\(\int \frac {b^3+a^3 x^3}{(-b^3+a^3 x^3) \sqrt {b^4+a^4 x^4}} \, dx\) [1538]

Optimal result

Integrand size = 42, antiderivative size = 106 \[ \int \frac {b^3+a^3 x^3}{\left (-b^3+a^3 x^3\right ) \sqrt {b^4+a^4 x^4}} \, dx=-\frac {4 \arctan \left (\frac {a b x}{b^2+a b x+a^2 x^2+\sqrt {b^4+a^4 x^4}}\right )}{3 a b}-\frac {\sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} a b x}{b^2-2 a b x+a^2 x^2+\sqrt {b^4+a^4 x^4}}\right )}{3 a b} \] Output:

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

Mathematica [A] (verified)

Time = 1.89 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.93 \[ \int \frac {b^3+a^3 x^3}{\left (-b^3+a^3 x^3\right ) \sqrt {b^4+a^4 x^4}} \, dx=-\frac {4 \arctan \left (\frac {a b x}{b^2+a b x+a^2 x^2+\sqrt {b^4+a^4 x^4}}\right )+\sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} a b x}{b^2-2 a b x+a^2 x^2+\sqrt {b^4+a^4 x^4}}\right )}{3 a b} \] Input:

Integrate[(b^3 + a^3*x^3)/((-b^3 + a^3*x^3)*Sqrt[b^4 + a^4*x^4]),x]
 

Output:

-1/3*(4*ArcTan[(a*b*x)/(b^2 + a*b*x + a^2*x^2 + Sqrt[b^4 + a^4*x^4])] + Sq 
rt[2]*ArcTanh[(Sqrt[2]*a*b*x)/(b^2 - 2*a*b*x + a^2*x^2 + 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 = 1.70 (sec) , antiderivative size = 523, normalized size of antiderivative = 4.93, 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.

\(\displaystyle \int \frac {a^3 x^3+b^3}{\left (a^3 x^3-b^3\right ) \sqrt {a^4 x^4+b^4}} \, dx\)

\(\Big \downarrow \) 7276

\(\displaystyle \int \left (\frac {1}{\sqrt {a^4 x^4+b^4}}+\frac {2 b^3}{\left (a^3 x^3-b^3\right ) \sqrt {a^4 x^4+b^4}}\right )dx\)

\(\Big \downarrow \) 2009

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

Input:

Int[(b^3 + a^3*x^3)/((-b^3 + a^3*x^3)*Sqrt[b^4 + a^4*x^4]),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) - ArcTanh[(b^2 + a^2*x^2)/(Sqrt[2]*S 
qrt[b^4 + a^4*x^4])]/(3*Sqrt[2]*a*b) + ((-1)^(1/3)*ArcTanh[((-1)^(2/3)*b^2 
 + a^2*x^2)/(Sqrt[1 - (-1)^(1/3)]*Sqrt[b^4 + a^4*x^4])])/(3*Sqrt[1 - (-1)^ 
(1/3)]*a*b) - ((-1)^(2/3)*ArcTanh[(2*(b^2 + (-1)^(2/3)*a^2*x^2))/((I - Sqr 
t[3])*Sqrt[b^4 + a^4*x^4])])/(3*Sqrt[1 + (-1)^(2/3)]*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 = 1.88 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.66

method	result	size
default	\(-\frac {\sqrt {2}\, \left (\ln \left (\frac {\sqrt {2}\, \sqrt {a^{2} b^{2}}\, a^{2} \sqrt {a^{4} x^{4}+b^{4}}+2 b \,a^{3} \left (a^{2} x^{2}-a b x +b^{2}\right )}{\left (a x -b \right )^{2}}\right )+\ln \left (2\right )\right ) \sqrt {-a^{2} b^{2}}+4 \left (\ln \left (-\frac {a^{2} \left (-\sqrt {-a^{2} b^{2}}\, \sqrt {a^{4} x^{4}+b^{4}}+b a \left (a x +b \right )^{2}\right )}{a^{2} x^{2}+a b x +b^{2}}\right )+\ln \left (2\right )\right ) \sqrt {a^{2} b^{2}}}{6 \sqrt {-a^{2} b^{2}}\, \sqrt {a^{2} b^{2}}}\)	\(176\)
pseudoelliptic	\(-\frac {\sqrt {2}\, \left (\ln \left (\frac {\sqrt {2}\, \sqrt {a^{2} b^{2}}\, a^{2} \sqrt {a^{4} x^{4}+b^{4}}+2 b \,a^{3} \left (a^{2} x^{2}-a b x +b^{2}\right )}{\left (a x -b \right )^{2}}\right )+\ln \left (2\right )\right ) \sqrt {-a^{2} b^{2}}+4 \left (\ln \left (-\frac {a^{2} \left (-\sqrt {-a^{2} b^{2}}\, \sqrt {a^{4} x^{4}+b^{4}}+b a \left (a x +b \right )^{2}\right )}{a^{2} x^{2}+a b x +b^{2}}\right )+\ln \left (2\right )\right ) \sqrt {a^{2} b^{2}}}{6 \sqrt {-a^{2} b^{2}}\, \sqrt {a^{2} b^{2}}}\)	\(176\)
elliptic	\(-a^{3} b^{3} \left (-\frac {2 b^{2} \sqrt {2}\, \ln \left (\frac {4 b^{4}+2 \left (x^{2}-\frac {b^{2}}{a^{2}}\right ) a^{2} b^{2}+2 \sqrt {2}\, \sqrt {b^{4}}\, \sqrt {\left (x^{2}-\frac {b^{2}}{a^{2}}\right )^{2} a^{4}+2 \left (x^{2}-\frac {b^{2}}{a^{2}}\right ) a^{2} b^{2}+2 b^{4}}}{x^{2}-\frac {b^{2}}{a^{2}}}\right )}{\left (-3 a^{2} b^{2}+\sqrt {-3 b^{4} a^{4}}\right ) \left (3 a^{2} b^{2}+\sqrt {-3 b^{4} a^{4}}\right ) \sqrt {b^{4}}}+\frac {\left (-a^{2} b^{2}+\sqrt {-3 b^{4} a^{4}}\right ) \sqrt {2}\, \ln \left (\frac {\frac {b^{2} \left (a^{2} b^{2}-\sqrt {-3 b^{4} a^{4}}\right )}{a^{2}}+\left (-a^{2} b^{2}+\sqrt {-3 b^{4} a^{4}}\right ) \left (x^{2}-\frac {-a^{2} b^{2}+\sqrt {-3 b^{4} a^{4}}}{2 a^{4}}\right )+\frac {\sqrt {2}\, \sqrt {\frac {b^{2} \left (a^{2} b^{2}-\sqrt {-3 b^{4} a^{4}}\right )}{a^{2}}}\, \sqrt {4 \left (x^{2}-\frac {-a^{2} b^{2}+\sqrt {-3 b^{4} a^{4}}}{2 a^{4}}\right )^{2} a^{4}+4 \left (-a^{2} b^{2}+\sqrt {-3 b^{4} a^{4}}\right ) \left (x^{2}-\frac {-a^{2} b^{2}+\sqrt {-3 b^{4} a^{4}}}{2 a^{4}}\right )+\frac {2 b^{2} \left (a^{2} b^{2}-\sqrt {-3 b^{4} a^{4}}\right )}{a^{2}}}}{2}}{x^{2}-\frac {-a^{2} b^{2}+\sqrt {-3 b^{4} a^{4}}}{2 a^{4}}}\right )}{a^{2} \left (-3 a^{2} b^{2}+\sqrt {-3 b^{4} a^{4}}\right ) \sqrt {-3 b^{4} a^{4}}\, \sqrt {\frac {b^{2} \left (a^{2} b^{2}-\sqrt {-3 b^{4} a^{4}}\right )}{a^{2}}}}-\frac {\left (a^{2} b^{2}+\sqrt {-3 b^{4} a^{4}}\right ) \sqrt {2}\, \ln \left (\frac {\frac {b^{2} \left (a^{2} b^{2}+\sqrt {-3 b^{4} a^{4}}\right )}{a^{2}}+\left (-a^{2} b^{2}-\sqrt {-3 b^{4} a^{4}}\right ) \left (x^{2}+\frac {a^{2} b^{2}+\sqrt {-3 b^{4} a^{4}}}{2 a^{4}}\right )+\frac {\sqrt {2}\, \sqrt {\frac {b^{2} \left (a^{2} b^{2}+\sqrt {-3 b^{4} a^{4}}\right )}{a^{2}}}\, \sqrt {4 \left (x^{2}+\frac {a^{2} b^{2}+\sqrt {-3 b^{4} a^{4}}}{2 a^{4}}\right )^{2} a^{4}+4 \left (-a^{2} b^{2}-\sqrt {-3 b^{4} a^{4}}\right ) \left (x^{2}+\frac {a^{2} b^{2}+\sqrt {-3 b^{4} a^{4}}}{2 a^{4}}\right )+\frac {2 b^{2} \left (a^{2} b^{2}+\sqrt {-3 b^{4} a^{4}}\right )}{a^{2}}}}{2}}{x^{2}+\frac {a^{2} b^{2}+\sqrt {-3 b^{4} a^{4}}}{2 a^{4}}}\right )}{a^{2} \left (3 a^{2} b^{2}+\sqrt {-3 b^{4} a^{4}}\right ) \sqrt {-3 b^{4} a^{4}}\, \sqrt {\frac {b^{2} \left (a^{2} b^{2}+\sqrt {-3 b^{4} a^{4}}\right )}{a^{2}}}}\right )+\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}\)	\(952\)

Input:

int((a^3*x^3+b^3)/(a^3*x^3-b^3)/(a^4*x^4+b^4)^(1/2),x,method=_RETURNVERBOS 
E)
 

Output:

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

Fricas [A] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.57 \[ \int \frac {b^3+a^3 x^3}{\left (-b^3+a^3 x^3\right ) \sqrt {b^4+a^4 x^4}} \, dx=\frac {\sqrt {2} \log \left (-\frac {3 \, a^{4} x^{4} - 4 \, a^{3} b x^{3} + 6 \, a^{2} b^{2} x^{2} - 4 \, a b^{3} x + 3 \, b^{4} - 2 \, \sqrt {2} \sqrt {a^{4} x^{4} + b^{4}} {\left (a^{2} x^{2} - a b x + b^{2}\right )}}{a^{4} x^{4} - 4 \, a^{3} b x^{3} + 6 \, a^{2} b^{2} x^{2} - 4 \, a b^{3} x + b^{4}}\right ) + 8 \, \arctan \left (\frac {\sqrt {a^{4} x^{4} + b^{4}}}{a^{2} x^{2} + 2 \, a b x + b^{2}}\right )}{12 \, a b} \] Input:

integrate((a^3*x^3+b^3)/(a^3*x^3-b^3)/(a^4*x^4+b^4)^(1/2),x, algorithm="fr 
icas")
 

Output:

1/12*(sqrt(2)*log(-(3*a^4*x^4 - 4*a^3*b*x^3 + 6*a^2*b^2*x^2 - 4*a*b^3*x + 
3*b^4 - 2*sqrt(2)*sqrt(a^4*x^4 + b^4)*(a^2*x^2 - a*b*x + b^2))/(a^4*x^4 - 
4*a^3*b*x^3 + 6*a^2*b^2*x^2 - 4*a*b^3*x + b^4)) + 8*arctan(sqrt(a^4*x^4 + 
b^4)/(a^2*x^2 + 2*a*b*x + b^2)))/(a*b)
 

Sympy [F]

\[ \int \frac {b^3+a^3 x^3}{\left (-b^3+a^3 x^3\right ) \sqrt {b^4+a^4 x^4}} \, dx=\int \frac {\left (a x + b\right ) \left (a^{2} x^{2} - a b x + b^{2}\right )}{\left (a x - b\right ) \sqrt {a^{4} x^{4} + b^{4}} \left (a^{2} x^{2} + a b x + b^{2}\right )}\, dx \] Input:

integrate((a**3*x**3+b**3)/(a**3*x**3-b**3)/(a**4*x**4+b**4)**(1/2),x)
 

Output:

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

Maxima [F]

\[ \int \frac {b^3+a^3 x^3}{\left (-b^3+a^3 x^3\right ) \sqrt {b^4+a^4 x^4}} \, dx=\int { \frac {a^{3} x^{3} + b^{3}}{\sqrt {a^{4} x^{4} + b^{4}} {\left (a^{3} x^{3} - b^{3}\right )}} \,d x } \] Input:

integrate((a^3*x^3+b^3)/(a^3*x^3-b^3)/(a^4*x^4+b^4)^(1/2),x, algorithm="ma 
xima")
 

Output:

integrate((a^3*x^3 + b^3)/(sqrt(a^4*x^4 + b^4)*(a^3*x^3 - b^3)), x)
 

Giac [F]

\[ \int \frac {b^3+a^3 x^3}{\left (-b^3+a^3 x^3\right ) \sqrt {b^4+a^4 x^4}} \, dx=\int { \frac {a^{3} x^{3} + b^{3}}{\sqrt {a^{4} x^{4} + b^{4}} {\left (a^{3} x^{3} - b^{3}\right )}} \,d x } \] Input:

integrate((a^3*x^3+b^3)/(a^3*x^3-b^3)/(a^4*x^4+b^4)^(1/2),x, algorithm="gi 
ac")
 

Output:

integrate((a^3*x^3 + b^3)/(sqrt(a^4*x^4 + b^4)*(a^3*x^3 - b^3)), x)
                                                                                    
                                                                                    
 

Mupad [F(-1)]

Timed out. \[ \int \frac {b^3+a^3 x^3}{\left (-b^3+a^3 x^3\right ) \sqrt {b^4+a^4 x^4}} \, dx=\int -\frac {a^3\,x^3+b^3}{\left (b^3-a^3\,x^3\right )\,\sqrt {a^4\,x^4+b^4}} \,d x \] Input:

int(-(b^3 + a^3*x^3)/((b^3 - a^3*x^3)*(b^4 + a^4*x^4)^(1/2)),x)
 

Output:

int(-(b^3 + a^3*x^3)/((b^3 - a^3*x^3)*(b^4 + a^4*x^4)^(1/2)), x)
 

Reduce [F]

\[ \int \frac {b^3+a^3 x^3}{\left (-b^3+a^3 x^3\right ) \sqrt {b^4+a^4 x^4}} \, dx=\left (\int \frac {\sqrt {a^{4} x^{4}+b^{4}}}{a^{7} x^{7}-a^{4} b^{3} x^{4}+a^{3} b^{4} x^{3}-b^{7}}d x \right ) b^{3}+\left (\int \frac {\sqrt {a^{4} x^{4}+b^{4}}\, x^{3}}{a^{7} x^{7}-a^{4} b^{3} x^{4}+a^{3} b^{4} x^{3}-b^{7}}d x \right ) a^{3} \] Input:

int((a^3*x^3+b^3)/(a^3*x^3-b^3)/(a^4*x^4+b^4)^(1/2),x)
 

Output:

int(sqrt(a**4*x**4 + b**4)/(a**7*x**7 - a**4*b**3*x**4 + a**3*b**4*x**3 - 
b**7),x)*b**3 + int((sqrt(a**4*x**4 + b**4)*x**3)/(a**7*x**7 - a**4*b**3*x 
**4 + a**3*b**4*x**3 - b**7),x)*a**3