\(\int \frac {b^6+a^6 x^6}{\sqrt {b^4+a^4 x^4} (-b^6+a^6 x^6)} \, dx\) [2113]

Optimal result

Integrand size = 42, antiderivative size = 153 \[ \int \frac {b^6+a^6 x^6}{\sqrt {b^4+a^4 x^4} \left (-b^6+a^6 x^6\right )} \, dx=-\frac {2 \arctan \left (\frac {a b x}{\sqrt {b^4+a^4 x^4}}\right )}{3 a b}+\frac {\text {arctanh}\left (\frac {\sqrt {6-4 \sqrt {2}} a b x}{b^2+a^2 x^2+\sqrt {b^4+a^4 x^4}}\right )}{3 \sqrt {2} a b}-\frac {\text {arctanh}\left (\frac {\sqrt {6+4 \sqrt {2}} a b x}{b^2+a^2 x^2+\sqrt {b^4+a^4 x^4}}\right )}{3 \sqrt {2} a b} \] Output:

-2/3*arctan(a*b*x/(a^4*x^4+b^4)^(1/2))/a/b+1/6*arctanh((2-2^(1/2))*a*b*x/( 
b^2+a^2*x^2+(a^4*x^4+b^4)^(1/2)))*2^(1/2)/a/b-1/6*arctanh((2+2^(1/2))*a*b* 
x/(b^2+a^2*x^2+(a^4*x^4+b^4)^(1/2)))*2^(1/2)/a/b
 

Mathematica [A] (verified)

Time = 1.38 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.42 \[ \int \frac {b^6+a^6 x^6}{\sqrt {b^4+a^4 x^4} \left (-b^6+a^6 x^6\right )} \, dx=-\frac {4 \arctan \left (\frac {a b x}{\sqrt {b^4+a^4 x^4}}\right )+\sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} a b x}{\sqrt {b^4+a^4 x^4}}\right )}{6 a b} \] Input:

Integrate[(b^6 + a^6*x^6)/(Sqrt[b^4 + a^4*x^4]*(-b^6 + a^6*x^6)),x]
 

Output:

-1/6*(4*ArcTan[(a*b*x)/Sqrt[b^4 + a^4*x^4]] + Sqrt[2]*ArcTanh[(Sqrt[2]*a*b 
*x)/Sqrt[b^4 + a^4*x^4]])/(a*b)
 

Rubi [C] (verified)

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

Time = 2.63 (sec) , antiderivative size = 327, normalized size of antiderivative = 2.14, 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.

\(\displaystyle \int \frac {a^6 x^6+b^6}{\sqrt {a^4 x^4+b^4} \left (a^6 x^6-b^6\right )} \, dx\)

\(\Big \downarrow \) 7276

\(\displaystyle \int \left (\frac {1}{\sqrt {a^4 x^4+b^4}}+\frac {2 b^6}{\sqrt {a^4 x^4+b^4} \left (a^6 x^6-b^6\right )}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {2 \arctan \left (\frac {a b x}{\sqrt {a^4 x^4+b^4}}\right )}{3 a b}-\frac {\text {arctanh}\left (\frac {\sqrt {2} a b x}{\sqrt {a^4 x^4+b^4}}\right )}{3 \sqrt {2} a b}-\frac {2 \left (a^2 x^2+b^2\right ) \sqrt {\frac {a^4 x^4+b^4}{\left (a^2 x^2+b^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {a x}{b}\right ),\frac {1}{2}\right )}{3 \left (1-i \sqrt {3}\right ) a b \sqrt {a^4 x^4+b^4}}-\frac {\left (1-\sqrt [3]{-1}\right ) \left (a^2 x^2+b^2\right ) \sqrt {\frac {a^4 x^4+b^4}{\left (a^2 x^2+b^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {a x}{b}\right ),\frac {1}{2}\right )}{3 a b \sqrt {a^4 x^4+b^4}}+\frac {\left (a^2 x^2+b^2\right ) \sqrt {\frac {a^4 x^4+b^4}{\left (a^2 x^2+b^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {a x}{b}\right ),\frac {1}{2}\right )}{3 a b \sqrt {a^4 x^4+b^4}}\)

Input:

Int[(b^6 + a^6*x^6)/(Sqrt[b^4 + a^4*x^4]*(-b^6 + a^6*x^6)),x]
 

Output:

(-2*ArcTan[(a*b*x)/Sqrt[b^4 + a^4*x^4]])/(3*a*b) - ArcTanh[(Sqrt[2]*a*b*x) 
/Sqrt[b^4 + a^4*x^4]]/(3*Sqrt[2]*a*b) + ((b^2 + a^2*x^2)*Sqrt[(b^4 + a^4*x 
^4)/(b^2 + a^2*x^2)^2]*EllipticF[2*ArcTan[(a*x)/b], 1/2])/(3*a*b*Sqrt[b^4 
+ a^4*x^4]) - ((1 - (-1)^(1/3))*(b^2 + a^2*x^2)*Sqrt[(b^4 + a^4*x^4)/(b^2 
+ a^2*x^2)^2]*EllipticF[2*ArcTan[(a*x)/b], 1/2])/(3*a*b*Sqrt[b^4 + a^4*x^4 
]) - (2*(b^2 + a^2*x^2)*Sqrt[(b^4 + a^4*x^4)/(b^2 + a^2*x^2)^2]*EllipticF[ 
2*ArcTan[(a*x)/b], 1/2])/(3*(1 - I*Sqrt[3])*a*b*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 [A] (verified)

Time = 3.40 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.73

method	result	size
elliptic	\(\frac {\left (\frac {\ln \left (-a b +\frac {\sqrt {a^{4} x^{4}+b^{4}}\, \sqrt {2}}{2 x}\right )}{6 a b}-\frac {\ln \left (a b +\frac {\sqrt {a^{4} x^{4}+b^{4}}\, \sqrt {2}}{2 x}\right )}{6 a b}+\frac {2 \sqrt {2}\, \arctan \left (\frac {\sqrt {a^{4} x^{4}+b^{4}}}{a b x}\right )}{3 a b}\right ) \sqrt {2}}{2}\)	\(111\)
default	\(-\frac {\left (\ln \left (\frac {\sqrt {2}\, \sqrt {a^{2} b^{2}}\, a^{2} \sqrt {a^{4} x^{4}+b^{4}}+2 b \,a^{3} \left (a^{2} x^{2}-a b x +b^{2}\right )}{\left (a x -b \right )^{2}}\right )+\ln \left (\frac {\sqrt {2}\, \sqrt {a^{2} b^{2}}\, a^{2} \sqrt {a^{4} x^{4}+b^{4}}-2 b \,a^{3} \left (a^{2} x^{2}+a b x +b^{2}\right )}{\left (a x +b \right )^{2}}\right )+2 \ln \left (2\right )\right ) \sqrt {2}\, \sqrt {-a^{2} b^{2}}+4 \left (\ln \left (\frac {\left (\sqrt {-a^{2} b^{2}}\, \sqrt {a^{4} x^{4}+b^{4}}+b a \left (a x -b \right )^{2}\right ) a^{2}}{a^{2} x^{2}-a b x +b^{2}}\right )+\ln \left (-\frac {\left (-\sqrt {-a^{2} b^{2}}\, \sqrt {a^{4} x^{4}+b^{4}}+b a \left (a x +b \right )^{2}\right ) a^{2}}{a^{2} x^{2}+a b x +b^{2}}\right )+2 \ln \left (2\right )\right ) \sqrt {a^{2} b^{2}}}{12 \sqrt {-a^{2} b^{2}}\, \sqrt {a^{2} b^{2}}}\)	\(300\)
pseudoelliptic	\(-\frac {\left (\ln \left (\frac {\sqrt {2}\, \sqrt {a^{2} b^{2}}\, a^{2} \sqrt {a^{4} x^{4}+b^{4}}+2 b \,a^{3} \left (a^{2} x^{2}-a b x +b^{2}\right )}{\left (a x -b \right )^{2}}\right )+\ln \left (\frac {\sqrt {2}\, \sqrt {a^{2} b^{2}}\, a^{2} \sqrt {a^{4} x^{4}+b^{4}}-2 b \,a^{3} \left (a^{2} x^{2}+a b x +b^{2}\right )}{\left (a x +b \right )^{2}}\right )+2 \ln \left (2\right )\right ) \sqrt {2}\, \sqrt {-a^{2} b^{2}}+4 \left (\ln \left (\frac {\left (\sqrt {-a^{2} b^{2}}\, \sqrt {a^{4} x^{4}+b^{4}}+b a \left (a x -b \right )^{2}\right ) a^{2}}{a^{2} x^{2}-a b x +b^{2}}\right )+\ln \left (-\frac {\left (-\sqrt {-a^{2} b^{2}}\, \sqrt {a^{4} x^{4}+b^{4}}+b a \left (a x +b \right )^{2}\right ) a^{2}}{a^{2} x^{2}+a b x +b^{2}}\right )+2 \ln \left (2\right )\right ) \sqrt {a^{2} b^{2}}}{12 \sqrt {-a^{2} b^{2}}\, \sqrt {a^{2} b^{2}}}\)	\(300\)

Input:

int((a^6*x^6+b^6)/(a^4*x^4+b^4)^(1/2)/(a^6*x^6-b^6),x,method=_RETURNVERBOS 
E)
 

Output:

1/2*(1/6/a/b*ln(-a*b+1/2*(a^4*x^4+b^4)^(1/2)*2^(1/2)/x)-1/6/a/b*ln(a*b+1/2 
*(a^4*x^4+b^4)^(1/2)*2^(1/2)/x)+2/3*2^(1/2)/a/b*arctan(1/a/b/x*(a^4*x^4+b^ 
4)^(1/2)))*2^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.83 \[ \int \frac {b^6+a^6 x^6}{\sqrt {b^4+a^4 x^4} \left (-b^6+a^6 x^6\right )} \, dx=\frac {\sqrt {2} \log \left (\frac {a^{4} x^{4} + 2 \, a^{2} b^{2} x^{2} + b^{4} - 2 \, \sqrt {2} \sqrt {a^{4} x^{4} + b^{4}} a b x}{a^{4} x^{4} - 2 \, a^{2} b^{2} x^{2} + b^{4}}\right ) - 4 \, \arctan \left (\frac {2 \, \sqrt {a^{4} x^{4} + b^{4}} a b x}{a^{4} x^{4} - a^{2} b^{2} x^{2} + b^{4}}\right )}{12 \, a b} \] Input:

integrate((a^6*x^6+b^6)/(a^4*x^4+b^4)^(1/2)/(a^6*x^6-b^6),x, algorithm="fr 
icas")
 

Output:

1/12*(sqrt(2)*log((a^4*x^4 + 2*a^2*b^2*x^2 + b^4 - 2*sqrt(2)*sqrt(a^4*x^4 
+ b^4)*a*b*x)/(a^4*x^4 - 2*a^2*b^2*x^2 + b^4)) - 4*arctan(2*sqrt(a^4*x^4 + 
 b^4)*a*b*x/(a^4*x^4 - a^2*b^2*x^2 + b^4)))/(a*b)
 

Sympy [F]

\[ \int \frac {b^6+a^6 x^6}{\sqrt {b^4+a^4 x^4} \left (-b^6+a^6 x^6\right )} \, dx=\int \frac {\left (a^{2} x^{2} + b^{2}\right ) \left (a^{4} x^{4} - a^{2} b^{2} x^{2} + b^{4}\right )}{\left (a x - b\right ) \left (a x + b\right ) \sqrt {a^{4} x^{4} + b^{4}} \left (a^{2} x^{2} - a b x + b^{2}\right ) \left (a^{2} x^{2} + a b x + b^{2}\right )}\, dx \] Input:

integrate((a**6*x**6+b**6)/(a**4*x**4+b**4)**(1/2)/(a**6*x**6-b**6),x)
 

Output:

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

Maxima [F]

\[ \int \frac {b^6+a^6 x^6}{\sqrt {b^4+a^4 x^4} \left (-b^6+a^6 x^6\right )} \, dx=\int { \frac {a^{6} x^{6} + b^{6}}{{\left (a^{6} x^{6} - b^{6}\right )} \sqrt {a^{4} x^{4} + b^{4}}} \,d x } \] Input:

integrate((a^6*x^6+b^6)/(a^4*x^4+b^4)^(1/2)/(a^6*x^6-b^6),x, algorithm="ma 
xima")
 

Output:

integrate((a^6*x^6 + b^6)/((a^6*x^6 - b^6)*sqrt(a^4*x^4 + b^4)), x)
 

Giac [F]

\[ \int \frac {b^6+a^6 x^6}{\sqrt {b^4+a^4 x^4} \left (-b^6+a^6 x^6\right )} \, dx=\int { \frac {a^{6} x^{6} + b^{6}}{{\left (a^{6} x^{6} - b^{6}\right )} \sqrt {a^{4} x^{4} + b^{4}}} \,d x } \] Input:

integrate((a^6*x^6+b^6)/(a^4*x^4+b^4)^(1/2)/(a^6*x^6-b^6),x, algorithm="gi 
ac")
 

Output:

integrate((a^6*x^6 + b^6)/((a^6*x^6 - b^6)*sqrt(a^4*x^4 + b^4)), x)
                                                                                    
                                                                                    
 

Mupad [F(-1)]

Timed out. \[ \int \frac {b^6+a^6 x^6}{\sqrt {b^4+a^4 x^4} \left (-b^6+a^6 x^6\right )} \, dx=\int -\frac {a^6\,x^6+b^6}{\sqrt {a^4\,x^4+b^4}\,\left (b^6-a^6\,x^6\right )} \,d x \] Input:

int(-(b^6 + a^6*x^6)/((b^4 + a^4*x^4)^(1/2)*(b^6 - a^6*x^6)),x)
 

Output:

int(-(b^6 + a^6*x^6)/((b^4 + a^4*x^4)^(1/2)*(b^6 - a^6*x^6)), x)
 

Reduce [F]

\[ \int \frac {b^6+a^6 x^6}{\sqrt {b^4+a^4 x^4} \left (-b^6+a^6 x^6\right )} \, dx=\left (\int \frac {\sqrt {a^{4} x^{4}+b^{4}}}{a^{10} x^{10}+a^{6} b^{4} x^{6}-a^{4} b^{6} x^{4}-b^{10}}d x \right ) b^{6}+\left (\int \frac {\sqrt {a^{4} x^{4}+b^{4}}\, x^{6}}{a^{10} x^{10}+a^{6} b^{4} x^{6}-a^{4} b^{6} x^{4}-b^{10}}d x \right ) a^{6} \] Input:

int((a^6*x^6+b^6)/(a^4*x^4+b^4)^(1/2)/(a^6*x^6-b^6),x)
 

Output:

int(sqrt(a**4*x**4 + b**4)/(a**10*x**10 + a**6*b**4*x**6 - a**4*b**6*x**4 
- b**10),x)*b**6 + int((sqrt(a**4*x**4 + b**4)*x**6)/(a**10*x**10 + a**6*b 
**4*x**6 - a**4*b**6*x**4 - b**10),x)*a**6