Integrand size = 42, antiderivative size = 196 \[ \int \frac {b^8+a^8 x^8}{\sqrt {b^4+a^4 x^4} \left (-b^8+a^8 x^8\right )} \, dx=-\frac {x}{2 \sqrt {b^4+a^4 x^4}}-\frac {\arctan \left (\frac {\sqrt {2} a b x}{b^2+a^2 x^2+\sqrt {b^4+a^4 x^4}}\right )}{2 \sqrt {2} a b}+\frac {\text {arctanh}\left (\frac {\sqrt {6-4 \sqrt {2}} a b x}{b^2+a^2 x^2+\sqrt {b^4+a^4 x^4}}\right )}{4 \sqrt {2} a b}-\frac {\text {arctanh}\left (\frac {\sqrt {6+4 \sqrt {2}} a b x}{b^2+a^2 x^2+\sqrt {b^4+a^4 x^4}}\right )}{4 \sqrt {2} a b} \]
-1/2*x/(a^4*x^4+b^4)^(1/2)-1/4*2^(1/2)*arctan(2^(1/2)*a*b*x/(b^2+a^2*x^2+( a^4*x^4+b^4)^(1/2)))/a/b+1/8*arctanh((2-2^(1/2))*a*b*x/(b^2+a^2*x^2+(a^4*x ^4+b^4)^(1/2)))*2^(1/2)/a/b-1/8*arctanh((2+2^(1/2))*a*b*x/(b^2+a^2*x^2+(a^ 4*x^4+b^4)^(1/2)))*2^(1/2)/a/b
Time = 0.70 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.55 \[ \int \frac {b^8+a^8 x^8}{\sqrt {b^4+a^4 x^4} \left (-b^8+a^8 x^8\right )} \, dx=-\frac {\frac {4 a b x}{\sqrt {b^4+a^4 x^4}}+2 \sqrt {2} \arctan \left (\frac {\sqrt {2} a b x}{b^2+a^2 x^2+\sqrt {b^4+a^4 x^4}}\right )+\sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} a b x}{\sqrt {b^4+a^4 x^4}}\right )}{8 a b} \]
-1/8*((4*a*b*x)/Sqrt[b^4 + a^4*x^4] + 2*Sqrt[2]*ArcTan[(Sqrt[2]*a*b*x)/(b^ 2 + a^2*x^2 + Sqrt[b^4 + a^4*x^4])] + Sqrt[2]*ArcTanh[(Sqrt[2]*a*b*x)/Sqrt [b^4 + a^4*x^4]])/(a*b)
Time = 0.87 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.52, 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 |
| \(\displaystyle \int \frac {a^8 x^8+b^8}{\left (a^4 x^4-b^4\right ) \left (a^4 x^4+b^4\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \int \left (\frac {b^4}{\left (a^4 x^4+b^4\right )^{3/2}}+\frac {a^4 x^4}{\left (a^4 x^4+b^4\right )^{3/2}}+\frac {2 b^8}{\left (a^4 x^4-b^4\right ) \left (a^4 x^4+b^4\right )^{3/2}}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\arctan \left (\frac {\sqrt {2} a b x}{\sqrt {a^4 x^4+b^4}}\right )}{4 \sqrt {2} a b}-\frac {\text {arctanh}\left (\frac {\sqrt {2} a b x}{\sqrt {a^4 x^4+b^4}}\right )}{4 \sqrt {2} a b}-\frac {x}{2 \sqrt {a^4 x^4+b^4}}\) |
-1/2*x/Sqrt[b^4 + a^4*x^4] - ArcTan[(Sqrt[2]*a*b*x)/Sqrt[b^4 + a^4*x^4]]/( 4*Sqrt[2]*a*b) - ArcTanh[(Sqrt[2]*a*b*x)/Sqrt[b^4 + a^4*x^4]]/(4*Sqrt[2]*a *b)
3.25.23.3.1 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 = 3.45 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.67
| method | result | size |
| elliptic | \(\frac {\left (\frac {\arctan \left (\frac {\sqrt {a^{4} x^{4}+b^{4}}\, \sqrt {2}}{2 x a b}\right )}{4 a b}-\frac {\ln \left (a b +\frac {\sqrt {a^{4} x^{4}+b^{4}}\, \sqrt {2}}{2 x}\right )}{8 a b}-\frac {\sqrt {2}\, x}{2 \sqrt {a^{4} x^{4}+b^{4}}}+\frac {\ln \left (-a b +\frac {\sqrt {a^{4} x^{4}+b^{4}}\, \sqrt {2}}{2 x}\right )}{8 a b}\right ) \sqrt {2}}{2}\) | \(131\) |
| default | \(-\frac {\sqrt {2}\, \left (\left (\sqrt {a^{4} x^{4}+b^{4}}\, \sqrt {2}\, \sqrt {a^{2} b^{2}}\, x +\frac {\left (a^{4} x^{4}+b^{4}\right ) \left (\ln \left (\frac {\sqrt {2}\, \sqrt {a^{2} b^{2}}\, a^{2} \sqrt {a^{4} x^{4}+b^{4}}+2 a^{3} b \left (a^{2} x^{2}-a b x +b^{2}\right )}{\left (a x -b \right )^{2}}\right )+\ln \left (\frac {\sqrt {2}\, \sqrt {a^{2} b^{2}}\, a^{2} \sqrt {a^{4} x^{4}+b^{4}}-2 a^{3} b \left (a^{2} x^{2}+a b x +b^{2}\right )}{\left (a x +b \right )^{2}}\right )+2 \ln \left (2\right )\right )}{4}\right ) \sqrt {-a^{2} b^{2}}+\frac {\sqrt {a^{2} b^{2}}\, \left (a^{4} x^{4}+b^{4}\right ) \left (\ln \left (2\right )+\ln \left (\frac {\left (-2 x \,a^{2} b^{2}+\sqrt {2}\, \sqrt {-a^{2} b^{2}}\, \sqrt {a^{4} x^{4}+b^{4}}\right ) a^{2}}{a^{2} x^{2}+b^{2}}\right )\right )}{2}\right )}{4 \sqrt {-a^{2} b^{2}}\, \sqrt {a^{2} b^{2}}\, \left (a^{2} x^{2}-\sqrt {2}\, \sqrt {a^{2} b^{2}}\, x +b^{2}\right ) \left (a^{2} x^{2}+\sqrt {2}\, \sqrt {a^{2} b^{2}}\, x +b^{2}\right )}\) | \(341\) |
| pseudoelliptic | \(-\frac {\sqrt {2}\, \left (\left (\sqrt {a^{4} x^{4}+b^{4}}\, \sqrt {2}\, \sqrt {a^{2} b^{2}}\, x +\frac {\left (a^{4} x^{4}+b^{4}\right ) \left (\ln \left (\frac {\sqrt {2}\, \sqrt {a^{2} b^{2}}\, a^{2} \sqrt {a^{4} x^{4}+b^{4}}+2 a^{3} b \left (a^{2} x^{2}-a b x +b^{2}\right )}{\left (a x -b \right )^{2}}\right )+\ln \left (\frac {\sqrt {2}\, \sqrt {a^{2} b^{2}}\, a^{2} \sqrt {a^{4} x^{4}+b^{4}}-2 a^{3} b \left (a^{2} x^{2}+a b x +b^{2}\right )}{\left (a x +b \right )^{2}}\right )+2 \ln \left (2\right )\right )}{4}\right ) \sqrt {-a^{2} b^{2}}+\frac {\sqrt {a^{2} b^{2}}\, \left (a^{4} x^{4}+b^{4}\right ) \left (\ln \left (2\right )+\ln \left (\frac {\left (-2 x \,a^{2} b^{2}+\sqrt {2}\, \sqrt {-a^{2} b^{2}}\, \sqrt {a^{4} x^{4}+b^{4}}\right ) a^{2}}{a^{2} x^{2}+b^{2}}\right )\right )}{2}\right )}{4 \sqrt {-a^{2} b^{2}}\, \sqrt {a^{2} b^{2}}\, \left (a^{2} x^{2}-\sqrt {2}\, \sqrt {a^{2} b^{2}}\, x +b^{2}\right ) \left (a^{2} x^{2}+\sqrt {2}\, \sqrt {a^{2} b^{2}}\, x +b^{2}\right )}\) | \(341\) |
1/2*(1/4/a/b*arctan(1/2*(a^4*x^4+b^4)^(1/2)*2^(1/2)/x/a/b)-1/8/a/b*ln(a*b+ 1/2*(a^4*x^4+b^4)^(1/2)*2^(1/2)/x)-1/2/(a^4*x^4+b^4)^(1/2)*2^(1/2)*x+1/8/a /b*ln(-a*b+1/2*(a^4*x^4+b^4)^(1/2)*2^(1/2)/x))*2^(1/2)
Time = 0.34 (sec) , antiderivative size = 159, normalized size of antiderivative = 0.81 \[ \int \frac {b^8+a^8 x^8}{\sqrt {b^4+a^4 x^4} \left (-b^8+a^8 x^8\right )} \, dx=-\frac {8 \, \sqrt {a^{4} x^{4} + b^{4}} a b x + 2 \, \sqrt {2} {\left (a^{4} x^{4} + b^{4}\right )} \arctan \left (\frac {\sqrt {2} a b x}{\sqrt {a^{4} x^{4} + b^{4}}}\right ) - \sqrt {2} {\left (a^{4} x^{4} + b^{4}\right )} \log \left (\frac {a^{4} x^{4} + 2 \, a^{2} b^{2} x^{2} + b^{4} - 2 \, \sqrt {2} \sqrt {a^{4} x^{4} + b^{4}} a b x}{a^{4} x^{4} - 2 \, a^{2} b^{2} x^{2} + b^{4}}\right )}{16 \, {\left (a^{5} b x^{4} + a b^{5}\right )}} \]
-1/16*(8*sqrt(a^4*x^4 + b^4)*a*b*x + 2*sqrt(2)*(a^4*x^4 + b^4)*arctan(sqrt (2)*a*b*x/sqrt(a^4*x^4 + b^4)) - sqrt(2)*(a^4*x^4 + b^4)*log((a^4*x^4 + 2* a^2*b^2*x^2 + b^4 - 2*sqrt(2)*sqrt(a^4*x^4 + b^4)*a*b*x)/(a^4*x^4 - 2*a^2* b^2*x^2 + b^4)))/(a^5*b*x^4 + a*b^5)
\[ \int \frac {b^8+a^8 x^8}{\sqrt {b^4+a^4 x^4} \left (-b^8+a^8 x^8\right )} \, dx=\int \frac {a^{8} x^{8} + b^{8}}{\left (a x - b\right ) \left (a x + b\right ) \left (a^{2} x^{2} + b^{2}\right ) \left (a^{4} x^{4} + b^{4}\right )^{\frac {3}{2}}}\, dx \]
Integral((a**8*x**8 + b**8)/((a*x - b)*(a*x + b)*(a**2*x**2 + b**2)*(a**4* x**4 + b**4)**(3/2)), 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. \[ \int \frac {b^8+a^8 x^8}{\sqrt {b^4+a^4 x^4} \left (-b^8+a^8 x^8\right )} \, dx=\int -\frac {a^8\,x^8+b^8}{\sqrt {a^4\,x^4+b^4}\,\left (b^8-a^8\,x^8\right )} \,d x \]