\(\int \frac {1}{x^2 (1+x^5)} \, dx\) [1305]

Optimal result

Integrand size = 11, antiderivative size = 190 \[ \int \frac {1}{x^2 \left (1+x^5\right )} \, dx=-\frac {1}{x}-\frac {1}{5} \sqrt {\frac {1}{2} \left (5+\sqrt {5}\right )} \arctan \left (\sqrt {\frac {1}{5} \left (5-2 \sqrt {5}\right )}+2 \sqrt {\frac {2}{5+\sqrt {5}}} x\right )+\frac {1}{5} \sqrt {\frac {1}{2} \left (5-\sqrt {5}\right )} \arctan \left (\sqrt {\frac {1}{5} \left (5+2 \sqrt {5}\right )}-\sqrt {\frac {2}{5} \left (5+\sqrt {5}\right )} x\right )+\frac {1}{5} \log (1+x)-\frac {1}{20} \left (1-\sqrt {5}\right ) \log \left (1-\frac {1}{2} \left (1-\sqrt {5}\right ) x+x^2\right )-\frac {1}{20} \left (1+\sqrt {5}\right ) \log \left (1-\frac {1}{2} \left (1+\sqrt {5}\right ) x+x^2\right ) \]

[Out]

-1/x+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*arctan(1/5*x*(50+10*5^(1/2))^(1/2)-1/5*(25+10*5^(1/2))^(1/2))*(10-2*5^(1/2))^(1/2)-1/10*arctan(1/5*(25-10*
5^(1/2))^(1/2)+2*x*2^(1/2)/(5+5^(1/2))^(1/2))*(10+2*5^(1/2))^(1/2)

Rubi [A] (verified)

Time = 0.23 (sec) , antiderivative size = 190, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.636, Rules used = {331, 299, 648, 632, 210, 642, 31} \[ \int \frac {1}{x^2 \left (1+x^5\right )} \, dx=-\frac {1}{5} \sqrt {\frac {1}{2} \left (5+\sqrt {5}\right )} \arctan \left (2 \sqrt {\frac {2}{5+\sqrt {5}}} x+\sqrt {\frac {1}{5} \left (5-2 \sqrt {5}\right )}\right )+\frac {1}{5} \sqrt {\frac {1}{2} \left (5-\sqrt {5}\right )} \arctan \left (\sqrt {\frac {1}{5} \left (5+2 \sqrt {5}\right )}-\sqrt {\frac {2}{5} \left (5+\sqrt {5}\right )} x\right )-\frac {1}{20} \left (1-\sqrt {5}\right ) \log \left (x^2-\frac {1}{2} \left (1-\sqrt {5}\right ) x+1\right )-\frac {1}{20} \left (1+\sqrt {5}\right ) \log \left (x^2-\frac {1}{2} \left (1+\sqrt {5}\right ) x+1\right )-\frac {1}{x}+\frac {1}{5} \log (x+1) \]

[In]

Int[1/(x^2*(1 + x^5)),x]

[Out]

-x^(-1) - (Sqrt[(5 + Sqrt[5])/2]*ArcTan[Sqrt[(5 - 2*Sqrt[5])/5] + 2*Sqrt[2/(5 + Sqrt[5])]*x])/5 + (Sqrt[(5 - S
qrt[5])/2]*ArcTan[Sqrt[(5 + 2*Sqrt[5])/5] - Sqrt[(2*(5 + Sqrt[5]))/5]*x])/5 + Log[1 + x]/5 - ((1 - Sqrt[5])*Lo
g[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 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 299

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Module[{r = Numerator[Rt[a/b, n]], s = Denominator[Rt[a/
b, n]], k, u}, Simp[u = Int[(r*Cos[(2*k - 1)*m*(Pi/n)] - s*Cos[(2*k - 1)*(m + 1)*(Pi/n)]*x)/(r^2 - 2*r*s*Cos[(
2*k - 1)*(Pi/n)]*x + s^2*x^2), x]; (-(-r)^(m + 1)/(a*n*s^m))*Int[1/(r + s*x), x] + Dist[2*(r^(m + 1)/(a*n*s^m)
), Sum[u, {k, 1, (n - 1)/2}], x], x]] /; FreeQ[{a, b}, x] && IGtQ[(n - 1)/2, 0] && IGtQ[m, 0] && LtQ[m, n - 1]
&& PosQ[a/b]

Rule 331

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*c
*(m + 1))), x] - Dist[b*((m + n*(p + 1) + 1)/(a*c^n*(m + 1))), Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; Free
Q[{a, b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, n, m, p, x]

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*} \text {integral}& = -\frac {1}{x}-\int \frac {x^3}{1+x^5} \, dx \\ & = -\frac {1}{x}-\frac {2}{5} \int \frac {\frac {1}{4} \left (1+\sqrt {5}\right )-\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 {\frac {1}{4} \left (1-\sqrt {5}\right )-\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}{x}+\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}{x}+\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}{x}+\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*}

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.78 \[ \int \frac {1}{x^2 \left (1+x^5\right )} \, dx=\frac {1}{20} \left (-\frac {20}{x}+2 \sqrt {10-2 \sqrt {5}} \arctan \left (\frac {1+\sqrt {5}-4 x}{\sqrt {10-2 \sqrt {5}}}\right )-2 \sqrt {2 \left (5+\sqrt {5}\right )} \arctan \left (\frac {-1+\sqrt {5}+4 x}{\sqrt {2 \left (5+\sqrt {5}\right )}}\right )+4 \log (1+x)+\left (-1+\sqrt {5}\right ) \log \left (1+\frac {1}{2} \left (-1+\sqrt {5}\right ) x+x^2\right )-\left (1+\sqrt {5}\right ) \log \left (1-\frac {1}{2} \left (1+\sqrt {5}\right ) x+x^2\right )\right ) \]

[In]

Integrate[1/(x^2*(1 + x^5)),x]

[Out]

(-20/x + 2*Sqrt[10 - 2*Sqrt[5]]*ArcTan[(1 + Sqrt[5] - 4*x)/Sqrt[10 - 2*Sqrt[5]]] - 2*Sqrt[2*(5 + Sqrt[5])]*Arc
Tan[(-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

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 4.44 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.26

method	result	size
risch	\(-\frac {1}{x}+\frac {\ln \left (1+x \right )}{5}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4}+\textit {\_Z}^{3}+\textit {\_Z}^{2}+\textit {\_Z} +1\right )}{\sum }\textit {\_R} \ln \left (-\textit {\_R}^{3}-\textit {\_R}^{2}-\textit {\_R} +x -1\right )\right )}{5}\)	\(50\)
default	\(\frac {\ln \left (1+x \right )}{5}-\frac {\left (\sqrt {5}+1\right ) \ln \left (-x \sqrt {5}+2 x^{2}-x +2\right )}{20}-\frac {2 \left (-\sqrt {5}+1-\frac {\left (\sqrt {5}+1\right ) \left (-\sqrt {5}-1\right )}{4}\right ) \arctan \left (\frac {-\sqrt {5}+4 x -1}{\sqrt {10-2 \sqrt {5}}}\right )}{5 \sqrt {10-2 \sqrt {5}}}+\frac {\left (\sqrt {5}-1\right ) \ln \left (x \sqrt {5}+2 x^{2}-x +2\right )}{20}+\frac {2 \left (-\sqrt {5}-1-\frac {\left (\sqrt {5}-1\right )^{2}}{4}\right ) \arctan \left (\frac {\sqrt {5}+4 x -1}{\sqrt {10+2 \sqrt {5}}}\right )}{5 \sqrt {10+2 \sqrt {5}}}-\frac {1}{x}\)	\(159\)
meijerg	\(-\frac {1}{x}-\frac {x^{4} \left (-\frac {\ln \left (1+\left (x^{5}\right )^{\frac {1}{5}}\right )}{\left (x^{5}\right )^{\frac {4}{5}}}+\frac {\cos \left (\frac {\pi }{5}\right ) \ln \left (1-2 \cos \left (\frac {\pi }{5}\right ) \left (x^{5}\right )^{\frac {1}{5}}+\left (x^{5}\right )^{\frac {2}{5}}\right )}{\left (x^{5}\right )^{\frac {4}{5}}}+\frac {2 \sin \left (\frac {\pi }{5}\right ) \arctan \left (\frac {\sin \left (\frac {\pi }{5}\right ) \left (x^{5}\right )^{\frac {1}{5}}}{1-\cos \left (\frac {\pi }{5}\right ) \left (x^{5}\right )^{\frac {1}{5}}}\right )}{\left (x^{5}\right )^{\frac {4}{5}}}-\frac {\cos \left (\frac {2 \pi }{5}\right ) \ln \left (1+2 \cos \left (\frac {2 \pi }{5}\right ) \left (x^{5}\right )^{\frac {1}{5}}+\left (x^{5}\right )^{\frac {2}{5}}\right )}{\left (x^{5}\right )^{\frac {4}{5}}}+\frac {2 \sin \left (\frac {2 \pi }{5}\right ) \arctan \left (\frac {\sin \left (\frac {2 \pi }{5}\right ) \left (x^{5}\right )^{\frac {1}{5}}}{1+\cos \left (\frac {2 \pi }{5}\right ) \left (x^{5}\right )^{\frac {1}{5}}}\right )}{\left (x^{5}\right )^{\frac {4}{5}}}\right )}{5}\)	\(160\)

[In]

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

[Out]

-1/x+1/5*ln(1+x)+1/5*sum(_R*ln(-_R^3-_R^2-_R+x-1),_R=RootOf(_Z^4+_Z^3+_Z^2+_Z+1))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1073 vs. \(2 (127) = 254\).

Time = 1.24 (sec) , antiderivative size = 1073, normalized size of antiderivative = 5.65 \[ \int \frac {1}{x^2 \left (1+x^5\right )} \, dx=\text {Too large to display} \]

[In]

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

[Out]

-1/40*(2*x*(2*sqrt(1/2)*sqrt(sqrt(5) - 5) + sqrt(5) + 1)*log(1/64*(2*sqrt(1/2)*sqrt(sqrt(5) - 5) + sqrt(5) + 1
)^3 - 1/16*(2*sqrt(1/2)*sqrt(sqrt(5) - 5) + sqrt(5) + 1)^2 + x + 1/2*sqrt(1/2)*sqrt(sqrt(5) - 5) + 1/4*sqrt(5)
- 3/4) - 2*x*(2*sqrt(1/2)*sqrt(sqrt(5) - 5) - sqrt(5) - 1)*log(-1/64*(2*sqrt(1/2)*sqrt(sqrt(5) - 5) + sqrt(5)
+ 1)^3 - 1/64*(2*sqrt(1/2)*sqrt(sqrt(5) - 5) + sqrt(5) + 1)*(2*sqrt(1/2)*sqrt(sqrt(5) - 5) - sqrt(5) - 1)^2 +
1/16*(2*sqrt(1/2)*sqrt(sqrt(5) - 5) + sqrt(5) + 1)^2 + 1/64*((2*sqrt(1/2)*sqrt(sqrt(5) - 5) + sqrt(5) + 1)^2
- 8*sqrt(1/2)*sqrt(sqrt(5) - 5) - 4*sqrt(5) - 4)*(2*sqrt(1/2)*sqrt(sqrt(5) - 5) - sqrt(5) - 1) + x - 1/2*sqrt(
1/2)*sqrt(sqrt(5) - 5) - 1/4*sqrt(5) - 1/4) - (x*(2*sqrt(1/2)*sqrt(sqrt(5) - 5) + sqrt(5) + 1) - x*(2*sqrt(1/2
)*sqrt(sqrt(5) - 5) - sqrt(5) - 1) + 4*sqrt(-3/16*(2*sqrt(1/2)*sqrt(sqrt(5) - 5) + sqrt(5) + 1)^2 + 1/8*(2*sqr
t(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)*x - 4*x)*log(1/64*(2*sqrt(1/
2)*sqrt(sqrt(5) - 5) + sqrt(5) + 1)*(2*sqrt(1/2)*sqrt(sqrt(5) - 5) - sqrt(5) - 1)^2 + 1/16*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)*(2*sqrt(1/2)*sqrt(sqrt(5) - 5) + sqrt(5) + 1)*(2*sqrt(1/2)*sqrt(sqrt(5) - 5) - sqrt(5)
- 1) - 1/64*((2*sqrt(1/2)*sqrt(sqrt(5) - 5) + sqrt(5) + 1)^2 - 8*sqrt(1/2)*sqrt(sqrt(5) - 5) - 4*sqrt(5) - 4)
*(2*sqrt(1/2)*sqrt(sqrt(5) - 5) - sqrt(5) - 1) + 2*x) - (x*(2*sqrt(1/2)*sqrt(sqrt(5) - 5) + sqrt(5) + 1) - x*(
2*sqrt(1/2)*sqrt(sqrt(5) - 5) - sqrt(5) - 1) - 4*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)*x - 4*x)*log(1/64*
(2*sqrt(1/2)*sqrt(sqrt(5) - 5) + sqrt(5) + 1)*(2*sqrt(1/2)*sqrt(sqrt(5) - 5) - sqrt(5) - 1)^2 - 1/16*sqrt(-3/1
6*(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)*(2*sqrt(1/2)*sqrt(sqrt(5) - 5) + sqrt(5) + 1)*(2*sqrt(1/2)*sqrt(sqrt(5) - 5)
- sqrt(5) - 1) - 1/64*((2*sqrt(1/2)*sqrt(sqrt(5) - 5) + sqrt(5) + 1)^2 - 8*sqrt(1/2)*sqrt(sqrt(5) - 5) - 4*sq
rt(5) - 4)*(2*sqrt(1/2)*sqrt(sqrt(5) - 5) - sqrt(5) - 1) + 2*x) - 8*x*log(x + 1) + 40)/x

Sympy [A] (verification not implemented)

Time = 0.90 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.21 \[ \int \frac {1}{x^2 \left (1+x^5\right )} \, dx=\frac {\log {\left (x + 1 \right )}}{5} + \operatorname {RootSum} {\left (625 t^{4} + 125 t^{3} + 25 t^{2} + 5 t + 1, \left ( t \mapsto t \log {\left (625 t^{4} + x \right )} \right )\right )} - \frac {1}{x} \]

[In]

integrate(1/x**2/(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(625*_t**4 + x))) - 1/x

Maxima [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.78 \[ \int \frac {1}{x^2 \left (1+x^5\right )} \, dx=-\frac {\sqrt {5} {\left (\sqrt {5} + 1\right )} \arctan \left (\frac {4 \, x + \sqrt {5} - 1}{\sqrt {2 \, \sqrt {5} + 10}}\right )}{5 \, \sqrt {2 \, \sqrt {5} + 10}} - \frac {\sqrt {5} {\left (\sqrt {5} - 1\right )} \arctan \left (\frac {4 \, x - \sqrt {5} - 1}{\sqrt {-2 \, \sqrt {5} + 10}}\right )}{5 \, \sqrt {-2 \, \sqrt {5} + 10}} - \frac {{\left (\sqrt {5} + 3\right )} \log \left (2 \, x^{2} - x {\left (\sqrt {5} + 1\right )} + 2\right )}{10 \, {\left (\sqrt {5} + 1\right )}} - \frac {{\left (\sqrt {5} - 3\right )} \log \left (2 \, x^{2} + x {\left (\sqrt {5} - 1\right )} + 2\right )}{10 \, {\left (\sqrt {5} - 1\right )}} - \frac {1}{x} + \frac {1}{5} \, \log \left (x + 1\right ) \]

[In]

integrate(1/x^2/(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)*lo
g(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/x + 1/5*log(x + 1)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.69 \[ \int \frac {1}{x^2 \left (1+x^5\right )} \, dx=-\frac {1}{10} \, \sqrt {2 \, \sqrt {5} + 10} \arctan \left (\frac {4 \, x + \sqrt {5} - 1}{\sqrt {2 \, \sqrt {5} + 10}}\right ) - \frac {1}{10} \, \sqrt {-2 \, \sqrt {5} + 10} \arctan \left (\frac {4 \, x - \sqrt {5} - 1}{\sqrt {-2 \, \sqrt {5} + 10}}\right ) - \frac {1}{20} \, \sqrt {5} \log \left (x^{2} - \frac {1}{2} \, x {\left (\sqrt {5} + 1\right )} + 1\right ) + \frac {1}{20} \, \sqrt {5} \log \left (x^{2} + \frac {1}{2} \, x {\left (\sqrt {5} - 1\right )} + 1\right ) - \frac {1}{x} - \frac {1}{20} \, \log \left (x^{4} - x^{3} + x^{2} - x + 1\right ) + \frac {1}{5} \, \log \left ({\left | x + 1 \right |}\right ) \]

[In]

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

[Out]

-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)) - 1/20*sqrt(5)*log(x^2 - 1/2*x*(sqrt(5) + 1) + 1) + 1/20*sqrt(5)*
log(x^2 + 1/2*x*(sqrt(5) - 1) + 1) - 1/x - 1/20*log(x^4 - x^3 + x^2 - x + 1) + 1/5*log(abs(x + 1))

Mupad [B] (verification not implemented)

Time = 5.74 (sec) , antiderivative size = 203, normalized size of antiderivative = 1.07 \[ \int \frac {1}{x^2 \left (1+x^5\right )} \, dx=\frac {\ln \left (x+1\right )}{5}-\ln \left (25\,x\,\left (\frac {\sqrt {2}\,\sqrt {-\sqrt {5}-5}}{20}-\frac {\sqrt {5}}{20}+\frac {1}{20}\right )-5\right )\,\left (\frac {\sqrt {2}\,\sqrt {-\sqrt {5}-5}}{20}-\frac {\sqrt {5}}{20}+\frac {1}{20}\right )+\ln \left (25\,x\,\left (\frac {\sqrt {2}\,\sqrt {-\sqrt {5}-5}}{20}+\frac {\sqrt {5}}{20}-\frac {1}{20}\right )+5\right )\,\left (\frac {\sqrt {2}\,\sqrt {-\sqrt {5}-5}}{20}+\frac {\sqrt {5}}{20}-\frac {1}{20}\right )-\frac {1}{x}-\ln \left (25\,x\,\left (\frac {\sqrt {5}}{20}-\frac {\sqrt {2}\,\sqrt {\sqrt {5}-5}}{20}+\frac {1}{20}\right )-5\right )\,\left (\frac {\sqrt {5}}{20}-\frac {\sqrt {2}\,\sqrt {\sqrt {5}-5}}{20}+\frac {1}{20}\right )-\ln \left (25\,x\,\left (\frac {\sqrt {5}}{20}+\frac {\sqrt {2}\,\sqrt {\sqrt {5}-5}}{20}+\frac {1}{20}\right )-5\right )\,\left (\frac {\sqrt {5}}{20}+\frac {\sqrt {2}\,\sqrt {\sqrt {5}-5}}{20}+\frac {1}{20}\right ) \]

[In]

int(1/(x^2*(x^5 + 1)),x)

[Out]

log(x + 1)/5 - log(25*x*((2^(1/2)*(- 5^(1/2) - 5)^(1/2))/20 - 5^(1/2)/20 + 1/20) - 5)*((2^(1/2)*(- 5^(1/2) - 5
)^(1/2))/20 - 5^(1/2)/20 + 1/20) + log(25*x*((2^(1/2)*(- 5^(1/2) - 5)^(1/2))/20 + 5^(1/2)/20 - 1/20) + 5)*((2^
(1/2)*(- 5^(1/2) - 5)^(1/2))/20 + 5^(1/2)/20 - 1/20) - 1/x - log(25*x*(5^(1/2)/20 - (2^(1/2)*(5^(1/2) - 5)^(1/
2))/20 + 1/20) - 5)*(5^(1/2)/20 - (2^(1/2)*(5^(1/2) - 5)^(1/2))/20 + 1/20) - log(25*x*(5^(1/2)/20 + (2^(1/2)*(
5^(1/2) - 5)^(1/2))/20 + 1/20) - 5)*(5^(1/2)/20 + (2^(1/2)*(5^(1/2) - 5)^(1/2))/20 + 1/20)