Integrand size = 50, antiderivative size = 229 \[ \int \frac {-b^8+a^8 x^8}{\sqrt {-b^4+a^4 x^4} \left (b^8-c x^4+a^8 x^8\right )} \, dx=-\frac {\arctan \left (\frac {\frac {b^4}{\sqrt {2} \sqrt [4]{2 a^4 b^4-c}}+\frac {\sqrt [4]{2 a^4 b^4-c} x^2}{\sqrt {2}}-\frac {a^4 x^4}{\sqrt {2} \sqrt [4]{2 a^4 b^4-c}}}{x \sqrt {-b^4+a^4 x^4}}\right )}{2 \sqrt {2} \sqrt [4]{2 a^4 b^4-c}}-\frac {\text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{2 a^4 b^4-c} x \sqrt {-b^4+a^4 x^4}}{-b^4+\sqrt {2 a^4 b^4-c} x^2+a^4 x^4}\right )}{2 \sqrt {2} \sqrt [4]{2 a^4 b^4-c}} \]
-1/4*arctan((1/2*b^4*2^(1/2)/(2*a^4*b^4-c)^(1/4)+1/2*(2*a^4*b^4-c)^(1/4)*x ^2*2^(1/2)-1/2*a^4*x^4*2^(1/2)/(2*a^4*b^4-c)^(1/4))/x/(a^4*x^4-b^4)^(1/2)) *2^(1/2)/(2*a^4*b^4-c)^(1/4)-1/4*arctanh(2^(1/2)*(2*a^4*b^4-c)^(1/4)*x*(a^ 4*x^4-b^4)^(1/2)/(-b^4+(2*a^4*b^4-c)^(1/2)*x^2+a^4*x^4))*2^(1/2)/(2*a^4*b^ 4-c)^(1/4)
Time = 1.07 (sec) , antiderivative size = 171, normalized size of antiderivative = 0.75 \[ \int \frac {-b^8+a^8 x^8}{\sqrt {-b^4+a^4 x^4} \left (b^8-c x^4+a^8 x^8\right )} \, dx=\frac {\arctan \left (\frac {\sqrt [4]{2 a^4 b^4-c} x \sqrt {-2 b^4+2 a^4 x^4}}{b^4+\sqrt {2 a^4 b^4-c} x^2-a^4 x^4}\right )-\text {arctanh}\left (\frac {-b^4+\sqrt {2 a^4 b^4-c} x^2+a^4 x^4}{\sqrt [4]{2 a^4 b^4-c} x \sqrt {-2 b^4+2 a^4 x^4}}\right )}{2 \sqrt {2} \sqrt [4]{2 a^4 b^4-c}} \]
(ArcTan[((2*a^4*b^4 - c)^(1/4)*x*Sqrt[-2*b^4 + 2*a^4*x^4])/(b^4 + Sqrt[2*a ^4*b^4 - c]*x^2 - a^4*x^4)] - ArcTanh[(-b^4 + Sqrt[2*a^4*b^4 - c]*x^2 + a^ 4*x^4)/((2*a^4*b^4 - c)^(1/4)*x*Sqrt[-2*b^4 + 2*a^4*x^4])])/(2*Sqrt[2]*(2* a^4*b^4 - c)^(1/4))
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 1.29 (sec) , antiderivative size = 508, normalized size of antiderivative = 2.22, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.060, Rules used = {1388, 7279, 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.
| \(\displaystyle \int \frac {a^8 x^8-b^8}{\sqrt {a^4 x^4-b^4} \left (a^8 x^8+b^8-c x^4\right )} \, dx\) |
| \(\Big \downarrow \) 1388 |
| \(\displaystyle \int \frac {\sqrt {a^4 x^4-b^4} \left (a^4 x^4+b^4\right )}{a^8 x^8+b^8-c x^4}dx\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle \int \left (\frac {\left (a^4-\frac {a^4 \left (2 a^4 b^4+c\right )}{\sqrt {c^2-4 a^8 b^8}}\right ) \sqrt {a^4 x^4-b^4}}{\sqrt {c^2-4 a^8 b^8}+2 a^8 x^4-c}+\frac {\left (a^4+\frac {a^4 \left (2 a^4 b^4+c\right )}{\sqrt {c^2-4 a^8 b^8}}\right ) \sqrt {a^4 x^4-b^4}}{-\sqrt {c^2-4 a^8 b^8}+2 a^8 x^4-c}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {b \left (1-\frac {2 a^4 b^4+c}{\sqrt {c^2-4 a^8 b^8}}\right ) \sqrt {1-\frac {a^4 x^4}{b^4}} \operatorname {EllipticF}\left (\arcsin \left (\frac {a x}{b}\right ),-1\right )}{2 a \sqrt {a^4 x^4-b^4}}+\frac {b \left (\frac {2 a^4 b^4+c}{\sqrt {c^2-4 a^8 b^8}}+1\right ) \sqrt {1-\frac {a^4 x^4}{b^4}} \operatorname {EllipticF}\left (\arcsin \left (\frac {a x}{b}\right ),-1\right )}{2 a \sqrt {a^4 x^4-b^4}}-\frac {b \sqrt {1-\frac {a^4 x^4}{b^4}} \operatorname {EllipticPi}\left (-\frac {\sqrt {2} a^2 b^2}{\sqrt {c-\sqrt {c^2-4 a^8 b^8}}},\arcsin \left (\frac {a x}{b}\right ),-1\right )}{2 a \sqrt {a^4 x^4-b^4}}-\frac {b \sqrt {1-\frac {a^4 x^4}{b^4}} \operatorname {EllipticPi}\left (\frac {\sqrt {2} a^2 b^2}{\sqrt {c-\sqrt {c^2-4 a^8 b^8}}},\arcsin \left (\frac {a x}{b}\right ),-1\right )}{2 a \sqrt {a^4 x^4-b^4}}-\frac {b \sqrt {1-\frac {a^4 x^4}{b^4}} \operatorname {EllipticPi}\left (-\frac {\sqrt {2} a^2 b^2}{\sqrt {c+\sqrt {c^2-4 a^8 b^8}}},\arcsin \left (\frac {a x}{b}\right ),-1\right )}{2 a \sqrt {a^4 x^4-b^4}}-\frac {b \sqrt {1-\frac {a^4 x^4}{b^4}} \operatorname {EllipticPi}\left (\frac {\sqrt {2} a^2 b^2}{\sqrt {c+\sqrt {c^2-4 a^8 b^8}}},\arcsin \left (\frac {a x}{b}\right ),-1\right )}{2 a \sqrt {a^4 x^4-b^4}}\) |
(b*(1 - (2*a^4*b^4 + c)/Sqrt[-4*a^8*b^8 + c^2])*Sqrt[1 - (a^4*x^4)/b^4]*El lipticF[ArcSin[(a*x)/b], -1])/(2*a*Sqrt[-b^4 + a^4*x^4]) + (b*(1 + (2*a^4* b^4 + c)/Sqrt[-4*a^8*b^8 + c^2])*Sqrt[1 - (a^4*x^4)/b^4]*EllipticF[ArcSin[ (a*x)/b], -1])/(2*a*Sqrt[-b^4 + a^4*x^4]) - (b*Sqrt[1 - (a^4*x^4)/b^4]*Ell ipticPi[-((Sqrt[2]*a^2*b^2)/Sqrt[c - Sqrt[-4*a^8*b^8 + c^2]]), ArcSin[(a*x )/b], -1])/(2*a*Sqrt[-b^4 + a^4*x^4]) - (b*Sqrt[1 - (a^4*x^4)/b^4]*Ellipti cPi[(Sqrt[2]*a^2*b^2)/Sqrt[c - Sqrt[-4*a^8*b^8 + c^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[2]*a^2*b^2)/Sqrt[c + Sqrt[-4*a^8*b^8 + c^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[2 ]*a^2*b^2)/Sqrt[c + Sqrt[-4*a^8*b^8 + c^2]], ArcSin[(a*x)/b], -1])/(2*a*Sq rt[-b^4 + a^4*x^4])
3.27.19.3.1 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_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[ {v = RationalFunctionExpand[u/(a + b*x^n + c*x^(2*n)), x]}, Int[v, x] /; Su mQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 3.01 (sec) , antiderivative size = 185, normalized size of antiderivative = 0.81
| method | result | size |
| pseudoelliptic | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}-8 i a^{2} b^{2} \textit {\_Z}^{6}+\left (8 a^{4} b^{4}-16 c \right ) \textit {\_Z}^{4}+32 i a^{6} b^{6} \textit {\_Z}^{2}+16 a^{8} b^{8}\right )}{\sum }\frac {\left (-8 i a^{6} b^{6}+4 \textit {\_R}^{2} a^{4} b^{4}-2 i \textit {\_R}^{4} a^{2} b^{2}+\textit {\_R}^{6}\right ) \ln \left (\frac {\sqrt {a^{4} x^{4}-b^{4}}+\left (-a^{2} x^{2}-i b^{2}\right ) \operatorname {csgn}\left (a^{2}\right )-\textit {\_R} x}{x}\right )}{\textit {\_R}^{7}-6 i a^{2} b^{2} \textit {\_R}^{5}+\left (4 a^{4} b^{4}-8 c \right ) \textit {\_R}^{3}+8 i a^{6} b^{6} \textit {\_R}}\right )}{4}\) | \(185\) |
| elliptic | \(\frac {\left (\ln \left (\frac {\frac {a^{4} x^{4}-b^{4}}{2 x^{2}}-\frac {\left (2 a^{4} b^{4}-c \right )^{\frac {1}{4}} \sqrt {a^{4} x^{4}-b^{4}}\, \sqrt {2}}{2 x}+\frac {\sqrt {2 a^{4} b^{4}-c}}{2}}{\frac {a^{4} x^{4}-b^{4}}{2 x^{2}}+\frac {\left (2 a^{4} b^{4}-c \right )^{\frac {1}{4}} \sqrt {a^{4} x^{4}-b^{4}}\, \sqrt {2}}{2 x}+\frac {\sqrt {2 a^{4} b^{4}-c}}{2}}\right )+2 \arctan \left (\frac {\sqrt {a^{4} x^{4}-b^{4}}\, \sqrt {2}}{\left (2 a^{4} b^{4}-c \right )^{\frac {1}{4}} x}+1\right )+2 \arctan \left (\frac {\sqrt {a^{4} x^{4}-b^{4}}\, \sqrt {2}}{\left (2 a^{4} b^{4}-c \right )^{\frac {1}{4}} x}-1\right )\right ) \sqrt {2}}{8 \left (2 a^{4} b^{4}-c \right )^{\frac {1}{4}}}\) | \(251\) |
| default | \(\frac {\sqrt {\frac {a^{2} x^{2}}{b^{2}}+1}\, \sqrt {1-\frac {a^{2} x^{2}}{b^{2}}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {a^{2}}{b^{2}}}, i\right )}{\sqrt {-\frac {a^{2}}{b^{2}}}\, \sqrt {a^{4} x^{4}-b^{4}}}+\frac {\left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (a^{8} \textit {\_Z}^{8}+b^{8}-c \,\textit {\_Z}^{4}\right )}{\sum }\frac {\left (-2 b^{8}+\underline {\hspace {1.25 ex}}\alpha ^{4} c \right ) \left (-\frac {\operatorname {arctanh}\left (\frac {\underline {\hspace {1.25 ex}}\alpha ^{2} \left (\underline {\hspace {1.25 ex}}\alpha ^{6} a^{8}+a^{4} b^{4} x^{2}-\underline {\hspace {1.25 ex}}\alpha ^{2} c \right )}{b^{4} \sqrt {\underline {\hspace {1.25 ex}}\alpha ^{4} a^{4}-b^{4}}\, \sqrt {a^{4} x^{4}-b^{4}}}\right )}{\sqrt {\underline {\hspace {1.25 ex}}\alpha ^{4} a^{4}-b^{4}}}+\frac {2 \underline {\hspace {1.25 ex}}\alpha ^{3} \left (\underline {\hspace {1.25 ex}}\alpha ^{4} a^{8}-c \right ) \sqrt {\frac {a^{2} x^{2}}{b^{2}}+1}\, \sqrt {1-\frac {a^{2} x^{2}}{b^{2}}}\, \operatorname {EllipticPi}\left (x \sqrt {-\frac {a^{2}}{b^{2}}}, \frac {\underline {\hspace {1.25 ex}}\alpha ^{2} \left (\underline {\hspace {1.25 ex}}\alpha ^{4} a^{8}-c \right )}{a^{2} b^{6}}, \frac {\sqrt {\frac {a^{2}}{b^{2}}}}{\sqrt {-\frac {a^{2}}{b^{2}}}}\right )}{\sqrt {-\frac {a^{2}}{b^{2}}}\, b^{8} \sqrt {a^{4} x^{4}-b^{4}}}\right )}{\underline {\hspace {1.25 ex}}\alpha ^{3} \left (2 \underline {\hspace {1.25 ex}}\alpha ^{4} a^{8}-c \right )}\right )}{8}\) | \(331\) |
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^7-6*I*a^2*b^2*_R^5+(4*a^ 4*b^4-8*c)*_R^3+8*I*a^6*b^6*_R),_R=RootOf(_Z^8-8*I*a^2*b^2*_Z^6+(8*a^4*b^4 -16*c)*_Z^4+32*I*a^6*b^6*_Z^2+16*a^8*b^8))
Result contains complex when optimal does not.
Time = 5.66 (sec) , antiderivative size = 873, normalized size of antiderivative = 3.81 \[ \int \frac {-b^8+a^8 x^8}{\sqrt {-b^4+a^4 x^4} \left (b^8-c x^4+a^8 x^8\right )} \, dx=\frac {1}{8} \, \left (-\frac {1}{2 \, a^{4} b^{4} - c}\right )^{\frac {1}{4}} \log \left (\frac {2 \, {\left ({\left (2 \, a^{8} b^{4} - a^{4} c\right )} x^{6} - {\left (2 \, a^{4} b^{8} - b^{4} c\right )} x^{2}\right )} \left (-\frac {1}{2 \, a^{4} b^{4} - c}\right )^{\frac {3}{4}} + 2 \, {\left (a^{4} x^{5} - b^{4} x - {\left (2 \, a^{4} b^{4} - c\right )} x^{3} \sqrt {-\frac {1}{2 \, a^{4} b^{4} - c}}\right )} \sqrt {a^{4} x^{4} - b^{4}} - {\left (a^{8} x^{8} + b^{8} - {\left (4 \, a^{4} b^{4} - c\right )} x^{4}\right )} \left (-\frac {1}{2 \, a^{4} b^{4} - c}\right )^{\frac {1}{4}}}{2 \, {\left (a^{8} x^{8} + b^{8} - c x^{4}\right )}}\right ) - \frac {1}{8} \, \left (-\frac {1}{2 \, a^{4} b^{4} - c}\right )^{\frac {1}{4}} \log \left (-\frac {2 \, {\left ({\left (2 \, a^{8} b^{4} - a^{4} c\right )} x^{6} - {\left (2 \, a^{4} b^{8} - b^{4} c\right )} x^{2}\right )} \left (-\frac {1}{2 \, a^{4} b^{4} - c}\right )^{\frac {3}{4}} - 2 \, {\left (a^{4} x^{5} - b^{4} x - {\left (2 \, a^{4} b^{4} - c\right )} x^{3} \sqrt {-\frac {1}{2 \, a^{4} b^{4} - c}}\right )} \sqrt {a^{4} x^{4} - b^{4}} - {\left (a^{8} x^{8} + b^{8} - {\left (4 \, a^{4} b^{4} - c\right )} x^{4}\right )} \left (-\frac {1}{2 \, a^{4} b^{4} - c}\right )^{\frac {1}{4}}}{2 \, {\left (a^{8} x^{8} + b^{8} - c x^{4}\right )}}\right ) + \frac {1}{8} i \, \left (-\frac {1}{2 \, a^{4} b^{4} - c}\right )^{\frac {1}{4}} \log \left (-\frac {2 \, {\left (i \, {\left (2 \, a^{8} b^{4} - a^{4} c\right )} x^{6} - i \, {\left (2 \, a^{4} b^{8} - b^{4} c\right )} x^{2}\right )} \left (-\frac {1}{2 \, a^{4} b^{4} - c}\right )^{\frac {3}{4}} - 2 \, {\left (a^{4} x^{5} - b^{4} x + {\left (2 \, a^{4} b^{4} - c\right )} x^{3} \sqrt {-\frac {1}{2 \, a^{4} b^{4} - c}}\right )} \sqrt {a^{4} x^{4} - b^{4}} + {\left (i \, a^{8} x^{8} + i \, b^{8} - i \, {\left (4 \, a^{4} b^{4} - c\right )} x^{4}\right )} \left (-\frac {1}{2 \, a^{4} b^{4} - c}\right )^{\frac {1}{4}}}{2 \, {\left (a^{8} x^{8} + b^{8} - c x^{4}\right )}}\right ) - \frac {1}{8} i \, \left (-\frac {1}{2 \, a^{4} b^{4} - c}\right )^{\frac {1}{4}} \log \left (-\frac {2 \, {\left (-i \, {\left (2 \, a^{8} b^{4} - a^{4} c\right )} x^{6} + i \, {\left (2 \, a^{4} b^{8} - b^{4} c\right )} x^{2}\right )} \left (-\frac {1}{2 \, a^{4} b^{4} - c}\right )^{\frac {3}{4}} - 2 \, {\left (a^{4} x^{5} - b^{4} x + {\left (2 \, a^{4} b^{4} - c\right )} x^{3} \sqrt {-\frac {1}{2 \, a^{4} b^{4} - c}}\right )} \sqrt {a^{4} x^{4} - b^{4}} + {\left (-i \, a^{8} x^{8} - i \, b^{8} + i \, {\left (4 \, a^{4} b^{4} - c\right )} x^{4}\right )} \left (-\frac {1}{2 \, a^{4} b^{4} - c}\right )^{\frac {1}{4}}}{2 \, {\left (a^{8} x^{8} + b^{8} - c x^{4}\right )}}\right ) \]
1/8*(-1/(2*a^4*b^4 - c))^(1/4)*log(1/2*(2*((2*a^8*b^4 - a^4*c)*x^6 - (2*a^ 4*b^8 - b^4*c)*x^2)*(-1/(2*a^4*b^4 - c))^(3/4) + 2*(a^4*x^5 - b^4*x - (2*a ^4*b^4 - c)*x^3*sqrt(-1/(2*a^4*b^4 - c)))*sqrt(a^4*x^4 - b^4) - (a^8*x^8 + b^8 - (4*a^4*b^4 - c)*x^4)*(-1/(2*a^4*b^4 - c))^(1/4))/(a^8*x^8 + b^8 - c *x^4)) - 1/8*(-1/(2*a^4*b^4 - c))^(1/4)*log(-1/2*(2*((2*a^8*b^4 - a^4*c)*x ^6 - (2*a^4*b^8 - b^4*c)*x^2)*(-1/(2*a^4*b^4 - c))^(3/4) - 2*(a^4*x^5 - b^ 4*x - (2*a^4*b^4 - c)*x^3*sqrt(-1/(2*a^4*b^4 - c)))*sqrt(a^4*x^4 - b^4) - (a^8*x^8 + b^8 - (4*a^4*b^4 - c)*x^4)*(-1/(2*a^4*b^4 - c))^(1/4))/(a^8*x^8 + b^8 - c*x^4)) + 1/8*I*(-1/(2*a^4*b^4 - c))^(1/4)*log(-1/2*(2*(I*(2*a^8* b^4 - a^4*c)*x^6 - I*(2*a^4*b^8 - b^4*c)*x^2)*(-1/(2*a^4*b^4 - c))^(3/4) - 2*(a^4*x^5 - b^4*x + (2*a^4*b^4 - c)*x^3*sqrt(-1/(2*a^4*b^4 - c)))*sqrt(a ^4*x^4 - b^4) + (I*a^8*x^8 + I*b^8 - I*(4*a^4*b^4 - c)*x^4)*(-1/(2*a^4*b^4 - c))^(1/4))/(a^8*x^8 + b^8 - c*x^4)) - 1/8*I*(-1/(2*a^4*b^4 - c))^(1/4)* log(-1/2*(2*(-I*(2*a^8*b^4 - a^4*c)*x^6 + I*(2*a^4*b^8 - b^4*c)*x^2)*(-1/( 2*a^4*b^4 - c))^(3/4) - 2*(a^4*x^5 - b^4*x + (2*a^4*b^4 - c)*x^3*sqrt(-1/( 2*a^4*b^4 - c)))*sqrt(a^4*x^4 - b^4) + (-I*a^8*x^8 - I*b^8 + I*(4*a^4*b^4 - c)*x^4)*(-1/(2*a^4*b^4 - c))^(1/4))/(a^8*x^8 + b^8 - c*x^4))
Timed out. \[ \int \frac {-b^8+a^8 x^8}{\sqrt {-b^4+a^4 x^4} \left (b^8-c x^4+a^8 x^8\right )} \, dx=\text {Timed out} \]
\[ \int \frac {-b^8+a^8 x^8}{\sqrt {-b^4+a^4 x^4} \left (b^8-c x^4+a^8 x^8\right )} \, dx=\int { \frac {a^{8} x^{8} - b^{8}}{{\left (a^{8} x^{8} + b^{8} - c x^{4}\right )} \sqrt {a^{4} x^{4} - b^{4}}} \,d x } \]
\[ \int \frac {-b^8+a^8 x^8}{\sqrt {-b^4+a^4 x^4} \left (b^8-c x^4+a^8 x^8\right )} \, dx=\int { \frac {a^{8} x^{8} - b^{8}}{{\left (a^{8} x^{8} + b^{8} - c x^{4}\right )} \sqrt {a^{4} x^{4} - b^{4}}} \,d x } \]
Timed out. \[ \int \frac {-b^8+a^8 x^8}{\sqrt {-b^4+a^4 x^4} \left (b^8-c x^4+a^8 x^8\right )} \, dx=\int -\frac {b^8-a^8\,x^8}{\sqrt {a^4\,x^4-b^4}\,\left (a^8\,x^8+b^8-c\,x^4\right )} \,d x \]