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

Optimal. Leaf size=126 \[ -\frac{1}{12 x^4}+\frac{1}{12} \log \left (x^2-x+1\right )-\frac{\log \left (x^2-\sqrt [3]{3} x+3^{2/3}\right )}{108 \sqrt [3]{3}}+\frac{4}{9 x}-\frac{1}{6} \log (x+1)+\frac{\log \left (x+\sqrt [3]{3}\right )}{54 \sqrt [3]{3}}-\frac{\tan ^{-1}\left (\frac{1-2 x}{\sqrt{3}}\right )}{2 \sqrt{3}}+\frac{\tan ^{-1}\left (\frac{\sqrt [3]{3}-2 x}{3^{5/6}}\right )}{18\ 3^{5/6}} \]

[Out]

-1/(12*x^4) + 4/(9*x) - ArcTan[(1 - 2*x)/Sqrt[3]]/(2*Sqrt[3]) + ArcTan[(3^(1/3) - 2*x)/3^(5/6)]/(18*3^(5/6)) -
Log[1 + x]/6 + Log[3^(1/3) + x]/(54*3^(1/3)) + Log[1 - x + x^2]/12 - Log[3^(2/3) - 3^(1/3)*x + x^2]/(108*3^(1
/3))

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Rubi [A]  time = 0.10298, antiderivative size = 126, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 10, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.625, Rules used = {1368, 1504, 1510, 292, 31, 634, 618, 204, 628, 617} \[ -\frac{1}{12 x^4}+\frac{1}{12} \log \left (x^2-x+1\right )-\frac{\log \left (x^2-\sqrt [3]{3} x+3^{2/3}\right )}{108 \sqrt [3]{3}}+\frac{4}{9 x}-\frac{1}{6} \log (x+1)+\frac{\log \left (x+\sqrt [3]{3}\right )}{54 \sqrt [3]{3}}-\frac{\tan ^{-1}\left (\frac{1-2 x}{\sqrt{3}}\right )}{2 \sqrt{3}}+\frac{\tan ^{-1}\left (\frac{\sqrt [3]{3}-2 x}{3^{5/6}}\right )}{18\ 3^{5/6}} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^5*(3 + 4*x^3 + x^6)),x]

[Out]

-1/(12*x^4) + 4/(9*x) - ArcTan[(1 - 2*x)/Sqrt[3]]/(2*Sqrt[3]) + ArcTan[(3^(1/3) - 2*x)/3^(5/6)]/(18*3^(5/6)) -
Log[1 + x]/6 + Log[3^(1/3) + x]/(54*3^(1/3)) + Log[1 - x + x^2]/12 - Log[3^(2/3) - 3^(1/3)*x + x^2]/(108*3^(1
/3))

Rule 1368

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] - Dist[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 1504

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] + Dist[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] && IntegerQ[p]

Rule 1510

Int[(((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(n_)))/((a_) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_)), x_Symbol] :> Wi
th[{q = Rt[b^2 - 4*a*c, 2]}, Dist[e/2 + (2*c*d - b*e)/(2*q), Int[(f*x)^m/(b/2 - q/2 + c*x^n), x], x] + Dist[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]

Rule 292

Int[(x_)/((a_) + (b_.)*(x_)^3), x_Symbol] :> -Dist[(3*Rt[a, 3]*Rt[b, 3])^(-1), Int[1/(Rt[a, 3] + Rt[b, 3]*x),
x], x] + Dist[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 31

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

Rule 634

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]

Rule 618

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 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 628

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 617

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

Rubi steps

\begin{align*} \int \frac{1}{x^5 \left (3+4 x^3+x^6\right )} \, dx &=-\frac{1}{12 x^4}+\frac{1}{12} \int \frac{-16-4 x^3}{x^2 \left (3+4 x^3+x^6\right )} \, dx\\ &=-\frac{1}{12 x^4}+\frac{4}{9 x}-\frac{1}{36} \int \frac{x \left (-52-16 x^3\right )}{3+4 x^3+x^6} \, dx\\ &=-\frac{1}{12 x^4}+\frac{4}{9 x}-\frac{1}{18} \int \frac{x}{3+x^3} \, dx+\frac{1}{2} \int \frac{x}{1+x^3} \, dx\\ &=-\frac{1}{12 x^4}+\frac{4}{9 x}-\frac{1}{6} \int \frac{1}{1+x} \, dx+\frac{1}{6} \int \frac{1+x}{1-x+x^2} \, dx+\frac{\int \frac{1}{\sqrt [3]{3}+x} \, dx}{54 \sqrt [3]{3}}-\frac{\int \frac{\sqrt [3]{3}+x}{3^{2/3}-\sqrt [3]{3} x+x^2} \, dx}{54 \sqrt [3]{3}}\\ &=-\frac{1}{12 x^4}+\frac{4}{9 x}-\frac{1}{6} \log (1+x)+\frac{\log \left (\sqrt [3]{3}+x\right )}{54 \sqrt [3]{3}}-\frac{1}{36} \int \frac{1}{3^{2/3}-\sqrt [3]{3} x+x^2} \, dx+\frac{1}{12} \int \frac{-1+2 x}{1-x+x^2} \, dx+\frac{1}{4} \int \frac{1}{1-x+x^2} \, dx-\frac{\int \frac{-\sqrt [3]{3}+2 x}{3^{2/3}-\sqrt [3]{3} x+x^2} \, dx}{108 \sqrt [3]{3}}\\ &=-\frac{1}{12 x^4}+\frac{4}{9 x}-\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}}-\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,-1+2 x\right )-\frac{\operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1-\frac{2 x}{\sqrt [3]{3}}\right )}{18 \sqrt [3]{3}}\\ &=-\frac{1}{12 x^4}+\frac{4}{9 x}-\frac{\tan ^{-1}\left (\frac{1-2 x}{\sqrt{3}}\right )}{2 \sqrt{3}}+\frac{\tan ^{-1}\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}}\\ \end{align*}

Mathematica [A]  time = 0.0510884, size = 118, normalized size = 0.94 \[ \frac{1}{324} \left (-\frac{27}{x^4}+27 \log \left (x^2-x+1\right )-3^{2/3} \log \left (\sqrt [3]{3} x^2-3^{2/3} x+3\right )+\frac{144}{x}-54 \log (x+1)+2\ 3^{2/3} \log \left (3^{2/3} x+3\right )+6 \sqrt [6]{3} \tan ^{-1}\left (\frac{\sqrt [3]{3}-2 x}{3^{5/6}}\right )+54 \sqrt{3} \tan ^{-1}\left (\frac{2 x-1}{\sqrt{3}}\right )\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-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

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Maple [A]  time = 0.008, size = 94, normalized size = 0.8 \begin{align*} -{\frac{1}{12\,{x}^{4}}}+{\frac{4}{9\,x}}+{\frac{\ln \left ({x}^{2}-x+1 \right ) }{12}}+{\frac{\sqrt{3}}{6}\arctan \left ({\frac{ \left ( 2\,x-1 \right ) \sqrt{3}}{3}} \right ) }+{\frac{{3}^{{\frac{2}{3}}}\ln \left ( \sqrt [3]{3}+x \right ) }{162}}-{\frac{{3}^{{\frac{2}{3}}}\ln \left ({3}^{{\frac{2}{3}}}-\sqrt [3]{3}x+{x}^{2} \right ) }{324}}-{\frac{\sqrt [6]{3}}{54}\arctan \left ({\frac{\sqrt{3}}{3} \left ({\frac{2\,{3}^{2/3}x}{3}}-1 \right ) } \right ) }-{\frac{\ln \left ( 1+x \right ) }{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.60797, size = 130, normalized size = 1.03 \begin{align*} -\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 ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-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)

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Fricas [A]  time = 1.47367, size = 339, normalized size = 2.69 \begin{align*} -\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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-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*arctan(-1/3*3^(1/6)*(2*x - 3^(1/3))) - 27*x^4*log(x^2 - x + 1) + 54*x^4*l
og(x + 1) - 144*x^3 + 27)/x^4

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Sympy [C]  time = 1.3051, size = 141, normalized size = 1.12 \begin{align*} - \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-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)

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: NotImplementedError