3.137 \(\int \frac{1}{x (2+3 \log ^3(6 x))} \, dx\)

Optimal. Leaf size=111 \[ -\frac{\log \left (3^{2/3} \log ^2(6 x)-\sqrt [3]{6} \log (6 x)+2^{2/3}\right )}{6\ 2^{2/3} \sqrt [3]{3}}+\frac{\log \left (\sqrt [3]{3} \log (6 x)+\sqrt [3]{2}\right )}{3\ 2^{2/3} \sqrt [3]{3}}-\frac{\tan ^{-1}\left (\frac{1}{\sqrt{3}}-\frac{2^{2/3} \log (6 x)}{\sqrt [6]{3}}\right )}{2^{2/3} 3^{5/6}} \]

[Out]

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

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Rubi [A]  time = 0.0998283, antiderivative size = 111, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {200, 31, 634, 617, 204, 628} \[ -\frac{\log \left (3^{2/3} \log ^2(6 x)-\sqrt [3]{6} \log (6 x)+2^{2/3}\right )}{6\ 2^{2/3} \sqrt [3]{3}}+\frac{\log \left (\sqrt [3]{3} \log (6 x)+\sqrt [3]{2}\right )}{3\ 2^{2/3} \sqrt [3]{3}}-\frac{\tan ^{-1}\left (\frac{1}{\sqrt{3}}-\frac{2^{2/3} \log (6 x)}{\sqrt [6]{3}}\right )}{2^{2/3} 3^{5/6}} \]

Antiderivative was successfully verified.

[In]

Int[1/(x*(2 + 3*Log[6*x]^3)),x]

[Out]

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

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 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 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]

Rubi steps

\begin{align*} \int \frac{1}{x \left (2+3 \log ^3(6 x)\right )} \, dx &=\operatorname{Subst}\left (\int \frac{1}{2+3 x^3} \, dx,x,\log (6 x)\right )\\ &=\frac{\operatorname{Subst}\left (\int \frac{1}{\sqrt [3]{2}+\sqrt [3]{3} x} \, dx,x,\log (6 x)\right )}{3\ 2^{2/3}}+\frac{\operatorname{Subst}\left (\int \frac{2 \sqrt [3]{2}-\sqrt [3]{3} x}{2^{2/3}-\sqrt [3]{6} x+3^{2/3} x^2} \, dx,x,\log (6 x)\right )}{3\ 2^{2/3}}\\ &=\frac{\log \left (\sqrt [3]{2}+\sqrt [3]{3} \log (6 x)\right )}{3\ 2^{2/3} \sqrt [3]{3}}+\frac{\operatorname{Subst}\left (\int \frac{1}{2^{2/3}-\sqrt [3]{6} x+3^{2/3} x^2} \, dx,x,\log (6 x)\right )}{2 \sqrt [3]{2}}-\frac{\operatorname{Subst}\left (\int \frac{-\sqrt [3]{6}+2\ 3^{2/3} x}{2^{2/3}-\sqrt [3]{6} x+3^{2/3} x^2} \, dx,x,\log (6 x)\right )}{6\ 2^{2/3} \sqrt [3]{3}}\\ &=\frac{\log \left (\sqrt [3]{2}+\sqrt [3]{3} \log (6 x)\right )}{3\ 2^{2/3} \sqrt [3]{3}}-\frac{\log \left (2^{2/3}-\sqrt [3]{6} \log (6 x)+3^{2/3} \log ^2(6 x)\right )}{6\ 2^{2/3} \sqrt [3]{3}}+\frac{\operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1-2^{2/3} \sqrt [3]{3} \log (6 x)\right )}{2^{2/3} \sqrt [3]{3}}\\ &=-\frac{\tan ^{-1}\left (\frac{1-2^{2/3} \sqrt [3]{3} \log (6 x)}{\sqrt{3}}\right )}{2^{2/3} 3^{5/6}}+\frac{\log \left (\sqrt [3]{2}+\sqrt [3]{3} \log (6 x)\right )}{3\ 2^{2/3} \sqrt [3]{3}}-\frac{\log \left (2^{2/3}-\sqrt [3]{6} \log (6 x)+3^{2/3} \log ^2(6 x)\right )}{6\ 2^{2/3} \sqrt [3]{3}}\\ \end{align*}

Mathematica [A]  time = 0.0709949, size = 106, normalized size = 0.95 \[ \frac{\sqrt{3} \left (2 \log \left (2^{2/3} \sqrt [3]{3} \log (6 x)+2\right )-\log \left (\sqrt [3]{2} 3^{2/3} \log ^2(6 x)-2^{2/3} \sqrt [3]{3} \log (6 x)+2\right )\right )+6 \tan ^{-1}\left (\frac{2^{2/3} \sqrt [3]{3} \log (6 x)-1}{\sqrt{3}}\right )}{6\ 2^{2/3} 3^{5/6}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x*(2 + 3*Log[6*x]^3)),x]

[Out]

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

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Maple [A]  time = 0.007, size = 87, normalized size = 0.8 \begin{align*}{\frac{\sqrt [3]{2}{3}^{{\frac{2}{3}}}}{18}\ln \left ( \ln \left ( 6\,x \right ) +{\frac{\sqrt [3]{2}{3}^{{\frac{2}{3}}}}{3}} \right ) }-{\frac{\sqrt [3]{2}{3}^{{\frac{2}{3}}}}{36}\ln \left ( \left ( \ln \left ( 6\,x \right ) \right ) ^{2}-{\frac{\sqrt [3]{2}{3}^{{\frac{2}{3}}}\ln \left ( 6\,x \right ) }{3}}+{\frac{{2}^{{\frac{2}{3}}}\sqrt [3]{3}}{3}} \right ) }+{\frac{\sqrt [3]{2}\sqrt [6]{3}}{6}\arctan \left ({\frac{\sqrt{3} \left ({2}^{{\frac{2}{3}}}\sqrt [3]{3}\ln \left ( 6\,x \right ) -1 \right ) }{3}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x/(2+3*ln(6*x)^3),x)

[Out]

1/18*2^(1/3)*3^(2/3)*ln(ln(6*x)+1/3*2^(1/3)*3^(2/3))-1/36*2^(1/3)*3^(2/3)*ln(ln(6*x)^2-1/3*2^(1/3)*3^(2/3)*ln(
6*x)+1/3*2^(2/3)*3^(1/3))+1/6*2^(1/3)*3^(1/6)*arctan(1/3*3^(1/2)*(2^(2/3)*3^(1/3)*ln(6*x)-1))

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Maxima [A]  time = 1.59835, size = 131, normalized size = 1.18 \begin{align*} -\frac{1}{36} \cdot 3^{\frac{2}{3}} 2^{\frac{1}{3}} \log \left (3^{\frac{2}{3}} \log \left (6 \, x\right )^{2} - 3^{\frac{1}{3}} 2^{\frac{1}{3}} \log \left (6 \, x\right ) + 2^{\frac{2}{3}}\right ) + \frac{1}{18} \cdot 3^{\frac{2}{3}} 2^{\frac{1}{3}} \log \left (\frac{1}{3} \cdot 3^{\frac{2}{3}}{\left (3^{\frac{1}{3}} \log \left (6 \, x\right ) + 2^{\frac{1}{3}}\right )}\right ) + \frac{1}{6} \cdot 3^{\frac{1}{6}} 2^{\frac{1}{3}} \arctan \left (\frac{1}{6} \cdot 3^{\frac{1}{6}} 2^{\frac{2}{3}}{\left (2 \cdot 3^{\frac{2}{3}} \log \left (6 \, x\right ) - 3^{\frac{1}{3}} 2^{\frac{1}{3}}\right )}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(2+3*log(6*x)^3),x, algorithm="maxima")

[Out]

-1/36*3^(2/3)*2^(1/3)*log(3^(2/3)*log(6*x)^2 - 3^(1/3)*2^(1/3)*log(6*x) + 2^(2/3)) + 1/18*3^(2/3)*2^(1/3)*log(
1/3*3^(2/3)*(3^(1/3)*log(6*x) + 2^(1/3))) + 1/6*3^(1/6)*2^(1/3)*arctan(1/6*3^(1/6)*2^(2/3)*(2*3^(2/3)*log(6*x)
- 3^(1/3)*2^(1/3)))

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Fricas [A]  time = 2.02954, size = 239, normalized size = 2.15 \begin{align*} -\frac{1}{72} \cdot 12^{\frac{2}{3}} \log \left (6 \, \log \left (6 \, x\right )^{2} - 12^{\frac{2}{3}} \log \left (6 \, x\right ) + 2 \cdot 12^{\frac{1}{3}}\right ) + \frac{1}{36} \cdot 12^{\frac{2}{3}} \log \left (12^{\frac{2}{3}} + 6 \, \log \left (6 \, x\right )\right ) + \frac{1}{6} \cdot 12^{\frac{1}{6}} \arctan \left (\frac{1}{6} \cdot 12^{\frac{1}{6}}{\left (12^{\frac{2}{3}} \log \left (6 \, x\right ) - 12^{\frac{1}{3}}\right )}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(2+3*log(6*x)^3),x, algorithm="fricas")

[Out]

-1/72*12^(2/3)*log(6*log(6*x)^2 - 12^(2/3)*log(6*x) + 2*12^(1/3)) + 1/36*12^(2/3)*log(12^(2/3) + 6*log(6*x)) +
1/6*12^(1/6)*arctan(1/6*12^(1/6)*(12^(2/3)*log(6*x) - 12^(1/3)))

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Sympy [A]  time = 0.164076, size = 17, normalized size = 0.15 \begin{align*} \operatorname{RootSum}{\left (324 z^{3} - 1, \left ( i \mapsto i \log{\left (6 i + \log{\left (6 x \right )} \right )} \right )\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(2+3*ln(6*x)**3),x)

[Out]

RootSum(324*_z**3 - 1, Lambda(_i, _i*log(6*_i + log(6*x))))

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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/(2+3*log(6*x)^3),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError