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

Optimal. Leaf size=17 \[ -1-x+x \left (x+\log \left (\frac {9}{4 x^2}\right )\right ) \]

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 17, normalized size of antiderivative = 1.00, 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 = {2332} \begin {gather*} x^2+x \log \left (\frac {9}{4 x^2}\right )-x \end {gather*}

Antiderivative was successfully verified.

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

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Mathematica [A]
time = 0.00, size = 17, normalized size = 1.00 \begin {gather*} -x+x^2+x \log \left (\frac {9}{4 x^2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-x + x^2 + x*Log[9/(4*x^2)]

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Maple [A]
time = 1.54, size = 24, normalized size = 1.41

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.28, size = 15, normalized size = 0.88 \begin {gather*} x^{2} + x \log \left (\frac {9}{4 \, x^{2}}\right ) - x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^2 + x*log(9/4/x^2) - x

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Fricas [A]
time = 0.40, size = 15, normalized size = 0.88 \begin {gather*} x^{2} + x \log \left (\frac {9}{4 \, x^{2}}\right ) - x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^2 + x*log(9/4/x^2) - x

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Sympy [A]
time = 0.04, size = 14, normalized size = 0.82 \begin {gather*} x^{2} + x \log {\left (\frac {9}{4 x^{2}} \right )} - x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x**2 + x*log(9/(4*x**2)) - x

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Giac [A]
time = 0.40, size = 15, normalized size = 0.88 \begin {gather*} x^{2} + x \log \left (\frac {9}{4 \, x^{2}}\right ) - x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^2 + x*log(9/4/x^2) - x

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Mupad [B]
time = 3.98, size = 14, normalized size = 0.82 \begin {gather*} x^2+x\,\left (\ln \left (\frac {9}{4\,x^2}\right )-1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(2*x + log(9/(4*x^2)) - 3,x)

[Out]

x^2 + x*(log(9/(4*x^2)) - 1)

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