\(\int \frac {-2 x+(-1-2 x) \log (3)}{x} \, dx\) [7990]

Optimal result

Integrand size = 16, antiderivative size = 20 \[ \int \frac {-2 x+(-1-2 x) \log (3)}{x} \, dx=-11-2 x-\log (3) \left (-7+2 x+\log \left (\frac {x}{3}\right )\right ) \]

[Out]

-(2*x-7+ln(1/3*x))*ln(3)-2*x-11

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.70, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {192, 45} \[ \int \frac {-2 x+(-1-2 x) \log (3)}{x} \, dx=-2 x (1+\log (3))-\log (3) \log (x) \]

[In]

Int[(-2*x + (-1 - 2*x)*Log[3])/x,x]

[Out]

-2*x*(1 + Log[3]) - Log[3]*Log[x]

Rule 45

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 192

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{align*} \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{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.70 \[ \int \frac {-2 x+(-1-2 x) \log (3)}{x} \, dx=-x (2+\log (9))-\log (3) \log (x) \]

[In]

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

[Out]

-(x*(2 + Log[9])) - Log[3]*Log[x]

Maple [A] (verified)

Time = 0.04 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.80

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

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.75 \[ \int \frac {-2 x+(-1-2 x) \log (3)}{x} \, dx=-2 \, x \log \left (3\right ) - \log \left (3\right ) \log \left (x\right ) - 2 \, x \]

[In]

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

[Out]

-2*x*log(3) - log(3)*log(x) - 2*x

Sympy [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.75 \[ \int \frac {-2 x+(-1-2 x) \log (3)}{x} \, dx=- x \left (2 + 2 \log {\left (3 \right )}\right ) - \log {\left (3 \right )} \log {\left (x \right )} \]

[In]

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

[Out]

-x*(2 + 2*log(3)) - log(3)*log(x)

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.70 \[ \int \frac {-2 x+(-1-2 x) \log (3)}{x} \, dx=-2 \, x {\left (\log \left (3\right ) + 1\right )} - \log \left (3\right ) \log \left (x\right ) \]

[In]

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

[Out]

-2*x*(log(3) + 1) - log(3)*log(x)

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.80 \[ \int \frac {-2 x+(-1-2 x) \log (3)}{x} \, dx=-2 \, x \log \left (3\right ) - \log \left (3\right ) \log \left ({\left | x \right |}\right ) - 2 \, x \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.70 \[ \int \frac {-2 x+(-1-2 x) \log (3)}{x} \, dx=-x\,\left (\ln \left (9\right )+2\right )-\ln \left (3\right )\,\ln \left (x\right ) \]

[In]

int(-(2*x + log(3)*(2*x + 1))/x,x)

[Out]

- x*(log(9) + 2) - log(3)*log(x)