3.148 \(\int \frac {1-4 \log (x)+\log ^2(x)}{x (-1+\log (x))^4} \, dx\)

Optimal. Leaf size=27 \[ \frac {1}{(\log (x)-1)^2}+\frac {1}{1-\log (x)}-\frac {2}{3 (1-\log (x))^3} \]

[Out]

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

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Rubi [A]  time = 0.05, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {698} \[ \frac {1}{(\log (x)-1)^2}+\frac {1}{1-\log (x)}-\frac {2}{3 (1-\log (x))^3} \]

Antiderivative was successfully verified.

[In]

Int[(1 - 4*Log[x] + Log[x]^2)/(x*(-1 + Log[x])^4),x]

[Out]

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

Rule 698

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d +
e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*
e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps

\begin {align*} \int \frac {1-4 \log (x)+\log ^2(x)}{x (-1+\log (x))^4} \, dx &=\operatorname {Subst}\left (\int \frac {1-4 x+x^2}{(-1+x)^4} \, dx,x,\log (x)\right )\\ &=\operatorname {Subst}\left (\int \left (-\frac {2}{(-1+x)^4}-\frac {2}{(-1+x)^3}+\frac {1}{(-1+x)^2}\right ) \, dx,x,\log (x)\right )\\ &=-\frac {2}{3 (1-\log (x))^3}+\frac {1}{1-\log (x)}+\frac {1}{(-1+\log (x))^2}\\ \end {align*}

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Mathematica [A]  time = 0.04, size = 22, normalized size = 0.81 \[ \frac {-3 \log ^2(x)+9 \log (x)-4}{3 (\log (x)-1)^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(1 - 4*Log[x] + Log[x]^2)/(x*(-1 + Log[x])^4),x]

[Out]

(-4 + 9*Log[x] - 3*Log[x]^2)/(3*(-1 + Log[x])^3)

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fricas [A]  time = 2.13, size = 32, normalized size = 1.19 \[ -\frac {3 \, \log \relax (x)^{2} - 9 \, \log \relax (x) + 4}{3 \, {\left (\log \relax (x)^{3} - 3 \, \log \relax (x)^{2} + 3 \, \log \relax (x) - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-4*log(x)+log(x)^2)/x/(-1+log(x))^4,x, algorithm="fricas")

[Out]

-1/3*(3*log(x)^2 - 9*log(x) + 4)/(log(x)^3 - 3*log(x)^2 + 3*log(x) - 1)

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giac [A]  time = 0.19, size = 20, normalized size = 0.74 \[ -\frac {3 \, \log \relax (x)^{2} - 9 \, \log \relax (x) + 4}{3 \, {\left (\log \relax (x) - 1\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-4*log(x)+log(x)^2)/x/(-1+log(x))^4,x, algorithm="giac")

[Out]

-1/3*(3*log(x)^2 - 9*log(x) + 4)/(log(x) - 1)^3

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maple [A]  time = 0.07, size = 24, normalized size = 0.89 \[ -\frac {1}{\ln \relax (x )-1}+\frac {1}{\left (\ln \relax (x )-1\right )^{2}}+\frac {2}{3 \left (\ln \relax (x )-1\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1-4*ln(x)+ln(x)^2)/x/(-1+ln(x))^4,x)

[Out]

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

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maxima [A]  time = 0.55, size = 32, normalized size = 1.19 \[ -\frac {3 \, \log \relax (x)^{2} - 9 \, \log \relax (x) + 4}{3 \, {\left (\log \relax (x)^{3} - 3 \, \log \relax (x)^{2} + 3 \, \log \relax (x) - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-4*log(x)+log(x)^2)/x/(-1+log(x))^4,x, algorithm="maxima")

[Out]

-1/3*(3*log(x)^2 - 9*log(x) + 4)/(log(x)^3 - 3*log(x)^2 + 3*log(x) - 1)

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mupad [B]  time = 0.42, size = 18, normalized size = 0.67 \[ -\frac {{\ln \relax (x)}^2-3\,\ln \relax (x)+\frac {4}{3}}{{\left (\ln \relax (x)-1\right )}^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(x)^2 - 4*log(x) + 1)/(x*(log(x) - 1)^4),x)

[Out]

-(log(x)^2 - 3*log(x) + 4/3)/(log(x) - 1)^3

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sympy [A]  time = 0.13, size = 32, normalized size = 1.19 \[ \frac {- 3 \log {\relax (x )}^{2} + 9 \log {\relax (x )} - 4}{3 \log {\relax (x )}^{3} - 9 \log {\relax (x )}^{2} + 9 \log {\relax (x )} - 3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-4*ln(x)+ln(x)**2)/x/(-1+ln(x))**4,x)

[Out]

(-3*log(x)**2 + 9*log(x) - 4)/(3*log(x)**3 - 9*log(x)**2 + 9*log(x) - 3)

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