3.6.11 \(\int \frac {36-108 x+216 x^2-126 x^3-72 x^5-18 x^6}{(3 x-12 x^2+36 x^3-42 x^4+43 x^5+24 x^6+3 x^7+x^5 \log (2)) \log ^2(\frac {3-12 x+36 x^2-42 x^3+43 x^4+24 x^5+3 x^6+x^4 \log (2)}{x^4})} \, dx\)

Optimal. Leaf size=29 \[ \frac {3}{\log \left (4+3 \left (1+\left (3+\frac {(1-x)^2}{x^2}+x\right )^2\right )+\log (2)\right )} \]

________________________________________________________________________________________

Rubi [A]  time = 0.21, antiderivative size = 45, normalized size of antiderivative = 1.55, number of steps used = 2, number of rules used = 2, integrand size = 111, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.018, Rules used = {6, 6686} \begin {gather*} \frac {3}{\log \left (\frac {3 x^6+24 x^5+43 x^4+x^4 \log (2)-42 x^3+36 x^2-12 x+3}{x^4}\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(36 - 108*x + 216*x^2 - 126*x^3 - 72*x^5 - 18*x^6)/((3*x - 12*x^2 + 36*x^3 - 42*x^4 + 43*x^5 + 24*x^6 + 3*
x^7 + x^5*Log[2])*Log[(3 - 12*x + 36*x^2 - 42*x^3 + 43*x^4 + 24*x^5 + 3*x^6 + x^4*Log[2])/x^4]^2),x]

[Out]

3/Log[(3 - 12*x + 36*x^2 - 42*x^3 + 43*x^4 + 24*x^5 + 3*x^6 + x^4*Log[2])/x^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 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} &=\int \frac {36-108 x+216 x^2-126 x^3-72 x^5-18 x^6}{\left (3 x-12 x^2+36 x^3-42 x^4+24 x^6+3 x^7+x^5 (43+\log (2))\right ) \log ^2\left (\frac {3-12 x+36 x^2-42 x^3+43 x^4+24 x^5+3 x^6+x^4 \log (2)}{x^4}\right )} \, dx\\ &=\frac {3}{\log \left (\frac {3-12 x+36 x^2-42 x^3+43 x^4+24 x^5+3 x^6+x^4 \log (2)}{x^4}\right )}\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.03, size = 37, normalized size = 1.28 \begin {gather*} \frac {3}{\log \left (43+\frac {3}{x^4}-\frac {12}{x^3}+\frac {36}{x^2}-\frac {42}{x}+24 x+3 x^2+\log (2)\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(36 - 108*x + 216*x^2 - 126*x^3 - 72*x^5 - 18*x^6)/((3*x - 12*x^2 + 36*x^3 - 42*x^4 + 43*x^5 + 24*x^
6 + 3*x^7 + x^5*Log[2])*Log[(3 - 12*x + 36*x^2 - 42*x^3 + 43*x^4 + 24*x^5 + 3*x^6 + x^4*Log[2])/x^4]^2),x]

[Out]

3/Log[43 + 3/x^4 - 12/x^3 + 36/x^2 - 42/x + 24*x + 3*x^2 + Log[2]]

________________________________________________________________________________________

fricas [A]  time = 0.69, size = 45, normalized size = 1.55 \begin {gather*} \frac {3}{\log \left (\frac {3 \, x^{6} + 24 \, x^{5} + x^{4} \log \relax (2) + 43 \, x^{4} - 42 \, x^{3} + 36 \, x^{2} - 12 \, x + 3}{x^{4}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-18*x^6-72*x^5-126*x^3+216*x^2-108*x+36)/(x^5*log(2)+3*x^7+24*x^6+43*x^5-42*x^4+36*x^3-12*x^2+3*x)/
log((x^4*log(2)+3*x^6+24*x^5+43*x^4-42*x^3+36*x^2-12*x+3)/x^4)^2,x, algorithm="fricas")

[Out]

3/log((3*x^6 + 24*x^5 + x^4*log(2) + 43*x^4 - 42*x^3 + 36*x^2 - 12*x + 3)/x^4)

________________________________________________________________________________________

giac [A]  time = 0.76, size = 48, normalized size = 1.66 \begin {gather*} \frac {3}{\log \left (3 \, x^{6} + 24 \, x^{5} + x^{4} \log \relax (2) + 43 \, x^{4} - 42 \, x^{3} + 36 \, x^{2} - 12 \, x + 3\right ) - \log \left (x^{4}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-18*x^6-72*x^5-126*x^3+216*x^2-108*x+36)/(x^5*log(2)+3*x^7+24*x^6+43*x^5-42*x^4+36*x^3-12*x^2+3*x)/
log((x^4*log(2)+3*x^6+24*x^5+43*x^4-42*x^3+36*x^2-12*x+3)/x^4)^2,x, algorithm="giac")

[Out]

3/(log(3*x^6 + 24*x^5 + x^4*log(2) + 43*x^4 - 42*x^3 + 36*x^2 - 12*x + 3) - log(x^4))

________________________________________________________________________________________

maple [A]  time = 0.35, size = 46, normalized size = 1.59

method	result	size
norman	\(\frac {3}{\ln \left (\frac {x^{4} \ln \relax (2)+3 x^{6}+24 x^{5}+43 x^{4}-42 x^{3}+36 x^{2}-12 x +3}{x^{4}}\right )}\)	\(46\)
risch	\(\frac {3}{\ln \left (\frac {x^{4} \ln \relax (2)+3 x^{6}+24 x^{5}+43 x^{4}-42 x^{3}+36 x^{2}-12 x +3}{x^{4}}\right )}\)	\(46\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-18*x^6-72*x^5-126*x^3+216*x^2-108*x+36)/(x^5*ln(2)+3*x^7+24*x^6+43*x^5-42*x^4+36*x^3-12*x^2+3*x)/ln((x^4
*ln(2)+3*x^6+24*x^5+43*x^4-42*x^3+36*x^2-12*x+3)/x^4)^2,x,method=_RETURNVERBOSE)

[Out]

3/ln((x^4*ln(2)+3*x^6+24*x^5+43*x^4-42*x^3+36*x^2-12*x+3)/x^4)

________________________________________________________________________________________

maxima [A]  time = 0.90, size = 43, normalized size = 1.48 \begin {gather*} \frac {3}{\log \left (3 \, x^{6} + 24 \, x^{5} + x^{4} {\left (\log \relax (2) + 43\right )} - 42 \, x^{3} + 36 \, x^{2} - 12 \, x + 3\right ) - 4 \, \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-18*x^6-72*x^5-126*x^3+216*x^2-108*x+36)/(x^5*log(2)+3*x^7+24*x^6+43*x^5-42*x^4+36*x^3-12*x^2+3*x)/
log((x^4*log(2)+3*x^6+24*x^5+43*x^4-42*x^3+36*x^2-12*x+3)/x^4)^2,x, algorithm="maxima")

[Out]

3/(log(3*x^6 + 24*x^5 + x^4*(log(2) + 43) - 42*x^3 + 36*x^2 - 12*x + 3) - 4*log(x))

________________________________________________________________________________________

mupad [B]  time = 0.79, size = 46, normalized size = 1.59 \begin {gather*} \frac {3}{\ln \left (\frac {1}{x^4}\right )+\ln \left (x^4\,\ln \relax (2)-12\,x+36\,x^2-42\,x^3+43\,x^4+24\,x^5+3\,x^6+3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(108*x - 216*x^2 + 126*x^3 + 72*x^5 + 18*x^6 - 36)/(log((x^4*log(2) - 12*x + 36*x^2 - 42*x^3 + 43*x^4 + 2
4*x^5 + 3*x^6 + 3)/x^4)^2*(3*x + x^5*log(2) - 12*x^2 + 36*x^3 - 42*x^4 + 43*x^5 + 24*x^6 + 3*x^7)),x)

[Out]

3/(log(1/x^4) + log(x^4*log(2) - 12*x + 36*x^2 - 42*x^3 + 43*x^4 + 24*x^5 + 3*x^6 + 3))

________________________________________________________________________________________

sympy [A]  time = 0.29, size = 42, normalized size = 1.45 \begin {gather*} \frac {3}{\log {\left (\frac {3 x^{6} + 24 x^{5} + x^{4} \log {\relax (2 )} + 43 x^{4} - 42 x^{3} + 36 x^{2} - 12 x + 3}{x^{4}} \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-18*x**6-72*x**5-126*x**3+216*x**2-108*x+36)/(x**5*ln(2)+3*x**7+24*x**6+43*x**5-42*x**4+36*x**3-12*
x**2+3*x)/ln((x**4*ln(2)+3*x**6+24*x**5+43*x**4-42*x**3+36*x**2-12*x+3)/x**4)**2,x)

[Out]

3/log((3*x**6 + 24*x**5 + x**4*log(2) + 43*x**4 - 42*x**3 + 36*x**2 - 12*x + 3)/x**4)

________________________________________________________________________________________