (\int \frac {\sqrt {a^{4} x^{4}-b^{4}}}{a^{16} x^{16}-a^{12} b^{4} x^{12}-a^{4} b^{12} x^{4}+b^{16}}d x \right ) b^{12}+\left (\int \frac {\sqrt {a^{4} x^{4}-b^{4}}\, x^{12}}{a^{16} x^{16}-a^{12} b^{4} x^{12}-a^{4} b^{12} x^{4}+b^{16}}d x \right ) a^{12} \] 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