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

Optimal. Leaf size=119 \[ -\frac {1}{6 x^2}+\frac {1}{12} \log \left (x^2-x+1\right )-\frac {\log \left (x^2-\sqrt [3]{3} x+3^{2/3}\right )}{36\ 3^{2/3}}-\frac {1}{6} \log (x+1)+\frac {\log \left (x+\sqrt [3]{3}\right )}{18\ 3^{2/3}}+\frac {\tan ^{-1}\left (\frac {1-2 x}{\sqrt {3}}\right )}{2 \sqrt {3}}-\frac {\tan ^{-1}\left (\frac {\sqrt [3]{3}-2 x}{3^{5/6}}\right )}{18 \sqrt [6]{3}} \]

[Out]

-1/6/x^2-1/54*3^(5/6)*arctan(1/3*(3^(1/3)-2*x)*3^(1/6))-1/6*ln(1+x)+1/54*3^(1/3)*ln(3^(1/3)+x)+1/12*ln(x^2-x+1
)-1/108*3^(1/3)*ln(3^(2/3)-3^(1/3)*x+x^2)+1/6*arctan(1/3*(1-2*x)*3^(1/2))*3^(1/2)

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Rubi [A]  time = 0.08, antiderivative size = 119, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 9, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.562, Rules used = {1368, 1422, 200, 31, 634, 618, 204, 628, 617} \[ -\frac {1}{6 x^2}+\frac {1}{12} \log \left (x^2-x+1\right )-\frac {\log \left (x^2-\sqrt [3]{3} x+3^{2/3}\right )}{36\ 3^{2/3}}-\frac {1}{6} \log (x+1)+\frac {\log \left (x+\sqrt [3]{3}\right )}{18\ 3^{2/3}}+\frac {\tan ^{-1}\left (\frac {1-2 x}{\sqrt {3}}\right )}{2 \sqrt {3}}-\frac {\tan ^{-1}\left (\frac {\sqrt [3]{3}-2 x}{3^{5/6}}\right )}{18 \sqrt [6]{3}} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^3*(3 + 4*x^3 + x^6)),x]

[Out]

-1/(6*x^2) + ArcTan[(1 - 2*x)/Sqrt[3]]/(2*Sqrt[3]) - ArcTan[(3^(1/3) - 2*x)/3^(5/6)]/(18*3^(1/6)) - Log[1 + x]
/6 + Log[3^(1/3) + x]/(18*3^(2/3)) + Log[1 - x + x^2]/12 - Log[3^(2/3) - 3^(1/3)*x + x^2]/(36*3^(2/3))

Rule 31

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

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 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 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]

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 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 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 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 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])

Rubi steps

\begin {align*} \int \frac {1}{x^3 \left (3+4 x^3+x^6\right )} \, dx &=-\frac {1}{6 x^2}+\frac {1}{6} \int \frac {-8-2 x^3}{3+4 x^3+x^6} \, dx\\ &=-\frac {1}{6 x^2}+\frac {1}{6} \int \frac {1}{3+x^3} \, dx-\frac {1}{2} \int \frac {1}{1+x^3} \, dx\\ &=-\frac {1}{6 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}{18\ 3^{2/3}}+\frac {\int \frac {2 \sqrt [3]{3}-x}{3^{2/3}-\sqrt [3]{3} x+x^2} \, dx}{18\ 3^{2/3}}\\ &=-\frac {1}{6 x^2}-\frac {1}{6} \log (1+x)+\frac {\log \left (\sqrt [3]{3}+x\right )}{18\ 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}{36\ 3^{2/3}}+\frac {\int \frac {1}{3^{2/3}-\sqrt [3]{3} x+x^2} \, dx}{12 \sqrt [3]{3}}\\ &=-\frac {1}{6 x^2}-\frac {1}{6} \log (1+x)+\frac {\log \left (\sqrt [3]{3}+x\right )}{18\ 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 )}{36\ 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 )}{6\ 3^{2/3}}\\ &=-\frac {1}{6 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 )}{18 \sqrt [6]{3}}-\frac {1}{6} \log (1+x)+\frac {\log \left (\sqrt [3]{3}+x\right )}{18\ 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 )}{36\ 3^{2/3}}\\ \end {align*}

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Mathematica [A]  time = 0.05, size = 113, normalized size = 0.95 \[ \frac {1}{108} \left (-\frac {18}{x^2}+9 \log \left (x^2-x+1\right )-\sqrt [3]{3} \log \left (\sqrt [3]{3} x^2-3^{2/3} x+3\right )-18 \log (x+1)+2 \sqrt [3]{3} \log \left (3^{2/3} x+3\right )-2\ 3^{5/6} \tan ^{-1}\left (\frac {\sqrt [3]{3}-2 x}{3^{5/6}}\right )-18 \sqrt {3} \tan ^{-1}\left (\frac {2 x-1}{\sqrt {3}}\right )\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 1.08, size = 126, normalized size = 1.06 \[ \frac {6 \cdot 9^{\frac {1}{6}} \sqrt {3} x^{2} \arctan \left (\frac {1}{27} \cdot 9^{\frac {1}{6}} {\left (2 \cdot 9^{\frac {2}{3}} \sqrt {3} x - 3 \cdot 9^{\frac {1}{3}} \sqrt {3}\right )}\right ) - 9^{\frac {2}{3}} x^{2} \log \left (3 \, x^{2} - 9^{\frac {2}{3}} x + 3 \cdot 9^{\frac {1}{3}}\right ) + 2 \cdot 9^{\frac {2}{3}} x^{2} \log \left (3 \, x + 9^{\frac {2}{3}}\right ) - 54 \, \sqrt {3} x^{2} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) + 27 \, x^{2} \log \left (x^{2} - x + 1\right ) - 54 \, x^{2} \log \left (x + 1\right ) - 54}{324 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/324*(6*9^(1/6)*sqrt(3)*x^2*arctan(1/27*9^(1/6)*(2*9^(2/3)*sqrt(3)*x - 3*9^(1/3)*sqrt(3))) - 9^(2/3)*x^2*log(
3*x^2 - 9^(2/3)*x + 3*9^(1/3)) + 2*9^(2/3)*x^2*log(3*x + 9^(2/3)) - 54*sqrt(3)*x^2*arctan(1/3*sqrt(3)*(2*x - 1
)) + 27*x^2*log(x^2 - x + 1) - 54*x^2*log(x + 1) - 54)/x^2

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giac [A]  time = 0.41, size = 91, normalized size = 0.76 \[ \frac {1}{54} \cdot 3^{\frac {5}{6}} \arctan \left (\frac {1}{3} \cdot 3^{\frac {1}{6}} {\left (2 \, x - 3^{\frac {1}{3}}\right )}\right ) - \frac {1}{6} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) - \frac {1}{108} \cdot 3^{\frac {1}{3}} \log \left (x^{2} - 3^{\frac {1}{3}} x + 3^{\frac {2}{3}}\right ) + \frac {1}{54} \cdot 3^{\frac {1}{3}} \log \left ({\left | x + 3^{\frac {1}{3}} \right |}\right ) - \frac {1}{6 \, x^{2}} + \frac {1}{12} \, \log \left (x^{2} - x + 1\right ) - \frac {1}{6} \, \log \left ({\left | x + 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/54*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)) - 1/108*3^(1/3)*l
og(x^2 - 3^(1/3)*x + 3^(2/3)) + 1/54*3^(1/3)*log(abs(x + 3^(1/3))) - 1/6/x^2 + 1/12*log(x^2 - x + 1) - 1/6*log
(abs(x + 1))

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maple [A]  time = 0.01, size = 89, normalized size = 0.75 \[ \frac {3^{\frac {5}{6}} \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \,3^{\frac {2}{3}} x}{3}-1\right )}{3}\right )}{54}-\frac {\sqrt {3}\, \arctan \left (\frac {\left (2 x -1\right ) \sqrt {3}}{3}\right )}{6}-\frac {\ln \left (x +1\right )}{6}+\frac {3^{\frac {1}{3}} \ln \left (x +3^{\frac {1}{3}}\right )}{54}-\frac {3^{\frac {1}{3}} \ln \left (x^{2}-3^{\frac {1}{3}} x +3^{\frac {2}{3}}\right )}{108}+\frac {\ln \left (x^{2}-x +1\right )}{12}-\frac {1}{6 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*ln(x+1)+1/54*3^(1/3)*ln(x+3^(1/3))-1/108*3^(1/3)*ln(x^2-3^(1/3)*x+3^(2/3))+1/54*3^(5/6)*arctan(1/3*3^(1/2
)*(2/3*3^(2/3)*x-1))-1/6/x^2+1/12*ln(x^2-x+1)-1/6*3^(1/2)*arctan(1/3*(2*x-1)*3^(1/2))

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maxima [A]  time = 1.27, size = 89, normalized size = 0.75 \[ \frac {1}{54} \cdot 3^{\frac {5}{6}} \arctan \left (\frac {1}{3} \cdot 3^{\frac {1}{6}} {\left (2 \, x - 3^{\frac {1}{3}}\right )}\right ) - \frac {1}{6} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) - \frac {1}{108} \cdot 3^{\frac {1}{3}} \log \left (x^{2} - 3^{\frac {1}{3}} x + 3^{\frac {2}{3}}\right ) + \frac {1}{54} \cdot 3^{\frac {1}{3}} \log \left (x + 3^{\frac {1}{3}}\right ) - \frac {1}{6 \, x^{2}} + \frac {1}{12} \, \log \left (x^{2} - x + 1\right ) - \frac {1}{6} \, \log \left (x + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/54*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)) - 1/108*3^(1/3)*l
og(x^2 - 3^(1/3)*x + 3^(2/3)) + 1/54*3^(1/3)*log(x + 3^(1/3)) - 1/6/x^2 + 1/12*log(x^2 - x + 1) - 1/6*log(x +
1)

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mupad [B]  time = 1.36, size = 118, normalized size = 0.99 \[ \frac {3^{1/3}\,\ln \left (x+3^{1/3}\right )}{54}-\frac {\ln \left (x+1\right )}{6}+\ln \left (x-\frac {1}{2}-\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{12}+\frac {\sqrt {3}\,1{}\mathrm {i}}{12}\right )-\ln \left (x-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (-\frac {1}{12}+\frac {\sqrt {3}\,1{}\mathrm {i}}{12}\right )-\frac {1}{6\,x^2}-\ln \left (x-\frac {3^{1/3}}{2}-\frac {3^{5/6}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {3^{1/3}}{108}+\frac {3^{5/6}\,1{}\mathrm {i}}{108}\right )-\ln \left (x-\frac {3^{1/3}}{2}+\frac {3^{5/6}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {3^{1/3}}{108}-\frac {3^{5/6}\,1{}\mathrm {i}}{108}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^3*(4*x^3 + x^6 + 3)),x)

[Out]

(3^(1/3)*log(x + 3^(1/3)))/54 - 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) - 1/(6*x^2) - log(x - 3^(1/3)/2 - (3^(5/6)*1i)/2)*(3^(1/3)/10
8 + (3^(5/6)*1i)/108) - log(x - 3^(1/3)/2 + (3^(5/6)*1i)/2)*(3^(1/3)/108 - (3^(5/6)*1i)/108)

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sympy [C]  time = 1.74, size = 128, normalized size = 1.08 \[ - \frac {\log {\left (x + 1 \right )}}{6} + \left (\frac {1}{12} - \frac {\sqrt {3} i}{12}\right ) \log {\left (x + \frac {1093}{244} - \frac {1093 \sqrt {3} i}{244} + \frac {787320 \left (\frac {1}{12} - \frac {\sqrt {3} i}{12}\right )^{4}}{61} \right )} + \left (\frac {1}{12} + \frac {\sqrt {3} i}{12}\right ) \log {\left (x + \frac {1093}{244} + \frac {787320 \left (\frac {1}{12} + \frac {\sqrt {3} i}{12}\right )^{4}}{61} + \frac {1093 \sqrt {3} i}{244} \right )} + \operatorname {RootSum} {\left (52488 t^{3} - 1, \left (t \mapsto t \log {\left (\frac {787320 t^{4}}{61} + \frac {3279 t}{61} + x \right )} \right )\right )} - \frac {1}{6 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(x + 1)/6 + (1/12 - sqrt(3)*I/12)*log(x + 1093/244 - 1093*sqrt(3)*I/244 + 787320*(1/12 - sqrt(3)*I/12)**4/
61) + (1/12 + sqrt(3)*I/12)*log(x + 1093/244 + 787320*(1/12 + sqrt(3)*I/12)**4/61 + 1093*sqrt(3)*I/244) + Root
Sum(52488*_t**3 - 1, Lambda(_t, _t*log(787320*_t**4/61 + 3279*_t/61 + x))) - 1/(6*x**2)

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