Integrand size = 42, antiderivative size = 107 \[ \int \frac {-b^3+a^3 x^3}{\left (b^3+a^3 x^3\right ) \sqrt {b^4+a^4 x^4}} \, dx=-\frac {4 \arctan \left (\frac {a b x}{b^2-a b x+a^2 x^2+\sqrt {b^4+a^4 x^4}}\right )}{3 a b}-\frac {\sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} a b x}{b^2+2 a b x+a^2 x^2+\sqrt {b^4+a^4 x^4}}\right )}{3 a b} \]
-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
Time = 1.92 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.93 \[ \int \frac {-b^3+a^3 x^3}{\left (b^3+a^3 x^3\right ) \sqrt {b^4+a^4 x^4}} \, dx=-\frac {4 \arctan \left (\frac {a b x}{b^2-a b x+a^2 x^2+\sqrt {b^4+a^4 x^4}}\right )+\sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} a b x}{b^2+2 a b x+a^2 x^2+\sqrt {b^4+a^4 x^4}}\right )}{3 a b} \]
-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)
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 1.61 (sec) , antiderivative size = 523, normalized size of antiderivative = 4.89, 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 |
| \(\displaystyle -\frac {2 \arctan \left (\frac {a b x}{\sqrt {a^4 x^4+b^4}}\right )}{3 a b}-\frac {\text {arctanh}\left (\frac {\sqrt {2} a b x}{\sqrt {a^4 x^4+b^4}}\right )}{3 \sqrt {2} a b}-\frac {2 \left (a^2 x^2+b^2\right ) \sqrt {\frac {a^4 x^4+b^4}{\left (a^2 x^2+b^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {a x}{b}\right ),\frac {1}{2}\right )}{3 \left (1-i \sqrt {3}\right ) a b \sqrt {a^4 x^4+b^4}}-\frac {\left (1-\sqrt [3]{-1}\right ) \left (a^2 x^2+b^2\right ) \sqrt {\frac {a^4 x^4+b^4}{\left (a^2 x^2+b^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {a x}{b}\right ),\frac {1}{2}\right )}{3 a b \sqrt {a^4 x^4+b^4}}+\frac {\left (a^2 x^2+b^2\right ) \sqrt {\frac {a^4 x^4+b^4}{\left (a^2 x^2+b^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {a x}{b}\right ),\frac {1}{2}\right )}{3 a b \sqrt {a^4 x^4+b^4}}+\frac {\text {arctanh}\left (\frac {a^2 x^2+b^2}{\sqrt {2} \sqrt {a^4 x^4+b^4}}\right )}{3 \sqrt {2} a b}-\frac {\sqrt [3]{-1} \text {arctanh}\left (\frac {a^2 x^2+(-1)^{2/3} b^2}{\sqrt {1-\sqrt [3]{-1}} \sqrt {a^4 x^4+b^4}}\right )}{3 \sqrt {1-\sqrt [3]{-1}} a b}+\frac {(-1)^{2/3} \text {arctanh}\left (\frac {2 \left ((-1)^{2/3} a^2 x^2+b^2\right )}{\left (-\sqrt {3}+i\right ) \sqrt {a^4 x^4+b^4}}\right )}{3 \sqrt {1+(-1)^{2/3}} a b}\) |
(-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])
3.16.58.3.1 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 = 1.14 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.63
| method | result | size |
| default | \(-\frac {\left (\ln \left (\frac {\sqrt {2}\, \sqrt {a^{2} b^{2}}\, a^{2} \sqrt {a^{4} x^{4}+b^{4}}-2 a^{3} b \left (a^{2} x^{2}+a b x +b^{2}\right )}{\left (a x +b \right )^{2}}\right )+\ln \left (2\right )\right ) \sqrt {2}\, \sqrt {-a^{2} b^{2}}+4 \sqrt {a^{2} b^{2}}\, \left (\ln \left (\frac {\left (\sqrt {-a^{2} b^{2}}\, \sqrt {a^{4} x^{4}+b^{4}}+a b \left (a x -b \right )^{2}\right ) a^{2}}{a^{2} x^{2}-a b x +b^{2}}\right )+\ln \left (2\right )\right )}{6 \sqrt {a^{2} b^{2}}\, \sqrt {-a^{2} b^{2}}}\) | \(174\) |
| pseudoelliptic | \(-\frac {\left (\ln \left (\frac {\sqrt {2}\, \sqrt {a^{2} b^{2}}\, a^{2} \sqrt {a^{4} x^{4}+b^{4}}-2 a^{3} b \left (a^{2} x^{2}+a b x +b^{2}\right )}{\left (a x +b \right )^{2}}\right )+\ln \left (2\right )\right ) \sqrt {2}\, \sqrt {-a^{2} b^{2}}+4 \sqrt {a^{2} b^{2}}\, \left (\ln \left (\frac {\left (\sqrt {-a^{2} b^{2}}\, \sqrt {a^{4} x^{4}+b^{4}}+a b \left (a x -b \right )^{2}\right ) a^{2}}{a^{2} x^{2}-a b x +b^{2}}\right )+\ln \left (2\right )\right )}{6 \sqrt {a^{2} b^{2}}\, \sqrt {-a^{2} b^{2}}}\) | \(174\) |
| elliptic | \(a^{3} b^{3} \left (-\frac {2 b^{2} \sqrt {2}\, \ln \left (\frac {4 b^{4}+2 a^{2} b^{2} \left (x^{2}-\frac {b^{2}}{a^{2}}\right )+2 \sqrt {2}\, \sqrt {b^{4}}\, \sqrt {\left (x^{2}-\frac {b^{2}}{a^{2}}\right )^{2} a^{4}+2 a^{2} b^{2} \left (x^{2}-\frac {b^{2}}{a^{2}}\right )+2 b^{4}}}{x^{2}-\frac {b^{2}}{a^{2}}}\right )}{\left (-3 a^{2} b^{2}+\sqrt {-3 a^{4} b^{4}}\right ) \left (3 a^{2} b^{2}+\sqrt {-3 a^{4} b^{4}}\right ) \sqrt {b^{4}}}-\frac {\left (a^{2} b^{2}+\sqrt {-3 a^{4} b^{4}}\right ) \sqrt {2}\, \ln \left (\frac {\frac {b^{2} \left (a^{2} b^{2}+\sqrt {-3 a^{4} b^{4}}\right )}{a^{2}}+\left (-a^{2} b^{2}-\sqrt {-3 a^{4} b^{4}}\right ) \left (x^{2}+\frac {a^{2} b^{2}+\sqrt {-3 a^{4} b^{4}}}{2 a^{4}}\right )+\frac {\sqrt {2}\, \sqrt {\frac {b^{2} \left (a^{2} b^{2}+\sqrt {-3 a^{4} b^{4}}\right )}{a^{2}}}\, \sqrt {4 \left (x^{2}+\frac {a^{2} b^{2}+\sqrt {-3 a^{4} b^{4}}}{2 a^{4}}\right )^{2} a^{4}+4 \left (-a^{2} b^{2}-\sqrt {-3 a^{4} b^{4}}\right ) \left (x^{2}+\frac {a^{2} b^{2}+\sqrt {-3 a^{4} b^{4}}}{2 a^{4}}\right )+\frac {2 b^{2} \left (a^{2} b^{2}+\sqrt {-3 a^{4} b^{4}}\right )}{a^{2}}}}{2}}{x^{2}+\frac {a^{2} b^{2}+\sqrt {-3 a^{4} b^{4}}}{2 a^{4}}}\right )}{a^{2} \left (3 a^{2} b^{2}+\sqrt {-3 a^{4} b^{4}}\right ) \sqrt {-3 a^{4} b^{4}}\, \sqrt {\frac {b^{2} \left (a^{2} b^{2}+\sqrt {-3 a^{4} b^{4}}\right )}{a^{2}}}}+\frac {\left (-a^{2} b^{2}+\sqrt {-3 a^{4} b^{4}}\right ) \sqrt {2}\, \ln \left (\frac {\frac {b^{2} \left (a^{2} b^{2}-\sqrt {-3 a^{4} b^{4}}\right )}{a^{2}}+\left (-a^{2} b^{2}+\sqrt {-3 a^{4} b^{4}}\right ) \left (x^{2}-\frac {-a^{2} b^{2}+\sqrt {-3 a^{4} b^{4}}}{2 a^{4}}\right )+\frac {\sqrt {2}\, \sqrt {\frac {b^{2} \left (a^{2} b^{2}-\sqrt {-3 a^{4} b^{4}}\right )}{a^{2}}}\, \sqrt {4 \left (x^{2}-\frac {-a^{2} b^{2}+\sqrt {-3 a^{4} b^{4}}}{2 a^{4}}\right )^{2} a^{4}+4 \left (-a^{2} b^{2}+\sqrt {-3 a^{4} b^{4}}\right ) \left (x^{2}-\frac {-a^{2} b^{2}+\sqrt {-3 a^{4} b^{4}}}{2 a^{4}}\right )+\frac {2 b^{2} \left (a^{2} b^{2}-\sqrt {-3 a^{4} b^{4}}\right )}{a^{2}}}}{2}}{x^{2}-\frac {-a^{2} b^{2}+\sqrt {-3 a^{4} b^{4}}}{2 a^{4}}}\right )}{a^{2} \left (-3 a^{2} b^{2}+\sqrt {-3 a^{4} b^{4}}\right ) \sqrt {-3 a^{4} b^{4}}\, \sqrt {\frac {b^{2} \left (a^{2} b^{2}-\sqrt {-3 a^{4} b^{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}\) | \(951\) |
-1/6*((ln((2^(1/2)*(a^2*b^2)^(1/2)*a^2*(a^4*x^4+b^4)^(1/2)-2*a^3*b*(a^2*x^ 2+a*b*x+b^2))/(a*x+b)^2)+ln(2))*2^(1/2)*(-a^2*b^2)^(1/2)+4*(a^2*b^2)^(1/2) *(ln(((-a^2*b^2)^(1/2)*(a^4*x^4+b^4)^(1/2)+a*b*(a*x-b)^2)*a^2/(a^2*x^2-a*b *x+b^2))+ln(2)))/(a^2*b^2)^(1/2)/(-a^2*b^2)^(1/2)
Time = 0.35 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.54 \[ \int \frac {-b^3+a^3 x^3}{\left (b^3+a^3 x^3\right ) \sqrt {b^4+a^4 x^4}} \, dx=\frac {\sqrt {2} \log \left (-\frac {3 \, a^{4} x^{4} + 4 \, a^{3} b x^{3} + 6 \, a^{2} b^{2} x^{2} + 4 \, a b^{3} x + 3 \, b^{4} + 2 \, \sqrt {2} \sqrt {a^{4} x^{4} + b^{4}} {\left (a^{2} x^{2} + a b x + b^{2}\right )}}{a^{4} x^{4} + 4 \, a^{3} b x^{3} + 6 \, a^{2} b^{2} x^{2} + 4 \, a b^{3} x + b^{4}}\right ) - 8 \, \arctan \left (\frac {\sqrt {a^{4} x^{4} + b^{4}}}{a^{2} x^{2} - 2 \, a b x + b^{2}}\right )}{12 \, a b} \]
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)
\[ \int \frac {-b^3+a^3 x^3}{\left (b^3+a^3 x^3\right ) \sqrt {b^4+a^4 x^4}} \, dx=\int \frac {\left (a x - b\right ) \left (a^{2} x^{2} + a b x + b^{2}\right )}{\left (a x + b\right ) \sqrt {a^{4} x^{4} + b^{4}} \left (a^{2} x^{2} - a b x + b^{2}\right )}\, dx \]
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)
\[ \int \frac {-b^3+a^3 x^3}{\left (b^3+a^3 x^3\right ) \sqrt {b^4+a^4 x^4}} \, dx=\int { \frac {a^{3} x^{3} - b^{3}}{\sqrt {a^{4} x^{4} + b^{4}} {\left (a^{3} x^{3} + b^{3}\right )}} \,d x } \]
\[ \int \frac {-b^3+a^3 x^3}{\left (b^3+a^3 x^3\right ) \sqrt {b^4+a^4 x^4}} \, dx=\int { \frac {a^{3} x^{3} - b^{3}}{\sqrt {a^{4} x^{4} + b^{4}} {\left (a^{3} x^{3} + b^{3}\right )}} \,d x } \]
Timed out. \[ \int \frac {-b^3+a^3 x^3}{\left (b^3+a^3 x^3\right ) \sqrt {b^4+a^4 x^4}} \, dx=\int -\frac {b^3-a^3\,x^3}{\left (a^3\,x^3+b^3\right )\,\sqrt {a^4\,x^4+b^4}} \,d x \]