Integrand size = 44, antiderivative size = 150 \[ \int \frac {-b^8+a^8 x^8}{\sqrt {-b^4+a^4 x^4} \left (b^8+a^8 x^8\right )} \, dx=-\frac {\arctan \left (\frac {\frac {b^3}{2^{3/4} a}+\frac {a b x^2}{\sqrt [4]{2}}-\frac {a^3 x^4}{2^{3/4} b}}{x \sqrt {-b^4+a^4 x^4}}\right )}{2\ 2^{3/4} a b}-\frac {\text {arctanh}\left (\frac {2^{3/4} a b x \sqrt {-b^4+a^4 x^4}}{-b^4+\sqrt {2} a^2 b^2 x^2+a^4 x^4}\right )}{2\ 2^{3/4} a b} \]
-1/4*arctan((1/2*b^3*2^(1/4)/a+1/2*a*b*x^2*2^(3/4)-1/2*a^3*x^4*2^(1/4)/b)/ x/(a^4*x^4-b^4)^(1/2))*2^(1/4)/a/b-1/4*arctanh(2^(3/4)*a*b*x*(a^4*x^4-b^4) ^(1/2)/(-b^4+2^(1/2)*a^2*b^2*x^2+a^4*x^4))*2^(1/4)/a/b
Time = 0.66 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.01 \[ \int \frac {-b^8+a^8 x^8}{\sqrt {-b^4+a^4 x^4} \left (b^8+a^8 x^8\right )} \, dx=\frac {\arctan \left (\frac {a b x}{a b x-\sqrt [4]{2} \sqrt {-b^4+a^4 x^4}}\right )-\arctan \left (\frac {a b x}{a b x+\sqrt [4]{2} \sqrt {-b^4+a^4 x^4}}\right )-\text {arctanh}\left (\frac {-b^4+\sqrt {2} a^2 b^2 x^2+a^4 x^4}{2^{3/4} a b x \sqrt {-b^4+a^4 x^4}}\right )}{2\ 2^{3/4} a b} \]
(ArcTan[(a*b*x)/(a*b*x - 2^(1/4)*Sqrt[-b^4 + a^4*x^4])] - ArcTan[(a*b*x)/( a*b*x + 2^(1/4)*Sqrt[-b^4 + a^4*x^4])] - ArcTanh[(-b^4 + Sqrt[2]*a^2*b^2*x ^2 + a^4*x^4)/(2^(3/4)*a*b*x*Sqrt[-b^4 + a^4*x^4])])/(2*2^(3/4)*a*b)
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 0.89 (sec) , antiderivative size = 400, normalized size of antiderivative = 2.67, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.068, 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 |
| \(\displaystyle \int \frac {\sqrt {a^4 x^4-b^4} \left (a^4 x^4+b^4\right )}{a^8 x^8+b^8}dx\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \int \left (\frac {\sqrt {-a^8} \left (a^4 b^4-\sqrt {-a^8} b^4\right ) \sqrt {a^4 x^4-b^4}}{2 a^8 b^4 \left (\sqrt {-a^8} x^4+b^4\right )}-\frac {\sqrt {-a^8} \left (\sqrt {-a^8} b^4+a^4 b^4\right ) \sqrt {a^4 x^4-b^4}}{2 a^8 b^4 \left (b^4-\sqrt {-a^8} x^4\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {b \sqrt {1-\frac {a^4 x^4}{b^4}} \operatorname {EllipticPi}\left (\frac {a^6}{\left (-a^8\right )^{3/4}},\arcsin \left (\frac {a x}{b}\right ),-1\right )}{2 a \sqrt {a^4 x^4-b^4}}+\frac {\left (a^4-\sqrt {-a^8}\right ) b \sqrt {1-\frac {a^4 x^4}{b^4}} \operatorname {EllipticF}\left (\arcsin \left (\frac {a x}{b}\right ),-1\right )}{2 a^5 \sqrt {a^4 x^4-b^4}}+\frac {\left (\sqrt {-a^8}+a^4\right ) b \sqrt {1-\frac {a^4 x^4}{b^4}} \operatorname {EllipticF}\left (\arcsin \left (\frac {a x}{b}\right ),-1\right )}{2 a^5 \sqrt {a^4 x^4-b^4}}-\frac {b \sqrt {1-\frac {a^4 x^4}{b^4}} \operatorname {EllipticPi}\left (\frac {\sqrt [4]{-a^8}}{a^2},\arcsin \left (\frac {a x}{b}\right ),-1\right )}{2 a \sqrt {a^4 x^4-b^4}}-\frac {b \sqrt {1-\frac {a^4 x^4}{b^4}} \operatorname {EllipticPi}\left (-\frac {\sqrt {-\sqrt {-a^8}}}{a^2},\arcsin \left (\frac {a x}{b}\right ),-1\right )}{2 a \sqrt {a^4 x^4-b^4}}-\frac {b \sqrt {1-\frac {a^4 x^4}{b^4}} \operatorname {EllipticPi}\left (\frac {\sqrt {-\sqrt {-a^8}}}{a^2},\arcsin \left (\frac {a x}{b}\right ),-1\right )}{2 a \sqrt {a^4 x^4-b^4}}\) |
((a^4 - Sqrt[-a^8])*b*Sqrt[1 - (a^4*x^4)/b^4]*EllipticF[ArcSin[(a*x)/b], - 1])/(2*a^5*Sqrt[-b^4 + a^4*x^4]) + ((a^4 + Sqrt[-a^8])*b*Sqrt[1 - (a^4*x^4 )/b^4]*EllipticF[ArcSin[(a*x)/b], -1])/(2*a^5*Sqrt[-b^4 + a^4*x^4]) - (b*S qrt[1 - (a^4*x^4)/b^4]*EllipticPi[a^6/(-a^8)^(3/4), ArcSin[(a*x)/b], -1])/ (2*a*Sqrt[-b^4 + a^4*x^4]) - (b*Sqrt[1 - (a^4*x^4)/b^4]*EllipticPi[(-a^8)^ (1/4)/a^2, ArcSin[(a*x)/b], -1])/(2*a*Sqrt[-b^4 + a^4*x^4]) - (b*Sqrt[1 - (a^4*x^4)/b^4]*EllipticPi[-(Sqrt[-Sqrt[-a^8]]/a^2), ArcSin[(a*x)/b], -1])/ (2*a*Sqrt[-b^4 + a^4*x^4]) - (b*Sqrt[1 - (a^4*x^4)/b^4]*EllipticPi[Sqrt[-S qrt[-a^8]]/a^2, ArcSin[(a*x)/b], -1])/(2*a*Sqrt[-b^4 + a^4*x^4])
3.21.83.3.1 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]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 2.94 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.21
| method | result | size |
| pseudoelliptic | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (16 a^{8} b^{8}+32 i a^{6} b^{6} \textit {\_Z}^{2}+8 b^{4} a^{4} \textit {\_Z}^{4}-8 i a^{2} b^{2} \textit {\_Z}^{6}+\textit {\_Z}^{8}\right )}{\sum }\frac {\left (8 i a^{6} b^{6}-4 \textit {\_R}^{2} a^{4} b^{4}+2 i \textit {\_R}^{4} a^{2} b^{2}-\textit {\_R}^{6}\right ) \ln \left (\frac {\sqrt {a^{4} x^{4}-b^{4}}+\left (-a^{2} x^{2}-i b^{2}\right ) \operatorname {csgn}\left (a^{2}\right )-\textit {\_R} x}{x}\right )}{\textit {\_R} \left (-8 i a^{6} b^{6}-4 \textit {\_R}^{2} a^{4} b^{4}+6 i \textit {\_R}^{4} a^{2} b^{2}-\textit {\_R}^{6}\right )}\right )}{4}\) | \(181\) |
| elliptic | \(\frac {\left (\ln \left (\frac {\frac {a^{4} x^{4}-b^{4}}{2 x^{2}}-\frac {\sqrt {\sqrt {2}\, \sqrt {a^{4} b^{4}}}\, \sqrt {a^{4} x^{4}-b^{4}}\, \sqrt {2}}{2 x}+\frac {\sqrt {2}\, \sqrt {a^{4} b^{4}}}{2}}{\frac {a^{4} x^{4}-b^{4}}{2 x^{2}}+\frac {\sqrt {\sqrt {2}\, \sqrt {a^{4} b^{4}}}\, \sqrt {a^{4} x^{4}-b^{4}}\, \sqrt {2}}{2 x}+\frac {\sqrt {2}\, \sqrt {a^{4} b^{4}}}{2}}\right )+2 \arctan \left (\frac {\sqrt {a^{4} x^{4}-b^{4}}\, \sqrt {2}}{\sqrt {\sqrt {2}\, \sqrt {a^{4} b^{4}}}\, x}+1\right )+2 \arctan \left (\frac {\sqrt {a^{4} x^{4}-b^{4}}\, \sqrt {2}}{\sqrt {\sqrt {2}\, \sqrt {a^{4} b^{4}}}\, x}-1\right )\right ) \sqrt {2}}{8 \sqrt {\sqrt {2}\, \sqrt {a^{4} b^{4}}}}\) | \(252\) |
| default | \(\frac {\sqrt {\frac {a^{2} x^{2}}{b^{2}}+1}\, \sqrt {1-\frac {a^{2} x^{2}}{b^{2}}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {a^{2}}{b^{2}}}, i\right )}{\sqrt {-\frac {a^{2}}{b^{2}}}\, \sqrt {a^{4} x^{4}-b^{4}}}-\frac {b^{8} \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (a^{8} \textit {\_Z}^{8}+b^{8}\right )}{\sum }\frac {-\frac {\operatorname {arctanh}\left (\frac {\underline {\hspace {1.25 ex}}\alpha ^{2} \left (a^{4} \underline {\hspace {1.25 ex}}\alpha ^{6}+b^{4} x^{2}\right ) a^{4}}{b^{4} \sqrt {a^{4} \underline {\hspace {1.25 ex}}\alpha ^{4}-b^{4}}\, \sqrt {a^{4} x^{4}-b^{4}}}\right )}{\sqrt {a^{4} \underline {\hspace {1.25 ex}}\alpha ^{4}-b^{4}}}+\frac {2 \underline {\hspace {1.25 ex}}\alpha ^{7} a^{8} \sqrt {\frac {a^{2} x^{2}}{b^{2}}+1}\, \sqrt {1-\frac {a^{2} x^{2}}{b^{2}}}\, \operatorname {EllipticPi}\left (x \sqrt {-\frac {a^{2}}{b^{2}}}, \frac {\underline {\hspace {1.25 ex}}\alpha ^{6} a^{6}}{b^{6}}, \frac {\sqrt {\frac {a^{2}}{b^{2}}}}{\sqrt {-\frac {a^{2}}{b^{2}}}}\right )}{\sqrt {-\frac {a^{2}}{b^{2}}}\, b^{8} \sqrt {a^{4} x^{4}-b^{4}}}}{\underline {\hspace {1.25 ex}}\alpha ^{7}}\right )}{8 a^{8}}\) | \(281\) |
1/4*sum((8*I*a^6*b^6-4*_R^2*a^4*b^4+2*I*_R^4*a^2*b^2-_R^6)*ln(((a^4*x^4-b^ 4)^(1/2)+(-a^2*x^2-I*b^2)*csgn(a^2)-_R*x)/x)/_R/(-8*I*a^6*b^6-4*_R^2*a^4*b ^4+6*I*_R^4*a^2*b^2-_R^6),_R=RootOf(16*a^8*b^8+32*I*a^6*b^6*_Z^2+8*b^4*a^4 *_Z^4-8*I*a^2*b^2*_Z^6+_Z^8))
Result contains complex when optimal does not.
Time = 1.15 (sec) , antiderivative size = 651, normalized size of antiderivative = 4.34 \[ \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)*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))
\[ \int \frac {-b^8+a^8 x^8}{\sqrt {-b^4+a^4 x^4} \left (b^8+a^8 x^8\right )} \, dx=\int \frac {\left (a x - b\right ) \left (a x + b\right ) \left (a^{2} x^{2} + b^{2}\right ) \left (a^{4} x^{4} + b^{4}\right )}{\sqrt {\left (a x - b\right ) \left (a x + b\right ) \left (a^{2} x^{2} + b^{2}\right )} \left (a^{8} x^{8} + b^{8}\right )}\, dx \]
Integral((a*x - b)*(a*x + b)*(a**2*x**2 + b**2)*(a**4*x**4 + b**4)/(sqrt(( a*x - b)*(a*x + b)*(a**2*x**2 + b**2))*(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. \[ \int \frac {-b^8+a^8 x^8}{\sqrt {-b^4+a^4 x^4} \left (b^8+a^8 x^8\right )} \, dx=\int -\frac {b^8-a^8\,x^8}{\sqrt {a^4\,x^4-b^4}\,\left (a^8\,x^8+b^8\right )} \,d x \]