Integrand size = 22, antiderivative size = 119 \[ \int \frac {1+x^{16}}{\sqrt {1+x^4} \left (-1+x^{16}\right )} \, dx=-\frac {x}{4 \sqrt {1+x^4}}-\frac {\arctan \left (\frac {\sqrt [4]{2} x}{\sqrt {1+x^4}}\right )}{4 \sqrt [4]{2}}-\frac {\arctan \left (\frac {\sqrt {2} x}{\sqrt {1+x^4}}\right )}{8 \sqrt {2}}-\frac {\text {arctanh}\left (\frac {\sqrt [4]{2} x}{\sqrt {1+x^4}}\right )}{4 \sqrt [4]{2}}-\frac {\text {arctanh}\left (\frac {\sqrt {2} x}{\sqrt {1+x^4}}\right )}{8 \sqrt {2}} \]
-1/4*x/(x^4+1)^(1/2)-1/8*arctan(2^(1/4)*x/(x^4+1)^(1/2))*2^(3/4)-1/16*arct an(2^(1/2)*x/(x^4+1)^(1/2))*2^(1/2)-1/8*arctanh(2^(1/4)*x/(x^4+1)^(1/2))*2 ^(3/4)-1/16*arctanh(2^(1/2)*x/(x^4+1)^(1/2))*2^(1/2)
Time = 0.53 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.95 \[ \int \frac {1+x^{16}}{\sqrt {1+x^4} \left (-1+x^{16}\right )} \, dx=\frac {1}{16} \left (-\frac {4 x}{\sqrt {1+x^4}}-2\ 2^{3/4} \arctan \left (\frac {\sqrt [4]{2} x}{\sqrt {1+x^4}}\right )-\sqrt {2} \arctan \left (\frac {\sqrt {2} x}{\sqrt {1+x^4}}\right )-2\ 2^{3/4} \text {arctanh}\left (\frac {\sqrt [4]{2} x}{\sqrt {1+x^4}}\right )-\sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} x}{\sqrt {1+x^4}}\right )\right ) \]
((-4*x)/Sqrt[1 + x^4] - 2*2^(3/4)*ArcTan[(2^(1/4)*x)/Sqrt[1 + x^4]] - Sqrt [2]*ArcTan[(Sqrt[2]*x)/Sqrt[1 + x^4]] - 2*2^(3/4)*ArcTanh[(2^(1/4)*x)/Sqrt [1 + x^4]] - Sqrt[2]*ArcTanh[(Sqrt[2]*x)/Sqrt[1 + x^4]])/16
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 1.14 (sec) , antiderivative size = 380, normalized size of antiderivative = 3.19, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, 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.
| \(\displaystyle \int \frac {x^{16}+1}{\sqrt {x^4+1} \left (x^{16}-1\right )} \, dx\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \int \left (\frac {1}{\sqrt {x^4+1}}+\frac {2}{\left (x^{16}-1\right ) \sqrt {x^4+1}}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\arctan \left (\frac {\sqrt [4]{2} x}{\sqrt {x^4+1}}\right )}{4 \sqrt [4]{2}}-\frac {\arctan \left (\frac {\sqrt {2} x}{\sqrt {x^4+1}}\right )}{8 \sqrt {2}}+\frac {i \left (\sqrt {2}+(1+i)\right ) \left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan (x),\frac {1}{2}\right )}{16 \sqrt {x^4+1}}+\frac {i \left (\sqrt {2}+(-1+i)\right ) \left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan (x),\frac {1}{2}\right )}{16 \sqrt {x^4+1}}-\frac {i \left (\sqrt {2}+(-1-i)\right ) \left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan (x),\frac {1}{2}\right )}{16 \sqrt {x^4+1}}+\frac {\left ((-1-i)-i \sqrt {2}\right ) \left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan (x),\frac {1}{2}\right )}{16 \sqrt {x^4+1}}+\frac {\left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan (x),\frac {1}{2}\right )}{4 \sqrt {x^4+1}}-\frac {\text {arctanh}\left (\frac {\sqrt [4]{2} x}{\sqrt {x^4+1}}\right )}{4 \sqrt [4]{2}}-\frac {\text {arctanh}\left (\frac {\sqrt {2} x}{\sqrt {x^4+1}}\right )}{8 \sqrt {2}}-\frac {x}{4 \sqrt {x^4+1}}\) |
-1/4*x/Sqrt[1 + x^4] - ArcTan[(2^(1/4)*x)/Sqrt[1 + x^4]]/(4*2^(1/4)) - Arc Tan[(Sqrt[2]*x)/Sqrt[1 + x^4]]/(8*Sqrt[2]) - ArcTanh[(2^(1/4)*x)/Sqrt[1 + x^4]]/(4*2^(1/4)) - ArcTanh[(Sqrt[2]*x)/Sqrt[1 + x^4]]/(8*Sqrt[2]) + ((1 + x^2)*Sqrt[(1 + x^4)/(1 + x^2)^2]*EllipticF[2*ArcTan[x], 1/2])/(4*Sqrt[1 + x^4]) + (((-1 - I) - I*Sqrt[2])*(1 + x^2)*Sqrt[(1 + x^4)/(1 + x^2)^2]*Ell ipticF[2*ArcTan[x], 1/2])/(16*Sqrt[1 + x^4]) - ((I/16)*((-1 - I) + Sqrt[2] )*(1 + x^2)*Sqrt[(1 + x^4)/(1 + x^2)^2]*EllipticF[2*ArcTan[x], 1/2])/Sqrt[ 1 + x^4] + ((I/16)*((-1 + I) + Sqrt[2])*(1 + x^2)*Sqrt[(1 + x^4)/(1 + x^2) ^2]*EllipticF[2*ArcTan[x], 1/2])/Sqrt[1 + x^4] + ((I/16)*((1 + I) + Sqrt[2 ])*(1 + x^2)*Sqrt[(1 + x^4)/(1 + x^2)^2]*EllipticF[2*ArcTan[x], 1/2])/Sqrt [1 + x^4]
3.18.72.3.1 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 = 8.95 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.17
| method | result | size |
| risch | \(-\frac {x}{4 \sqrt {x^{4}+1}}-\frac {\arctan \left (\frac {\sqrt {2}\, x}{\sqrt {x^{4}+1}}\right ) \sqrt {2}}{16}-\frac {\sqrt {2}\, \operatorname {arctanh}\left (\frac {\left (x^{2}-x +1\right ) \sqrt {2}}{\sqrt {x^{4}+1}}\right )}{32}+\frac {\sqrt {2}\, \operatorname {arctanh}\left (\frac {\left (x^{2}+x +1\right ) \sqrt {2}}{\sqrt {x^{4}+1}}\right )}{32}+\frac {\arctan \left (\frac {2^{\frac {3}{4}} \sqrt {x^{4}+1}}{2 x}\right ) 2^{\frac {3}{4}}}{8}-\frac {\ln \left (\frac {-2^{\frac {1}{4}} x -\sqrt {x^{4}+1}}{2^{\frac {1}{4}} x -\sqrt {x^{4}+1}}\right ) 2^{\frac {3}{4}}}{16}\) | \(139\) |
| elliptic | \(\frac {\left (-\frac {\sqrt {2}\, x}{4 \sqrt {x^{4}+1}}-\frac {\ln \left (1+\frac {\sqrt {2}\, \sqrt {x^{4}+1}}{2 x}\right )}{16}+\frac {\ln \left (-1+\frac {\sqrt {2}\, \sqrt {x^{4}+1}}{2 x}\right )}{16}+\frac {2^{\frac {1}{4}} \left (2 \arctan \left (\frac {2^{\frac {3}{4}} \sqrt {x^{4}+1}}{2 x}\right )-\ln \left (\frac {\frac {\sqrt {2}\, \sqrt {x^{4}+1}}{2 x}+\frac {2^{\frac {3}{4}}}{2}}{\frac {\sqrt {2}\, \sqrt {x^{4}+1}}{2 x}-\frac {2^{\frac {3}{4}}}{2}}\right )\right )}{8}+\frac {\arctan \left (\frac {\sqrt {2}\, \sqrt {x^{4}+1}}{2 x}\right )}{8}\right ) \sqrt {2}}{2}\) | \(150\) |
| default | \(\frac {-8 \sqrt {x^{4}+1}\, x -\left (x^{4}+1\right ) \left (\left (2 \arctan \left (\frac {\sqrt {2}\, x}{\sqrt {x^{4}+1}}\right )+\operatorname {arctanh}\left (\frac {\left (x^{2}-x +1\right ) \sqrt {2}}{\sqrt {x^{4}+1}}\right )-\operatorname {arctanh}\left (\frac {\left (x^{2}+x +1\right ) \sqrt {2}}{\sqrt {x^{4}+1}}\right )\right ) \sqrt {2}-2 \,2^{\frac {3}{4}} \left (\arctan \left (\frac {\left (\sqrt {2-\sqrt {2}}\, x^{2}+\sqrt {2-\sqrt {2}}-2 x \right ) 2^{\frac {3}{4}}}{2 \sqrt {x^{4}+1}}\right )-\arctan \left (\frac {\left (\sqrt {2-\sqrt {2}}\, x^{2}+\sqrt {2-\sqrt {2}}+2 x \right ) 2^{\frac {3}{4}}}{2 \sqrt {x^{4}+1}}\right )-\operatorname {arctanh}\left (\frac {\left (\sqrt {2+\sqrt {2}}\, x^{2}+\sqrt {2+\sqrt {2}}-2 x \right ) 2^{\frac {3}{4}}}{2 \sqrt {x^{4}+1}}\right )+\operatorname {arctanh}\left (\frac {\left (\sqrt {2+\sqrt {2}}\, x^{2}+\sqrt {2+\sqrt {2}}+2 x \right ) 2^{\frac {3}{4}}}{2 \sqrt {x^{4}+1}}\right )\right )\right )}{32 \left (-x \sqrt {2}+x^{2}+1\right ) \left (x \sqrt {2}+x^{2}+1\right )}\) | \(265\) |
| pseudoelliptic | \(\frac {-8 \sqrt {x^{4}+1}\, x -\left (x^{4}+1\right ) \left (\left (2 \arctan \left (\frac {\sqrt {2}\, x}{\sqrt {x^{4}+1}}\right )+\operatorname {arctanh}\left (\frac {\left (x^{2}-x +1\right ) \sqrt {2}}{\sqrt {x^{4}+1}}\right )-\operatorname {arctanh}\left (\frac {\left (x^{2}+x +1\right ) \sqrt {2}}{\sqrt {x^{4}+1}}\right )\right ) \sqrt {2}-2 \,2^{\frac {3}{4}} \left (\arctan \left (\frac {\left (\sqrt {2-\sqrt {2}}\, x^{2}+\sqrt {2-\sqrt {2}}-2 x \right ) 2^{\frac {3}{4}}}{2 \sqrt {x^{4}+1}}\right )-\arctan \left (\frac {\left (\sqrt {2-\sqrt {2}}\, x^{2}+\sqrt {2-\sqrt {2}}+2 x \right ) 2^{\frac {3}{4}}}{2 \sqrt {x^{4}+1}}\right )-\operatorname {arctanh}\left (\frac {\left (\sqrt {2+\sqrt {2}}\, x^{2}+\sqrt {2+\sqrt {2}}-2 x \right ) 2^{\frac {3}{4}}}{2 \sqrt {x^{4}+1}}\right )+\operatorname {arctanh}\left (\frac {\left (\sqrt {2+\sqrt {2}}\, x^{2}+\sqrt {2+\sqrt {2}}+2 x \right ) 2^{\frac {3}{4}}}{2 \sqrt {x^{4}+1}}\right )\right )\right )}{32 \left (-x \sqrt {2}+x^{2}+1\right ) \left (x \sqrt {2}+x^{2}+1\right )}\) | \(265\) |
| trager | \(-\frac {x}{4 \sqrt {x^{4}+1}}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) \ln \left (\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) x -\sqrt {x^{4}+1}}{\left (1+x \right ) \left (-1+x \right )}\right )}{16}-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right ) \ln \left (\frac {-\operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right ) x +\sqrt {x^{4}+1}}{x^{2}+1}\right )}{16}-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+2 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right )\right ) \ln \left (\frac {-x^{4} \operatorname {RootOf}\left (\textit {\_Z}^{2}+2 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right )\right )+x^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+2 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right )\right ) \operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right )-4 \sqrt {x^{4}+1}\, x -\operatorname {RootOf}\left (\textit {\_Z}^{2}+2 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right )\right )}{x^{4}+\operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) x^{2}+1}\right )}{16}-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) \operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right ) \operatorname {RootOf}\left (\textit {\_Z}^{2}+2 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right )\right ) \ln \left (-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) \operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right ) \operatorname {RootOf}\left (\textit {\_Z}^{2}+2 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right )\right ) x^{4}+\operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right ) \operatorname {RootOf}\left (\textit {\_Z}^{2}+2 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right )\right ) x^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) \operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right ) \operatorname {RootOf}\left (\textit {\_Z}^{2}+2 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right )\right )+8 \sqrt {x^{4}+1}\, x}{-x^{4}+\operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) x^{2}-1}\right )}{32}\) | \(331\) |
-1/4*x/(x^4+1)^(1/2)-1/16*arctan(2^(1/2)*x/(x^4+1)^(1/2))*2^(1/2)-1/32*2^( 1/2)*arctanh((x^2-x+1)*2^(1/2)/(x^4+1)^(1/2))+1/32*2^(1/2)*arctanh((x^2+x+ 1)*2^(1/2)/(x^4+1)^(1/2))+1/8*arctan(1/2*2^(3/4)/x*(x^4+1)^(1/2))*2^(3/4)- 1/16*ln((-2^(1/4)*x-(x^4+1)^(1/2))/(2^(1/4)*x-(x^4+1)^(1/2)))*2^(3/4)
Result contains complex when optimal does not.
Time = 0.39 (sec) , antiderivative size = 376, normalized size of antiderivative = 3.16 \[ \int \frac {1+x^{16}}{\sqrt {1+x^4} \left (-1+x^{16}\right )} \, dx=-\frac {2^{\frac {3}{4}} {\left (x^{4} + 1\right )} \log \left (-\frac {2^{\frac {3}{4}} {\left (x^{8} + 4 \, x^{4} + 1\right )} + 4 \, {\left (x^{5} + \sqrt {2} x^{3} + x\right )} \sqrt {x^{4} + 1} + 4 \cdot 2^{\frac {1}{4}} {\left (x^{6} + x^{2}\right )}}{x^{8} + 1}\right ) - 2^{\frac {3}{4}} {\left (x^{4} + 1\right )} \log \left (\frac {2^{\frac {3}{4}} {\left (x^{8} + 4 \, x^{4} + 1\right )} - 4 \, {\left (x^{5} + \sqrt {2} x^{3} + x\right )} \sqrt {x^{4} + 1} + 4 \cdot 2^{\frac {1}{4}} {\left (x^{6} + x^{2}\right )}}{x^{8} + 1}\right ) - 2^{\frac {3}{4}} {\left (i \, x^{4} + i\right )} \log \left (\frac {2^{\frac {3}{4}} {\left (i \, x^{8} + 4 i \, x^{4} + i\right )} - 4 \, {\left (x^{5} - \sqrt {2} x^{3} + x\right )} \sqrt {x^{4} + 1} - 4 \cdot 2^{\frac {1}{4}} {\left (i \, x^{6} + i \, x^{2}\right )}}{x^{8} + 1}\right ) - 2^{\frac {3}{4}} {\left (-i \, x^{4} - i\right )} \log \left (\frac {2^{\frac {3}{4}} {\left (-i \, x^{8} - 4 i \, x^{4} - i\right )} - 4 \, {\left (x^{5} - \sqrt {2} x^{3} + x\right )} \sqrt {x^{4} + 1} - 4 \cdot 2^{\frac {1}{4}} {\left (-i \, x^{6} - i \, x^{2}\right )}}{x^{8} + 1}\right ) + 2 \, \sqrt {2} {\left (x^{4} + 1\right )} \arctan \left (\frac {\sqrt {2} x}{\sqrt {x^{4} + 1}}\right ) - \sqrt {2} {\left (x^{4} + 1\right )} \log \left (\frac {x^{4} - 2 \, \sqrt {2} \sqrt {x^{4} + 1} x + 2 \, x^{2} + 1}{x^{4} - 2 \, x^{2} + 1}\right ) + 8 \, \sqrt {x^{4} + 1} x}{32 \, {\left (x^{4} + 1\right )}} \]
-1/32*(2^(3/4)*(x^4 + 1)*log(-(2^(3/4)*(x^8 + 4*x^4 + 1) + 4*(x^5 + sqrt(2 )*x^3 + x)*sqrt(x^4 + 1) + 4*2^(1/4)*(x^6 + x^2))/(x^8 + 1)) - 2^(3/4)*(x^ 4 + 1)*log((2^(3/4)*(x^8 + 4*x^4 + 1) - 4*(x^5 + sqrt(2)*x^3 + x)*sqrt(x^4 + 1) + 4*2^(1/4)*(x^6 + x^2))/(x^8 + 1)) - 2^(3/4)*(I*x^4 + I)*log((2^(3/ 4)*(I*x^8 + 4*I*x^4 + I) - 4*(x^5 - sqrt(2)*x^3 + x)*sqrt(x^4 + 1) - 4*2^( 1/4)*(I*x^6 + I*x^2))/(x^8 + 1)) - 2^(3/4)*(-I*x^4 - I)*log((2^(3/4)*(-I*x ^8 - 4*I*x^4 - I) - 4*(x^5 - sqrt(2)*x^3 + x)*sqrt(x^4 + 1) - 4*2^(1/4)*(- I*x^6 - I*x^2))/(x^8 + 1)) + 2*sqrt(2)*(x^4 + 1)*arctan(sqrt(2)*x/sqrt(x^4 + 1)) - sqrt(2)*(x^4 + 1)*log((x^4 - 2*sqrt(2)*sqrt(x^4 + 1)*x + 2*x^2 + 1)/(x^4 - 2*x^2 + 1)) + 8*sqrt(x^4 + 1)*x)/(x^4 + 1)
Timed out. \[ \int \frac {1+x^{16}}{\sqrt {1+x^4} \left (-1+x^{16}\right )} \, dx=\text {Timed out} \]
\[ \int \frac {1+x^{16}}{\sqrt {1+x^4} \left (-1+x^{16}\right )} \, dx=\int { \frac {x^{16} + 1}{{\left (x^{16} - 1\right )} \sqrt {x^{4} + 1}} \,d x } \]
\[ \int \frac {1+x^{16}}{\sqrt {1+x^4} \left (-1+x^{16}\right )} \, dx=\int { \frac {x^{16} + 1}{{\left (x^{16} - 1\right )} \sqrt {x^{4} + 1}} \,d x } \]
Timed out. \[ \int \frac {1+x^{16}}{\sqrt {1+x^4} \left (-1+x^{16}\right )} \, dx=\int \frac {x^{16}+1}{\sqrt {x^4+1}\,\left (x^{16}-1\right )} \,d x \]