3.14.4 \(\int \frac {1}{1+x^5} \, dx\) [1304]

Optimal. Leaf size=168 \[ -\frac {1}{5} \sqrt {\frac {1}{2} \left (5+\sqrt {5}\right )} \tan ^{-1}\left (\frac {1-\sqrt {5}-4 x}{\sqrt {2 \left (5+\sqrt {5}\right )}}\right )-\frac {1}{5} \sqrt {\frac {1}{2} \left (5-\sqrt {5}\right )} \tan ^{-1}\left (\frac {1}{2} \sqrt {\frac {1}{10} \left (5+\sqrt {5}\right )} \left (1+\sqrt {5}-4 x\right )\right )+\frac {1}{5} \log (1+x)-\frac {1}{20} \left (1-\sqrt {5}\right ) \log \left (1-\frac {1}{2} \left (1-\sqrt {5}\right ) x+x^2\right )-\frac {1}{20} \left (1+\sqrt {5}\right ) \log \left (1-\frac {1}{2} \left (1+\sqrt {5}\right ) x+x^2\right ) \]

[Out]

1/5*ln(1+x)-1/20*ln(1+x^2-1/2*x*(-5^(1/2)+1))*(-5^(1/2)+1)-1/20*ln(1+x^2-1/2*x*(5^(1/2)+1))*(5^(1/2)+1)-1/10*a
rctan(1/20*(1-4*x+5^(1/2))*(50+10*5^(1/2))^(1/2))*(10-2*5^(1/2))^(1/2)-1/10*arctan((1-4*x-5^(1/2))/(10+2*5^(1/
2))^(1/2))*(10+2*5^(1/2))^(1/2)

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Rubi [A]
time = 0.16, antiderivative size = 185, normalized size of antiderivative = 1.10, number of steps used = 6, number of rules used = 6, integrand size = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.857, Rules used = {207, 648, 632, 210, 642, 31} \begin {gather*} \frac {1}{5} \sqrt {\frac {1}{2} \left (5+\sqrt {5}\right )} \text {ArcTan}\left (2 \sqrt {\frac {2}{5+\sqrt {5}}} x+\sqrt {\frac {1}{5} \left (5-2 \sqrt {5}\right )}\right )-\frac {1}{5} \sqrt {\frac {1}{2} \left (5-\sqrt {5}\right )} \text {ArcTan}\left (\sqrt {\frac {1}{5} \left (5+2 \sqrt {5}\right )}-\sqrt {\frac {2}{5} \left (5+\sqrt {5}\right )} x\right )-\frac {1}{20} \left (1-\sqrt {5}\right ) \log \left (x^2-\frac {1}{2} \left (1-\sqrt {5}\right ) x+1\right )-\frac {1}{20} \left (1+\sqrt {5}\right ) \log \left (x^2-\frac {1}{2} \left (1+\sqrt {5}\right ) x+1\right )+\frac {1}{5} \log (x+1) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 + x^5)^(-1),x]

[Out]

(Sqrt[(5 + Sqrt[5])/2]*ArcTan[Sqrt[(5 - 2*Sqrt[5])/5] + 2*Sqrt[2/(5 + Sqrt[5])]*x])/5 - (Sqrt[(5 - Sqrt[5])/2]
*ArcTan[Sqrt[(5 + 2*Sqrt[5])/5] - Sqrt[(2*(5 + Sqrt[5]))/5]*x])/5 + Log[1 + x]/5 - ((1 - Sqrt[5])*Log[1 - ((1
- Sqrt[5])*x)/2 + x^2])/20 - ((1 + Sqrt[5])*Log[1 - ((1 + Sqrt[5])*x)/2 + x^2])/20

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 207

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] + Dist[2*(r/(a*n)), Sum[u, {k, 1, (n - 1)/2}], x], x]] /; FreeQ[{a, b}, x] && IGtQ[
(n - 3)/2, 0] && PosQ[a/b]

Rule 210

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])

Rule 632

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 642

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[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]

Rule 648

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rubi steps

\begin {align*} \int \frac {1}{1+x^5} \, dx &=\frac {2}{5} \int \frac {1-\frac {1}{4} \left (1-\sqrt {5}\right ) x}{1-\frac {1}{2} \left (1-\sqrt {5}\right ) x+x^2} \, dx+\frac {2}{5} \int \frac {1-\frac {1}{4} \left (1+\sqrt {5}\right ) x}{1-\frac {1}{2} \left (1+\sqrt {5}\right ) x+x^2} \, dx+\frac {1}{5} \int \frac {1}{1+x} \, dx\\ &=\frac {1}{5} \log (1+x)+\frac {1}{20} \left (-1-\sqrt {5}\right ) \int \frac {\frac {1}{2} \left (-1-\sqrt {5}\right )+2 x}{1+\frac {1}{2} \left (-1-\sqrt {5}\right ) x+x^2} \, dx+\frac {1}{20} \left (5-\sqrt {5}\right ) \int \frac {1}{1+\frac {1}{2} \left (-1-\sqrt {5}\right ) x+x^2} \, dx+\frac {1}{20} \left (-1+\sqrt {5}\right ) \int \frac {\frac {1}{2} \left (-1+\sqrt {5}\right )+2 x}{1+\frac {1}{2} \left (-1+\sqrt {5}\right ) x+x^2} \, dx+\frac {1}{20} \left (5+\sqrt {5}\right ) \int \frac {1}{1+\frac {1}{2} \left (-1+\sqrt {5}\right ) x+x^2} \, dx\\ &=\frac {1}{5} \log (1+x)-\frac {1}{20} \left (1+\sqrt {5}\right ) \log \left (2-x-\sqrt {5} x+2 x^2\right )-\frac {1}{20} \left (1-\sqrt {5}\right ) \log \left (2-x+\sqrt {5} x+2 x^2\right )+\frac {1}{10} \left (-5+\sqrt {5}\right ) \text {Subst}\left (\int \frac {1}{\frac {1}{2} \left (-5+\sqrt {5}\right )-x^2} \, dx,x,\frac {1}{2} \left (-1-\sqrt {5}\right )+2 x\right )-\frac {1}{10} \left (5+\sqrt {5}\right ) \text {Subst}\left (\int \frac {1}{\frac {1}{2} \left (-5-\sqrt {5}\right )-x^2} \, dx,x,\frac {1}{2} \left (-1+\sqrt {5}\right )+2 x\right )\\ &=-\frac {1}{5} \sqrt {\frac {1}{2} \left (5+\sqrt {5}\right )} \tan ^{-1}\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 )} \tan ^{-1}\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 (2-x-\sqrt {5} x+2 x^2\right )-\frac {1}{20} \left (1-\sqrt {5}\right ) \log \left (2-x+\sqrt {5} x+2 x^2\right )\\ \end {align*}

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Mathematica [A]
time = 0.03, size = 144, normalized size = 0.86 \begin {gather*} \frac {1}{20} \left (-2 \sqrt {10-2 \sqrt {5}} \tan ^{-1}\left (\frac {1+\sqrt {5}-4 x}{\sqrt {10-2 \sqrt {5}}}\right )+2 \sqrt {2 \left (5+\sqrt {5}\right )} \tan ^{-1}\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 ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 + x^5)^(-1),x]

[Out]

(-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

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Maple [A]
time = 0.18, size = 144, normalized size = 0.86

method	result	size
risch	\(\frac {\ln \left (x +1\right )}{5}+\frac {\left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{4}+\textit {\_Z}^{3}+\textit {\_Z}^{2}+\textit {\_Z} +1\right )}{\sum }\textit {\_R} \ln \left (\textit {\_R} +x \right )\right )}{5}\)	\(32\)
default	\(\frac {\ln \left (x +1\right )}{5}+\frac {\left (\sqrt {5}-1\right ) \ln \left (x \sqrt {5}+2 x^{2}-x +2\right )}{20}+\frac {2 \left (4-\frac {\left (\sqrt {5}-1\right )^{2}}{4}\right ) \arctan \left (\frac {\sqrt {5}+4 x -1}{\sqrt {10+2 \sqrt {5}}}\right )}{5 \sqrt {10+2 \sqrt {5}}}-\frac {\left (\sqrt {5}+1\right ) \ln \left (-x \sqrt {5}+2 x^{2}-x +2\right )}{20}-\frac {2 \left (-4-\frac {\left (\sqrt {5}+1\right ) \left (-\sqrt {5}-1\right )}{4}\right ) \arctan \left (\frac {-\sqrt {5}+4 x -1}{\sqrt {10-2 \sqrt {5}}}\right )}{5 \sqrt {10-2 \sqrt {5}}}\)	\(144\)
meijerg	\(\frac {x \ln \left (1+\left (x^{5}\right )^{\frac {1}{5}}\right )}{5 \left (x^{5}\right )^{\frac {1}{5}}}-\frac {x \cos \left (\frac {\pi }{5}\right ) \ln \left (1-2 \cos \left (\frac {\pi }{5}\right ) \left (x^{5}\right )^{\frac {1}{5}}+\left (x^{5}\right )^{\frac {2}{5}}\right )}{5 \left (x^{5}\right )^{\frac {1}{5}}}+\frac {2 x \sin \left (\frac {\pi }{5}\right ) \arctan \left (\frac {\sin \left (\frac {\pi }{5}\right ) \left (x^{5}\right )^{\frac {1}{5}}}{1-\cos \left (\frac {\pi }{5}\right ) \left (x^{5}\right )^{\frac {1}{5}}}\right )}{5 \left (x^{5}\right )^{\frac {1}{5}}}+\frac {x \cos \left (\frac {2 \pi }{5}\right ) \ln \left (1+2 \cos \left (\frac {2 \pi }{5}\right ) \left (x^{5}\right )^{\frac {1}{5}}+\left (x^{5}\right )^{\frac {2}{5}}\right )}{5 \left (x^{5}\right )^{\frac {1}{5}}}+\frac {2 x \sin \left (\frac {2 \pi }{5}\right ) \arctan \left (\frac {\sin \left (\frac {2 \pi }{5}\right ) \left (x^{5}\right )^{\frac {1}{5}}}{1+\cos \left (\frac {2 \pi }{5}\right ) \left (x^{5}\right )^{\frac {1}{5}}}\right )}{5 \left (x^{5}\right )^{\frac {1}{5}}}\)	\(155\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*ln(x+1)+1/20*(5^(1/2)-1)*ln(x*5^(1/2)+2*x^2-x+2)+2/5*(4-1/4*(5^(1/2)-1)^2)/(10+2*5^(1/2))^(1/2)*arctan((5^
(1/2)+4*x-1)/(10+2*5^(1/2))^(1/2))-1/20*(5^(1/2)+1)*ln(-x*5^(1/2)+2*x^2-x+2)-2/5*(-4-1/4*(5^(1/2)+1)*(-5^(1/2)
-1))/(10-2*5^(1/2))^(1/2)*arctan((-5^(1/2)+4*x-1)/(10-2*5^(1/2))^(1/2))

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Maxima [A]
time = 0.50, size = 144, normalized size = 0.86 \begin {gather*} \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 ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 539 vs. \(2 (113) = 226\).
time = 1.11, size = 539, normalized size = 3.21 \begin {gather*} \frac {1}{20} \, {\left (\sqrt {5} + 2 \, \sqrt {-\frac {3}{16} \, {\left (2 \, \sqrt {\frac {1}{2}} \sqrt {\sqrt {5} - 5} + \sqrt {5} + 1\right )}^{2} + \frac {1}{8} \, {\left (2 \, \sqrt {\frac {1}{2}} \sqrt {\sqrt {5} - 5} + \sqrt {5} - 3\right )} {\left (2 \, \sqrt {\frac {1}{2}} \sqrt {\sqrt {5} - 5} - \sqrt {5} - 1\right )} - \frac {3}{16} \, {\left (2 \, \sqrt {\frac {1}{2}} \sqrt {\sqrt {5} - 5} - \sqrt {5} - 1\right )}^{2} + \sqrt {\frac {1}{2}} \sqrt {\sqrt {5} - 5} + \frac {1}{2} \, \sqrt {5} - \frac {5}{2}} - 1\right )} \log \left (2 \, x + \frac {1}{2} \, \sqrt {5} + \sqrt {-\frac {3}{16} \, {\left (2 \, \sqrt {\frac {1}{2}} \sqrt {\sqrt {5} - 5} + \sqrt {5} + 1\right )}^{2} + \frac {1}{8} \, {\left (2 \, \sqrt {\frac {1}{2}} \sqrt {\sqrt {5} - 5} + \sqrt {5} - 3\right )} {\left (2 \, \sqrt {\frac {1}{2}} \sqrt {\sqrt {5} - 5} - \sqrt {5} - 1\right )} - \frac {3}{16} \, {\left (2 \, \sqrt {\frac {1}{2}} \sqrt {\sqrt {5} - 5} - \sqrt {5} - 1\right )}^{2} + \sqrt {\frac {1}{2}} \sqrt {\sqrt {5} - 5} + \frac {1}{2} \, \sqrt {5} - \frac {5}{2}} - \frac {1}{2}\right ) + \frac {1}{20} \, {\left (\sqrt {5} - 2 \, \sqrt {-\frac {3}{16} \, {\left (2 \, \sqrt {\frac {1}{2}} \sqrt {\sqrt {5} - 5} + \sqrt {5} + 1\right )}^{2} + \frac {1}{8} \, {\left (2 \, \sqrt {\frac {1}{2}} \sqrt {\sqrt {5} - 5} + \sqrt {5} - 3\right )} {\left (2 \, \sqrt {\frac {1}{2}} \sqrt {\sqrt {5} - 5} - \sqrt {5} - 1\right )} - \frac {3}{16} \, {\left (2 \, \sqrt {\frac {1}{2}} \sqrt {\sqrt {5} - 5} - \sqrt {5} - 1\right )}^{2} + \sqrt {\frac {1}{2}} \sqrt {\sqrt {5} - 5} + \frac {1}{2} \, \sqrt {5} - \frac {5}{2}} - 1\right )} \log \left (2 \, x + \frac {1}{2} \, \sqrt {5} - \sqrt {-\frac {3}{16} \, {\left (2 \, \sqrt {\frac {1}{2}} \sqrt {\sqrt {5} - 5} + \sqrt {5} + 1\right )}^{2} + \frac {1}{8} \, {\left (2 \, \sqrt {\frac {1}{2}} \sqrt {\sqrt {5} - 5} + \sqrt {5} - 3\right )} {\left (2 \, \sqrt {\frac {1}{2}} \sqrt {\sqrt {5} - 5} - \sqrt {5} - 1\right )} - \frac {3}{16} \, {\left (2 \, \sqrt {\frac {1}{2}} \sqrt {\sqrt {5} - 5} - \sqrt {5} - 1\right )}^{2} + \sqrt {\frac {1}{2}} \sqrt {\sqrt {5} - 5} + \frac {1}{2} \, \sqrt {5} - \frac {5}{2}} - \frac {1}{2}\right ) + \frac {1}{20} \, {\left (2 \, \sqrt {\frac {1}{2}} \sqrt {\sqrt {5} - 5} - \sqrt {5} - 1\right )} \log \left (x + \frac {1}{2} \, \sqrt {\frac {1}{2}} \sqrt {\sqrt {5} - 5} - \frac {1}{4} \, \sqrt {5} - \frac {1}{4}\right ) - \frac {1}{20} \, {\left (2 \, \sqrt {\frac {1}{2}} \sqrt {\sqrt {5} - 5} + \sqrt {5} + 1\right )} \log \left (x - \frac {1}{2} \, \sqrt {\frac {1}{2}} \sqrt {\sqrt {5} - 5} - \frac {1}{4} \, \sqrt {5} - \frac {1}{4}\right ) + \frac {1}{5} \, \log \left (x + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/20*(sqrt(5) + 2*sqrt(-3/16*(2*sqrt(1/2)*sqrt(sqrt(5) - 5) + sqrt(5) + 1)^2 + 1/8*(2*sqrt(1/2)*sqrt(sqrt(5) -
5) + sqrt(5) - 3)*(2*sqrt(1/2)*sqrt(sqrt(5) - 5) - sqrt(5) - 1) - 3/16*(2*sqrt(1/2)*sqrt(sqrt(5) - 5) - sqrt(
5) - 1)^2 + sqrt(1/2)*sqrt(sqrt(5) - 5) + 1/2*sqrt(5) - 5/2) - 1)*log(2*x + 1/2*sqrt(5) + sqrt(-3/16*(2*sqrt(1
/2)*sqrt(sqrt(5) - 5) + sqrt(5) + 1)^2 + 1/8*(2*sqrt(1/2)*sqrt(sqrt(5) - 5) + sqrt(5) - 3)*(2*sqrt(1/2)*sqrt(s
qrt(5) - 5) - sqrt(5) - 1) - 3/16*(2*sqrt(1/2)*sqrt(sqrt(5) - 5) - sqrt(5) - 1)^2 + sqrt(1/2)*sqrt(sqrt(5) - 5
) + 1/2*sqrt(5) - 5/2) - 1/2) + 1/20*(sqrt(5) - 2*sqrt(-3/16*(2*sqrt(1/2)*sqrt(sqrt(5) - 5) + sqrt(5) + 1)^2 +
1/8*(2*sqrt(1/2)*sqrt(sqrt(5) - 5) + sqrt(5) - 3)*(2*sqrt(1/2)*sqrt(sqrt(5) - 5) - sqrt(5) - 1) - 3/16*(2*sqr
t(1/2)*sqrt(sqrt(5) - 5) - sqrt(5) - 1)^2 + sqrt(1/2)*sqrt(sqrt(5) - 5) + 1/2*sqrt(5) - 5/2) - 1)*log(2*x + 1/
2*sqrt(5) - sqrt(-3/16*(2*sqrt(1/2)*sqrt(sqrt(5) - 5) + sqrt(5) + 1)^2 + 1/8*(2*sqrt(1/2)*sqrt(sqrt(5) - 5) +
sqrt(5) - 3)*(2*sqrt(1/2)*sqrt(sqrt(5) - 5) - sqrt(5) - 1) - 3/16*(2*sqrt(1/2)*sqrt(sqrt(5) - 5) - sqrt(5) - 1
)^2 + sqrt(1/2)*sqrt(sqrt(5) - 5) + 1/2*sqrt(5) - 5/2) - 1/2) + 1/20*(2*sqrt(1/2)*sqrt(sqrt(5) - 5) - sqrt(5)
- 1)*log(x + 1/2*sqrt(1/2)*sqrt(sqrt(5) - 5) - 1/4*sqrt(5) - 1/4) - 1/20*(2*sqrt(1/2)*sqrt(sqrt(5) - 5) + sqrt
(5) + 1)*log(x - 1/2*sqrt(1/2)*sqrt(sqrt(5) - 5) - 1/4*sqrt(5) - 1/4) + 1/5*log(x + 1)

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Sympy [A]
time = 0.83, size = 34, normalized size = 0.20 \begin {gather*} \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 )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(x**5+1),x)

[Out]

log(x + 1)/5 + RootSum(625*_t**4 + 125*_t**3 + 25*_t**2 + 5*_t + 1, Lambda(_t, _t*log(5*_t + x)))

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Giac [A]
time = 1.82, size = 112, normalized size = 0.67 \begin {gather*} -\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 ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-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)*arc
tan((4*x - sqrt(5) - 1)/sqrt(-2*sqrt(5) + 10)) + 1/5*log(abs(x + 1))

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Mupad [B]
time = 1.17, size = 182, normalized size = 1.08 \begin {gather*} \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 ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^5 + 1),x)

[Out]

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)

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