Integrand size = 42, antiderivative size = 81 \[ \int \frac {-b^8+a^8 x^8}{\sqrt {b^4+a^4 x^4} \left (b^8+a^8 x^8\right )} \, dx=-\frac {\arctan \left (\frac {\sqrt [4]{2} a b x}{\sqrt {b^4+a^4 x^4}}\right )}{2 \sqrt [4]{2} a b}-\frac {\text {arctanh}\left (\frac {\sqrt [4]{2} a b x}{\sqrt {b^4+a^4 x^4}}\right )}{2 \sqrt [4]{2} a b} \]
-1/4*arctan(2^(1/4)*a*b*x/(a^4*x^4+b^4)^(1/2))*2^(3/4)/a/b-1/4*arctanh(2^( 1/4)*a*b*x/(a^4*x^4+b^4)^(1/2))*2^(3/4)/a/b
Time = 0.47 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.81 \[ \int \frac {-b^8+a^8 x^8}{\sqrt {b^4+a^4 x^4} \left (b^8+a^8 x^8\right )} \, dx=-\frac {\arctan \left (\frac {\sqrt [4]{2} a b x}{\sqrt {b^4+a^4 x^4}}\right )+\text {arctanh}\left (\frac {\sqrt [4]{2} a b x}{\sqrt {b^4+a^4 x^4}}\right )}{2 \sqrt [4]{2} a b} \]
-1/2*(ArcTan[(2^(1/4)*a*b*x)/Sqrt[b^4 + a^4*x^4]] + ArcTanh[(2^(1/4)*a*b*x )/Sqrt[b^4 + a^4*x^4]])/(2^(1/4)*a*b)
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 3.05 (sec) , antiderivative size = 1632, normalized size of antiderivative = 20.15, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {1388, 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.
| \(\displaystyle \int \frac {a^8 x^8-b^8}{\sqrt {a^4 x^4+b^4} \left (a^8 x^8+b^8\right )} \, dx\) |
| \(\Big \downarrow \) 1388 |
| \(\displaystyle \int \frac {\left (a^4 x^4-b^4\right ) \sqrt {a^4 x^4+b^4}}{a^8 x^8+b^8}dx\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \int \left (\frac {\sqrt {-a^8} \left (\sqrt {-a^8} b^4+a^4 b^4\right ) \sqrt {a^4 x^4+b^4}}{2 a^8 b^4 \left (\sqrt {-a^8} x^4+b^4\right )}-\frac {\sqrt {-a^8} \left (a^4 b^4-\sqrt {-a^8} b^4\right ) \sqrt {a^4 x^4+b^4}}{2 a^8 b^4 \left (b^4-\sqrt {-a^8} x^4\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\left (a^2+\sqrt [4]{-a^8}\right ) \left (b^2+a^2 x^2\right ) \sqrt {\frac {b^4+a^4 x^4}{\left (b^2+a^2 x^2\right )^2}} \operatorname {EllipticPi}\left (\frac {\left (a^2+\sqrt [4]{-a^8}\right )^2}{4 a^2 \sqrt [4]{-a^8}},2 \arctan \left (\frac {a x}{b}\right ),\frac {1}{2}\right ) \left (a^2-\sqrt [4]{-a^8}\right )^3}{16 a^5 \sqrt {-a^8} b \sqrt {b^4+a^4 x^4}}-\frac {\left (a^4+\sqrt {-a^8}\right ) \left (b^2+a^2 x^2\right ) \sqrt {\frac {b^4+a^4 x^4}{\left (b^2+a^2 x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {a x}{b}\right ),\frac {1}{2}\right ) \left (a^2-\sqrt [4]{-a^8}\right )}{8 a^7 b \sqrt {b^4+a^4 x^4}}+\frac {\left (-\sqrt {-a^8}\right )^{5/4} \left (a^4-\sqrt {-a^8}\right )^{3/2} \arctan \left (\frac {\sqrt {a^4-\sqrt {-a^8}} b x}{\sqrt [4]{-\sqrt {-a^8}} \sqrt {b^4+a^4 x^4}}\right )}{8 a^{12} b}-\frac {\left (a^4+\sqrt {-a^8}\right )^{3/2} \arctan \left (\frac {\sqrt {a^4+\sqrt {-a^8}} b x}{\sqrt [8]{-a^8} \sqrt {b^4+a^4 x^4}}\right )}{8 a^4 \left (-a^8\right )^{3/8} b}+\frac {\left (-\sqrt {-a^8}\right )^{5/4} \left (a^4-\sqrt {-a^8}\right )^{3/2} \text {arctanh}\left (\frac {\sqrt {a^4-\sqrt {-a^8}} b x}{\sqrt [4]{-\sqrt {-a^8}} \sqrt {b^4+a^4 x^4}}\right )}{8 a^{12} b}-\frac {\left (a^4+\sqrt {-a^8}\right )^{3/2} \text {arctanh}\left (\frac {\sqrt {a^4+\sqrt {-a^8}} b x}{\sqrt [8]{-a^8} \sqrt {b^4+a^4 x^4}}\right )}{8 a^4 \left (-a^8\right )^{3/8} b}-\frac {\sqrt {-a^8} \left (a^2+\sqrt [4]{-a^8}\right ) \left (a^4-\sqrt {-a^8}\right ) \left (b^2+a^2 x^2\right ) \sqrt {\frac {b^4+a^4 x^4}{\left (b^2+a^2 x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {a x}{b}\right ),\frac {1}{2}\right )}{8 a^{11} b \sqrt {b^4+a^4 x^4}}+\frac {\left (a^4-\sqrt {-a^8}\right ) \left (b^2+a^2 x^2\right ) \sqrt {\frac {b^4+a^4 x^4}{\left (b^2+a^2 x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {a x}{b}\right ),\frac {1}{2}\right )}{4 a^5 b \sqrt {b^4+a^4 x^4}}+\frac {\left (a^4+\sqrt {-a^8}\right ) \left (b^2+a^2 x^2\right ) \sqrt {\frac {b^4+a^4 x^4}{\left (b^2+a^2 x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {a x}{b}\right ),\frac {1}{2}\right )}{4 a^5 b \sqrt {b^4+a^4 x^4}}+\frac {\sqrt {-a^8} \left (a^4+\sqrt {-a^8}\right ) \left (a^2-\sqrt {-\sqrt {-a^8}}\right ) \left (b^2+a^2 x^2\right ) \sqrt {\frac {b^4+a^4 x^4}{\left (b^2+a^2 x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {a x}{b}\right ),\frac {1}{2}\right )}{8 a^{11} b \sqrt {b^4+a^4 x^4}}+\frac {\sqrt {-a^8} \left (a^4+\sqrt {-a^8}\right ) \left (a^2+\sqrt {-\sqrt {-a^8}}\right ) \left (b^2+a^2 x^2\right ) \sqrt {\frac {b^4+a^4 x^4}{\left (b^2+a^2 x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {a x}{b}\right ),\frac {1}{2}\right )}{8 a^{11} b \sqrt {b^4+a^4 x^4}}+\frac {\sqrt {-a^8} \left (a^2+\sqrt [4]{-a^8}\right )^2 \left (a^4-\sqrt {-a^8}\right ) \left (b^2+a^2 x^2\right ) \sqrt {\frac {b^4+a^4 x^4}{\left (b^2+a^2 x^2\right )^2}} \operatorname {EllipticPi}\left (\frac {a^6 \left (a^2-\sqrt [4]{-a^8}\right )^2}{4 \left (-a^8\right )^{5/4}},2 \arctan \left (\frac {a x}{b}\right ),\frac {1}{2}\right )}{16 a^{13} b \sqrt {b^4+a^4 x^4}}-\frac {\sqrt {-a^8} \left (a^4+\sqrt {-a^8}\right ) \left (a^2+\sqrt {-\sqrt {-a^8}}\right )^2 \left (b^2+a^2 x^2\right ) \sqrt {\frac {b^4+a^4 x^4}{\left (b^2+a^2 x^2\right )^2}} \operatorname {EllipticPi}\left (-\frac {\left (a^2-\sqrt {-\sqrt {-a^8}}\right )^2}{4 a^2 \sqrt {-\sqrt {-a^8}}},2 \arctan \left (\frac {a x}{b}\right ),\frac {1}{2}\right )}{16 a^{13} b \sqrt {b^4+a^4 x^4}}-\frac {\sqrt {-a^8} \left (a^4+\sqrt {-a^8}\right ) \left (a^2-\sqrt {-\sqrt {-a^8}}\right )^2 \left (b^2+a^2 x^2\right ) \sqrt {\frac {b^4+a^4 x^4}{\left (b^2+a^2 x^2\right )^2}} \operatorname {EllipticPi}\left (\frac {\left (a^2+\sqrt {-\sqrt {-a^8}}\right )^2}{4 a^2 \sqrt {-\sqrt {-a^8}}},2 \arctan \left (\frac {a x}{b}\right ),\frac {1}{2}\right )}{16 a^{13} b \sqrt {b^4+a^4 x^4}}\) |
((-Sqrt[-a^8])^(5/4)*(a^4 - Sqrt[-a^8])^(3/2)*ArcTan[(Sqrt[a^4 - Sqrt[-a^8 ]]*b*x)/((-Sqrt[-a^8])^(1/4)*Sqrt[b^4 + a^4*x^4])])/(8*a^12*b) - ((a^4 + S qrt[-a^8])^(3/2)*ArcTan[(Sqrt[a^4 + Sqrt[-a^8]]*b*x)/((-a^8)^(1/8)*Sqrt[b^ 4 + a^4*x^4])])/(8*a^4*(-a^8)^(3/8)*b) + ((-Sqrt[-a^8])^(5/4)*(a^4 - Sqrt[ -a^8])^(3/2)*ArcTanh[(Sqrt[a^4 - Sqrt[-a^8]]*b*x)/((-Sqrt[-a^8])^(1/4)*Sqr t[b^4 + a^4*x^4])])/(8*a^12*b) - ((a^4 + Sqrt[-a^8])^(3/2)*ArcTanh[(Sqrt[a ^4 + Sqrt[-a^8]]*b*x)/((-a^8)^(1/8)*Sqrt[b^4 + a^4*x^4])])/(8*a^4*(-a^8)^( 3/8)*b) + ((a^4 - Sqrt[-a^8])*(b^2 + a^2*x^2)*Sqrt[(b^4 + a^4*x^4)/(b^2 + a^2*x^2)^2]*EllipticF[2*ArcTan[(a*x)/b], 1/2])/(4*a^5*b*Sqrt[b^4 + a^4*x^4 ]) - (Sqrt[-a^8]*(a^2 + (-a^8)^(1/4))*(a^4 - Sqrt[-a^8])*(b^2 + a^2*x^2)*S qrt[(b^4 + a^4*x^4)/(b^2 + a^2*x^2)^2]*EllipticF[2*ArcTan[(a*x)/b], 1/2])/ (8*a^11*b*Sqrt[b^4 + a^4*x^4]) + ((a^4 + Sqrt[-a^8])*(b^2 + a^2*x^2)*Sqrt[ (b^4 + a^4*x^4)/(b^2 + a^2*x^2)^2]*EllipticF[2*ArcTan[(a*x)/b], 1/2])/(4*a ^5*b*Sqrt[b^4 + a^4*x^4]) - ((a^2 - (-a^8)^(1/4))*(a^4 + Sqrt[-a^8])*(b^2 + a^2*x^2)*Sqrt[(b^4 + a^4*x^4)/(b^2 + a^2*x^2)^2]*EllipticF[2*ArcTan[(a*x )/b], 1/2])/(8*a^7*b*Sqrt[b^4 + a^4*x^4]) + (Sqrt[-a^8]*(a^4 + Sqrt[-a^8]) *(a^2 - Sqrt[-Sqrt[-a^8]])*(b^2 + a^2*x^2)*Sqrt[(b^4 + a^4*x^4)/(b^2 + a^2 *x^2)^2]*EllipticF[2*ArcTan[(a*x)/b], 1/2])/(8*a^11*b*Sqrt[b^4 + a^4*x^4]) + (Sqrt[-a^8]*(a^4 + Sqrt[-a^8])*(a^2 + Sqrt[-Sqrt[-a^8]])*(b^2 + a^2*x^2 )*Sqrt[(b^4 + a^4*x^4)/(b^2 + a^2*x^2)^2]*EllipticF[2*ArcTan[(a*x)/b], ...
3.11.95.3.1 Defintions of rubi rules used
Int[(u_.)*((a_) + (c_.)*(x_)^(n2_.))^(p_.)*((d_) + (e_.)*(x_)^(n_))^(q_.), x_Symbol] :> Int[u*(d + e*x^n)^(p + q)*(a/d + (c/e)*x^n)^p, x] /; FreeQ[{a, c, d, e, n, p, q}, x] && EqQ[n2, 2*n] && EqQ[c*d^2 + a*e^2, 0] && (Integer Q[p] || (GtQ[a, 0] && GtQ[d, 0]))
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]
Time = 9.40 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.41
| method | result | size |
| default | \(-\frac {\left (-2 \arctan \left (\frac {\sqrt {a^{4} x^{4}+b^{4}}\, 2^{\frac {3}{4}}}{2 x \left (a^{4} b^{4}\right )^{\frac {1}{4}}}\right )+\ln \left (\frac {-2^{\frac {1}{4}} \left (a^{4} b^{4}\right )^{\frac {1}{4}} x -\sqrt {a^{4} x^{4}+b^{4}}}{2^{\frac {1}{4}} \left (a^{4} b^{4}\right )^{\frac {1}{4}} x -\sqrt {a^{4} x^{4}+b^{4}}}\right )\right ) 2^{\frac {3}{4}}}{8 \left (a^{4} b^{4}\right )^{\frac {1}{4}}}\) | \(114\) |
| pseudoelliptic | \(-\frac {\left (-2 \arctan \left (\frac {\sqrt {a^{4} x^{4}+b^{4}}\, 2^{\frac {3}{4}}}{2 x \left (a^{4} b^{4}\right )^{\frac {1}{4}}}\right )+\ln \left (\frac {-2^{\frac {1}{4}} \left (a^{4} b^{4}\right )^{\frac {1}{4}} x -\sqrt {a^{4} x^{4}+b^{4}}}{2^{\frac {1}{4}} \left (a^{4} b^{4}\right )^{\frac {1}{4}} x -\sqrt {a^{4} x^{4}+b^{4}}}\right )\right ) 2^{\frac {3}{4}}}{8 \left (a^{4} b^{4}\right )^{\frac {1}{4}}}\) | \(114\) |
| elliptic | \(\frac {2 \arctan \left (\frac {\sqrt {a^{4} x^{4}+b^{4}}}{x \sqrt {\sqrt {2}\, \sqrt {a^{4} b^{4}}}}\right )-\ln \left (\frac {\frac {\sqrt {a^{4} x^{4}+b^{4}}\, \sqrt {2}}{2 x}+\frac {\sqrt {2}\, \sqrt {\sqrt {2}\, \sqrt {a^{4} b^{4}}}}{2}}{\frac {\sqrt {a^{4} x^{4}+b^{4}}\, \sqrt {2}}{2 x}-\frac {\sqrt {2}\, \sqrt {\sqrt {2}\, \sqrt {a^{4} b^{4}}}}{2}}\right )}{4 \sqrt {\sqrt {2}\, \sqrt {a^{4} b^{4}}}}\) | \(144\) |
-1/8*(-2*arctan(1/2*(a^4*x^4+b^4)^(1/2)/x*2^(3/4)/(a^4*b^4)^(1/4))+ln((-2^ (1/4)*(a^4*b^4)^(1/4)*x-(a^4*x^4+b^4)^(1/2))/(2^(1/4)*(a^4*b^4)^(1/4)*x-(a ^4*x^4+b^4)^(1/2))))*2^(3/4)/(a^4*b^4)^(1/4)
Result contains complex when optimal does not.
Time = 1.14 (sec) , antiderivative size = 623, normalized size of antiderivative = 7.69 \[ \int \frac {-b^8+a^8 x^8}{\sqrt {b^4+a^4 x^4} \left (b^8+a^8 x^8\right )} \, dx=-\frac {1}{8} \, \left (\frac {1}{2}\right )^{\frac {1}{4}} \left (\frac {1}{a^{4} b^{4}}\right )^{\frac {1}{4}} \log \left (-\frac {4 \, \left (\frac {1}{2}\right )^{\frac {3}{4}} {\left (a^{8} b^{4} x^{6} + a^{4} b^{8} x^{2}\right )} \left (\frac {1}{a^{4} b^{4}}\right )^{\frac {3}{4}} + 2 \, {\left (2 \, \sqrt {\frac {1}{2}} a^{4} b^{4} x^{3} \sqrt {\frac {1}{a^{4} b^{4}}} + a^{4} x^{5} + b^{4} x\right )} \sqrt {a^{4} x^{4} + b^{4}} + \left (\frac {1}{2}\right )^{\frac {1}{4}} {\left (a^{8} x^{8} + 4 \, a^{4} b^{4} x^{4} + b^{8}\right )} \left (\frac {1}{a^{4} b^{4}}\right )^{\frac {1}{4}}}{2 \, {\left (a^{8} x^{8} + b^{8}\right )}}\right ) + \frac {1}{8} \, \left (\frac {1}{2}\right )^{\frac {1}{4}} \left (\frac {1}{a^{4} b^{4}}\right )^{\frac {1}{4}} \log \left (\frac {4 \, \left (\frac {1}{2}\right )^{\frac {3}{4}} {\left (a^{8} b^{4} x^{6} + a^{4} b^{8} x^{2}\right )} \left (\frac {1}{a^{4} b^{4}}\right )^{\frac {3}{4}} - 2 \, {\left (2 \, \sqrt {\frac {1}{2}} a^{4} b^{4} x^{3} \sqrt {\frac {1}{a^{4} b^{4}}} + a^{4} x^{5} + b^{4} x\right )} \sqrt {a^{4} x^{4} + b^{4}} + \left (\frac {1}{2}\right )^{\frac {1}{4}} {\left (a^{8} x^{8} + 4 \, a^{4} b^{4} x^{4} + b^{8}\right )} \left (\frac {1}{a^{4} b^{4}}\right )^{\frac {1}{4}}}{2 \, {\left (a^{8} x^{8} + b^{8}\right )}}\right ) + \frac {1}{8} i \, \left (\frac {1}{2}\right )^{\frac {1}{4}} \left (\frac {1}{a^{4} b^{4}}\right )^{\frac {1}{4}} \log \left (-\frac {4 \, \left (\frac {1}{2}\right )^{\frac {3}{4}} {\left (i \, a^{8} b^{4} x^{6} + i \, a^{4} b^{8} x^{2}\right )} \left (\frac {1}{a^{4} b^{4}}\right )^{\frac {3}{4}} - 2 \, {\left (2 \, \sqrt {\frac {1}{2}} a^{4} b^{4} x^{3} \sqrt {\frac {1}{a^{4} b^{4}}} - a^{4} x^{5} - b^{4} x\right )} \sqrt {a^{4} x^{4} + b^{4}} + \left (\frac {1}{2}\right )^{\frac {1}{4}} {\left (-i \, a^{8} x^{8} - 4 i \, a^{4} b^{4} x^{4} - i \, b^{8}\right )} \left (\frac {1}{a^{4} b^{4}}\right )^{\frac {1}{4}}}{2 \, {\left (a^{8} x^{8} + b^{8}\right )}}\right ) - \frac {1}{8} i \, \left (\frac {1}{2}\right )^{\frac {1}{4}} \left (\frac {1}{a^{4} b^{4}}\right )^{\frac {1}{4}} \log \left (-\frac {4 \, \left (\frac {1}{2}\right )^{\frac {3}{4}} {\left (-i \, a^{8} b^{4} x^{6} - i \, a^{4} b^{8} x^{2}\right )} \left (\frac {1}{a^{4} b^{4}}\right )^{\frac {3}{4}} - 2 \, {\left (2 \, \sqrt {\frac {1}{2}} a^{4} b^{4} x^{3} \sqrt {\frac {1}{a^{4} b^{4}}} - a^{4} x^{5} - b^{4} x\right )} \sqrt {a^{4} x^{4} + b^{4}} + \left (\frac {1}{2}\right )^{\frac {1}{4}} {\left (i \, a^{8} x^{8} + 4 i \, a^{4} b^{4} x^{4} + i \, b^{8}\right )} \left (\frac {1}{a^{4} b^{4}}\right )^{\frac {1}{4}}}{2 \, {\left (a^{8} x^{8} + b^{8}\right )}}\right ) \]
-1/8*(1/2)^(1/4)*(1/(a^4*b^4))^(1/4)*log(-1/2*(4*(1/2)^(3/4)*(a^8*b^4*x^6 + a^4*b^8*x^2)*(1/(a^4*b^4))^(3/4) + 2*(2*sqrt(1/2)*a^4*b^4*x^3*sqrt(1/(a^ 4*b^4)) + a^4*x^5 + b^4*x)*sqrt(a^4*x^4 + b^4) + (1/2)^(1/4)*(a^8*x^8 + 4* a^4*b^4*x^4 + b^8)*(1/(a^4*b^4))^(1/4))/(a^8*x^8 + b^8)) + 1/8*(1/2)^(1/4) *(1/(a^4*b^4))^(1/4)*log(1/2*(4*(1/2)^(3/4)*(a^8*b^4*x^6 + a^4*b^8*x^2)*(1 /(a^4*b^4))^(3/4) - 2*(2*sqrt(1/2)*a^4*b^4*x^3*sqrt(1/(a^4*b^4)) + a^4*x^5 + b^4*x)*sqrt(a^4*x^4 + b^4) + (1/2)^(1/4)*(a^8*x^8 + 4*a^4*b^4*x^4 + b^8 )*(1/(a^4*b^4))^(1/4))/(a^8*x^8 + b^8)) + 1/8*I*(1/2)^(1/4)*(1/(a^4*b^4))^ (1/4)*log(-1/2*(4*(1/2)^(3/4)*(I*a^8*b^4*x^6 + I*a^4*b^8*x^2)*(1/(a^4*b^4) )^(3/4) - 2*(2*sqrt(1/2)*a^4*b^4*x^3*sqrt(1/(a^4*b^4)) - a^4*x^5 - b^4*x)* sqrt(a^4*x^4 + b^4) + (1/2)^(1/4)*(-I*a^8*x^8 - 4*I*a^4*b^4*x^4 - I*b^8)*( 1/(a^4*b^4))^(1/4))/(a^8*x^8 + b^8)) - 1/8*I*(1/2)^(1/4)*(1/(a^4*b^4))^(1/ 4)*log(-1/2*(4*(1/2)^(3/4)*(-I*a^8*b^4*x^6 - I*a^4*b^8*x^2)*(1/(a^4*b^4))^ (3/4) - 2*(2*sqrt(1/2)*a^4*b^4*x^3*sqrt(1/(a^4*b^4)) - a^4*x^5 - b^4*x)*sq rt(a^4*x^4 + b^4) + (1/2)^(1/4)*(I*a^8*x^8 + 4*I*a^4*b^4*x^4 + I*b^8)*(1/( a^4*b^4))^(1/4))/(a^8*x^8 + b^8))
\[ \int \frac {-b^8+a^8 x^8}{\sqrt {b^4+a^4 x^4} \left (b^8+a^8 x^8\right )} \, dx=\int \frac {\left (a x - b\right ) \left (a x + b\right ) \left (a^{2} x^{2} + b^{2}\right ) \sqrt {a^{4} x^{4} + b^{4}}}{a^{8} x^{8} + b^{8}}\, dx \]
Integral((a*x - b)*(a*x + b)*(a**2*x**2 + b**2)*sqrt(a**4*x**4 + b**4)/(a* *8*x**8 + b**8), x)
\[ \int \frac {-b^8+a^8 x^8}{\sqrt {b^4+a^4 x^4} \left (b^8+a^8 x^8\right )} \, dx=\int { \frac {a^{8} x^{8} - b^{8}}{{\left (a^{8} x^{8} + b^{8}\right )} \sqrt {a^{4} x^{4} + b^{4}}} \,d x } \]
\[ \int \frac {-b^8+a^8 x^8}{\sqrt {b^4+a^4 x^4} \left (b^8+a^8 x^8\right )} \, dx=\int { \frac {a^{8} x^{8} - b^{8}}{{\left (a^{8} x^{8} + b^{8}\right )} \sqrt {a^{4} x^{4} + b^{4}}} \,d x } \]
Timed out. \[ \int \frac {-b^8+a^8 x^8}{\sqrt {b^4+a^4 x^4} \left (b^8+a^8 x^8\right )} \, dx=\int -\frac {b^8-a^8\,x^8}{\sqrt {a^4\,x^4+b^4}\,\left (a^8\,x^8+b^8\right )} \,d x \]