\(\int \frac {1+x^5}{\sqrt {1+x^4} (-1+x^5)} \, dx\) [2232]

Optimal result

Integrand size = 22, antiderivative size = 166 \[ \int \frac {1+x^5}{\sqrt {1+x^4} \left (-1+x^5\right )} \, dx=-\frac {1}{5} \sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} x}{1-2 x+x^2+\sqrt {1+x^4}}\right )+\frac {4}{5} \text {RootSum}\left [4+4 \text {$\#$1}+2 \text {$\#$1}^3+\text {$\#$1}^4\&,\frac {-2 \log (x)+2 \log \left (1+x^2+\sqrt {1+x^4}-x \text {$\#$1}\right )-\log (x) \text {$\#$1}+\log \left (1+x^2+\sqrt {1+x^4}-x \text {$\#$1}\right ) \text {$\#$1}-\log (x) \text {$\#$1}^2+\log \left (1+x^2+\sqrt {1+x^4}-x \text {$\#$1}\right ) \text {$\#$1}^2}{2+3 \text {$\#$1}^2+2 \text {$\#$1}^3}\&\right ] \]

[Out]

Unintegrable

Rubi [F]

\[ \int \frac {1+x^5}{\sqrt {1+x^4} \left (-1+x^5\right )} \, dx=\int \frac {1+x^5}{\sqrt {1+x^4} \left (-1+x^5\right )} \, dx \]

[In]

Int[(1 + x^5)/(Sqrt[1 + x^4]*(-1 + x^5)),x]

[Out]

-1/5*ArcTanh[(Sqrt[2]*x)/Sqrt[1 + x^4]]/Sqrt[2] - ArcTanh[(1 + x^2)/(Sqrt[2]*Sqrt[1 + x^4])]/(5*Sqrt[2]) + (2*
(1 + x^2)*Sqrt[(1 + x^4)/(1 + x^2)^2]*EllipticF[2*ArcTan[x], 1/2])/(5*Sqrt[1 + x^4]) - (8*Defer[Int][1/(Sqrt[1
+ x^4]*(1 + x + x^2 + x^3 + x^4)), x])/5 - (6*Defer[Int][x/(Sqrt[1 + x^4]*(1 + x + x^2 + x^3 + x^4)), x])/5 -
(4*Defer[Int][x^2/(Sqrt[1 + x^4]*(1 + x + x^2 + x^3 + x^4)), x])/5 - (2*Defer[Int][x^3/(Sqrt[1 + x^4]*(1 + x
+ x^2 + x^3 + x^4)), x])/5

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {1}{\sqrt {1+x^4}}+\frac {2}{\sqrt {1+x^4} \left (-1+x^5\right )}\right ) \, dx \\ & = 2 \int \frac {1}{\sqrt {1+x^4} \left (-1+x^5\right )} \, dx+\int \frac {1}{\sqrt {1+x^4}} \, dx \\ & = \frac {\left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan (x),\frac {1}{2}\right )}{2 \sqrt {1+x^4}}+2 \int \left (\frac {1}{5 (-1+x) \sqrt {1+x^4}}+\frac {-4-3 x-2 x^2-x^3}{5 \sqrt {1+x^4} \left (1+x+x^2+x^3+x^4\right )}\right ) \, dx \\ & = \frac {\left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan (x),\frac {1}{2}\right )}{2 \sqrt {1+x^4}}+\frac {2}{5} \int \frac {1}{(-1+x) \sqrt {1+x^4}} \, dx+\frac {2}{5} \int \frac {-4-3 x-2 x^2-x^3}{\sqrt {1+x^4} \left (1+x+x^2+x^3+x^4\right )} \, dx \\ & = \frac {\left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan (x),\frac {1}{2}\right )}{2 \sqrt {1+x^4}}-\frac {2}{5} \int \frac {1}{\left (1-x^2\right ) \sqrt {1+x^4}} \, dx-\frac {2}{5} \int \frac {x}{\left (1-x^2\right ) \sqrt {1+x^4}} \, dx+\frac {2}{5} \int \left (-\frac {4}{\sqrt {1+x^4} \left (1+x+x^2+x^3+x^4\right )}-\frac {3 x}{\sqrt {1+x^4} \left (1+x+x^2+x^3+x^4\right )}-\frac {2 x^2}{\sqrt {1+x^4} \left (1+x+x^2+x^3+x^4\right )}-\frac {x^3}{\sqrt {1+x^4} \left (1+x+x^2+x^3+x^4\right )}\right ) \, dx \\ & = \frac {\left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan (x),\frac {1}{2}\right )}{2 \sqrt {1+x^4}}-\frac {1}{5} \int \frac {1}{\sqrt {1+x^4}} \, dx-\frac {1}{5} \int \frac {1+x^2}{\left (1-x^2\right ) \sqrt {1+x^4}} \, dx-\frac {1}{5} \text {Subst}\left (\int \frac {1}{(1-x) \sqrt {1+x^2}} \, dx,x,x^2\right )-\frac {2}{5} \int \frac {x^3}{\sqrt {1+x^4} \left (1+x+x^2+x^3+x^4\right )} \, dx-\frac {4}{5} \int \frac {x^2}{\sqrt {1+x^4} \left (1+x+x^2+x^3+x^4\right )} \, dx-\frac {6}{5} \int \frac {x}{\sqrt {1+x^4} \left (1+x+x^2+x^3+x^4\right )} \, dx-\frac {8}{5} \int \frac {1}{\sqrt {1+x^4} \left (1+x+x^2+x^3+x^4\right )} \, dx \\ & = \frac {2 \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan (x),\frac {1}{2}\right )}{5 \sqrt {1+x^4}}-\frac {1}{5} \text {Subst}\left (\int \frac {1}{1-2 x^2} \, dx,x,\frac {x}{\sqrt {1+x^4}}\right )+\frac {1}{5} \text {Subst}\left (\int \frac {1}{2-x^2} \, dx,x,\frac {-1-x^2}{\sqrt {1+x^4}}\right )-\frac {2}{5} \int \frac {x^3}{\sqrt {1+x^4} \left (1+x+x^2+x^3+x^4\right )} \, dx-\frac {4}{5} \int \frac {x^2}{\sqrt {1+x^4} \left (1+x+x^2+x^3+x^4\right )} \, dx-\frac {6}{5} \int \frac {x}{\sqrt {1+x^4} \left (1+x+x^2+x^3+x^4\right )} \, dx-\frac {8}{5} \int \frac {1}{\sqrt {1+x^4} \left (1+x+x^2+x^3+x^4\right )} \, dx \\ & = -\frac {\text {arctanh}\left (\frac {\sqrt {2} x}{\sqrt {1+x^4}}\right )}{5 \sqrt {2}}-\frac {\text {arctanh}\left (\frac {1+x^2}{\sqrt {2} \sqrt {1+x^4}}\right )}{5 \sqrt {2}}+\frac {2 \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan (x),\frac {1}{2}\right )}{5 \sqrt {1+x^4}}-\frac {2}{5} \int \frac {x^3}{\sqrt {1+x^4} \left (1+x+x^2+x^3+x^4\right )} \, dx-\frac {4}{5} \int \frac {x^2}{\sqrt {1+x^4} \left (1+x+x^2+x^3+x^4\right )} \, dx-\frac {6}{5} \int \frac {x}{\sqrt {1+x^4} \left (1+x+x^2+x^3+x^4\right )} \, dx-\frac {8}{5} \int \frac {1}{\sqrt {1+x^4} \left (1+x+x^2+x^3+x^4\right )} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.00 \[ \int \frac {1+x^5}{\sqrt {1+x^4} \left (-1+x^5\right )} \, dx=-\frac {1}{5} \sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} x}{1-2 x+x^2+\sqrt {1+x^4}}\right )+\frac {4}{5} \text {RootSum}\left [4+4 \text {$\#$1}+2 \text {$\#$1}^3+\text {$\#$1}^4\&,\frac {-2 \log (x)+2 \log \left (1+x^2+\sqrt {1+x^4}-x \text {$\#$1}\right )-\log (x) \text {$\#$1}+\log \left (1+x^2+\sqrt {1+x^4}-x \text {$\#$1}\right ) \text {$\#$1}-\log (x) \text {$\#$1}^2+\log \left (1+x^2+\sqrt {1+x^4}-x \text {$\#$1}\right ) \text {$\#$1}^2}{2+3 \text {$\#$1}^2+2 \text {$\#$1}^3}\&\right ] \]

[In]

Integrate[(1 + x^5)/(Sqrt[1 + x^4]*(-1 + x^5)),x]

[Out]

-1/5*(Sqrt[2]*ArcTanh[(Sqrt[2]*x)/(1 - 2*x + x^2 + Sqrt[1 + x^4])]) + (4*RootSum[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

Maple [N/A] (verified)

Time = 4.28 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.74

method	result	size
default	\(-\frac {\sqrt {2}\, \operatorname {arctanh}\left (\frac {\left (x^{2}-x +1\right ) \sqrt {2}}{\sqrt {x^{4}+1}}\right )}{10}+\frac {\operatorname {arctanh}\left (\frac {\sqrt {5}\, x^{2}+x^{2}+\sqrt {5}+4 x +1}{\sqrt {-2+2 \sqrt {5}}\, \sqrt {x^{4}+1}}\right ) \sqrt {2+2 \sqrt {5}}}{5}+\frac {\sqrt {-2+2 \sqrt {5}}\, \arctan \left (\frac {\sqrt {5}\, x^{2}-x^{2}+\sqrt {5}-4 x -1}{\sqrt {2+2 \sqrt {5}}\, \sqrt {x^{4}+1}}\right )}{5}\)	\(123\)
pseudoelliptic	\(-\frac {\sqrt {2}\, \operatorname {arctanh}\left (\frac {\left (x^{2}-x +1\right ) \sqrt {2}}{\sqrt {x^{4}+1}}\right )}{10}+\frac {\operatorname {arctanh}\left (\frac {\sqrt {5}\, x^{2}+x^{2}+\sqrt {5}+4 x +1}{\sqrt {-2+2 \sqrt {5}}\, \sqrt {x^{4}+1}}\right ) \sqrt {2+2 \sqrt {5}}}{5}+\frac {\sqrt {-2+2 \sqrt {5}}\, \arctan \left (\frac {\sqrt {5}\, x^{2}-x^{2}+\sqrt {5}-4 x -1}{\sqrt {2+2 \sqrt {5}}\, \sqrt {x^{4}+1}}\right )}{5}\)	\(123\)
elliptic	\(-\frac {\sqrt {2}\, \operatorname {arctanh}\left (\frac {\left (2 x^{2}+2\right ) \sqrt {2}}{4 \sqrt {\left (x^{2}-1\right )^{2}+2 x^{2}}}\right )}{10}-\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4}-\textit {\_Z}^{2}-1\right )}{\sum }\textit {\_R} \ln \left (\left (-x^{2}+\sqrt {x^{4}+1}\right )^{2}+\left (\textit {\_R}^{2}-\textit {\_R} -1\right ) \left (-x^{2}+\sqrt {x^{4}+1}\right )+\textit {\_R}^{2}-\textit {\_R} \right )\right )}{5}+\frac {\left (-\frac {\ln \left (1+\frac {\sqrt {2}\, \sqrt {x^{4}+1}}{2 x}\right )}{10}-\frac {4 \,\operatorname {arctanh}\left (\frac {\sqrt {2}\, \sqrt {x^{4}+1}}{x \sqrt {\sqrt {5}-1}}\right )}{5 \sqrt {\sqrt {5}-1}}+\frac {4 \arctan \left (\frac {\sqrt {2}\, \sqrt {x^{4}+1}}{x \sqrt {\sqrt {5}+1}}\right )}{5 \sqrt {\sqrt {5}+1}}+\frac {\ln \left (-1+\frac {\sqrt {2}\, \sqrt {x^{4}+1}}{2 x}\right )}{10}\right ) \sqrt {2}}{2}\)	\(206\)
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}-\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 \operatorname {RootOf}\left (625 \textit {\_Z}^{4}-100 \textit {\_Z}^{2}-16\right )^{2} \sqrt {x^{4}+1}+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 )}{25 \operatorname {RootOf}\left (625 \textit {\_Z}^{4}-100 \textit {\_Z}^{2}-16\right )^{2} x -4 x^{2}-4 x -4}\right )}{5}-\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}+30 \operatorname {RootOf}\left (625 \textit {\_Z}^{4}-100 \textit {\_Z}^{2}-16\right )^{2} \sqrt {x^{4}+1}+50 \operatorname {RootOf}\left (625 \textit {\_Z}^{4}-100 \textit {\_Z}^{2}-16\right )^{3}-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 -8 \sqrt {x^{4}+1}-12 \operatorname {RootOf}\left (625 \textit {\_Z}^{4}-100 \textit {\_Z}^{2}-16\right )}{4 x^{2}+25 \operatorname {RootOf}\left (625 \textit {\_Z}^{4}-100 \textit {\_Z}^{2}-16\right )^{2} x +4}\right )\)	\(561\)

[In]

int((x^5+1)/(x^4+1)^(1/2)/(x^5-1),x,method=_RETURNVERBOSE)

[Out]

-1/10*2^(1/2)*arctanh((x^2-x+1)*2^(1/2)/(x^4+1)^(1/2))+1/5*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))*(2+2*5^(1/2))^(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))

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 3 vs. order 1.

Time = 0.36 (sec) , antiderivative size = 542, normalized size of antiderivative = 3.27 \[ \int \frac {1+x^5}{\sqrt {1+x^4} \left (-1+x^5\right )} \, dx=-\frac {1}{10} \, \sqrt {2} \sqrt {\sqrt {5} + 1} \log \left (-\frac {2 \, {\left (\sqrt {2} {\left (4 \, x^{4} - 7 \, x^{3} - 14 \, x^{2} - \sqrt {5} {\left (2 \, x^{4} - 3 \, x^{3} - 6 \, x^{2} - 3 \, x + 2\right )} - 7 \, x + 4\right )} \sqrt {\sqrt {5} + 1} + 2 \, \sqrt {x^{4} + 1} {\left (7 \, x^{2} - \sqrt {5} {\left (3 \, x^{2} + 5 \, x + 3\right )} + 11 \, x + 7\right )}\right )}}{x^{4} + x^{3} + x^{2} + x + 1}\right ) + \frac {1}{10} \, \sqrt {2} \sqrt {\sqrt {5} + 1} \log \left (\frac {2 \, {\left (\sqrt {2} {\left (4 \, x^{4} - 7 \, x^{3} - 14 \, x^{2} - \sqrt {5} {\left (2 \, x^{4} - 3 \, x^{3} - 6 \, x^{2} - 3 \, x + 2\right )} - 7 \, x + 4\right )} \sqrt {\sqrt {5} + 1} - 2 \, \sqrt {x^{4} + 1} {\left (7 \, x^{2} - \sqrt {5} {\left (3 \, x^{2} + 5 \, x + 3\right )} + 11 \, x + 7\right )}\right )}}{x^{4} + x^{3} + x^{2} + x + 1}\right ) + \frac {1}{20} \, \sqrt {2} \log \left (-\frac {3 \, x^{4} - 4 \, x^{3} - 2 \, \sqrt {2} \sqrt {x^{4} + 1} {\left (x^{2} - x + 1\right )} + 6 \, x^{2} - 4 \, x + 3}{x^{4} - 4 \, x^{3} + 6 \, x^{2} - 4 \, x + 1}\right ) - \frac {1}{20} \, \sqrt {-8 \, \sqrt {5} + 8} \log \left (-\frac {4 \, \sqrt {x^{4} + 1} {\left (7 \, x^{2} + \sqrt {5} {\left (3 \, x^{2} + 5 \, x + 3\right )} + 11 \, x + 7\right )} + {\left (4 \, x^{4} - 7 \, x^{3} - 14 \, x^{2} + \sqrt {5} {\left (2 \, x^{4} - 3 \, x^{3} - 6 \, x^{2} - 3 \, x + 2\right )} - 7 \, x + 4\right )} \sqrt {-8 \, \sqrt {5} + 8}}{x^{4} + x^{3} + x^{2} + x + 1}\right ) + \frac {1}{20} \, \sqrt {-8 \, \sqrt {5} + 8} \log \left (-\frac {4 \, \sqrt {x^{4} + 1} {\left (7 \, x^{2} + \sqrt {5} {\left (3 \, x^{2} + 5 \, x + 3\right )} + 11 \, x + 7\right )} - {\left (4 \, x^{4} - 7 \, x^{3} - 14 \, x^{2} + \sqrt {5} {\left (2 \, x^{4} - 3 \, x^{3} - 6 \, x^{2} - 3 \, x + 2\right )} - 7 \, x + 4\right )} \sqrt {-8 \, \sqrt {5} + 8}}{x^{4} + x^{3} + x^{2} + x + 1}\right ) \]

[In]

integrate((x^5+1)/(x^4+1)^(1/2)/(x^5-1),x, algorithm="fricas")

[Out]

-1/10*sqrt(2)*sqrt(sqrt(5) + 1)*log(-2*(sqrt(2)*(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(sqrt(5) + 1) + 2*sqrt(x^4 + 1)*(7*x^2 - sqrt(5)*(3*x^2 + 5*x + 3) + 11*x + 7))/(x^4 + x^
3 + x^2 + x + 1)) + 1/10*sqrt(2)*sqrt(sqrt(5) + 1)*log(2*(sqrt(2)*(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(sqrt(5) + 1) - 2*sqrt(x^4 + 1)*(7*x^2 - sqrt(5)*(3*x^2 + 5*x + 3) + 11
*x + 7))/(x^4 + x^3 + x^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)) - 1/20*sqrt(-8*sqrt(5) + 8)*log(-(4*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(-8*sqrt(5) + 8))/(x^4 + x^3 + x^2 + x + 1)) + 1/20*sqrt(-8*sqrt(5) + 8)*log(-(4*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(-8*sqrt(5) + 8))/(x^4 + x^3 + x^2 + x + 1))

Sympy [N/A]

Not integrable

Time = 24.94 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.25 \[ \int \frac {1+x^5}{\sqrt {1+x^4} \left (-1+x^5\right )} \, dx=\int \frac {\left (x + 1\right ) \left (x^{4} - x^{3} + x^{2} - x + 1\right )}{\left (x - 1\right ) \sqrt {x^{4} + 1} \left (x^{4} + x^{3} + x^{2} + x + 1\right )}\, dx \]

[In]

integrate((x**5+1)/(x**4+1)**(1/2)/(x**5-1),x)

[Out]

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)

Maxima [N/A]

Not integrable

Time = 0.30 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.13 \[ \int \frac {1+x^5}{\sqrt {1+x^4} \left (-1+x^5\right )} \, dx=\int { \frac {x^{5} + 1}{{\left (x^{5} - 1\right )} \sqrt {x^{4} + 1}} \,d x } \]

[In]

integrate((x^5+1)/(x^4+1)^(1/2)/(x^5-1),x, algorithm="maxima")

[Out]

integrate((x^5 + 1)/((x^5 - 1)*sqrt(x^4 + 1)), x)

Giac [N/A]

Not integrable

Time = 0.30 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.13 \[ \int \frac {1+x^5}{\sqrt {1+x^4} \left (-1+x^5\right )} \, dx=\int { \frac {x^{5} + 1}{{\left (x^{5} - 1\right )} \sqrt {x^{4} + 1}} \,d x } \]

[In]

integrate((x^5+1)/(x^4+1)^(1/2)/(x^5-1),x, algorithm="giac")

[Out]

integrate((x^5 + 1)/((x^5 - 1)*sqrt(x^4 + 1)), x)

Mupad [N/A]

Not integrable

Time = 0.00 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.13 \[ \int \frac {1+x^5}{\sqrt {1+x^4} \left (-1+x^5\right )} \, dx=\int \frac {x^5+1}{\sqrt {x^4+1}\,\left (x^5-1\right )} \,d x \]

[In]

int((x^5 + 1)/((x^4 + 1)^(1/2)*(x^5 - 1)),x)

[Out]

int((x^5 + 1)/((x^4 + 1)^(1/2)*(x^5 - 1)), x)