Integrand size = 39, antiderivative size = 140 \[ \int \frac {\left (1+x^2\right ) \left (1+x^8\right ) \sqrt {1+x^2+x^4+x^6+x^8}}{x^7 \left (-1+x^2\right )} \, dx=\frac {\sqrt {1+x^2+x^4+x^6+x^8} \left (8+26 x^2+65 x^4+26 x^6+8 x^8\right )}{48 x^6}+2 \sqrt {5} \text {arctanh}\left (\frac {\sqrt {5} x^2}{1-2 x^2+x^4-\sqrt {1+x^2+x^4+x^6+x^8}}\right )+\frac {65 \log (x)}{16}-\frac {65}{32} \log \left (-2-x^2-2 x^4+2 \sqrt {1+x^2+x^4+x^6+x^8}\right ) \]
1/48*(x^8+x^6+x^4+x^2+1)^(1/2)*(8*x^8+26*x^6+65*x^4+26*x^2+8)/x^6+2*5^(1/2 )*arctanh(5^(1/2)*x^2/(1-2*x^2+x^4-(x^8+x^6+x^4+x^2+1)^(1/2)))+65/16*ln(x) -65/32*ln(-2-x^2-2*x^4+2*(x^8+x^6+x^4+x^2+1)^(1/2))
Time = 0.32 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.00 \[ \int \frac {\left (1+x^2\right ) \left (1+x^8\right ) \sqrt {1+x^2+x^4+x^6+x^8}}{x^7 \left (-1+x^2\right )} \, dx=\frac {\sqrt {1+x^2+x^4+x^6+x^8} \left (8+26 x^2+65 x^4+26 x^6+8 x^8\right )}{48 x^6}+2 \sqrt {5} \text {arctanh}\left (\frac {\sqrt {5} x^2}{1-2 x^2+x^4-\sqrt {1+x^2+x^4+x^6+x^8}}\right )+\frac {65 \log (x)}{16}-\frac {65}{32} \log \left (-2-x^2-2 x^4+2 \sqrt {1+x^2+x^4+x^6+x^8}\right ) \]
(Sqrt[1 + x^2 + x^4 + x^6 + x^8]*(8 + 26*x^2 + 65*x^4 + 26*x^6 + 8*x^8))/( 48*x^6) + 2*Sqrt[5]*ArcTanh[(Sqrt[5]*x^2)/(1 - 2*x^2 + x^4 - Sqrt[1 + x^2 + x^4 + x^6 + x^8])] + (65*Log[x])/16 - (65*Log[-2 - x^2 - 2*x^4 + 2*Sqrt[ 1 + x^2 + x^4 + x^6 + x^8]])/32
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.
| \(\displaystyle \int \frac {\left (x^2+1\right ) \left (x^8+1\right ) \sqrt {x^8+x^6+x^4+x^2+1}}{x^7 \left (x^2-1\right )} \, dx\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \int \left (\frac {4 \sqrt {x^8+x^6+x^4+x^2+1} x}{x^2-1}+\sqrt {x^8+x^6+x^4+x^2+1} x-\frac {2 \sqrt {x^8+x^6+x^4+x^2+1}}{x}-\frac {\sqrt {x^8+x^6+x^4+x^2+1}}{x^7}-\frac {2 \sqrt {x^8+x^6+x^4+x^2+1}}{x^5}-\frac {2 \sqrt {x^8+x^6+x^4+x^2+1}}{x^3}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{2} \text {Subst}\left (\int \sqrt {x^4+x^3+x^2+x+1}dx,x,x^2\right )+2 \text {Subst}\left (\int \frac {\sqrt {x^4+x^3+x^2+x+1}}{x-1}dx,x,x^2\right )-\text {Subst}\left (\int \frac {\sqrt {x^4+x^3+x^2+x+1}}{x}dx,x,x^2\right )-\int \frac {\sqrt {x^8+x^6+x^4+x^2+1}}{x^7}dx-2 \int \frac {\sqrt {x^8+x^6+x^4+x^2+1}}{x^5}dx-2 \int \frac {\sqrt {x^8+x^6+x^4+x^2+1}}{x^3}dx\) |
3.20.81.3.1 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 = 2.62 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.96
| method | result | size |
| pseudoelliptic | \(\frac {195 \ln \left (\frac {2 x^{4}+2 \sqrt {\frac {x^{8}+x^{6}+x^{4}+x^{2}+1}{x^{2}}}\, x +x^{2}+2}{x^{2}}\right ) x^{5}-96 \sqrt {5}\, \operatorname {arctanh}\left (\frac {\left (x^{4}+1\right ) \sqrt {5}}{2 x \sqrt {\frac {x^{8}+x^{6}+x^{4}+x^{2}+1}{x^{2}}}}\right ) x^{5}+16 \left (x^{8}+\frac {13}{4} x^{6}+\frac {65}{8} x^{4}+\frac {13}{4} x^{2}+1\right ) \sqrt {\frac {x^{8}+x^{6}+x^{4}+x^{2}+1}{x^{2}}}}{96 x^{5}}\) | \(134\) |
| trager | \(\frac {\sqrt {x^{8}+x^{6}+x^{4}+x^{2}+1}\, \left (8 x^{8}+26 x^{6}+65 x^{4}+26 x^{2}+8\right )}{48 x^{6}}+\frac {65 \ln \left (\frac {2 x^{4}+x^{2}+2 \sqrt {x^{8}+x^{6}+x^{4}+x^{2}+1}+2}{x^{2}}\right )}{32}-\operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right ) \ln \left (\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right ) x^{4}+\operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right )+2 \sqrt {x^{8}+x^{6}+x^{4}+x^{2}+1}}{\left (1+x \right )^{2} \left (-1+x \right )^{2}}\right )\) | \(135\) |
| risch | \(\frac {65 x^{12}+91 x^{10}+99 x^{8}+99 x^{6}+99 x^{4}+34 x^{2}+8}{48 x^{6} \sqrt {x^{8}+x^{6}+x^{4}+x^{2}+1}}+\frac {\left (8 x^{2}+26\right ) \sqrt {x^{8}+x^{6}+x^{4}+x^{2}+1}}{48}-\frac {65 \ln \left (\frac {-2-x^{2}-2 x^{4}+2 \sqrt {x^{8}+x^{6}+x^{4}+x^{2}+1}}{x^{2}}\right )}{32}-\operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right ) \ln \left (\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right ) x^{4}+\operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right )+2 \sqrt {x^{8}+x^{6}+x^{4}+x^{2}+1}}{\left (1+x \right )^{2} \left (-1+x \right )^{2}}\right )\) | \(172\) |
1/96*(195*ln((2*x^4+2*(1/x^2*(x^8+x^6+x^4+x^2+1))^(1/2)*x+x^2+2)/x^2)*x^5- 96*5^(1/2)*arctanh(1/2*(x^4+1)/x*5^(1/2)/(1/x^2*(x^8+x^6+x^4+x^2+1))^(1/2) )*x^5+16*(x^8+13/4*x^6+65/8*x^4+13/4*x^2+1)*(1/x^2*(x^8+x^6+x^4+x^2+1))^(1 /2))/x^5
Time = 0.31 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.19 \[ \int \frac {\left (1+x^2\right ) \left (1+x^8\right ) \sqrt {1+x^2+x^4+x^6+x^8}}{x^7 \left (-1+x^2\right )} \, dx=\frac {48 \, \sqrt {5} x^{6} \log \left (-\frac {9 \, x^{8} + 4 \, x^{6} + 14 \, x^{4} - 4 \, \sqrt {5} \sqrt {x^{8} + x^{6} + x^{4} + x^{2} + 1} {\left (x^{4} + 1\right )} + 4 \, x^{2} + 9}{x^{8} - 4 \, x^{6} + 6 \, x^{4} - 4 \, x^{2} + 1}\right ) + 195 \, x^{6} \log \left (-\frac {2 \, x^{4} + x^{2} + 2 \, \sqrt {x^{8} + x^{6} + x^{4} + x^{2} + 1} + 2}{x^{2}}\right ) + 2 \, {\left (8 \, x^{8} + 26 \, x^{6} + 65 \, x^{4} + 26 \, x^{2} + 8\right )} \sqrt {x^{8} + x^{6} + x^{4} + x^{2} + 1}}{96 \, x^{6}} \]
1/96*(48*sqrt(5)*x^6*log(-(9*x^8 + 4*x^6 + 14*x^4 - 4*sqrt(5)*sqrt(x^8 + x ^6 + x^4 + x^2 + 1)*(x^4 + 1) + 4*x^2 + 9)/(x^8 - 4*x^6 + 6*x^4 - 4*x^2 + 1)) + 195*x^6*log(-(2*x^4 + x^2 + 2*sqrt(x^8 + x^6 + x^4 + x^2 + 1) + 2)/x ^2) + 2*(8*x^8 + 26*x^6 + 65*x^4 + 26*x^2 + 8)*sqrt(x^8 + x^6 + x^4 + x^2 + 1))/x^6
\[ \int \frac {\left (1+x^2\right ) \left (1+x^8\right ) \sqrt {1+x^2+x^4+x^6+x^8}}{x^7 \left (-1+x^2\right )} \, dx=\int \frac {\sqrt {\left (x^{4} - x^{3} + x^{2} - x + 1\right ) \left (x^{4} + x^{3} + x^{2} + x + 1\right )} \left (x^{2} + 1\right ) \left (x^{8} + 1\right )}{x^{7} \left (x - 1\right ) \left (x + 1\right )}\, dx \]
Integral(sqrt((x**4 - x**3 + x**2 - x + 1)*(x**4 + x**3 + x**2 + x + 1))*( x**2 + 1)*(x**8 + 1)/(x**7*(x - 1)*(x + 1)), x)
\[ \int \frac {\left (1+x^2\right ) \left (1+x^8\right ) \sqrt {1+x^2+x^4+x^6+x^8}}{x^7 \left (-1+x^2\right )} \, dx=\int { \frac {\sqrt {x^{8} + x^{6} + x^{4} + x^{2} + 1} {\left (x^{8} + 1\right )} {\left (x^{2} + 1\right )}}{{\left (x^{2} - 1\right )} x^{7}} \,d x } \]
\[ \int \frac {\left (1+x^2\right ) \left (1+x^8\right ) \sqrt {1+x^2+x^4+x^6+x^8}}{x^7 \left (-1+x^2\right )} \, dx=\int { \frac {\sqrt {x^{8} + x^{6} + x^{4} + x^{2} + 1} {\left (x^{8} + 1\right )} {\left (x^{2} + 1\right )}}{{\left (x^{2} - 1\right )} x^{7}} \,d x } \]
Timed out. \[ \int \frac {\left (1+x^2\right ) \left (1+x^8\right ) \sqrt {1+x^2+x^4+x^6+x^8}}{x^7 \left (-1+x^2\right )} \, dx=\int \frac {\left (x^2+1\right )\,\left (x^8+1\right )\,\sqrt {x^8+x^6+x^4+x^2+1}}{x^7\,\left (x^2-1\right )} \,d x \]