Integrand size = 42, antiderivative size = 366 \[ \int \frac {-b^{10}+a^{10} x^{10}}{\sqrt {b^4+a^4 x^4} \left (b^{10}+a^{10} x^{10}\right )} \, dx=-\frac {\sqrt {2} \arctan \left (\frac {\sqrt {2} a b x}{b^2+a^2 x^2+\sqrt {b^4+a^4 x^4}}\right )}{5 a b}+\frac {1}{5} \text {RootSum}\left [16 a^8 b^8-48 a^6 b^6 \text {$\#$1}^2+24 a^4 b^4 \text {$\#$1}^4-12 a^2 b^2 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {8 a^6 b^6 \log (x)-8 a^6 b^6 \log \left (b^2+a^2 x^2+\sqrt {b^4+a^4 x^4}-x \text {$\#$1}\right )-8 a^4 b^4 \log (x) \text {$\#$1}^2+8 a^4 b^4 \log \left (b^2+a^2 x^2+\sqrt {b^4+a^4 x^4}-x \text {$\#$1}\right ) \text {$\#$1}^2-4 a^2 b^2 \log (x) \text {$\#$1}^4+4 a^2 b^2 \log \left (b^2+a^2 x^2+\sqrt {b^4+a^4 x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4+\log (x) \text {$\#$1}^6-\log \left (b^2+a^2 x^2+\sqrt {b^4+a^4 x^4}-x \text {$\#$1}\right ) \text {$\#$1}^6}{12 a^6 b^6 \text {$\#$1}-12 a^4 b^4 \text {$\#$1}^3+9 a^2 b^2 \text {$\#$1}^5-\text {$\#$1}^7}\&\right ] \]
Time = 0.00 (sec) , antiderivative size = 364, normalized size of antiderivative = 0.99 \[ \int \frac {-b^{10}+a^{10} x^{10}}{\sqrt {b^4+a^4 x^4} \left (b^{10}+a^{10} x^{10}\right )} \, dx=-\frac {\sqrt {2} \arctan \left (\frac {\sqrt {2} a b x}{b^2+a^2 x^2+\sqrt {b^4+a^4 x^4}}\right )}{5 a b}+\frac {1}{5} \text {RootSum}\left [16 a^8 b^8-48 a^6 b^6 \text {$\#$1}^2+24 a^4 b^4 \text {$\#$1}^4-12 a^2 b^2 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {-8 a^6 b^6 \log (x)+8 a^6 b^6 \log \left (b^2+a^2 x^2+\sqrt {b^4+a^4 x^4}-x \text {$\#$1}\right )+8 a^4 b^4 \log (x) \text {$\#$1}^2-8 a^4 b^4 \log \left (b^2+a^2 x^2+\sqrt {b^4+a^4 x^4}-x \text {$\#$1}\right ) \text {$\#$1}^2+4 a^2 b^2 \log (x) \text {$\#$1}^4-4 a^2 b^2 \log \left (b^2+a^2 x^2+\sqrt {b^4+a^4 x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4-\log (x) \text {$\#$1}^6+\log \left (b^2+a^2 x^2+\sqrt {b^4+a^4 x^4}-x \text {$\#$1}\right ) \text {$\#$1}^6}{-12 a^6 b^6 \text {$\#$1}+12 a^4 b^4 \text {$\#$1}^3-9 a^2 b^2 \text {$\#$1}^5+\text {$\#$1}^7}\&\right ] \]
-1/5*(Sqrt[2]*ArcTan[(Sqrt[2]*a*b*x)/(b^2 + a^2*x^2 + Sqrt[b^4 + a^4*x^4]) ])/(a*b) + RootSum[16*a^8*b^8 - 48*a^6*b^6*#1^2 + 24*a^4*b^4*#1^4 - 12*a^2 *b^2*#1^6 + #1^8 & , (-8*a^6*b^6*Log[x] + 8*a^6*b^6*Log[b^2 + a^2*x^2 + Sq rt[b^4 + a^4*x^4] - x*#1] + 8*a^4*b^4*Log[x]*#1^2 - 8*a^4*b^4*Log[b^2 + a^ 2*x^2 + Sqrt[b^4 + a^4*x^4] - x*#1]*#1^2 + 4*a^2*b^2*Log[x]*#1^4 - 4*a^2*b ^2*Log[b^2 + a^2*x^2 + Sqrt[b^4 + a^4*x^4] - x*#1]*#1^4 - Log[x]*#1^6 + Lo g[b^2 + a^2*x^2 + Sqrt[b^4 + a^4*x^4] - x*#1]*#1^6)/(-12*a^6*b^6*#1 + 12*a ^4*b^4*#1^3 - 9*a^2*b^2*#1^5 + #1^7) & ]/5
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.
| \(\displaystyle \int \frac {a^{10} x^{10}-b^{10}}{\sqrt {a^4 x^4+b^4} \left (a^{10} x^{10}+b^{10}\right )} \, dx\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \int \left (\frac {1}{\sqrt {a^4 x^4+b^4}}-\frac {2 b^{10}}{\sqrt {a^4 x^4+b^4} \left (a^{10} x^{10}+b^{10}\right )}\right )dx\) |
| \(\Big \downarrow \) 7299 |
| \(\displaystyle \int \left (\frac {1}{\sqrt {a^4 x^4+b^4}}-\frac {2 b^{10}}{\sqrt {a^4 x^4+b^4} \left (a^{10} x^{10}+b^{10}\right )}\right )dx\) |
3.30.65.3.1 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 = 0.00 (sec) , antiderivative size = 187, normalized size of antiderivative = 0.51
| method | result | size |
| elliptic | \(\frac {\left (\frac {4 \arctan \left (\frac {\sqrt {a^{4} x^{4}+b^{4}}\, \sqrt {2}}{x \sqrt {-a^{2} b^{2}-\sqrt {5}\, \sqrt {a^{4} b^{4}}}}\right )}{5 \sqrt {-a^{2} b^{2}-\sqrt {5}\, \sqrt {a^{4} b^{4}}}}+\frac {4 \arctan \left (\frac {\sqrt {a^{4} x^{4}+b^{4}}\, \sqrt {2}}{x \sqrt {-a^{2} b^{2}+\sqrt {5}\, \sqrt {a^{4} b^{4}}}}\right )}{5 \sqrt {-a^{2} b^{2}+\sqrt {5}\, \sqrt {a^{4} b^{4}}}}+\frac {\arctan \left (\frac {\sqrt {a^{4} x^{4}+b^{4}}\, \sqrt {2}}{2 x a b}\right )}{5 a b}\right ) \sqrt {2}}{2}\) | \(187\) |
| default | \(-\frac {\sqrt {2}\, \left (\ln \left (2\right )+\ln \left (\frac {\left (-2 a^{2} b^{2} x +\sqrt {2}\, \sqrt {-a^{2} b^{2}}\, \sqrt {a^{4} x^{4}+b^{4}}\right ) a^{2}}{a^{2} x^{2}+b^{2}}\right )\right )}{10 \sqrt {-a^{2} b^{2}}}+\frac {4 \arctan \left (\frac {2 \sqrt {a^{4} x^{4}+b^{4}}}{x \sqrt {-2 a^{2} b^{2}-2 \sqrt {5}\, \sqrt {a^{4} b^{4}}}}\right )}{5 \sqrt {-2 a^{2} b^{2}-2 \sqrt {5}\, \sqrt {a^{4} b^{4}}}}+\frac {4 \arctan \left (\frac {2 \sqrt {a^{4} x^{4}+b^{4}}}{x \sqrt {-2 a^{2} b^{2}+2 \sqrt {5}\, \sqrt {a^{4} b^{4}}}}\right )}{5 \sqrt {-2 a^{2} b^{2}+2 \sqrt {5}\, \sqrt {a^{4} b^{4}}}}\) | \(217\) |
| pseudoelliptic | \(-\frac {\sqrt {2}\, \left (\ln \left (2\right )+\ln \left (\frac {\left (-2 a^{2} b^{2} x +\sqrt {2}\, \sqrt {-a^{2} b^{2}}\, \sqrt {a^{4} x^{4}+b^{4}}\right ) a^{2}}{a^{2} x^{2}+b^{2}}\right )\right )}{10 \sqrt {-a^{2} b^{2}}}+\frac {4 \arctan \left (\frac {2 \sqrt {a^{4} x^{4}+b^{4}}}{x \sqrt {-2 a^{2} b^{2}-2 \sqrt {5}\, \sqrt {a^{4} b^{4}}}}\right )}{5 \sqrt {-2 a^{2} b^{2}-2 \sqrt {5}\, \sqrt {a^{4} b^{4}}}}+\frac {4 \arctan \left (\frac {2 \sqrt {a^{4} x^{4}+b^{4}}}{x \sqrt {-2 a^{2} b^{2}+2 \sqrt {5}\, \sqrt {a^{4} b^{4}}}}\right )}{5 \sqrt {-2 a^{2} b^{2}+2 \sqrt {5}\, \sqrt {a^{4} b^{4}}}}\) | \(217\) |
1/2*(4/5/(-a^2*b^2-5^(1/2)*(a^4*b^4)^(1/2))^(1/2)*arctan((a^4*x^4+b^4)^(1/ 2)*2^(1/2)/x/(-a^2*b^2-5^(1/2)*(a^4*b^4)^(1/2))^(1/2))+4/5/(-a^2*b^2+5^(1/ 2)*(a^4*b^4)^(1/2))^(1/2)*arctan((a^4*x^4+b^4)^(1/2)*2^(1/2)/x/(-a^2*b^2+5 ^(1/2)*(a^4*b^4)^(1/2))^(1/2))+1/5/a/b*arctan(1/2*(a^4*x^4+b^4)^(1/2)*2^(1 /2)/x/a/b))*2^(1/2)
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 1.23 (sec) , antiderivative size = 1303, normalized size of antiderivative = 3.56 \[ \int \frac {-b^{10}+a^{10} x^{10}}{\sqrt {b^4+a^4 x^4} \left (b^{10}+a^{10} x^{10}\right )} \, dx=\text {Too large to display} \]
-1/20*(sqrt(2)*a*b*sqrt(-(5*sqrt(1/5)*a^2*b^2*sqrt(1/(a^4*b^4)) + 1)/(a^2* b^2))*log(-(sqrt(2)*(3*a^8*x^8 + 5*a^6*b^2*x^6 + 9*a^4*b^4*x^4 + 5*a^2*b^6 *x^2 + 3*b^8 - 5*sqrt(1/5)*(a^10*b^2*x^8 + 3*a^8*b^4*x^6 + 3*a^6*b^6*x^4 + 3*a^4*b^8*x^2 + a^2*b^10)*sqrt(1/(a^4*b^4)))*sqrt(-(5*sqrt(1/5)*a^2*b^2*s qrt(1/(a^4*b^4)) + 1)/(a^2*b^2)) + 4*(3*a^4*x^5 + a^2*b^2*x^3 + 3*b^4*x - 5*sqrt(1/5)*(a^6*b^2*x^5 + a^4*b^4*x^3 + a^2*b^6*x)*sqrt(1/(a^4*b^4)))*sqr t(a^4*x^4 + b^4))/(a^8*x^8 - a^6*b^2*x^6 + a^4*b^4*x^4 - a^2*b^6*x^2 + b^8 )) - sqrt(2)*a*b*sqrt(-(5*sqrt(1/5)*a^2*b^2*sqrt(1/(a^4*b^4)) + 1)/(a^2*b^ 2))*log((sqrt(2)*(3*a^8*x^8 + 5*a^6*b^2*x^6 + 9*a^4*b^4*x^4 + 5*a^2*b^6*x^ 2 + 3*b^8 - 5*sqrt(1/5)*(a^10*b^2*x^8 + 3*a^8*b^4*x^6 + 3*a^6*b^6*x^4 + 3* a^4*b^8*x^2 + a^2*b^10)*sqrt(1/(a^4*b^4)))*sqrt(-(5*sqrt(1/5)*a^2*b^2*sqrt (1/(a^4*b^4)) + 1)/(a^2*b^2)) - 4*(3*a^4*x^5 + a^2*b^2*x^3 + 3*b^4*x - 5*s qrt(1/5)*(a^6*b^2*x^5 + a^4*b^4*x^3 + a^2*b^6*x)*sqrt(1/(a^4*b^4)))*sqrt(a ^4*x^4 + b^4))/(a^8*x^8 - a^6*b^2*x^6 + a^4*b^4*x^4 - a^2*b^6*x^2 + b^8)) + sqrt(2)*a*b*sqrt((5*sqrt(1/5)*a^2*b^2*sqrt(1/(a^4*b^4)) - 1)/(a^2*b^2))* log(-(sqrt(2)*(3*a^8*x^8 + 5*a^6*b^2*x^6 + 9*a^4*b^4*x^4 + 5*a^2*b^6*x^2 + 3*b^8 + 5*sqrt(1/5)*(a^10*b^2*x^8 + 3*a^8*b^4*x^6 + 3*a^6*b^6*x^4 + 3*a^4 *b^8*x^2 + a^2*b^10)*sqrt(1/(a^4*b^4)))*sqrt((5*sqrt(1/5)*a^2*b^2*sqrt(1/( a^4*b^4)) - 1)/(a^2*b^2)) + 4*(3*a^4*x^5 + a^2*b^2*x^3 + 3*b^4*x + 5*sqrt( 1/5)*(a^6*b^2*x^5 + a^4*b^4*x^3 + a^2*b^6*x)*sqrt(1/(a^4*b^4)))*sqrt(a^...
Timed out. \[ \int \frac {-b^{10}+a^{10} x^{10}}{\sqrt {b^4+a^4 x^4} \left (b^{10}+a^{10} x^{10}\right )} \, dx=\text {Timed out} \]
Not integrable
Time = 0.29 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.11 \[ \int \frac {-b^{10}+a^{10} x^{10}}{\sqrt {b^4+a^4 x^4} \left (b^{10}+a^{10} x^{10}\right )} \, dx=\int { \frac {a^{10} x^{10} - b^{10}}{{\left (a^{10} x^{10} + b^{10}\right )} \sqrt {a^{4} x^{4} + b^{4}}} \,d x } \]
Not integrable
Time = 0.40 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.11 \[ \int \frac {-b^{10}+a^{10} x^{10}}{\sqrt {b^4+a^4 x^4} \left (b^{10}+a^{10} x^{10}\right )} \, dx=\int { \frac {a^{10} x^{10} - b^{10}}{{\left (a^{10} x^{10} + b^{10}\right )} \sqrt {a^{4} x^{4} + b^{4}}} \,d x } \]
Not integrable
Time = 0.00 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.11 \[ \int \frac {-b^{10}+a^{10} x^{10}}{\sqrt {b^4+a^4 x^4} \left (b^{10}+a^{10} x^{10}\right )} \, dx=\int -\frac {b^{10}-a^{10}\,x^{10}}{\sqrt {a^4\,x^4+b^4}\,\left (a^{10}\,x^{10}+b^{10}\right )} \,d x \]