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

Optimal. Leaf size=41 \[ \frac{1}{2} \log \left (\log ^2(3 x)-\log (3 x)+1\right )+\frac{\tan ^{-1}\left (\frac{1-2 \log (3 x)}{\sqrt{3}}\right )}{\sqrt{3}} \]

[Out]

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

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Rubi [A]  time = 0.0575292, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {634, 618, 204, 628} \[ \frac{1}{2} \log \left (\log ^2(3 x)-\log (3 x)+1\right )+\frac{\tan ^{-1}\left (\frac{1-2 \log (3 x)}{\sqrt{3}}\right )}{\sqrt{3}} \]

Antiderivative was successfully verified.

[In]

Int[(-1 + Log[3*x])/(x*(1 - Log[3*x] + Log[3*x]^2)),x]

[Out]

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

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]

Rubi steps

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

Mathematica [A]  time = 0.0890602, size = 42, normalized size = 1.02 \[ \frac{1}{2} \log \left (\log ^2(3 x)-\log (3 x)+1\right )-\frac{\tan ^{-1}\left (\frac{2 \log (3 x)-1}{\sqrt{3}}\right )}{\sqrt{3}} \]

Antiderivative was successfully verified.

[In]

Integrate[(-1 + Log[3*x])/(x*(1 - Log[3*x] + Log[3*x]^2)),x]

[Out]

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

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Maple [A]  time = 0.005, size = 38, normalized size = 0.9 \begin{align*}{\frac{\ln \left ( 1-\ln \left ( 3\,x \right ) + \left ( \ln \left ( 3\,x \right ) \right ) ^{2} \right ) }{2}}-{\frac{\sqrt{3}}{3}\arctan \left ({\frac{ \left ( -1+2\,\ln \left ( 3\,x \right ) \right ) \sqrt{3}}{3}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.52494, size = 50, normalized size = 1.22 \begin{align*} -\frac{1}{3} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, \log \left (3 \, x\right ) - 1\right )}\right ) + \frac{1}{2} \, \log \left (\log \left (3 \, x\right )^{2} - \log \left (3 \, x\right ) + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*sqrt(3)*arctan(1/3*sqrt(3)*(2*log(3*x) - 1)) + 1/2*log(log(3*x)^2 - log(3*x) + 1)

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Fricas [A]  time = 1.84896, size = 127, normalized size = 3.1 \begin{align*} -\frac{1}{3} \, \sqrt{3} \arctan \left (\frac{2}{3} \, \sqrt{3} \log \left (3 \, x\right ) - \frac{1}{3} \, \sqrt{3}\right ) + \frac{1}{2} \, \log \left (\log \left (3 \, x\right )^{2} - \log \left (3 \, x\right ) + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*sqrt(3)*arctan(2/3*sqrt(3)*log(3*x) - 1/3*sqrt(3)) + 1/2*log(log(3*x)^2 - log(3*x) + 1)

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Sympy [A]  time = 0.178734, size = 22, normalized size = 0.54 \begin{align*} \operatorname{RootSum}{\left (3 z^{2} - 3 z + 1, \left ( i \mapsto i \log{\left (- 3 i + \log{\left (3 x \right )} + 1 \right )} \right )\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

RootSum(3*_z**2 - 3*_z + 1, Lambda(_i, _i*log(-3*_i + log(3*x) + 1)))

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Giac [A]  time = 1.23167, size = 50, normalized size = 1.22 \begin{align*} -\frac{1}{3} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, \log \left (3 \, x\right ) - 1\right )}\right ) + \frac{1}{2} \, \log \left (\log \left (3 \, x\right )^{2} - \log \left (3 \, x\right ) + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*sqrt(3)*arctan(1/3*sqrt(3)*(2*log(3*x) - 1)) + 1/2*log(log(3*x)^2 - log(3*x) + 1)