Integrand size = 138, antiderivative size = 31 \[ \int \frac {392-49 x+9 x^3 \log ^2(3)+e^{2 x} \left (72-81 x+18 x^2\right ) \log ^2(3)+e^x \left (\left (336-210 x+42 x^2\right ) \log (3)+\left (72-45 x+9 x^2\right ) \log ^2(3)\right )}{-196 x+49 x^2+9 x^4 \log ^2(3)+e^{2 x} \left (-36 x+9 x^2\right ) \log ^2(3)+e^x \left (\left (-168 x+42 x^2\right ) \log (3)+\left (-36 x+9 x^2\right ) \log ^2(3)\right )} \, dx=\log \left (x-\frac {(4-x) \left (e^x+\left (e^x+\frac {7}{3 \log (3)}\right )^2\right )}{x^2}\right ) \] Output:
ln(x-(4-x)/x^2*((exp(x)+7/3/ln(3))^2+exp(x)))
Leaf count is larger than twice the leaf count of optimal. \(77\) vs. \(2(31)=62\).
Time = 0.15 (sec) , antiderivative size = 77, normalized size of antiderivative = 2.48 \[ \int \frac {392-49 x+9 x^3 \log ^2(3)+e^{2 x} \left (72-81 x+18 x^2\right ) \log ^2(3)+e^x \left (\left (336-210 x+42 x^2\right ) \log (3)+\left (72-45 x+9 x^2\right ) \log ^2(3)\right )}{-196 x+49 x^2+9 x^4 \log ^2(3)+e^{2 x} \left (-36 x+9 x^2\right ) \log ^2(3)+e^x \left (\left (-168 x+42 x^2\right ) \log (3)+\left (-36 x+9 x^2\right ) \log ^2(3)\right )} \, dx=-2 \log (x)+\log \left (196-49 x+168 e^x \log (3)-42 e^x x \log (3)+36 e^{2 x} \log ^2(3)-9 e^{2 x} x \log ^2(3)-9 x^3 \log ^2(3)+12 e^x \log (3) \log (27)-3 e^x x \log (3) \log (27)\right ) \] Input:
Integrate[(392 - 49*x + 9*x^3*Log[3]^2 + E^(2*x)*(72 - 81*x + 18*x^2)*Log[ 3]^2 + E^x*((336 - 210*x + 42*x^2)*Log[3] + (72 - 45*x + 9*x^2)*Log[3]^2)) /(-196*x + 49*x^2 + 9*x^4*Log[3]^2 + E^(2*x)*(-36*x + 9*x^2)*Log[3]^2 + E^ x*((-168*x + 42*x^2)*Log[3] + (-36*x + 9*x^2)*Log[3]^2)),x]
Output:
-2*Log[x] + Log[196 - 49*x + 168*E^x*Log[3] - 42*E^x*x*Log[3] + 36*E^(2*x) *Log[3]^2 - 9*E^(2*x)*x*Log[3]^2 - 9*x^3*Log[3]^2 + 12*E^x*Log[3]*Log[27] - 3*E^x*x*Log[3]*Log[27]]
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.
| \(\displaystyle \int \frac {9 x^3 \log ^2(3)+e^x \left (\left (9 x^2-45 x+72\right ) \log ^2(3)+\left (42 x^2-210 x+336\right ) \log (3)\right )+e^{2 x} \left (18 x^2-81 x+72\right ) \log ^2(3)-49 x+392}{9 x^4 \log ^2(3)+49 x^2+e^x \left (\left (9 x^2-36 x\right ) \log ^2(3)+\left (42 x^2-168 x\right ) \log (3)\right )+e^{2 x} \left (9 x^2-36 x\right ) \log ^2(3)-196 x} \, dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {2 x^2-9 x+8}{(x-4) x}+\frac {-18 x^4 \log ^2(3)+90 x^3 \log ^2(3)-98 x^2 \left (1+\frac {54 \log ^2(3)}{49}\right )-42 e^x x^2 \left (1+\frac {3 \log (3)}{14}\right ) \log (3)+784 x+336 e^x x \left (1+\frac {3 \log (3)}{14}\right ) \log (3)-672 e^x \left (1+\frac {3 \log (3)}{14}\right ) \log (3)-1568}{(4-x) \left (-9 x^3 \log ^2(3)-49 x-9 e^{2 x} x \log ^2(3)+36 e^{2 x} \log ^2(3)-42 e^x x \left (1+\frac {3 \log (3)}{14}\right ) \log (3)+168 e^x \left (1+\frac {3 \log (3)}{14}\right ) \log (3)+196\right )}\right )dx\) |
| \(\Big \downarrow \) 7299 |
| \(\displaystyle \int \left (\frac {2 x^2-9 x+8}{(x-4) x}+\frac {-18 x^4 \log ^2(3)+90 x^3 \log ^2(3)-98 x^2 \left (1+\frac {54 \log ^2(3)}{49}\right )-42 e^x x^2 \left (1+\frac {3 \log (3)}{14}\right ) \log (3)+784 x+336 e^x x \left (1+\frac {3 \log (3)}{14}\right ) \log (3)-672 e^x \left (1+\frac {3 \log (3)}{14}\right ) \log (3)-1568}{(4-x) \left (-9 x^3 \log ^2(3)-49 x-9 e^{2 x} x \log ^2(3)+36 e^{2 x} \log ^2(3)-42 e^x x \left (1+\frac {3 \log (3)}{14}\right ) \log (3)+168 e^x \left (1+\frac {3 \log (3)}{14}\right ) \log (3)+196\right )}\right )dx\) |
Input:
Int[(392 - 49*x + 9*x^3*Log[3]^2 + E^(2*x)*(72 - 81*x + 18*x^2)*Log[3]^2 + E^x*((336 - 210*x + 42*x^2)*Log[3] + (72 - 45*x + 9*x^2)*Log[3]^2))/(-196 *x + 49*x^2 + 9*x^4*Log[3]^2 + E^(2*x)*(-36*x + 9*x^2)*Log[3]^2 + E^x*((-1 68*x + 42*x^2)*Log[3] + (-36*x + 9*x^2)*Log[3]^2)),x]
Output:
$Aborted
Time = 0.62 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.77
| method | result | size |
| risch | \(\ln \left (x -4\right )-2 \ln \left (x \right )+\ln \left ({\mathrm e}^{2 x}+\frac {\left (3 \ln \left (3\right )+14\right ) {\mathrm e}^{x}}{3 \ln \left (3\right )}+\frac {9 x^{3} \ln \left (3\right )^{2}+49 x -196}{9 \ln \left (3\right )^{2} \left (x -4\right )}\right )\) | \(55\) |
| norman | \(-2 \ln \left (x \right )+\ln \left (9 x^{3} \ln \left (3\right )^{2}+9 \,{\mathrm e}^{x} \ln \left (3\right )^{2} x +9 x \ln \left (3\right )^{2} {\mathrm e}^{2 x}-36 \ln \left (3\right )^{2} {\mathrm e}^{x}+42 x \ln \left (3\right ) {\mathrm e}^{x}-36 \ln \left (3\right )^{2} {\mathrm e}^{2 x}-168 \ln \left (3\right ) {\mathrm e}^{x}+49 x -196\right )\) | \(72\) |
| parallelrisch | \(-2 \ln \left (x \right )+\ln \left (\frac {9 x^{3} \ln \left (3\right )^{2}+9 \,{\mathrm e}^{x} \ln \left (3\right )^{2} x +9 x \ln \left (3\right )^{2} {\mathrm e}^{2 x}-36 \ln \left (3\right )^{2} {\mathrm e}^{x}+42 x \ln \left (3\right ) {\mathrm e}^{x}-36 \ln \left (3\right )^{2} {\mathrm e}^{2 x}-168 \ln \left (3\right ) {\mathrm e}^{x}+49 x -196}{9 \ln \left (3\right )^{2}}\right )\) | \(78\) |
Input:
int(((18*x^2-81*x+72)*ln(3)^2*exp(x)^2+((9*x^2-45*x+72)*ln(3)^2+(42*x^2-21 0*x+336)*ln(3))*exp(x)+9*x^3*ln(3)^2-49*x+392)/((9*x^2-36*x)*ln(3)^2*exp(x )^2+((9*x^2-36*x)*ln(3)^2+(42*x^2-168*x)*ln(3))*exp(x)+9*x^4*ln(3)^2+49*x^ 2-196*x),x,method=_RETURNVERBOSE)
Output:
ln(x-4)-2*ln(x)+ln(exp(2*x)+1/3/ln(3)*(3*ln(3)+14)*exp(x)+1/9*(9*x^3*ln(3) ^2+49*x-196)/ln(3)^2/(x-4))
Leaf count of result is larger than twice the leaf count of optimal. 64 vs. \(2 (29) = 58\).
Time = 0.10 (sec) , antiderivative size = 64, normalized size of antiderivative = 2.06 \[ \int \frac {392-49 x+9 x^3 \log ^2(3)+e^{2 x} \left (72-81 x+18 x^2\right ) \log ^2(3)+e^x \left (\left (336-210 x+42 x^2\right ) \log (3)+\left (72-45 x+9 x^2\right ) \log ^2(3)\right )}{-196 x+49 x^2+9 x^4 \log ^2(3)+e^{2 x} \left (-36 x+9 x^2\right ) \log ^2(3)+e^x \left (\left (-168 x+42 x^2\right ) \log (3)+\left (-36 x+9 x^2\right ) \log ^2(3)\right )} \, dx=\log \left (x - 4\right ) - 2 \, \log \left (x\right ) + \log \left (\frac {9 \, x^{3} \log \left (3\right )^{2} + 9 \, {\left (x - 4\right )} e^{\left (2 \, x\right )} \log \left (3\right )^{2} + 3 \, {\left (3 \, {\left (x - 4\right )} \log \left (3\right )^{2} + 14 \, {\left (x - 4\right )} \log \left (3\right )\right )} e^{x} + 49 \, x - 196}{x - 4}\right ) \] Input:
integrate(((18*x^2-81*x+72)*log(3)^2*exp(x)^2+((9*x^2-45*x+72)*log(3)^2+(4 2*x^2-210*x+336)*log(3))*exp(x)+9*x^3*log(3)^2-49*x+392)/((9*x^2-36*x)*log (3)^2*exp(x)^2+((9*x^2-36*x)*log(3)^2+(42*x^2-168*x)*log(3))*exp(x)+9*x^4* log(3)^2+49*x^2-196*x),x, algorithm="fricas")
Output:
log(x - 4) - 2*log(x) + log((9*x^3*log(3)^2 + 9*(x - 4)*e^(2*x)*log(3)^2 + 3*(3*(x - 4)*log(3)^2 + 14*(x - 4)*log(3))*e^x + 49*x - 196)/(x - 4))
Leaf count of result is larger than twice the leaf count of optimal. 61 vs. \(2 (24) = 48\).
Time = 0.63 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.97 \[ \int \frac {392-49 x+9 x^3 \log ^2(3)+e^{2 x} \left (72-81 x+18 x^2\right ) \log ^2(3)+e^x \left (\left (336-210 x+42 x^2\right ) \log (3)+\left (72-45 x+9 x^2\right ) \log ^2(3)\right )}{-196 x+49 x^2+9 x^4 \log ^2(3)+e^{2 x} \left (-36 x+9 x^2\right ) \log ^2(3)+e^x \left (\left (-168 x+42 x^2\right ) \log (3)+\left (-36 x+9 x^2\right ) \log ^2(3)\right )} \, dx=- 2 \log {\left (x \right )} + \log {\left (x - 4 \right )} + \log {\left (e^{2 x} + \frac {\left (3 \log {\left (3 \right )} + 14\right ) e^{x}}{3 \log {\left (3 \right )}} + \frac {9 x^{3} \log {\left (3 \right )}^{2} + 49 x - 196}{9 x \log {\left (3 \right )}^{2} - 36 \log {\left (3 \right )}^{2}} \right )} \] Input:
integrate(((18*x**2-81*x+72)*ln(3)**2*exp(x)**2+((9*x**2-45*x+72)*ln(3)**2 +(42*x**2-210*x+336)*ln(3))*exp(x)+9*x**3*ln(3)**2-49*x+392)/((9*x**2-36*x )*ln(3)**2*exp(x)**2+((9*x**2-36*x)*ln(3)**2+(42*x**2-168*x)*ln(3))*exp(x) +9*x**4*ln(3)**2+49*x**2-196*x),x)
Output:
-2*log(x) + log(x - 4) + log(exp(2*x) + (3*log(3) + 14)*exp(x)/(3*log(3)) + (9*x**3*log(3)**2 + 49*x - 196)/(9*x*log(3)**2 - 36*log(3)**2))
Leaf count of result is larger than twice the leaf count of optimal. 88 vs. \(2 (29) = 58\).
Time = 0.18 (sec) , antiderivative size = 88, normalized size of antiderivative = 2.84 \[ \int \frac {392-49 x+9 x^3 \log ^2(3)+e^{2 x} \left (72-81 x+18 x^2\right ) \log ^2(3)+e^x \left (\left (336-210 x+42 x^2\right ) \log (3)+\left (72-45 x+9 x^2\right ) \log ^2(3)\right )}{-196 x+49 x^2+9 x^4 \log ^2(3)+e^{2 x} \left (-36 x+9 x^2\right ) \log ^2(3)+e^x \left (\left (-168 x+42 x^2\right ) \log (3)+\left (-36 x+9 x^2\right ) \log ^2(3)\right )} \, dx=\log \left (x - 4\right ) - 2 \, \log \left (x\right ) + \log \left (\frac {9 \, x^{3} \log \left (3\right )^{2} + 9 \, {\left (x \log \left (3\right )^{2} - 4 \, \log \left (3\right )^{2}\right )} e^{\left (2 \, x\right )} + 3 \, {\left ({\left (3 \, \log \left (3\right )^{2} + 14 \, \log \left (3\right )\right )} x - 12 \, \log \left (3\right )^{2} - 56 \, \log \left (3\right )\right )} e^{x} + 49 \, x - 196}{9 \, {\left (x \log \left (3\right )^{2} - 4 \, \log \left (3\right )^{2}\right )}}\right ) \] Input:
integrate(((18*x^2-81*x+72)*log(3)^2*exp(x)^2+((9*x^2-45*x+72)*log(3)^2+(4 2*x^2-210*x+336)*log(3))*exp(x)+9*x^3*log(3)^2-49*x+392)/((9*x^2-36*x)*log (3)^2*exp(x)^2+((9*x^2-36*x)*log(3)^2+(42*x^2-168*x)*log(3))*exp(x)+9*x^4* log(3)^2+49*x^2-196*x),x, algorithm="maxima")
Output:
log(x - 4) - 2*log(x) + log(1/9*(9*x^3*log(3)^2 + 9*(x*log(3)^2 - 4*log(3) ^2)*e^(2*x) + 3*((3*log(3)^2 + 14*log(3))*x - 12*log(3)^2 - 56*log(3))*e^x + 49*x - 196)/(x*log(3)^2 - 4*log(3)^2))
Leaf count of result is larger than twice the leaf count of optimal. 71 vs. \(2 (29) = 58\).
Time = 0.20 (sec) , antiderivative size = 71, normalized size of antiderivative = 2.29 \[ \int \frac {392-49 x+9 x^3 \log ^2(3)+e^{2 x} \left (72-81 x+18 x^2\right ) \log ^2(3)+e^x \left (\left (336-210 x+42 x^2\right ) \log (3)+\left (72-45 x+9 x^2\right ) \log ^2(3)\right )}{-196 x+49 x^2+9 x^4 \log ^2(3)+e^{2 x} \left (-36 x+9 x^2\right ) \log ^2(3)+e^x \left (\left (-168 x+42 x^2\right ) \log (3)+\left (-36 x+9 x^2\right ) \log ^2(3)\right )} \, dx=\log \left (9 \, x^{3} \log \left (3\right )^{2} + 9 \, x e^{\left (2 \, x\right )} \log \left (3\right )^{2} + 9 \, x e^{x} \log \left (3\right )^{2} + 42 \, x e^{x} \log \left (3\right ) - 36 \, e^{\left (2 \, x\right )} \log \left (3\right )^{2} - 36 \, e^{x} \log \left (3\right )^{2} - 168 \, e^{x} \log \left (3\right ) + 49 \, x - 196\right ) - 2 \, \log \left (x\right ) \] Input:
integrate(((18*x^2-81*x+72)*log(3)^2*exp(x)^2+((9*x^2-45*x+72)*log(3)^2+(4 2*x^2-210*x+336)*log(3))*exp(x)+9*x^3*log(3)^2-49*x+392)/((9*x^2-36*x)*log (3)^2*exp(x)^2+((9*x^2-36*x)*log(3)^2+(42*x^2-168*x)*log(3))*exp(x)+9*x^4* log(3)^2+49*x^2-196*x),x, algorithm="giac")
Output:
log(9*x^3*log(3)^2 + 9*x*e^(2*x)*log(3)^2 + 9*x*e^x*log(3)^2 + 42*x*e^x*lo g(3) - 36*e^(2*x)*log(3)^2 - 36*e^x*log(3)^2 - 168*e^x*log(3) + 49*x - 196 ) - 2*log(x)
Timed out. \[ \int \frac {392-49 x+9 x^3 \log ^2(3)+e^{2 x} \left (72-81 x+18 x^2\right ) \log ^2(3)+e^x \left (\left (336-210 x+42 x^2\right ) \log (3)+\left (72-45 x+9 x^2\right ) \log ^2(3)\right )}{-196 x+49 x^2+9 x^4 \log ^2(3)+e^{2 x} \left (-36 x+9 x^2\right ) \log ^2(3)+e^x \left (\left (-168 x+42 x^2\right ) \log (3)+\left (-36 x+9 x^2\right ) \log ^2(3)\right )} \, dx=-\int \frac {9\,x^3\,{\ln \left (3\right )}^2-49\,x+{\mathrm {e}}^x\,\left (\ln \left (3\right )\,\left (42\,x^2-210\,x+336\right )+{\ln \left (3\right )}^2\,\left (9\,x^2-45\,x+72\right )\right )+{\mathrm {e}}^{2\,x}\,{\ln \left (3\right )}^2\,\left (18\,x^2-81\,x+72\right )+392}{196\,x-9\,x^4\,{\ln \left (3\right )}^2+{\mathrm {e}}^x\,\left (\ln \left (3\right )\,\left (168\,x-42\,x^2\right )+{\ln \left (3\right )}^2\,\left (36\,x-9\,x^2\right )\right )-49\,x^2+{\mathrm {e}}^{2\,x}\,{\ln \left (3\right )}^2\,\left (36\,x-9\,x^2\right )} \,d x \] Input:
int(-(9*x^3*log(3)^2 - 49*x + exp(x)*(log(3)*(42*x^2 - 210*x + 336) + log( 3)^2*(9*x^2 - 45*x + 72)) + exp(2*x)*log(3)^2*(18*x^2 - 81*x + 72) + 392)/ (196*x - 9*x^4*log(3)^2 + exp(x)*(log(3)*(168*x - 42*x^2) + log(3)^2*(36*x - 9*x^2)) - 49*x^2 + exp(2*x)*log(3)^2*(36*x - 9*x^2)),x)
Output:
-int((9*x^3*log(3)^2 - 49*x + exp(x)*(log(3)*(42*x^2 - 210*x + 336) + log( 3)^2*(9*x^2 - 45*x + 72)) + exp(2*x)*log(3)^2*(18*x^2 - 81*x + 72) + 392)/ (196*x - 9*x^4*log(3)^2 + exp(x)*(log(3)*(168*x - 42*x^2) + log(3)^2*(36*x - 9*x^2)) - 49*x^2 + exp(2*x)*log(3)^2*(36*x - 9*x^2)), x)
\[ \int \frac {392-49 x+9 x^3 \log ^2(3)+e^{2 x} \left (72-81 x+18 x^2\right ) \log ^2(3)+e^x \left (\left (336-210 x+42 x^2\right ) \log (3)+\left (72-45 x+9 x^2\right ) \log ^2(3)\right )}{-196 x+49 x^2+9 x^4 \log ^2(3)+e^{2 x} \left (-36 x+9 x^2\right ) \log ^2(3)+e^x \left (\left (-168 x+42 x^2\right ) \log (3)+\left (-36 x+9 x^2\right ) \log ^2(3)\right )} \, dx =\text {Too large to display} \] Input:
int(((18*x^2-81*x+72)*log(3)^2*exp(x)^2+((9*x^2-45*x+72)*log(3)^2+(42*x^2- 210*x+336)*log(3))*exp(x)+9*x^3*log(3)^2-49*x+392)/((9*x^2-36*x)*log(3)^2* exp(x)^2+((9*x^2-36*x)*log(3)^2+(42*x^2-168*x)*log(3))*exp(x)+9*x^4*log(3) ^2+49*x^2-196*x),x)
Output:
72*int(e**(2*x)/(9*e**(2*x)*log(3)**2*x**2 - 36*e**(2*x)*log(3)**2*x + 9*e **x*log(3)**2*x**2 - 36*e**x*log(3)**2*x + 42*e**x*log(3)*x**2 - 168*e**x* log(3)*x + 9*log(3)**2*x**4 + 49*x**2 - 196*x),x)*log(3)**2 - 81*int(e**(2 *x)/(9*e**(2*x)*log(3)**2*x - 36*e**(2*x)*log(3)**2 + 9*e**x*log(3)**2*x - 36*e**x*log(3)**2 + 42*e**x*log(3)*x - 168*e**x*log(3) + 9*log(3)**2*x**3 + 49*x - 196),x)*log(3)**2 + 72*int(e**x/(9*e**(2*x)*log(3)**2*x**2 - 36* e**(2*x)*log(3)**2*x + 9*e**x*log(3)**2*x**2 - 36*e**x*log(3)**2*x + 42*e* *x*log(3)*x**2 - 168*e**x*log(3)*x + 9*log(3)**2*x**4 + 49*x**2 - 196*x),x )*log(3)**2 + 336*int(e**x/(9*e**(2*x)*log(3)**2*x**2 - 36*e**(2*x)*log(3) **2*x + 9*e**x*log(3)**2*x**2 - 36*e**x*log(3)**2*x + 42*e**x*log(3)*x**2 - 168*e**x*log(3)*x + 9*log(3)**2*x**4 + 49*x**2 - 196*x),x)*log(3) - 45*i nt(e**x/(9*e**(2*x)*log(3)**2*x - 36*e**(2*x)*log(3)**2 + 9*e**x*log(3)**2 *x - 36*e**x*log(3)**2 + 42*e**x*log(3)*x - 168*e**x*log(3) + 9*log(3)**2* x**3 + 49*x - 196),x)*log(3)**2 - 210*int(e**x/(9*e**(2*x)*log(3)**2*x - 3 6*e**(2*x)*log(3)**2 + 9*e**x*log(3)**2*x - 36*e**x*log(3)**2 + 42*e**x*lo g(3)*x - 168*e**x*log(3) + 9*log(3)**2*x**3 + 49*x - 196),x)*log(3) + 9*in t(x**2/(9*e**(2*x)*log(3)**2*x - 36*e**(2*x)*log(3)**2 + 9*e**x*log(3)**2* x - 36*e**x*log(3)**2 + 42*e**x*log(3)*x - 168*e**x*log(3) + 9*log(3)**2*x **3 + 49*x - 196),x)*log(3)**2 + 18*int((e**(2*x)*x)/(9*e**(2*x)*log(3)**2 *x - 36*e**(2*x)*log(3)**2 + 9*e**x*log(3)**2*x - 36*e**x*log(3)**2 + 4...