Integrand size = 99, antiderivative size = 32 \[ \int \frac {84+198 x+152 x^2+60 x^3+10 x^4+\left (72+132 x+96 x^2+32 x^3+4 x^4\right ) \log \left (\frac {12+12 x+4 x^2}{12+16 x+7 x^2+x^3}\right )}{18-3 x-24 x^2-7 x^3+9 x^4+6 x^5+x^6} \, dx=\frac {2+4 \left (x+\log \left (\frac {3+\frac {x^2}{(2+x)^2}}{3+x}\right )\right )}{1-x} \] Output:
(2+4*x+4*ln((3+1/(2+x)^2*x^2)/(3+x)))/(1-x)
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 7.16 (sec) , antiderivative size = 8222, normalized size of antiderivative = 256.94 \[ \int \frac {84+198 x+152 x^2+60 x^3+10 x^4+\left (72+132 x+96 x^2+32 x^3+4 x^4\right ) \log \left (\frac {12+12 x+4 x^2}{12+16 x+7 x^2+x^3}\right )}{18-3 x-24 x^2-7 x^3+9 x^4+6 x^5+x^6} \, dx=\text {Result too large to show} \] Input:
Integrate[(84 + 198*x + 152*x^2 + 60*x^3 + 10*x^4 + (72 + 132*x + 96*x^2 + 32*x^3 + 4*x^4)*Log[(12 + 12*x + 4*x^2)/(12 + 16*x + 7*x^2 + x^3)])/(18 - 3*x - 24*x^2 - 7*x^3 + 9*x^4 + 6*x^5 + x^6),x]
Output:
Result too large to show
Time = 6.41 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.25, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.020, Rules used = {2463, 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.
\(\displaystyle \int \frac {10 x^4+60 x^3+152 x^2+\left (4 x^4+32 x^3+96 x^2+132 x+72\right ) \log \left (\frac {4 x^2+12 x+12}{x^3+7 x^2+16 x+12}\right )+198 x+84}{x^6+6 x^5+9 x^4-7 x^3-24 x^2-3 x+18} \, dx\) |
\(\Big \downarrow \) 2463 |
\(\displaystyle \int \left (-\frac {109 \left (10 x^4+60 x^3+152 x^2+\left (4 x^4+32 x^3+96 x^2+132 x+72\right ) \log \left (\frac {4 x^2+12 x+12}{x^3+7 x^2+16 x+12}\right )+198 x+84\right )}{7056 (x-1)}+\frac {10 x^4+60 x^3+152 x^2+\left (4 x^4+32 x^3+96 x^2+132 x+72\right ) \log \left (\frac {4 x^2+12 x+12}{x^3+7 x^2+16 x+12}\right )+198 x+84}{9 (x+2)}-\frac {10 x^4+60 x^3+152 x^2+\left (4 x^4+32 x^3+96 x^2+132 x+72\right ) \log \left (\frac {4 x^2+12 x+12}{x^3+7 x^2+16 x+12}\right )+198 x+84}{48 (x+3)}+\frac {(-11 x-9) \left (10 x^4+60 x^3+152 x^2+\left (4 x^4+32 x^3+96 x^2+132 x+72\right ) \log \left (\frac {4 x^2+12 x+12}{x^3+7 x^2+16 x+12}\right )+198 x+84\right )}{147 \left (x^2+3 x+3\right )}+\frac {10 x^4+60 x^3+152 x^2+\left (4 x^4+32 x^3+96 x^2+132 x+72\right ) \log \left (\frac {4 x^2+12 x+12}{x^3+7 x^2+16 x+12}\right )+198 x+84}{84 (x-1)^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {4 \log \left (\frac {4 \left (x^2+3 x+3\right )}{(x+2)^2 (x+3)}\right )}{1-x}+\frac {6}{1-x}\) |
Input:
Int[(84 + 198*x + 152*x^2 + 60*x^3 + 10*x^4 + (72 + 132*x + 96*x^2 + 32*x^ 3 + 4*x^4)*Log[(12 + 12*x + 4*x^2)/(12 + 16*x + 7*x^2 + x^3)])/(18 - 3*x - 24*x^2 - 7*x^3 + 9*x^4 + 6*x^5 + x^6),x]
Output:
6/(1 - x) + (4*Log[(4*(3 + 3*x + x^2))/((2 + x)^2*(3 + x))])/(1 - x)
Int[(u_.)*(Px_)^(p_), x_Symbol] :> With[{Qx = Factor[Px]}, Int[ExpandIntegr and[u, Qx^p, x], x] /; !SumQ[NonfreeFactors[Qx, x]]] /; PolyQ[Px, x] && Gt Q[Expon[Px, x], 2] && !BinomialQ[Px, x] && !TrinomialQ[Px, x] && ILtQ[p, 0]
Time = 1.20 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.19
method | result | size |
norman | \(\frac {-4 \ln \left (\frac {4 x^{2}+12 x +12}{x^{3}+7 x^{2}+16 x +12}\right )-6}{-1+x}\) | \(38\) |
parallelrisch | \(-\frac {6+4 \ln \left (\frac {4 x^{2}+12 x +12}{x^{3}+7 x^{2}+16 x +12}\right )}{-1+x}\) | \(38\) |
risch | \(-\frac {4 \ln \left (\frac {4 x^{2}+12 x +12}{x^{3}+7 x^{2}+16 x +12}\right )}{-1+x}-\frac {6}{-1+x}\) | \(43\) |
orering | \(-\frac {\left (3 x^{7}+31 x^{6}+142 x^{5}+394 x^{4}+732 x^{3}+870 x^{2}+558 x +126\right ) \left (\left (4 x^{4}+32 x^{3}+96 x^{2}+132 x +72\right ) \ln \left (\frac {4 x^{2}+12 x +12}{x^{3}+7 x^{2}+16 x +12}\right )+10 x^{4}+60 x^{3}+152 x^{2}+198 x +84\right )}{\left (x^{6}+8 x^{5}+26 x^{4}+54 x^{3}+102 x^{2}+144 x +90\right ) \left (x^{6}+6 x^{5}+9 x^{4}-7 x^{3}-24 x^{2}-3 x +18\right )}-\frac {\left (x^{3}+4 x^{2}+6 x +6\right ) \left (-1+x \right ) \left (2+x \right ) \left (x^{2}+3 x +3\right ) \left (3+x \right ) \left (\frac {\left (16 x^{3}+96 x^{2}+192 x +132\right ) \ln \left (\frac {4 x^{2}+12 x +12}{x^{3}+7 x^{2}+16 x +12}\right )+\frac {\left (4 x^{4}+32 x^{3}+96 x^{2}+132 x +72\right ) \left (\frac {8 x +12}{x^{3}+7 x^{2}+16 x +12}-\frac {\left (4 x^{2}+12 x +12\right ) \left (3 x^{2}+14 x +16\right )}{\left (x^{3}+7 x^{2}+16 x +12\right )^{2}}\right ) \left (x^{3}+7 x^{2}+16 x +12\right )}{4 x^{2}+12 x +12}+40 x^{3}+180 x^{2}+304 x +198}{x^{6}+6 x^{5}+9 x^{4}-7 x^{3}-24 x^{2}-3 x +18}-\frac {\left (\left (4 x^{4}+32 x^{3}+96 x^{2}+132 x +72\right ) \ln \left (\frac {4 x^{2}+12 x +12}{x^{3}+7 x^{2}+16 x +12}\right )+10 x^{4}+60 x^{3}+152 x^{2}+198 x +84\right ) \left (6 x^{5}+30 x^{4}+36 x^{3}-21 x^{2}-48 x -3\right )}{\left (x^{6}+6 x^{5}+9 x^{4}-7 x^{3}-24 x^{2}-3 x +18\right )^{2}}\right )}{x^{6}+8 x^{5}+26 x^{4}+54 x^{3}+102 x^{2}+144 x +90}\) | \(549\) |
Input:
int(((4*x^4+32*x^3+96*x^2+132*x+72)*ln((4*x^2+12*x+12)/(x^3+7*x^2+16*x+12) )+10*x^4+60*x^3+152*x^2+198*x+84)/(x^6+6*x^5+9*x^4-7*x^3-24*x^2-3*x+18),x, method=_RETURNVERBOSE)
Output:
(-4*ln((4*x^2+12*x+12)/(x^3+7*x^2+16*x+12))-6)/(-1+x)
Time = 0.07 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.16 \[ \int \frac {84+198 x+152 x^2+60 x^3+10 x^4+\left (72+132 x+96 x^2+32 x^3+4 x^4\right ) \log \left (\frac {12+12 x+4 x^2}{12+16 x+7 x^2+x^3}\right )}{18-3 x-24 x^2-7 x^3+9 x^4+6 x^5+x^6} \, dx=-\frac {2 \, {\left (2 \, \log \left (\frac {4 \, {\left (x^{2} + 3 \, x + 3\right )}}{x^{3} + 7 \, x^{2} + 16 \, x + 12}\right ) + 3\right )}}{x - 1} \] Input:
integrate(((4*x^4+32*x^3+96*x^2+132*x+72)*log((4*x^2+12*x+12)/(x^3+7*x^2+1 6*x+12))+10*x^4+60*x^3+152*x^2+198*x+84)/(x^6+6*x^5+9*x^4-7*x^3-24*x^2-3*x +18),x, algorithm="fricas")
Output:
-2*(2*log(4*(x^2 + 3*x + 3)/(x^3 + 7*x^2 + 16*x + 12)) + 3)/(x - 1)
Time = 0.13 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.12 \[ \int \frac {84+198 x+152 x^2+60 x^3+10 x^4+\left (72+132 x+96 x^2+32 x^3+4 x^4\right ) \log \left (\frac {12+12 x+4 x^2}{12+16 x+7 x^2+x^3}\right )}{18-3 x-24 x^2-7 x^3+9 x^4+6 x^5+x^6} \, dx=- \frac {4 \log {\left (\frac {4 x^{2} + 12 x + 12}{x^{3} + 7 x^{2} + 16 x + 12} \right )}}{x - 1} - \frac {6}{x - 1} \] Input:
integrate(((4*x**4+32*x**3+96*x**2+132*x+72)*ln((4*x**2+12*x+12)/(x**3+7*x **2+16*x+12))+10*x**4+60*x**3+152*x**2+198*x+84)/(x**6+6*x**5+9*x**4-7*x** 3-24*x**2-3*x+18),x)
Output:
-4*log((4*x**2 + 12*x + 12)/(x**3 + 7*x**2 + 16*x + 12))/(x - 1) - 6/(x - 1)
Leaf count of result is larger than twice the leaf count of optimal. 77 vs. \(2 (32) = 64\).
Time = 1.37 (sec) , antiderivative size = 77, normalized size of antiderivative = 2.41 \[ \int \frac {84+198 x+152 x^2+60 x^3+10 x^4+\left (72+132 x+96 x^2+32 x^3+4 x^4\right ) \log \left (\frac {12+12 x+4 x^2}{12+16 x+7 x^2+x^3}\right )}{18-3 x-24 x^2-7 x^3+9 x^4+6 x^5+x^6} \, dx=-\frac {6 \, {\left (5 \, x + 9\right )} \log \left (x^{2} + 3 \, x + 3\right ) - 21 \, {\left (x + 3\right )} \log \left (x + 3\right ) - 56 \, {\left (x + 2\right )} \log \left (x + 2\right ) + 168 \, \log \left (2\right )}{21 \, {\left (x - 1\right )}} - \frac {6}{x - 1} + \frac {10}{7} \, \log \left (x^{2} + 3 \, x + 3\right ) - \log \left (x + 3\right ) - \frac {8}{3} \, \log \left (x + 2\right ) \] Input:
integrate(((4*x^4+32*x^3+96*x^2+132*x+72)*log((4*x^2+12*x+12)/(x^3+7*x^2+1 6*x+12))+10*x^4+60*x^3+152*x^2+198*x+84)/(x^6+6*x^5+9*x^4-7*x^3-24*x^2-3*x +18),x, algorithm="maxima")
Output:
-1/21*(6*(5*x + 9)*log(x^2 + 3*x + 3) - 21*(x + 3)*log(x + 3) - 56*(x + 2) *log(x + 2) + 168*log(2))/(x - 1) - 6/(x - 1) + 10/7*log(x^2 + 3*x + 3) - log(x + 3) - 8/3*log(x + 2)
Time = 0.16 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.28 \[ \int \frac {84+198 x+152 x^2+60 x^3+10 x^4+\left (72+132 x+96 x^2+32 x^3+4 x^4\right ) \log \left (\frac {12+12 x+4 x^2}{12+16 x+7 x^2+x^3}\right )}{18-3 x-24 x^2-7 x^3+9 x^4+6 x^5+x^6} \, dx=-\frac {4 \, \log \left (\frac {4 \, {\left (x^{2} + 3 \, x + 3\right )}}{x^{3} + 7 \, x^{2} + 16 \, x + 12}\right )}{x - 1} - \frac {6}{x - 1} \] Input:
integrate(((4*x^4+32*x^3+96*x^2+132*x+72)*log((4*x^2+12*x+12)/(x^3+7*x^2+1 6*x+12))+10*x^4+60*x^3+152*x^2+198*x+84)/(x^6+6*x^5+9*x^4-7*x^3-24*x^2-3*x +18),x, algorithm="giac")
Output:
-4*log(4*(x^2 + 3*x + 3)/(x^3 + 7*x^2 + 16*x + 12))/(x - 1) - 6/(x - 1)
Time = 0.84 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.31 \[ \int \frac {84+198 x+152 x^2+60 x^3+10 x^4+\left (72+132 x+96 x^2+32 x^3+4 x^4\right ) \log \left (\frac {12+12 x+4 x^2}{12+16 x+7 x^2+x^3}\right )}{18-3 x-24 x^2-7 x^3+9 x^4+6 x^5+x^6} \, dx=-\frac {6}{x-1}-\frac {4\,\ln \left (\frac {4\,x^2+12\,x+12}{x^3+7\,x^2+16\,x+12}\right )}{x-1} \] Input:
int((198*x + log((12*x + 4*x^2 + 12)/(16*x + 7*x^2 + x^3 + 12))*(132*x + 9 6*x^2 + 32*x^3 + 4*x^4 + 72) + 152*x^2 + 60*x^3 + 10*x^4 + 84)/(9*x^4 - 24 *x^2 - 7*x^3 - 3*x + 6*x^5 + x^6 + 18),x)
Output:
- 6/(x - 1) - (4*log((12*x + 4*x^2 + 12)/(16*x + 7*x^2 + x^3 + 12)))/(x - 1)
Time = 0.33 (sec) , antiderivative size = 90, normalized size of antiderivative = 2.81 \[ \int \frac {84+198 x+152 x^2+60 x^3+10 x^4+\left (72+132 x+96 x^2+32 x^3+4 x^4\right ) \log \left (\frac {12+12 x+4 x^2}{12+16 x+7 x^2+x^3}\right )}{18-3 x-24 x^2-7 x^3+9 x^4+6 x^5+x^6} \, dx=\frac {4 \,\mathrm {log}\left (x^{2}+3 x +3\right ) x -4 \,\mathrm {log}\left (x^{2}+3 x +3\right )-4 \,\mathrm {log}\left (x +3\right ) x +4 \,\mathrm {log}\left (x +3\right )-8 \,\mathrm {log}\left (x +2\right ) x +8 \,\mathrm {log}\left (x +2\right )-4 \,\mathrm {log}\left (\frac {4 x^{2}+12 x +12}{x^{3}+7 x^{2}+16 x +12}\right ) x -6 x}{x -1} \] Input:
int(((4*x^4+32*x^3+96*x^2+132*x+72)*log((4*x^2+12*x+12)/(x^3+7*x^2+16*x+12 ))+10*x^4+60*x^3+152*x^2+198*x+84)/(x^6+6*x^5+9*x^4-7*x^3-24*x^2-3*x+18),x )
Output:
(2*(2*log(x**2 + 3*x + 3)*x - 2*log(x**2 + 3*x + 3) - 2*log(x + 3)*x + 2*l og(x + 3) - 4*log(x + 2)*x + 4*log(x + 2) - 2*log((4*x**2 + 12*x + 12)/(x* *3 + 7*x**2 + 16*x + 12))*x - 3*x))/(x - 1)