3.71.16 \(\int \frac {(-864-288 x^2-336 x^4+256 x^6+64 x^8+16 x^{10}) \log (\frac {9+6 x^2+2 x^4+x^6}{6 x+2 x^3+x^5})}{54 x+54 x^3+33 x^5+16 x^7+4 x^9+x^{11}+(216 x+216 x^3+132 x^5+64 x^7+16 x^9+4 x^{11}) \log ^2(\frac {9+6 x^2+2 x^4+x^6}{6 x+2 x^3+x^5})} \, dx\)

Optimal. Leaf size=28 \[ 2 \log \left (\frac {1}{4}+\log ^2\left (x+\frac {9}{x \left (5+\left (1+x^2\right )^2\right )}\right )\right ) \]

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Rubi [A]  time = 29.28, antiderivative size = 40, normalized size of antiderivative = 1.43, number of steps used = 4, number of rules used = 7, integrand size = 151, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.046, Rules used = {6741, 12, 6742, 6728, 6725, 6708, 31} \begin {gather*} 2 \log \left (4 \log ^2\left (\frac {x^6+2 x^4+6 x^2+9}{x^5+2 x^3+6 x}\right )+1\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((-864 - 288*x^2 - 336*x^4 + 256*x^6 + 64*x^8 + 16*x^10)*Log[(9 + 6*x^2 + 2*x^4 + x^6)/(6*x + 2*x^3 + x^5)
])/(54*x + 54*x^3 + 33*x^5 + 16*x^7 + 4*x^9 + x^11 + (216*x + 216*x^3 + 132*x^5 + 64*x^7 + 16*x^9 + 4*x^11)*Lo
g[(9 + 6*x^2 + 2*x^4 + x^6)/(6*x + 2*x^3 + x^5)]^2),x]

[Out]

2*Log[1 + 4*Log[(9 + 6*x^2 + 2*x^4 + x^6)/(6*x + 2*x^3 + x^5)]^2]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 6708

Int[(u_)*((a_) + (b_.)*(v_)^(p_.)*(w_)^(p_.))^(m_.), x_Symbol] :> With[{c = Simplify[u/(w*D[v, x] + v*D[w, x])
]}, Dist[c, Subst[Int[(a + b*x^p)^m, x], x, v*w], x] /; FreeQ[c, x]] /; FreeQ[{a, b, m, p}, x] && IntegerQ[p]

Rule 6725

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

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 {16 \left (-54-18 x^2-21 x^4+16 x^6+4 x^8+x^{10}\right ) \log \left (\frac {9+6 x^2+2 x^4+x^6}{x \left (6+2 x^2+x^4\right )}\right )}{x \left (54+54 x^2+33 x^4+16 x^6+4 x^8+x^{10}\right ) \left (1+4 \log ^2\left (\frac {9+6 x^2+2 x^4+x^6}{6 x+2 x^3+x^5}\right )\right )} \, dx\\ &=16 \int \frac {\left (-54-18 x^2-21 x^4+16 x^6+4 x^8+x^{10}\right ) \log \left (\frac {9+6 x^2+2 x^4+x^6}{x \left (6+2 x^2+x^4\right )}\right )}{x \left (54+54 x^2+33 x^4+16 x^6+4 x^8+x^{10}\right ) \left (1+4 \log ^2\left (\frac {9+6 x^2+2 x^4+x^6}{6 x+2 x^3+x^5}\right )\right )} \, dx\\ &=2 \operatorname {Subst}\left (\int \frac {1}{1+x} \, dx,x,4 \log ^2\left (\frac {9+6 x^2+2 x^4+x^6}{6 x+2 x^3+x^5}\right )\right )\\ &=2 \log \left (1+4 \log ^2\left (\frac {9+6 x^2+2 x^4+x^6}{6 x+2 x^3+x^5}\right )\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.05, size = 40, normalized size = 1.43 \begin {gather*} 2 \log \left (1+4 \log ^2\left (\frac {9+6 x^2+2 x^4+x^6}{6 x+2 x^3+x^5}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((-864 - 288*x^2 - 336*x^4 + 256*x^6 + 64*x^8 + 16*x^10)*Log[(9 + 6*x^2 + 2*x^4 + x^6)/(6*x + 2*x^3
+ x^5)])/(54*x + 54*x^3 + 33*x^5 + 16*x^7 + 4*x^9 + x^11 + (216*x + 216*x^3 + 132*x^5 + 64*x^7 + 16*x^9 + 4*x^
11)*Log[(9 + 6*x^2 + 2*x^4 + x^6)/(6*x + 2*x^3 + x^5)]^2),x]

[Out]

2*Log[1 + 4*Log[(9 + 6*x^2 + 2*x^4 + x^6)/(6*x + 2*x^3 + x^5)]^2]

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fricas [A]  time = 0.82, size = 40, normalized size = 1.43 \begin {gather*} 2 \, \log \left (4 \, \log \left (\frac {x^{6} + 2 \, x^{4} + 6 \, x^{2} + 9}{x^{5} + 2 \, x^{3} + 6 \, x}\right )^{2} + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((16*x^10+64*x^8+256*x^6-336*x^4-288*x^2-864)*log((x^6+2*x^4+6*x^2+9)/(x^5+2*x^3+6*x))/((4*x^11+16*x^
9+64*x^7+132*x^5+216*x^3+216*x)*log((x^6+2*x^4+6*x^2+9)/(x^5+2*x^3+6*x))^2+x^11+4*x^9+16*x^7+33*x^5+54*x^3+54*
x),x, algorithm="fricas")

[Out]

2*log(4*log((x^6 + 2*x^4 + 6*x^2 + 9)/(x^5 + 2*x^3 + 6*x))^2 + 1)

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giac [A]  time = 0.75, size = 40, normalized size = 1.43 \begin {gather*} 2 \, \log \left (4 \, \log \left (\frac {x^{6} + 2 \, x^{4} + 6 \, x^{2} + 9}{x^{5} + 2 \, x^{3} + 6 \, x}\right )^{2} + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((16*x^10+64*x^8+256*x^6-336*x^4-288*x^2-864)*log((x^6+2*x^4+6*x^2+9)/(x^5+2*x^3+6*x))/((4*x^11+16*x^
9+64*x^7+132*x^5+216*x^3+216*x)*log((x^6+2*x^4+6*x^2+9)/(x^5+2*x^3+6*x))^2+x^11+4*x^9+16*x^7+33*x^5+54*x^3+54*
x),x, algorithm="giac")

[Out]

2*log(4*log((x^6 + 2*x^4 + 6*x^2 + 9)/(x^5 + 2*x^3 + 6*x))^2 + 1)

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maple [A]  time = 0.04, size = 39, normalized size = 1.39

method	result	size
risch	\(2 \ln \left (\ln \left (\frac {x^{6}+2 x^{4}+6 x^{2}+9}{x^{5}+2 x^{3}+6 x}\right )^{2}+\frac {1}{4}\right )\)	\(39\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((16*x^10+64*x^8+256*x^6-336*x^4-288*x^2-864)*ln((x^6+2*x^4+6*x^2+9)/(x^5+2*x^3+6*x))/((4*x^11+16*x^9+64*x^
7+132*x^5+216*x^3+216*x)*ln((x^6+2*x^4+6*x^2+9)/(x^5+2*x^3+6*x))^2+x^11+4*x^9+16*x^7+33*x^5+54*x^3+54*x),x,met
hod=_RETURNVERBOSE)

[Out]

2*ln(ln((x^6+2*x^4+6*x^2+9)/(x^5+2*x^3+6*x))^2+1/4)

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maxima [B]  time = 0.53, size = 87, normalized size = 3.11 \begin {gather*} 2 \, \log \left (-2 \, {\left (\log \left (x^{4} + 2 \, x^{2} + 6\right ) + \log \relax (x)\right )} \log \left (x^{6} + 2 \, x^{4} + 6 \, x^{2} + 9\right ) + \log \left (x^{6} + 2 \, x^{4} + 6 \, x^{2} + 9\right )^{2} + \log \left (x^{4} + 2 \, x^{2} + 6\right )^{2} + 2 \, \log \left (x^{4} + 2 \, x^{2} + 6\right ) \log \relax (x) + \log \relax (x)^{2} + \frac {1}{4}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((16*x^10+64*x^8+256*x^6-336*x^4-288*x^2-864)*log((x^6+2*x^4+6*x^2+9)/(x^5+2*x^3+6*x))/((4*x^11+16*x^
9+64*x^7+132*x^5+216*x^3+216*x)*log((x^6+2*x^4+6*x^2+9)/(x^5+2*x^3+6*x))^2+x^11+4*x^9+16*x^7+33*x^5+54*x^3+54*
x),x, algorithm="maxima")

[Out]

2*log(-2*(log(x^4 + 2*x^2 + 6) + log(x))*log(x^6 + 2*x^4 + 6*x^2 + 9) + log(x^6 + 2*x^4 + 6*x^2 + 9)^2 + log(x
^4 + 2*x^2 + 6)^2 + 2*log(x^4 + 2*x^2 + 6)*log(x) + log(x)^2 + 1/4)

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mupad [B]  time = 5.16, size = 38, normalized size = 1.36 \begin {gather*} 2\,\ln \left ({\ln \left (\frac {x^6+2\,x^4+6\,x^2+9}{x^5+2\,x^3+6\,x}\right )}^2+\frac {1}{4}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(log((6*x^2 + 2*x^4 + x^6 + 9)/(6*x + 2*x^3 + x^5))*(288*x^2 + 336*x^4 - 256*x^6 - 64*x^8 - 16*x^10 + 864
))/(54*x + log((6*x^2 + 2*x^4 + x^6 + 9)/(6*x + 2*x^3 + x^5))^2*(216*x + 216*x^3 + 132*x^5 + 64*x^7 + 16*x^9 +
4*x^11) + 54*x^3 + 33*x^5 + 16*x^7 + 4*x^9 + x^11),x)

[Out]

2*log(log((6*x^2 + 2*x^4 + x^6 + 9)/(6*x + 2*x^3 + x^5))^2 + 1/4)

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sympy [A]  time = 0.82, size = 36, normalized size = 1.29 \begin {gather*} 2 \log {\left (\log {\left (\frac {x^{6} + 2 x^{4} + 6 x^{2} + 9}{x^{5} + 2 x^{3} + 6 x} \right )}^{2} + \frac {1}{4} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((16*x**10+64*x**8+256*x**6-336*x**4-288*x**2-864)*ln((x**6+2*x**4+6*x**2+9)/(x**5+2*x**3+6*x))/((4*x
**11+16*x**9+64*x**7+132*x**5+216*x**3+216*x)*ln((x**6+2*x**4+6*x**2+9)/(x**5+2*x**3+6*x))**2+x**11+4*x**9+16*
x**7+33*x**5+54*x**3+54*x),x)

[Out]

2*log(log((x**6 + 2*x**4 + 6*x**2 + 9)/(x**5 + 2*x**3 + 6*x))**2 + 1/4)

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