\(\int \frac {x^2+12 x^3+(-12 x+2 x^2+24 x^3) \log (x)+(-1-12 x) \log (\frac {1}{3} (1+12 x))}{3 x+36 x^2} \, dx\) [3561]

Optimal result

Integrand size = 54, antiderivative size = 20 \[ \int \frac {x^2+12 x^3+\left (-12 x+2 x^2+24 x^3\right ) \log (x)+(-1-12 x) \log \left (\frac {1}{3} (1+12 x)\right )}{3 x+36 x^2} \, dx=\frac {1}{3} \log (x) \left (x^2-\log \left (\frac {1}{3}+4 x\right )\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.16 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.50, number of steps used = 11, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {1607, 6820, 12, 2404, 2341, 2354, 2438, 2439} \[ \int \frac {x^2+12 x^3+\left (-12 x+2 x^2+24 x^3\right ) \log (x)+(-1-12 x) \log \left (\frac {1}{3} (1+12 x)\right )}{3 x+36 x^2} \, dx=\frac {1}{3} x^2 \log (x)+\frac {1}{3} \log (3) \log (x)-\frac {1}{3} \log (x) \log (12 x+1) \]

[In]

Int[(x^2 + 12*x^3 + (-12*x + 2*x^2 + 24*x^3)*Log[x] + (-1 - 12*x)*Log[(1 + 12*x)/3])/(3*x + 36*x^2),x]

[Out]

(x^2*Log[x])/3 + (Log[3]*Log[x])/3 - (Log[x]*Log[1 + 12*x])/3

Rule 12

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

Rule 1607

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^(q - p))^n, x] /; F
reeQ[{a, b, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rule 2341

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*Log[c*x^
n])/(d*(m + 1))), x] - Simp[b*n*((d*x)^(m + 1)/(d*(m + 1)^2)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1
]

Rule 2354

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 2404

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 2438

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 2439

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

Rule 6820

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

Rubi steps \begin{align*} \text {integral}& = \int \frac {x^2+12 x^3+\left (-12 x+2 x^2+24 x^3\right ) \log (x)+(-1-12 x) \log \left (\frac {1}{3} (1+12 x)\right )}{x (3+36 x)} \, dx \\ & = \int \frac {1}{3} \left (x+\frac {2 \left (-6+x+12 x^2\right ) \log (x)}{1+12 x}-\frac {\log \left (\frac {1}{3}+4 x\right )}{x}\right ) \, dx \\ & = \frac {1}{3} \int \left (x+\frac {2 \left (-6+x+12 x^2\right ) \log (x)}{1+12 x}-\frac {\log \left (\frac {1}{3}+4 x\right )}{x}\right ) \, dx \\ & = \frac {x^2}{6}-\frac {1}{3} \int \frac {\log \left (\frac {1}{3}+4 x\right )}{x} \, dx+\frac {2}{3} \int \frac {\left (-6+x+12 x^2\right ) \log (x)}{1+12 x} \, dx \\ & = \frac {x^2}{6}+\frac {1}{3} \log (3) \log (x)-\frac {1}{3} \int \frac {\log (1+12 x)}{x} \, dx+\frac {2}{3} \int \left (x \log (x)-\frac {6 \log (x)}{1+12 x}\right ) \, dx \\ & = \frac {x^2}{6}+\frac {1}{3} \log (3) \log (x)+\frac {\text {Li}_2(-12 x)}{3}+\frac {2}{3} \int x \log (x) \, dx-4 \int \frac {\log (x)}{1+12 x} \, dx \\ & = \frac {1}{3} x^2 \log (x)+\frac {1}{3} \log (3) \log (x)-\frac {1}{3} \log (x) \log (1+12 x)+\frac {\text {Li}_2(-12 x)}{3}+\frac {1}{3} \int \frac {\log (1+12 x)}{x} \, dx \\ & = \frac {1}{3} x^2 \log (x)+\frac {1}{3} \log (3) \log (x)-\frac {1}{3} \log (x) \log (1+12 x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.30 \[ \int \frac {x^2+12 x^3+\left (-12 x+2 x^2+24 x^3\right ) \log (x)+(-1-12 x) \log \left (\frac {1}{3} (1+12 x)\right )}{3 x+36 x^2} \, dx=\frac {1}{3} \left (x^2 \log (x)+\log (3) \log (x)-\log (x) \log (1+12 x)\right ) \]

[In]

Integrate[(x^2 + 12*x^3 + (-12*x + 2*x^2 + 24*x^3)*Log[x] + (-1 - 12*x)*Log[(1 + 12*x)/3])/(3*x + 36*x^2),x]

[Out]

(x^2*Log[x] + Log[3]*Log[x] - Log[x]*Log[1 + 12*x])/3

Maple [A] (verified)

Time = 2.03 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.95

method	result	size
risch	\(-\frac {\ln \left (4 x +\frac {1}{3}\right ) \ln \left (x \right )}{3}+\frac {x^{2} \ln \left (x \right )}{3}\)	\(19\)
parallelrisch	\(-\frac {\ln \left (4 x +\frac {1}{3}\right ) \ln \left (x \right )}{3}+\frac {x^{2} \ln \left (x \right )}{3}\)	\(19\)
default	\(\frac {\ln \left (3\right ) \ln \left (x \right )}{3}+\frac {x^{2} \ln \left (x \right )}{3}-\frac {\ln \left (x \right ) \ln \left (12 x +1\right )}{3}\)	\(25\)
parts	\(-\frac {\left (\ln \left (4 x +\frac {1}{3}\right )-\ln \left (12 x +1\right )\right ) \ln \left (-12 x \right )}{3}+\frac {x^{2} \ln \left (x \right )}{3}-\frac {\ln \left (x \right ) \ln \left (12 x +1\right )}{3}\)	\(40\)

[In]

int(((-12*x-1)*ln(4*x+1/3)+(24*x^3+2*x^2-12*x)*ln(x)+12*x^3+x^2)/(36*x^2+3*x),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.90 \[ \int \frac {x^2+12 x^3+\left (-12 x+2 x^2+24 x^3\right ) \log (x)+(-1-12 x) \log \left (\frac {1}{3} (1+12 x)\right )}{3 x+36 x^2} \, dx=\frac {1}{3} \, x^{2} \log \left (x\right ) - \frac {1}{3} \, \log \left (4 \, x + \frac {1}{3}\right ) \log \left (x\right ) \]

[In]

integrate(((-12*x-1)*log(4*x+1/3)+(24*x^3+2*x^2-12*x)*log(x)+12*x^3+x^2)/(36*x^2+3*x),x, algorithm="fricas")

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {x^2+12 x^3+\left (-12 x+2 x^2+24 x^3\right ) \log (x)+(-1-12 x) \log \left (\frac {1}{3} (1+12 x)\right )}{3 x+36 x^2} \, dx=\frac {x^{2} \log {\left (x \right )}}{3} - \frac {\log {\left (x \right )} \log {\left (4 x + \frac {1}{3} \right )}}{3} \]

[In]

integrate(((-12*x-1)*ln(4*x+1/3)+(24*x**3+2*x**2-12*x)*ln(x)+12*x**3+x**2)/(36*x**2+3*x),x)

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.33 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.05 \[ \int \frac {x^2+12 x^3+\left (-12 x+2 x^2+24 x^3\right ) \log (x)+(-1-12 x) \log \left (\frac {1}{3} (1+12 x)\right )}{3 x+36 x^2} \, dx=\frac {1}{3} \, {\left (x^{2} + \log \left (3\right )\right )} \log \left (x\right ) - \frac {1}{3} \, \log \left (12 \, x + 1\right ) \log \left (x\right ) \]

[In]

integrate(((-12*x-1)*log(4*x+1/3)+(24*x^3+2*x^2-12*x)*log(x)+12*x^3+x^2)/(36*x^2+3*x),x, algorithm="maxima")

[Out]

1/3*(x^2 + log(3))*log(x) - 1/3*log(12*x + 1)*log(x)

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.20 \[ \int \frac {x^2+12 x^3+\left (-12 x+2 x^2+24 x^3\right ) \log (x)+(-1-12 x) \log \left (\frac {1}{3} (1+12 x)\right )}{3 x+36 x^2} \, dx=\frac {1}{3} \, x^{2} \log \left (x\right ) + \frac {1}{3} \, \log \left (3\right ) \log \left (x\right ) - \frac {1}{3} \, \log \left (12 \, x + 1\right ) \log \left (x\right ) \]

[In]

integrate(((-12*x-1)*log(4*x+1/3)+(24*x^3+2*x^2-12*x)*log(x)+12*x^3+x^2)/(36*x^2+3*x),x, algorithm="giac")

[Out]

1/3*x^2*log(x) + 1/3*log(3)*log(x) - 1/3*log(12*x + 1)*log(x)

Mupad [B] (verification not implemented)

Time = 9.26 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.80 \[ \int \frac {x^2+12 x^3+\left (-12 x+2 x^2+24 x^3\right ) \log (x)+(-1-12 x) \log \left (\frac {1}{3} (1+12 x)\right )}{3 x+36 x^2} \, dx=-\frac {\ln \left (x\right )\,\left (\ln \left (4\,x+\frac {1}{3}\right )-x^2\right )}{3} \]

[In]

int((x^2 - log(4*x + 1/3)*(12*x + 1) + 12*x^3 + log(x)*(2*x^2 - 12*x + 24*x^3))/(3*x + 36*x^2),x)

[Out]

-(log(x)*(log(4*x + 1/3) - x^2))/3