Integrand size = 85, antiderivative size = 25 \[ \int \frac {\left (-8 x-4 x^2\right ) \log \left (\frac {2+x}{x}\right )+\left (6+8 x+\left (-6-11 x-4 x^2\right ) \log \left (\frac {2+x}{x}\right )\right ) \log (3+4 x)}{\left (6 x+11 x^2+4 x^3\right ) \log \left (\frac {2+x}{x}\right ) \log (3+4 x)} \, dx=\log \left (\frac {\log (2)}{x \log \left (1+\frac {2}{x}\right ) \log (3+4 x)}\right ) \] Output:
ln(1/x/ln(2/x+1)/ln(3+4*x)*ln(2))
Time = 0.27 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \frac {\left (-8 x-4 x^2\right ) \log \left (\frac {2+x}{x}\right )+\left (6+8 x+\left (-6-11 x-4 x^2\right ) \log \left (\frac {2+x}{x}\right )\right ) \log (3+4 x)}{\left (6 x+11 x^2+4 x^3\right ) \log \left (\frac {2+x}{x}\right ) \log (3+4 x)} \, dx=-\log (x)-\log \left (\log \left (\frac {2+x}{x}\right )\right )-\log (\log (3+4 x)) \] Input:
Integrate[((-8*x - 4*x^2)*Log[(2 + x)/x] + (6 + 8*x + (-6 - 11*x - 4*x^2)* Log[(2 + x)/x])*Log[3 + 4*x])/((6*x + 11*x^2 + 4*x^3)*Log[(2 + x)/x]*Log[3 + 4*x]),x]
Output:
-Log[x] - Log[Log[(2 + x)/x]] - Log[Log[3 + 4*x]]
Time = 0.81 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.92, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.035, Rules used = {2026, 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.
\(\displaystyle \int \frac {\left (-4 x^2-8 x\right ) \log \left (\frac {x+2}{x}\right )+\left (\left (-4 x^2-11 x-6\right ) \log \left (\frac {x+2}{x}\right )+8 x+6\right ) \log (4 x+3)}{\left (4 x^3+11 x^2+6 x\right ) \log \left (\frac {x+2}{x}\right ) \log (4 x+3)} \, dx\) |
\(\Big \downarrow \) 2026 |
\(\displaystyle \int \frac {\left (-4 x^2-8 x\right ) \log \left (\frac {x+2}{x}\right )+\left (\left (-4 x^2-11 x-6\right ) \log \left (\frac {x+2}{x}\right )+8 x+6\right ) \log (4 x+3)}{x \left (4 x^2+11 x+6\right ) \log \left (\frac {x+2}{x}\right ) \log (4 x+3)}dx\) |
\(\Big \downarrow \) 7279 |
\(\displaystyle \int \left (\frac {-x \log \left (\frac {x+2}{x}\right )-2 \log \left (\frac {x+2}{x}\right )+2}{x (x+2) \log \left (\frac {2}{x}+1\right )}-\frac {4}{(4 x+3) \log (4 x+3)}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\log \left (x \log \left (\frac {2}{x}+1\right )\right )-\log (\log (4 x+3))\) |
Input:
Int[((-8*x - 4*x^2)*Log[(2 + x)/x] + (6 + 8*x + (-6 - 11*x - 4*x^2)*Log[(2 + x)/x])*Log[3 + 4*x])/((6*x + 11*x^2 + 4*x^3)*Log[(2 + x)/x]*Log[3 + 4*x ]),x]
Output:
-Log[x*Log[1 + 2/x]] - Log[Log[3 + 4*x]]
Int[(Fx_.)*(Px_)^(p_.), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Int[x^(p *r)*ExpandToSum[Px/x^r, x]^p*Fx, x] /; IGtQ[r, 0]] /; PolyQ[Px, x] && Integ erQ[p] && !MonomialQ[Px, x] && (ILtQ[p, 0] || !PolyQ[u, 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 = 3.19 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.04
method | result | size |
default | \(-\ln \left (x \right )-\ln \left (\ln \left (3+4 x \right )\right )-\ln \left (\ln \left (\frac {2+x}{x}\right )\right )\) | \(26\) |
parallelrisch | \(-\ln \left (x \right )-\ln \left (\ln \left (3+4 x \right )\right )-\ln \left (\ln \left (\frac {2+x}{x}\right )\right )\) | \(26\) |
parts | \(-\ln \left (x \right )-\ln \left (\ln \left (3+4 x \right )\right )-\ln \left (\ln \left (\frac {2+x}{x}\right )\right )\) | \(26\) |
risch | \(-\ln \left (x \right )-\ln \left (\ln \left (2+x \right )-\frac {i \left (\pi \,\operatorname {csgn}\left (i \left (2+x \right )\right ) \operatorname {csgn}\left (\frac {i}{x}\right ) \operatorname {csgn}\left (\frac {i \left (2+x \right )}{x}\right )-\pi \,\operatorname {csgn}\left (i \left (2+x \right )\right ) \operatorname {csgn}\left (\frac {i \left (2+x \right )}{x}\right )^{2}-\pi \,\operatorname {csgn}\left (\frac {i}{x}\right ) \operatorname {csgn}\left (\frac {i \left (2+x \right )}{x}\right )^{2}+\pi \operatorname {csgn}\left (\frac {i \left (2+x \right )}{x}\right )^{3}-2 i \ln \left (x \right )\right )}{2}\right )-\ln \left (\ln \left (3+4 x \right )\right )\) | \(116\) |
Input:
int((((-4*x^2-11*x-6)*ln((2+x)/x)+8*x+6)*ln(3+4*x)+(-4*x^2-8*x)*ln((2+x)/x ))/(4*x^3+11*x^2+6*x)/ln((2+x)/x)/ln(3+4*x),x,method=_RETURNVERBOSE)
Output:
-ln(x)-ln(ln(3+4*x))-ln(ln((2+x)/x))
Time = 0.10 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \frac {\left (-8 x-4 x^2\right ) \log \left (\frac {2+x}{x}\right )+\left (6+8 x+\left (-6-11 x-4 x^2\right ) \log \left (\frac {2+x}{x}\right )\right ) \log (3+4 x)}{\left (6 x+11 x^2+4 x^3\right ) \log \left (\frac {2+x}{x}\right ) \log (3+4 x)} \, dx=-\log \left (x\right ) - \log \left (\log \left (4 \, x + 3\right )\right ) - \log \left (\log \left (\frac {x + 2}{x}\right )\right ) \] Input:
integrate((((-4*x^2-11*x-6)*log((2+x)/x)+8*x+6)*log(3+4*x)+(-4*x^2-8*x)*lo g((2+x)/x))/(4*x^3+11*x^2+6*x)/log((2+x)/x)/log(3+4*x),x, algorithm="frica s")
Output:
-log(x) - log(log(4*x + 3)) - log(log((x + 2)/x))
Time = 0.13 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.80 \[ \int \frac {\left (-8 x-4 x^2\right ) \log \left (\frac {2+x}{x}\right )+\left (6+8 x+\left (-6-11 x-4 x^2\right ) \log \left (\frac {2+x}{x}\right )\right ) \log (3+4 x)}{\left (6 x+11 x^2+4 x^3\right ) \log \left (\frac {2+x}{x}\right ) \log (3+4 x)} \, dx=- \log {\left (x \right )} - \log {\left (\log {\left (\frac {x + 2}{x} \right )} \right )} - \log {\left (\log {\left (4 x + 3 \right )} \right )} \] Input:
integrate((((-4*x**2-11*x-6)*ln((2+x)/x)+8*x+6)*ln(3+4*x)+(-4*x**2-8*x)*ln ((2+x)/x))/(4*x**3+11*x**2+6*x)/ln((2+x)/x)/ln(3+4*x),x)
Output:
-log(x) - log(log((x + 2)/x)) - log(log(4*x + 3))
Time = 0.09 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.04 \[ \int \frac {\left (-8 x-4 x^2\right ) \log \left (\frac {2+x}{x}\right )+\left (6+8 x+\left (-6-11 x-4 x^2\right ) \log \left (\frac {2+x}{x}\right )\right ) \log (3+4 x)}{\left (6 x+11 x^2+4 x^3\right ) \log \left (\frac {2+x}{x}\right ) \log (3+4 x)} \, dx=-\log \left (x\right ) - \log \left (\log \left (x + 2\right ) - \log \left (x\right )\right ) - \log \left (\log \left (4 \, x + 3\right )\right ) \] Input:
integrate((((-4*x^2-11*x-6)*log((2+x)/x)+8*x+6)*log(3+4*x)+(-4*x^2-8*x)*lo g((2+x)/x))/(4*x^3+11*x^2+6*x)/log((2+x)/x)/log(3+4*x),x, algorithm="maxim a")
Output:
-log(x) - log(log(x + 2) - log(x)) - log(log(4*x + 3))
Time = 0.14 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.04 \[ \int \frac {\left (-8 x-4 x^2\right ) \log \left (\frac {2+x}{x}\right )+\left (6+8 x+\left (-6-11 x-4 x^2\right ) \log \left (\frac {2+x}{x}\right )\right ) \log (3+4 x)}{\left (6 x+11 x^2+4 x^3\right ) \log \left (\frac {2+x}{x}\right ) \log (3+4 x)} \, dx=-\log \left (x\right ) - \log \left (\log \left (x + 2\right ) - \log \left (x\right )\right ) - \log \left (\log \left (4 \, x + 3\right )\right ) \] Input:
integrate((((-4*x^2-11*x-6)*log((2+x)/x)+8*x+6)*log(3+4*x)+(-4*x^2-8*x)*lo g((2+x)/x))/(4*x^3+11*x^2+6*x)/log((2+x)/x)/log(3+4*x),x, algorithm="giac" )
Output:
-log(x) - log(log(x + 2) - log(x)) - log(log(4*x + 3))
Time = 3.12 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \frac {\left (-8 x-4 x^2\right ) \log \left (\frac {2+x}{x}\right )+\left (6+8 x+\left (-6-11 x-4 x^2\right ) \log \left (\frac {2+x}{x}\right )\right ) \log (3+4 x)}{\left (6 x+11 x^2+4 x^3\right ) \log \left (\frac {2+x}{x}\right ) \log (3+4 x)} \, dx=-\ln \left (\ln \left (4\,x+3\right )\right )-\ln \left (x\right )-\ln \left (\ln \left (\frac {x+2}{x}\right )\right ) \] Input:
int(-(log((x + 2)/x)*(8*x + 4*x^2) - log(4*x + 3)*(8*x - log((x + 2)/x)*(1 1*x + 4*x^2 + 6) + 6))/(log((x + 2)/x)*log(4*x + 3)*(6*x + 11*x^2 + 4*x^3) ),x)
Output:
- log(log(4*x + 3)) - log(x) - log(log((x + 2)/x))
Time = 0.18 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \frac {\left (-8 x-4 x^2\right ) \log \left (\frac {2+x}{x}\right )+\left (6+8 x+\left (-6-11 x-4 x^2\right ) \log \left (\frac {2+x}{x}\right )\right ) \log (3+4 x)}{\left (6 x+11 x^2+4 x^3\right ) \log \left (\frac {2+x}{x}\right ) \log (3+4 x)} \, dx=-\mathrm {log}\left (\mathrm {log}\left (4 x +3\right )\right )-\mathrm {log}\left (\mathrm {log}\left (\frac {x +2}{x}\right )\right )-\mathrm {log}\left (x \right ) \] Input:
int((((-4*x^2-11*x-6)*log((2+x)/x)+8*x+6)*log(3+4*x)+(-4*x^2-8*x)*log((2+x )/x))/(4*x^3+11*x^2+6*x)/log((2+x)/x)/log(3+4*x),x)
Output:
- (log(log(4*x + 3)) + log(log((x + 2)/x)) + log(x))