3.61.3 \(\int \frac {128-192 x-16 x^3}{(16 x-48 x^2+21 x^3+4 x^4) \log ^2(\frac {-16+48 x-21 x^2-4 x^3}{9 x^2})} \, dx\)

Optimal. Leaf size=32 \[ \frac {4}{\log \left (\frac {1}{3} \left (5+\frac {1}{3} \left (-\left (5+\frac {-4+x}{x}\right )^2-4 x\right )\right )\right )} \]

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Rubi [A]  time = 0.09, antiderivative size = 27, normalized size of antiderivative = 0.84, number of steps used = 1, number of rules used = 1, integrand size = 57, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.018, Rules used = {6686} \begin {gather*} \frac {4}{\log \left (-\frac {4 x^3+21 x^2-48 x+16}{9 x^2}\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(128 - 192*x - 16*x^3)/((16*x - 48*x^2 + 21*x^3 + 4*x^4)*Log[(-16 + 48*x - 21*x^2 - 4*x^3)/(9*x^2)]^2),x]

[Out]

4/Log[-1/9*(16 - 48*x + 21*x^2 + 4*x^3)/x^2]

Rule 6686

Int[(u_)*(y_)^(m_.), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[(q*y^(m + 1))/(m + 1), x] /;  !F
alseQ[q]] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {4}{\log \left (-\frac {16-48 x+21 x^2+4 x^3}{9 x^2}\right )}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.02, size = 27, normalized size = 0.84 \begin {gather*} \frac {4}{\log \left (-\frac {16-48 x+21 x^2+4 x^3}{9 x^2}\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(128 - 192*x - 16*x^3)/((16*x - 48*x^2 + 21*x^3 + 4*x^4)*Log[(-16 + 48*x - 21*x^2 - 4*x^3)/(9*x^2)]^
2),x]

[Out]

4/Log[-1/9*(16 - 48*x + 21*x^2 + 4*x^3)/x^2]

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fricas [A]  time = 0.62, size = 25, normalized size = 0.78 \begin {gather*} \frac {4}{\log \left (-\frac {4 \, x^{3} + 21 \, x^{2} - 48 \, x + 16}{9 \, x^{2}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-16*x^3-192*x+128)/(4*x^4+21*x^3-48*x^2+16*x)/log(1/9*(-4*x^3-21*x^2+48*x-16)/x^2)^2,x, algorithm="
fricas")

[Out]

4/log(-1/9*(4*x^3 + 21*x^2 - 48*x + 16)/x^2)

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giac [A]  time = 0.18, size = 25, normalized size = 0.78 \begin {gather*} \frac {4}{\log \left (-\frac {4 \, x^{3} + 21 \, x^{2} - 48 \, x + 16}{9 \, x^{2}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-16*x^3-192*x+128)/(4*x^4+21*x^3-48*x^2+16*x)/log(1/9*(-4*x^3-21*x^2+48*x-16)/x^2)^2,x, algorithm="
giac")

[Out]

4/log(-1/9*(4*x^3 + 21*x^2 - 48*x + 16)/x^2)

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maple [A]  time = 0.04, size = 26, normalized size = 0.81

method	result	size
norman	\(\frac {4}{\ln \left (\frac {-4 x^{3}-21 x^{2}+48 x -16}{9 x^{2}}\right )}\)	\(26\)
risch	\(\frac {4}{\ln \left (\frac {-4 x^{3}-21 x^{2}+48 x -16}{9 x^{2}}\right )}\)	\(26\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-16*x^3-192*x+128)/(4*x^4+21*x^3-48*x^2+16*x)/ln(1/9*(-4*x^3-21*x^2+48*x-16)/x^2)^2,x,method=_RETURNVERBO
SE)

[Out]

4/ln(1/9*(-4*x^3-21*x^2+48*x-16)/x^2)

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maxima [A]  time = 0.47, size = 31, normalized size = 0.97 \begin {gather*} -\frac {4}{2 \, \log \relax (3) - \log \left (-4 \, x^{3} - 21 \, x^{2} + 48 \, x - 16\right ) + 2 \, \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-16*x^3-192*x+128)/(4*x^4+21*x^3-48*x^2+16*x)/log(1/9*(-4*x^3-21*x^2+48*x-16)/x^2)^2,x, algorithm="
maxima")

[Out]

-4/(2*log(3) - log(-4*x^3 - 21*x^2 + 48*x - 16) + 2*log(x))

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mupad [B]  time = 4.52, size = 25, normalized size = 0.78 \begin {gather*} \frac {4}{\ln \left (-\frac {4\,x^3+21\,x^2-48\,x+16}{9\,x^2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(192*x + 16*x^3 - 128)/(log(-((7*x^2)/3 - (16*x)/3 + (4*x^3)/9 + 16/9)/x^2)^2*(16*x - 48*x^2 + 21*x^3 + 4
*x^4)),x)

[Out]

4/log(-(21*x^2 - 48*x + 4*x^3 + 16)/(9*x^2))

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sympy [A]  time = 0.24, size = 27, normalized size = 0.84 \begin {gather*} \frac {4}{\log {\left (\frac {- \frac {4 x^{3}}{9} - \frac {7 x^{2}}{3} + \frac {16 x}{3} - \frac {16}{9}}{x^{2}} \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-16*x**3-192*x+128)/(4*x**4+21*x**3-48*x**2+16*x)/ln(1/9*(-4*x**3-21*x**2+48*x-16)/x**2)**2,x)

[Out]

4/log((-4*x**3/9 - 7*x**2/3 + 16*x/3 - 16/9)/x**2)

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