\(\int \frac {1}{x^5 (3+4 x^3+x^6)} \, dx\) [153]

Optimal result

Integrand size = 16, antiderivative size = 126 \[ \int \frac {1}{x^5 \left (3+4 x^3+x^6\right )} \, dx=-\frac {1}{12 x^4}+\frac {4}{9 x}-\frac {\arctan \left (\frac {1-2 x}{\sqrt {3}}\right )}{2 \sqrt {3}}+\frac {\arctan \left (\frac {\sqrt [3]{3}-2 x}{3^{5/6}}\right )}{18\ 3^{5/6}}-\frac {1}{6} \log (1+x)+\frac {\log \left (\sqrt [3]{3}+x\right )}{54 \sqrt [3]{3}}+\frac {1}{12} \log \left (1-x+x^2\right )-\frac {\log \left (3^{2/3}-\sqrt [3]{3} x+x^2\right )}{108 \sqrt [3]{3}} \] Output:

-1/12/x^4+4/9/x-1/6*arctan(1/3*(1-2*x)*3^(1/2))*3^(1/2)+1/54*3^(1/6)*arcta 
n(1/3*(3^(1/3)-2*x)*3^(1/6))-1/6*ln(1+x)+1/162*3^(2/3)*ln(3^(1/3)+x)+1/12* 
ln(x^2-x+1)-1/324*3^(2/3)*ln(3^(2/3)-3^(1/3)*x+x^2)
 

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.94 \[ \int \frac {1}{x^5 \left (3+4 x^3+x^6\right )} \, dx=\frac {1}{324} \left (-\frac {27}{x^4}+\frac {144}{x}+6 \sqrt [6]{3} \arctan \left (\frac {\sqrt [3]{3}-2 x}{3^{5/6}}\right )+54 \sqrt {3} \arctan \left (\frac {-1+2 x}{\sqrt {3}}\right )-54 \log (1+x)+2\ 3^{2/3} \log \left (3+3^{2/3} x\right )+27 \log \left (1-x+x^2\right )-3^{2/3} \log \left (3-3^{2/3} x+\sqrt [3]{3} x^2\right )\right ) \] Input:

Integrate[1/(x^5*(3 + 4*x^3 + x^6)),x]
 

Output:

(-27/x^4 + 144/x + 6*3^(1/6)*ArcTan[(3^(1/3) - 2*x)/3^(5/6)] + 54*Sqrt[3]* 
ArcTan[(-1 + 2*x)/Sqrt[3]] - 54*Log[1 + x] + 2*3^(2/3)*Log[3 + 3^(2/3)*x] 
+ 27*Log[1 - x + x^2] - 3^(2/3)*Log[3 - 3^(2/3)*x + 3^(1/3)*x^2])/324
 

Rubi [A] (verified)

Time = 0.39 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.21, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.812, Rules used = {1704, 27, 1828, 1834, 821, 16, 1142, 25, 1082, 217, 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.

\(\displaystyle \int \frac {1}{x^5 \left (x^6+4 x^3+3\right )} \, dx\)

\(\Big \downarrow \) 1704

\(\displaystyle \frac {1}{12} \int -\frac {4 \left (x^3+4\right )}{x^2 \left (x^6+4 x^3+3\right )}dx-\frac {1}{12 x^4}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {1}{3} \int \frac {x^3+4}{x^2 \left (x^6+4 x^3+3\right )}dx-\frac {1}{12 x^4}\)

\(\Big \downarrow \) 1828

\(\displaystyle \frac {1}{3} \left (\frac {1}{3} \int \frac {x \left (4 x^3+13\right )}{x^6+4 x^3+3}dx+\frac {4}{3 x}\right )-\frac {1}{12 x^4}\)

\(\Big \downarrow \) 1834

\(\displaystyle \frac {1}{3} \left (\frac {1}{3} \left (\frac {9}{2} \int \frac {x}{x^3+1}dx-\frac {1}{2} \int \frac {x}{x^3+3}dx\right )+\frac {4}{3 x}\right )-\frac {1}{12 x^4}\)

\(\Big \downarrow \) 821

\(\displaystyle \frac {1}{3} \left (\frac {1}{3} \left (\frac {9}{2} \left (\frac {1}{3} \int \frac {x+1}{x^2-x+1}dx-\frac {1}{3} \int \frac {1}{x+1}dx\right )+\frac {1}{2} \left (\frac {\int \frac {1}{x+\sqrt [3]{3}}dx}{3 \sqrt [3]{3}}-\frac {\int \frac {x+\sqrt [3]{3}}{x^2-\sqrt [3]{3} x+3^{2/3}}dx}{3 \sqrt [3]{3}}\right )\right )+\frac {4}{3 x}\right )-\frac {1}{12 x^4}\)

\(\Big \downarrow \) 16

\(\displaystyle \frac {1}{3} \left (\frac {1}{3} \left (\frac {9}{2} \left (\frac {1}{3} \int \frac {x+1}{x^2-x+1}dx-\frac {1}{3} \log (x+1)\right )+\frac {1}{2} \left (\frac {\log \left (x+\sqrt [3]{3}\right )}{3 \sqrt [3]{3}}-\frac {\int \frac {x+\sqrt [3]{3}}{x^2-\sqrt [3]{3} x+3^{2/3}}dx}{3 \sqrt [3]{3}}\right )\right )+\frac {4}{3 x}\right )-\frac {1}{12 x^4}\)

\(\Big \downarrow \) 1142

\(\displaystyle \frac {1}{3} \left (\frac {1}{3} \left (\frac {9}{2} \left (\frac {1}{3} \left (\frac {3}{2} \int \frac {1}{x^2-x+1}dx+\frac {1}{2} \int -\frac {1-2 x}{x^2-x+1}dx\right )-\frac {1}{3} \log (x+1)\right )+\frac {1}{2} \left (\frac {\log \left (x+\sqrt [3]{3}\right )}{3 \sqrt [3]{3}}-\frac {\frac {3}{2} \sqrt [3]{3} \int \frac {1}{x^2-\sqrt [3]{3} x+3^{2/3}}dx+\frac {1}{2} \int -\frac {\sqrt [3]{3}-2 x}{x^2-\sqrt [3]{3} x+3^{2/3}}dx}{3 \sqrt [3]{3}}\right )\right )+\frac {4}{3 x}\right )-\frac {1}{12 x^4}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {1}{3} \left (\frac {1}{3} \left (\frac {9}{2} \left (\frac {1}{3} \left (\frac {3}{2} \int \frac {1}{x^2-x+1}dx-\frac {1}{2} \int \frac {1-2 x}{x^2-x+1}dx\right )-\frac {1}{3} \log (x+1)\right )+\frac {1}{2} \left (\frac {\log \left (x+\sqrt [3]{3}\right )}{3 \sqrt [3]{3}}-\frac {\frac {3}{2} \sqrt [3]{3} \int \frac {1}{x^2-\sqrt [3]{3} x+3^{2/3}}dx-\frac {1}{2} \int \frac {\sqrt [3]{3}-2 x}{x^2-\sqrt [3]{3} x+3^{2/3}}dx}{3 \sqrt [3]{3}}\right )\right )+\frac {4}{3 x}\right )-\frac {1}{12 x^4}\)

\(\Big \downarrow \) 1082

\(\displaystyle \frac {1}{3} \left (\frac {1}{3} \left (\frac {9}{2} \left (\frac {1}{3} \left (\frac {3}{2} \int \frac {1}{x^2-x+1}dx-\frac {1}{2} \int \frac {1-2 x}{x^2-x+1}dx\right )-\frac {1}{3} \log (x+1)\right )+\frac {1}{2} \left (\frac {\log \left (x+\sqrt [3]{3}\right )}{3 \sqrt [3]{3}}-\frac {3 \int \frac {1}{-\left (1-\frac {2 x}{\sqrt [3]{3}}\right )^2-3}d\left (1-\frac {2 x}{\sqrt [3]{3}}\right )-\frac {1}{2} \int \frac {\sqrt [3]{3}-2 x}{x^2-\sqrt [3]{3} x+3^{2/3}}dx}{3 \sqrt [3]{3}}\right )\right )+\frac {4}{3 x}\right )-\frac {1}{12 x^4}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {1}{3} \left (\frac {1}{3} \left (\frac {1}{2} \left (\frac {\log \left (x+\sqrt [3]{3}\right )}{3 \sqrt [3]{3}}-\frac {-\frac {1}{2} \int \frac {\sqrt [3]{3}-2 x}{x^2-\sqrt [3]{3} x+3^{2/3}}dx-\sqrt {3} \arctan \left (\frac {1-\frac {2 x}{\sqrt [3]{3}}}{\sqrt {3}}\right )}{3 \sqrt [3]{3}}\right )+\frac {9}{2} \left (\frac {1}{3} \left (\frac {3}{2} \int \frac {1}{x^2-x+1}dx-\frac {1}{2} \int \frac {1-2 x}{x^2-x+1}dx\right )-\frac {1}{3} \log (x+1)\right )\right )+\frac {4}{3 x}\right )-\frac {1}{12 x^4}\)

\(\Big \downarrow \) 1083

\(\displaystyle \frac {1}{3} \left (\frac {1}{3} \left (\frac {1}{2} \left (\frac {\log \left (x+\sqrt [3]{3}\right )}{3 \sqrt [3]{3}}-\frac {-\frac {1}{2} \int \frac {\sqrt [3]{3}-2 x}{x^2-\sqrt [3]{3} x+3^{2/3}}dx-\sqrt {3} \arctan \left (\frac {1-\frac {2 x}{\sqrt [3]{3}}}{\sqrt {3}}\right )}{3 \sqrt [3]{3}}\right )+\frac {9}{2} \left (\frac {1}{3} \left (-\frac {1}{2} \int \frac {1-2 x}{x^2-x+1}dx-3 \int \frac {1}{-(2 x-1)^2-3}d(2 x-1)\right )-\frac {1}{3} \log (x+1)\right )\right )+\frac {4}{3 x}\right )-\frac {1}{12 x^4}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {1}{3} \left (\frac {1}{3} \left (\frac {9}{2} \left (\frac {1}{3} \left (\sqrt {3} \arctan \left (\frac {2 x-1}{\sqrt {3}}\right )-\frac {1}{2} \int \frac {1-2 x}{x^2-x+1}dx\right )-\frac {1}{3} \log (x+1)\right )+\frac {1}{2} \left (\frac {\log \left (x+\sqrt [3]{3}\right )}{3 \sqrt [3]{3}}-\frac {-\frac {1}{2} \int \frac {\sqrt [3]{3}-2 x}{x^2-\sqrt [3]{3} x+3^{2/3}}dx-\sqrt {3} \arctan \left (\frac {1-\frac {2 x}{\sqrt [3]{3}}}{\sqrt {3}}\right )}{3 \sqrt [3]{3}}\right )\right )+\frac {4}{3 x}\right )-\frac {1}{12 x^4}\)

\(\Big \downarrow \) 1103

\(\displaystyle \frac {1}{3} \left (\frac {1}{3} \left (\frac {9}{2} \left (\frac {1}{3} \left (\sqrt {3} \arctan \left (\frac {2 x-1}{\sqrt {3}}\right )+\frac {1}{2} \log \left (x^2-x+1\right )\right )-\frac {1}{3} \log (x+1)\right )+\frac {1}{2} \left (\frac {\log \left (x+\sqrt [3]{3}\right )}{3 \sqrt [3]{3}}-\frac {\frac {1}{2} \log \left (x^2-\sqrt [3]{3} x+3^{2/3}\right )-\sqrt {3} \arctan \left (\frac {1-\frac {2 x}{\sqrt [3]{3}}}{\sqrt {3}}\right )}{3 \sqrt [3]{3}}\right )\right )+\frac {4}{3 x}\right )-\frac {1}{12 x^4}\)

Input:

Int[1/(x^5*(3 + 4*x^3 + x^6)),x]
 

Output:

-1/12*1/x^4 + (4/(3*x) + ((9*(-1/3*Log[1 + x] + (Sqrt[3]*ArcTan[(-1 + 2*x) 
/Sqrt[3]] + Log[1 - x + x^2]/2)/3))/2 + (Log[3^(1/3) + x]/(3*3^(1/3)) - (- 
(Sqrt[3]*ArcTan[(1 - (2*x)/3^(1/3))/Sqrt[3]]) + Log[3^(2/3) - 3^(1/3)*x + 
x^2]/2)/(3*3^(1/3)))/2)/3)/3
 

Defintions of rubi rules used

rule 16

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

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 217

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 821

Int[(x_)/((a_) + (b_.)*(x_)^3), x_Symbol] :> Simp[-(3*Rt[a, 3]*Rt[b, 3])^(- 
1)   Int[1/(Rt[a, 3] + Rt[b, 3]*x), x], x] + Simp[1/(3*Rt[a, 3]*Rt[b, 3]) 
 Int[(Rt[a, 3] + Rt[b, 3]*x)/(Rt[a, 3]^2 - Rt[a, 3]*Rt[b, 3]*x + Rt[b, 3]^2 
*x^2), x], x] /; FreeQ[{a, b}, x]
 

rule 1082

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S 
implify[a*(c/b^2)]}, Simp[-2/b   Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b 
)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /; Fre 
eQ[{a, b, c}, x]
 

rule 1083

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]
 

rule 1103

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]
 

rule 1142

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]
 

rule 1704

Int[((d_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_), x_ 
Symbol] :> Simp[(d*x)^(m + 1)*((a + b*x^n + c*x^(2*n))^(p + 1)/(a*d*(m + 1) 
)), x] - Simp[1/(a*d^n*(m + 1))   Int[(d*x)^(m + n)*(b*(m + n*(p + 1) + 1) 
+ c*(m + 2*n*(p + 1) + 1)*x^n)*(a + b*x^n + c*x^(2*n))^p, x], x] /; FreeQ[{ 
a, b, c, d, p}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && 
LtQ[m, -1] && IntegerQ[p]
 

rule 1828

Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(n_))*((a_) + (b_.)*(x_)^(n_) + ( 
c_.)*(x_)^(n2_))^(p_), x_Symbol] :> Simp[d*(f*x)^(m + 1)*((a + b*x^n + c*x^ 
(2*n))^(p + 1)/(a*f*(m + 1))), x] + Simp[1/(a*f^n*(m + 1))   Int[(f*x)^(m + 
 n)*(a + b*x^n + c*x^(2*n))^p*Simp[a*e*(m + 1) - b*d*(m + n*(p + 1) + 1) - 
c*d*(m + 2*n*(p + 1) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x 
] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && LtQ[m, -1] && Int 
egerQ[p]
 

rule 1834

Int[(((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(n_)))/((a_) + (b_.)*(x_)^(n_) + 
 (c_.)*(x_)^(n2_)), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[(e/2 + 
 (2*c*d - b*e)/(2*q))   Int[(f*x)^m/(b/2 - q/2 + c*x^n), x], x] + Simp[(e/2 
 - (2*c*d - b*e)/(2*q))   Int[(f*x)^m/(b/2 + q/2 + c*x^n), x], x]] /; FreeQ 
[{a, b, c, d, e, f, m}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n 
, 0]
 

Maple [C] (verified)

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

Time = 0.05 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.56

method	result	size
risch	\(\frac {\frac {4 x^{3}}{9}-\frac {1}{12}}{x^{4}}+\frac {\ln \left (4 x^{2}-4 x +4\right )}{12}+\frac {\sqrt {3}\, \arctan \left (\frac {\left (2 x -1\right ) \sqrt {3}}{3}\right )}{6}-\frac {\ln \left (x +1\right )}{6}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (3 \textit {\_Z}^{3}-1\right )}{\sum }\textit {\_R} \ln \left (3 \textit {\_R}^{2}+x \right )\right )}{54}\)	\(71\)
default	\(\frac {\ln \left (x^{2}-x +1\right )}{12}+\frac {\sqrt {3}\, \arctan \left (\frac {\left (2 x -1\right ) \sqrt {3}}{3}\right )}{6}+\frac {3^{\frac {2}{3}} \ln \left (3^{\frac {1}{3}}+x \right )}{162}-\frac {3^{\frac {2}{3}} \ln \left (3^{\frac {2}{3}}-3^{\frac {1}{3}} x +x^{2}\right )}{324}-\frac {3^{\frac {1}{6}} \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \,3^{\frac {2}{3}} x}{3}-1\right )}{3}\right )}{54}-\frac {\ln \left (x +1\right )}{6}-\frac {1}{12 x^{4}}+\frac {4}{9 x}\)	\(94\)

Input:

int(1/x^5/(x^6+4*x^3+3),x,method=_RETURNVERBOSE)
 

Output:

(4/9*x^3-1/12)/x^4+1/12*ln(4*x^2-4*x+4)+1/6*3^(1/2)*arctan(1/3*(2*x-1)*3^( 
1/2))-1/6*ln(x+1)+1/54*sum(_R*ln(3*_R^2+x),_R=RootOf(3*_Z^3-1))
 

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.89 \[ \int \frac {1}{x^5 \left (3+4 x^3+x^6\right )} \, dx=-\frac {3^{\frac {2}{3}} x^{4} \log \left (x^{2} - 3^{\frac {1}{3}} x + 3^{\frac {2}{3}}\right ) - 2 \cdot 3^{\frac {2}{3}} x^{4} \log \left (x + 3^{\frac {1}{3}}\right ) - 54 \, \sqrt {3} x^{4} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) - 6 \cdot 3^{\frac {1}{6}} x^{4} \arctan \left (-\frac {1}{3} \cdot 3^{\frac {1}{6}} {\left (2 \, x - 3^{\frac {1}{3}}\right )}\right ) - 27 \, x^{4} \log \left (x^{2} - x + 1\right ) + 54 \, x^{4} \log \left (x + 1\right ) - 144 \, x^{3} + 27}{324 \, x^{4}} \] Input:

integrate(1/x^5/(x^6+4*x^3+3),x, algorithm="fricas")
 

Output:

-1/324*(3^(2/3)*x^4*log(x^2 - 3^(1/3)*x + 3^(2/3)) - 2*3^(2/3)*x^4*log(x + 
 3^(1/3)) - 54*sqrt(3)*x^4*arctan(1/3*sqrt(3)*(2*x - 1)) - 6*3^(1/6)*x^4*a 
rctan(-1/3*3^(1/6)*(2*x - 3^(1/3))) - 27*x^4*log(x^2 - x + 1) + 54*x^4*log 
(x + 1) - 144*x^3 + 27)/x^4
 

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 1.28 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.12 \[ \int \frac {1}{x^5 \left (3+4 x^3+x^6\right )} \, dx=- \frac {\log {\left (x + 1 \right )}}{6} + \left (\frac {1}{12} - \frac {\sqrt {3} i}{12}\right ) \log {\left (x + \frac {4782978 \left (\frac {1}{12} - \frac {\sqrt {3} i}{12}\right )^{2}}{547} + \frac {1028869776 \left (\frac {1}{12} - \frac {\sqrt {3} i}{12}\right )^{5}}{547} \right )} + \left (\frac {1}{12} + \frac {\sqrt {3} i}{12}\right ) \log {\left (x + \frac {1028869776 \left (\frac {1}{12} + \frac {\sqrt {3} i}{12}\right )^{5}}{547} + \frac {4782978 \left (\frac {1}{12} + \frac {\sqrt {3} i}{12}\right )^{2}}{547} \right )} + \operatorname {RootSum} {\left (472392 t^{3} - 1, \left ( t \mapsto t \log {\left (\frac {1028869776 t^{5}}{547} + \frac {4782978 t^{2}}{547} + x \right )} \right )\right )} + \frac {16 x^{3} - 3}{36 x^{4}} \] Input:

integrate(1/x**5/(x**6+4*x**3+3),x)
 

Output:

-log(x + 1)/6 + (1/12 - sqrt(3)*I/12)*log(x + 4782978*(1/12 - sqrt(3)*I/12 
)**2/547 + 1028869776*(1/12 - sqrt(3)*I/12)**5/547) + (1/12 + sqrt(3)*I/12 
)*log(x + 1028869776*(1/12 + sqrt(3)*I/12)**5/547 + 4782978*(1/12 + sqrt(3 
)*I/12)**2/547) + RootSum(472392*_t**3 - 1, Lambda(_t, _t*log(1028869776*_ 
t**5/547 + 4782978*_t**2/547 + x))) + (16*x**3 - 3)/(36*x**4)
 

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.76 \[ \int \frac {1}{x^5 \left (3+4 x^3+x^6\right )} \, dx=-\frac {1}{324} \cdot 3^{\frac {2}{3}} \log \left (x^{2} - 3^{\frac {1}{3}} x + 3^{\frac {2}{3}}\right ) + \frac {1}{162} \cdot 3^{\frac {2}{3}} \log \left (x + 3^{\frac {1}{3}}\right ) + \frac {1}{6} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) - \frac {1}{54} \cdot 3^{\frac {1}{6}} \arctan \left (\frac {1}{3} \cdot 3^{\frac {1}{6}} {\left (2 \, x - 3^{\frac {1}{3}}\right )}\right ) + \frac {16 \, x^{3} - 3}{36 \, x^{4}} + \frac {1}{12} \, \log \left (x^{2} - x + 1\right ) - \frac {1}{6} \, \log \left (x + 1\right ) \] Input:

integrate(1/x^5/(x^6+4*x^3+3),x, algorithm="maxima")
 

Output:

-1/324*3^(2/3)*log(x^2 - 3^(1/3)*x + 3^(2/3)) + 1/162*3^(2/3)*log(x + 3^(1 
/3)) + 1/6*sqrt(3)*arctan(1/3*sqrt(3)*(2*x - 1)) - 1/54*3^(1/6)*arctan(1/3 
*3^(1/6)*(2*x - 3^(1/3))) + 1/36*(16*x^3 - 3)/x^4 + 1/12*log(x^2 - x + 1) 
- 1/6*log(x + 1)
 

Giac [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.78 \[ \int \frac {1}{x^5 \left (3+4 x^3+x^6\right )} \, dx=-\frac {1}{324} \cdot 3^{\frac {2}{3}} \log \left (x^{2} - 3^{\frac {1}{3}} x + 3^{\frac {2}{3}}\right ) + \frac {1}{162} \cdot 3^{\frac {2}{3}} \log \left ({\left | x + 3^{\frac {1}{3}} \right |}\right ) + \frac {1}{6} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) - \frac {1}{54} \cdot 3^{\frac {1}{6}} \arctan \left (\frac {1}{3} \cdot 3^{\frac {1}{6}} {\left (2 \, x - 3^{\frac {1}{3}}\right )}\right ) + \frac {16 \, x^{3} - 3}{36 \, x^{4}} + \frac {1}{12} \, \log \left (x^{2} - x + 1\right ) - \frac {1}{6} \, \log \left ({\left | x + 1 \right |}\right ) \] Input:

integrate(1/x^5/(x^6+4*x^3+3),x, algorithm="giac")
 

Output:

-1/324*3^(2/3)*log(x^2 - 3^(1/3)*x + 3^(2/3)) + 1/162*3^(2/3)*log(abs(x + 
3^(1/3))) + 1/6*sqrt(3)*arctan(1/3*sqrt(3)*(2*x - 1)) - 1/54*3^(1/6)*arcta 
n(1/3*3^(1/6)*(2*x - 3^(1/3))) + 1/36*(16*x^3 - 3)/x^4 + 1/12*log(x^2 - x 
+ 1) - 1/6*log(abs(x + 1))
 

Mupad [B] (verification not implemented)

Time = 19.93 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.98 \[ \int \frac {1}{x^5 \left (3+4 x^3+x^6\right )} \, dx=\frac {3^{2/3}\,\ln \left (x+3^{1/3}\right )}{162}-\frac {\ln \left (x+1\right )}{6}-\ln \left (x-\frac {1}{2}-\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (-\frac {1}{12}+\frac {\sqrt {3}\,1{}\mathrm {i}}{12}\right )+\ln \left (x-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{12}+\frac {\sqrt {3}\,1{}\mathrm {i}}{12}\right )+\frac {\frac {4\,x^3}{9}-\frac {1}{12}}{x^4}-\ln \left (x-\frac {3^{1/3}}{2}-\frac {3^{5/6}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {3^{2/3}}{324}-\frac {3^{1/6}\,1{}\mathrm {i}}{108}\right )-\ln \left (x-\frac {3^{1/3}}{2}+\frac {3^{5/6}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {3^{2/3}}{324}+\frac {3^{1/6}\,1{}\mathrm {i}}{108}\right ) \] Input:

int(1/(x^5*(4*x^3 + x^6 + 3)),x)
 

Output:

(3^(2/3)*log(x + 3^(1/3)))/162 - log(x + 1)/6 - log(x - (3^(1/2)*1i)/2 - 1 
/2)*((3^(1/2)*1i)/12 - 1/12) + log(x + (3^(1/2)*1i)/2 - 1/2)*((3^(1/2)*1i) 
/12 + 1/12) + ((4*x^3)/9 - 1/12)/x^4 - log(x - 3^(1/3)/2 - (3^(5/6)*1i)/2) 
*(3^(2/3)/324 - (3^(1/6)*1i)/108) - log(x - 3^(1/3)/2 + (3^(5/6)*1i)/2)*(3 
^(2/3)/324 + (3^(1/6)*1i)/108)
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.98 \[ \int \frac {1}{x^5 \left (3+4 x^3+x^6\right )} \, dx=\frac {\left (2 \sqrt {3}\, \mathit {atan} \left (\frac {\left (3^{\frac {1}{3}}-2 x \right ) 3^{\frac {1}{6}}}{3}\right ) x^{4}+18 \sqrt {3}\, 3^{\frac {1}{3}} \mathit {atan} \left (\frac {2 x -1}{\sqrt {3}}\right ) x^{4}+9 \,3^{\frac {1}{3}} \mathrm {log}\left (x^{2}-x +1\right ) x^{4}-18 \,3^{\frac {1}{3}} \mathrm {log}\left (x +1\right ) x^{4}+48 \,3^{\frac {1}{3}} x^{3}-9 \,3^{\frac {1}{3}}-\mathrm {log}\left (3^{\frac {2}{3}}-3^{\frac {1}{3}} x +x^{2}\right ) x^{4}+2 \,\mathrm {log}\left (3^{\frac {1}{3}}+x \right ) x^{4}\right ) 3^{\frac {2}{3}}}{324 x^{4}} \] Input:

int(1/x^5/(x^6+4*x^3+3),x)
 

Output:

(2*3**(1/3)*3**(1/6)*atan((3**(1/3) - 2*x)/(3**(2/3)*3**(1/6)))*x**4 + 18* 
sqrt(3)*3**(1/3)*atan((2*x - 1)/sqrt(3))*x**4 + 9*3**(1/3)*log(x**2 - x + 
1)*x**4 - 18*3**(1/3)*log(x + 1)*x**4 + 48*3**(1/3)*x**3 - 9*3**(1/3) - lo 
g(3**(2/3) - 3**(1/3)*x + x**2)*x**4 + 2*log(3**(1/3) + x)*x**4)/(108*3**( 
1/3)*x**4)