\(\int \frac {2-4 x-4 x \log (2 x)}{x} \, dx\) [7687]

Optimal result

Integrand size = 16, antiderivative size = 13 \[ \int \frac {2-4 x-4 x \log (2 x)}{x} \, dx=-\frac {2}{3} (-3+6 x) \log (2 x) \]

[Out]

-2/3*(-3+6*x)*ln(2*x)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.92, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {14, 45, 2332} \[ \int \frac {2-4 x-4 x \log (2 x)}{x} \, dx=2 \log (x)-4 x \log (2 x) \]

[In]

Int[(2 - 4*x - 4*x*Log[2*x])/x,x]

[Out]

2*Log[x] - 4*x*Log[2*x]

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 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 2332

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{c, n}, x]

Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {2 (-1+2 x)}{x}-4 \log (2 x)\right ) \, dx \\ & = -\left (2 \int \frac {-1+2 x}{x} \, dx\right )-4 \int \log (2 x) \, dx \\ & = 4 x-4 x \log (2 x)-2 \int \left (2-\frac {1}{x}\right ) \, dx \\ & = 2 \log (x)-4 x \log (2 x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.92 \[ \int \frac {2-4 x-4 x \log (2 x)}{x} \, dx=2 \log (x)-4 x \log (2 x) \]

[In]

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

[Out]

2*Log[x] - 4*x*Log[2*x]

Maple [A] (verified)

Time = 0.04 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00

method	result	size
risch	\(-4 x \ln \left (2 x \right )+2 \ln \left (x \right )\)	\(13\)
parts	\(-4 x \ln \left (2 x \right )+2 \ln \left (x \right )\)	\(13\)
derivativedivides	\(-4 x \ln \left (2 x \right )+2 \ln \left (2 x \right )\)	\(15\)
default	\(-4 x \ln \left (2 x \right )+2 \ln \left (2 x \right )\)	\(15\)
norman	\(-4 x \ln \left (2 x \right )+2 \ln \left (2 x \right )\)	\(15\)
parallelrisch	\(-4 x \ln \left (2 x \right )+2 \ln \left (2 x \right )\)	\(15\)

[In]

int((-4*x*ln(2*x)-4*x+2)/x,x,method=_RETURNVERBOSE)

[Out]

-4*x*ln(2*x)+2*ln(x)

Fricas [A] (verification not implemented)

none

Time = 0.38 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.85 \[ \int \frac {2-4 x-4 x \log (2 x)}{x} \, dx=-2 \, {\left (2 \, x - 1\right )} \log \left (2 \, x\right ) \]

[In]

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

[Out]

-2*(2*x - 1)*log(2*x)

Sympy [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.92 \[ \int \frac {2-4 x-4 x \log (2 x)}{x} \, dx=- 4 x \log {\left (2 x \right )} + 2 \log {\left (x \right )} \]

[In]

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

[Out]

-4*x*log(2*x) + 2*log(x)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.92 \[ \int \frac {2-4 x-4 x \log (2 x)}{x} \, dx=-4 \, x \log \left (2 \, x\right ) + 2 \, \log \left (x\right ) \]

[In]

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

[Out]

-4*x*log(2*x) + 2*log(x)

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.92 \[ \int \frac {2-4 x-4 x \log (2 x)}{x} \, dx=-4 \, x \log \left (2 \, x\right ) + 2 \, \log \left (x\right ) \]

[In]

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

[Out]

-4*x*log(2*x) + 2*log(x)

Mupad [B] (verification not implemented)

Time = 13.34 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.15 \[ \int \frac {2-4 x-4 x \log (2 x)}{x} \, dx=2\,\ln \left (x\right )-4\,x\,\ln \left (2\right )-4\,x\,\ln \left (x\right ) \]

[In]

int(-(4*x + 4*x*log(2*x) - 2)/x,x)

[Out]

2*log(x) - 4*x*log(2) - 4*x*log(x)