3.49.65 \(\int \frac {32-72 x+72 x^2-30 x^3+4 x^4+(8 x-20 x^2+2 x^3+2 x^4) \log (x^2)}{(16 x-36 x^2+36 x^3-15 x^4+2 x^5) \log (x^2)} \, dx\)

Optimal. Leaf size=27 \[ \log \left (\frac {(-4+x)^2 \log \left (x^2\right )}{1-\frac {4}{2-x}+2 x}\right ) \]

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Rubi [A]  time = 0.77, antiderivative size = 33, normalized size of antiderivative = 1.22, number of steps used = 8, number of rules used = 6, integrand size = 77, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.078, Rules used = {6741, 6742, 6728, 628, 2302, 29} \begin {gather*} -\log \left (2 x^2-3 x+2\right )+\log \left (\log \left (x^2\right )\right )+\log (2-x)+2 \log (4-x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(32 - 72*x + 72*x^2 - 30*x^3 + 4*x^4 + (8*x - 20*x^2 + 2*x^3 + 2*x^4)*Log[x^2])/((16*x - 36*x^2 + 36*x^3 -
15*x^4 + 2*x^5)*Log[x^2]),x]

[Out]

Log[2 - x] + 2*Log[4 - x] - Log[2 - 3*x + 2*x^2] + Log[Log[x^2]]

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], 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]

Rule 2302

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/(x_), x_Symbol] :> Dist[1/(b*n), Subst[Int[x^p, x], x, a + b*L
og[c*x^n]], x] /; FreeQ[{a, b, c, n, p}, x]

Rule 6728

Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a +
b*x^n + c*x^(2*n)), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]

Rule 6741

Int[u_, x_Symbol] :> With[{v = NormalizeIntegrand[u, x]}, Int[v, x] /; v =!= u]

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {32-72 x+72 x^2-30 x^3+4 x^4+\left (8 x-20 x^2+2 x^3+2 x^4\right ) \log \left (x^2\right )}{x \left (16-36 x+36 x^2-15 x^3+2 x^4\right ) \log \left (x^2\right )} \, dx\\ &=\int \left (\frac {2 \left (4-10 x+x^2+x^3\right )}{(-4+x) (-2+x) \left (2-3 x+2 x^2\right )}+\frac {2}{x \log \left (x^2\right )}\right ) \, dx\\ &=2 \int \frac {4-10 x+x^2+x^3}{(-4+x) (-2+x) \left (2-3 x+2 x^2\right )} \, dx+2 \int \frac {1}{x \log \left (x^2\right )} \, dx\\ &=2 \int \left (\frac {1}{-4+x}+\frac {1}{2 (-2+x)}+\frac {3-4 x}{2 \left (2-3 x+2 x^2\right )}\right ) \, dx+\operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,\log \left (x^2\right )\right )\\ &=\log (2-x)+2 \log (4-x)+\log \left (\log \left (x^2\right )\right )+\int \frac {3-4 x}{2-3 x+2 x^2} \, dx\\ &=\log (2-x)+2 \log (4-x)-\log \left (2-3 x+2 x^2\right )+\log \left (\log \left (x^2\right )\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.10, size = 33, normalized size = 1.22 \begin {gather*} \log (2-x)+2 \log (4-x)-\log \left (2-3 x+2 x^2\right )+\log \left (\log \left (x^2\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(32 - 72*x + 72*x^2 - 30*x^3 + 4*x^4 + (8*x - 20*x^2 + 2*x^3 + 2*x^4)*Log[x^2])/((16*x - 36*x^2 + 36
*x^3 - 15*x^4 + 2*x^5)*Log[x^2]),x]

[Out]

Log[2 - x] + 2*Log[4 - x] - Log[2 - 3*x + 2*x^2] + Log[Log[x^2]]

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fricas [A]  time = 0.58, size = 29, normalized size = 1.07 \begin {gather*} -\log \left (2 \, x^{2} - 3 \, x + 2\right ) + \log \left (x - 2\right ) + 2 \, \log \left (x - 4\right ) + \log \left (\log \left (x^{2}\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^4+2*x^3-20*x^2+8*x)*log(x^2)+4*x^4-30*x^3+72*x^2-72*x+32)/(2*x^5-15*x^4+36*x^3-36*x^2+16*x)/lo
g(x^2),x, algorithm="fricas")

[Out]

-log(2*x^2 - 3*x + 2) + log(x - 2) + 2*log(x - 4) + log(log(x^2))

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giac [A]  time = 0.16, size = 29, normalized size = 1.07 \begin {gather*} -\log \left (2 \, x^{2} - 3 \, x + 2\right ) + \log \left (x - 2\right ) + 2 \, \log \left (x - 4\right ) + \log \left (\log \left (x^{2}\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^4+2*x^3-20*x^2+8*x)*log(x^2)+4*x^4-30*x^3+72*x^2-72*x+32)/(2*x^5-15*x^4+36*x^3-36*x^2+16*x)/lo
g(x^2),x, algorithm="giac")

[Out]

-log(2*x^2 - 3*x + 2) + log(x - 2) + 2*log(x - 4) + log(log(x^2))

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maple [A]  time = 0.06, size = 30, normalized size = 1.11

method	result	size
default	\(\ln \left (\ln \left (x^{2}\right )\right )+\ln \left (x -2\right )-\ln \left (2 x^{2}-3 x +2\right )+2 \ln \left (x -4\right )\)	\(30\)
norman	\(\ln \left (\ln \left (x^{2}\right )\right )+\ln \left (x -2\right )-\ln \left (2 x^{2}-3 x +2\right )+2 \ln \left (x -4\right )\)	\(30\)
risch	\(\ln \left (\ln \left (x^{2}\right )\right )+\ln \left (x -2\right )-\ln \left (2 x^{2}-3 x +2\right )+2 \ln \left (x -4\right )\)	\(30\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((2*x^4+2*x^3-20*x^2+8*x)*ln(x^2)+4*x^4-30*x^3+72*x^2-72*x+32)/(2*x^5-15*x^4+36*x^3-36*x^2+16*x)/ln(x^2),x
,method=_RETURNVERBOSE)

[Out]

ln(ln(x^2))+ln(x-2)-ln(2*x^2-3*x+2)+2*ln(x-4)

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maxima [A]  time = 0.41, size = 27, normalized size = 1.00 \begin {gather*} -\log \left (2 \, x^{2} - 3 \, x + 2\right ) + \log \left (x - 2\right ) + 2 \, \log \left (x - 4\right ) + \log \left (\log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^4+2*x^3-20*x^2+8*x)*log(x^2)+4*x^4-30*x^3+72*x^2-72*x+32)/(2*x^5-15*x^4+36*x^3-36*x^2+16*x)/lo
g(x^2),x, algorithm="maxima")

[Out]

-log(2*x^2 - 3*x + 2) + log(x - 2) + 2*log(x - 4) + log(log(x))

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mupad [B]  time = 3.42, size = 27, normalized size = 1.00 \begin {gather*} \ln \left (x-2\right )+2\,\ln \left (x-4\right )-\ln \left (x^2-\frac {3\,x}{2}+1\right )+\ln \left (\ln \left (x^2\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(x^2)*(8*x - 20*x^2 + 2*x^3 + 2*x^4) - 72*x + 72*x^2 - 30*x^3 + 4*x^4 + 32)/(log(x^2)*(16*x - 36*x^2 +
36*x^3 - 15*x^4 + 2*x^5)),x)

[Out]

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

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sympy [A]  time = 0.20, size = 29, normalized size = 1.07 \begin {gather*} 2 \log {\left (x - 4 \right )} + \log {\left (x - 2 \right )} - \log {\left (2 x^{2} - 3 x + 2 \right )} + \log {\left (\log {\left (x^{2} \right )} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x**4+2*x**3-20*x**2+8*x)*ln(x**2)+4*x**4-30*x**3+72*x**2-72*x+32)/(2*x**5-15*x**4+36*x**3-36*x**
2+16*x)/ln(x**2),x)

[Out]

2*log(x - 4) + log(x - 2) - log(2*x**2 - 3*x + 2) + log(log(x**2))

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