Integrand size = 39, antiderivative size = 26 \[ \int \frac {6 x+4 x^2+(-6-4 x) \log \left (\frac {1}{-4-8 x+(1+2 x) \log (8)}\right )}{1+2 x} \, dx=5+\left (-x+\log \left (\frac {x}{\left (x+2 x^2\right ) (-4+\log (8))}\right )\right )^2 \]
Time = 0.02 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.81 \[ \int \frac {6 x+4 x^2+(-6-4 x) \log \left (\frac {1}{-4-8 x+(1+2 x) \log (8)}\right )}{1+2 x} \, dx=\left (x-\log \left (\frac {1}{(1+2 x) (-4+\log (8))}\right )\right )^2 \]
Time = 0.31 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.92, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {7292, 27, 7237}
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.
\(\displaystyle \int \frac {4 x^2+6 x+(-4 x-6) \log \left (\frac {1}{-8 x+(2 x+1) \log (8)-4}\right )}{2 x+1} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {2 (2 x+3) \left (x-\log \left (\frac {1}{(2 x+1) (\log (8)-4)}\right )\right )}{2 x+1}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle 2 \int \frac {(2 x+3) \left (x-\log \left (-\frac {1}{(2 x+1) (4-\log (8))}\right )\right )}{2 x+1}dx\) |
\(\Big \downarrow \) 7237 |
\(\displaystyle \left (x-\log \left (-\frac {1}{(2 x+1) (4-\log (8))}\right )\right )^2\) |
3.23.78.3.1 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_)*(y_)^(m_.), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Si mp[q*(y^(m + 1)/(m + 1)), x] /; !FalseQ[q]] /; FreeQ[m, x] && NeQ[m, -1]
Time = 0.52 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.69
method | result | size |
norman | \(x^{2}+\ln \left (\frac {1}{3 \left (1+2 x \right ) \ln \left (2\right )-8 x -4}\right )^{2}-2 x \ln \left (\frac {1}{3 \left (1+2 x \right ) \ln \left (2\right )-8 x -4}\right )\) | \(44\) |
parallelrisch | \(x^{2}-2 \ln \left (\frac {1}{6 x \ln \left (2\right )+3 \ln \left (2\right )-8 x -4}\right ) x +\ln \left (\frac {1}{6 x \ln \left (2\right )+3 \ln \left (2\right )-8 x -4}\right )^{2}\) | \(44\) |
parts | \(x^{2}+2 x -\ln \left (1+2 x \right )+\frac {3 \ln \left (2\right ) \ln \left (\frac {1}{\left (1+2 x \right ) \left (3 \ln \left (2\right )-4\right )}\right )^{2}}{3 \ln \left (2\right )-4}-\frac {4 \ln \left (\frac {1}{\left (1+2 x \right ) \left (3 \ln \left (2\right )-4\right )}\right )^{2}}{3 \ln \left (2\right )-4}+\frac {-\left (1+2 x \right ) \left (3 \ln \left (2\right )-4\right ) \ln \left (\frac {1}{\left (1+2 x \right ) \left (3 \ln \left (2\right )-4\right )}\right )-\left (1+2 x \right ) \left (3 \ln \left (2\right )-4\right )}{3 \ln \left (2\right )-4}\) | \(129\) |
risch | \(x^{2}+2 x -\ln \left (1+2 x \right )+\frac {3 \ln \left (2\right ) \ln \left (\frac {1}{\left (1+2 x \right ) \left (3 \ln \left (2\right )-4\right )}\right )^{2}}{3 \ln \left (2\right )-4}-\frac {4 \ln \left (\frac {1}{\left (1+2 x \right ) \left (3 \ln \left (2\right )-4\right )}\right )^{2}}{3 \ln \left (2\right )-4}-\frac {6 \ln \left (2\right ) \ln \left (\frac {1}{\left (1+2 x \right ) \left (3 \ln \left (2\right )-4\right )}\right ) x}{3 \ln \left (2\right )-4}-\frac {3 \ln \left (2\right ) \ln \left (\frac {1}{\left (1+2 x \right ) \left (3 \ln \left (2\right )-4\right )}\right )}{3 \ln \left (2\right )-4}-\frac {6 x \ln \left (2\right )}{3 \ln \left (2\right )-4}+\frac {8 \ln \left (\frac {1}{\left (1+2 x \right ) \left (3 \ln \left (2\right )-4\right )}\right ) x}{3 \ln \left (2\right )-4}-\frac {3 \ln \left (2\right )}{3 \ln \left (2\right )-4}+\frac {4 \ln \left (\frac {1}{\left (1+2 x \right ) \left (3 \ln \left (2\right )-4\right )}\right )}{3 \ln \left (2\right )-4}+\frac {8 x}{3 \ln \left (2\right )-4}+\frac {4}{3 \ln \left (2\right )-4}\) | \(236\) |
derivativedivides | \(\frac {9 \ln \left (2\right )^{2} \ln \left (\frac {1}{\left (1+2 x \right ) \left (3 \ln \left (2\right )-4\right )}\right )^{2}}{\left (3 \ln \left (2\right )-4\right )^{2}}-\frac {24 \ln \left (2\right ) \ln \left (\frac {1}{\left (1+2 x \right ) \left (3 \ln \left (2\right )-4\right )}\right )^{2}}{\left (3 \ln \left (2\right )-4\right )^{2}}+\frac {9 \ln \left (2\right )^{2} \ln \left (\frac {1}{\left (1+2 x \right ) \left (3 \ln \left (2\right )-4\right )}\right )}{\left (3 \ln \left (2\right )-4\right )^{2}}+\frac {3 \ln \left (2\right ) \left (-\left (1+2 x \right ) \left (3 \ln \left (2\right )-4\right ) \ln \left (\frac {1}{\left (1+2 x \right ) \left (3 \ln \left (2\right )-4\right )}\right )-\left (1+2 x \right ) \left (3 \ln \left (2\right )-4\right )\right )}{\left (3 \ln \left (2\right )-4\right )^{2}}+\frac {16 \ln \left (\frac {1}{\left (1+2 x \right ) \left (3 \ln \left (2\right )-4\right )}\right )^{2}}{\left (3 \ln \left (2\right )-4\right )^{2}}-\frac {24 \ln \left (2\right ) \ln \left (\frac {1}{\left (1+2 x \right ) \left (3 \ln \left (2\right )-4\right )}\right )}{\left (3 \ln \left (2\right )-4\right )^{2}}-\frac {4 \left (-\left (1+2 x \right ) \left (3 \ln \left (2\right )-4\right ) \ln \left (\frac {1}{\left (1+2 x \right ) \left (3 \ln \left (2\right )-4\right )}\right )-\left (1+2 x \right ) \left (3 \ln \left (2\right )-4\right )\right )}{\left (3 \ln \left (2\right )-4\right )^{2}}+\frac {3 \left (1+2 x \right ) \ln \left (2\right )}{2 \left (3 \ln \left (2\right )-4\right )}+\frac {16 \ln \left (\frac {1}{\left (1+2 x \right ) \left (3 \ln \left (2\right )-4\right )}\right )}{\left (3 \ln \left (2\right )-4\right )^{2}}-\frac {2 \left (1+2 x \right )}{3 \ln \left (2\right )-4}+\frac {\left (1+2 x \right )^{2}}{4}\) | \(333\) |
default | \(\frac {9 \ln \left (2\right )^{2} \ln \left (\frac {1}{\left (1+2 x \right ) \left (3 \ln \left (2\right )-4\right )}\right )^{2}}{\left (3 \ln \left (2\right )-4\right )^{2}}-\frac {24 \ln \left (2\right ) \ln \left (\frac {1}{\left (1+2 x \right ) \left (3 \ln \left (2\right )-4\right )}\right )^{2}}{\left (3 \ln \left (2\right )-4\right )^{2}}+\frac {9 \ln \left (2\right )^{2} \ln \left (\frac {1}{\left (1+2 x \right ) \left (3 \ln \left (2\right )-4\right )}\right )}{\left (3 \ln \left (2\right )-4\right )^{2}}+\frac {3 \ln \left (2\right ) \left (-\left (1+2 x \right ) \left (3 \ln \left (2\right )-4\right ) \ln \left (\frac {1}{\left (1+2 x \right ) \left (3 \ln \left (2\right )-4\right )}\right )-\left (1+2 x \right ) \left (3 \ln \left (2\right )-4\right )\right )}{\left (3 \ln \left (2\right )-4\right )^{2}}+\frac {16 \ln \left (\frac {1}{\left (1+2 x \right ) \left (3 \ln \left (2\right )-4\right )}\right )^{2}}{\left (3 \ln \left (2\right )-4\right )^{2}}-\frac {24 \ln \left (2\right ) \ln \left (\frac {1}{\left (1+2 x \right ) \left (3 \ln \left (2\right )-4\right )}\right )}{\left (3 \ln \left (2\right )-4\right )^{2}}-\frac {4 \left (-\left (1+2 x \right ) \left (3 \ln \left (2\right )-4\right ) \ln \left (\frac {1}{\left (1+2 x \right ) \left (3 \ln \left (2\right )-4\right )}\right )-\left (1+2 x \right ) \left (3 \ln \left (2\right )-4\right )\right )}{\left (3 \ln \left (2\right )-4\right )^{2}}+\frac {3 \left (1+2 x \right ) \ln \left (2\right )}{2 \left (3 \ln \left (2\right )-4\right )}+\frac {16 \ln \left (\frac {1}{\left (1+2 x \right ) \left (3 \ln \left (2\right )-4\right )}\right )}{\left (3 \ln \left (2\right )-4\right )^{2}}-\frac {2 \left (1+2 x \right )}{3 \ln \left (2\right )-4}+\frac {\left (1+2 x \right )^{2}}{4}\) | \(333\) |
Time = 0.27 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.65 \[ \int \frac {6 x+4 x^2+(-6-4 x) \log \left (\frac {1}{-4-8 x+(1+2 x) \log (8)}\right )}{1+2 x} \, dx=x^{2} - 2 \, x \log \left (\frac {1}{3 \, {\left (2 \, x + 1\right )} \log \left (2\right ) - 8 \, x - 4}\right ) + \log \left (\frac {1}{3 \, {\left (2 \, x + 1\right )} \log \left (2\right ) - 8 \, x - 4}\right )^{2} \]
Leaf count of result is larger than twice the leaf count of optimal. 41 vs. \(2 (20) = 40\).
Time = 0.07 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.58 \[ \int \frac {6 x+4 x^2+(-6-4 x) \log \left (\frac {1}{-4-8 x+(1+2 x) \log (8)}\right )}{1+2 x} \, dx=x^{2} - 2 x \log {\left (\frac {1}{- 8 x + \left (6 x + 3\right ) \log {\left (2 \right )} - 4} \right )} + \log {\left (\frac {1}{- 8 x + \left (6 x + 3\right ) \log {\left (2 \right )} - 4} \right )}^{2} \]
Leaf count of result is larger than twice the leaf count of optimal. 57 vs. \(2 (28) = 56\).
Time = 0.28 (sec) , antiderivative size = 57, normalized size of antiderivative = 2.19 \[ \int \frac {6 x+4 x^2+(-6-4 x) \log \left (\frac {1}{-4-8 x+(1+2 x) \log (8)}\right )}{1+2 x} \, dx=x^{2} + {\left (2 \, x - \log \left (2 \, x + 1\right )\right )} \log \left (6 \, x \log \left (2\right ) - 8 \, x + 3 \, \log \left (2\right ) - 4\right ) + 2 \, \log \left (2 \, x + 1\right )^{2} + 3 \, \log \left (2 \, x + 1\right ) \log \left (3 \, \log \left (2\right ) - 4\right ) \]
x^2 + (2*x - log(2*x + 1))*log(6*x*log(2) - 8*x + 3*log(2) - 4) + 2*log(2* x + 1)^2 + 3*log(2*x + 1)*log(3*log(2) - 4)
Leaf count of result is larger than twice the leaf count of optimal. 405 vs. \(2 (28) = 56\).
Time = 1.54 (sec) , antiderivative size = 405, normalized size of antiderivative = 15.58 \[ \int \frac {6 x+4 x^2+(-6-4 x) \log \left (\frac {1}{-4-8 x+(1+2 x) \log (8)}\right )}{1+2 x} \, dx =\text {Too large to display} \]
-1/144*(24*(9*log(2)/(6*x*log(2) - 8*x + 3*log(2) - 4) - 12/(6*x*log(2) - 8*x + 3*log(2) - 4) + 1)*log(6*x*log(2) - 8*x + 3*log(2) - 4)/(81*log(2)^4 /(6*x*log(2) - 8*x + 3*log(2) - 4)^3 - 432*log(2)^3/(6*x*log(2) - 8*x + 3* log(2) - 4)^3 + 864*log(2)^2/(6*x*log(2) - 8*x + 3*log(2) - 4)^3 - 768*log (2)/(6*x*log(2) - 8*x + 3*log(2) - 4)^3 + 256/(6*x*log(2) - 8*x + 3*log(2) - 4)^3) + (12*log(2)/(6*x*log(2) - 8*x + 3*log(2) - 4) - 648*log(2)^2/(6* x*log(2) - 8*x + 3*log(2) - 4)^2 - 16/(6*x*log(2) - 8*x + 3*log(2) - 4) + 1728*log(2)/(6*x*log(2) - 8*x + 3*log(2) - 4)^2 - 1152/(6*x*log(2) - 8*x + 3*log(2) - 4)^2 + 9)/(243*log(2)^5/(6*x*log(2) - 8*x + 3*log(2) - 4)^4 - 1620*log(2)^4/(6*x*log(2) - 8*x + 3*log(2) - 4)^4 + 4320*log(2)^3/(6*x*log (2) - 8*x + 3*log(2) - 4)^4 - 5760*log(2)^2/(6*x*log(2) - 8*x + 3*log(2) - 4)^4 + 3840*log(2)/(6*x*log(2) - 8*x + 3*log(2) - 4)^4 - 1024/(6*x*log(2) - 8*x + 3*log(2) - 4)^4))*(3*log(2) - 4)^2
Time = 17.17 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.96 \[ \int \frac {6 x+4 x^2+(-6-4 x) \log \left (\frac {1}{-4-8 x+(1+2 x) \log (8)}\right )}{1+2 x} \, dx={\left (x-\ln \left (-\frac {1}{8\,x-3\,\ln \left (2\right )\,\left (2\,x+1\right )+4}\right )\right )}^2 \]