3.6.74 \(\int \frac {-8 x-4 x^2+(8+4 x) \log (3)+(-8 x-4 x \log (3)) \log (x)+(-4 x-2 x^2+(4+2 x) \log (3)) \log (\frac {x^2-2 x \log (3)+\log ^2(3)}{4+4 x+x^2})}{-2 x^2-x^3+(2 x+x^2) \log (3)} \, dx\)

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

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Rubi [C]  time = 0.41, antiderivative size = 153, normalized size of antiderivative = 5.28, number of steps used = 15, number of rules used = 7, integrand size = 96, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.073, Rules used = {6688, 2357, 2317, 2391, 2316, 2315, 2524} \begin {gather*} 4 \text {Li}_2\left (-\frac {x}{2}\right )-\frac {(8+\log (81)) \text {Li}_2\left (-\frac {x}{2}\right )}{2+\log (3)}-\frac {(8+\log (81)) \text {Li}_2\left (1-\frac {x}{\log (3)}\right )}{2+\log (3)}+4 \text {Li}_2\left (1-\frac {x}{\log (3)}\right )-\frac {(8+\log (81)) \log \left (\frac {x}{2}+1\right ) \log (x)}{2+\log (3)}+4 \log \left (\frac {x}{2}+1\right ) \log (x)+2 \left (\log \left (\frac {(x-\log (3))^2}{(x+2)^2}\right )+2\right ) \log (x)+\frac {(8+\log (81)) \log (\log (3)) \log (x-\log (3))}{2+\log (3)}-4 \log (\log (3)) \log (x-\log (3)) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-8*x - 4*x^2 + (8 + 4*x)*Log[3] + (-8*x - 4*x*Log[3])*Log[x] + (-4*x - 2*x^2 + (4 + 2*x)*Log[3])*Log[(x^2
- 2*x*Log[3] + Log[3]^2)/(4 + 4*x + x^2)])/(-2*x^2 - x^3 + (2*x + x^2)*Log[3]),x]

[Out]

4*Log[1 + x/2]*Log[x] - ((8 + Log[81])*Log[1 + x/2]*Log[x])/(2 + Log[3]) + 2*Log[x]*(2 + Log[(x - Log[3])^2/(2
+ x)^2]) - 4*Log[x - Log[3]]*Log[Log[3]] + ((8 + Log[81])*Log[x - Log[3]]*Log[Log[3]])/(2 + Log[3]) + 4*PolyL
og[2, -1/2*x] - ((8 + Log[81])*PolyLog[2, -1/2*x])/(2 + Log[3]) + 4*PolyLog[2, 1 - x/Log[3]] - ((8 + Log[81])*
PolyLog[2, 1 - x/Log[3]])/(2 + Log[3])

Rule 2315

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> -Simp[PolyLog[2, 1 - c*x]/e, x] /; FreeQ[{c, d, e}, x] &
& EqQ[e + c*d, 0]

Rule 2316

Int[((a_.) + Log[(c_.)*(x_)]*(b_.))/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[((a + b*Log[-((c*d)/e)])*Log[d + e*
x])/e, x] + Dist[b, Int[Log[-((e*x)/d)]/(d + e*x), x], x] /; FreeQ[{a, b, c, d, e}, x] && GtQ[-((c*d)/e), 0]

Rule 2317

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(Log[1 + (e*x)/d]*(a +
b*Log[c*x^n])^p)/e, x] - Dist[(b*n*p)/e, Int[(Log[1 + (e*x)/d]*(a + b*Log[c*x^n])^(p - 1))/x, x], x] /; FreeQ[
{a, b, c, d, e, n}, x] && IGtQ[p, 0]

Rule 2357

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(RFx_), x_Symbol] :> With[{u = ExpandIntegrand[(a + b*Log[c*x^
n])^p, RFx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, n}, x] && RationalFunctionQ[RFx, x] && IGtQ[p, 0]

Rule 2391

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> -Simp[PolyLog[2, -(c*e*x^n)]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 2524

Int[((a_.) + Log[(c_.)*(RFx_)^(p_.)]*(b_.))^(n_.)/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[(Log[d + e*x]*(a + b
*Log[c*RFx^p])^n)/e, x] - Dist[(b*n*p)/e, Int[(Log[d + e*x]*(a + b*Log[c*RFx^p])^(n - 1)*D[RFx, x])/RFx, x], x
] /; FreeQ[{a, b, c, d, e, p}, x] && RationalFunctionQ[RFx, x] && IGtQ[n, 0]

Rule 6688

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {(8+\log (81)) \log (x)}{(2+x) (x-\log (3))}+\frac {2 \left (2+\log \left (\frac {(x-\log (3))^2}{(2+x)^2}\right )\right )}{x}\right ) \, dx\\ &=2 \int \frac {2+\log \left (\frac {(x-\log (3))^2}{(2+x)^2}\right )}{x} \, dx+(8+\log (81)) \int \frac {\log (x)}{(2+x) (x-\log (3))} \, dx\\ &=2 \log (x) \left (2+\log \left (\frac {(x-\log (3))^2}{(2+x)^2}\right )\right )-2 \int \frac {(2+x)^2 \left (\frac {2 (x-\log (3))}{(2+x)^2}-\frac {2 (x-\log (3))^2}{(2+x)^3}\right ) \log (x)}{(x-\log (3))^2} \, dx+(8+\log (81)) \int \left (-\frac {\log (x)}{(2+x) (2+\log (3))}+\frac {\log (x)}{(x-\log (3)) (2+\log (3))}\right ) \, dx\\ &=2 \log (x) \left (2+\log \left (\frac {(x-\log (3))^2}{(2+x)^2}\right )\right )-2 \int \left (-\frac {2 \log (x)}{2+x}+\frac {2 \log (x)}{x-\log (3)}\right ) \, dx-\frac {(8+\log (81)) \int \frac {\log (x)}{2+x} \, dx}{2+\log (3)}+\frac {(8+\log (81)) \int \frac {\log (x)}{x-\log (3)} \, dx}{2+\log (3)}\\ &=-\frac {(8+\log (81)) \log \left (1+\frac {x}{2}\right ) \log (x)}{2+\log (3)}+2 \log (x) \left (2+\log \left (\frac {(x-\log (3))^2}{(2+x)^2}\right )\right )+\frac {(8+\log (81)) \log (x-\log (3)) \log (\log (3))}{2+\log (3)}+4 \int \frac {\log (x)}{2+x} \, dx-4 \int \frac {\log (x)}{x-\log (3)} \, dx+\frac {(8+\log (81)) \int \frac {\log \left (1+\frac {x}{2}\right )}{x} \, dx}{2+\log (3)}+\frac {(8+\log (81)) \int \frac {\log \left (\frac {x}{\log (3)}\right )}{x-\log (3)} \, dx}{2+\log (3)}\\ &=4 \log \left (1+\frac {x}{2}\right ) \log (x)-\frac {(8+\log (81)) \log \left (1+\frac {x}{2}\right ) \log (x)}{2+\log (3)}+2 \log (x) \left (2+\log \left (\frac {(x-\log (3))^2}{(2+x)^2}\right )\right )-4 \log (x-\log (3)) \log (\log (3))+\frac {(8+\log (81)) \log (x-\log (3)) \log (\log (3))}{2+\log (3)}-\frac {(8+\log (81)) \text {Li}_2\left (-\frac {x}{2}\right )}{2+\log (3)}-\frac {(8+\log (81)) \text {Li}_2\left (1-\frac {x}{\log (3)}\right )}{2+\log (3)}-4 \int \frac {\log \left (1+\frac {x}{2}\right )}{x} \, dx-4 \int \frac {\log \left (\frac {x}{\log (3)}\right )}{x-\log (3)} \, dx\\ &=4 \log \left (1+\frac {x}{2}\right ) \log (x)-\frac {(8+\log (81)) \log \left (1+\frac {x}{2}\right ) \log (x)}{2+\log (3)}+2 \log (x) \left (2+\log \left (\frac {(x-\log (3))^2}{(2+x)^2}\right )\right )-4 \log (x-\log (3)) \log (\log (3))+\frac {(8+\log (81)) \log (x-\log (3)) \log (\log (3))}{2+\log (3)}+4 \text {Li}_2\left (-\frac {x}{2}\right )-\frac {(8+\log (81)) \text {Li}_2\left (-\frac {x}{2}\right )}{2+\log (3)}+4 \text {Li}_2\left (1-\frac {x}{\log (3)}\right )-\frac {(8+\log (81)) \text {Li}_2\left (1-\frac {x}{\log (3)}\right )}{2+\log (3)}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.07, size = 38, normalized size = 1.31 \begin {gather*} \frac {\log (x) \left (8+\log (81)+(4+\log (9)) \log \left (\frac {x^2+\log ^2(3)-x \log (9)}{(2+x)^2}\right )\right )}{2+\log (3)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-8*x - 4*x^2 + (8 + 4*x)*Log[3] + (-8*x - 4*x*Log[3])*Log[x] + (-4*x - 2*x^2 + (4 + 2*x)*Log[3])*Lo
g[(x^2 - 2*x*Log[3] + Log[3]^2)/(4 + 4*x + x^2)])/(-2*x^2 - x^3 + (2*x + x^2)*Log[3]),x]

[Out]

(Log[x]*(8 + Log[81] + (4 + Log[9])*Log[(x^2 + Log[3]^2 - x*Log[9])/(2 + x)^2]))/(2 + Log[3])

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fricas [A]  time = 0.60, size = 34, normalized size = 1.17 \begin {gather*} 2 \, \log \relax (x) \log \left (\frac {x^{2} - 2 \, x \log \relax (3) + \log \relax (3)^{2}}{x^{2} + 4 \, x + 4}\right ) + 4 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((2*x+4)*log(3)-2*x^2-4*x)*log((log(3)^2-2*x*log(3)+x^2)/(x^2+4*x+4))+(-4*x*log(3)-8*x)*log(x)+(4*x
+8)*log(3)-4*x^2-8*x)/((x^2+2*x)*log(3)-x^3-2*x^2),x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.58, size = 36, normalized size = 1.24 \begin {gather*} 2 \, \log \left (x^{2} - 2 \, x \log \relax (3) + \log \relax (3)^{2}\right ) \log \relax (x) - 2 \, \log \left (x^{2} + 4 \, x + 4\right ) \log \relax (x) + 4 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((2*x+4)*log(3)-2*x^2-4*x)*log((log(3)^2-2*x*log(3)+x^2)/(x^2+4*x+4))+(-4*x*log(3)-8*x)*log(x)+(4*x
+8)*log(3)-4*x^2-8*x)/((x^2+2*x)*log(3)-x^3-2*x^2),x, algorithm="giac")

[Out]

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

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maple [C]  time = 0.34, size = 310, normalized size = 10.69

method	result	size
risch	\(4 \ln \relax (x ) \ln \left (\ln \relax (3)-x \right )-i \pi \ln \relax (x ) \mathrm {csgn}\left (\frac {i}{\left (2+x \right )^{2}}\right ) \mathrm {csgn}\left (i \left (\ln \relax (3)-x \right )^{2}\right ) \mathrm {csgn}\left (\frac {i \left (\ln \relax (3)-x \right )^{2}}{\left (2+x \right )^{2}}\right )+i \pi \ln \relax (x ) \mathrm {csgn}\left (\frac {i}{\left (2+x \right )^{2}}\right ) \mathrm {csgn}\left (\frac {i \left (\ln \relax (3)-x \right )^{2}}{\left (2+x \right )^{2}}\right )^{2}+i \pi \ln \relax (x ) \mathrm {csgn}\left (i \left (2+x \right )\right )^{2} \mathrm {csgn}\left (i \left (2+x \right )^{2}\right )-2 i \pi \ln \relax (x ) \mathrm {csgn}\left (i \left (2+x \right )\right ) \mathrm {csgn}\left (i \left (2+x \right )^{2}\right )^{2}+i \pi \ln \relax (x ) \mathrm {csgn}\left (i \left (2+x \right )^{2}\right )^{3}-i \pi \ln \relax (x ) \mathrm {csgn}\left (i \left (\ln \relax (3)-x \right )\right )^{2} \mathrm {csgn}\left (i \left (\ln \relax (3)-x \right )^{2}\right )+2 i \pi \ln \relax (x ) \mathrm {csgn}\left (i \left (\ln \relax (3)-x \right )\right ) \mathrm {csgn}\left (i \left (\ln \relax (3)-x \right )^{2}\right )^{2}-i \pi \ln \relax (x ) \mathrm {csgn}\left (i \left (\ln \relax (3)-x \right )^{2}\right )^{3}+i \pi \ln \relax (x ) \mathrm {csgn}\left (i \left (\ln \relax (3)-x \right )^{2}\right ) \mathrm {csgn}\left (\frac {i \left (\ln \relax (3)-x \right )^{2}}{\left (2+x \right )^{2}}\right )^{2}-i \pi \ln \relax (x ) \mathrm {csgn}\left (\frac {i \left (\ln \relax (3)-x \right )^{2}}{\left (2+x \right )^{2}}\right )^{3}-4 \ln \relax (x ) \ln \left (2+x \right )+4 \ln \relax (x )\)	\(310\)
default	\(4 \ln \relax (x )-2 \ln \left (\frac {1}{2+x}\right ) \ln \left (\frac {\ln \relax (3)^{2}}{\left (2+x \right )^{2}}+\frac {4 \ln \relax (3)}{\left (2+x \right )^{2}}-\frac {2 \ln \relax (3)}{2+x}+\frac {4}{\left (2+x \right )^{2}}-\frac {4}{2+x}+1\right )+\frac {8 \ln \left (-\frac {2+\ln \relax (3)}{2+x}+1\right ) \ln \left (\frac {1}{2+x}\right )}{2+\ln \relax (3)}-\frac {8 \ln \left (-\frac {2+\ln \relax (3)}{2+x}+1\right ) \ln \left (\frac {2+\ln \relax (3)}{2+x}\right )}{2+\ln \relax (3)}-\frac {8 \dilog \left (\frac {2+\ln \relax (3)}{2+x}\right )}{2+\ln \relax (3)}+\frac {4 \ln \relax (3) \ln \left (-\frac {2+\ln \relax (3)}{2+x}+1\right ) \ln \left (\frac {1}{2+x}\right )}{2+\ln \relax (3)}-\frac {4 \ln \relax (3) \ln \left (-\frac {2+\ln \relax (3)}{2+x}+1\right ) \ln \left (\frac {2+\ln \relax (3)}{2+x}\right )}{2+\ln \relax (3)}-\frac {4 \ln \relax (3) \dilog \left (\frac {2+\ln \relax (3)}{2+x}\right )}{2+\ln \relax (3)}+2 \ln \left (-1+\frac {2}{2+x}\right ) \ln \left (\frac {\ln \relax (3)^{2}}{\left (2+x \right )^{2}}+\frac {4 \ln \relax (3)}{\left (2+x \right )^{2}}-\frac {2 \ln \relax (3)}{2+x}+\frac {4}{\left (2+x \right )^{2}}-\frac {4}{2+x}+1\right )-\frac {4 \ln \relax (3) \dilog \left (\frac {\left (2+\ln \relax (3)\right ) \left (-1+\frac {2}{2+x}\right )+\ln \relax (3)}{\ln \relax (3)}\right )}{2+\ln \relax (3)}-\frac {4 \ln \relax (3) \ln \left (-1+\frac {2}{2+x}\right ) \ln \left (\frac {\left (2+\ln \relax (3)\right ) \left (-1+\frac {2}{2+x}\right )+\ln \relax (3)}{\ln \relax (3)}\right )}{2+\ln \relax (3)}-\frac {8 \dilog \left (\frac {\left (2+\ln \relax (3)\right ) \left (-1+\frac {2}{2+x}\right )+\ln \relax (3)}{\ln \relax (3)}\right )}{2+\ln \relax (3)}-\frac {8 \ln \left (-1+\frac {2}{2+x}\right ) \ln \left (\frac {\left (2+\ln \relax (3)\right ) \left (-1+\frac {2}{2+x}\right )+\ln \relax (3)}{\ln \relax (3)}\right )}{2+\ln \relax (3)}-\frac {4 \ln \relax (3) \ln \relax (x ) \ln \left (1+\frac {x}{2}\right )}{2+\ln \relax (3)}-\frac {4 \ln \relax (3) \dilog \left (1+\frac {x}{2}\right )}{2+\ln \relax (3)}-\frac {4 \ln \relax (3) \ln \left (\frac {\ln \relax (3)-x}{\ln \relax (3)}\right ) \ln \left (\frac {x}{\ln \relax (3)}\right )}{2+\ln \relax (3)}+\frac {4 \ln \relax (3) \ln \left (\frac {\ln \relax (3)-x}{\ln \relax (3)}\right ) \ln \relax (x )}{2+\ln \relax (3)}-\frac {4 \ln \relax (3) \dilog \left (\frac {x}{\ln \relax (3)}\right )}{2+\ln \relax (3)}-\frac {8 \ln \relax (x ) \ln \left (1+\frac {x}{2}\right )}{2+\ln \relax (3)}-\frac {8 \dilog \left (1+\frac {x}{2}\right )}{2+\ln \relax (3)}-\frac {8 \ln \left (\frac {\ln \relax (3)-x}{\ln \relax (3)}\right ) \ln \left (\frac {x}{\ln \relax (3)}\right )}{2+\ln \relax (3)}+\frac {8 \ln \left (\frac {\ln \relax (3)-x}{\ln \relax (3)}\right ) \ln \relax (x )}{2+\ln \relax (3)}-\frac {8 \dilog \left (\frac {x}{\ln \relax (3)}\right )}{2+\ln \relax (3)}\)	\(628\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((2*x+4)*ln(3)-2*x^2-4*x)*ln((ln(3)^2-2*x*ln(3)+x^2)/(x^2+4*x+4))+(-4*x*ln(3)-8*x)*ln(x)+(4*x+8)*ln(3)-4*
x^2-8*x)/((x^2+2*x)*ln(3)-x^3-2*x^2),x,method=_RETURNVERBOSE)

[Out]

4*ln(x)*ln(ln(3)-x)-I*Pi*ln(x)*csgn(I/(2+x)^2)*csgn(I*(ln(3)-x)^2)*csgn(I/(2+x)^2*(ln(3)-x)^2)+I*Pi*ln(x)*csgn
(I/(2+x)^2)*csgn(I/(2+x)^2*(ln(3)-x)^2)^2+I*Pi*ln(x)*csgn(I*(2+x))^2*csgn(I*(2+x)^2)-2*I*Pi*ln(x)*csgn(I*(2+x)
)*csgn(I*(2+x)^2)^2+I*Pi*ln(x)*csgn(I*(2+x)^2)^3-I*Pi*ln(x)*csgn(I*(ln(3)-x))^2*csgn(I*(ln(3)-x)^2)+2*I*Pi*ln(
x)*csgn(I*(ln(3)-x))*csgn(I*(ln(3)-x)^2)^2-I*Pi*ln(x)*csgn(I*(ln(3)-x)^2)^3+I*Pi*ln(x)*csgn(I*(ln(3)-x)^2)*csg
n(I/(2+x)^2*(ln(3)-x)^2)^2-I*Pi*ln(x)*csgn(I/(2+x)^2*(ln(3)-x)^2)^3-4*ln(x)*ln(2+x)+4*ln(x)

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maxima [B]  time = 0.71, size = 127, normalized size = 4.38 \begin {gather*} -4 \, {\left (\frac {2 \, \log \left (x - \log \relax (3)\right )}{\log \relax (3)^{2} + 2 \, \log \relax (3)} + \frac {\log \left (x + 2\right )}{\log \relax (3) + 2} - \frac {\log \relax (x)}{\log \relax (3)}\right )} \log \relax (3) - 4 \, {\left (\frac {\log \left (x - \log \relax (3)\right )}{\log \relax (3) + 2} - \frac {\log \left (x + 2\right )}{\log \relax (3) + 2}\right )} \log \relax (3) + 4 \, \log \left (x - \log \relax (3)\right ) \log \relax (x) - 4 \, \log \left (x + 2\right ) \log \relax (x) + \frac {4 \, \log \relax (3) \log \left (x - \log \relax (3)\right )}{\log \relax (3) + 2} + \frac {8 \, \log \left (x - \log \relax (3)\right )}{\log \relax (3) + 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((2*x+4)*log(3)-2*x^2-4*x)*log((log(3)^2-2*x*log(3)+x^2)/(x^2+4*x+4))+(-4*x*log(3)-8*x)*log(x)+(4*x
+8)*log(3)-4*x^2-8*x)/((x^2+2*x)*log(3)-x^3-2*x^2),x, algorithm="maxima")

[Out]

-4*(2*log(x - log(3))/(log(3)^2 + 2*log(3)) + log(x + 2)/(log(3) + 2) - log(x)/log(3))*log(3) - 4*(log(x - log
(3))/(log(3) + 2) - log(x + 2)/(log(3) + 2))*log(3) + 4*log(x - log(3))*log(x) - 4*log(x + 2)*log(x) + 4*log(3
)*log(x - log(3))/(log(3) + 2) + 8*log(x - log(3))/(log(3) + 2)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {8\,x-\ln \relax (3)\,\left (4\,x+8\right )+\ln \left (\frac {x^2-2\,\ln \relax (3)\,x+{\ln \relax (3)}^2}{x^2+4\,x+4}\right )\,\left (4\,x-\ln \relax (3)\,\left (2\,x+4\right )+2\,x^2\right )+\ln \relax (x)\,\left (8\,x+4\,x\,\ln \relax (3)\right )+4\,x^2}{2\,x^2+x^3-\ln \relax (3)\,\left (x^2+2\,x\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((8*x - log(3)*(4*x + 8) + log((log(3)^2 - 2*x*log(3) + x^2)/(4*x + x^2 + 4))*(4*x - log(3)*(2*x + 4) + 2*x
^2) + log(x)*(8*x + 4*x*log(3)) + 4*x^2)/(2*x^2 + x^3 - log(3)*(2*x + x^2)),x)

[Out]

int((8*x - log(3)*(4*x + 8) + log((log(3)^2 - 2*x*log(3) + x^2)/(4*x + x^2 + 4))*(4*x - log(3)*(2*x + 4) + 2*x
^2) + log(x)*(8*x + 4*x*log(3)) + 4*x^2)/(2*x^2 + x^3 - log(3)*(2*x + x^2)), x)

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sympy [A]  time = 0.44, size = 34, normalized size = 1.17 \begin {gather*} 2 \log {\relax (x )} \log {\left (\frac {x^{2} - 2 x \log {\relax (3 )} + \log {\relax (3 )}^{2}}{x^{2} + 4 x + 4} \right )} + 4 \log {\relax (x )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((2*x+4)*ln(3)-2*x**2-4*x)*ln((ln(3)**2-2*x*ln(3)+x**2)/(x**2+4*x+4))+(-4*x*ln(3)-8*x)*ln(x)+(4*x+8
)*ln(3)-4*x**2-8*x)/((x**2+2*x)*ln(3)-x**3-2*x**2),x)

[Out]

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

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