Integrand size = 22, antiderivative size = 166 \[ \int \frac {-1+x^5}{\sqrt {1+x^4} \left (1+x^5\right )} \, dx=-\frac {1}{5} \sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} x}{1+2 x+x^2+\sqrt {1+x^4}}\right )+\frac {4}{5} \text {RootSum}\left [4-4 \text {$\#$1}-2 \text {$\#$1}^3+\text {$\#$1}^4\&,\frac {-2 \log (x)+2 \log \left (1+x^2+\sqrt {1+x^4}-x \text {$\#$1}\right )+\log (x) \text {$\#$1}-\log \left (1+x^2+\sqrt {1+x^4}-x \text {$\#$1}\right ) \text {$\#$1}-\log (x) \text {$\#$1}^2+\log \left (1+x^2+\sqrt {1+x^4}-x \text {$\#$1}\right ) \text {$\#$1}^2}{-2-3 \text {$\#$1}^2+2 \text {$\#$1}^3}\&\right ] \] Output:
Unintegrable
Time = 0.00 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.00 \[ \int \frac {-1+x^5}{\sqrt {1+x^4} \left (1+x^5\right )} \, dx=-\frac {1}{5} \sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} x}{1+2 x+x^2+\sqrt {1+x^4}}\right )+\frac {4}{5} \text {RootSum}\left [4-4 \text {$\#$1}-2 \text {$\#$1}^3+\text {$\#$1}^4\&,\frac {-2 \log (x)+2 \log \left (1+x^2+\sqrt {1+x^4}-x \text {$\#$1}\right )+\log (x) \text {$\#$1}-\log \left (1+x^2+\sqrt {1+x^4}-x \text {$\#$1}\right ) \text {$\#$1}-\log (x) \text {$\#$1}^2+\log \left (1+x^2+\sqrt {1+x^4}-x \text {$\#$1}\right ) \text {$\#$1}^2}{-2-3 \text {$\#$1}^2+2 \text {$\#$1}^3}\&\right ] \] Input:
Integrate[(-1 + x^5)/(Sqrt[1 + x^4]*(1 + x^5)),x]
Output:
-1/5*(Sqrt[2]*ArcTanh[(Sqrt[2]*x)/(1 + 2*x + x^2 + Sqrt[1 + x^4])]) + (4*R ootSum[4 - 4*#1 - 2*#1^3 + #1^4 & , (-2*Log[x] + 2*Log[1 + x^2 + Sqrt[1 + x^4] - x*#1] + Log[x]*#1 - Log[1 + x^2 + Sqrt[1 + x^4] - x*#1]*#1 - Log[x] *#1^2 + Log[1 + x^2 + Sqrt[1 + x^4] - x*#1]*#1^2)/(-2 - 3*#1^2 + 2*#1^3) & ])/5
Leaf count is larger than twice the leaf count of optimal. \(1127\) vs. \(2(166)=332\).
Time = 1.83 (sec) , antiderivative size = 1127, normalized size of antiderivative = 6.79, 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.
| \(\displaystyle \int \frac {x^5-1}{\sqrt {x^4+1} \left (x^5+1\right )} \, dx\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \int \left (\frac {1}{\sqrt {x^4+1}}-\frac {2}{\sqrt {x^4+1} \left (x^5+1\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {(-1)^{4/5} \left (1+\sqrt [5]{-1}\right ) \left (1+(-1)^{4/5}\right ) \arctan \left (\frac {\sqrt {\sqrt [5]{-1}-(-1)^{4/5}} x}{\sqrt {x^4+1}}\right )}{5 \left (1+(-1)^{3/5}\right ) \sqrt {\sqrt [5]{-1}-(-1)^{4/5}}}-\frac {\arctan \left (\frac {\sqrt {\sqrt [5]{-1}-(-1)^{4/5}} x}{\sqrt {x^4+1}}\right )}{5 \sqrt {\sqrt [5]{-1}-(-1)^{4/5}}}-\frac {\text {arctanh}\left (\frac {\sqrt {2} x}{\sqrt {x^4+1}}\right )}{5 \sqrt {2}}-\frac {\left (1-\sqrt [5]{-1}+(-1)^{2/5}\right ) \left (1-(-1)^{3/5}\right ) \text {arctanh}\left (\frac {\sqrt {(-1)^{2/5}-(-1)^{3/5}} x}{\sqrt {x^4+1}}\right )}{5 \sqrt {(-1)^{2/5}-(-1)^{3/5}}}-\frac {\text {arctanh}\left (\frac {\sqrt {(-1)^{2/5}-(-1)^{3/5}} x}{\sqrt {x^4+1}}\right )}{5 \sqrt {(-1)^{2/5}-(-1)^{3/5}}}+\frac {\text {arctanh}\left (\frac {x^2+1}{\sqrt {2} \sqrt {x^4+1}}\right )}{5 \sqrt {2}}-\frac {\sqrt [5]{-1} \text {arctanh}\left (\frac {x^2+(-1)^{2/5}}{\sqrt {1+(-1)^{4/5}} \sqrt {x^4+1}}\right )}{5 \sqrt {1+(-1)^{4/5}}}+\frac {(-1)^{2/5} \text {arctanh}\left (\frac {x^2+(-1)^{4/5}}{\sqrt {1-(-1)^{3/5}} \sqrt {x^4+1}}\right )}{5 \sqrt {1-(-1)^{3/5}}}-\frac {(-1)^{4/5} \text {arctanh}\left (\frac {(-1)^{3/5} \left ((-1)^{2/5} x^2+1\right )}{\sqrt {1-\sqrt [5]{-1}} \sqrt {x^4+1}}\right )}{5 \sqrt {1-\sqrt [5]{-1}}}+\frac {(-1)^{3/5} \text {arctanh}\left (\frac {\sqrt [5]{-1} \left ((-1)^{4/5} x^2+1\right )}{\sqrt {1+(-1)^{2/5}} \sqrt {x^4+1}}\right )}{5 \sqrt {1+(-1)^{2/5}}}-\frac {\left (1-(-1)^{4/5}\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 )}{5 \left (1+(-1)^{3/5}\right ) \sqrt {x^4+1}}-\frac {\left (1-\sqrt [5]{-1}+(-1)^{2/5}\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 )}{5 \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 )}{5 \left (1+(-1)^{2/5}\right ) \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 )}{5 \left (1-\sqrt [5]{-1}\right ) \sqrt {x^4+1}}+\frac {2 \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 )}{5 \sqrt {x^4+1}}+\frac {\sqrt [5]{-1} \left (1+\sqrt [5]{-1}\right ) \left (1+(-1)^{2/5}\right ) \left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} \operatorname {EllipticPi}\left (\frac {1}{4} (-1)^{4/5} \left (1-\sqrt [5]{-1}\right )^2,2 \arctan (x),\frac {1}{2}\right )}{10 \sqrt {x^4+1}}+\frac {\left (1-(-1)^{2/5}\right ) \left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} \operatorname {EllipticPi}\left (-\frac {1}{4} (-1)^{3/5} \left (1+(-1)^{2/5}\right )^2,2 \arctan (x),\frac {1}{2}\right )}{10 \left (1+(-1)^{2/5}\right ) \sqrt {x^4+1}}+\frac {(-1)^{3/5} \left (1-\sqrt [5]{-1}\right ) \left (1+(-1)^{3/5}\right ) \left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} \operatorname {EllipticPi}\left (\frac {1}{4} (-1)^{2/5} \left (1-(-1)^{3/5}\right )^2,2 \arctan (x),\frac {1}{2}\right )}{10 \sqrt {x^4+1}}+\frac {\left (1-(-1)^{4/5}\right )^2 \left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} \operatorname {EllipticPi}\left (-\frac {1}{4} \sqrt [5]{-1} \left (1+(-1)^{4/5}\right )^2,2 \arctan (x),\frac {1}{2}\right )}{10 \left (1+(-1)^{3/5}\right ) \sqrt {x^4+1}}\) |
Input:
Int[(-1 + x^5)/(Sqrt[1 + x^4]*(1 + x^5)),x]
Output:
-1/5*ArcTan[(Sqrt[(-1)^(1/5) - (-1)^(4/5)]*x)/Sqrt[1 + x^4]]/Sqrt[(-1)^(1/ 5) - (-1)^(4/5)] + ((-1)^(4/5)*(1 + (-1)^(1/5))*(1 + (-1)^(4/5))*ArcTan[(S qrt[(-1)^(1/5) - (-1)^(4/5)]*x)/Sqrt[1 + x^4]])/(5*(1 + (-1)^(3/5))*Sqrt[( -1)^(1/5) - (-1)^(4/5)]) - ArcTanh[(Sqrt[2]*x)/Sqrt[1 + x^4]]/(5*Sqrt[2]) - ArcTanh[(Sqrt[(-1)^(2/5) - (-1)^(3/5)]*x)/Sqrt[1 + x^4]]/(5*Sqrt[(-1)^(2 /5) - (-1)^(3/5)]) - ((1 - (-1)^(1/5) + (-1)^(2/5))*(1 - (-1)^(3/5))*ArcTa nh[(Sqrt[(-1)^(2/5) - (-1)^(3/5)]*x)/Sqrt[1 + x^4]])/(5*Sqrt[(-1)^(2/5) - (-1)^(3/5)]) + ArcTanh[(1 + x^2)/(Sqrt[2]*Sqrt[1 + x^4])]/(5*Sqrt[2]) - (( -1)^(1/5)*ArcTanh[((-1)^(2/5) + x^2)/(Sqrt[1 + (-1)^(4/5)]*Sqrt[1 + x^4])] )/(5*Sqrt[1 + (-1)^(4/5)]) + ((-1)^(2/5)*ArcTanh[((-1)^(4/5) + x^2)/(Sqrt[ 1 - (-1)^(3/5)]*Sqrt[1 + x^4])])/(5*Sqrt[1 - (-1)^(3/5)]) - ((-1)^(4/5)*Ar cTanh[((-1)^(3/5)*(1 + (-1)^(2/5)*x^2))/(Sqrt[1 - (-1)^(1/5)]*Sqrt[1 + x^4 ])])/(5*Sqrt[1 - (-1)^(1/5)]) + ((-1)^(3/5)*ArcTanh[((-1)^(1/5)*(1 + (-1)^ (4/5)*x^2))/(Sqrt[1 + (-1)^(2/5)]*Sqrt[1 + x^4])])/(5*Sqrt[1 + (-1)^(2/5)] ) + (2*(1 + x^2)*Sqrt[(1 + x^4)/(1 + x^2)^2]*EllipticF[2*ArcTan[x], 1/2])/ (5*Sqrt[1 + x^4]) - ((1 + x^2)*Sqrt[(1 + x^4)/(1 + x^2)^2]*EllipticF[2*Arc Tan[x], 1/2])/(5*(1 - (-1)^(1/5))*Sqrt[1 + x^4]) - ((1 + x^2)*Sqrt[(1 + x^ 4)/(1 + x^2)^2]*EllipticF[2*ArcTan[x], 1/2])/(5*(1 + (-1)^(2/5))*Sqrt[1 + x^4]) - ((1 - (-1)^(1/5) + (-1)^(2/5))*(1 + x^2)*Sqrt[(1 + x^4)/(1 + x^2)^ 2]*EllipticF[2*ArcTan[x], 1/2])/(5*Sqrt[1 + x^4]) - ((1 - (-1)^(4/5))*(...
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 = 2.64 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.73
| method | result | size |
| default | \(\frac {\sqrt {2}\, \operatorname {arctanh}\left (\frac {\left (x^{2}+x +1\right ) \sqrt {2}}{\sqrt {x^{4}+1}}\right )}{10}-\frac {\sqrt {2+2 \sqrt {5}}\, \operatorname {arctanh}\left (\frac {\sqrt {5}\, x^{2}+x^{2}+\sqrt {5}-4 x +1}{\sqrt {-2+2 \sqrt {5}}\, \sqrt {x^{4}+1}}\right )}{5}-\frac {\sqrt {-2+2 \sqrt {5}}\, \arctan \left (\frac {\sqrt {5}\, x^{2}-x^{2}+\sqrt {5}+4 x -1}{\sqrt {2+2 \sqrt {5}}\, \sqrt {x^{4}+1}}\right )}{5}\) | \(121\) |
| pseudoelliptic | \(\frac {\sqrt {2}\, \operatorname {arctanh}\left (\frac {\left (x^{2}+x +1\right ) \sqrt {2}}{\sqrt {x^{4}+1}}\right )}{10}-\frac {\sqrt {2+2 \sqrt {5}}\, \operatorname {arctanh}\left (\frac {\sqrt {5}\, x^{2}+x^{2}+\sqrt {5}-4 x +1}{\sqrt {-2+2 \sqrt {5}}\, \sqrt {x^{4}+1}}\right )}{5}-\frac {\sqrt {-2+2 \sqrt {5}}\, \arctan \left (\frac {\sqrt {5}\, x^{2}-x^{2}+\sqrt {5}+4 x -1}{\sqrt {2+2 \sqrt {5}}\, \sqrt {x^{4}+1}}\right )}{5}\) | \(121\) |
| elliptic | \(\frac {\sqrt {2}\, \operatorname {arctanh}\left (\frac {\left (2 x^{2}+2\right ) \sqrt {2}}{4 \sqrt {\left (x^{2}-1\right )^{2}+2 x^{2}}}\right )}{10}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4}-\textit {\_Z}^{2}-1\right )}{\sum }\textit {\_R} \ln \left (\left (-x^{2}+\sqrt {x^{4}+1}\right )^{2}+\left (\textit {\_R}^{2}-\textit {\_R} -1\right ) \left (-x^{2}+\sqrt {x^{4}+1}\right )+\textit {\_R}^{2}-\textit {\_R} \right )\right )}{5}+\frac {\left (\frac {4 \arctan \left (\frac {\sqrt {2}\, \sqrt {x^{4}+1}}{x \sqrt {\sqrt {5}+1}}\right )}{5 \sqrt {\sqrt {5}+1}}-\frac {4 \,\operatorname {arctanh}\left (\frac {\sqrt {2}\, \sqrt {x^{4}+1}}{x \sqrt {\sqrt {5}-1}}\right )}{5 \sqrt {\sqrt {5}-1}}-\frac {\ln \left (1+\frac {\sqrt {2}\, \sqrt {x^{4}+1}}{2 x}\right )}{10}+\frac {\ln \left (-1+\frac {\sqrt {2}\, \sqrt {x^{4}+1}}{2 x}\right )}{10}\right ) \sqrt {2}}{2}\) | \(206\) |
| trager | \(-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) \ln \left (\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) x^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) x +\operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right )-\sqrt {x^{4}+1}}{\left (1+x \right )^{2}}\right )}{10}+\operatorname {RootOf}\left (625 \textit {\_Z}^{4}-100 \textit {\_Z}^{2}-16\right ) \ln \left (-\frac {625 \operatorname {RootOf}\left (625 \textit {\_Z}^{4}-100 \textit {\_Z}^{2}-16\right )^{5} x -50 x^{2} \operatorname {RootOf}\left (625 \textit {\_Z}^{4}-100 \textit {\_Z}^{2}-16\right )^{3}-250 x \operatorname {RootOf}\left (625 \textit {\_Z}^{4}-100 \textit {\_Z}^{2}-16\right )^{3}-50 \operatorname {RootOf}\left (625 \textit {\_Z}^{4}-100 \textit {\_Z}^{2}-16\right )^{3}-30 \sqrt {x^{4}+1}\, \operatorname {RootOf}\left (625 \textit {\_Z}^{4}-100 \textit {\_Z}^{2}-16\right )^{2}+12 \operatorname {RootOf}\left (625 \textit {\_Z}^{4}-100 \textit {\_Z}^{2}-16\right ) x^{2}+24 \operatorname {RootOf}\left (625 \textit {\_Z}^{4}-100 \textit {\_Z}^{2}-16\right ) x +12 \operatorname {RootOf}\left (625 \textit {\_Z}^{4}-100 \textit {\_Z}^{2}-16\right )+8 \sqrt {x^{4}+1}}{25 \operatorname {RootOf}\left (625 \textit {\_Z}^{4}-100 \textit {\_Z}^{2}-16\right )^{2} x -4 x^{2}-4}\right )+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+25 \operatorname {RootOf}\left (625 \textit {\_Z}^{4}-100 \textit {\_Z}^{2}-16\right )^{2}-4\right ) \ln \left (-\frac {625 \operatorname {RootOf}\left (\textit {\_Z}^{2}+25 \operatorname {RootOf}\left (625 \textit {\_Z}^{4}-100 \textit {\_Z}^{2}-16\right )^{2}-4\right ) \operatorname {RootOf}\left (625 \textit {\_Z}^{4}-100 \textit {\_Z}^{2}-16\right )^{4} x +50 \operatorname {RootOf}\left (\textit {\_Z}^{2}+25 \operatorname {RootOf}\left (625 \textit {\_Z}^{4}-100 \textit {\_Z}^{2}-16\right )^{2}-4\right ) \operatorname {RootOf}\left (625 \textit {\_Z}^{4}-100 \textit {\_Z}^{2}-16\right )^{2} x^{2}+50 \operatorname {RootOf}\left (\textit {\_Z}^{2}+25 \operatorname {RootOf}\left (625 \textit {\_Z}^{4}-100 \textit {\_Z}^{2}-16\right )^{2}-4\right ) \operatorname {RootOf}\left (625 \textit {\_Z}^{4}-100 \textit {\_Z}^{2}-16\right )^{2} x +150 \sqrt {x^{4}+1}\, \operatorname {RootOf}\left (625 \textit {\_Z}^{4}-100 \textit {\_Z}^{2}-16\right )^{2}+50 \operatorname {RootOf}\left (625 \textit {\_Z}^{4}-100 \textit {\_Z}^{2}-16\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+25 \operatorname {RootOf}\left (625 \textit {\_Z}^{4}-100 \textit {\_Z}^{2}-16\right )^{2}-4\right )+4 \operatorname {RootOf}\left (\textit {\_Z}^{2}+25 \operatorname {RootOf}\left (625 \textit {\_Z}^{4}-100 \textit {\_Z}^{2}-16\right )^{2}-4\right ) x^{2}+16 \sqrt {x^{4}+1}+4 \operatorname {RootOf}\left (\textit {\_Z}^{2}+25 \operatorname {RootOf}\left (625 \textit {\_Z}^{4}-100 \textit {\_Z}^{2}-16\right )^{2}-4\right )}{4 x^{2}-4 x +25 \operatorname {RootOf}\left (625 \textit {\_Z}^{4}-100 \textit {\_Z}^{2}-16\right )^{2} x +4}\right )}{5}\) | \(561\) |
Input:
int((x^5-1)/(x^4+1)^(1/2)/(x^5+1),x,method=_RETURNVERBOSE)
Output:
1/10*2^(1/2)*arctanh((x^2+x+1)*2^(1/2)/(x^4+1)^(1/2))-1/5*(2+2*5^(1/2))^(1 /2)*arctanh((5^(1/2)*x^2+x^2+5^(1/2)-4*x+1)/(-2+2*5^(1/2))^(1/2)/(x^4+1)^( 1/2))-1/5*(-2+2*5^(1/2))^(1/2)*arctan((5^(1/2)*x^2-x^2+5^(1/2)+4*x-1)/(2+2 *5^(1/2))^(1/2)/(x^4+1)^(1/2))
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 0.22 (sec) , antiderivative size = 384, normalized size of antiderivative = 2.31 \[ \int \frac {-1+x^5}{\sqrt {1+x^4} \left (1+x^5\right )} \, dx=\frac {1}{5} \, \sqrt {2 \, \sqrt {5} - 2} \arctan \left (\frac {\sqrt {x^{4} + 1} {\left (3 \, x^{2} + \sqrt {5} {\left (x^{2} - 2 \, x + 1\right )} - 2 \, x + 3\right )} \sqrt {2 \, \sqrt {5} - 2}}{4 \, {\left (x^{4} + 2 \, x^{3} - 2 \, x^{2} + 2 \, x + 1\right )}}\right ) + \frac {1}{20} \, \sqrt {2} \log \left (-\frac {3 \, x^{4} + 4 \, x^{3} + 2 \, \sqrt {2} \sqrt {x^{4} + 1} {\left (x^{2} + x + 1\right )} + 6 \, x^{2} + 4 \, x + 3}{x^{4} + 4 \, x^{3} + 6 \, x^{2} + 4 \, x + 1}\right ) + \frac {1}{10} \, \sqrt {2 \, \sqrt {5} + 2} \log \left (-\frac {2 \, \sqrt {x^{4} + 1} {\left (7 \, x^{2} - \sqrt {5} {\left (3 \, x^{2} - 5 \, x + 3\right )} - 11 \, x + 7\right )} + {\left (4 \, x^{4} + 7 \, x^{3} - 14 \, x^{2} - \sqrt {5} {\left (2 \, x^{4} + 3 \, x^{3} - 6 \, x^{2} + 3 \, x + 2\right )} + 7 \, x + 4\right )} \sqrt {2 \, \sqrt {5} + 2}}{x^{4} - x^{3} + x^{2} - x + 1}\right ) - \frac {1}{10} \, \sqrt {2 \, \sqrt {5} + 2} \log \left (-\frac {2 \, \sqrt {x^{4} + 1} {\left (7 \, x^{2} - \sqrt {5} {\left (3 \, x^{2} - 5 \, x + 3\right )} - 11 \, x + 7\right )} - {\left (4 \, x^{4} + 7 \, x^{3} - 14 \, x^{2} - \sqrt {5} {\left (2 \, x^{4} + 3 \, x^{3} - 6 \, x^{2} + 3 \, x + 2\right )} + 7 \, x + 4\right )} \sqrt {2 \, \sqrt {5} + 2}}{x^{4} - x^{3} + x^{2} - x + 1}\right ) \] Input:
integrate((x^5-1)/(x^4+1)^(1/2)/(x^5+1),x, algorithm="fricas")
Output:
1/5*sqrt(2*sqrt(5) - 2)*arctan(1/4*sqrt(x^4 + 1)*(3*x^2 + sqrt(5)*(x^2 - 2 *x + 1) - 2*x + 3)*sqrt(2*sqrt(5) - 2)/(x^4 + 2*x^3 - 2*x^2 + 2*x + 1)) + 1/20*sqrt(2)*log(-(3*x^4 + 4*x^3 + 2*sqrt(2)*sqrt(x^4 + 1)*(x^2 + x + 1) + 6*x^2 + 4*x + 3)/(x^4 + 4*x^3 + 6*x^2 + 4*x + 1)) + 1/10*sqrt(2*sqrt(5) + 2)*log(-(2*sqrt(x^4 + 1)*(7*x^2 - sqrt(5)*(3*x^2 - 5*x + 3) - 11*x + 7) + (4*x^4 + 7*x^3 - 14*x^2 - sqrt(5)*(2*x^4 + 3*x^3 - 6*x^2 + 3*x + 2) + 7*x + 4)*sqrt(2*sqrt(5) + 2))/(x^4 - x^3 + x^2 - x + 1)) - 1/10*sqrt(2*sqrt(5 ) + 2)*log(-(2*sqrt(x^4 + 1)*(7*x^2 - sqrt(5)*(3*x^2 - 5*x + 3) - 11*x + 7 ) - (4*x^4 + 7*x^3 - 14*x^2 - sqrt(5)*(2*x^4 + 3*x^3 - 6*x^2 + 3*x + 2) + 7*x + 4)*sqrt(2*sqrt(5) + 2))/(x^4 - x^3 + x^2 - x + 1))
Not integrable
Time = 18.40 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.25 \[ \int \frac {-1+x^5}{\sqrt {1+x^4} \left (1+x^5\right )} \, dx=\int \frac {\left (x - 1\right ) \left (x^{4} + x^{3} + x^{2} + x + 1\right )}{\left (x + 1\right ) \sqrt {x^{4} + 1} \left (x^{4} - x^{3} + x^{2} - x + 1\right )}\, dx \] Input:
integrate((x**5-1)/(x**4+1)**(1/2)/(x**5+1),x)
Output:
Integral((x - 1)*(x**4 + x**3 + x**2 + x + 1)/((x + 1)*sqrt(x**4 + 1)*(x** 4 - x**3 + x**2 - x + 1)), x)
Not integrable
Time = 0.15 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.13 \[ \int \frac {-1+x^5}{\sqrt {1+x^4} \left (1+x^5\right )} \, dx=\int { \frac {x^{5} - 1}{{\left (x^{5} + 1\right )} \sqrt {x^{4} + 1}} \,d x } \] Input:
integrate((x^5-1)/(x^4+1)^(1/2)/(x^5+1),x, algorithm="maxima")
Output:
integrate((x^5 - 1)/((x^5 + 1)*sqrt(x^4 + 1)), x)
Not integrable
Time = 0.24 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.13 \[ \int \frac {-1+x^5}{\sqrt {1+x^4} \left (1+x^5\right )} \, dx=\int { \frac {x^{5} - 1}{{\left (x^{5} + 1\right )} \sqrt {x^{4} + 1}} \,d x } \] Input:
integrate((x^5-1)/(x^4+1)^(1/2)/(x^5+1),x, algorithm="giac")
Output:
integrate((x^5 - 1)/((x^5 + 1)*sqrt(x^4 + 1)), x)
Not integrable
Time = 0.00 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.13 \[ \int \frac {-1+x^5}{\sqrt {1+x^4} \left (1+x^5\right )} \, dx=\int \frac {x^5-1}{\sqrt {x^4+1}\,\left (x^5+1\right )} \,d x \] Input:
int((x^5 - 1)/((x^4 + 1)^(1/2)*(x^5 + 1)),x)
Output:
int((x^5 - 1)/((x^4 + 1)^(1/2)*(x^5 + 1)), x)
Not integrable
Time = 0.43 (sec) , antiderivative size = 233, normalized size of antiderivative = 1.40 \[ \int \frac {-1+x^5}{\sqrt {1+x^4} \left (1+x^5\right )} \, dx=-\frac {\sqrt {2}\, \mathrm {log}\left (x^{2}+2 x +1\right )}{10}+\frac {\sqrt {2}\, \mathrm {log}\left (\sqrt {x^{4}+1}\, \sqrt {2}+2 x^{2}+2 x +2\right )}{10}-\frac {4 \left (\int \frac {\sqrt {x^{4}+1}}{x^{8}-x^{7}+x^{6}-x^{5}+2 x^{4}-x^{3}+x^{2}-x +1}d x \right )}{5}+\frac {4 \left (\int \frac {\sqrt {x^{4}+1}\, x^{4}}{x^{8}-x^{7}+x^{6}-x^{5}+2 x^{4}-x^{3}+x^{2}-x +1}d x \right )}{5}-\frac {2 \left (\int \frac {\sqrt {x^{4}+1}\, x^{3}}{x^{8}-x^{7}+x^{6}-x^{5}+2 x^{4}-x^{3}+x^{2}-x +1}d x \right )}{5}+\frac {2 \left (\int \frac {\sqrt {x^{4}+1}\, x}{x^{8}-x^{7}+x^{6}-x^{5}+2 x^{4}-x^{3}+x^{2}-x +1}d x \right )}{5} \] Input:
int((x^5-1)/(x^4+1)^(1/2)/(x^5+1),x)
Output:
( - sqrt(2)*log(x**2 + 2*x + 1) + sqrt(2)*log(sqrt(x**4 + 1)*sqrt(2) + 2*x **2 + 2*x + 2) - 8*int(sqrt(x**4 + 1)/(x**8 - x**7 + x**6 - x**5 + 2*x**4 - x**3 + x**2 - x + 1),x) + 8*int((sqrt(x**4 + 1)*x**4)/(x**8 - x**7 + x** 6 - x**5 + 2*x**4 - x**3 + x**2 - x + 1),x) - 4*int((sqrt(x**4 + 1)*x**3)/ (x**8 - x**7 + x**6 - x**5 + 2*x**4 - x**3 + x**2 - x + 1),x) + 4*int((sqr t(x**4 + 1)*x)/(x**8 - x**7 + x**6 - x**5 + 2*x**4 - x**3 + x**2 - x + 1), x))/10