Integrand size = 42, antiderivative size = 166 \[ \int \frac {-b^6+a^6 x^6}{\sqrt {b^4+a^4 x^4} \left (b^6+a^6 x^6\right )} \, dx=-\frac {\sqrt {2} \arctan \left (\frac {\sqrt {2} a b x}{b^2+a^2 x^2+\sqrt {b^4+a^4 x^4}}\right )}{3 a b}+\frac {2 \text {arctanh}\left (\frac {\sqrt {4-2 \sqrt {3}} a b x}{b^2+a^2 x^2+\sqrt {b^4+a^4 x^4}}\right )}{3 a b}-\frac {2 \text {arctanh}\left (\frac {\sqrt {4+2 \sqrt {3}} a b x}{b^2+a^2 x^2+\sqrt {b^4+a^4 x^4}}\right )}{3 a b} \]
-1/3*2^(1/2)*arctan(2^(1/2)*a*b*x/(b^2+a^2*x^2+(a^4*x^4+b^4)^(1/2)))/a/b+2 /3*arctanh((3^(1/2)-1)*a*b*x/(b^2+a^2*x^2+(a^4*x^4+b^4)^(1/2)))/a/b-2/3*ar ctanh((1+3^(1/2))*a*b*x/(b^2+a^2*x^2+(a^4*x^4+b^4)^(1/2)))/a/b
Time = 1.19 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.46 \[ \int \frac {-b^6+a^6 x^6}{\sqrt {b^4+a^4 x^4} \left (b^6+a^6 x^6\right )} \, dx=-\frac {\sqrt {2} \arctan \left (\frac {\sqrt {2} a b x}{b^2+a^2 x^2+\sqrt {b^4+a^4 x^4}}\right )+2 \text {arctanh}\left (\frac {a b x}{\sqrt {b^4+a^4 x^4}}\right )}{3 a b} \]
-1/3*(Sqrt[2]*ArcTan[(Sqrt[2]*a*b*x)/(b^2 + a^2*x^2 + Sqrt[b^4 + a^4*x^4]) ] + 2*ArcTanh[(a*b*x)/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 = 6.10 (sec) , antiderivative size = 1192, normalized size of antiderivative = 7.18, 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 {(-1)^{5/6} a \arctan \left (\frac {\sqrt [6]{-1} \sqrt [6]{-a^6} \sqrt {a^2+\sqrt [3]{-1} \sqrt [3]{-a^6}} b x}{a \sqrt {b^4+a^4 x^4}}\right )}{3 \sqrt [6]{-a^6} \sqrt {a^2+\sqrt [3]{-1} \sqrt [3]{-a^6}} b}+\frac {\sqrt [6]{-1} \sqrt [6]{-a^6} \left ((-1)^{2/3} a^4-\left (-a^6\right )^{2/3}\right ) \arctan \left (\frac {\sqrt [6]{-1} \sqrt {a^4-\sqrt [3]{-1} \left (-a^6\right )^{2/3}} b x}{\sqrt [6]{-a^6} \sqrt {b^4+a^4 x^4}}\right )}{3 \sqrt {a^4-\sqrt [3]{-1} \left (-a^6\right )^{2/3}} \left (a^4+\sqrt [3]{-1} \left (-a^6\right )^{2/3}\right ) b}-\frac {\sqrt [6]{-a^6} \text {arctanh}\left (\frac {\sqrt {a^4+\left (-a^6\right )^{2/3}} b x}{\sqrt [6]{-a^6} \sqrt {b^4+a^4 x^4}}\right )}{3 \sqrt {a^4+\left (-a^6\right )^{2/3}} b}+\frac {\left (b^2+a^2 x^2\right ) \sqrt {\frac {b^4+a^4 x^4}{\left (b^2+a^2 x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {a x}{b}\right ),\frac {1}{2}\right )}{2 a b \sqrt {b^4+a^4 x^4}}-\frac {a \left (b^2+a^2 x^2\right ) \sqrt {\frac {b^4+a^4 x^4}{\left (b^2+a^2 x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {a x}{b}\right ),\frac {1}{2}\right )}{3 \left (a^2+\sqrt [3]{-a^6}\right ) b \sqrt {b^4+a^4 x^4}}-\frac {a \left (b^2+a^2 x^2\right ) \sqrt {\frac {b^4+a^4 x^4}{\left (b^2+a^2 x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {a x}{b}\right ),\frac {1}{2}\right )}{3 \left (a^2-\sqrt [3]{-1} \sqrt [3]{-a^6}\right ) b \sqrt {b^4+a^4 x^4}}-\frac {a \left (a^2-(-1)^{2/3} \sqrt [3]{-a^6}\right ) \left (b^2+a^2 x^2\right ) \sqrt {\frac {b^4+a^4 x^4}{\left (b^2+a^2 x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {a x}{b}\right ),\frac {1}{2}\right )}{3 \left (a^4+\sqrt [3]{-1} \left (-a^6\right )^{2/3}\right ) b \sqrt {b^4+a^4 x^4}}+\frac {\left (a^2-\sqrt [3]{-a^6}\right ) \left (b^2+a^2 x^2\right ) \sqrt {\frac {b^4+a^4 x^4}{\left (b^2+a^2 x^2\right )^2}} \operatorname {EllipticPi}\left (\frac {\left (a^2+\sqrt [3]{-a^6}\right )^2}{4 a^2 \sqrt [3]{-a^6}},2 \arctan \left (\frac {a x}{b}\right ),\frac {1}{2}\right )}{6 a \left (a^2+\sqrt [3]{-a^6}\right ) b \sqrt {b^4+a^4 x^4}}+\frac {\left (a^2+\sqrt [3]{-1} \sqrt [3]{-a^6}\right ) \left (b^2+a^2 x^2\right ) \sqrt {\frac {b^4+a^4 x^4}{\left (b^2+a^2 x^2\right )^2}} \operatorname {EllipticPi}\left (\frac {(-1)^{2/3} \left (a^2-\sqrt [3]{-1} \sqrt [3]{-a^6}\right )^2}{4 a^2 \sqrt [3]{-a^6}},2 \arctan \left (\frac {a x}{b}\right ),\frac {1}{2}\right )}{6 a \left (a^2-\sqrt [3]{-1} \sqrt [3]{-a^6}\right ) b \sqrt {b^4+a^4 x^4}}+\frac {\left (a^4-2 (-1)^{2/3} \sqrt [3]{-a^6} a^2-\sqrt [3]{-1} \left (-a^6\right )^{2/3}\right ) \left (b^2+a^2 x^2\right ) \sqrt {\frac {b^4+a^4 x^4}{\left (b^2+a^2 x^2\right )^2}} \operatorname {EllipticPi}\left (\frac {\sqrt [3]{-1} a^4 \left (a^2+(-1)^{2/3} \sqrt [3]{-a^6}\right )^2}{4 \left (-a^6\right )^{4/3}},2 \arctan \left (\frac {a x}{b}\right ),\frac {1}{2}\right )}{6 a \left (a^4+\sqrt [3]{-1} \left (-a^6\right )^{2/3}\right ) b \sqrt {b^4+a^4 x^4}}\) |
((-1)^(5/6)*a*ArcTan[((-1)^(1/6)*(-a^6)^(1/6)*Sqrt[a^2 + (-1)^(1/3)*(-a^6) ^(1/3)]*b*x)/(a*Sqrt[b^4 + a^4*x^4])])/(3*(-a^6)^(1/6)*Sqrt[a^2 + (-1)^(1/ 3)*(-a^6)^(1/3)]*b) + ((-1)^(1/6)*(-a^6)^(1/6)*((-1)^(2/3)*a^4 - (-a^6)^(2 /3))*ArcTan[((-1)^(1/6)*Sqrt[a^4 - (-1)^(1/3)*(-a^6)^(2/3)]*b*x)/((-a^6)^( 1/6)*Sqrt[b^4 + a^4*x^4])])/(3*Sqrt[a^4 - (-1)^(1/3)*(-a^6)^(2/3)]*(a^4 + (-1)^(1/3)*(-a^6)^(2/3))*b) - ((-a^6)^(1/6)*ArcTanh[(Sqrt[a^4 + (-a^6)^(2/ 3)]*b*x)/((-a^6)^(1/6)*Sqrt[b^4 + a^4*x^4])])/(3*Sqrt[a^4 + (-a^6)^(2/3)]* 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])/(2*a*b*Sqrt[b^4 + a^4*x^4]) - (a*(b^2 + a^2*x^2)*Sq rt[(b^4 + a^4*x^4)/(b^2 + a^2*x^2)^2]*EllipticF[2*ArcTan[(a*x)/b], 1/2])/( 3*(a^2 + (-a^6)^(1/3))*b*Sqrt[b^4 + a^4*x^4]) - (a*(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^ 2 - (-1)^(1/3)*(-a^6)^(1/3))*b*Sqrt[b^4 + a^4*x^4]) - (a*(a^2 - (-1)^(2/3) *(-a^6)^(1/3))*(b^2 + a^2*x^2)*Sqrt[(b^4 + a^4*x^4)/(b^2 + a^2*x^2)^2]*Ell ipticF[2*ArcTan[(a*x)/b], 1/2])/(3*(a^4 + (-1)^(1/3)*(-a^6)^(2/3))*b*Sqrt[ b^4 + a^4*x^4]) + ((a^2 - (-a^6)^(1/3))*(b^2 + a^2*x^2)*Sqrt[(b^4 + a^4*x^ 4)/(b^2 + a^2*x^2)^2]*EllipticPi[(a^2 + (-a^6)^(1/3))^2/(4*a^2*(-a^6)^(1/3 )), 2*ArcTan[(a*x)/b], 1/2])/(6*a*(a^2 + (-a^6)^(1/3))*b*Sqrt[b^4 + a^4*x^ 4]) + ((a^2 + (-1)^(1/3)*(-a^6)^(1/3))*(b^2 + a^2*x^2)*Sqrt[(b^4 + a^4*x^4 )/(b^2 + a^2*x^2)^2]*EllipticPi[((-1)^(2/3)*(a^2 - (-1)^(1/3)*(-a^6)^(1...
3.23.33.3.1 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 = 2.95 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.47
| method | result | size |
| elliptic | \(\frac {\left (-\frac {2 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a^{4} x^{4}+b^{4}}}{a b x}\right )}{3 a b}+\frac {\arctan \left (\frac {\sqrt {a^{4} x^{4}+b^{4}}\, \sqrt {2}}{2 x a b}\right )}{3 a b}\right ) \sqrt {2}}{2}\) | \(78\) |
| default | \(-\frac {\left (\sqrt {a^{2} b^{2}}\, \ln \left (2\right )+2 \sqrt {2}\, \sqrt {-a^{2} b^{2}}\, \ln \left (2\right )+\sqrt {a^{2} b^{2}}\, \ln \left (\frac {\left (-2 x \,a^{2} b^{2}+\sqrt {2}\, \sqrt {-a^{2} b^{2}}\, \sqrt {a^{4} x^{4}+b^{4}}\right ) a^{2}}{a^{2} x^{2}+b^{2}}\right )+\sqrt {2}\, \sqrt {-a^{2} b^{2}}\, \ln \left (-\frac {2 \left (-\frac {\sqrt {a^{2} b^{2}}\, \sqrt {a^{4} x^{4}+b^{4}}}{2}+\frac {\sqrt {3}\, \left (a^{2} x^{2}+b^{2}\right ) \sqrt {a^{2} b^{2}}}{2}+x \,a^{2} b^{2}\right ) a^{2}}{a^{2} x^{2}+\sqrt {3}\, \sqrt {a^{2} b^{2}}\, x +b^{2}}\right )+\sqrt {2}\, \sqrt {-a^{2} b^{2}}\, \ln \left (-\frac {2 a^{2} \left (-\frac {\sqrt {a^{2} b^{2}}\, \sqrt {a^{4} x^{4}+b^{4}}}{2}-\frac {\sqrt {3}\, \left (a^{2} x^{2}+b^{2}\right ) \sqrt {a^{2} b^{2}}}{2}+x \,a^{2} b^{2}\right )}{a^{2} x^{2}-\sqrt {3}\, \sqrt {a^{2} b^{2}}\, x +b^{2}}\right )\right ) \sqrt {2}}{6 \sqrt {-a^{2} b^{2}}\, \sqrt {a^{2} b^{2}}}\) | \(331\) |
| pseudoelliptic | \(-\frac {\left (\sqrt {a^{2} b^{2}}\, \ln \left (2\right )+2 \sqrt {2}\, \sqrt {-a^{2} b^{2}}\, \ln \left (2\right )+\sqrt {a^{2} b^{2}}\, \ln \left (\frac {\left (-2 x \,a^{2} b^{2}+\sqrt {2}\, \sqrt {-a^{2} b^{2}}\, \sqrt {a^{4} x^{4}+b^{4}}\right ) a^{2}}{a^{2} x^{2}+b^{2}}\right )+\sqrt {2}\, \sqrt {-a^{2} b^{2}}\, \ln \left (-\frac {2 \left (-\frac {\sqrt {a^{2} b^{2}}\, \sqrt {a^{4} x^{4}+b^{4}}}{2}+\frac {\sqrt {3}\, \left (a^{2} x^{2}+b^{2}\right ) \sqrt {a^{2} b^{2}}}{2}+x \,a^{2} b^{2}\right ) a^{2}}{a^{2} x^{2}+\sqrt {3}\, \sqrt {a^{2} b^{2}}\, x +b^{2}}\right )+\sqrt {2}\, \sqrt {-a^{2} b^{2}}\, \ln \left (-\frac {2 a^{2} \left (-\frac {\sqrt {a^{2} b^{2}}\, \sqrt {a^{4} x^{4}+b^{4}}}{2}-\frac {\sqrt {3}\, \left (a^{2} x^{2}+b^{2}\right ) \sqrt {a^{2} b^{2}}}{2}+x \,a^{2} b^{2}\right )}{a^{2} x^{2}-\sqrt {3}\, \sqrt {a^{2} b^{2}}\, x +b^{2}}\right )\right ) \sqrt {2}}{6 \sqrt {-a^{2} b^{2}}\, \sqrt {a^{2} b^{2}}}\) | \(331\) |
1/2*(-2/3*2^(1/2)/a/b*arctanh(1/a/b/x*(a^4*x^4+b^4)^(1/2))+1/3/a/b*arctan( 1/2*(a^4*x^4+b^4)^(1/2)*2^(1/2)/x/a/b))*2^(1/2)
Time = 0.32 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.61 \[ \int \frac {-b^6+a^6 x^6}{\sqrt {b^4+a^4 x^4} \left (b^6+a^6 x^6\right )} \, dx=-\frac {\sqrt {2} \arctan \left (\frac {\sqrt {2} a b x}{\sqrt {a^{4} x^{4} + b^{4}}}\right ) - 2 \, \log \left (\frac {a^{4} x^{4} + a^{2} b^{2} x^{2} + b^{4} - 2 \, \sqrt {a^{4} x^{4} + b^{4}} a b x}{a^{4} x^{4} - a^{2} b^{2} x^{2} + b^{4}}\right )}{6 \, a b} \]
-1/6*(sqrt(2)*arctan(sqrt(2)*a*b*x/sqrt(a^4*x^4 + b^4)) - 2*log((a^4*x^4 + a^2*b^2*x^2 + b^4 - 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)
\[ \int \frac {-b^6+a^6 x^6}{\sqrt {b^4+a^4 x^4} \left (b^6+a^6 x^6\right )} \, dx=\int \frac {\left (a x - b\right ) \left (a x + b\right ) \left (a^{2} x^{2} - a b x + b^{2}\right ) \left (a^{2} x^{2} + a b x + b^{2}\right )}{\left (a^{2} x^{2} + b^{2}\right ) \sqrt {a^{4} x^{4} + b^{4}} \left (a^{4} x^{4} - a^{2} b^{2} x^{2} + b^{4}\right )}\, dx \]
Integral((a*x - b)*(a*x + b)*(a**2*x**2 - a*b*x + b**2)*(a**2*x**2 + a*b*x + b**2)/((a**2*x**2 + b**2)*sqrt(a**4*x**4 + b**4)*(a**4*x**4 - a**2*b**2 *x**2 + b**4)), x)
\[ \int \frac {-b^6+a^6 x^6}{\sqrt {b^4+a^4 x^4} \left (b^6+a^6 x^6\right )} \, dx=\int { \frac {a^{6} x^{6} - b^{6}}{{\left (a^{6} x^{6} + b^{6}\right )} \sqrt {a^{4} x^{4} + b^{4}}} \,d x } \]
\[ \int \frac {-b^6+a^6 x^6}{\sqrt {b^4+a^4 x^4} \left (b^6+a^6 x^6\right )} \, dx=\int { \frac {a^{6} x^{6} - b^{6}}{{\left (a^{6} x^{6} + b^{6}\right )} \sqrt {a^{4} x^{4} + b^{4}}} \,d x } \]
Timed out. \[ \int \frac {-b^6+a^6 x^6}{\sqrt {b^4+a^4 x^4} \left (b^6+a^6 x^6\right )} \, dx=\int -\frac {b^6-a^6\,x^6}{\sqrt {a^4\,x^4+b^4}\,\left (a^6\,x^6+b^6\right )} \,d x \]