3.31.1 \(\int \frac {-27 x-18 x^2+18 x^3+(-6+12 x^2) \log (3)+x \log ^2(3)+(36 x+(6-6 x) \log (3)) \log (x)-9 x \log ^2(x)}{9 x^3+6 x^2 \log (3)+x \log ^2(3)} \, dx\)

Optimal. Leaf size=29 \[ x+\frac {5 (1+x-\log (x))^2}{x \left (5+\frac {5 \log (3)}{3 x}\right )} \]

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Rubi [C]  time = 0.76, antiderivative size = 263, normalized size of antiderivative = 9.07, number of steps used = 21, number of rules used = 13, integrand size = 74, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {6, 1594, 27, 6742, 43, 894, 2357, 2301, 2314, 31, 2317, 2391, 2318} \begin {gather*} -\frac {18 \text {Li}_2\left (-\frac {x \log (27)}{\log ^2(3)}\right )}{\log (27)}+\frac {6 \text {Li}_2\left (-\frac {3 x}{\log (3)}\right )}{\log (3)}+2 x-\frac {9 x \log ^2(x)}{\log (3) (3 x+\log (3))}+\frac {3 \log ^2(x)}{\log (3)}-\frac {18 \log \left (\frac {x \log (27)}{\log ^2(3)}+1\right ) \log (x)}{\log (27)}+\frac {2}{3} \left (2+\frac {9}{\log ^2(3)}\right ) \log (3) \log (3 x+\log (3))-\frac {2 \log ^2(3)}{3 (3 x+\log (3))}+\frac {27-\log ^2(3)}{3 (3 x+\log (3))}-\frac {2 \left (9-2 \log ^2(3)\right )}{3 (3 x+\log (3))}+\frac {6 \log \left (\frac {3 x}{\log (3)}+1\right ) \log (x)}{\log (3)}+\frac {6 x (3-\log (3)) \log (x)}{\log (3) (3 x+\log (3))}-\frac {6 \log (x)}{\log (3)}-\frac {4}{3} \log (3) \log (3 x+\log (3))-\frac {2 (3-\log (3)) \log (3 x+\log (3))}{\log (3)}-2 \log (9 x+\log (27))-\frac {2 \log (3)}{3 x+\log (3)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-27*x - 18*x^2 + 18*x^3 + (-6 + 12*x^2)*Log[3] + x*Log[3]^2 + (36*x + (6 - 6*x)*Log[3])*Log[x] - 9*x*Log[
x]^2)/(9*x^3 + 6*x^2*Log[3] + x*Log[3]^2),x]

[Out]

2*x - (2*Log[3])/(3*x + Log[3]) - (2*Log[3]^2)/(3*(3*x + Log[3])) - (2*(9 - 2*Log[3]^2))/(3*(3*x + Log[3])) +
(27 - Log[3]^2)/(3*(3*x + Log[3])) - (6*Log[x])/Log[3] + (6*x*(3 - Log[3])*Log[x])/(Log[3]*(3*x + Log[3])) + (
3*Log[x]^2)/Log[3] - (9*x*Log[x]^2)/(Log[3]*(3*x + Log[3])) + (6*Log[x]*Log[1 + (3*x)/Log[3]])/Log[3] - (2*(3
- Log[3])*Log[3*x + Log[3]])/Log[3] - (4*Log[3]*Log[3*x + Log[3]])/3 + (2*(2 + 9/Log[3]^2)*Log[3]*Log[3*x + Lo
g[3]])/3 - 2*Log[9*x + Log[27]] - (18*Log[x]*Log[1 + (x*Log[27])/Log[3]^2])/Log[27] + (6*PolyLog[2, (-3*x)/Log
[3]])/Log[3] - (18*PolyLog[2, -((x*Log[27])/Log[3]^2)])/Log[27]

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 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 31

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

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 894

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIn
tegrand[(d + e*x)^m*(f + g*x)^n*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] &&
NeQ[c*d^2 + a*e^2, 0] && IntegerQ[p] && ((EqQ[p, 1] && IntegersQ[m, n]) || (ILtQ[m, 0] && ILtQ[n, 0]))

Rule 1594

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 2301

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/(x_), x_Symbol] :> Simp[(a + b*Log[c*x^n])^2/(2*b*n), x] /; FreeQ[{a
, b, c, n}, x]

Rule 2314

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_) + (e_.)*(x_)^(r_.))^(q_), x_Symbol] :> Simp[(x*(d + e*x^r)^(q
+ 1)*(a + b*Log[c*x^n]))/d, x] - Dist[(b*n)/d, Int[(d + e*x^r)^(q + 1), x], x] /; FreeQ[{a, b, c, d, e, n, q,
r}, x] && EqQ[r*(q + 1) + 1, 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 2318

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((d_) + (e_.)*(x_))^2, x_Symbol] :> Simp[(x*(a + b*Log[c*x^n])
^p)/(d*(d + e*x)), x] - Dist[(b*n*p)/d, Int[(a + b*Log[c*x^n])^(p - 1)/(d + e*x), x], x] /; FreeQ[{a, b, c, d,
e, n, p}, x] && GtQ[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 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 {-18 x^2+18 x^3+\left (-6+12 x^2\right ) \log (3)+x \left (-27+\log ^2(3)\right )+(36 x+(6-6 x) \log (3)) \log (x)-9 x \log ^2(x)}{9 x^3+6 x^2 \log (3)+x \log ^2(3)} \, dx\\ &=\int \frac {-18 x^2+18 x^3+\left (-6+12 x^2\right ) \log (3)+x \left (-27+\log ^2(3)\right )+(36 x+(6-6 x) \log (3)) \log (x)-9 x \log ^2(x)}{x \left (9 x^2+6 x \log (3)+\log ^2(3)\right )} \, dx\\ &=\int \frac {-18 x^2+18 x^3+\left (-6+12 x^2\right ) \log (3)+x \left (-27+\log ^2(3)\right )+(36 x+(6-6 x) \log (3)) \log (x)-9 x \log ^2(x)}{x (3 x+\log (3))^2} \, dx\\ &=\int \left (-\frac {18 x}{(3 x+\log (3))^2}+\frac {18 x^2}{(3 x+\log (3))^2}+\frac {6 \left (-1+2 x^2\right ) \log (3)}{x (3 x+\log (3))^2}+\frac {-27+\log ^2(3)}{(3 x+\log (3))^2}+\frac {6 (x (6-\log (3))+\log (3)) \log (x)}{x (3 x+\log (3))^2}-\frac {9 \log ^2(x)}{(3 x+\log (3))^2}\right ) \, dx\\ &=\frac {27-\log ^2(3)}{3 (3 x+\log (3))}+6 \int \frac {(x (6-\log (3))+\log (3)) \log (x)}{x (3 x+\log (3))^2} \, dx-9 \int \frac {\log ^2(x)}{(3 x+\log (3))^2} \, dx-18 \int \frac {x}{(3 x+\log (3))^2} \, dx+18 \int \frac {x^2}{(3 x+\log (3))^2} \, dx+(6 \log (3)) \int \frac {-1+2 x^2}{x (3 x+\log (3))^2} \, dx\\ &=\frac {27-\log ^2(3)}{3 (3 x+\log (3))}-\frac {9 x \log ^2(x)}{\log (3) (3 x+\log (3))}+6 \int \left (\frac {\log (x)}{x \log (3)}+\frac {(3-\log (3)) \log (x)}{(3 x+\log (3))^2}-\frac {3 \log (x)}{\log ^2(3)+x \log (27)}\right ) \, dx+18 \int \left (\frac {1}{9}+\frac {\log ^2(3)}{9 (3 x+\log (3))^2}-\frac {2 \log (3)}{9 (3 x+\log (3))}\right ) \, dx-18 \int \left (-\frac {\log (3)}{3 (3 x+\log (3))^2}+\frac {1}{9 x+\log (27)}\right ) \, dx+\frac {18 \int \frac {\log (x)}{3 x+\log (3)} \, dx}{\log (3)}+(6 \log (3)) \int \left (-\frac {1}{x \log ^2(3)}+\frac {9-2 \log ^2(3)}{3 \log (3) (3 x+\log (3))^2}+\frac {9+2 \log ^2(3)}{3 \log ^2(3) (3 x+\log (3))}\right ) \, dx\\ &=2 x-\frac {2 \log (3)}{3 x+\log (3)}-\frac {2 \log ^2(3)}{3 (3 x+\log (3))}-\frac {2 \left (9-2 \log ^2(3)\right )}{3 (3 x+\log (3))}+\frac {27-\log ^2(3)}{3 (3 x+\log (3))}-\frac {6 \log (x)}{\log (3)}-\frac {9 x \log ^2(x)}{\log (3) (3 x+\log (3))}+\frac {6 \log (x) \log \left (1+\frac {3 x}{\log (3)}\right )}{\log (3)}-\frac {4}{3} \log (3) \log (3 x+\log (3))+\frac {2}{3} \left (2+\frac {9}{\log ^2(3)}\right ) \log (3) \log (3 x+\log (3))-2 \log (9 x+\log (27))-18 \int \frac {\log (x)}{\log ^2(3)+x \log (27)} \, dx+(6 (3-\log (3))) \int \frac {\log (x)}{(3 x+\log (3))^2} \, dx+\frac {6 \int \frac {\log (x)}{x} \, dx}{\log (3)}-\frac {6 \int \frac {\log \left (1+\frac {3 x}{\log (3)}\right )}{x} \, dx}{\log (3)}\\ &=2 x-\frac {2 \log (3)}{3 x+\log (3)}-\frac {2 \log ^2(3)}{3 (3 x+\log (3))}-\frac {2 \left (9-2 \log ^2(3)\right )}{3 (3 x+\log (3))}+\frac {27-\log ^2(3)}{3 (3 x+\log (3))}-\frac {6 \log (x)}{\log (3)}+\frac {6 x (3-\log (3)) \log (x)}{\log (3) (3 x+\log (3))}+\frac {3 \log ^2(x)}{\log (3)}-\frac {9 x \log ^2(x)}{\log (3) (3 x+\log (3))}+\frac {6 \log (x) \log \left (1+\frac {3 x}{\log (3)}\right )}{\log (3)}-\frac {4}{3} \log (3) \log (3 x+\log (3))+\frac {2}{3} \left (2+\frac {9}{\log ^2(3)}\right ) \log (3) \log (3 x+\log (3))-2 \log (9 x+\log (27))-\frac {18 \log (x) \log \left (1+\frac {x \log (27)}{\log ^2(3)}\right )}{\log (27)}+\frac {6 \text {Li}_2\left (-\frac {3 x}{\log (3)}\right )}{\log (3)}-\frac {(6 (3-\log (3))) \int \frac {1}{3 x+\log (3)} \, dx}{\log (3)}+\frac {18 \int \frac {\log \left (1+\frac {x \log (27)}{\log ^2(3)}\right )}{x} \, dx}{\log (27)}\\ &=2 x-\frac {2 \log (3)}{3 x+\log (3)}-\frac {2 \log ^2(3)}{3 (3 x+\log (3))}-\frac {2 \left (9-2 \log ^2(3)\right )}{3 (3 x+\log (3))}+\frac {27-\log ^2(3)}{3 (3 x+\log (3))}-\frac {6 \log (x)}{\log (3)}+\frac {6 x (3-\log (3)) \log (x)}{\log (3) (3 x+\log (3))}+\frac {3 \log ^2(x)}{\log (3)}-\frac {9 x \log ^2(x)}{\log (3) (3 x+\log (3))}+\frac {6 \log (x) \log \left (1+\frac {3 x}{\log (3)}\right )}{\log (3)}-\frac {2 (3-\log (3)) \log (3 x+\log (3))}{\log (3)}-\frac {4}{3} \log (3) \log (3 x+\log (3))+\frac {2}{3} \left (2+\frac {9}{\log ^2(3)}\right ) \log (3) \log (3 x+\log (3))-2 \log (9 x+\log (27))-\frac {18 \log (x) \log \left (1+\frac {x \log (27)}{\log ^2(3)}\right )}{\log (27)}+\frac {6 \text {Li}_2\left (-\frac {3 x}{\log (3)}\right )}{\log (3)}-\frac {18 \text {Li}_2\left (-\frac {x \log (27)}{\log ^2(3)}\right )}{\log (27)}\\ \end {aligned} \end {gather*}

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Mathematica [C]  time = 0.47, size = 194, normalized size = 6.69 \begin {gather*} 2 x-\frac {-3+\log ^2(3)+\log (9)-\frac {\log ^3(9)}{\log (729)}}{3 x+\log (3)}+\frac {3 (-1+\log (x))^2}{\log (3)}+\frac {2 (-3+\log (3)) \log (x)}{3 x+\log (3)}+\frac {3 \log ^2(x)}{3 x+\log (3)}-\frac {2 \left (-9+2 \log ^2(3)-\log (3) \log (9)+\log (27)+9 \log (x)\right ) \log \left (1+\frac {3 x}{\log (3)}\right )}{\log (27)}-\frac {2 (-3+\log (3)) (\log (x)-\log (3 x+\log (3)))}{\log (3)}+3 \log (x) \left (-\frac {\log (x)}{\log (3)}+\frac {6 \log \left (1+\frac {x \log (27)}{\log ^2(3)}\right )}{\log (27)}\right )-\frac {6 \text {Li}_2\left (-\frac {3 x}{\log (3)}\right )}{\log (3)}+\frac {18 \text {Li}_2\left (-\frac {x \log (27)}{\log ^2(3)}\right )}{\log (27)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-27*x - 18*x^2 + 18*x^3 + (-6 + 12*x^2)*Log[3] + x*Log[3]^2 + (36*x + (6 - 6*x)*Log[3])*Log[x] - 9*
x*Log[x]^2)/(9*x^3 + 6*x^2*Log[3] + x*Log[3]^2),x]

[Out]

2*x - (-3 + Log[3]^2 + Log[9] - Log[9]^3/Log[729])/(3*x + Log[3]) + (3*(-1 + Log[x])^2)/Log[3] + (2*(-3 + Log[
3])*Log[x])/(3*x + Log[3]) + (3*Log[x]^2)/(3*x + Log[3]) - (2*(-9 + 2*Log[3]^2 - Log[3]*Log[9] + Log[27] + 9*L
og[x])*Log[1 + (3*x)/Log[3]])/Log[27] - (2*(-3 + Log[3])*(Log[x] - Log[3*x + Log[3]]))/Log[3] + 3*Log[x]*(-(Lo
g[x]/Log[3]) + (6*Log[1 + (x*Log[27])/Log[3]^2])/Log[27]) - (6*PolyLog[2, (-3*x)/Log[3]])/Log[3] + (18*PolyLog
[2, -((x*Log[27])/Log[3]^2)])/Log[27]

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fricas [A]  time = 1.09, size = 41, normalized size = 1.41 \begin {gather*} \frac {18 \, x^{2} + 6 \, {\left (x - 1\right )} \log \relax (3) + \log \relax (3)^{2} - 18 \, {\left (x + 1\right )} \log \relax (x) + 9 \, \log \relax (x)^{2} + 9}{3 \, {\left (3 \, x + \log \relax (3)\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-9*x*log(x)^2+((6-6*x)*log(3)+36*x)*log(x)+x*log(3)^2+(12*x^2-6)*log(3)+18*x^3-18*x^2-27*x)/(x*log(
3)^2+6*x^2*log(3)+9*x^3),x, algorithm="fricas")

[Out]

1/3*(18*x^2 + 6*(x - 1)*log(3) + log(3)^2 - 18*(x + 1)*log(x) + 9*log(x)^2 + 9)/(3*x + log(3))

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giac [B]  time = 0.27, size = 58, normalized size = 2.00 \begin {gather*} 2 \, x + \frac {2 \, {\left (\log \relax (3) - 3\right )} \log \relax (x)}{3 \, x + \log \relax (3)} + \frac {3 \, \log \relax (x)^{2}}{3 \, x + \log \relax (3)} + \frac {\log \relax (3)^{2} - 6 \, \log \relax (3) + 9}{3 \, {\left (3 \, x + \log \relax (3)\right )}} - 2 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-9*x*log(x)^2+((6-6*x)*log(3)+36*x)*log(x)+x*log(3)^2+(12*x^2-6)*log(3)+18*x^3-18*x^2-27*x)/(x*log(
3)^2+6*x^2*log(3)+9*x^3),x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.15, size = 42, normalized size = 1.45

method	result	size
norman	\(\frac {-6 \ln \relax (x )-6 x \ln \relax (x )+6 x^{2}+3 \ln \relax (x )^{2}-\frac {\ln \relax (3)^{2}}{3}+3-2 \ln \relax (3)}{\ln \relax (3)+3 x}\)	\(42\)
risch	\(\frac {3 \ln \relax (x )^{2}}{\ln \relax (3)+3 x}+\frac {2 \left (\ln \relax (3)-3\right ) \ln \relax (x )}{\ln \relax (3)+3 x}-\frac {6 \ln \relax (3) \ln \relax (x )+18 x \ln \relax (x )-\ln \relax (3)^{2}-6 x \ln \relax (3)-18 x^{2}+6 \ln \relax (3)-9}{3 \left (\ln \relax (3)+3 x \right )}\)	\(75\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-9*x*ln(x)^2+((6-6*x)*ln(3)+36*x)*ln(x)+x*ln(3)^2+(12*x^2-6)*ln(3)+18*x^3-18*x^2-27*x)/(x*ln(3)^2+6*x^2*l
n(3)+9*x^3),x,method=_RETURNVERBOSE)

[Out]

(-6*ln(x)-6*x*ln(x)+6*x^2+3*ln(x)^2-1/3*ln(3)^2+3-2*ln(3))/(ln(3)+3*x)

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maxima [B]  time = 0.83, size = 173, normalized size = 5.97 \begin {gather*} \frac {4}{3} \, {\left (\frac {\log \relax (3)}{3 \, x + \log \relax (3)} + \log \left (3 \, x + \log \relax (3)\right )\right )} \log \relax (3) - 6 \, {\left (\frac {1}{3 \, x \log \relax (3) + \log \relax (3)^{2}} - \frac {\log \left (3 \, x + \log \relax (3)\right )}{\log \relax (3)^{2}} + \frac {\log \relax (x)}{\log \relax (3)^{2}}\right )} \log \relax (3) - \frac {4}{3} \, \log \relax (3) \log \left (3 \, x + \log \relax (3)\right ) + 2 \, x - \frac {\log \relax (3)^{2}}{3 \, x + \log \relax (3)} + \frac {2 \, {\left (\log \relax (3) - 3\right )} \log \left (3 \, x + \log \relax (3)\right )}{\log \relax (3)} - \frac {2 \, {\left (\log \relax (3) - 3\right )} \log \relax (x)}{\log \relax (3)} + \frac {2 \, {\left (\log \relax (3) - 3\right )} \log \relax (x) + 3 \, \log \relax (x)^{2}}{3 \, x + \log \relax (3)} - \frac {2 \, \log \relax (3)}{3 \, x + \log \relax (3)} + \frac {9}{3 \, x + \log \relax (3)} - 2 \, \log \left (3 \, x + \log \relax (3)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-9*x*log(x)^2+((6-6*x)*log(3)+36*x)*log(x)+x*log(3)^2+(12*x^2-6)*log(3)+18*x^3-18*x^2-27*x)/(x*log(
3)^2+6*x^2*log(3)+9*x^3),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 1.92, size = 40, normalized size = 1.38 \begin {gather*} \frac {6\,x\,\ln \relax (3)-6\,\ln \relax (3)\,\ln \relax (x)}{3\,\ln \relax (3)}+\frac {{\left (\ln \relax (3)+3\,\ln \relax (x)-3\right )}^2}{3\,\left (3\,x+\ln \relax (3)\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(3)*(12*x^2 - 6) - 9*x*log(x)^2 - 27*x + log(x)*(36*x - log(3)*(6*x - 6)) + x*log(3)^2 - 18*x^2 + 18*x
^3)/(x*log(3)^2 + 6*x^2*log(3) + 9*x^3),x)

[Out]

(6*x*log(3) - 6*log(3)*log(x))/(3*log(3)) + (log(3) + 3*log(x) - 3)^2/(3*(3*x + log(3)))

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sympy [B]  time = 0.47, size = 58, normalized size = 2.00 \begin {gather*} 2 x - 2 \log {\relax (x )} + \frac {- 6 \log {\relax (3 )} + \log {\relax (3 )}^{2} + 9}{9 x + 3 \log {\relax (3 )}} + \frac {3 \log {\relax (x )}^{2}}{3 x + \log {\relax (3 )}} + \frac {\left (-6 + 2 \log {\relax (3 )}\right ) \log {\relax (x )}}{3 x + \log {\relax (3 )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-9*x*ln(x)**2+((6-6*x)*ln(3)+36*x)*ln(x)+x*ln(3)**2+(12*x**2-6)*ln(3)+18*x**3-18*x**2-27*x)/(x*ln(3
)**2+6*x**2*ln(3)+9*x**3),x)

[Out]

2*x - 2*log(x) + (-6*log(3) + log(3)**2 + 9)/(9*x + 3*log(3)) + 3*log(x)**2/(3*x + log(3)) + (-6 + 2*log(3))*l
og(x)/(3*x + log(3))

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