Integrand size = 120, antiderivative size = 35 \[ \int \frac {88 x-21 x^2-10 x^3+2 x^4+\left (-96+19 x+11 x^2-2 x^3\right ) \log (3+x)+\left (-12 x-x^2+x^3+\left (12+x-x^2\right ) \log (3+x)\right ) \log \left (\frac {x-\log (3+x)}{\left (-4 x+x^2\right ) \log (5)}\right )}{60 x+5 x^2-5 x^3+\left (-60-5 x+5 x^2\right ) \log (3+x)} \, dx=-5+x-\frac {1}{5} x \left (-2+x+\log \left (\frac {-x+\log (3+x)}{(4-x) x \log (5)}\right )\right ) \]
Time = 0.09 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.09 \[ \int \frac {88 x-21 x^2-10 x^3+2 x^4+\left (-96+19 x+11 x^2-2 x^3\right ) \log (3+x)+\left (-12 x-x^2+x^3+\left (12+x-x^2\right ) \log (3+x)\right ) \log \left (\frac {x-\log (3+x)}{\left (-4 x+x^2\right ) \log (5)}\right )}{60 x+5 x^2-5 x^3+\left (-60-5 x+5 x^2\right ) \log (3+x)} \, dx=\frac {1}{5} \left (7 x-x^2-x \log \left (\frac {x-\log (3+x)}{(-4+x) x \log (5)}\right )\right ) \]
Integrate[(88*x - 21*x^2 - 10*x^3 + 2*x^4 + (-96 + 19*x + 11*x^2 - 2*x^3)* Log[3 + x] + (-12*x - x^2 + x^3 + (12 + x - x^2)*Log[3 + x])*Log[(x - Log[ 3 + x])/((-4*x + x^2)*Log[5])])/(60*x + 5*x^2 - 5*x^3 + (-60 - 5*x + 5*x^2 )*Log[3 + x]),x]
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.
\(\displaystyle \int \frac {2 x^4-10 x^3-21 x^2+\left (-2 x^3+11 x^2+19 x-96\right ) \log (x+3)+\left (x^3-x^2+\left (-x^2+x+12\right ) \log (x+3)-12 x\right ) \log \left (\frac {x-\log (x+3)}{\left (x^2-4 x\right ) \log (5)}\right )+88 x}{-5 x^3+5 x^2+\left (5 x^2-5 x-60\right ) \log (x+3)+60 x} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {2 x^4-10 x^3-21 x^2+\left (-2 x^3+11 x^2+19 x-96\right ) \log (x+3)+\left (x^3-x^2+\left (-x^2+x+12\right ) \log (x+3)-12 x\right ) \log \left (\frac {x-\log (x+3)}{\left (x^2-4 x\right ) \log (5)}\right )+88 x}{5 \left (-x^2+x+12\right ) (x-\log (x+3))}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{5} \int \frac {2 x^4-10 x^3-21 x^2+88 x-\left (2 x^3-11 x^2-19 x+96\right ) \log (x+3)-\left (-x^3+x^2+12 x-\left (-x^2+x+12\right ) \log (x+3)\right ) \log \left (-\frac {x-\log (x+3)}{\left (4 x-x^2\right ) \log (5)}\right )}{\left (-x^2+x+12\right ) (x-\log (x+3))}dx\) |
\(\Big \downarrow \) 7279 |
\(\displaystyle \frac {1}{5} \int \left (-\frac {2 x^4}{(x-4) (x+3) (x-\log (x+3))}+\frac {10 x^3}{(x-4) (x+3) (x-\log (x+3))}+\frac {21 x^2}{(x-4) (x+3) (x-\log (x+3))}-\frac {88 x}{(x-4) (x+3) (x-\log (x+3))}+\frac {\left (2 x^2-17 x+32\right ) \log (x+3)}{(x-4) (x-\log (x+3))}-\log \left (\frac {x-\log (x+3)}{(x-4) x \log (5)}\right )\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{5} \left (\int \frac {1}{x-\log (x+3)}dx+\int \frac {1}{\log (x+3)-x}dx-x^2+7 x-x \log \left (-\frac {x-\log (x+3)}{(4-x) x \log (5)}\right )\right )\) |
Int[(88*x - 21*x^2 - 10*x^3 + 2*x^4 + (-96 + 19*x + 11*x^2 - 2*x^3)*Log[3 + x] + (-12*x - x^2 + x^3 + (12 + x - x^2)*Log[3 + x])*Log[(x - Log[3 + x] )/((-4*x + x^2)*Log[5])])/(60*x + 5*x^2 - 5*x^3 + (-60 - 5*x + 5*x^2)*Log[ 3 + x]),x]
3.4.37.3.1 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 = 1.01 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.06
method | result | size |
parallelrisch | \(-\frac {9}{5}-\frac {x^{2}}{5}-\frac {\ln \left (-\frac {\ln \left (3+x \right )-x}{\left (x -4\right ) x \ln \left (5\right )}\right ) x}{5}+\frac {7 x}{5}\) | \(37\) |
risch | \(-\frac {x \ln \left (-\ln \left (3+x \right )+x \right )}{5}+\frac {x \ln \left (x -4\right )}{5}+\frac {x \ln \left (x \right )}{5}-\frac {i \pi x \,\operatorname {csgn}\left (\frac {i}{x -4}\right ) \operatorname {csgn}\left (\frac {i \left (\ln \left (3+x \right )-x \right )}{x -4}\right )^{2}}{10}+\frac {i \pi x \,\operatorname {csgn}\left (i \left (\ln \left (3+x \right )-x \right )\right ) \operatorname {csgn}\left (\frac {i \left (\ln \left (3+x \right )-x \right )}{x -4}\right )^{2}}{10}+\frac {i \pi x \,\operatorname {csgn}\left (\frac {i}{x -4}\right ) \operatorname {csgn}\left (i \left (\ln \left (3+x \right )-x \right )\right ) \operatorname {csgn}\left (\frac {i \left (\ln \left (3+x \right )-x \right )}{x -4}\right )}{10}-\frac {i \pi x \operatorname {csgn}\left (\frac {i \left (\ln \left (3+x \right )-x \right )}{\left (x -4\right ) x}\right )^{3}}{10}+\frac {i \pi x \,\operatorname {csgn}\left (\frac {i}{x}\right ) \operatorname {csgn}\left (\frac {i \left (\ln \left (3+x \right )-x \right )}{x -4}\right ) \operatorname {csgn}\left (\frac {i \left (\ln \left (3+x \right )-x \right )}{\left (x -4\right ) x}\right )}{10}+\frac {i \pi x \,\operatorname {csgn}\left (\frac {i \left (\ln \left (3+x \right )-x \right )}{x -4}\right ) \operatorname {csgn}\left (\frac {i \left (\ln \left (3+x \right )-x \right )}{\left (x -4\right ) x}\right )^{2}}{10}-\frac {i \pi x \,\operatorname {csgn}\left (\frac {i}{x}\right ) \operatorname {csgn}\left (\frac {i \left (\ln \left (3+x \right )-x \right )}{\left (x -4\right ) x}\right )^{2}}{10}-\frac {i \pi x \operatorname {csgn}\left (\frac {i \left (\ln \left (3+x \right )-x \right )}{x -4}\right )^{3}}{10}+\frac {x \ln \left (\ln \left (5\right )\right )}{5}-\frac {x^{2}}{5}+\frac {7 x}{5}\) | \(330\) |
int((((-x^2+x+12)*ln(3+x)+x^3-x^2-12*x)*ln((-ln(3+x)+x)/(x^2-4*x)/ln(5))+( -2*x^3+11*x^2+19*x-96)*ln(3+x)+2*x^4-10*x^3-21*x^2+88*x)/((5*x^2-5*x-60)*l n(3+x)-5*x^3+5*x^2+60*x),x,method=_RETURNVERBOSE)
Time = 0.24 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.00 \[ \int \frac {88 x-21 x^2-10 x^3+2 x^4+\left (-96+19 x+11 x^2-2 x^3\right ) \log (3+x)+\left (-12 x-x^2+x^3+\left (12+x-x^2\right ) \log (3+x)\right ) \log \left (\frac {x-\log (3+x)}{\left (-4 x+x^2\right ) \log (5)}\right )}{60 x+5 x^2-5 x^3+\left (-60-5 x+5 x^2\right ) \log (3+x)} \, dx=-\frac {1}{5} \, x^{2} - \frac {1}{5} \, x \log \left (\frac {x - \log \left (x + 3\right )}{{\left (x^{2} - 4 \, x\right )} \log \left (5\right )}\right ) + \frac {7}{5} \, x \]
integrate((((-x^2+x+12)*log(3+x)+x^3-x^2-12*x)*log((-log(3+x)+x)/(x^2-4*x) /log(5))+(-2*x^3+11*x^2+19*x-96)*log(3+x)+2*x^4-10*x^3-21*x^2+88*x)/((5*x^ 2-5*x-60)*log(3+x)-5*x^3+5*x^2+60*x),x, algorithm=\
Leaf count of result is larger than twice the leaf count of optimal. 54 vs. \(2 (26) = 52\).
Time = 0.54 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.54 \[ \int \frac {88 x-21 x^2-10 x^3+2 x^4+\left (-96+19 x+11 x^2-2 x^3\right ) \log (3+x)+\left (-12 x-x^2+x^3+\left (12+x-x^2\right ) \log (3+x)\right ) \log \left (\frac {x-\log (3+x)}{\left (-4 x+x^2\right ) \log (5)}\right )}{60 x+5 x^2-5 x^3+\left (-60-5 x+5 x^2\right ) \log (3+x)} \, dx=- \frac {x^{2}}{5} + \frac {7 x}{5} + \left (\frac {1}{60} - \frac {x}{5}\right ) \log {\left (\frac {x - \log {\left (x + 3 \right )}}{\left (x^{2} - 4 x\right ) \log {\left (5 \right )}} \right )} - \frac {\log {\left (- x + \log {\left (x + 3 \right )} \right )}}{60} + \frac {\log {\left (x^{2} - 4 x \right )}}{60} \]
integrate((((-x**2+x+12)*ln(3+x)+x**3-x**2-12*x)*ln((-ln(3+x)+x)/(x**2-4*x )/ln(5))+(-2*x**3+11*x**2+19*x-96)*ln(3+x)+2*x**4-10*x**3-21*x**2+88*x)/(( 5*x**2-5*x-60)*ln(3+x)-5*x**3+5*x**2+60*x),x)
-x**2/5 + 7*x/5 + (1/60 - x/5)*log((x - log(x + 3))/((x**2 - 4*x)*log(5))) - log(-x + log(x + 3))/60 + log(x**2 - 4*x)/60
Time = 0.30 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.09 \[ \int \frac {88 x-21 x^2-10 x^3+2 x^4+\left (-96+19 x+11 x^2-2 x^3\right ) \log (3+x)+\left (-12 x-x^2+x^3+\left (12+x-x^2\right ) \log (3+x)\right ) \log \left (\frac {x-\log (3+x)}{\left (-4 x+x^2\right ) \log (5)}\right )}{60 x+5 x^2-5 x^3+\left (-60-5 x+5 x^2\right ) \log (3+x)} \, dx=-\frac {1}{5} \, x^{2} + \frac {1}{5} \, x {\left (\log \left (\log \left (5\right )\right ) + 7\right )} - \frac {1}{5} \, x \log \left (x - \log \left (x + 3\right )\right ) + \frac {1}{5} \, x \log \left (x - 4\right ) + \frac {1}{5} \, x \log \left (x\right ) \]
integrate((((-x^2+x+12)*log(3+x)+x^3-x^2-12*x)*log((-log(3+x)+x)/(x^2-4*x) /log(5))+(-2*x^3+11*x^2+19*x-96)*log(3+x)+2*x^4-10*x^3-21*x^2+88*x)/((5*x^ 2-5*x-60)*log(3+x)-5*x^3+5*x^2+60*x),x, algorithm=\
-1/5*x^2 + 1/5*x*(log(log(5)) + 7) - 1/5*x*log(x - log(x + 3)) + 1/5*x*log (x - 4) + 1/5*x*log(x)
Time = 0.32 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.06 \[ \int \frac {88 x-21 x^2-10 x^3+2 x^4+\left (-96+19 x+11 x^2-2 x^3\right ) \log (3+x)+\left (-12 x-x^2+x^3+\left (12+x-x^2\right ) \log (3+x)\right ) \log \left (\frac {x-\log (3+x)}{\left (-4 x+x^2\right ) \log (5)}\right )}{60 x+5 x^2-5 x^3+\left (-60-5 x+5 x^2\right ) \log (3+x)} \, dx=-\frac {1}{5} \, x^{2} + \frac {1}{5} \, x \log \left (x^{2} \log \left (5\right ) - 4 \, x \log \left (5\right )\right ) - \frac {1}{5} \, x \log \left (x - \log \left (x + 3\right )\right ) + \frac {7}{5} \, x \]
integrate((((-x^2+x+12)*log(3+x)+x^3-x^2-12*x)*log((-log(3+x)+x)/(x^2-4*x) /log(5))+(-2*x^3+11*x^2+19*x-96)*log(3+x)+2*x^4-10*x^3-21*x^2+88*x)/((5*x^ 2-5*x-60)*log(3+x)-5*x^3+5*x^2+60*x),x, algorithm=\
Time = 9.02 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.91 \[ \int \frac {88 x-21 x^2-10 x^3+2 x^4+\left (-96+19 x+11 x^2-2 x^3\right ) \log (3+x)+\left (-12 x-x^2+x^3+\left (12+x-x^2\right ) \log (3+x)\right ) \log \left (\frac {x-\log (3+x)}{\left (-4 x+x^2\right ) \log (5)}\right )}{60 x+5 x^2-5 x^3+\left (-60-5 x+5 x^2\right ) \log (3+x)} \, dx=-\frac {x\,\left (x+\ln \left (-\frac {x-\ln \left (x+3\right )}{\ln \left (5\right )\,\left (4\,x-x^2\right )}\right )-7\right )}{5} \]