Integrand size = 86, antiderivative size = 24 \[ \int \frac {e^{2 x} \left ((-32+32 x) \log ^2(6)-96 \log ^3(6)\right )+e^x \left (\left (-144 x+144 x^2\right ) \log (6)+(432-864 x) \log ^2(6)+1296 \log ^3(6)\right )}{-x^3+9 x^2 \log (6)-27 x \log ^2(6)+27 \log ^3(6)} \, dx=5-\left (18-\frac {4 e^x}{3-\frac {x}{\log (6)}}\right )^2 \] Output:
5-(18-4/(3-x/ln(6))*exp(x))^2
Time = 0.21 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.12 \[ \int \frac {e^{2 x} \left ((-32+32 x) \log ^2(6)-96 \log ^3(6)\right )+e^x \left (\left (-144 x+144 x^2\right ) \log (6)+(432-864 x) \log ^2(6)+1296 \log ^3(6)\right )}{-x^3+9 x^2 \log (6)-27 x \log ^2(6)+27 \log ^3(6)} \, dx=-\frac {16 e^x \log (6) \left (9 x+\left (-27+e^x\right ) \log (6)\right )}{(x-3 \log (6))^2} \] Input:
Integrate[(E^(2*x)*((-32 + 32*x)*Log[6]^2 - 96*Log[6]^3) + E^x*((-144*x + 144*x^2)*Log[6] + (432 - 864*x)*Log[6]^2 + 1296*Log[6]^3))/(-x^3 + 9*x^2*L og[6] - 27*x*Log[6]^2 + 27*Log[6]^3),x]
Output:
(-16*E^x*Log[6]*(9*x + (-27 + E^x)*Log[6]))/(x - 3*Log[6])^2
Time = 0.89 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.46, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.058, Rules used = {2007, 7239, 27, 7293, 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.
| \(\displaystyle \int \frac {e^x \left (\left (144 x^2-144 x\right ) \log (6)+(432-864 x) \log ^2(6)+1296 \log ^3(6)\right )+e^{2 x} \left ((32 x-32) \log ^2(6)-96 \log ^3(6)\right )}{-x^3+9 x^2 \log (6)-27 x \log ^2(6)+27 \log ^3(6)} \, dx\) |
| \(\Big \downarrow \) 2007 |
| \(\displaystyle \int \frac {e^x \left (\left (144 x^2-144 x\right ) \log (6)+(432-864 x) \log ^2(6)+1296 \log ^3(6)\right )+e^{2 x} \left ((32 x-32) \log ^2(6)-96 \log ^3(6)\right )}{(3 \log (6)-x)^3}dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {16 e^x \log (6) (-x+1+3 \log (6)) \left (9 x+\left (2 e^x-27\right ) \log (6)\right )}{(x-3 \log (6))^3}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 16 \log (6) \int \frac {e^x \left (9 x-\left (27-2 e^x\right ) \log (6)\right ) (-x+\log (216)+1)}{(x-3 \log (6))^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 16 \log (6) \int \left (-\frac {9 e^x (x-\log (216)-1)}{(x-3 \log (6))^2}-\frac {2 e^{2 x} \log (6) (x-\log (216)-1)}{(x-3 \log (6))^3}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 16 \log (6) \left (-\frac {9 e^x}{x-3 \log (6)}-\frac {e^{2 x} \log (6)}{(x-3 \log (6))^2}\right )\) |
Input:
Int[(E^(2*x)*((-32 + 32*x)*Log[6]^2 - 96*Log[6]^3) + E^x*((-144*x + 144*x^ 2)*Log[6] + (432 - 864*x)*Log[6]^2 + 1296*Log[6]^3))/(-x^3 + 9*x^2*Log[6] - 27*x*Log[6]^2 + 27*Log[6]^3),x]
Output:
16*Log[6]*((-9*E^x)/(x - 3*Log[6]) - (E^(2*x)*Log[6])/(x - 3*Log[6])^2)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(u_.)*(Px_)^(p_), x_Symbol] :> With[{a = Rt[Coeff[Px, x, 0], Expon[Px, x]], b = Rt[Coeff[Px, x, Expon[Px, x]], Expon[Px, x]]}, Int[u*(a + b*x)^(Ex pon[Px, x]*p), x] /; EqQ[Px, (a + b*x)^Expon[Px, x]]] /; IntegerQ[p] && Pol yQ[Px, x] && GtQ[Expon[Px, x], 1] && NeQ[Coeff[Px, x, 0], 0]
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Time = 0.97 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.42
| method | result | size |
| parts | \(-\frac {144 \ln \left (6\right ) {\mathrm e}^{x}}{-3 \ln \left (6\right )+x}-\frac {16 \ln \left (6\right )^{2} {\mathrm e}^{2 x}}{\left (-3 \ln \left (6\right )+x \right )^{2}}\) | \(34\) |
| norman | \(\frac {432 \ln \left (6\right )^{2} {\mathrm e}^{x}-16 \ln \left (6\right )^{2} {\mathrm e}^{2 x}-144 \ln \left (6\right ) {\mathrm e}^{x} x}{\left (3 \ln \left (6\right )-x \right )^{2}}\) | \(38\) |
| parallelrisch | \(-\frac {16 \ln \left (6\right )^{2} {\mathrm e}^{2 x}-432 \ln \left (6\right )^{2} {\mathrm e}^{x}+144 \ln \left (6\right ) {\mathrm e}^{x} x}{9 \ln \left (6\right )^{2}-6 x \ln \left (6\right )+x^{2}}\) | \(46\) |
| orering | \(\frac {\left (81 \ln \left (3\right )^{3}+243 \ln \left (2\right ) \ln \left (3\right )^{2}-81 x \ln \left (3\right )^{2}+243 \ln \left (3\right ) \ln \left (2\right )^{2}-162 x \ln \left (2\right ) \ln \left (3\right )+27 x^{2} \ln \left (3\right )+81 \ln \left (2\right )^{3}-81 x \ln \left (2\right )^{2}+27 x^{2} \ln \left (2\right )-3 x^{3}+54 \ln \left (3\right )^{2}+108 \ln \left (2\right ) \ln \left (3\right )-36 x \ln \left (3\right )+54 \ln \left (2\right )^{2}-36 x \ln \left (2\right )+6 x^{2}+12 \ln \left (3\right )+12 \ln \left (2\right )-4 x \right ) \left (\left (-96 \ln \left (6\right )^{3}+\left (32 x -32\right ) \ln \left (6\right )^{2}\right ) {\mathrm e}^{2 x}+\left (1296 \ln \left (6\right )^{3}+\left (-864 x +432\right ) \ln \left (6\right )^{2}+\left (144 x^{2}-144 x \right ) \ln \left (6\right )\right ) {\mathrm e}^{x}\right )}{2 \left (3 \ln \left (2\right )+3 \ln \left (3\right )-x +1\right )^{3} \left (27 \ln \left (6\right )^{3}-27 x \ln \left (6\right )^{2}+9 x^{2} \ln \left (6\right )-x^{3}\right )}-\frac {\left (3 \ln \left (3\right )+3 \ln \left (2\right )-x \right )^{2} \left (\frac {32 \ln \left (6\right )^{2} {\mathrm e}^{2 x}+2 \left (-96 \ln \left (6\right )^{3}+\left (32 x -32\right ) \ln \left (6\right )^{2}\right ) {\mathrm e}^{2 x}+\left (-864 \ln \left (6\right )^{2}+\left (288 x -144\right ) \ln \left (6\right )\right ) {\mathrm e}^{x}+\left (1296 \ln \left (6\right )^{3}+\left (-864 x +432\right ) \ln \left (6\right )^{2}+\left (144 x^{2}-144 x \right ) \ln \left (6\right )\right ) {\mathrm e}^{x}}{27 \ln \left (6\right )^{3}-27 x \ln \left (6\right )^{2}+9 x^{2} \ln \left (6\right )-x^{3}}-\frac {\left (\left (-96 \ln \left (6\right )^{3}+\left (32 x -32\right ) \ln \left (6\right )^{2}\right ) {\mathrm e}^{2 x}+\left (1296 \ln \left (6\right )^{3}+\left (-864 x +432\right ) \ln \left (6\right )^{2}+\left (144 x^{2}-144 x \right ) \ln \left (6\right )\right ) {\mathrm e}^{x}\right ) \left (-27 \ln \left (6\right )^{2}+18 x \ln \left (6\right )-3 x^{2}\right )}{\left (27 \ln \left (6\right )^{3}-27 x \ln \left (6\right )^{2}+9 x^{2} \ln \left (6\right )-x^{3}\right )^{2}}\right )}{2 \left (3 \ln \left (2\right )+3 \ln \left (3\right )-x +1\right )^{2}}\) | \(462\) |
| default | \(-96 \ln \left (6\right )^{3} \left (\frac {{\mathrm e}^{2 x}}{2 \left (-3 \ln \left (6\right )+x \right )^{2}}+\frac {{\mathrm e}^{2 x}}{-3 \ln \left (6\right )+x}+93312 \,\operatorname {expIntegral}_{1}\left (-2 x +6 \ln \left (6\right )\right )\right )+432 \ln \left (6\right )^{2} \left (\frac {{\mathrm e}^{x}}{2 \left (-3 \ln \left (6\right )+x \right )^{2}}+\frac {{\mathrm e}^{x}}{2 x -6 \ln \left (6\right )}+108 \,\operatorname {expIntegral}_{1}\left (3 \ln \left (6\right )-x \right )\right )-32 \ln \left (6\right )^{2} \left (\frac {{\mathrm e}^{2 x}}{2 \left (-3 \ln \left (6\right )+x \right )^{2}}+\frac {{\mathrm e}^{2 x}}{-3 \ln \left (6\right )+x}+93312 \,\operatorname {expIntegral}_{1}\left (-2 x +6 \ln \left (6\right )\right )\right )+1296 \ln \left (6\right )^{3} \left (\frac {{\mathrm e}^{x}}{2 \left (-3 \ln \left (6\right )+x \right )^{2}}+\frac {{\mathrm e}^{x}}{2 x -6 \ln \left (6\right )}+108 \,\operatorname {expIntegral}_{1}\left (3 \ln \left (6\right )-x \right )\right )+32 \ln \left (6\right )^{2} \left (-3 \ln \left (6\right ) \left (-\frac {{\mathrm e}^{2 x}}{2 \left (-3 \ln \left (6\right )+x \right )^{2}}-\frac {{\mathrm e}^{2 x}}{-3 \ln \left (6\right )+x}-93312 \,\operatorname {expIntegral}_{1}\left (-2 x +6 \ln \left (6\right )\right )\right )+\frac {{\mathrm e}^{2 x}}{-3 \ln \left (6\right )+x}+93312 \,\operatorname {expIntegral}_{1}\left (-2 x +6 \ln \left (6\right )\right )\right )-144 \ln \left (6\right ) \left (-3 \ln \left (6\right ) \left (-\frac {{\mathrm e}^{x}}{2 \left (-3 \ln \left (6\right )+x \right )^{2}}-\frac {{\mathrm e}^{x}}{2 \left (-3 \ln \left (6\right )+x \right )}-108 \,\operatorname {expIntegral}_{1}\left (3 \ln \left (6\right )-x \right )\right )+\frac {{\mathrm e}^{x}}{-3 \ln \left (6\right )+x}+216 \,\operatorname {expIntegral}_{1}\left (3 \ln \left (6\right )-x \right )\right )+144 \ln \left (6\right ) \left (-9 \ln \left (6\right )^{2} \left (-\frac {{\mathrm e}^{x}}{2 \left (-3 \ln \left (6\right )+x \right )^{2}}-\frac {{\mathrm e}^{x}}{2 \left (-3 \ln \left (6\right )+x \right )}-108 \,\operatorname {expIntegral}_{1}\left (3 \ln \left (6\right )-x \right )\right )+216 \,\operatorname {expIntegral}_{1}\left (3 \ln \left (6\right )-x \right )-6 \ln \left (6\right ) \left (-\frac {{\mathrm e}^{x}}{-3 \ln \left (6\right )+x}-216 \,\operatorname {expIntegral}_{1}\left (3 \ln \left (6\right )-x \right )\right )\right )-864 \ln \left (6\right )^{2} \left (-3 \ln \left (6\right ) \left (-\frac {{\mathrm e}^{x}}{2 \left (-3 \ln \left (6\right )+x \right )^{2}}-\frac {{\mathrm e}^{x}}{2 \left (-3 \ln \left (6\right )+x \right )}-108 \,\operatorname {expIntegral}_{1}\left (3 \ln \left (6\right )-x \right )\right )+\frac {{\mathrm e}^{x}}{-3 \ln \left (6\right )+x}+216 \,\operatorname {expIntegral}_{1}\left (3 \ln \left (6\right )-x \right )\right )\) | \(486\) |
Input:
int(((-96*ln(6)^3+(32*x-32)*ln(6)^2)*exp(x)^2+(1296*ln(6)^3+(-864*x+432)*l n(6)^2+(144*x^2-144*x)*ln(6))*exp(x))/(27*ln(6)^3-27*x*ln(6)^2+9*x^2*ln(6) -x^3),x,method=_RETURNVERBOSE)
Output:
-144*ln(6)*exp(x)/(-3*ln(6)+x)-16*ln(6)^2*exp(x)^2/(-3*ln(6)+x)^2
Time = 0.09 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.83 \[ \int \frac {e^{2 x} \left ((-32+32 x) \log ^2(6)-96 \log ^3(6)\right )+e^x \left (\left (-144 x+144 x^2\right ) \log (6)+(432-864 x) \log ^2(6)+1296 \log ^3(6)\right )}{-x^3+9 x^2 \log (6)-27 x \log ^2(6)+27 \log ^3(6)} \, dx=-\frac {16 \, {\left (e^{\left (2 \, x\right )} \log \left (6\right )^{2} + 9 \, {\left (x \log \left (6\right ) - 3 \, \log \left (6\right )^{2}\right )} e^{x}\right )}}{x^{2} - 6 \, x \log \left (6\right ) + 9 \, \log \left (6\right )^{2}} \] Input:
integrate(((-96*log(6)^3+(32*x-32)*log(6)^2)*exp(x)^2+(1296*log(6)^3+(-864 *x+432)*log(6)^2+(144*x^2-144*x)*log(6))*exp(x))/(27*log(6)^3-27*x*log(6)^ 2+9*x^2*log(6)-x^3),x, algorithm="fricas")
Output:
-16*(e^(2*x)*log(6)^2 + 9*(x*log(6) - 3*log(6)^2)*e^x)/(x^2 - 6*x*log(6) + 9*log(6)^2)
Leaf count of result is larger than twice the leaf count of optimal. 73 vs. \(2 (15) = 30\).
Time = 0.11 (sec) , antiderivative size = 73, normalized size of antiderivative = 3.04 \[ \int \frac {e^{2 x} \left ((-32+32 x) \log ^2(6)-96 \log ^3(6)\right )+e^x \left (\left (-144 x+144 x^2\right ) \log (6)+(432-864 x) \log ^2(6)+1296 \log ^3(6)\right )}{-x^3+9 x^2 \log (6)-27 x \log ^2(6)+27 \log ^3(6)} \, dx=\frac {\left (- 16 x \log {\left (6 \right )}^{2} + 48 \log {\left (6 \right )}^{3}\right ) e^{2 x} + \left (- 144 x^{2} \log {\left (6 \right )} + 864 x \log {\left (6 \right )}^{2} - 1296 \log {\left (6 \right )}^{3}\right ) e^{x}}{x^{3} - 9 x^{2} \log {\left (6 \right )} + 27 x \log {\left (6 \right )}^{2} - 27 \log {\left (6 \right )}^{3}} \] Input:
integrate(((-96*ln(6)**3+(32*x-32)*ln(6)**2)*exp(x)**2+(1296*ln(6)**3+(-86 4*x+432)*ln(6)**2+(144*x**2-144*x)*ln(6))*exp(x))/(27*ln(6)**3-27*x*ln(6)* *2+9*x**2*ln(6)-x**3),x)
Output:
((-16*x*log(6)**2 + 48*log(6)**3)*exp(2*x) + (-144*x**2*log(6) + 864*x*log (6)**2 - 1296*log(6)**3)*exp(x))/(x**3 - 9*x**2*log(6) + 27*x*log(6)**2 - 27*log(6)**3)
\[ \int \frac {e^{2 x} \left ((-32+32 x) \log ^2(6)-96 \log ^3(6)\right )+e^x \left (\left (-144 x+144 x^2\right ) \log (6)+(432-864 x) \log ^2(6)+1296 \log ^3(6)\right )}{-x^3+9 x^2 \log (6)-27 x \log ^2(6)+27 \log ^3(6)} \, dx=\int { -\frac {16 \, {\left (2 \, {\left ({\left (x - 1\right )} \log \left (6\right )^{2} - 3 \, \log \left (6\right )^{3}\right )} e^{\left (2 \, x\right )} - 9 \, {\left (3 \, {\left (2 \, x - 1\right )} \log \left (6\right )^{2} - 9 \, \log \left (6\right )^{3} - {\left (x^{2} - x\right )} \log \left (6\right )\right )} e^{x}\right )}}{x^{3} - 9 \, x^{2} \log \left (6\right ) + 27 \, x \log \left (6\right )^{2} - 27 \, \log \left (6\right )^{3}} \,d x } \] Input:
integrate(((-96*log(6)^3+(32*x-32)*log(6)^2)*exp(x)^2+(1296*log(6)^3+(-864 *x+432)*log(6)^2+(144*x^2-144*x)*log(6))*exp(x))/(27*log(6)^3-27*x*log(6)^ 2+9*x^2*log(6)-x^3),x, algorithm="maxima")
Output:
279936*exp_integral_e(3, -x + 3*log(6))*log(6)^3/(x - 3*log(6))^2 + 93312* exp_integral_e(3, -x + 3*log(6))*log(6)^2/(x - 3*log(6))^2 + 16*((3*log(3) ^3 + 9*log(3)^2*log(2) + 9*log(3)*log(2)^2 + 3*log(2)^3 - (log(3)^2 + 2*lo g(3)*log(2) + log(2)^2)*x)*e^(2*x) - 9*(x^2*(log(3) + log(2)) - 6*(log(3)^ 2 + 2*log(3)*log(2) + log(2)^2)*x)*e^x)/(x^3 - 9*x^2*(log(3) + log(2)) - 2 7*log(3)^3 - 81*log(3)^2*log(2) - 81*log(3)*log(2)^2 - 27*log(2)^3 + 27*(l og(3)^2 + 2*log(3)*log(2) + log(2)^2)*x) + 16*integrate(27*(6*log(3)^3 + 1 8*log(3)^2*log(2) + 18*log(3)*log(2)^2 + 6*log(2)^3 + (log(3)^2 + 2*log(3) *log(2) + log(2)^2)*x)*e^x/(x^4 - 12*x^3*(log(3) + log(2)) + 81*log(3)^4 + 324*log(3)^3*log(2) + 486*log(3)^2*log(2)^2 + 324*log(3)*log(2)^3 + 81*lo g(2)^4 + 54*(log(3)^2 + 2*log(3)*log(2) + log(2)^2)*x^2 - 108*(log(3)^3 + 3*log(3)^2*log(2) + 3*log(3)*log(2)^2 + log(2)^3)*x), x)
Time = 0.12 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.83 \[ \int \frac {e^{2 x} \left ((-32+32 x) \log ^2(6)-96 \log ^3(6)\right )+e^x \left (\left (-144 x+144 x^2\right ) \log (6)+(432-864 x) \log ^2(6)+1296 \log ^3(6)\right )}{-x^3+9 x^2 \log (6)-27 x \log ^2(6)+27 \log ^3(6)} \, dx=-\frac {16 \, {\left (9 \, x e^{x} \log \left (6\right ) + e^{\left (2 \, x\right )} \log \left (6\right )^{2} - 27 \, e^{x} \log \left (6\right )^{2}\right )}}{x^{2} - 6 \, x \log \left (6\right ) + 9 \, \log \left (6\right )^{2}} \] Input:
integrate(((-96*log(6)^3+(32*x-32)*log(6)^2)*exp(x)^2+(1296*log(6)^3+(-864 *x+432)*log(6)^2+(144*x^2-144*x)*log(6))*exp(x))/(27*log(6)^3-27*x*log(6)^ 2+9*x^2*log(6)-x^3),x, algorithm="giac")
Output:
-16*(9*x*e^x*log(6) + e^(2*x)*log(6)^2 - 27*e^x*log(6)^2)/(x^2 - 6*x*log(6 ) + 9*log(6)^2)
Time = 0.27 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.12 \[ \int \frac {e^{2 x} \left ((-32+32 x) \log ^2(6)-96 \log ^3(6)\right )+e^x \left (\left (-144 x+144 x^2\right ) \log (6)+(432-864 x) \log ^2(6)+1296 \log ^3(6)\right )}{-x^3+9 x^2 \log (6)-27 x \log ^2(6)+27 \log ^3(6)} \, dx=-\frac {16\,{\mathrm {e}}^x\,\ln \left (6\right )\,\left (9\,x-27\,\ln \left (6\right )+{\mathrm {e}}^x\,\ln \left (6\right )\right )}{{\left (x-3\,\ln \left (6\right )\right )}^2} \] Input:
int((exp(x)*(log(6)*(144*x - 144*x^2) + log(6)^2*(864*x - 432) - 1296*log( 6)^3) - exp(2*x)*(log(6)^2*(32*x - 32) - 96*log(6)^3))/(27*x*log(6)^2 - 9* x^2*log(6) - 27*log(6)^3 + x^3),x)
Output:
-(16*exp(x)*log(6)*(9*x - 27*log(6) + exp(x)*log(6)))/(x - 3*log(6))^2
Time = 0.17 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.62 \[ \int \frac {e^{2 x} \left ((-32+32 x) \log ^2(6)-96 \log ^3(6)\right )+e^x \left (\left (-144 x+144 x^2\right ) \log (6)+(432-864 x) \log ^2(6)+1296 \log ^3(6)\right )}{-x^3+9 x^2 \log (6)-27 x \log ^2(6)+27 \log ^3(6)} \, dx=\frac {16 e^{x} \mathrm {log}\left (6\right ) \left (-e^{x} \mathrm {log}\left (6\right )+27 \,\mathrm {log}\left (6\right )-9 x \right )}{9 \mathrm {log}\left (6\right )^{2}-6 \,\mathrm {log}\left (6\right ) x +x^{2}} \] Input:
int(((-96*log(6)^3+(32*x-32)*log(6)^2)*exp(x)^2+(1296*log(6)^3+(-864*x+432 )*log(6)^2+(144*x^2-144*x)*log(6))*exp(x))/(27*log(6)^3-27*x*log(6)^2+9*x^ 2*log(6)-x^3),x)
Output:
(16*e**x*log(6)*( - e**x*log(6) + 27*log(6) - 9*x))/(9*log(6)**2 - 6*log(6 )*x + x**2)