3.71.49 \(\int \frac {-18+x \log (3)-2 \log (x)}{x} \, dx\)

Optimal. Leaf size=16 \[ 4+(4+x) \log (3)-(9+\log (x))^2 \]

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Rubi [A]  time = 0.02, antiderivative size = 15, normalized size of antiderivative = 0.94, number of steps used = 5, number of rules used = 3, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {14, 43, 2301} \begin {gather*} -\log ^2(x)-18 \log (x)+x \log (3) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-18 + x*Log[3] - 2*Log[x])/x,x]

[Out]

x*Log[3] - 18*Log[x] - Log[x]^2

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 2301

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/(x_), x_Symbol] :> Simp[(a + b*Log[c*x^n])^2/(2*b*n), x] /; FreeQ[{a
, b, c, n}, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {-18+x \log (3)}{x}-\frac {2 \log (x)}{x}\right ) \, dx\\ &=-\left (2 \int \frac {\log (x)}{x} \, dx\right )+\int \frac {-18+x \log (3)}{x} \, dx\\ &=-\log ^2(x)+\int \left (-\frac {18}{x}+\log (3)\right ) \, dx\\ &=x \log (3)-18 \log (x)-\log ^2(x)\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 15, normalized size = 0.94 \begin {gather*} x \log (3)-18 \log (x)-\log ^2(x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-18 + x*Log[3] - 2*Log[x])/x,x]

[Out]

x*Log[3] - 18*Log[x] - Log[x]^2

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fricas [A]  time = 0.57, size = 15, normalized size = 0.94 \begin {gather*} x \log \relax (3) - \log \relax (x)^{2} - 18 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*log(x)+x*log(3)-18)/x,x, algorithm="fricas")

[Out]

x*log(3) - log(x)^2 - 18*log(x)

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giac [A]  time = 0.19, size = 15, normalized size = 0.94 \begin {gather*} x \log \relax (3) - \log \relax (x)^{2} - 18 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*log(x)+x*log(3)-18)/x,x, algorithm="giac")

[Out]

x*log(3) - log(x)^2 - 18*log(x)

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maple [A]  time = 0.02, size = 16, normalized size = 1.00

method	result	size
default	\(x \ln \relax (3)-\ln \relax (x )^{2}-18 \ln \relax (x )\)	\(16\)
norman	\(x \ln \relax (3)-\ln \relax (x )^{2}-18 \ln \relax (x )\)	\(16\)
risch	\(x \ln \relax (3)-\ln \relax (x )^{2}-18 \ln \relax (x )\)	\(16\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-2*ln(x)+x*ln(3)-18)/x,x,method=_RETURNVERBOSE)

[Out]

x*ln(3)-ln(x)^2-18*ln(x)

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maxima [A]  time = 0.37, size = 15, normalized size = 0.94 \begin {gather*} x \log \relax (3) - \log \relax (x)^{2} - 18 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*log(x)+x*log(3)-18)/x,x, algorithm="maxima")

[Out]

x*log(3) - log(x)^2 - 18*log(x)

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mupad [B]  time = 4.15, size = 15, normalized size = 0.94 \begin {gather*} -{\ln \relax (x)}^2-18\,\ln \relax (x)+x\,\ln \relax (3) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(2*log(x) - x*log(3) + 18)/x,x)

[Out]

x*log(3) - 18*log(x) - log(x)^2

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sympy [A]  time = 0.11, size = 14, normalized size = 0.88 \begin {gather*} x \log {\relax (3 )} - \log {\relax (x )}^{2} - 18 \log {\relax (x )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*ln(x)+x*ln(3)-18)/x,x)

[Out]

x*log(3) - log(x)**2 - 18*log(x)

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