3.91.36 \(\int \frac {3-2 x-\log (13)}{(-3 x+x^2+x \log (13)) \log ^2(\frac {1}{3} (3 x-x^2-x \log (13)))} \, dx\)

Optimal. Leaf size=14 \[ \frac {1}{\log \left (x-\frac {1}{3} x (x+\log (13))\right )} \]

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Rubi [A]  time = 0.15, antiderivative size = 21, normalized size of antiderivative = 1.50, number of steps used = 3, number of rules used = 3, integrand size = 44, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.068, Rules used = {6, 1593, 6686} \begin {gather*} \frac {1}{\log \left (\frac {1}{3} \left (-x^2+3 x-x \log (13)\right )\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(3 - 2*x - Log[13])/((-3*x + x^2 + x*Log[13])*Log[(3*x - x^2 - x*Log[13])/3]^2),x]

[Out]

Log[(3*x - x^2 - x*Log[13])/3]^(-1)

Rule 6

Int[(u_.)*((w_.) + (a_.)*(v_) + (b_.)*(v_))^(p_.), x_Symbol] :> Int[u*((a + b)*v + w)^p, x] /; FreeQ[{a, b}, x
] &&  !FreeQ[v, x]

Rule 1593

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^(q - p))^n, x] /; F
reeQ[{a, b, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rule 6686

Int[(u_)*(y_)^(m_.), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[(q*y^(m + 1))/(m + 1), x] /;  !F
alseQ[q]] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {3-2 x-\log (13)}{\left (x^2+x (-3+\log (13))\right ) \log ^2\left (\frac {1}{3} \left (3 x-x^2-x \log (13)\right )\right )} \, dx\\ &=\int \frac {3-2 x-\log (13)}{x (-3+x+\log (13)) \log ^2\left (\frac {1}{3} \left (3 x-x^2-x \log (13)\right )\right )} \, dx\\ &=\frac {1}{\log \left (\frac {1}{3} \left (3 x-x^2-x \log (13)\right )\right )}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.02, size = 13, normalized size = 0.93 \begin {gather*} \frac {1}{\log \left (-\frac {1}{3} x (-3+x+\log (13))\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(3 - 2*x - Log[13])/((-3*x + x^2 + x*Log[13])*Log[(3*x - x^2 - x*Log[13])/3]^2),x]

[Out]

Log[-1/3*(x*(-3 + x + Log[13]))]^(-1)

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fricas [A]  time = 0.47, size = 15, normalized size = 1.07 \begin {gather*} \frac {1}{\log \left (-\frac {1}{3} \, x^{2} - \frac {1}{3} \, x \log \left (13\right ) + x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-log(13)+3-2*x)/(x*log(13)+x^2-3*x)/log(-1/3*x*log(13)-1/3*x^2+x)^2,x, algorithm="fricas")

[Out]

1/log(-1/3*x^2 - 1/3*x*log(13) + x)

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giac [A]  time = 0.20, size = 24, normalized size = 1.71 \begin {gather*} -\frac {1}{\log \relax (3) - \log \left (-x^{2} - x \log \left (13\right ) + 3 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-log(13)+3-2*x)/(x*log(13)+x^2-3*x)/log(-1/3*x*log(13)-1/3*x^2+x)^2,x, algorithm="giac")

[Out]

-1/(log(3) - log(-x^2 - x*log(13) + 3*x))

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maple [A]  time = 0.50, size = 16, normalized size = 1.14

method	result	size
norman	\(\frac {1}{\ln \left (-\frac {x \ln \left (13\right )}{3}-\frac {x^{2}}{3}+x \right )}\)	\(16\)
risch	\(\frac {1}{\ln \left (-\frac {x \ln \left (13\right )}{3}-\frac {x^{2}}{3}+x \right )}\)	\(16\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-ln(13)+3-2*x)/(x*ln(13)+x^2-3*x)/ln(-1/3*x*ln(13)-1/3*x^2+x)^2,x,method=_RETURNVERBOSE)

[Out]

1/ln(-1/3*x*ln(13)-1/3*x^2+x)

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maxima [A]  time = 0.47, size = 23, normalized size = 1.64 \begin {gather*} -\frac {1}{\log \relax (3) - \log \relax (x) - \log \left (-x - \log \left (13\right ) + 3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-log(13)+3-2*x)/(x*log(13)+x^2-3*x)/log(-1/3*x*log(13)-1/3*x^2+x)^2,x, algorithm="maxima")

[Out]

-1/(log(3) - log(x) - log(-x - log(13) + 3))

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mupad [B]  time = 9.20, size = 15, normalized size = 1.07 \begin {gather*} \frac {1}{\ln \left (x-\frac {x\,\ln \left (13\right )}{3}-\frac {x^2}{3}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(2*x + log(13) - 3)/(log(x - (x*log(13))/3 - x^2/3)^2*(x*log(13) - 3*x + x^2)),x)

[Out]

1/log(x - (x*log(13))/3 - x^2/3)

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sympy [A]  time = 0.15, size = 15, normalized size = 1.07 \begin {gather*} \frac {1}{\log {\left (- \frac {x^{2}}{3} - \frac {x \log {\left (13 \right )}}{3} + x \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-ln(13)+3-2*x)/(x*ln(13)+x**2-3*x)/ln(-1/3*x*ln(13)-1/3*x**2+x)**2,x)

[Out]

1/log(-x**2/3 - x*log(13)/3 + x)

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