\(\int (-3+2 x+\log (\frac {9}{4 x^2})) \, dx\) [6733]

Optimal result

Integrand size = 13, antiderivative size = 17 \[ \int \left (-3+2 x+\log \left (\frac {9}{4 x^2}\right )\right ) \, dx=-1-x+x \left (x+\log \left (\frac {9}{4 x^2}\right )\right ) \]

[Out]

-1+(ln(9/4/x^2)+x)*x-x

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {2332} \[ \int \left (-3+2 x+\log \left (\frac {9}{4 x^2}\right )\right ) \, dx=x^2+x \log \left (\frac {9}{4 x^2}\right )-x \]

[In]

Int[-3 + 2*x + Log[9/(4*x^2)],x]

[Out]

-x + x^2 + x*Log[9/(4*x^2)]

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}& = -3 x+x^2+\int \log \left (\frac {9}{4 x^2}\right ) \, dx \\ & = -x+x^2+x \log \left (\frac {9}{4 x^2}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00 \[ \int \left (-3+2 x+\log \left (\frac {9}{4 x^2}\right )\right ) \, dx=-x+x^2+x \log \left (\frac {9}{4 x^2}\right ) \]

[In]

Integrate[-3 + 2*x + Log[9/(4*x^2)],x]

[Out]

-x + x^2 + x*Log[9/(4*x^2)]

Maple [A] (verified)

Time = 0.32 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.94

method	result	size
norman	\(x^{2}+x \ln \left (\frac {9}{4 x^{2}}\right )-x\)	\(16\)
risch	\(x^{2}+x \ln \left (\frac {9}{4 x^{2}}\right )-x\)	\(16\)
parallelrisch	\(x^{2}+x \ln \left (\frac {9}{4 x^{2}}\right )-x\)	\(16\)
derivativedivides	\(x \ln \left (\frac {1}{x^{2}}\right )-x -2 x \ln \left (2\right )+2 x \ln \left (3\right )+x^{2}\)	\(24\)
default	\(x \ln \left (\frac {1}{x^{2}}\right )-x -2 x \ln \left (2\right )+2 x \ln \left (3\right )+x^{2}\)	\(24\)
parts	\(x \ln \left (\frac {1}{x^{2}}\right )-x -2 x \ln \left (2\right )+2 x \ln \left (3\right )+x^{2}\)	\(24\)

[In]

int(ln(9/4/x^2)+2*x-3,x,method=_RETURNVERBOSE)

[Out]

x^2+x*ln(9/4/x^2)-x

Fricas [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.88 \[ \int \left (-3+2 x+\log \left (\frac {9}{4 x^2}\right )\right ) \, dx=x^{2} + x \log \left (\frac {9}{4 \, x^{2}}\right ) - x \]

[In]

integrate(log(9/4/x^2)+2*x-3,x, algorithm="fricas")

[Out]

x^2 + x*log(9/4/x^2) - x

Sympy [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.82 \[ \int \left (-3+2 x+\log \left (\frac {9}{4 x^2}\right )\right ) \, dx=x^{2} + x \log {\left (\frac {9}{4 x^{2}} \right )} - x \]

[In]

integrate(ln(9/4/x**2)+2*x-3,x)

[Out]

x**2 + x*log(9/(4*x**2)) - x

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.88 \[ \int \left (-3+2 x+\log \left (\frac {9}{4 x^2}\right )\right ) \, dx=x^{2} + x \log \left (\frac {9}{4 \, x^{2}}\right ) - x \]

[In]

integrate(log(9/4/x^2)+2*x-3,x, algorithm="maxima")

[Out]

x^2 + x*log(9/4/x^2) - x

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.88 \[ \int \left (-3+2 x+\log \left (\frac {9}{4 x^2}\right )\right ) \, dx=x^{2} + x \log \left (\frac {9}{4 \, x^{2}}\right ) - x \]

[In]

integrate(log(9/4/x^2)+2*x-3,x, algorithm="giac")

[Out]

x^2 + x*log(9/4/x^2) - x

Mupad [B] (verification not implemented)

Time = 17.84 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.82 \[ \int \left (-3+2 x+\log \left (\frac {9}{4 x^2}\right )\right ) \, dx=x^2+x\,\left (\ln \left (\frac {9}{4\,x^2}\right )-1\right ) \]

[In]

int(2*x + log(9/(4*x^2)) - 3,x)

[Out]

x^2 + x*(log(9/(4*x^2)) - 1)