3.87.34 \(\int \frac {3-2 x+2 x \log (4)}{3 x-x^2+x^2 \log (4)} \, dx\)

Optimal. Leaf size=12 \[ \log (x (3-x+x \log (4))) \]

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Rubi [A]  time = 0.01, antiderivative size = 16, normalized size of antiderivative = 1.33, number of steps used = 3, number of rules used = 2, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {6, 628} \begin {gather*} \log \left (3 x-x^2 (1-\log (4))\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

Log[3*x - x^2*(1 - Log[4])]

Rule 6

Int[(u_.)*((w_.) + (a_.)*(v_) + (b_.)*(v_))^(p_.), x_Symbol] :> Int[u*((a + b)*v + w)^p, x] /; FreeQ[{a, b}, x
] &&  !FreeQ[v, x]

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {3+x (-2+2 \log (4))}{3 x-x^2+x^2 \log (4)} \, dx\\ &=\int \frac {3+x (-2+2 \log (4))}{3 x+x^2 (-1+\log (4))} \, dx\\ &=\log \left (3 x-x^2 (1-\log (4))\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 13, normalized size = 1.08 \begin {gather*} \log (x)+\log (3-x+x \log (4)) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

Log[x] + Log[3 - x + x*Log[4]]

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fricas [A]  time = 0.73, size = 17, normalized size = 1.42 \begin {gather*} \log \left (2 \, x^{2} \log \relax (2) - x^{2} + 3 \, x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(2*x^2*log(2) - x^2 + 3*x)

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giac [A]  time = 0.14, size = 18, normalized size = 1.50 \begin {gather*} \log \left ({\left | 2 \, x^{2} \log \relax (2) - x^{2} + 3 \, x \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(abs(2*x^2*log(2) - x^2 + 3*x))

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maple [A]  time = 0.40, size = 14, normalized size = 1.17

method	result	size
default	\(\ln \left (\left (3+2 x \ln \relax (2)-x \right ) x \right )\)	\(14\)
norman	\(\ln \relax (x )+\ln \left (3+2 x \ln \relax (2)-x \right )\)	\(15\)
risch	\(\ln \left (\left (2 \ln \relax (2)-1\right ) x^{2}+3 x \right )\)	\(16\)
derivativedivides	\(\ln \left (2 x^{2} \ln \relax (2)-x^{2}+3 x \right )\)	\(18\)
meijerg	\(\frac {3 \left (\frac {2 \ln \relax (2)}{3}-\frac {1}{3}\right ) \left (\ln \relax (x )-\ln \relax (3)+\ln \left (2 \ln \relax (2)-1\right )-\ln \left (1+\frac {\left (2 \ln \relax (2)-1\right ) x}{3}\right )\right )}{2 \ln \relax (2)-1}+\frac {3 \left (\frac {4 \ln \relax (2)}{3}-\frac {2}{3}\right ) \ln \left (1+\frac {\left (2 \ln \relax (2)-1\right ) x}{3}\right )}{2 \ln \relax (2)-1}\)	\(74\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.36, size = 17, normalized size = 1.42 \begin {gather*} \log \left (2 \, x^{2} \log \relax (2) - x^{2} + 3 \, x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(2*x^2*log(2) - x^2 + 3*x)

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mupad [B]  time = 5.29, size = 28, normalized size = 2.33 \begin {gather*} \ln \relax (x)+\ln \left (2\,x\,\ln \relax (2)-x+3\right )\,\left (\frac {\ln \left (16\right )-2}{\ln \relax (4)-1}-1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4*x*log(2) - 2*x + 3)/(3*x + 2*x^2*log(2) - x^2),x)

[Out]

log(x) + log(2*x*log(2) - x + 3)*((log(16) - 2)/(log(4) - 1) - 1)

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sympy [A]  time = 0.18, size = 14, normalized size = 1.17 \begin {gather*} \log {\left (x^{2} \left (-1 + 2 \log {\relax (2 )}\right ) + 3 x \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(x**2*(-1 + 2*log(2)) + 3*x)

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