\(\int \frac {-b^8+a^8 x^8}{\sqrt {b^4+a^4 x^4} (b^8+a^8 x^8)} \, dx\) [1095]

Optimal result

Integrand size = 42, antiderivative size = 81 \[ \int \frac {-b^8+a^8 x^8}{\sqrt {b^4+a^4 x^4} \left (b^8+a^8 x^8\right )} \, dx=-\frac {\arctan \left (\frac {\sqrt [4]{2} a b x}{\sqrt {b^4+a^4 x^4}}\right )}{2 \sqrt [4]{2} a b}-\frac {\text {arctanh}\left (\frac {\sqrt [4]{2} a b x}{\sqrt {b^4+a^4 x^4}}\right )}{2 \sqrt [4]{2} a b} \] Output:

-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
 

Mathematica [A] (verified)

Time = 0.49 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.81 \[ \int \frac {-b^8+a^8 x^8}{\sqrt {b^4+a^4 x^4} \left (b^8+a^8 x^8\right )} \, dx=-\frac {\arctan \left (\frac {\sqrt [4]{2} a b x}{\sqrt {b^4+a^4 x^4}}\right )+\text {arctanh}\left (\frac {\sqrt [4]{2} a b x}{\sqrt {b^4+a^4 x^4}}\right )}{2 \sqrt [4]{2} a b} \] Input:

Integrate[(-b^8 + a^8*x^8)/(Sqrt[b^4 + a^4*x^4]*(b^8 + a^8*x^8)),x]
 

Output:

-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)
 

Rubi [C] (verified)

Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.

Time = 2.95 (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}}\)

Input:

Int[(-b^8 + a^8*x^8)/(Sqrt[b^4 + a^4*x^4]*(b^8 + a^8*x^8)),x]
 

Output:

((-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], ...
 

Defintions of rubi rules used

rule 1388

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]))
 

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 [A] (verified)

Time = 2.52 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.41

method	result	size
default	\(-\frac {2^{\frac {3}{4}} \left (-2 \arctan \left (\frac {\sqrt {a^{4} x^{4}+b^{4}}\, 2^{\frac {3}{4}}}{2 x \left (b^{4} a^{4}\right )^{\frac {1}{4}}}\right )+\ln \left (\frac {-2^{\frac {1}{4}} \left (b^{4} a^{4}\right )^{\frac {1}{4}} x -\sqrt {a^{4} x^{4}+b^{4}}}{2^{\frac {1}{4}} \left (b^{4} a^{4}\right )^{\frac {1}{4}} x -\sqrt {a^{4} x^{4}+b^{4}}}\right )\right )}{8 \left (b^{4} a^{4}\right )^{\frac {1}{4}}}\)	\(114\)
pseudoelliptic	\(-\frac {2^{\frac {3}{4}} \left (-2 \arctan \left (\frac {\sqrt {a^{4} x^{4}+b^{4}}\, 2^{\frac {3}{4}}}{2 x \left (b^{4} a^{4}\right )^{\frac {1}{4}}}\right )+\ln \left (\frac {-2^{\frac {1}{4}} \left (b^{4} a^{4}\right )^{\frac {1}{4}} x -\sqrt {a^{4} x^{4}+b^{4}}}{2^{\frac {1}{4}} \left (b^{4} a^{4}\right )^{\frac {1}{4}} x -\sqrt {a^{4} x^{4}+b^{4}}}\right )\right )}{8 \left (b^{4} a^{4}\right )^{\frac {1}{4}}}\)	\(114\)
elliptic	\(\frac {2 \arctan \left (\frac {\sqrt {a^{4} x^{4}+b^{4}}}{x \sqrt {\sqrt {2}\, \sqrt {b^{4} a^{4}}}}\right )-\ln \left (\frac {\frac {\sqrt {a^{4} x^{4}+b^{4}}\, \sqrt {2}}{2 x}+\frac {\sqrt {2}\, \sqrt {\sqrt {2}\, \sqrt {b^{4} a^{4}}}}{2}}{\frac {\sqrt {a^{4} x^{4}+b^{4}}\, \sqrt {2}}{2 x}-\frac {\sqrt {2}\, \sqrt {\sqrt {2}\, \sqrt {b^{4} a^{4}}}}{2}}\right )}{4 \sqrt {\sqrt {2}\, \sqrt {b^{4} a^{4}}}}\)	\(144\)

Input:

int((a^8*x^8-b^8)/(a^4*x^4+b^4)^(1/2)/(a^8*x^8+b^8),x,method=_RETURNVERBOS 
E)
 

Output:

-1/8*2^(3/4)*(-2*arctan(1/2*(a^4*x^4+b^4)^(1/2)/x*2^(3/4)/(b^4*a^4)^(1/4)) 
+ln((-2^(1/4)*(b^4*a^4)^(1/4)*x-(a^4*x^4+b^4)^(1/2))/(2^(1/4)*(b^4*a^4)^(1 
/4)*x-(a^4*x^4+b^4)^(1/2))))/(b^4*a^4)^(1/4)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 308 vs. \(2 (65) = 130\).

Time = 0.99 (sec) , antiderivative size = 308, normalized size of antiderivative = 3.80 \[ \int \frac {-b^8+a^8 x^8}{\sqrt {b^4+a^4 x^4} \left (b^8+a^8 x^8\right )} \, dx=-\frac {2 \cdot 2^{\frac {3}{4}} \arctan \left (\frac {2 \, {\left (2^{\frac {3}{4}} a^{3} b^{3} x^{3} + 2^{\frac {1}{4}} {\left (a^{5} b x^{5} + a b^{5} x\right )}\right )} \sqrt {a^{4} x^{4} + b^{4}}}{a^{8} x^{8} + b^{8}}\right ) + 2^{\frac {3}{4}} \log \left (-\frac {2^{\frac {3}{4}} {\left (a^{8} x^{8} + 4 \, a^{4} b^{4} x^{4} + b^{8}\right )} + 4 \, {\left (a^{5} b x^{5} + \sqrt {2} a^{3} b^{3} x^{3} + a b^{5} x\right )} \sqrt {a^{4} x^{4} + b^{4}} + 4 \cdot 2^{\frac {1}{4}} {\left (a^{6} b^{2} x^{6} + a^{2} b^{6} x^{2}\right )}}{a^{8} x^{8} + b^{8}}\right ) - 2^{\frac {3}{4}} \log \left (\frac {2^{\frac {3}{4}} {\left (a^{8} x^{8} + 4 \, a^{4} b^{4} x^{4} + b^{8}\right )} - 4 \, {\left (a^{5} b x^{5} + \sqrt {2} a^{3} b^{3} x^{3} + a b^{5} x\right )} \sqrt {a^{4} x^{4} + b^{4}} + 4 \cdot 2^{\frac {1}{4}} {\left (a^{6} b^{2} x^{6} + a^{2} b^{6} x^{2}\right )}}{a^{8} x^{8} + b^{8}}\right )}{16 \, a b} \] Input:

integrate((a^8*x^8-b^8)/(a^4*x^4+b^4)^(1/2)/(a^8*x^8+b^8),x, algorithm="fr 
icas")
 

Output:

-1/16*(2*2^(3/4)*arctan(2*(2^(3/4)*a^3*b^3*x^3 + 2^(1/4)*(a^5*b*x^5 + a*b^ 
5*x))*sqrt(a^4*x^4 + b^4)/(a^8*x^8 + b^8)) + 2^(3/4)*log(-(2^(3/4)*(a^8*x^ 
8 + 4*a^4*b^4*x^4 + b^8) + 4*(a^5*b*x^5 + sqrt(2)*a^3*b^3*x^3 + a*b^5*x)*s 
qrt(a^4*x^4 + b^4) + 4*2^(1/4)*(a^6*b^2*x^6 + a^2*b^6*x^2))/(a^8*x^8 + b^8 
)) - 2^(3/4)*log((2^(3/4)*(a^8*x^8 + 4*a^4*b^4*x^4 + b^8) - 4*(a^5*b*x^5 + 
 sqrt(2)*a^3*b^3*x^3 + a*b^5*x)*sqrt(a^4*x^4 + b^4) + 4*2^(1/4)*(a^6*b^2*x 
^6 + a^2*b^6*x^2))/(a^8*x^8 + b^8)))/(a*b)
 

Sympy [F]

\[ \int \frac {-b^8+a^8 x^8}{\sqrt {b^4+a^4 x^4} \left (b^8+a^8 x^8\right )} \, dx=\int \frac {\left (a x - b\right ) \left (a x + b\right ) \left (a^{2} x^{2} + b^{2}\right ) \sqrt {a^{4} x^{4} + b^{4}}}{a^{8} x^{8} + b^{8}}\, dx \] Input:

integrate((a**8*x**8-b**8)/(a**4*x**4+b**4)**(1/2)/(a**8*x**8+b**8),x)
 

Output:

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)
 

Maxima [F]

\[ \int \frac {-b^8+a^8 x^8}{\sqrt {b^4+a^4 x^4} \left (b^8+a^8 x^8\right )} \, dx=\int { \frac {a^{8} x^{8} - b^{8}}{{\left (a^{8} x^{8} + b^{8}\right )} \sqrt {a^{4} x^{4} + b^{4}}} \,d x } \] Input:

integrate((a^8*x^8-b^8)/(a^4*x^4+b^4)^(1/2)/(a^8*x^8+b^8),x, algorithm="ma 
xima")
 

Output:

integrate((a^8*x^8 - b^8)/((a^8*x^8 + b^8)*sqrt(a^4*x^4 + b^4)), x)
 

Giac [F]

\[ \int \frac {-b^8+a^8 x^8}{\sqrt {b^4+a^4 x^4} \left (b^8+a^8 x^8\right )} \, dx=\int { \frac {a^{8} x^{8} - b^{8}}{{\left (a^{8} x^{8} + b^{8}\right )} \sqrt {a^{4} x^{4} + b^{4}}} \,d x } \] Input:

integrate((a^8*x^8-b^8)/(a^4*x^4+b^4)^(1/2)/(a^8*x^8+b^8),x, algorithm="gi 
ac")
 

Output:

integrate((a^8*x^8 - b^8)/((a^8*x^8 + b^8)*sqrt(a^4*x^4 + b^4)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {-b^8+a^8 x^8}{\sqrt {b^4+a^4 x^4} \left (b^8+a^8 x^8\right )} \, dx=\int -\frac {b^8-a^8\,x^8}{\sqrt {a^4\,x^4+b^4}\,\left (a^8\,x^8+b^8\right )} \,d x \] Input:

int(-(b^8 - a^8*x^8)/((b^4 + a^4*x^4)^(1/2)*(b^8 + a^8*x^8)),x)
 

Output:

int(-(b^8 - a^8*x^8)/((b^4 + a^4*x^4)^(1/2)*(b^8 + a^8*x^8)), x)
 

Reduce [F]

\[ \int \frac {-b^8+a^8 x^8}{\sqrt {b^4+a^4 x^4} \left (b^8+a^8 x^8\right )} \, dx=-\left (\int \frac {\sqrt {a^{4} x^{4}+b^{4}}}{a^{8} x^{8}+b^{8}}d x \right ) b^{4}+\left (\int \frac {\sqrt {a^{4} x^{4}+b^{4}}\, x^{4}}{a^{8} x^{8}+b^{8}}d x \right ) a^{4} \] Input:

int((a^8*x^8-b^8)/(a^4*x^4+b^4)^(1/2)/(a^8*x^8+b^8),x)
 

Output:

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