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

Optimal. Leaf size=126 \[ \frac{2}{9 x^2}-\frac{1}{15 x^5}-\frac{1}{12} \log \left (x^2-x+1\right )+\frac{\log \left (x^2-\sqrt [3]{3} x+3^{2/3}\right )}{108\ 3^{2/3}}+\frac{1}{6} \log (x+1)-\frac{\log \left (x+\sqrt [3]{3}\right )}{54\ 3^{2/3}}-\frac{\tan ^{-1}\left (\frac{1-2 x}{\sqrt{3}}\right )}{2 \sqrt{3}}+\frac{\tan ^{-1}\left (\frac{\sqrt [3]{3}-2 x}{3^{5/6}}\right )}{54 \sqrt [6]{3}} \]

[Out]

-1/(15*x^5) + 2/(9*x^2) - ArcTan[(1 - 2*x)/Sqrt[3]]/(2*Sqrt[3]) + ArcTan[(3^(1/3) - 2*x)/3^(5/6)]/(54*3^(1/6))
+ Log[1 + x]/6 - Log[3^(1/3) + x]/(54*3^(2/3)) - Log[1 - x + x^2]/12 + Log[3^(2/3) - 3^(1/3)*x + x^2]/(108*3^
(2/3))

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Rubi [A]  time = 0.0985897, 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, 1422, 200, 31, 634, 618, 204, 628, 617} \[ \frac{2}{9 x^2}-\frac{1}{15 x^5}-\frac{1}{12} \log \left (x^2-x+1\right )+\frac{\log \left (x^2-\sqrt [3]{3} x+3^{2/3}\right )}{108\ 3^{2/3}}+\frac{1}{6} \log (x+1)-\frac{\log \left (x+\sqrt [3]{3}\right )}{54\ 3^{2/3}}-\frac{\tan ^{-1}\left (\frac{1-2 x}{\sqrt{3}}\right )}{2 \sqrt{3}}+\frac{\tan ^{-1}\left (\frac{\sqrt [3]{3}-2 x}{3^{5/6}}\right )}{54 \sqrt [6]{3}} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^6*(3 + 4*x^3 + x^6)),x]

[Out]

-1/(15*x^5) + 2/(9*x^2) - ArcTan[(1 - 2*x)/Sqrt[3]]/(2*Sqrt[3]) + ArcTan[(3^(1/3) - 2*x)/3^(5/6)]/(54*3^(1/6))
+ Log[1 + x]/6 - Log[3^(1/3) + x]/(54*3^(2/3)) - Log[1 - x + x^2]/12 + Log[3^(2/3) - 3^(1/3)*x + x^2]/(108*3^
(2/3))

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 1422

Int[((d_) + (e_.)*(x_)^(n_))/((a_) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_)), x_Symbol] :> With[{q = Rt[b^2 - 4*a*
c, 2]}, Dist[e/2 + (2*c*d - b*e)/(2*q), Int[1/(b/2 - q/2 + c*x^n), x], x] + Dist[e/2 - (2*c*d - b*e)/(2*q), In
t[1/(b/2 + q/2 + c*x^n), x], x]] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && NeQ
[c*d^2 - b*d*e + a*e^2, 0] && (PosQ[b^2 - 4*a*c] ||  !IGtQ[n/2, 0])

Rule 200

Int[((a_) + (b_.)*(x_)^3)^(-1), x_Symbol] :> Dist[1/(3*Rt[a, 3]^2), Int[1/(Rt[a, 3] + Rt[b, 3]*x), x], x] + Di
st[1/(3*Rt[a, 3]^2), Int[(2*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^6 \left (3+4 x^3+x^6\right )} \, dx &=-\frac{1}{15 x^5}+\frac{1}{15} \int \frac{-20-5 x^3}{x^3 \left (3+4 x^3+x^6\right )} \, dx\\ &=-\frac{1}{15 x^5}+\frac{2}{9 x^2}-\frac{1}{90} \int \frac{-130-40 x^3}{3+4 x^3+x^6} \, dx\\ &=-\frac{1}{15 x^5}+\frac{2}{9 x^2}-\frac{1}{18} \int \frac{1}{3+x^3} \, dx+\frac{1}{2} \int \frac{1}{1+x^3} \, dx\\ &=-\frac{1}{15 x^5}+\frac{2}{9 x^2}+\frac{1}{6} \int \frac{1}{1+x} \, dx+\frac{1}{6} \int \frac{2-x}{1-x+x^2} \, dx-\frac{\int \frac{1}{\sqrt [3]{3}+x} \, dx}{54\ 3^{2/3}}-\frac{\int \frac{2 \sqrt [3]{3}-x}{3^{2/3}-\sqrt [3]{3} x+x^2} \, dx}{54\ 3^{2/3}}\\ &=-\frac{1}{15 x^5}+\frac{2}{9 x^2}+\frac{1}{6} \log (1+x)-\frac{\log \left (\sqrt [3]{3}+x\right )}{54\ 3^{2/3}}-\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\ 3^{2/3}}-\frac{\int \frac{1}{3^{2/3}-\sqrt [3]{3} x+x^2} \, dx}{36 \sqrt [3]{3}}\\ &=-\frac{1}{15 x^5}+\frac{2}{9 x^2}+\frac{1}{6} \log (1+x)-\frac{\log \left (\sqrt [3]{3}+x\right )}{54\ 3^{2/3}}-\frac{1}{12} \log \left (1-x+x^2\right )+\frac{\log \left (3^{2/3}-\sqrt [3]{3} x+x^2\right )}{108\ 3^{2/3}}-\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\ 3^{2/3}}\\ &=-\frac{1}{15 x^5}+\frac{2}{9 x^2}-\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 )}{54 \sqrt [6]{3}}+\frac{1}{6} \log (1+x)-\frac{\log \left (\sqrt [3]{3}+x\right )}{54\ 3^{2/3}}-\frac{1}{12} \log \left (1-x+x^2\right )+\frac{\log \left (3^{2/3}-\sqrt [3]{3} x+x^2\right )}{108\ 3^{2/3}}\\ \end{align*}

Mathematica [A]  time = 0.0612248, size = 118, normalized size = 0.94 \[ \frac{\frac{360}{x^2}-\frac{108}{x^5}-135 \log \left (x^2-x+1\right )+5 \sqrt [3]{3} \log \left (\sqrt [3]{3} x^2-3^{2/3} x+3\right )+270 \log (x+1)-10 \sqrt [3]{3} \log \left (3^{2/3} x+3\right )+10\ 3^{5/6} \tan ^{-1}\left (\frac{\sqrt [3]{3}-2 x}{3^{5/6}}\right )+270 \sqrt{3} \tan ^{-1}\left (\frac{2 x-1}{\sqrt{3}}\right )}{1620} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-108/x^5 + 360/x^2 + 10*3^(5/6)*ArcTan[(3^(1/3) - 2*x)/3^(5/6)] + 270*Sqrt[3]*ArcTan[(-1 + 2*x)/Sqrt[3]] + 27
0*Log[1 + x] - 10*3^(1/3)*Log[3 + 3^(2/3)*x] - 135*Log[1 - x + x^2] + 5*3^(1/3)*Log[3 - 3^(2/3)*x + 3^(1/3)*x^
2])/1620

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Maple [A]  time = 0.01, size = 94, normalized size = 0.8 \begin{align*} -{\frac{1}{15\,{x}^{5}}}+{\frac{2}{9\,{x}^{2}}}-{\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{\sqrt [3]{3}\ln \left ( \sqrt [3]{3}+x \right ) }{162}}+{\frac{\sqrt [3]{3}\ln \left ({3}^{{\frac{2}{3}}}-\sqrt [3]{3}x+{x}^{2} \right ) }{324}}-{\frac{{3}^{{\frac{5}{6}}}}{162}\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^6/(x^6+4*x^3+3),x)

[Out]

-1/15/x^5+2/9/x^2-1/12*ln(x^2-x+1)+1/6*3^(1/2)*arctan(1/3*(2*x-1)*3^(1/2))-1/162*3^(1/3)*ln(3^(1/3)+x)+1/324*3
^(1/3)*ln(3^(2/3)-3^(1/3)*x+x^2)-1/162*3^(5/6)*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 = 1.6985, size = 130, normalized size = 1.03 \begin{align*} -\frac{1}{162} \cdot 3^{\frac{5}{6}} \arctan \left (\frac{1}{3} \cdot 3^{\frac{1}{6}}{\left (2 \, 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}{324} \cdot 3^{\frac{1}{3}} \log \left (x^{2} - 3^{\frac{1}{3}} x + 3^{\frac{2}{3}}\right ) - \frac{1}{162} \cdot 3^{\frac{1}{3}} \log \left (x + 3^{\frac{1}{3}}\right ) + \frac{10 \, x^{3} - 3}{45 \, x^{5}} - \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^6/(x^6+4*x^3+3),x, algorithm="maxima")

[Out]

-1/162*3^(5/6)*arctan(1/3*3^(1/6)*(2*x - 3^(1/3))) + 1/6*sqrt(3)*arctan(1/3*sqrt(3)*(2*x - 1)) + 1/324*3^(1/3)
*log(x^2 - 3^(1/3)*x + 3^(2/3)) - 1/162*3^(1/3)*log(x + 3^(1/3)) + 1/45*(10*x^3 - 3)/x^5 - 1/12*log(x^2 - x +
1) + 1/6*log(x + 1)

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Fricas [A]  time = 1.52307, size = 510, normalized size = 4.05 \begin{align*} \frac{30 \cdot 9^{\frac{1}{6}} \sqrt{3} \left (-1\right )^{\frac{1}{3}} x^{5} \arctan \left (\frac{1}{27} \cdot 9^{\frac{1}{6}}{\left (2 \cdot 9^{\frac{2}{3}} \sqrt{3} \left (-1\right )^{\frac{2}{3}} x - 3 \cdot 9^{\frac{1}{3}} \sqrt{3}\right )}\right ) - 5 \cdot 9^{\frac{2}{3}} \left (-1\right )^{\frac{1}{3}} x^{5} \log \left (9^{\frac{2}{3}} \left (-1\right )^{\frac{1}{3}} x + 3 \, x^{2} + 3 \cdot 9^{\frac{1}{3}} \left (-1\right )^{\frac{2}{3}}\right ) + 10 \cdot 9^{\frac{2}{3}} \left (-1\right )^{\frac{1}{3}} x^{5} \log \left (-9^{\frac{2}{3}} \left (-1\right )^{\frac{1}{3}} + 3 \, x\right ) + 810 \, \sqrt{3} x^{5} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right ) - 405 \, x^{5} \log \left (x^{2} - x + 1\right ) + 810 \, x^{5} \log \left (x + 1\right ) + 1080 \, x^{3} - 324}{4860 \, x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4860*(30*9^(1/6)*sqrt(3)*(-1)^(1/3)*x^5*arctan(1/27*9^(1/6)*(2*9^(2/3)*sqrt(3)*(-1)^(2/3)*x - 3*9^(1/3)*sqrt
(3))) - 5*9^(2/3)*(-1)^(1/3)*x^5*log(9^(2/3)*(-1)^(1/3)*x + 3*x^2 + 3*9^(1/3)*(-1)^(2/3)) + 10*9^(2/3)*(-1)^(1
/3)*x^5*log(-9^(2/3)*(-1)^(1/3) + 3*x) + 810*sqrt(3)*x^5*arctan(1/3*sqrt(3)*(2*x - 1)) - 405*x^5*log(x^2 - x +
1) + 810*x^5*log(x + 1) + 1080*x^3 - 324)/x^5

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Sympy [C]  time = 1.28467, size = 136, normalized size = 1.08 \begin{align*} \frac{\log{\left (x + 1 \right )}}{6} + \left (- \frac{1}{12} + \frac{\sqrt{3} i}{12}\right ) \log{\left (x + \frac{88573}{6562} - \frac{88573 \sqrt{3} i}{6562} + \frac{119042784 \left (- \frac{1}{12} + \frac{\sqrt{3} i}{12}\right )^{4}}{3281} \right )} + \left (- \frac{1}{12} - \frac{\sqrt{3} i}{12}\right ) \log{\left (x + \frac{88573}{6562} + \frac{119042784 \left (- \frac{1}{12} - \frac{\sqrt{3} i}{12}\right )^{4}}{3281} + \frac{88573 \sqrt{3} i}{6562} \right )} + \operatorname{RootSum}{\left (1417176 t^{3} + 1, \left ( t \mapsto t \log{\left (\frac{119042784 t^{4}}{3281} - \frac{531438 t}{3281} + x \right )} \right )\right )} + \frac{10 x^{3} - 3}{45 x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(x + 1)/6 + (-1/12 + sqrt(3)*I/12)*log(x + 88573/6562 - 88573*sqrt(3)*I/6562 + 119042784*(-1/12 + sqrt(3)*I
/12)**4/3281) + (-1/12 - sqrt(3)*I/12)*log(x + 88573/6562 + 119042784*(-1/12 - sqrt(3)*I/12)**4/3281 + 88573*s
qrt(3)*I/6562) + RootSum(1417176*_t**3 + 1, Lambda(_t, _t*log(119042784*_t**4/3281 - 531438*_t/3281 + x))) + (
10*x**3 - 3)/(45*x**5)

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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^6/(x^6+4*x^3+3),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError