Integrand size = 44, antiderivative size = 297 \[ \int \frac {b^{16}+a^{16} x^{16}}{\sqrt {-b^4+a^4 x^4} \left (-b^{16}+a^{16} x^{16}\right )} \, dx=-\frac {x}{4 \sqrt {-b^4+a^4 x^4}}+\frac {\arctan \left (\frac {2^{3/4} a b x \sqrt {-b^4+a^4 x^4}}{b^4+\sqrt {2} a^2 b^2 x^2-a^4 x^4}\right )}{4\ 2^{3/4} a b}-\frac {\arctan \left (\frac {\frac {b^3}{2 a}+a b x^2-\frac {a^3 x^4}{2 b}}{x \sqrt {-b^4+a^4 x^4}}\right )}{16 a b}+\frac {\text {arctanh}\left (\frac {\frac {b^3}{2 a}-a b x^2-\frac {a^3 x^4}{2 b}}{x \sqrt {-b^4+a^4 x^4}}\right )}{16 a b}+\frac {\text {arctanh}\left (\frac {\frac {b^3}{2^{3/4} a}-\frac {a b x^2}{\sqrt [4]{2}}-\frac {a^3 x^4}{2^{3/4} b}}{x \sqrt {-b^4+a^4 x^4}}\right )}{4\ 2^{3/4} a b} \]
-1/4*x/(a^4*x^4-b^4)^(1/2)+1/8*arctan(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-1/16*arctan((1/2*b^3/a+a*b*x^ 2-1/2*a^3*x^4/b)/x/(a^4*x^4-b^4)^(1/2))/a/b+1/16*arctanh((1/2*b^3/a-a*b*x^ 2-1/2*a^3*x^4/b)/x/(a^4*x^4-b^4)^(1/2))/a/b+1/8*arctanh((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
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 15.34 (sec) , antiderivative size = 373, normalized size of antiderivative = 1.26 \[ \int \frac {b^{16}+a^{16} x^{16}}{\sqrt {-b^4+a^4 x^4} \left (-b^{16}+a^{16} x^{16}\right )} \, dx=\frac {-\sqrt {-\frac {a^2}{b^2}} x-3 i \sqrt {1-\frac {a^4 x^4}{b^4}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {a^2}{b^2}} x\right ),-1\right )+i \sqrt {1-\frac {a^4 x^4}{b^4}} \operatorname {EllipticPi}\left (-i,i \text {arcsinh}\left (\sqrt {-\frac {a^2}{b^2}} x\right ),-1\right )+i \sqrt {1-\frac {a^4 x^4}{b^4}} \operatorname {EllipticPi}\left (i,i \text {arcsinh}\left (\sqrt {-\frac {a^2}{b^2}} x\right ),-1\right )+i \sqrt {1-\frac {a^4 x^4}{b^4}} \operatorname {EllipticPi}\left (-\sqrt [4]{-1},i \text {arcsinh}\left (\sqrt {-\frac {a^2}{b^2}} x\right ),-1\right )+i \sqrt {1-\frac {a^4 x^4}{b^4}} \operatorname {EllipticPi}\left (\sqrt [4]{-1},i \text {arcsinh}\left (\sqrt {-\frac {a^2}{b^2}} x\right ),-1\right )+i \sqrt {1-\frac {a^4 x^4}{b^4}} \operatorname {EllipticPi}\left (-(-1)^{3/4},i \text {arcsinh}\left (\sqrt {-\frac {a^2}{b^2}} x\right ),-1\right )+i \sqrt {1-\frac {a^4 x^4}{b^4}} \operatorname {EllipticPi}\left ((-1)^{3/4},i \text {arcsinh}\left (\sqrt {-\frac {a^2}{b^2}} x\right ),-1\right )}{4 \sqrt {-\frac {a^2}{b^2}} \sqrt {-b^4+a^4 x^4}} \]
(-(Sqrt[-(a^2/b^2)]*x) - (3*I)*Sqrt[1 - (a^4*x^4)/b^4]*EllipticF[I*ArcSinh [Sqrt[-(a^2/b^2)]*x], -1] + I*Sqrt[1 - (a^4*x^4)/b^4]*EllipticPi[-I, I*Arc Sinh[Sqrt[-(a^2/b^2)]*x], -1] + I*Sqrt[1 - (a^4*x^4)/b^4]*EllipticPi[I, I* ArcSinh[Sqrt[-(a^2/b^2)]*x], -1] + I*Sqrt[1 - (a^4*x^4)/b^4]*EllipticPi[-( -1)^(1/4), I*ArcSinh[Sqrt[-(a^2/b^2)]*x], -1] + I*Sqrt[1 - (a^4*x^4)/b^4]* EllipticPi[(-1)^(1/4), I*ArcSinh[Sqrt[-(a^2/b^2)]*x], -1] + I*Sqrt[1 - (a^ 4*x^4)/b^4]*EllipticPi[-(-1)^(3/4), I*ArcSinh[Sqrt[-(a^2/b^2)]*x], -1] + I *Sqrt[1 - (a^4*x^4)/b^4]*EllipticPi[(-1)^(3/4), I*ArcSinh[Sqrt[-(a^2/b^2)] *x], -1])/(4*Sqrt[-(a^2/b^2)]*Sqrt[-b^4 + a^4*x^4])
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 1.41 (sec) , antiderivative size = 455, normalized size of antiderivative = 1.53, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {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.
| \(\displaystyle \int \frac {a^{16} x^{16}+b^{16}}{\sqrt {a^4 x^4-b^4} \left (a^{16} x^{16}-b^{16}\right )} \, dx\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \int \left (\frac {1}{\sqrt {a^4 x^4-b^4}}+\frac {2 b^{16}}{\sqrt {a^4 x^4-b^4} \left (a^{16} x^{16}-b^{16}\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {b \sqrt {1-\frac {a^4 x^4}{b^4}} \operatorname {EllipticF}\left (\arcsin \left (\frac {a x}{b}\right ),-1\right )}{2 a \sqrt {a^4 x^4-b^4}}-\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{-a^4} b x}{\sqrt {a^4 x^4-b^4}}\right )}{8 \sqrt {2} \sqrt [4]{-a^4} b}-\frac {\text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{-a^4} b x}{\sqrt {a^4 x^4-b^4}}\right )}{8 \sqrt {2} \sqrt [4]{-a^4} b}-\frac {x}{4 \sqrt {a^4 x^4-b^4}}-\frac {b \sqrt {1-\frac {a^4 x^4}{b^4}} \operatorname {EllipticPi}\left (\frac {a^6}{\left (-a^8\right )^{3/4}},\arcsin \left (\frac {a x}{b}\right ),-1\right )}{4 a \sqrt {a^4 x^4-b^4}}-\frac {b \sqrt {1-\frac {a^4 x^4}{b^4}} \operatorname {EllipticPi}\left (\frac {\sqrt [4]{-a^8}}{a^2},\arcsin \left (\frac {a x}{b}\right ),-1\right )}{4 a \sqrt {a^4 x^4-b^4}}-\frac {b \sqrt {1-\frac {a^4 x^4}{b^4}} \operatorname {EllipticPi}\left (-\frac {\sqrt {-\sqrt {-a^8}}}{a^2},\arcsin \left (\frac {a x}{b}\right ),-1\right )}{4 a \sqrt {a^4 x^4-b^4}}-\frac {b \sqrt {1-\frac {a^4 x^4}{b^4}} \operatorname {EllipticPi}\left (\frac {\sqrt {-\sqrt {-a^8}}}{a^2},\arcsin \left (\frac {a x}{b}\right ),-1\right )}{4 a \sqrt {a^4 x^4-b^4}}\) |
-1/4*x/Sqrt[-b^4 + a^4*x^4] - ArcTan[(Sqrt[2]*(-a^4)^(1/4)*b*x)/Sqrt[-b^4 + a^4*x^4]]/(8*Sqrt[2]*(-a^4)^(1/4)*b) - ArcTanh[(Sqrt[2]*(-a^4)^(1/4)*b*x )/Sqrt[-b^4 + a^4*x^4]]/(8*Sqrt[2]*(-a^4)^(1/4)*b) + (b*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]*EllipticPi[a^6/(-a^8)^(3/4), ArcSin[(a*x)/b], -1])/(4* 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])/(4*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])/(4* 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])/(4*a*Sqrt[-b^4 + a^4*x^4])
3.29.56.3.1 Defintions of rubi rules used
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 = 7.36 (sec) , antiderivative size = 474, normalized size of antiderivative = 1.60
| method | result | size |
| elliptic | \(\frac {\left (\frac {\ln \left (\frac {\frac {a^{4} x^{4}-b^{4}}{2 x^{2}}-\frac {\sqrt {\sqrt {2}\, \sqrt {a^{4} b^{4}}}\, \sqrt {a^{4} x^{4}-b^{4}}\, \sqrt {2}}{2 x}+\frac {\sqrt {2}\, \sqrt {a^{4} b^{4}}}{2}}{\frac {a^{4} x^{4}-b^{4}}{2 x^{2}}+\frac {\sqrt {\sqrt {2}\, \sqrt {a^{4} b^{4}}}\, \sqrt {a^{4} x^{4}-b^{4}}\, \sqrt {2}}{2 x}+\frac {\sqrt {2}\, \sqrt {a^{4} b^{4}}}{2}}\right )+2 \arctan \left (\frac {\sqrt {a^{4} x^{4}-b^{4}}\, \sqrt {2}}{\sqrt {\sqrt {2}\, \sqrt {a^{4} b^{4}}}\, x}+1\right )+2 \arctan \left (\frac {\sqrt {a^{4} x^{4}-b^{4}}\, \sqrt {2}}{\sqrt {\sqrt {2}\, \sqrt {a^{4} b^{4}}}\, x}-1\right )}{8 \sqrt {\sqrt {2}\, \sqrt {a^{4} b^{4}}}}-\frac {\sqrt {2}\, x}{4 \sqrt {a^{4} x^{4}-b^{4}}}+\frac {\sqrt {2}\, \left (\ln \left (\frac {\frac {a^{4} x^{4}-b^{4}}{2 x^{2}}-\frac {\left (a^{4} b^{4}\right )^{\frac {1}{4}} \sqrt {a^{4} x^{4}-b^{4}}}{x}+\sqrt {a^{4} b^{4}}}{\frac {a^{4} x^{4}-b^{4}}{2 x^{2}}+\frac {\left (a^{4} b^{4}\right )^{\frac {1}{4}} \sqrt {a^{4} x^{4}-b^{4}}}{x}+\sqrt {a^{4} b^{4}}}\right )+2 \arctan \left (\frac {\sqrt {a^{4} x^{4}-b^{4}}}{\left (a^{4} b^{4}\right )^{\frac {1}{4}} x}+1\right )+2 \arctan \left (\frac {\sqrt {a^{4} x^{4}-b^{4}}}{\left (a^{4} b^{4}\right )^{\frac {1}{4}} x}-1\right )\right )}{32 \left (a^{4} b^{4}\right )^{\frac {1}{4}}}\right ) \sqrt {2}}{2}\) | \(474\) |
| pseudoelliptic | \(-\frac {i a^{2} \left (2 \left (-a^{4} x^{4}+b^{4}\right ) \sqrt {i a^{2} b^{2}}\, \sqrt {-i a^{2} b^{2}}\, \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (16 a^{8} b^{8}+32 i a^{6} b^{6} \textit {\_Z}^{2}+8 b^{4} \textit {\_Z}^{4} a^{4}-8 i a^{2} b^{2} \textit {\_Z}^{6}+\textit {\_Z}^{8}\right )}{\sum }\frac {\left (8 i a^{6} b^{6}-4 a^{4} b^{4} \textit {\_R}^{2}+2 i a^{2} b^{2} \textit {\_R}^{4}-\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 a^{4} b^{4} \textit {\_R}^{2}+6 i a^{2} b^{2} \textit {\_R}^{4}-\textit {\_R}^{6}\right )}\right )+\frac {\sqrt {2}\, \sqrt {-i a^{2} b^{2}}\, \left (a^{4} x^{4}-b^{4}\right ) \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 )}{2}+\frac {\sqrt {2}\, \sqrt {-i a^{2} b^{2}}\, \left (a^{4} x^{4}-b^{4}\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}+\sqrt {i a^{2} b^{2}}\, \sqrt {2}\, \left (a^{4} x^{4}-b^{4}\right ) \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 )+4 \sqrt {a^{4} x^{4}-b^{4}}\, \sqrt {i a^{2} b^{2}}\, \sqrt {-i a^{2} b^{2}}\, x +\ln \left (2\right ) \sqrt {2}\, \left (a x -b \right ) \left (a x +b \right ) \left (a^{2} x^{2}+b^{2}\right ) \left (\sqrt {i a^{2} b^{2}}+\sqrt {-i a^{2} b^{2}}\right )\right ) b^{2}}{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 ) \left (-a x +i b \right ) \left (a x -b \right ) \left (a x +b \right ) \left (-2 \sqrt {i a^{2} b^{2}}+\left (1+i\right ) a b \right ) \left (a x +i b \right )}\) | \(784\) |
| default | \(\text {Expression too large to display}\) | \(1190\) |
1/2*(1/8/(2^(1/2)*(a^4*b^4)^(1/2))^(1/2)*(ln((1/2*(a^4*x^4-b^4)/x^2-1/2*(2 ^(1/2)*(a^4*b^4)^(1/2))^(1/2)*(a^4*x^4-b^4)^(1/2)*2^(1/2)/x+1/2*2^(1/2)*(a ^4*b^4)^(1/2))/(1/2*(a^4*x^4-b^4)/x^2+1/2*(2^(1/2)*(a^4*b^4)^(1/2))^(1/2)* (a^4*x^4-b^4)^(1/2)*2^(1/2)/x+1/2*2^(1/2)*(a^4*b^4)^(1/2)))+2*arctan(1/(2^ (1/2)*(a^4*b^4)^(1/2))^(1/2)*(a^4*x^4-b^4)^(1/2)*2^(1/2)/x+1)+2*arctan(1/( 2^(1/2)*(a^4*b^4)^(1/2))^(1/2)*(a^4*x^4-b^4)^(1/2)*2^(1/2)/x-1))-1/4/(a^4* x^4-b^4)^(1/2)*2^(1/2)*x+1/32/(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*arcta n(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)
Result contains complex when optimal does not.
Time = 2.12 (sec) , antiderivative size = 871, normalized size of antiderivative = 2.93 \[ \int \frac {b^{16}+a^{16} x^{16}}{\sqrt {-b^4+a^4 x^4} \left (-b^{16}+a^{16} x^{16}\right )} \, dx=\text {Too large to display} \]
-1/16*(4*sqrt(a^4*x^4 - b^4)*a*b*x + (1/2)^(1/4)*(a^5*b*x^4 - a*b^5)*(-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)) - (1/2)^(1/4)*(a^5*b*x^4 - a*b^5)*(- 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)) + (1/2)^(1/4)*(I*a^5*b*x^4 - I*a *b^5)*(-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)) + (1/2)^(1/4)* (-I*a^5*b*x^4 + I*a*b^5)*(-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)) - 2*(a^4*x^4 - b^4)*arctan(sqrt(a^4*x^4 - b^4)*a*x/(a^2*b*x^2 + b^3 )) - (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^5*b*x^4 - a*b^5)
Timed out. \[ \int \frac {b^{16}+a^{16} x^{16}}{\sqrt {-b^4+a^4 x^4} \left (-b^{16}+a^{16} x^{16}\right )} \, dx=\text {Timed out} \]
\[ \int \frac {b^{16}+a^{16} x^{16}}{\sqrt {-b^4+a^4 x^4} \left (-b^{16}+a^{16} x^{16}\right )} \, dx=\int { \frac {a^{16} x^{16} + b^{16}}{{\left (a^{16} x^{16} - b^{16}\right )} \sqrt {a^{4} x^{4} - b^{4}}} \,d x } \]
\[ \int \frac {b^{16}+a^{16} x^{16}}{\sqrt {-b^4+a^4 x^4} \left (-b^{16}+a^{16} x^{16}\right )} \, dx=\int { \frac {a^{16} x^{16} + b^{16}}{{\left (a^{16} x^{16} - b^{16}\right )} \sqrt {a^{4} x^{4} - b^{4}}} \,d x } \]
Timed out. \[ \int \frac {b^{16}+a^{16} x^{16}}{\sqrt {-b^4+a^4 x^4} \left (-b^{16}+a^{16} x^{16}\right )} \, dx=\int -\frac {a^{16}\,x^{16}+b^{16}}{\sqrt {a^4\,x^4-b^4}\,\left (b^{16}-a^{16}\,x^{16}\right )} \,d x \]