Integrand size = 7, antiderivative size = 168 \[ \int \frac {1}{1+x^5} \, dx=-\frac {1}{5} \sqrt {\frac {1}{2} \left (5+\sqrt {5}\right )} \arctan \left (\frac {1-\sqrt {5}-4 x}{\sqrt {2 \left (5+\sqrt {5}\right )}}\right )-\frac {1}{5} \sqrt {\frac {1}{2} \left (5-\sqrt {5}\right )} \arctan \left (\frac {1}{2} \sqrt {\frac {1}{10} \left (5+\sqrt {5}\right )} \left (1+\sqrt {5}-4 x\right )\right )+\frac {1}{5} \log (1+x)-\frac {1}{20} \left (1-\sqrt {5}\right ) \log \left (1-\frac {1}{2} \left (1-\sqrt {5}\right ) x+x^2\right )-\frac {1}{20} \left (1+\sqrt {5}\right ) \log \left (1-\frac {1}{2} \left (1+\sqrt {5}\right ) x+x^2\right ) \] Output:
-1/10*(10+2*5^(1/2))^(1/2)*arctan((1-5^(1/2)-4*x)/(10+2*5^(1/2))^(1/2))-1/ 10*(10-2*5^(1/2))^(1/2)*arctan(1/20*(50+10*5^(1/2))^(1/2)*(1+5^(1/2)-4*x)) +1/5*ln(1+x)-1/20*(-5^(1/2)+1)*ln(1-1/2*x*(-5^(1/2)+1)+x^2)-1/20*(5^(1/2)+ 1)*ln(1-1/2*(5^(1/2)+1)*x+x^2)
Time = 0.08 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.86 \[ \int \frac {1}{1+x^5} \, dx=\frac {1}{20} \left (-2 \sqrt {10-2 \sqrt {5}} \arctan \left (\frac {1+\sqrt {5}-4 x}{\sqrt {10-2 \sqrt {5}}}\right )+2 \sqrt {2 \left (5+\sqrt {5}\right )} \arctan \left (\frac {-1+\sqrt {5}+4 x}{\sqrt {2 \left (5+\sqrt {5}\right )}}\right )+4 \log (1+x)+\left (-1+\sqrt {5}\right ) \log \left (1+\frac {1}{2} \left (-1+\sqrt {5}\right ) x+x^2\right )-\left (1+\sqrt {5}\right ) \log \left (1-\frac {1}{2} \left (1+\sqrt {5}\right ) x+x^2\right )\right ) \] Input:
Integrate[(1 + x^5)^(-1),x]
Output:
(-2*Sqrt[10 - 2*Sqrt[5]]*ArcTan[(1 + Sqrt[5] - 4*x)/Sqrt[10 - 2*Sqrt[5]]] + 2*Sqrt[2*(5 + Sqrt[5])]*ArcTan[(-1 + Sqrt[5] + 4*x)/Sqrt[2*(5 + Sqrt[5]) ]] + 4*Log[1 + x] + (-1 + Sqrt[5])*Log[1 + ((-1 + Sqrt[5])*x)/2 + x^2] - ( 1 + Sqrt[5])*Log[1 - ((1 + Sqrt[5])*x)/2 + x^2])/20
Time = 0.56 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.01, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.143, Rules used = {751, 16, 27, 1142, 25, 1083, 217, 1103}
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.
| \(\displaystyle \int \frac {1}{x^5+1} \, dx\) |
| \(\Big \downarrow \) 751 |
| \(\displaystyle \frac {2}{5} \int \frac {4-\left (1-\sqrt {5}\right ) x}{2 \left (2 x^2-\left (1-\sqrt {5}\right ) x+2\right )}dx+\frac {2}{5} \int \frac {4-\left (1+\sqrt {5}\right ) x}{2 \left (2 x^2-\left (1+\sqrt {5}\right ) x+2\right )}dx+\frac {1}{5} \int \frac {1}{x+1}dx\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {2}{5} \int \frac {4-\left (1-\sqrt {5}\right ) x}{2 \left (2 x^2-\left (1-\sqrt {5}\right ) x+2\right )}dx+\frac {2}{5} \int \frac {4-\left (1+\sqrt {5}\right ) x}{2 \left (2 x^2-\left (1+\sqrt {5}\right ) x+2\right )}dx+\frac {1}{5} \log (x+1)\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{5} \int \frac {4-\left (1-\sqrt {5}\right ) x}{2 x^2-\left (1-\sqrt {5}\right ) x+2}dx+\frac {1}{5} \int \frac {4-\left (1+\sqrt {5}\right ) x}{2 x^2-\left (1+\sqrt {5}\right ) x+2}dx+\frac {1}{5} \log (x+1)\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{2} \left (5+\sqrt {5}\right ) \int \frac {1}{2 x^2-\left (1-\sqrt {5}\right ) x+2}dx-\frac {1}{4} \left (1-\sqrt {5}\right ) \int -\frac {-4 x-\sqrt {5}+1}{2 x^2-\left (1-\sqrt {5}\right ) x+2}dx\right )+\frac {1}{5} \left (\frac {1}{2} \left (5-\sqrt {5}\right ) \int \frac {1}{2 x^2-\left (1+\sqrt {5}\right ) x+2}dx-\frac {1}{4} \left (1+\sqrt {5}\right ) \int -\frac {-4 x+\sqrt {5}+1}{2 x^2-\left (1+\sqrt {5}\right ) x+2}dx\right )+\frac {1}{5} \log (x+1)\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{2} \left (5+\sqrt {5}\right ) \int \frac {1}{2 x^2-\left (1-\sqrt {5}\right ) x+2}dx+\frac {1}{4} \left (1-\sqrt {5}\right ) \int \frac {-4 x-\sqrt {5}+1}{2 x^2-\left (1-\sqrt {5}\right ) x+2}dx\right )+\frac {1}{5} \left (\frac {1}{2} \left (5-\sqrt {5}\right ) \int \frac {1}{2 x^2-\left (1+\sqrt {5}\right ) x+2}dx+\frac {1}{4} \left (1+\sqrt {5}\right ) \int \frac {-4 x+\sqrt {5}+1}{2 x^2-\left (1+\sqrt {5}\right ) x+2}dx\right )+\frac {1}{5} \log (x+1)\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{4} \left (1+\sqrt {5}\right ) \int \frac {-4 x+\sqrt {5}+1}{2 x^2-\left (1+\sqrt {5}\right ) x+2}dx-\left (5-\sqrt {5}\right ) \int \frac {1}{-\left (4 x-\sqrt {5}-1\right )^2-2 \left (5-\sqrt {5}\right )}d\left (4 x-\sqrt {5}-1\right )\right )+\frac {1}{5} \left (\frac {1}{4} \left (1-\sqrt {5}\right ) \int \frac {-4 x-\sqrt {5}+1}{2 x^2-\left (1-\sqrt {5}\right ) x+2}dx-\left (5+\sqrt {5}\right ) \int \frac {1}{-\left (4 x+\sqrt {5}-1\right )^2-2 \left (5+\sqrt {5}\right )}d\left (4 x+\sqrt {5}-1\right )\right )+\frac {1}{5} \log (x+1)\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{4} \left (1-\sqrt {5}\right ) \int \frac {-4 x-\sqrt {5}+1}{2 x^2-\left (1-\sqrt {5}\right ) x+2}dx+\sqrt {\frac {1}{2} \left (5+\sqrt {5}\right )} \arctan \left (\frac {4 x+\sqrt {5}-1}{\sqrt {2 \left (5+\sqrt {5}\right )}}\right )\right )+\frac {1}{5} \left (\frac {1}{4} \left (1+\sqrt {5}\right ) \int \frac {-4 x+\sqrt {5}+1}{2 x^2-\left (1+\sqrt {5}\right ) x+2}dx+\sqrt {\frac {1}{2} \left (5-\sqrt {5}\right )} \arctan \left (\frac {4 x-\sqrt {5}-1}{\sqrt {2 \left (5-\sqrt {5}\right )}}\right )\right )+\frac {1}{5} \log (x+1)\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {1}{5} \left (\sqrt {\frac {1}{2} \left (5+\sqrt {5}\right )} \arctan \left (\frac {4 x+\sqrt {5}-1}{\sqrt {2 \left (5+\sqrt {5}\right )}}\right )-\frac {1}{4} \left (1-\sqrt {5}\right ) \log \left (2 x^2-\left (1-\sqrt {5}\right ) x+2\right )\right )+\frac {1}{5} \left (\sqrt {\frac {1}{2} \left (5-\sqrt {5}\right )} \arctan \left (\frac {4 x-\sqrt {5}-1}{\sqrt {2 \left (5-\sqrt {5}\right )}}\right )-\frac {1}{4} \left (1+\sqrt {5}\right ) \log \left (2 x^2-\left (1+\sqrt {5}\right ) x+2\right )\right )+\frac {1}{5} \log (x+1)\) |
Input:
Int[(1 + x^5)^(-1),x]
Output:
Log[1 + x]/5 + (Sqrt[(5 + Sqrt[5])/2]*ArcTan[(-1 + Sqrt[5] + 4*x)/Sqrt[2*( 5 + Sqrt[5])]] - ((1 - Sqrt[5])*Log[2 - (1 - Sqrt[5])*x + 2*x^2])/4)/5 + ( Sqrt[(5 - Sqrt[5])/2]*ArcTan[(-1 - Sqrt[5] + 4*x)/Sqrt[2*(5 - Sqrt[5])]] - ((1 + Sqrt[5])*Log[2 - (1 + Sqrt[5])*x + 2*x^2])/4)/5
Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_)^(n_))^(-1), x_Symbol] :> Module[{r = Numerator[Rt[a/ b, n]], s = Denominator[Rt[a/b, n]], k, u}, Simp[u = Int[(r - s*Cos[(2*k - 1)*(Pi/n)]*x)/(r^2 - 2*r*s*Cos[(2*k - 1)*(Pi/n)]*x + s^2*x^2), x]; r/(a*n) Int[1/(r + s*x), x] + 2*(r/(a*n)) Sum[u, {k, 1, (n - 1)/2}], x]] /; Fre eQ[{a, b}, x] && IGtQ[(n - 3)/2, 0] && PosQ[a/b]
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2 Subst[I nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[(2*c*d - b*e)/(2*c) Int[1/(a + b*x + c*x^2), x], x] + Simp[e/(2*c) Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.40 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.19
| method | result | size |
| risch | \(\frac {\ln \left (1+x \right )}{5}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4}+\textit {\_Z}^{3}+\textit {\_Z}^{2}+\textit {\_Z} +1\right )}{\sum }\textit {\_R} \ln \left (\textit {\_R} +x \right )\right )}{5}\) | \(32\) |
| default | \(-\frac {\left (\sqrt {5}+1\right ) \ln \left (-x \sqrt {5}+2 x^{2}-x +2\right )}{20}-\frac {2 \left (-4-\frac {\left (\sqrt {5}+1\right ) \left (-\sqrt {5}-1\right )}{4}\right ) \arctan \left (\frac {-\sqrt {5}+4 x -1}{\sqrt {10-2 \sqrt {5}}}\right )}{5 \sqrt {10-2 \sqrt {5}}}+\frac {\left (\sqrt {5}-1\right ) \ln \left (x \sqrt {5}+2 x^{2}-x +2\right )}{20}+\frac {2 \left (4-\frac {\left (\sqrt {5}-1\right )^{2}}{4}\right ) \arctan \left (\frac {\sqrt {5}+4 x -1}{\sqrt {10+2 \sqrt {5}}}\right )}{5 \sqrt {10+2 \sqrt {5}}}+\frac {\ln \left (1+x \right )}{5}\) | \(144\) |
| meijerg | \(\frac {x \ln \left (1+\left (x^{5}\right )^{\frac {1}{5}}\right )}{5 \left (x^{5}\right )^{\frac {1}{5}}}-\frac {x \cos \left (\frac {\pi }{5}\right ) \ln \left (1-2 \cos \left (\frac {\pi }{5}\right ) \left (x^{5}\right )^{\frac {1}{5}}+\left (x^{5}\right )^{\frac {2}{5}}\right )}{5 \left (x^{5}\right )^{\frac {1}{5}}}+\frac {2 x \sin \left (\frac {\pi }{5}\right ) \arctan \left (\frac {\sin \left (\frac {\pi }{5}\right ) \left (x^{5}\right )^{\frac {1}{5}}}{1-\cos \left (\frac {\pi }{5}\right ) \left (x^{5}\right )^{\frac {1}{5}}}\right )}{5 \left (x^{5}\right )^{\frac {1}{5}}}+\frac {x \cos \left (\frac {2 \pi }{5}\right ) \ln \left (1+2 \cos \left (\frac {2 \pi }{5}\right ) \left (x^{5}\right )^{\frac {1}{5}}+\left (x^{5}\right )^{\frac {2}{5}}\right )}{5 \left (x^{5}\right )^{\frac {1}{5}}}+\frac {2 x \sin \left (\frac {2 \pi }{5}\right ) \arctan \left (\frac {\sin \left (\frac {2 \pi }{5}\right ) \left (x^{5}\right )^{\frac {1}{5}}}{1+\cos \left (\frac {2 \pi }{5}\right ) \left (x^{5}\right )^{\frac {1}{5}}}\right )}{5 \left (x^{5}\right )^{\frac {1}{5}}}\) | \(155\) |
Input:
int(1/(x^5+1),x,method=_RETURNVERBOSE)
Output:
1/5*ln(1+x)+1/5*sum(_R*ln(_R+x),_R=RootOf(_Z^4+_Z^3+_Z^2+_Z+1))
Time = 0.08 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.76 \[ \int \frac {1}{1+x^5} \, dx=\frac {1}{20} \, {\left (\sqrt {5} - 1\right )} \log \left (2 \, x^{2} + \sqrt {5} x - x + 2\right ) - \frac {1}{20} \, {\left (\sqrt {5} + 1\right )} \log \left (2 \, x^{2} - \sqrt {5} x - x + 2\right ) - \frac {1}{5} \, \sqrt {\frac {1}{2} \, \sqrt {5} + \frac {5}{2}} \arctan \left (\frac {1}{10} \, {\left (\sqrt {5} {\left (2 \, x - 3\right )} - 10 \, x + 5\right )} \sqrt {\frac {1}{2} \, \sqrt {5} + \frac {5}{2}}\right ) + \frac {1}{5} \, \sqrt {-\frac {1}{2} \, \sqrt {5} + \frac {5}{2}} \arctan \left (\frac {1}{10} \, {\left (\sqrt {5} {\left (2 \, x - 3\right )} + 10 \, x - 5\right )} \sqrt {-\frac {1}{2} \, \sqrt {5} + \frac {5}{2}}\right ) + \frac {1}{5} \, \log \left (x + 1\right ) \] Input:
integrate(1/(x^5+1),x, algorithm="fricas")
Output:
1/20*(sqrt(5) - 1)*log(2*x^2 + sqrt(5)*x - x + 2) - 1/20*(sqrt(5) + 1)*log (2*x^2 - sqrt(5)*x - x + 2) - 1/5*sqrt(1/2*sqrt(5) + 5/2)*arctan(1/10*(sqr t(5)*(2*x - 3) - 10*x + 5)*sqrt(1/2*sqrt(5) + 5/2)) + 1/5*sqrt(-1/2*sqrt(5 ) + 5/2)*arctan(1/10*(sqrt(5)*(2*x - 3) + 10*x - 5)*sqrt(-1/2*sqrt(5) + 5/ 2)) + 1/5*log(x + 1)
Time = 0.90 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.20 \[ \int \frac {1}{1+x^5} \, dx=\frac {\log {\left (x + 1 \right )}}{5} + \operatorname {RootSum} {\left (625 t^{4} + 125 t^{3} + 25 t^{2} + 5 t + 1, \left ( t \mapsto t \log {\left (5 t + x \right )} \right )\right )} \] Input:
integrate(1/(x**5+1),x)
Output:
log(x + 1)/5 + RootSum(625*_t**4 + 125*_t**3 + 25*_t**2 + 5*_t + 1, Lambda (_t, _t*log(5*_t + x)))
Time = 0.11 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.86 \[ \int \frac {1}{1+x^5} \, dx=\frac {\sqrt {5} {\left (\sqrt {5} + 1\right )} \arctan \left (\frac {4 \, x + \sqrt {5} - 1}{\sqrt {2 \, \sqrt {5} + 10}}\right )}{5 \, \sqrt {2 \, \sqrt {5} + 10}} + \frac {\sqrt {5} {\left (\sqrt {5} - 1\right )} \arctan \left (\frac {4 \, x - \sqrt {5} - 1}{\sqrt {-2 \, \sqrt {5} + 10}}\right )}{5 \, \sqrt {-2 \, \sqrt {5} + 10}} - \frac {{\left (\sqrt {5} + 3\right )} \log \left (2 \, x^{2} - x {\left (\sqrt {5} + 1\right )} + 2\right )}{10 \, {\left (\sqrt {5} + 1\right )}} - \frac {{\left (\sqrt {5} - 3\right )} \log \left (2 \, x^{2} + x {\left (\sqrt {5} - 1\right )} + 2\right )}{10 \, {\left (\sqrt {5} - 1\right )}} + \frac {1}{5} \, \log \left (x + 1\right ) \] Input:
integrate(1/(x^5+1),x, algorithm="maxima")
Output:
1/5*sqrt(5)*(sqrt(5) + 1)*arctan((4*x + sqrt(5) - 1)/sqrt(2*sqrt(5) + 10)) /sqrt(2*sqrt(5) + 10) + 1/5*sqrt(5)*(sqrt(5) - 1)*arctan((4*x - sqrt(5) - 1)/sqrt(-2*sqrt(5) + 10))/sqrt(-2*sqrt(5) + 10) - 1/10*(sqrt(5) + 3)*log(2 *x^2 - x*(sqrt(5) + 1) + 2)/(sqrt(5) + 1) - 1/10*(sqrt(5) - 3)*log(2*x^2 + x*(sqrt(5) - 1) + 2)/(sqrt(5) - 1) + 1/5*log(x + 1)
Time = 0.13 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.67 \[ \int \frac {1}{1+x^5} \, dx=-\frac {1}{20} \, {\left (\sqrt {5} + 1\right )} \log \left (x^{2} - \frac {1}{2} \, x {\left (\sqrt {5} + 1\right )} + 1\right ) + \frac {1}{20} \, {\left (\sqrt {5} - 1\right )} \log \left (x^{2} + \frac {1}{2} \, x {\left (\sqrt {5} - 1\right )} + 1\right ) + \frac {1}{10} \, \sqrt {2 \, \sqrt {5} + 10} \arctan \left (\frac {4 \, x + \sqrt {5} - 1}{\sqrt {2 \, \sqrt {5} + 10}}\right ) + \frac {1}{10} \, \sqrt {-2 \, \sqrt {5} + 10} \arctan \left (\frac {4 \, x - \sqrt {5} - 1}{\sqrt {-2 \, \sqrt {5} + 10}}\right ) + \frac {1}{5} \, \log \left ({\left | x + 1 \right |}\right ) \] Input:
integrate(1/(x^5+1),x, algorithm="giac")
Output:
-1/20*(sqrt(5) + 1)*log(x^2 - 1/2*x*(sqrt(5) + 1) + 1) + 1/20*(sqrt(5) - 1 )*log(x^2 + 1/2*x*(sqrt(5) - 1) + 1) + 1/10*sqrt(2*sqrt(5) + 10)*arctan((4 *x + sqrt(5) - 1)/sqrt(2*sqrt(5) + 10)) + 1/10*sqrt(-2*sqrt(5) + 10)*arcta n((4*x - sqrt(5) - 1)/sqrt(-2*sqrt(5) + 10)) + 1/5*log(abs(x + 1))
Time = 0.18 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.08 \[ \int \frac {1}{1+x^5} \, dx=\frac {\ln \left (x+1\right )}{5}-\ln \left (x-\frac {\sqrt {5}}{4}-\frac {\sqrt {2}\,\sqrt {\sqrt {5}-5}}{4}-\frac {1}{4}\right )\,\left (\frac {\sqrt {5}}{20}+\frac {\sqrt {2}\,\sqrt {\sqrt {5}-5}}{20}+\frac {1}{20}\right )-\ln \left (x-\frac {\sqrt {5}}{4}+\frac {\sqrt {2}\,\sqrt {\sqrt {5}-5}}{4}-\frac {1}{4}\right )\,\left (\frac {\sqrt {5}}{20}-\frac {\sqrt {2}\,\sqrt {\sqrt {5}-5}}{20}+\frac {1}{20}\right )-\ln \left (x-\frac {\sqrt {2}\,\sqrt {-\sqrt {5}-5}}{4}+\frac {\sqrt {5}}{4}-\frac {1}{4}\right )\,\left (\frac {\sqrt {2}\,\sqrt {-\sqrt {5}-5}}{20}-\frac {\sqrt {5}}{20}+\frac {1}{20}\right )+\ln \left (x+\frac {\sqrt {2}\,\sqrt {-\sqrt {5}-5}}{4}+\frac {\sqrt {5}}{4}-\frac {1}{4}\right )\,\left (\frac {\sqrt {2}\,\sqrt {-\sqrt {5}-5}}{20}+\frac {\sqrt {5}}{20}-\frac {1}{20}\right ) \] Input:
int(1/(x^5 + 1),x)
Output:
log(x + 1)/5 - log(x - 5^(1/2)/4 - (2^(1/2)*(5^(1/2) - 5)^(1/2))/4 - 1/4)* (5^(1/2)/20 + (2^(1/2)*(5^(1/2) - 5)^(1/2))/20 + 1/20) - log(x - 5^(1/2)/4 + (2^(1/2)*(5^(1/2) - 5)^(1/2))/4 - 1/4)*(5^(1/2)/20 - (2^(1/2)*(5^(1/2) - 5)^(1/2))/20 + 1/20) - log(x - (2^(1/2)*(- 5^(1/2) - 5)^(1/2))/4 + 5^(1/ 2)/4 - 1/4)*((2^(1/2)*(- 5^(1/2) - 5)^(1/2))/20 - 5^(1/2)/20 + 1/20) + log (x + (2^(1/2)*(- 5^(1/2) - 5)^(1/2))/4 + 5^(1/2)/4 - 1/4)*((2^(1/2)*(- 5^( 1/2) - 5)^(1/2))/20 + 5^(1/2)/20 - 1/20)
\[ \int \frac {1}{1+x^5} \, dx=\int \frac {1}{x^{5}+1}d x \] Input:
int(1/(x^5+1),x)
Output:
int(1/(x**5 + 1),x)