Integrand size = 47, antiderivative size = 299 \[ \int \frac {b^8+x^4+a^8 x^8}{\sqrt {-b^4+a^4 x^4} \left (-b^8+a^8 x^8\right )} \, dx=\frac {\left (1+2 a^4 b^4\right ) x \sqrt {-b^4+a^4 x^4}}{4 a^4 b^4 \left (b^4-a^4 x^4\right )}-\frac {\left (\frac {1}{8}-\frac {i}{8}\right ) \left (-1+2 a^4 b^4\right ) \arctan \left (\frac {(1+i) a b x}{i b^2+a^2 x^2+\sqrt {-b^4+a^4 x^4}}\right )}{a^5 b^5}+\frac {\left (\frac {1}{16}-\frac {i}{16}\right ) \left (-1+2 a^4 b^4\right ) \text {arctanh}\left (\frac {(1+i) b^2+(1-i) a^2 x^2+(1-i) \sqrt {-b^4+a^4 x^4}}{2 \sqrt {3-2 \sqrt {2}} a b x}\right )}{a^5 b^5}-\frac {\left (\frac {1}{16}-\frac {i}{16}\right ) \left (-1+2 a^4 b^4\right ) \text {arctanh}\left (\frac {(1+i) b^2+(1-i) a^2 x^2+(1-i) \sqrt {-b^4+a^4 x^4}}{2 \sqrt {3+2 \sqrt {2}} a b x}\right )}{a^5 b^5} \]
1/4*(2*a^4*b^4+1)*x*(a^4*x^4-b^4)^(1/2)/a^4/b^4/(-a^4*x^4+b^4)+(-1/8+1/8*I )*(2*a^4*b^4-1)*arctan((1+I)*a*b*x/(I*b^2+a^2*x^2+(a^4*x^4-b^4)^(1/2)))/a^ 5/b^5+(1/16-1/16*I)*(2*a^4*b^4-1)*arctanh(1/2*((1+I)*b^2+(1-I)*a^2*x^2+(1- I)*(a^4*x^4-b^4)^(1/2))/(2^(1/2)-1)/a/b/x)/a^5/b^5+(-1/16+1/16*I)*(2*a^4*b ^4-1)*arctanh(1/2*((1+I)*b^2+(1-I)*a^2*x^2+(1-I)*(a^4*x^4-b^4)^(1/2))/(1+2 ^(1/2))/a/b/x)/a^5/b^5
Time = 1.16 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.51 \[ \int \frac {b^8+x^4+a^8 x^8}{\sqrt {-b^4+a^4 x^4} \left (-b^8+a^8 x^8\right )} \, dx=\frac {\left (\frac {1}{16}+\frac {i}{16}\right ) \left (-\frac {(2-2 i) a b \left (1+2 a^4 b^4\right ) x}{\sqrt {-b^4+a^4 x^4}}+\left (-1+2 a^4 b^4\right ) \arctan \left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {-b^4+a^4 x^4}}{a b x}\right )+2 i \left (-1+2 a^4 b^4\right ) \arctan \left (\frac {(1+i) a b x}{i b^2+a^2 x^2+\sqrt {-b^4+a^4 x^4}}\right )\right )}{a^5 b^5} \]
((1/16 + I/16)*(((-2 + 2*I)*a*b*(1 + 2*a^4*b^4)*x)/Sqrt[-b^4 + a^4*x^4] + (-1 + 2*a^4*b^4)*ArcTan[((1/2 + I/2)*Sqrt[-b^4 + a^4*x^4])/(a*b*x)] + (2*I )*(-1 + 2*a^4*b^4)*ArcTan[((1 + I)*a*b*x)/(I*b^2 + a^2*x^2 + Sqrt[-b^4 + a ^4*x^4])]))/(a^5*b^5)
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 1.08 (sec) , antiderivative size = 397, normalized size of antiderivative = 1.33, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.085, Rules used = {1388, 7276, 6, 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.
| \(\displaystyle \int \frac {a^8 x^8+b^8+x^4}{\sqrt {a^4 x^4-b^4} \left (a^8 x^8-b^8\right )} \, dx\) |
| \(\Big \downarrow \) 1388 |
| \(\displaystyle \int \frac {a^8 x^8+b^8+x^4}{\left (a^4 x^4-b^4\right )^{3/2} \left (a^4 x^4+b^4\right )}dx\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \int \left (-\frac {b^4}{\left (a^4 x^4-b^4\right )^{3/2}}+\frac {a^4 x^4}{\left (a^4 x^4-b^4\right )^{3/2}}+\frac {1}{a^4 \left (a^4 x^4-b^4\right )^{3/2}}+\frac {2 a^4 b^8-b^4}{a^4 \left (a^4 x^4-b^4\right )^{3/2} \left (a^4 x^4+b^4\right )}\right )dx\) |
| \(\Big \downarrow \) 6 |
| \(\displaystyle \int \left (\frac {a^4 x^4}{\left (a^4 x^4-b^4\right )^{3/2}}+\frac {\frac {1}{a^4}-b^4}{\left (a^4 x^4-b^4\right )^{3/2}}+\frac {2 a^4 b^8-b^4}{a^4 \left (a^4 x^4-b^4\right )^{3/2} \left (a^4 x^4+b^4\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {b \sqrt {1-\frac {a^4 x^4}{b^4}} \operatorname {EllipticF}\left (\arcsin \left (\frac {a x}{b}\right ),-1\right )}{2 a \sqrt {a^4 x^4-b^4}}-\frac {\left (\frac {1}{a^4}-b^4\right ) \sqrt {1-\frac {a^4 x^4}{b^4}} \operatorname {EllipticF}\left (\arcsin \left (\frac {a x}{b}\right ),-1\right )}{2 a b^3 \sqrt {a^4 x^4-b^4}}-\frac {\left (1-2 a^4 b^4\right ) \arctan \left (\frac {\sqrt {2} \sqrt [4]{-a^4} b x}{\sqrt {a^4 x^4-b^4}}\right )}{8 \sqrt {2} \left (-a^4\right )^{5/4} b^5}-\frac {\left (1-2 a^4 b^4\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{-a^4} b x}{\sqrt {a^4 x^4-b^4}}\right )}{8 \sqrt {2} \left (-a^4\right )^{5/4} b^5}-\frac {x \left (2-\frac {1}{a^4 b^4}\right )}{4 \sqrt {a^4 x^4-b^4}}-\frac {x \left (\frac {1}{a^4}-b^4\right )}{2 b^4 \sqrt {a^4 x^4-b^4}}-\frac {x}{2 \sqrt {a^4 x^4-b^4}}+\frac {\left (1-2 a^4 b^4\right ) \sqrt {1-\frac {a^4 x^4}{b^4}} \operatorname {EllipticF}\left (\arcsin \left (\frac {a x}{b}\right ),-1\right )}{2 a^5 b^3 \sqrt {a^4 x^4-b^4}}\) |
-1/2*x/Sqrt[-b^4 + a^4*x^4] - ((2 - 1/(a^4*b^4))*x)/(4*Sqrt[-b^4 + a^4*x^4 ]) - ((a^(-4) - b^4)*x)/(2*b^4*Sqrt[-b^4 + a^4*x^4]) - ((1 - 2*a^4*b^4)*Ar cTan[(Sqrt[2]*(-a^4)^(1/4)*b*x)/Sqrt[-b^4 + a^4*x^4]])/(8*Sqrt[2]*(-a^4)^( 5/4)*b^5) - ((1 - 2*a^4*b^4)*ArcTanh[(Sqrt[2]*(-a^4)^(1/4)*b*x)/Sqrt[-b^4 + a^4*x^4]])/(8*Sqrt[2]*(-a^4)^(5/4)*b^5) + (b*Sqrt[1 - (a^4*x^4)/b^4]*Ell ipticF[ArcSin[(a*x)/b], -1])/(2*a*Sqrt[-b^4 + a^4*x^4]) - ((a^(-4) - b^4)* Sqrt[1 - (a^4*x^4)/b^4]*EllipticF[ArcSin[(a*x)/b], -1])/(2*a*b^3*Sqrt[-b^4 + a^4*x^4]) + ((1 - 2*a^4*b^4)*Sqrt[1 - (a^4*x^4)/b^4]*EllipticF[ArcSin[( a*x)/b], -1])/(2*a^5*b^3*Sqrt[-b^4 + a^4*x^4])
3.29.59.3.1 Defintions of rubi rules used
Int[(u_.)*((v_.) + (a_.)*(Fx_) + (b_.)*(Fx_))^(p_.), x_Symbol] :> Int[u*(v + (a + b)*Fx)^p, x] /; FreeQ[{a, b}, x] && !FreeQ[Fx, x]
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 = 5.02 (sec) , antiderivative size = 258, normalized size of antiderivative = 0.86
| method | result | size |
| elliptic | \(\frac {\left (-\frac {\left (2 a^{4} b^{4}+1\right ) \sqrt {2}\, x}{4 a^{4} b^{4} \sqrt {a^{4} x^{4}-b^{4}}}+\frac {\left (2 a^{4} b^{4}-1\right ) \sqrt {2}\, \left (\ln \left (\frac {\frac {a^{4} x^{4}-b^{4}}{2 x^{2}}-\frac {\left (a^{4} b^{4}\right )^{\frac {1}{4}} \sqrt {a^{4} x^{4}-b^{4}}}{x}+\sqrt {a^{4} b^{4}}}{\frac {a^{4} x^{4}-b^{4}}{2 x^{2}}+\frac {\left (a^{4} b^{4}\right )^{\frac {1}{4}} \sqrt {a^{4} x^{4}-b^{4}}}{x}+\sqrt {a^{4} b^{4}}}\right )+2 \arctan \left (\frac {\sqrt {a^{4} x^{4}-b^{4}}}{\left (a^{4} b^{4}\right )^{\frac {1}{4}} x}+1\right )+2 \arctan \left (\frac {\sqrt {a^{4} x^{4}-b^{4}}}{\left (a^{4} b^{4}\right )^{\frac {1}{4}} x}-1\right )\right )}{32 a^{4} b^{4} \left (a^{4} b^{4}\right )^{\frac {1}{4}}}\right ) \sqrt {2}}{2}\) | \(258\) |
| default | \(-\frac {i \left (\left (4 x \sqrt {a^{4} x^{4}-b^{4}}\, \left (2 a^{4} b^{4}+1\right ) \sqrt {i a^{2} b^{2}}+\left (\ln \left (\frac {a^{2} \left (-2 i a^{2} b^{2} x +2 \sqrt {i a^{2} b^{2}}\, a^{2} x^{2}+2 i \sqrt {i a^{2} b^{2}}\, b^{2}+\sqrt {2}\, \sqrt {i a^{2} b^{2}}\, \sqrt {a^{4} x^{4}-b^{4}}\right )}{a^{2} x^{2}+i b^{2}-2 \sqrt {i a^{2} b^{2}}\, x}\right )+\ln \left (-\frac {2 a^{2} \left (-\frac {\sqrt {2}\, \sqrt {i a^{2} b^{2}}\, \sqrt {a^{4} x^{4}-b^{4}}}{2}+\left (a^{2} x^{2}+i b^{2}\right ) \sqrt {i a^{2} b^{2}}+i a^{2} b^{2} x \right )}{a^{2} x^{2}+i b^{2}+2 \sqrt {i a^{2} b^{2}}\, x}\right )+2 \ln \left (2\right )\right ) \left (a^{2} x^{2}+b^{2}\right ) \left (a x -b \right ) \sqrt {2}\, \left (a x +b \right ) \left (a^{4} b^{4}-\frac {1}{2}\right )\right ) \sqrt {-i a^{2} b^{2}}+2 \left (a^{2} x^{2}+b^{2}\right ) \left (a x -b \right ) \sqrt {2}\, \left (a x +b \right ) \sqrt {i a^{2} b^{2}}\, \left (\ln \left (\frac {\left (-2 i a^{2} b^{2} x +\sqrt {2}\, \sqrt {-i a^{2} b^{2}}\, \sqrt {a^{4} x^{4}-b^{4}}\right ) a^{2}}{a^{2} x^{2}+i b^{2}}\right )+\ln \left (2\right )\right ) \left (a^{4} b^{4}-\frac {1}{2}\right )\right )}{8 \sqrt {i a^{2} b^{2}}\, \sqrt {-i a^{2} b^{2}}\, \left (-2 \sqrt {i a^{2} b^{2}}+\left (1+i\right ) a b \right ) a^{2} \left (2 \sqrt {i a^{2} b^{2}}+\left (1+i\right ) a b \right ) \left (-a x +i b \right ) \left (a x -b \right ) \left (a x +b \right ) b^{2} \left (a x +i b \right )}\) | \(536\) |
| pseudoelliptic | \(-\frac {i \left (\left (4 x \sqrt {a^{4} x^{4}-b^{4}}\, \left (2 a^{4} b^{4}+1\right ) \sqrt {i a^{2} b^{2}}+\left (\ln \left (\frac {a^{2} \left (-2 i a^{2} b^{2} x +2 \sqrt {i a^{2} b^{2}}\, a^{2} x^{2}+2 i \sqrt {i a^{2} b^{2}}\, b^{2}+\sqrt {2}\, \sqrt {i a^{2} b^{2}}\, \sqrt {a^{4} x^{4}-b^{4}}\right )}{a^{2} x^{2}+i b^{2}-2 \sqrt {i a^{2} b^{2}}\, x}\right )+\ln \left (-\frac {2 a^{2} \left (-\frac {\sqrt {2}\, \sqrt {i a^{2} b^{2}}\, \sqrt {a^{4} x^{4}-b^{4}}}{2}+\left (a^{2} x^{2}+i b^{2}\right ) \sqrt {i a^{2} b^{2}}+i a^{2} b^{2} x \right )}{a^{2} x^{2}+i b^{2}+2 \sqrt {i a^{2} b^{2}}\, x}\right )+2 \ln \left (2\right )\right ) \left (a^{2} x^{2}+b^{2}\right ) \left (a x -b \right ) \sqrt {2}\, \left (a x +b \right ) \left (a^{4} b^{4}-\frac {1}{2}\right )\right ) \sqrt {-i a^{2} b^{2}}+2 \left (a^{2} x^{2}+b^{2}\right ) \left (a x -b \right ) \sqrt {2}\, \left (a x +b \right ) \sqrt {i a^{2} b^{2}}\, \left (\ln \left (\frac {\left (-2 i a^{2} b^{2} x +\sqrt {2}\, \sqrt {-i a^{2} b^{2}}\, \sqrt {a^{4} x^{4}-b^{4}}\right ) a^{2}}{a^{2} x^{2}+i b^{2}}\right )+\ln \left (2\right )\right ) \left (a^{4} b^{4}-\frac {1}{2}\right )\right )}{8 \sqrt {i a^{2} b^{2}}\, \sqrt {-i a^{2} b^{2}}\, \left (-2 \sqrt {i a^{2} b^{2}}+\left (1+i\right ) a b \right ) a^{2} \left (2 \sqrt {i a^{2} b^{2}}+\left (1+i\right ) a b \right ) \left (-a x +i b \right ) \left (a x -b \right ) \left (a x +b \right ) b^{2} \left (a x +i b \right )}\) | \(536\) |
1/2*(-1/4/a^4*(2*a^4*b^4+1)/b^4/(a^4*x^4-b^4)^(1/2)*2^(1/2)*x+1/32*(2*a^4* b^4-1)/a^4/b^4/(a^4*b^4)^(1/4)*2^(1/2)*(ln((1/2*(a^4*x^4-b^4)/x^2-(a^4*b^4 )^(1/4)*(a^4*x^4-b^4)^(1/2)/x+(a^4*b^4)^(1/2))/(1/2*(a^4*x^4-b^4)/x^2+(a^4 *b^4)^(1/4)*(a^4*x^4-b^4)^(1/2)/x+(a^4*b^4)^(1/2)))+2*arctan(1/(a^4*b^4)^( 1/4)*(a^4*x^4-b^4)^(1/2)/x+1)+2*arctan(1/(a^4*b^4)^(1/4)*(a^4*x^4-b^4)^(1/ 2)/x-1)))*2^(1/2)
Time = 1.10 (sec) , antiderivative size = 215, normalized size of antiderivative = 0.72 \[ \int \frac {b^8+x^4+a^8 x^8}{\sqrt {-b^4+a^4 x^4} \left (-b^8+a^8 x^8\right )} \, dx=-\frac {4 \, {\left (2 \, a^{5} b^{5} + a b\right )} \sqrt {a^{4} x^{4} - b^{4}} x + 2 \, {\left (2 \, a^{4} b^{8} - {\left (2 \, a^{8} b^{4} - a^{4}\right )} x^{4} - b^{4}\right )} \arctan \left (\frac {\sqrt {a^{4} x^{4} - b^{4}} a x}{a^{2} b x^{2} + b^{3}}\right ) + {\left (2 \, a^{4} b^{8} - {\left (2 \, a^{8} b^{4} - a^{4}\right )} x^{4} - b^{4}\right )} \log \left (\frac {a^{4} x^{4} + 2 \, a^{2} b^{2} x^{2} - b^{4} - 2 \, \sqrt {a^{4} x^{4} - b^{4}} a b x}{a^{4} x^{4} + b^{4}}\right )}{16 \, {\left (a^{9} b^{5} x^{4} - a^{5} b^{9}\right )}} \]
-1/16*(4*(2*a^5*b^5 + a*b)*sqrt(a^4*x^4 - b^4)*x + 2*(2*a^4*b^8 - (2*a^8*b ^4 - a^4)*x^4 - b^4)*arctan(sqrt(a^4*x^4 - b^4)*a*x/(a^2*b*x^2 + b^3)) + ( 2*a^4*b^8 - (2*a^8*b^4 - a^4)*x^4 - b^4)*log((a^4*x^4 + 2*a^2*b^2*x^2 - b^ 4 - 2*sqrt(a^4*x^4 - b^4)*a*b*x)/(a^4*x^4 + b^4)))/(a^9*b^5*x^4 - a^5*b^9)
\[ \int \frac {b^8+x^4+a^8 x^8}{\sqrt {-b^4+a^4 x^4} \left (-b^8+a^8 x^8\right )} \, dx=\int \frac {a^{8} x^{8} + b^{8} + x^{4}}{\sqrt {\left (a x - b\right ) \left (a x + b\right ) \left (a^{2} x^{2} + b^{2}\right )} \left (a x - b\right ) \left (a x + b\right ) \left (a^{2} x^{2} + b^{2}\right ) \left (a^{4} x^{4} + b^{4}\right )}\, dx \]
Integral((a**8*x**8 + b**8 + x**4)/(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**4*x**4 + b**4)), x)
\[ \int \frac {b^8+x^4+a^8 x^8}{\sqrt {-b^4+a^4 x^4} \left (-b^8+a^8 x^8\right )} \, dx=\int { \frac {a^{8} x^{8} + b^{8} + x^{4}}{{\left (a^{8} x^{8} - b^{8}\right )} \sqrt {a^{4} x^{4} - b^{4}}} \,d x } \]
\[ \int \frac {b^8+x^4+a^8 x^8}{\sqrt {-b^4+a^4 x^4} \left (-b^8+a^8 x^8\right )} \, dx=\int { \frac {a^{8} x^{8} + b^{8} + x^{4}}{{\left (a^{8} x^{8} - b^{8}\right )} \sqrt {a^{4} x^{4} - b^{4}}} \,d x } \]
Timed out. \[ \int \frac {b^8+x^4+a^8 x^8}{\sqrt {-b^4+a^4 x^4} \left (-b^8+a^8 x^8\right )} \, dx=\int -\frac {a^8\,x^8+b^8+x^4}{\sqrt {a^4\,x^4-b^4}\,\left (b^8-a^8\,x^8\right )} \,d x \]