3.16.51 \(\int (-3-6 x-3 \log (\frac {x}{3})) \, dx\)

Optimal. Leaf size=13 \[ 4-3 x \left (x+\log \left (\frac {x}{3}\right )\right ) \]

________________________________________________________________________________________

Rubi [A]  time = 0.00, antiderivative size = 15, normalized size of antiderivative = 1.15, number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {2295} \begin {gather*} -3 x^2-3 x \log \left (\frac {x}{3}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[-3 - 6*x - 3*Log[x/3],x]

[Out]

-3*x^2 - 3*x*Log[x/3]

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

________________________________________________________________________________________

Mathematica [A]  time = 0.00, size = 15, normalized size = 1.15 \begin {gather*} -3 x^2-3 x \log \left (\frac {x}{3}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[-3 - 6*x - 3*Log[x/3],x]

[Out]

-3*x^2 - 3*x*Log[x/3]

________________________________________________________________________________________

fricas [A]  time = 1.15, size = 13, normalized size = 1.00 \begin {gather*} -3 \, x^{2} - 3 \, x \log \left (\frac {1}{3} \, x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-3*log(1/3*x)-6*x-3,x, algorithm="fricas")

[Out]

-3*x^2 - 3*x*log(1/3*x)

________________________________________________________________________________________

giac [A]  time = 0.14, size = 13, normalized size = 1.00 \begin {gather*} -3 \, x^{2} - 3 \, x \log \left (\frac {1}{3} \, x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-3*log(1/3*x)-6*x-3,x, algorithm="giac")

[Out]

-3*x^2 - 3*x*log(1/3*x)

________________________________________________________________________________________

maple [A]  time = 0.02, size = 14, normalized size = 1.08

method	result	size
derivativedivides	\(-3 x^{2}-3 x \ln \left (\frac {x}{3}\right )\)	\(14\)
default	\(-3 x^{2}-3 x \ln \left (\frac {x}{3}\right )\)	\(14\)
norman	\(-3 x^{2}-3 x \ln \left (\frac {x}{3}\right )\)	\(14\)
risch	\(-3 x^{2}-3 x \ln \left (\frac {x}{3}\right )\)	\(14\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-3*ln(1/3*x)-6*x-3,x,method=_RETURNVERBOSE)

[Out]

-3*x^2-3*x*ln(1/3*x)

________________________________________________________________________________________

maxima [A]  time = 0.54, size = 13, normalized size = 1.00 \begin {gather*} -3 \, x^{2} - 3 \, x \log \left (\frac {1}{3} \, x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-3*log(1/3*x)-6*x-3,x, algorithm="maxima")

[Out]

-3*x^2 - 3*x*log(1/3*x)

________________________________________________________________________________________

mupad [B]  time = 1.00, size = 9, normalized size = 0.69 \begin {gather*} -3\,x\,\left (x+\ln \left (\frac {x}{3}\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(- 6*x - 3*log(x/3) - 3,x)

[Out]

-3*x*(x + log(x/3))

________________________________________________________________________________________

sympy [A]  time = 0.08, size = 14, normalized size = 1.08 \begin {gather*} - 3 x^{2} - 3 x \log {\left (\frac {x}{3} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-3*ln(1/3*x)-6*x-3,x)

[Out]

-3*x**2 - 3*x*log(x/3)

________________________________________________________________________________________