Integrand size = 143, antiderivative size = 29 \[ \int \frac {3 x+3 x^2+\left (9+12 x+3 x^2\right ) \log (3+x)+\left (-6 x^2-2 x^3\right ) \log (3+x) \log (x \log (3+x))+\left (-9+6 x+3 x^2\right ) \log (3+x) \log (x \log (3+x)) \log (\log (x \log (3+x)))}{\left (-12 x^2-16 x^3-4 x^4\right ) \log (3+x) \log (x \log (3+x))+\left (36 x+48 x^2+12 x^3\right ) \log (3+x) \log (x \log (3+x)) \log (\log (x \log (3+x)))} \, dx=\frac {1}{4} \log \left (\frac {5 (1+x)^2 (-x+3 \log (\log (x \log (3+x))))}{x}\right ) \]
Time = 0.11 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.97 \[ \int \frac {3 x+3 x^2+\left (9+12 x+3 x^2\right ) \log (3+x)+\left (-6 x^2-2 x^3\right ) \log (3+x) \log (x \log (3+x))+\left (-9+6 x+3 x^2\right ) \log (3+x) \log (x \log (3+x)) \log (\log (x \log (3+x)))}{\left (-12 x^2-16 x^3-4 x^4\right ) \log (3+x) \log (x \log (3+x))+\left (36 x+48 x^2+12 x^3\right ) \log (3+x) \log (x \log (3+x)) \log (\log (x \log (3+x)))} \, dx=\frac {1}{4} (-\log (x)+2 \log (1+x)+\log (x-3 \log (\log (x \log (3+x))))) \]
Integrate[(3*x + 3*x^2 + (9 + 12*x + 3*x^2)*Log[3 + x] + (-6*x^2 - 2*x^3)* Log[3 + x]*Log[x*Log[3 + x]] + (-9 + 6*x + 3*x^2)*Log[3 + x]*Log[x*Log[3 + x]]*Log[Log[x*Log[3 + x]]])/((-12*x^2 - 16*x^3 - 4*x^4)*Log[3 + x]*Log[x* Log[3 + x]] + (36*x + 48*x^2 + 12*x^3)*Log[3 + x]*Log[x*Log[3 + x]]*Log[Lo g[x*Log[3 + x]]]),x]
Time = 3.38 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.97, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.035, Rules used = {7292, 27, 25, 7279, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {3 x^2+\left (3 x^2+12 x+9\right ) \log (x+3)+\left (3 x^2+6 x-9\right ) \log (x+3) \log (x \log (x+3)) \log (\log (x \log (x+3)))+\left (-2 x^3-6 x^2\right ) \log (x+3) \log (x \log (x+3))+3 x}{\left (12 x^3+48 x^2+36 x\right ) \log (x+3) \log (\log (x \log (x+3))) \log (x \log (x+3))+\left (-4 x^4-16 x^3-12 x^2\right ) \log (x+3) \log (x \log (x+3))} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {-3 x^2-\left (3 x^2+12 x+9\right ) \log (x+3)-\left (3 x^2+6 x-9\right ) \log (x+3) \log (x \log (x+3)) \log (\log (x \log (x+3)))-\left (-2 x^3-6 x^2\right ) \log (x+3) \log (x \log (x+3))-3 x}{4 x \left (x^2+4 x+3\right ) \log (x+3) \log (x \log (x+3)) (x-3 \log (\log (x \log (x+3))))}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{4} \int -\frac {3 x^2+3 x+3 \left (x^2+4 x+3\right ) \log (x+3)-2 \left (x^3+3 x^2\right ) \log (x+3) \log (x \log (x+3))-3 \left (-x^2-2 x+3\right ) \log (x+3) \log (x \log (x+3)) \log (\log (x \log (x+3)))}{x \left (x^2+4 x+3\right ) \log (x+3) \log (x \log (x+3)) (x-3 \log (\log (x \log (x+3))))}dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {1}{4} \int \frac {3 x^2+3 x+3 \left (x^2+4 x+3\right ) \log (x+3)-2 \left (x^3+3 x^2\right ) \log (x+3) \log (x \log (x+3))-3 \left (-x^2-2 x+3\right ) \log (x+3) \log (x \log (x+3)) \log (\log (x \log (x+3)))}{x \left (x^2+4 x+3\right ) \log (x+3) \log (x \log (x+3)) (x-3 \log (\log (x \log (x+3))))}dx\) |
\(\Big \downarrow \) 7279 |
\(\displaystyle -\frac {1}{4} \int \left (\frac {1-x}{x (x+1)}+\frac {-\log (x+3) \log (x \log (x+3)) x^2+3 \log (x+3) x-3 \log (x+3) \log (x \log (x+3)) x+3 x+9 \log (x+3)}{x (x+3) \log (x+3) \log (x \log (x+3)) (x-3 \log (\log (x \log (x+3))))}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{4} (-\log (x)+2 \log (x+1)+\log (x-3 \log (\log (x \log (x+3)))))\) |
Int[(3*x + 3*x^2 + (9 + 12*x + 3*x^2)*Log[3 + x] + (-6*x^2 - 2*x^3)*Log[3 + x]*Log[x*Log[3 + x]] + (-9 + 6*x + 3*x^2)*Log[3 + x]*Log[x*Log[3 + x]]*L og[Log[x*Log[3 + x]]])/((-12*x^2 - 16*x^3 - 4*x^4)*Log[3 + x]*Log[x*Log[3 + x]] + (36*x + 48*x^2 + 12*x^3)*Log[3 + x]*Log[x*Log[3 + x]]*Log[Log[x*Lo g[3 + x]]]),x]
3.8.79.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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] /; Su mQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]
Time = 32.61 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.93
method | result | size |
parallelrisch | \(-\frac {\ln \left (x \right )}{4}+\frac {\ln \left (1+x \right )}{2}+\frac {\ln \left (x -3 \ln \left (\ln \left (x \ln \left (3+x \right )\right )\right )\right )}{4}\) | \(27\) |
risch | \(-\frac {\ln \left (x \right )}{4}+\frac {\ln \left (1+x \right )}{2}+\frac {\ln \left (-\frac {x}{3}+\ln \left (\ln \left (x \right )+\ln \left (\ln \left (3+x \right )\right )-\frac {i \pi \,\operatorname {csgn}\left (i x \ln \left (3+x \right )\right ) \left (-\operatorname {csgn}\left (i x \ln \left (3+x \right )\right )+\operatorname {csgn}\left (i x \right )\right ) \left (-\operatorname {csgn}\left (i x \ln \left (3+x \right )\right )+\operatorname {csgn}\left (i \ln \left (3+x \right )\right )\right )}{2}\right )\right )}{4}\) | \(78\) |
int(((3*x^2+6*x-9)*ln(3+x)*ln(x*ln(3+x))*ln(ln(x*ln(3+x)))+(-2*x^3-6*x^2)* ln(3+x)*ln(x*ln(3+x))+(3*x^2+12*x+9)*ln(3+x)+3*x^2+3*x)/((12*x^3+48*x^2+36 *x)*ln(3+x)*ln(x*ln(3+x))*ln(ln(x*ln(3+x)))+(-4*x^4-16*x^3-12*x^2)*ln(3+x) *ln(x*ln(3+x))),x,method=_RETURNVERBOSE)
Time = 0.25 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.97 \[ \int \frac {3 x+3 x^2+\left (9+12 x+3 x^2\right ) \log (3+x)+\left (-6 x^2-2 x^3\right ) \log (3+x) \log (x \log (3+x))+\left (-9+6 x+3 x^2\right ) \log (3+x) \log (x \log (3+x)) \log (\log (x \log (3+x)))}{\left (-12 x^2-16 x^3-4 x^4\right ) \log (3+x) \log (x \log (3+x))+\left (36 x+48 x^2+12 x^3\right ) \log (3+x) \log (x \log (3+x)) \log (\log (x \log (3+x)))} \, dx=\frac {1}{2} \, \log \left (x + 1\right ) - \frac {1}{4} \, \log \left (x\right ) + \frac {1}{4} \, \log \left (-x + 3 \, \log \left (\log \left (x \log \left (x + 3\right )\right )\right )\right ) \]
integrate(((3*x^2+6*x-9)*log(3+x)*log(x*log(3+x))*log(log(x*log(3+x)))+(-2 *x^3-6*x^2)*log(3+x)*log(x*log(3+x))+(3*x^2+12*x+9)*log(3+x)+3*x^2+3*x)/(( 12*x^3+48*x^2+36*x)*log(3+x)*log(x*log(3+x))*log(log(x*log(3+x)))+(-4*x^4- 16*x^3-12*x^2)*log(3+x)*log(x*log(3+x))),x, algorithm=\
Time = 0.72 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.93 \[ \int \frac {3 x+3 x^2+\left (9+12 x+3 x^2\right ) \log (3+x)+\left (-6 x^2-2 x^3\right ) \log (3+x) \log (x \log (3+x))+\left (-9+6 x+3 x^2\right ) \log (3+x) \log (x \log (3+x)) \log (\log (x \log (3+x)))}{\left (-12 x^2-16 x^3-4 x^4\right ) \log (3+x) \log (x \log (3+x))+\left (36 x+48 x^2+12 x^3\right ) \log (3+x) \log (x \log (3+x)) \log (\log (x \log (3+x)))} \, dx=- \frac {\log {\left (x \right )}}{4} + \frac {\log {\left (- \frac {x}{3} + \log {\left (\log {\left (x \log {\left (x + 3 \right )} \right )} \right )} \right )}}{4} + \frac {\log {\left (x + 1 \right )}}{2} \]
integrate(((3*x**2+6*x-9)*ln(3+x)*ln(x*ln(3+x))*ln(ln(x*ln(3+x)))+(-2*x**3 -6*x**2)*ln(3+x)*ln(x*ln(3+x))+(3*x**2+12*x+9)*ln(3+x)+3*x**2+3*x)/((12*x* *3+48*x**2+36*x)*ln(3+x)*ln(x*ln(3+x))*ln(ln(x*ln(3+x)))+(-4*x**4-16*x**3- 12*x**2)*ln(3+x)*ln(x*ln(3+x))),x)
Time = 0.24 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.93 \[ \int \frac {3 x+3 x^2+\left (9+12 x+3 x^2\right ) \log (3+x)+\left (-6 x^2-2 x^3\right ) \log (3+x) \log (x \log (3+x))+\left (-9+6 x+3 x^2\right ) \log (3+x) \log (x \log (3+x)) \log (\log (x \log (3+x)))}{\left (-12 x^2-16 x^3-4 x^4\right ) \log (3+x) \log (x \log (3+x))+\left (36 x+48 x^2+12 x^3\right ) \log (3+x) \log (x \log (3+x)) \log (\log (x \log (3+x)))} \, dx=\frac {1}{2} \, \log \left (x + 1\right ) - \frac {1}{4} \, \log \left (x\right ) + \frac {1}{4} \, \log \left (-\frac {1}{3} \, x + \log \left (\log \left (x\right ) + \log \left (\log \left (x + 3\right )\right )\right )\right ) \]
integrate(((3*x^2+6*x-9)*log(3+x)*log(x*log(3+x))*log(log(x*log(3+x)))+(-2 *x^3-6*x^2)*log(3+x)*log(x*log(3+x))+(3*x^2+12*x+9)*log(3+x)+3*x^2+3*x)/(( 12*x^3+48*x^2+36*x)*log(3+x)*log(x*log(3+x))*log(log(x*log(3+x)))+(-4*x^4- 16*x^3-12*x^2)*log(3+x)*log(x*log(3+x))),x, algorithm=\
Time = 1.00 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.90 \[ \int \frac {3 x+3 x^2+\left (9+12 x+3 x^2\right ) \log (3+x)+\left (-6 x^2-2 x^3\right ) \log (3+x) \log (x \log (3+x))+\left (-9+6 x+3 x^2\right ) \log (3+x) \log (x \log (3+x)) \log (\log (x \log (3+x)))}{\left (-12 x^2-16 x^3-4 x^4\right ) \log (3+x) \log (x \log (3+x))+\left (36 x+48 x^2+12 x^3\right ) \log (3+x) \log (x \log (3+x)) \log (\log (x \log (3+x)))} \, dx=\frac {1}{4} \, \log \left (x - 3 \, \log \left (\log \left (x \log \left (x + 3\right )\right )\right )\right ) + \frac {1}{2} \, \log \left (x + 1\right ) - \frac {1}{4} \, \log \left (x\right ) \]
integrate(((3*x^2+6*x-9)*log(3+x)*log(x*log(3+x))*log(log(x*log(3+x)))+(-2 *x^3-6*x^2)*log(3+x)*log(x*log(3+x))+(3*x^2+12*x+9)*log(3+x)+3*x^2+3*x)/(( 12*x^3+48*x^2+36*x)*log(3+x)*log(x*log(3+x))*log(log(x*log(3+x)))+(-4*x^4- 16*x^3-12*x^2)*log(3+x)*log(x*log(3+x))),x, algorithm=\
Time = 14.07 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.90 \[ \int \frac {3 x+3 x^2+\left (9+12 x+3 x^2\right ) \log (3+x)+\left (-6 x^2-2 x^3\right ) \log (3+x) \log (x \log (3+x))+\left (-9+6 x+3 x^2\right ) \log (3+x) \log (x \log (3+x)) \log (\log (x \log (3+x)))}{\left (-12 x^2-16 x^3-4 x^4\right ) \log (3+x) \log (x \log (3+x))+\left (36 x+48 x^2+12 x^3\right ) \log (3+x) \log (x \log (3+x)) \log (\log (x \log (3+x)))} \, dx=\frac {\ln \left (x+1\right )}{2}+\frac {\ln \left (\ln \left (\ln \left (x\,\ln \left (x+3\right )\right )\right )-\frac {x}{3}\right )}{4}-\frac {\ln \left (x\right )}{4} \]
int(-(3*x + log(x + 3)*(12*x + 3*x^2 + 9) + 3*x^2 - log(x + 3)*log(x*log(x + 3))*(6*x^2 + 2*x^3) + log(x + 3)*log(x*log(x + 3))*log(log(x*log(x + 3) ))*(6*x + 3*x^2 - 9))/(log(x + 3)*log(x*log(x + 3))*(12*x^2 + 16*x^3 + 4*x ^4) - log(x + 3)*log(x*log(x + 3))*log(log(x*log(x + 3)))*(36*x + 48*x^2 + 12*x^3)),x)