3.77.87 \(\int \frac {2-4 x-4 x \log (2 x)}{x} \, dx\)

Optimal. Leaf size=13 \[ -\frac {2}{3} (-3+6 x) \log (2 x) \]

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Rubi [A]  time = 0.01, antiderivative size = 12, normalized size of antiderivative = 0.92, number of steps used = 5, number of rules used = 3, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {14, 43, 2295} \begin {gather*} 2 \log (x)-4 x \log (2 x) \end {gather*}

Antiderivative was successfully verified.

[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 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 2295

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 {gather*} \begin {aligned} \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 {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 12, normalized size = 0.92 \begin {gather*} 2 \log (x)-4 x \log (2 x) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

2*Log[x] - 4*x*Log[2*x]

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fricas [A]  time = 0.67, size = 11, normalized size = 0.85 \begin {gather*} -2 \, {\left (2 \, x - 1\right )} \log \left (2 \, x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*(2*x - 1)*log(2*x)

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giac [A]  time = 0.24, size = 12, normalized size = 0.92 \begin {gather*} -4 \, x \log \left (2 \, x\right ) + 2 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4*x*log(2*x) + 2*log(x)

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maple [A]  time = 0.02, size = 13, normalized size = 1.00

method	result	size
risch	\(-4 x \ln \left (2 x \right )+2 \ln \relax (x )\)	\(13\)
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	\(-4 x \ln \left (2 x \right )+2 \ln \left (2 x \right )\)	\(15\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.36, size = 12, normalized size = 0.92 \begin {gather*} -4 \, x \log \left (2 \, x\right ) + 2 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4*x*log(2*x) + 2*log(x)

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mupad [B]  time = 4.95, size = 15, normalized size = 1.15 \begin {gather*} 2\,\ln \relax (x)-4\,x\,\ln \relax (2)-4\,x\,\ln \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(4*x + 4*x*log(2*x) - 2)/x,x)

[Out]

2*log(x) - 4*x*log(2) - 4*x*log(x)

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sympy [A]  time = 0.09, size = 12, normalized size = 0.92 \begin {gather*} - 4 x \log {\left (2 x \right )} + 2 \log {\relax (x )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4*x*log(2*x) + 2*log(x)

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