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

Optimal. Leaf size=20 \[ -11-2 x-\log (3) \left (-7+2 x+\log \left (\frac {x}{3}\right )\right ) \]

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Rubi [A]  time = 0.01, antiderivative size = 14, normalized size of antiderivative = 0.70, number of steps used = 3, number of rules used = 2, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {186, 43} \begin {gather*} -2 x (1+\log (3))-\log (3) \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-2*x + (-1 - 2*x)*Log[3])/x,x]

[Out]

-2*x*(1 + Log[3]) - Log[3]*Log[x]

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 186

Int[(u_)^(m_.)*(v_)^(n_.), x_Symbol] :> Int[ExpandToSum[u, x]^m*ExpandToSum[v, x]^n, x] /; FreeQ[{m, n}, x] &&
LinearQ[{u, v}, x] &&  !LinearMatchQ[{u, v}, x]

Rubi steps

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

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Mathematica [A]  time = 0.01, size = 14, normalized size = 0.70 \begin {gather*} -x (2+\log (9))-\log (3) \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-(x*(2 + Log[9])) - Log[3]*Log[x]

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fricas [A]  time = 0.56, size = 15, normalized size = 0.75 \begin {gather*} -2 \, x \log \relax (3) - \log \relax (3) \log \relax (x) - 2 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*x*log(3) - log(3)*log(x) - 2*x

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giac [A]  time = 0.19, size = 16, normalized size = 0.80 \begin {gather*} -2 \, x \log \relax (3) - \log \relax (3) \log \left ({\left | x \right |}\right ) - 2 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*x*log(3) - log(3)*log(abs(x)) - 2*x

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maple [A]  time = 0.03, size = 16, normalized size = 0.80

method	result	size
default	\(-2 x \ln \relax (3)-2 x -\ln \relax (3) \ln \relax (x )\)	\(16\)
norman	\(\left (-2 \ln \relax (3)-2\right ) x -\ln \relax (3) \ln \relax (x )\)	\(16\)
risch	\(-2 x \ln \relax (3)-2 x -\ln \relax (3) \ln \relax (x )\)	\(16\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.37, size = 14, normalized size = 0.70 \begin {gather*} -2 \, x {\left (\log \relax (3) + 1\right )} - \log \relax (3) \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*x*(log(3) + 1) - log(3)*log(x)

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mupad [B]  time = 0.05, size = 14, normalized size = 0.70 \begin {gather*} -x\,\left (\ln \relax (9)+2\right )-\ln \relax (3)\,\ln \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(2*x + log(3)*(2*x + 1))/x,x)

[Out]

- x*(log(9) + 2) - log(3)*log(x)

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sympy [A]  time = 0.09, size = 15, normalized size = 0.75 \begin {gather*} - x \left (2 + 2 \log {\relax (3 )}\right ) - \log {\relax (3 )} \log {\relax (x )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x*(2 + 2*log(3)) - log(3)*log(x)

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