Integrand size = 182, antiderivative size = 28 \[ \int \frac {e^4 \left (4-4 x+9 x^2-9 x^3\right )+\left (36 x^3-36 x^2 \log (x)\right ) \log \left (48+108 x^2\right )+\left (4-4 x+9 x^2-9 x^3\right ) \log ^2\left (48+108 x^2\right )}{e^8 \left (4 x+9 x^3\right )+e^4 \left (-4 x^2-9 x^4\right )+e^4 \left (4 x+9 x^3\right ) \log (x)+\left (-4 x^2-9 x^4+e^4 \left (8 x+18 x^3\right )+\left (4 x+9 x^3\right ) \log (x)\right ) \log ^2\left (48+108 x^2\right )+\left (4 x+9 x^3\right ) \log ^4\left (48+108 x^2\right )} \, dx=\log \left (-1+\frac {x-\log (x)}{e^4+\log ^2\left (12 \left (4+9 x^2\right )\right )}\right ) \]
Time = 0.12 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.50 \[ \int \frac {e^4 \left (4-4 x+9 x^2-9 x^3\right )+\left (36 x^3-36 x^2 \log (x)\right ) \log \left (48+108 x^2\right )+\left (4-4 x+9 x^2-9 x^3\right ) \log ^2\left (48+108 x^2\right )}{e^8 \left (4 x+9 x^3\right )+e^4 \left (-4 x^2-9 x^4\right )+e^4 \left (4 x+9 x^3\right ) \log (x)+\left (-4 x^2-9 x^4+e^4 \left (8 x+18 x^3\right )+\left (4 x+9 x^3\right ) \log (x)\right ) \log ^2\left (48+108 x^2\right )+\left (4 x+9 x^3\right ) \log ^4\left (48+108 x^2\right )} \, dx=-\log \left (e^4+\log ^2\left (12 \left (4+9 x^2\right )\right )\right )+\log \left (e^4-x+\log (x)+\log ^2\left (12 \left (4+9 x^2\right )\right )\right ) \]
Integrate[(E^4*(4 - 4*x + 9*x^2 - 9*x^3) + (36*x^3 - 36*x^2*Log[x])*Log[48 + 108*x^2] + (4 - 4*x + 9*x^2 - 9*x^3)*Log[48 + 108*x^2]^2)/(E^8*(4*x + 9 *x^3) + E^4*(-4*x^2 - 9*x^4) + E^4*(4*x + 9*x^3)*Log[x] + (-4*x^2 - 9*x^4 + E^4*(8*x + 18*x^3) + (4*x + 9*x^3)*Log[x])*Log[48 + 108*x^2]^2 + (4*x + 9*x^3)*Log[48 + 108*x^2]^4),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.
| \(\displaystyle \int \frac {e^4 \left (-9 x^3+9 x^2-4 x+4\right )+\left (-9 x^3+9 x^2-4 x+4\right ) \log ^2\left (108 x^2+48\right )+\left (36 x^3-36 x^2 \log (x)\right ) \log \left (108 x^2+48\right )}{e^8 \left (9 x^3+4 x\right )+e^4 \left (9 x^3+4 x\right ) \log (x)+e^4 \left (-9 x^4-4 x^2\right )+\left (9 x^3+4 x\right ) \log ^4\left (108 x^2+48\right )+\left (-9 x^4+e^4 \left (18 x^3+8 x\right )+\left (9 x^3+4 x\right ) \log (x)-4 x^2\right ) \log ^2\left (108 x^2+48\right )} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {e^4 \left (-9 x^3+9 x^2-4 x+4\right )+\left (-9 x^3+9 x^2-4 x+4\right ) \log ^2\left (108 x^2+48\right )+\left (36 x^3-36 x^2 \log (x)\right ) \log \left (108 x^2+48\right )}{x \left (9 x^2+4\right ) \left (\log ^2\left (12 \left (9 x^2+4\right )\right )+e^4\right ) \left (\log ^2\left (12 \left (9 x^2+4\right )\right )-x+\log (x)+e^4\right )}dx\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \int \left (\frac {(1-x) \log ^2\left (108 x^2+48\right )}{x \left (\log ^2\left (12 \left (9 x^2+4\right )\right )+e^4\right ) \left (\log ^2\left (12 \left (9 x^2+4\right )\right )-x+\log (x)+e^4\right )}+\frac {36 x (x-\log (x)) \log \left (108 x^2+48\right )}{\left (9 x^2+4\right ) \left (\log ^2\left (12 \left (9 x^2+4\right )\right )+e^4\right ) \left (\log ^2\left (12 \left (9 x^2+4\right )\right )-x+\log (x)+e^4\right )}+\frac {e^4 (x-1)}{x \left (-\log ^2\left (12 \left (9 x^2+4\right )\right )+x-\log (x)-e^4\right ) \left (\log ^2\left (12 \left (9 x^2+4\right )\right )+e^4\right )}\right )dx\) |
| \(\Big \downarrow \) 7299 |
| \(\displaystyle \int \left (\frac {(1-x) \log ^2\left (108 x^2+48\right )}{x \left (\log ^2\left (12 \left (9 x^2+4\right )\right )+e^4\right ) \left (\log ^2\left (12 \left (9 x^2+4\right )\right )-x+\log (x)+e^4\right )}+\frac {36 x (x-\log (x)) \log \left (108 x^2+48\right )}{\left (9 x^2+4\right ) \left (\log ^2\left (12 \left (9 x^2+4\right )\right )+e^4\right ) \left (\log ^2\left (12 \left (9 x^2+4\right )\right )-x+\log (x)+e^4\right )}+\frac {e^4 (x-1)}{x \left (-\log ^2\left (12 \left (9 x^2+4\right )\right )+x-\log (x)-e^4\right ) \left (\log ^2\left (12 \left (9 x^2+4\right )\right )+e^4\right )}\right )dx\) |
Int[(E^4*(4 - 4*x + 9*x^2 - 9*x^3) + (36*x^3 - 36*x^2*Log[x])*Log[48 + 108 *x^2] + (4 - 4*x + 9*x^2 - 9*x^3)*Log[48 + 108*x^2]^2)/(E^8*(4*x + 9*x^3) + E^4*(-4*x^2 - 9*x^4) + E^4*(4*x + 9*x^3)*Log[x] + (-4*x^2 - 9*x^4 + E^4* (8*x + 18*x^3) + (4*x + 9*x^3)*Log[x])*Log[48 + 108*x^2]^2 + (4*x + 9*x^3) *Log[48 + 108*x^2]^4),x]
3.16.92.3.1 Defintions of rubi rules used
Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionE xpand[u/(a + b*x^n), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ [n, 0]
Time = 30.79 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.32
| method | result | size |
| risch | \(\ln \left (\ln \left (108 x^{2}+48\right )^{2}+{\mathrm e}^{4}-x +\ln \left (x \right )\right )-\ln \left (\ln \left (108 x^{2}+48\right )^{2}+{\mathrm e}^{4}\right )\) | \(37\) |
| parallelrisch | \(-\ln \left (\ln \left (108 x^{2}+48\right )^{2}+{\mathrm e}^{4}\right )+\ln \left (-\ln \left (108 x^{2}+48\right )^{2}-{\mathrm e}^{4}+x -\ln \left (x \right )\right )\) | \(41\) |
| default | \(-\ln \left (\ln \left (9 x^{2}+4\right )^{2}+\left (2 \ln \left (3\right )+4 \ln \left (2\right )\right ) \ln \left (9 x^{2}+4\right )+\ln \left (3\right )^{2}+4 \ln \left (2\right ) \ln \left (3\right )+4 \ln \left (2\right )^{2}+{\mathrm e}^{4}\right )+\ln \left (\ln \left (9 x^{2}+4\right )^{2}+\left (2 \ln \left (3\right )+4 \ln \left (2\right )\right ) \ln \left (9 x^{2}+4\right )+\ln \left (3\right )^{2}+4 \ln \left (2\right ) \ln \left (3\right )+4 \ln \left (2\right )^{2}+{\mathrm e}^{4}+\ln \left (x \right )-x \right )\) | \(105\) |
int(((-9*x^3+9*x^2-4*x+4)*ln(108*x^2+48)^2+(-36*x^2*ln(x)+36*x^3)*ln(108*x ^2+48)+(-9*x^3+9*x^2-4*x+4)*exp(4))/((9*x^3+4*x)*ln(108*x^2+48)^4+((9*x^3+ 4*x)*ln(x)+(18*x^3+8*x)*exp(4)-9*x^4-4*x^2)*ln(108*x^2+48)^2+(9*x^3+4*x)*e xp(4)*ln(x)+(9*x^3+4*x)*exp(4)^2+(-9*x^4-4*x^2)*exp(4)),x,method=_RETURNVE RBOSE)
Time = 0.37 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.29 \[ \int \frac {e^4 \left (4-4 x+9 x^2-9 x^3\right )+\left (36 x^3-36 x^2 \log (x)\right ) \log \left (48+108 x^2\right )+\left (4-4 x+9 x^2-9 x^3\right ) \log ^2\left (48+108 x^2\right )}{e^8 \left (4 x+9 x^3\right )+e^4 \left (-4 x^2-9 x^4\right )+e^4 \left (4 x+9 x^3\right ) \log (x)+\left (-4 x^2-9 x^4+e^4 \left (8 x+18 x^3\right )+\left (4 x+9 x^3\right ) \log (x)\right ) \log ^2\left (48+108 x^2\right )+\left (4 x+9 x^3\right ) \log ^4\left (48+108 x^2\right )} \, dx=\log \left (\log \left (108 \, x^{2} + 48\right )^{2} - x + e^{4} + \log \left (x\right )\right ) - \log \left (\log \left (108 \, x^{2} + 48\right )^{2} + e^{4}\right ) \]
integrate(((-9*x^3+9*x^2-4*x+4)*log(108*x^2+48)^2+(-36*x^2*log(x)+36*x^3)* log(108*x^2+48)+(-9*x^3+9*x^2-4*x+4)*exp(4))/((9*x^3+4*x)*log(108*x^2+48)^ 4+((9*x^3+4*x)*log(x)+(18*x^3+8*x)*exp(4)-9*x^4-4*x^2)*log(108*x^2+48)^2+( 9*x^3+4*x)*exp(4)*log(x)+(9*x^3+4*x)*exp(4)^2+(-9*x^4-4*x^2)*exp(4)),x, al gorithm=\
Exception generated. \[ \int \frac {e^4 \left (4-4 x+9 x^2-9 x^3\right )+\left (36 x^3-36 x^2 \log (x)\right ) \log \left (48+108 x^2\right )+\left (4-4 x+9 x^2-9 x^3\right ) \log ^2\left (48+108 x^2\right )}{e^8 \left (4 x+9 x^3\right )+e^4 \left (-4 x^2-9 x^4\right )+e^4 \left (4 x+9 x^3\right ) \log (x)+\left (-4 x^2-9 x^4+e^4 \left (8 x+18 x^3\right )+\left (4 x+9 x^3\right ) \log (x)\right ) \log ^2\left (48+108 x^2\right )+\left (4 x+9 x^3\right ) \log ^4\left (48+108 x^2\right )} \, dx=\text {Exception raised: PolynomialError} \]
integrate(((-9*x**3+9*x**2-4*x+4)*ln(108*x**2+48)**2+(-36*x**2*ln(x)+36*x* *3)*ln(108*x**2+48)+(-9*x**3+9*x**2-4*x+4)*exp(4))/((9*x**3+4*x)*ln(108*x* *2+48)**4+((9*x**3+4*x)*ln(x)+(18*x**3+8*x)*exp(4)-9*x**4-4*x**2)*ln(108*x **2+48)**2+(9*x**3+4*x)*exp(4)*ln(x)+(9*x**3+4*x)*exp(4)**2+(-9*x**4-4*x** 2)*exp(4)),x)
Exception raised: PolynomialError >> 1/(81*x**6 + 72*x**4 + 16*x**2) conta ins an element of the set of generators.
Leaf count of result is larger than twice the leaf count of optimal. 102 vs. \(2 (25) = 50\).
Time = 0.34 (sec) , antiderivative size = 102, normalized size of antiderivative = 3.64 \[ \int \frac {e^4 \left (4-4 x+9 x^2-9 x^3\right )+\left (36 x^3-36 x^2 \log (x)\right ) \log \left (48+108 x^2\right )+\left (4-4 x+9 x^2-9 x^3\right ) \log ^2\left (48+108 x^2\right )}{e^8 \left (4 x+9 x^3\right )+e^4 \left (-4 x^2-9 x^4\right )+e^4 \left (4 x+9 x^3\right ) \log (x)+\left (-4 x^2-9 x^4+e^4 \left (8 x+18 x^3\right )+\left (4 x+9 x^3\right ) \log (x)\right ) \log ^2\left (48+108 x^2\right )+\left (4 x+9 x^3\right ) \log ^4\left (48+108 x^2\right )} \, dx=\log \left (\log \left (3\right )^{2} + 4 \, \log \left (3\right ) \log \left (2\right ) + 4 \, \log \left (2\right )^{2} + 2 \, {\left (\log \left (3\right ) + 2 \, \log \left (2\right )\right )} \log \left (9 \, x^{2} + 4\right ) + \log \left (9 \, x^{2} + 4\right )^{2} - x + e^{4} + \log \left (x\right )\right ) - \log \left (\log \left (3\right )^{2} + 4 \, \log \left (3\right ) \log \left (2\right ) + 4 \, \log \left (2\right )^{2} + 2 \, {\left (\log \left (3\right ) + 2 \, \log \left (2\right )\right )} \log \left (9 \, x^{2} + 4\right ) + \log \left (9 \, x^{2} + 4\right )^{2} + e^{4}\right ) \]
integrate(((-9*x^3+9*x^2-4*x+4)*log(108*x^2+48)^2+(-36*x^2*log(x)+36*x^3)* log(108*x^2+48)+(-9*x^3+9*x^2-4*x+4)*exp(4))/((9*x^3+4*x)*log(108*x^2+48)^ 4+((9*x^3+4*x)*log(x)+(18*x^3+8*x)*exp(4)-9*x^4-4*x^2)*log(108*x^2+48)^2+( 9*x^3+4*x)*exp(4)*log(x)+(9*x^3+4*x)*exp(4)^2+(-9*x^4-4*x^2)*exp(4)),x, al gorithm=\
log(log(3)^2 + 4*log(3)*log(2) + 4*log(2)^2 + 2*(log(3) + 2*log(2))*log(9* x^2 + 4) + log(9*x^2 + 4)^2 - x + e^4 + log(x)) - log(log(3)^2 + 4*log(3)* log(2) + 4*log(2)^2 + 2*(log(3) + 2*log(2))*log(9*x^2 + 4) + log(9*x^2 + 4 )^2 + e^4)
Time = 0.97 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.29 \[ \int \frac {e^4 \left (4-4 x+9 x^2-9 x^3\right )+\left (36 x^3-36 x^2 \log (x)\right ) \log \left (48+108 x^2\right )+\left (4-4 x+9 x^2-9 x^3\right ) \log ^2\left (48+108 x^2\right )}{e^8 \left (4 x+9 x^3\right )+e^4 \left (-4 x^2-9 x^4\right )+e^4 \left (4 x+9 x^3\right ) \log (x)+\left (-4 x^2-9 x^4+e^4 \left (8 x+18 x^3\right )+\left (4 x+9 x^3\right ) \log (x)\right ) \log ^2\left (48+108 x^2\right )+\left (4 x+9 x^3\right ) \log ^4\left (48+108 x^2\right )} \, dx=\log \left (\log \left (108 \, x^{2} + 48\right )^{2} - x + e^{4} + \log \left (x\right )\right ) - \log \left (\log \left (108 \, x^{2} + 48\right )^{2} + e^{4}\right ) \]
integrate(((-9*x^3+9*x^2-4*x+4)*log(108*x^2+48)^2+(-36*x^2*log(x)+36*x^3)* log(108*x^2+48)+(-9*x^3+9*x^2-4*x+4)*exp(4))/((9*x^3+4*x)*log(108*x^2+48)^ 4+((9*x^3+4*x)*log(x)+(18*x^3+8*x)*exp(4)-9*x^4-4*x^2)*log(108*x^2+48)^2+( 9*x^3+4*x)*exp(4)*log(x)+(9*x^3+4*x)*exp(4)^2+(-9*x^4-4*x^2)*exp(4)),x, al gorithm=\
Timed out. \[ \int \frac {e^4 \left (4-4 x+9 x^2-9 x^3\right )+\left (36 x^3-36 x^2 \log (x)\right ) \log \left (48+108 x^2\right )+\left (4-4 x+9 x^2-9 x^3\right ) \log ^2\left (48+108 x^2\right )}{e^8 \left (4 x+9 x^3\right )+e^4 \left (-4 x^2-9 x^4\right )+e^4 \left (4 x+9 x^3\right ) \log (x)+\left (-4 x^2-9 x^4+e^4 \left (8 x+18 x^3\right )+\left (4 x+9 x^3\right ) \log (x)\right ) \log ^2\left (48+108 x^2\right )+\left (4 x+9 x^3\right ) \log ^4\left (48+108 x^2\right )} \, dx=-\int \frac {\left (9\,x^3-9\,x^2+4\,x-4\right )\,{\ln \left (108\,x^2+48\right )}^2+\left (36\,x^2\,\ln \left (x\right )-36\,x^3\right )\,\ln \left (108\,x^2+48\right )+{\mathrm {e}}^4\,\left (9\,x^3-9\,x^2+4\,x-4\right )}{\left (9\,x^3+4\,x\right )\,{\ln \left (108\,x^2+48\right )}^4+\left ({\mathrm {e}}^4\,\left (18\,x^3+8\,x\right )+\ln \left (x\right )\,\left (9\,x^3+4\,x\right )-4\,x^2-9\,x^4\right )\,{\ln \left (108\,x^2+48\right )}^2+{\mathrm {e}}^8\,\left (9\,x^3+4\,x\right )-{\mathrm {e}}^4\,\left (9\,x^4+4\,x^2\right )+{\mathrm {e}}^4\,\ln \left (x\right )\,\left (9\,x^3+4\,x\right )} \,d x \]
int(-(log(108*x^2 + 48)*(36*x^2*log(x) - 36*x^3) + log(108*x^2 + 48)^2*(4* x - 9*x^2 + 9*x^3 - 4) + exp(4)*(4*x - 9*x^2 + 9*x^3 - 4))/(log(108*x^2 + 48)^4*(4*x + 9*x^3) + exp(8)*(4*x + 9*x^3) + log(108*x^2 + 48)^2*(exp(4)*( 8*x + 18*x^3) + log(x)*(4*x + 9*x^3) - 4*x^2 - 9*x^4) - exp(4)*(4*x^2 + 9* x^4) + exp(4)*log(x)*(4*x + 9*x^3)),x)
-int((log(108*x^2 + 48)*(36*x^2*log(x) - 36*x^3) + log(108*x^2 + 48)^2*(4* x - 9*x^2 + 9*x^3 - 4) + exp(4)*(4*x - 9*x^2 + 9*x^3 - 4))/(log(108*x^2 + 48)^4*(4*x + 9*x^3) + exp(8)*(4*x + 9*x^3) + log(108*x^2 + 48)^2*(exp(4)*( 8*x + 18*x^3) + log(x)*(4*x + 9*x^3) - 4*x^2 - 9*x^4) - exp(4)*(4*x^2 + 9* x^4) + exp(4)*log(x)*(4*x + 9*x^3)), x)