\(\int \frac {-144+72 x^2}{(48 x-71 x^2+24 x^3) \log ^2(\frac {2 x}{144-213 x+72 x^2})} \, dx\) [7466]

Optimal result

Integrand size = 42, antiderivative size = 26 \[ \int \frac {-144+72 x^2}{\left (48 x-71 x^2+24 x^3\right ) \log ^2\left (\frac {2 x}{144-213 x+72 x^2}\right )} \, dx=\frac {3}{\log \left (\frac {1}{\frac {3}{2}+\frac {36 (1-x) (2-x)}{x}}\right )} \]

[Out]

3/ln(1/(3/2+36*(1-x)/x*(2-x)))

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.85, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {1608, 6818} \[ \int \frac {-144+72 x^2}{\left (48 x-71 x^2+24 x^3\right ) \log ^2\left (\frac {2 x}{144-213 x+72 x^2}\right )} \, dx=\frac {3}{\log \left (\frac {2 x}{3 \left (24 x^2-71 x+48\right )}\right )} \]

[In]

Int[(-144 + 72*x^2)/((48*x - 71*x^2 + 24*x^3)*Log[(2*x)/(144 - 213*x + 72*x^2)]^2),x]

[Out]

3/Log[(2*x)/(3*(48 - 71*x + 24*x^2))]

Rule 1608

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.) + (c_.)*(x_)^(r_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^
(q - p) + c*x^(r - p))^n, x] /; FreeQ[{a, b, c, p, q, r}, x] && IntegerQ[n] && PosQ[q - p] && PosQ[r - p]

Rule 6818

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{align*} \text {integral}& = \int \frac {-144+72 x^2}{x \left (48-71 x+24 x^2\right ) \log ^2\left (\frac {2 x}{144-213 x+72 x^2}\right )} \, dx \\ & = \frac {3}{\log \left (\frac {2 x}{3 \left (48-71 x+24 x^2\right )}\right )} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.77 \[ \int \frac {-144+72 x^2}{\left (48 x-71 x^2+24 x^3\right ) \log ^2\left (\frac {2 x}{144-213 x+72 x^2}\right )} \, dx=\frac {3}{\log \left (\frac {2 x}{144-213 x+72 x^2}\right )} \]

[In]

Integrate[(-144 + 72*x^2)/((48*x - 71*x^2 + 24*x^3)*Log[(2*x)/(144 - 213*x + 72*x^2)]^2),x]

[Out]

3/Log[(2*x)/(144 - 213*x + 72*x^2)]

Maple [A] (verified)

Time = 0.06 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.81

method	result	size
norman	\(\frac {3}{\ln \left (\frac {2 x}{72 x^{2}-213 x +144}\right )}\)	\(21\)
risch	\(\frac {3}{\ln \left (\frac {2 x}{72 x^{2}-213 x +144}\right )}\)	\(21\)
parallelrisch	\(\frac {3}{\ln \left (\frac {2 x}{3 \left (24 x^{2}-71 x +48\right )}\right )}\)	\(21\)
default	\(\frac {3}{\ln \left (2\right )-\ln \left (3\right )+\ln \left (\frac {x}{24 x^{2}-71 x +48}\right )}\)	\(27\)

[In]

int((72*x^2-144)/(24*x^3-71*x^2+48*x)/ln(2*x/(72*x^2-213*x+144))^2,x,method=_RETURNVERBOSE)

[Out]

3/ln(2*x/(72*x^2-213*x+144))

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.77 \[ \int \frac {-144+72 x^2}{\left (48 x-71 x^2+24 x^3\right ) \log ^2\left (\frac {2 x}{144-213 x+72 x^2}\right )} \, dx=\frac {3}{\log \left (\frac {2 \, x}{3 \, {\left (24 \, x^{2} - 71 \, x + 48\right )}}\right )} \]

[In]

integrate((72*x^2-144)/(24*x^3-71*x^2+48*x)/log(2*x/(72*x^2-213*x+144))^2,x, algorithm="fricas")

[Out]

3/log(2/3*x/(24*x^2 - 71*x + 48))

Sympy [A] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.58 \[ \int \frac {-144+72 x^2}{\left (48 x-71 x^2+24 x^3\right ) \log ^2\left (\frac {2 x}{144-213 x+72 x^2}\right )} \, dx=\frac {3}{\log {\left (\frac {2 x}{72 x^{2} - 213 x + 144} \right )}} \]

[In]

integrate((72*x**2-144)/(24*x**3-71*x**2+48*x)/ln(2*x/(72*x**2-213*x+144))**2,x)

[Out]

3/log(2*x/(72*x**2 - 213*x + 144))

Maxima [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \frac {-144+72 x^2}{\left (48 x-71 x^2+24 x^3\right ) \log ^2\left (\frac {2 x}{144-213 x+72 x^2}\right )} \, dx=-\frac {3}{\log \left (3\right ) - \log \left (2\right ) + \log \left (24 \, x^{2} - 71 \, x + 48\right ) - \log \left (x\right )} \]

[In]

integrate((72*x^2-144)/(24*x^3-71*x^2+48*x)/log(2*x/(72*x^2-213*x+144))^2,x, algorithm="maxima")

[Out]

-3/(log(3) - log(2) + log(24*x^2 - 71*x + 48) - log(x))

Giac [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.77 \[ \int \frac {-144+72 x^2}{\left (48 x-71 x^2+24 x^3\right ) \log ^2\left (\frac {2 x}{144-213 x+72 x^2}\right )} \, dx=\frac {3}{\log \left (\frac {2 \, x}{3 \, {\left (24 \, x^{2} - 71 \, x + 48\right )}}\right )} \]

[In]

integrate((72*x^2-144)/(24*x^3-71*x^2+48*x)/log(2*x/(72*x^2-213*x+144))^2,x, algorithm="giac")

[Out]

3/log(2/3*x/(24*x^2 - 71*x + 48))

Mupad [B] (verification not implemented)

Time = 12.27 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.77 \[ \int \frac {-144+72 x^2}{\left (48 x-71 x^2+24 x^3\right ) \log ^2\left (\frac {2 x}{144-213 x+72 x^2}\right )} \, dx=\frac {3}{\ln \left (\frac {2\,x}{72\,x^2-213\,x+144}\right )} \]

[In]

int((72*x^2 - 144)/(log((2*x)/(72*x^2 - 213*x + 144))^2*(48*x - 71*x^2 + 24*x^3)),x)

[Out]

3/log((2*x)/(72*x^2 - 213*x + 144))