\(\int \frac {\sqrt {-b^4+a^4 x^4} (b^4+a^4 x^4)}{b^8+a^8 x^8} \, dx\) [2082]

Optimal result

Integrand size = 42, antiderivative size = 150 \[ \int \frac {\sqrt {-b^4+a^4 x^4} \left (b^4+a^4 x^4\right )}{b^8+a^8 x^8} \, dx=-\frac {\arctan \left (\frac {\frac {b^3}{2^{3/4} a}+\frac {a b x^2}{\sqrt [4]{2}}-\frac {a^3 x^4}{2^{3/4} b}}{x \sqrt {-b^4+a^4 x^4}}\right )}{2\ 2^{3/4} a b}-\frac {\text {arctanh}\left (\frac {2^{3/4} a b x \sqrt {-b^4+a^4 x^4}}{-b^4+\sqrt {2} a^2 b^2 x^2+a^4 x^4}\right )}{2\ 2^{3/4} a b} \] Output:

-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
 

Mathematica [A] (verified)

Time = 0.67 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.01 \[ \int \frac {\sqrt {-b^4+a^4 x^4} \left (b^4+a^4 x^4\right )}{b^8+a^8 x^8} \, dx=\frac {\arctan \left (\frac {a b x}{a b x-\sqrt [4]{2} \sqrt {-b^4+a^4 x^4}}\right )-\arctan \left (\frac {a b x}{a b x+\sqrt [4]{2} \sqrt {-b^4+a^4 x^4}}\right )-\text {arctanh}\left (\frac {-b^4+\sqrt {2} a^2 b^2 x^2+a^4 x^4}{2^{3/4} a b x \sqrt {-b^4+a^4 x^4}}\right )}{2\ 2^{3/4} a b} \] Input:

Integrate[(Sqrt[-b^4 + a^4*x^4]*(b^4 + a^4*x^4))/(b^8 + a^8*x^8),x]
 

Output:

(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)
 

Rubi [C] (verified)

Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.

Time = 0.92 (sec) , antiderivative size = 400, normalized size of antiderivative = 2.67, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, 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.

\(\displaystyle \int \frac {\sqrt {a^4 x^4-b^4} \left (a^4 x^4+b^4\right )}{a^8 x^8+b^8} \, dx\)

\(\Big \downarrow \) 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

\(\displaystyle -\frac {b \sqrt {1-\frac {a^4 x^4}{b^4}} \operatorname {EllipticPi}\left (\frac {a^6}{\left (-a^8\right )^{3/4}},\arcsin \left (\frac {a x}{b}\right ),-1\right )}{2 a \sqrt {a^4 x^4-b^4}}+\frac {\left (a^4-\sqrt {-a^8}\right ) b \sqrt {1-\frac {a^4 x^4}{b^4}} \operatorname {EllipticF}\left (\arcsin \left (\frac {a x}{b}\right ),-1\right )}{2 a^5 \sqrt {a^4 x^4-b^4}}+\frac {\left (\sqrt {-a^8}+a^4\right ) b \sqrt {1-\frac {a^4 x^4}{b^4}} \operatorname {EllipticF}\left (\arcsin \left (\frac {a x}{b}\right ),-1\right )}{2 a^5 \sqrt {a^4 x^4-b^4}}-\frac {b \sqrt {1-\frac {a^4 x^4}{b^4}} \operatorname {EllipticPi}\left (\frac {\sqrt [4]{-a^8}}{a^2},\arcsin \left (\frac {a x}{b}\right ),-1\right )}{2 a \sqrt {a^4 x^4-b^4}}-\frac {b \sqrt {1-\frac {a^4 x^4}{b^4}} \operatorname {EllipticPi}\left (-\frac {\sqrt {-\sqrt {-a^8}}}{a^2},\arcsin \left (\frac {a x}{b}\right ),-1\right )}{2 a \sqrt {a^4 x^4-b^4}}-\frac {b \sqrt {1-\frac {a^4 x^4}{b^4}} \operatorname {EllipticPi}\left (\frac {\sqrt {-\sqrt {-a^8}}}{a^2},\arcsin \left (\frac {a x}{b}\right ),-1\right )}{2 a \sqrt {a^4 x^4-b^4}}\)

Input:

Int[(Sqrt[-b^4 + a^4*x^4]*(b^4 + a^4*x^4))/(b^8 + a^8*x^8),x]
 

Output:

((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])
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 7276

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]
 

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 0.62 (sec) , antiderivative size = 183, normalized size of antiderivative = 1.22

method	result	size
pseudoelliptic	\(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (16 a^{8} b^{8}+32 i a^{6} b^{6} \textit {\_Z}^{2}+8 b^{4} a^{4} \textit {\_Z}^{4}-8 i a^{2} b^{2} \textit {\_Z}^{6}+\textit {\_Z}^{8}\right )}{\sum }\frac {\ln \left (\frac {-\operatorname {csgn}\left (a^{2}\right ) a^{2} x^{2}-i \operatorname {csgn}\left (a^{2}\right ) b^{2}-\textit {\_R} x +\sqrt {a^{4} x^{4}-b^{4}}}{x}\right ) \left (8 a^{6} b^{6}+4 i \textit {\_R}^{2} a^{4} b^{4}+2 \textit {\_R}^{4} a^{2} b^{2}+i \textit {\_R}^{6}\right )}{\textit {\_R} \left (-8 a^{6} b^{6}+4 i \textit {\_R}^{2} a^{4} b^{4}+6 \textit {\_R}^{4} a^{2} b^{2}+i \textit {\_R}^{6}\right )}\right )}{4}\)	\(183\)
default	\(\frac {\left (\ln \left (\frac {\frac {a^{4} x^{4}-b^{4}}{2 x^{2}}-\frac {\sqrt {\sqrt {2}\, \sqrt {b^{4} a^{4}}}\, \sqrt {a^{4} x^{4}-b^{4}}\, \sqrt {2}}{2 x}+\frac {\sqrt {2}\, \sqrt {b^{4} a^{4}}}{2}}{\frac {a^{4} x^{4}-b^{4}}{2 x^{2}}+\frac {\sqrt {\sqrt {2}\, \sqrt {b^{4} a^{4}}}\, \sqrt {a^{4} x^{4}-b^{4}}\, \sqrt {2}}{2 x}+\frac {\sqrt {2}\, \sqrt {b^{4} a^{4}}}{2}}\right )+2 \arctan \left (\frac {\sqrt {a^{4} x^{4}-b^{4}}\, \sqrt {2}}{\sqrt {\sqrt {2}\, \sqrt {b^{4} a^{4}}}\, x}+1\right )+2 \arctan \left (\frac {\sqrt {a^{4} x^{4}-b^{4}}\, \sqrt {2}}{\sqrt {\sqrt {2}\, \sqrt {b^{4} a^{4}}}\, x}-1\right )\right ) \sqrt {2}}{8 \sqrt {\sqrt {2}\, \sqrt {b^{4} a^{4}}}}\)	\(252\)
elliptic	\(\frac {\left (\ln \left (\frac {\frac {a^{4} x^{4}-b^{4}}{2 x^{2}}-\frac {\sqrt {\sqrt {2}\, \sqrt {b^{4} a^{4}}}\, \sqrt {a^{4} x^{4}-b^{4}}\, \sqrt {2}}{2 x}+\frac {\sqrt {2}\, \sqrt {b^{4} a^{4}}}{2}}{\frac {a^{4} x^{4}-b^{4}}{2 x^{2}}+\frac {\sqrt {\sqrt {2}\, \sqrt {b^{4} a^{4}}}\, \sqrt {a^{4} x^{4}-b^{4}}\, \sqrt {2}}{2 x}+\frac {\sqrt {2}\, \sqrt {b^{4} a^{4}}}{2}}\right )+2 \arctan \left (\frac {\sqrt {a^{4} x^{4}-b^{4}}\, \sqrt {2}}{\sqrt {\sqrt {2}\, \sqrt {b^{4} a^{4}}}\, x}+1\right )+2 \arctan \left (\frac {\sqrt {a^{4} x^{4}-b^{4}}\, \sqrt {2}}{\sqrt {\sqrt {2}\, \sqrt {b^{4} a^{4}}}\, x}-1\right )\right ) \sqrt {2}}{8 \sqrt {\sqrt {2}\, \sqrt {b^{4} a^{4}}}}\)	\(252\)

Input:

int((a^4*x^4-b^4)^(1/2)*(a^4*x^4+b^4)/(a^8*x^8+b^8),x,method=_RETURNVERBOS 
E)
 

Output:

1/4*sum(ln((-csgn(a^2)*a^2*x^2-I*csgn(a^2)*b^2-_R*x+(a^4*x^4-b^4)^(1/2))/x 
)/_R*(8*a^6*b^6+4*I*_R^2*a^4*b^4+2*_R^4*a^2*b^2+I*_R^6)/(-8*a^6*b^6+4*I*_R 
^2*a^4*b^4+6*_R^4*a^2*b^2+I*_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))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 673 vs. \(2 (130) = 260\).

Time = 6.54 (sec) , antiderivative size = 673, normalized size of antiderivative = 4.49 \[ \int \frac {\sqrt {-b^4+a^4 x^4} \left (b^4+a^4 x^4\right )}{b^8+a^8 x^8} \, dx=-\frac {2 \cdot 8^{\frac {3}{4}} \arctan \left (\frac {a^{16} x^{16} + 2 \, a^{8} b^{8} x^{8} + b^{16} + 4 \, \sqrt {2} {\left (a^{14} b^{2} x^{14} - a^{10} b^{6} x^{10} + a^{6} b^{10} x^{6} - a^{2} b^{14} x^{2}\right )} + \sqrt {a^{4} x^{4} - b^{4}} {\left (8^{\frac {3}{4}} {\left (3 \, a^{11} b^{3} x^{11} - 8 \, a^{7} b^{7} x^{7} + 3 \, a^{3} b^{11} x^{3}\right )} + 2 \cdot 8^{\frac {1}{4}} {\left (a^{13} b x^{13} - 9 \, a^{9} b^{5} x^{9} + 9 \, a^{5} b^{9} x^{5} - a b^{13} x\right )}\right )}}{a^{16} x^{16} - 32 \, a^{12} b^{4} x^{12} + 66 \, a^{8} b^{8} x^{8} - 32 \, a^{4} b^{12} x^{4} + b^{16}}\right ) + 2 \cdot 8^{\frac {3}{4}} \arctan \left (-\frac {a^{16} x^{16} + 2 \, a^{8} b^{8} x^{8} + b^{16} + 4 \, \sqrt {2} {\left (a^{14} b^{2} x^{14} - a^{10} b^{6} x^{10} + a^{6} b^{10} x^{6} - a^{2} b^{14} x^{2}\right )} - \sqrt {a^{4} x^{4} - b^{4}} {\left (8^{\frac {3}{4}} {\left (3 \, a^{11} b^{3} x^{11} - 8 \, a^{7} b^{7} x^{7} + 3 \, a^{3} b^{11} x^{3}\right )} + 2 \cdot 8^{\frac {1}{4}} {\left (a^{13} b x^{13} - 9 \, a^{9} b^{5} x^{9} + 9 \, a^{5} b^{9} x^{5} - a b^{13} x\right )}\right )}}{a^{16} x^{16} - 32 \, a^{12} b^{4} x^{12} + 66 \, a^{8} b^{8} x^{8} - 32 \, a^{4} b^{12} x^{4} + b^{16}}\right ) + 8^{\frac {3}{4}} \log \left (\frac {2 \, {\left (8 \, a^{6} b^{2} x^{6} - 8 \, a^{2} b^{6} x^{2} + \sqrt {2} {\left (a^{8} x^{8} + b^{8}\right )} + {\left (4 \cdot 8^{\frac {1}{4}} a^{3} b^{3} x^{3} + 8^{\frac {3}{4}} {\left (a^{5} b x^{5} - a b^{5} x\right )}\right )} \sqrt {a^{4} x^{4} - b^{4}}\right )}}{a^{8} x^{8} + b^{8}}\right ) - 8^{\frac {3}{4}} \log \left (\frac {2 \, {\left (8 \, a^{6} b^{2} x^{6} - 8 \, a^{2} b^{6} x^{2} + \sqrt {2} {\left (a^{8} x^{8} + b^{8}\right )} - {\left (4 \cdot 8^{\frac {1}{4}} a^{3} b^{3} x^{3} + 8^{\frac {3}{4}} {\left (a^{5} b x^{5} - a b^{5} x\right )}\right )} \sqrt {a^{4} x^{4} - b^{4}}\right )}}{a^{8} x^{8} + b^{8}}\right )}{64 \, a b} \] Input:

integrate((a^4*x^4-b^4)^(1/2)*(a^4*x^4+b^4)/(a^8*x^8+b^8),x, algorithm="fr 
icas")
 

Output:

-1/64*(2*8^(3/4)*arctan((a^16*x^16 + 2*a^8*b^8*x^8 + b^16 + 4*sqrt(2)*(a^1 
4*b^2*x^14 - a^10*b^6*x^10 + a^6*b^10*x^6 - a^2*b^14*x^2) + sqrt(a^4*x^4 - 
 b^4)*(8^(3/4)*(3*a^11*b^3*x^11 - 8*a^7*b^7*x^7 + 3*a^3*b^11*x^3) + 2*8^(1 
/4)*(a^13*b*x^13 - 9*a^9*b^5*x^9 + 9*a^5*b^9*x^5 - a*b^13*x)))/(a^16*x^16 
- 32*a^12*b^4*x^12 + 66*a^8*b^8*x^8 - 32*a^4*b^12*x^4 + b^16)) + 2*8^(3/4) 
*arctan(-(a^16*x^16 + 2*a^8*b^8*x^8 + b^16 + 4*sqrt(2)*(a^14*b^2*x^14 - a^ 
10*b^6*x^10 + a^6*b^10*x^6 - a^2*b^14*x^2) - sqrt(a^4*x^4 - b^4)*(8^(3/4)* 
(3*a^11*b^3*x^11 - 8*a^7*b^7*x^7 + 3*a^3*b^11*x^3) + 2*8^(1/4)*(a^13*b*x^1 
3 - 9*a^9*b^5*x^9 + 9*a^5*b^9*x^5 - a*b^13*x)))/(a^16*x^16 - 32*a^12*b^4*x 
^12 + 66*a^8*b^8*x^8 - 32*a^4*b^12*x^4 + b^16)) + 8^(3/4)*log(2*(8*a^6*b^2 
*x^6 - 8*a^2*b^6*x^2 + sqrt(2)*(a^8*x^8 + b^8) + (4*8^(1/4)*a^3*b^3*x^3 + 
8^(3/4)*(a^5*b*x^5 - a*b^5*x))*sqrt(a^4*x^4 - b^4))/(a^8*x^8 + b^8)) - 8^( 
3/4)*log(2*(8*a^6*b^2*x^6 - 8*a^2*b^6*x^2 + sqrt(2)*(a^8*x^8 + b^8) - (4*8 
^(1/4)*a^3*b^3*x^3 + 8^(3/4)*(a^5*b*x^5 - a*b^5*x))*sqrt(a^4*x^4 - b^4))/( 
a^8*x^8 + b^8)))/(a*b)
 

Sympy [F]

\[ \int \frac {\sqrt {-b^4+a^4 x^4} \left (b^4+a^4 x^4\right )}{b^8+a^8 x^8} \, dx=\int \frac {\sqrt {\left (a x - b\right ) \left (a x + b\right ) \left (a^{2} x^{2} + b^{2}\right )} \left (a^{4} x^{4} + b^{4}\right )}{a^{8} x^{8} + b^{8}}\, dx \] Input:

integrate((a**4*x**4-b**4)**(1/2)*(a**4*x**4+b**4)/(a**8*x**8+b**8),x)
 

Output:

Integral(sqrt((a*x - b)*(a*x + b)*(a**2*x**2 + b**2))*(a**4*x**4 + b**4)/( 
a**8*x**8 + b**8), x)
 

Maxima [F]

\[ \int \frac {\sqrt {-b^4+a^4 x^4} \left (b^4+a^4 x^4\right )}{b^8+a^8 x^8} \, dx=\int { \frac {{\left (a^{4} x^{4} + b^{4}\right )} \sqrt {a^{4} x^{4} - b^{4}}}{a^{8} x^{8} + b^{8}} \,d x } \] Input:

integrate((a^4*x^4-b^4)^(1/2)*(a^4*x^4+b^4)/(a^8*x^8+b^8),x, algorithm="ma 
xima")
 

Output:

integrate((a^4*x^4 + b^4)*sqrt(a^4*x^4 - b^4)/(a^8*x^8 + b^8), x)
 

Giac [F]

\[ \int \frac {\sqrt {-b^4+a^4 x^4} \left (b^4+a^4 x^4\right )}{b^8+a^8 x^8} \, dx=\int { \frac {{\left (a^{4} x^{4} + b^{4}\right )} \sqrt {a^{4} x^{4} - b^{4}}}{a^{8} x^{8} + b^{8}} \,d x } \] Input:

integrate((a^4*x^4-b^4)^(1/2)*(a^4*x^4+b^4)/(a^8*x^8+b^8),x, algorithm="gi 
ac")
 

Output:

integrate((a^4*x^4 + b^4)*sqrt(a^4*x^4 - b^4)/(a^8*x^8 + b^8), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {-b^4+a^4 x^4} \left (b^4+a^4 x^4\right )}{b^8+a^8 x^8} \, dx=\int \frac {\left (a^4\,x^4+b^4\right )\,\sqrt {a^4\,x^4-b^4}}{a^8\,x^8+b^8} \,d x \] Input:

int(((b^4 + a^4*x^4)*(a^4*x^4 - b^4)^(1/2))/(b^8 + a^8*x^8),x)
 

Output:

int(((b^4 + a^4*x^4)*(a^4*x^4 - b^4)^(1/2))/(b^8 + a^8*x^8), x)
 

Reduce [F]

\[ \int \frac {\sqrt {-b^4+a^4 x^4} \left (b^4+a^4 x^4\right )}{b^8+a^8 x^8} \, dx=\left (\int \frac {\sqrt {a^{4} x^{4}-b^{4}}}{a^{8} x^{8}+b^{8}}d x \right ) b^{4}+\left (\int \frac {\sqrt {a^{4} x^{4}-b^{4}}\, x^{4}}{a^{8} x^{8}+b^{8}}d x \right ) a^{4} \] Input:

int((a^4*x^4-b^4)^(1/2)*(a^4*x^4+b^4)/(a^8*x^8+b^8),x)
 

Output:

int(sqrt(a**4*x**4 - b**4)/(a**8*x**8 + b**8),x)*b**4 + int((sqrt(a**4*x** 
4 - b**4)*x**4)/(a**8*x**8 + b**8),x)*a**4