Integrand size = 270, antiderivative size = 33 \begin {dmath*} \int \frac {e^{16} x-e^{4+x} x+\left (-98 e^{16}-112 e^{16} x \log (3)-60 e^{16} x^2 \log ^2(3)-16 e^{16} x^3 \log ^3(3)-2 e^{16} x^4 \log ^4(3)+e^{4+x} \left (98+98 x+\left (112 x+112 x^2\right ) \log (3)+\left (60 x^2+60 x^3\right ) \log ^2(3)+\left (16 x^3+16 x^4\right ) \log ^3(3)+\left (2 x^4+2 x^5\right ) \log ^4(3)\right )\right ) \log \left (e^{16} x-e^{4+x} x\right )+\left (-56 e^{16} x \log (3)-60 e^{16} x^2 \log ^2(3)-24 e^{16} x^3 \log ^3(3)-4 e^{16} x^4 \log ^4(3)+e^{4+x} \left (56 x \log (3)+60 x^2 \log ^2(3)+24 x^3 \log ^3(3)+4 x^4 \log ^4(3)\right )\right ) \log ^2\left (e^{16} x-e^{4+x} x\right )}{-e^{16} x+e^{4+x} x} \, dx=-x+\left (3+(2+x \log (3))^2\right )^2 \log ^2\left (\left (e^{16}-e^{4+x}\right ) x\right ) \end {dmath*}
\begin {dmath*} \int \frac {e^{16} x-e^{4+x} x+\left (-98 e^{16}-112 e^{16} x \log (3)-60 e^{16} x^2 \log ^2(3)-16 e^{16} x^3 \log ^3(3)-2 e^{16} x^4 \log ^4(3)+e^{4+x} \left (98+98 x+\left (112 x+112 x^2\right ) \log (3)+\left (60 x^2+60 x^3\right ) \log ^2(3)+\left (16 x^3+16 x^4\right ) \log ^3(3)+\left (2 x^4+2 x^5\right ) \log ^4(3)\right )\right ) \log \left (e^{16} x-e^{4+x} x\right )+\left (-56 e^{16} x \log (3)-60 e^{16} x^2 \log ^2(3)-24 e^{16} x^3 \log ^3(3)-4 e^{16} x^4 \log ^4(3)+e^{4+x} \left (56 x \log (3)+60 x^2 \log ^2(3)+24 x^3 \log ^3(3)+4 x^4 \log ^4(3)\right )\right ) \log ^2\left (e^{16} x-e^{4+x} x\right )}{-e^{16} x+e^{4+x} x} \, dx=\int \frac {e^{16} x-e^{4+x} x+\left (-98 e^{16}-112 e^{16} x \log (3)-60 e^{16} x^2 \log ^2(3)-16 e^{16} x^3 \log ^3(3)-2 e^{16} x^4 \log ^4(3)+e^{4+x} \left (98+98 x+\left (112 x+112 x^2\right ) \log (3)+\left (60 x^2+60 x^3\right ) \log ^2(3)+\left (16 x^3+16 x^4\right ) \log ^3(3)+\left (2 x^4+2 x^5\right ) \log ^4(3)\right )\right ) \log \left (e^{16} x-e^{4+x} x\right )+\left (-56 e^{16} x \log (3)-60 e^{16} x^2 \log ^2(3)-24 e^{16} x^3 \log ^3(3)-4 e^{16} x^4 \log ^4(3)+e^{4+x} \left (56 x \log (3)+60 x^2 \log ^2(3)+24 x^3 \log ^3(3)+4 x^4 \log ^4(3)\right )\right ) \log ^2\left (e^{16} x-e^{4+x} x\right )}{-e^{16} x+e^{4+x} x} \, dx \end {dmath*}
Integrate[(E^16*x - E^(4 + x)*x + (-98*E^16 - 112*E^16*x*Log[3] - 60*E^16* x^2*Log[3]^2 - 16*E^16*x^3*Log[3]^3 - 2*E^16*x^4*Log[3]^4 + E^(4 + x)*(98 + 98*x + (112*x + 112*x^2)*Log[3] + (60*x^2 + 60*x^3)*Log[3]^2 + (16*x^3 + 16*x^4)*Log[3]^3 + (2*x^4 + 2*x^5)*Log[3]^4))*Log[E^16*x - E^(4 + x)*x] + (-56*E^16*x*Log[3] - 60*E^16*x^2*Log[3]^2 - 24*E^16*x^3*Log[3]^3 - 4*E^16 *x^4*Log[3]^4 + E^(4 + x)*(56*x*Log[3] + 60*x^2*Log[3]^2 + 24*x^3*Log[3]^3 + 4*x^4*Log[3]^4))*Log[E^16*x - E^(4 + x)*x]^2)/(-(E^16*x) + E^(4 + x)*x) ,x]
Integrate[(E^16*x - E^(4 + x)*x + (-98*E^16 - 112*E^16*x*Log[3] - 60*E^16* x^2*Log[3]^2 - 16*E^16*x^3*Log[3]^3 - 2*E^16*x^4*Log[3]^4 + E^(4 + x)*(98 + 98*x + (112*x + 112*x^2)*Log[3] + (60*x^2 + 60*x^3)*Log[3]^2 + (16*x^3 + 16*x^4)*Log[3]^3 + (2*x^4 + 2*x^5)*Log[3]^4))*Log[E^16*x - E^(4 + x)*x] + (-56*E^16*x*Log[3] - 60*E^16*x^2*Log[3]^2 - 24*E^16*x^3*Log[3]^3 - 4*E^16 *x^4*Log[3]^4 + E^(4 + x)*(56*x*Log[3] + 60*x^2*Log[3]^2 + 24*x^3*Log[3]^3 + 4*x^4*Log[3]^4))*Log[E^16*x - E^(4 + x)*x]^2)/(-(E^16*x) + E^(4 + x)*x) , 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.
| \(\displaystyle \int \frac {\left (-4 e^{16} x^4 \log ^4(3)-24 e^{16} x^3 \log ^3(3)-60 e^{16} x^2 \log ^2(3)+e^{x+4} \left (4 x^4 \log ^4(3)+24 x^3 \log ^3(3)+60 x^2 \log ^2(3)+56 x \log (3)\right )-56 e^{16} x \log (3)\right ) \log ^2\left (e^{16} x-e^{x+4} x\right )+\left (-2 e^{16} x^4 \log ^4(3)-16 e^{16} x^3 \log ^3(3)-60 e^{16} x^2 \log ^2(3)+e^{x+4} \left (\left (112 x^2+112 x\right ) \log (3)+\left (2 x^5+2 x^4\right ) \log ^4(3)+\left (16 x^4+16 x^3\right ) \log ^3(3)+\left (60 x^3+60 x^2\right ) \log ^2(3)+98 x+98\right )-112 e^{16} x \log (3)-98 e^{16}\right ) \log \left (e^{16} x-e^{x+4} x\right )-e^{x+4} x+e^{16} x}{e^{x+4} x-e^{16} x} \, dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {-2 \left (e^x (x+1)-e^{12}\right ) \log \left (e^{16} x-e^{x+4} x\right ) \left (x^2 \log ^2(3)+x \log (81)+7\right )^2-4 \left (e^x-e^{12}\right ) x \log (3) \left (x^3 \log ^3(3)+6 x^2 \log ^2(3)+15 x \log (3)+14\right ) \log ^2\left (e^{16} x-e^{x+4} x\right )+e^x x-e^{12} x}{\left (e^{12}-e^x\right ) x}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {2 e^{12} \log \left (e^4 \left (e^{12}-e^x\right ) x\right ) \left (x^2 \log ^2(3)+x \log (81)+7\right )^2}{e^x-e^{12}}+\frac {2 x^5 \log ^4(3) \log \left (e^{16} x-e^{x+4} x\right )+4 x^4 \log ^4(3) \log ^2\left (e^{16} x-e^{x+4} x\right )+2 x^4 \log ^4(3) \left (1+\frac {2 \log (81)}{\log ^2(3)}\right ) \log \left (e^{16} x-e^{x+4} x\right )+28 x^3 \log ^2(3) \left (1+\frac {1}{7} \log (81) \left (1+\frac {\log (81)}{2 \log ^2(3)}\right )\right ) \log \left (e^{16} x-e^{x+4} x\right )+24 x^3 \log ^3(3) \log ^2\left (e^{16} x-e^{x+4} x\right )+60 x^2 \log ^2(3) \log ^2\left (e^{16} x-e^{x+4} x\right )+28 x^2 \log ^2(3) \left (1+\frac {(7+\log (9)) \log (81)}{7 \log ^2(3)}\right ) \log \left (e^{16} x-e^{x+4} x\right )-x+56 x \log (3) \log ^2\left (e^{16} x-e^{x+4} x\right )+98 x \left (1+\frac {2 \log (81)}{7}\right ) \log \left (e^{16} x-e^{x+4} x\right )+98 \log \left (e^{16} x-e^{x+4} x\right )}{x}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {2}{5} \log ^4(3) \log \left (1-e^{x-12}\right ) x^5+\frac {2}{5} \log ^4(3) \log \left (e^4 \left (e^{12}-e^x\right ) x\right ) x^5-\frac {2}{25} \log ^4(3) x^5-\frac {1}{2} \log ^2(3) \left (\log ^2(3)+\log (6561)\right ) \log \left (1-e^{x-12}\right ) x^4+\frac {1}{2} \log ^2(3) \left (\log ^2(3)+\log (6561)\right ) \log \left (e^4 \left (e^{12}-e^x\right ) x\right ) x^4-2 \log ^4(3) \operatorname {PolyLog}\left (2,e^{x-12}\right ) x^4-\frac {1}{8} \log ^2(3) \left (\log ^2(3)+\log (6561)\right ) x^4-\frac {2}{3} \left (\log ^2(81)+2 \log ^2(3) (7+\log (81))\right ) \log \left (1-e^{x-12}\right ) x^3+\frac {2}{3} \left (\log ^2(81)+2 \log ^2(3) (7+\log (81))\right ) \log \left (e^4 \left (e^{12}-e^x\right ) x\right ) x^3-2 \log ^2(3) \left (\log ^2(3)+\log (6561)\right ) \operatorname {PolyLog}\left (2,e^{x-12}\right ) x^3+8 \log ^4(3) \operatorname {PolyLog}\left (3,e^{x-12}\right ) x^3-\frac {2}{9} \left (\log ^2(81)+2 \log ^2(3) (7+\log (81))\right ) x^3-\left (14 \log ^2(3)+\log (81) (14+\log (81))\right ) \log \left (1-e^{x-12}\right ) x^2+\left (14 \log ^2(3)+\log (81) (14+\log (81))\right ) \log \left (e^4 \left (e^{12}-e^x\right ) x\right ) x^2-2 \left (\log ^2(81)+2 \log ^2(3) (7+\log (81))\right ) \operatorname {PolyLog}\left (2,e^{x-12}\right ) x^2+6 \log ^2(3) \left (\log ^2(3)+\log (6561)\right ) \operatorname {PolyLog}\left (3,e^{x-12}\right ) x^2-24 \log ^4(3) \operatorname {PolyLog}\left (4,e^{x-12}\right ) x^2-\frac {1}{2} \left (14 \log ^2(3)+\log (81) (14+\log (81))\right ) x^2-14 (7+\log (6561)) \log \left (1-e^{x-12}\right ) x+14 (7+\log (6561)) \log \left (e^4 \left (e^{12}-e^x\right ) x\right ) x-2 \left (14 \log ^2(3)+\log (81) (14+\log (81))\right ) \operatorname {PolyLog}\left (2,e^{x-12}\right ) x+4 \left (\log ^2(81)+2 \log ^2(3) (7+\log (81))\right ) \operatorname {PolyLog}\left (3,e^{x-12}\right ) x-12 \log ^2(3) \left (\log ^2(3)+\log (6561)\right ) \operatorname {PolyLog}\left (4,e^{x-12}\right ) x+48 \log ^4(3) \operatorname {PolyLog}\left (5,e^{x-12}\right ) x-14 (7+\log (6561)) x-x-14 (7+\log (6561)) \operatorname {PolyLog}\left (2,e^{x-12}\right )+2 \left (14 \log ^2(3)+\log (81) (14+\log (81))\right ) \operatorname {PolyLog}\left (3,e^{x-12}\right )-4 \left (\log ^2(81)+2 \log ^2(3) (7+\log (81))\right ) \operatorname {PolyLog}\left (4,e^{x-12}\right )+12 \log ^2(3) \left (\log ^2(3)+\log (6561)\right ) \operatorname {PolyLog}\left (5,e^{x-12}\right )-48 \log ^4(3) \operatorname {PolyLog}\left (6,e^{x-12}\right )+98 e^{12} \int \frac {\log \left (e^4 \left (e^{12}-e^x\right ) x\right )}{-e^{12}+e^x}dx+98 \int \frac {\log \left (e^4 \left (e^{12}-e^x\right ) x\right )}{x}dx+28 e^{12} \log (81) \int \frac {x \log \left (e^4 \left (e^{12}-e^x\right ) x\right )}{-e^{12}+e^x}dx+2 e^{12} \left (14 \log ^2(3)+\log ^2(81)\right ) \int \frac {x^2 \log \left (e^4 \left (e^{12}-e^x\right ) x\right )}{-e^{12}+e^x}dx+4 e^{12} \log ^2(3) \log (81) \int \frac {x^3 \log \left (e^4 \left (e^{12}-e^x\right ) x\right )}{-e^{12}+e^x}dx+2 e^{12} \log ^4(3) \int \frac {x^4 \log \left (e^4 \left (e^{12}-e^x\right ) x\right )}{-e^{12}+e^x}dx+56 \log (3) \int \log ^2\left (e^4 \left (e^{12}-e^x\right ) x\right )dx+60 \log ^2(3) \int x \log ^2\left (e^4 \left (e^{12}-e^x\right ) x\right )dx+24 \log ^3(3) \int x^2 \log ^2\left (e^4 \left (e^{12}-e^x\right ) x\right )dx+4 \log ^4(3) \int x^3 \log ^2\left (e^4 \left (e^{12}-e^x\right ) x\right )dx\) |
Int[(E^16*x - E^(4 + x)*x + (-98*E^16 - 112*E^16*x*Log[3] - 60*E^16*x^2*Lo g[3]^2 - 16*E^16*x^3*Log[3]^3 - 2*E^16*x^4*Log[3]^4 + E^(4 + x)*(98 + 98*x + (112*x + 112*x^2)*Log[3] + (60*x^2 + 60*x^3)*Log[3]^2 + (16*x^3 + 16*x^ 4)*Log[3]^3 + (2*x^4 + 2*x^5)*Log[3]^4))*Log[E^16*x - E^(4 + x)*x] + (-56* E^16*x*Log[3] - 60*E^16*x^2*Log[3]^2 - 24*E^16*x^3*Log[3]^3 - 4*E^16*x^4*L og[3]^4 + E^(4 + x)*(56*x*Log[3] + 60*x^2*Log[3]^2 + 24*x^3*Log[3]^3 + 4*x ^4*Log[3]^4))*Log[E^16*x - E^(4 + x)*x]^2)/(-(E^16*x) + E^(4 + x)*x),x]
3.1.71.3.1 Defintions of rubi rules used
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Leaf count of result is larger than twice the leaf count of optimal. \(112\) vs. \(2(31)=62\).
Time = 2.33 (sec) , antiderivative size = 113, normalized size of antiderivative = 3.42
| method | result | size |
| parallelrisch | \(\ln \left (3\right )^{4} x^{4} {\ln \left (-x \left ({\mathrm e}^{4+x}-{\mathrm e}^{16}\right )\right )}^{2}+8 \ln \left (3\right )^{3} x^{3} {\ln \left (-x \left ({\mathrm e}^{4+x}-{\mathrm e}^{16}\right )\right )}^{2}+30 \ln \left (3\right )^{2} x^{2} {\ln \left (-x \left ({\mathrm e}^{4+x}-{\mathrm e}^{16}\right )\right )}^{2}+56 \ln \left (3\right ) x {\ln \left (-x \left ({\mathrm e}^{4+x}-{\mathrm e}^{16}\right )\right )}^{2}+49 {\ln \left (-x \left ({\mathrm e}^{4+x}-{\mathrm e}^{16}\right )\right )}^{2}-x\) | \(113\) |
| risch | \(\text {Expression too large to display}\) | \(2968\) |
int((((4*x^4*ln(3)^4+24*x^3*ln(3)^3+60*x^2*ln(3)^2+56*x*ln(3))*exp(4+x)-4* x^4*exp(16)*ln(3)^4-24*x^3*exp(16)*ln(3)^3-60*x^2*exp(16)*ln(3)^2-56*x*exp (16)*ln(3))*ln(-x*exp(4+x)+x*exp(16))^2+(((2*x^5+2*x^4)*ln(3)^4+(16*x^4+16 *x^3)*ln(3)^3+(60*x^3+60*x^2)*ln(3)^2+(112*x^2+112*x)*ln(3)+98*x+98)*exp(4 +x)-2*x^4*exp(16)*ln(3)^4-16*x^3*exp(16)*ln(3)^3-60*x^2*exp(16)*ln(3)^2-11 2*x*exp(16)*ln(3)-98*exp(16))*ln(-x*exp(4+x)+x*exp(16))-x*exp(4+x)+x*exp(1 6))/(x*exp(4+x)-x*exp(16)),x,method=_RETURNVERBOSE)
ln(3)^4*x^4*ln(-x*(exp(4+x)-exp(16)))^2+8*ln(3)^3*x^3*ln(-x*(exp(4+x)-exp( 16)))^2+30*ln(3)^2*x^2*ln(-x*(exp(4+x)-exp(16)))^2+56*ln(3)*x*ln(-x*(exp(4 +x)-exp(16)))^2+49*ln(-x*(exp(4+x)-exp(16)))^2-x
Time = 0.27 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.61 \begin {dmath*} \int \frac {e^{16} x-e^{4+x} x+\left (-98 e^{16}-112 e^{16} x \log (3)-60 e^{16} x^2 \log ^2(3)-16 e^{16} x^3 \log ^3(3)-2 e^{16} x^4 \log ^4(3)+e^{4+x} \left (98+98 x+\left (112 x+112 x^2\right ) \log (3)+\left (60 x^2+60 x^3\right ) \log ^2(3)+\left (16 x^3+16 x^4\right ) \log ^3(3)+\left (2 x^4+2 x^5\right ) \log ^4(3)\right )\right ) \log \left (e^{16} x-e^{4+x} x\right )+\left (-56 e^{16} x \log (3)-60 e^{16} x^2 \log ^2(3)-24 e^{16} x^3 \log ^3(3)-4 e^{16} x^4 \log ^4(3)+e^{4+x} \left (56 x \log (3)+60 x^2 \log ^2(3)+24 x^3 \log ^3(3)+4 x^4 \log ^4(3)\right )\right ) \log ^2\left (e^{16} x-e^{4+x} x\right )}{-e^{16} x+e^{4+x} x} \, dx={\left (x^{4} \log \left (3\right )^{4} + 8 \, x^{3} \log \left (3\right )^{3} + 30 \, x^{2} \log \left (3\right )^{2} + 56 \, x \log \left (3\right ) + 49\right )} \log \left (x e^{16} - x e^{\left (x + 4\right )}\right )^{2} - x \end {dmath*}
integrate((((4*x^4*log(3)^4+24*x^3*log(3)^3+60*x^2*log(3)^2+56*x*log(3))*e xp(4+x)-4*x^4*exp(16)*log(3)^4-24*x^3*exp(16)*log(3)^3-60*x^2*exp(16)*log( 3)^2-56*x*exp(16)*log(3))*log(-x*exp(4+x)+x*exp(16))^2+(((2*x^5+2*x^4)*log (3)^4+(16*x^4+16*x^3)*log(3)^3+(60*x^3+60*x^2)*log(3)^2+(112*x^2+112*x)*lo g(3)+98*x+98)*exp(4+x)-2*x^4*exp(16)*log(3)^4-16*x^3*exp(16)*log(3)^3-60*x ^2*exp(16)*log(3)^2-112*x*exp(16)*log(3)-98*exp(16))*log(-x*exp(4+x)+x*exp (16))-x*exp(4+x)+x*exp(16))/(x*exp(4+x)-x*exp(16)),x, algorithm=\
(x^4*log(3)^4 + 8*x^3*log(3)^3 + 30*x^2*log(3)^2 + 56*x*log(3) + 49)*log(x *e^16 - x*e^(x + 4))^2 - x
Leaf count of result is larger than twice the leaf count of optimal. 53 vs. \(2 (26) = 52\).
Time = 0.25 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.61 \begin {dmath*} \int \frac {e^{16} x-e^{4+x} x+\left (-98 e^{16}-112 e^{16} x \log (3)-60 e^{16} x^2 \log ^2(3)-16 e^{16} x^3 \log ^3(3)-2 e^{16} x^4 \log ^4(3)+e^{4+x} \left (98+98 x+\left (112 x+112 x^2\right ) \log (3)+\left (60 x^2+60 x^3\right ) \log ^2(3)+\left (16 x^3+16 x^4\right ) \log ^3(3)+\left (2 x^4+2 x^5\right ) \log ^4(3)\right )\right ) \log \left (e^{16} x-e^{4+x} x\right )+\left (-56 e^{16} x \log (3)-60 e^{16} x^2 \log ^2(3)-24 e^{16} x^3 \log ^3(3)-4 e^{16} x^4 \log ^4(3)+e^{4+x} \left (56 x \log (3)+60 x^2 \log ^2(3)+24 x^3 \log ^3(3)+4 x^4 \log ^4(3)\right )\right ) \log ^2\left (e^{16} x-e^{4+x} x\right )}{-e^{16} x+e^{4+x} x} \, dx=- x + \left (x^{4} \log {\left (3 \right )}^{4} + 8 x^{3} \log {\left (3 \right )}^{3} + 30 x^{2} \log {\left (3 \right )}^{2} + 56 x \log {\left (3 \right )} + 49\right ) \log {\left (- x e^{x + 4} + x e^{16} \right )}^{2} \end {dmath*}
integrate((((4*x**4*ln(3)**4+24*x**3*ln(3)**3+60*x**2*ln(3)**2+56*x*ln(3)) *exp(4+x)-4*x**4*exp(16)*ln(3)**4-24*x**3*exp(16)*ln(3)**3-60*x**2*exp(16) *ln(3)**2-56*x*exp(16)*ln(3))*ln(-x*exp(4+x)+x*exp(16))**2+(((2*x**5+2*x** 4)*ln(3)**4+(16*x**4+16*x**3)*ln(3)**3+(60*x**3+60*x**2)*ln(3)**2+(112*x** 2+112*x)*ln(3)+98*x+98)*exp(4+x)-2*x**4*exp(16)*ln(3)**4-16*x**3*exp(16)*l n(3)**3-60*x**2*exp(16)*ln(3)**2-112*x*exp(16)*ln(3)-98*exp(16))*ln(-x*exp (4+x)+x*exp(16))-x*exp(4+x)+x*exp(16))/(x*exp(4+x)-x*exp(16)),x)
-x + (x**4*log(3)**4 + 8*x**3*log(3)**3 + 30*x**2*log(3)**2 + 56*x*log(3) + 49)*log(-x*exp(x + 4) + x*exp(16))**2
Leaf count of result is larger than twice the leaf count of optimal. 263 vs. \(2 (31) = 62\).
Time = 0.36 (sec) , antiderivative size = 263, normalized size of antiderivative = 7.97 \begin {dmath*} \int \frac {e^{16} x-e^{4+x} x+\left (-98 e^{16}-112 e^{16} x \log (3)-60 e^{16} x^2 \log ^2(3)-16 e^{16} x^3 \log ^3(3)-2 e^{16} x^4 \log ^4(3)+e^{4+x} \left (98+98 x+\left (112 x+112 x^2\right ) \log (3)+\left (60 x^2+60 x^3\right ) \log ^2(3)+\left (16 x^3+16 x^4\right ) \log ^3(3)+\left (2 x^4+2 x^5\right ) \log ^4(3)\right )\right ) \log \left (e^{16} x-e^{4+x} x\right )+\left (-56 e^{16} x \log (3)-60 e^{16} x^2 \log ^2(3)-24 e^{16} x^3 \log ^3(3)-4 e^{16} x^4 \log ^4(3)+e^{4+x} \left (56 x \log (3)+60 x^2 \log ^2(3)+24 x^3 \log ^3(3)+4 x^4 \log ^4(3)\right )\right ) \log ^2\left (e^{16} x-e^{4+x} x\right )}{-e^{16} x+e^{4+x} x} \, dx=16 \, x^{4} \log \left (3\right )^{4} + 128 \, x^{3} \log \left (3\right )^{3} + 480 \, x^{2} \log \left (3\right )^{2} + {\left (x^{4} \log \left (3\right )^{4} + 8 \, x^{3} \log \left (3\right )^{3} + 30 \, x^{2} \log \left (3\right )^{2} + 56 \, x \log \left (3\right ) + 49\right )} \log \left (x\right )^{2} + {\left (x^{4} \log \left (3\right )^{4} + 8 \, x^{3} \log \left (3\right )^{3} + 30 \, x^{2} \log \left (3\right )^{2} + 56 \, x \log \left (3\right ) + 49\right )} \log \left (e^{12} - e^{x}\right )^{2} - {\left (x e^{\left (-16\right )} - e^{\left (-16\right )} \log \left (-e^{12} + e^{x}\right )\right )} e^{16} + 896 \, x \log \left (3\right ) + 8 \, {\left (x^{4} \log \left (3\right )^{4} + 8 \, x^{3} \log \left (3\right )^{3} + 30 \, x^{2} \log \left (3\right )^{2} + 56 \, x \log \left (3\right ) + 49\right )} \log \left (x\right ) + 2 \, {\left (4 \, x^{4} \log \left (3\right )^{4} + 32 \, x^{3} \log \left (3\right )^{3} + 120 \, x^{2} \log \left (3\right )^{2} + 224 \, x \log \left (3\right ) + {\left (x^{4} \log \left (3\right )^{4} + 8 \, x^{3} \log \left (3\right )^{3} + 30 \, x^{2} \log \left (3\right )^{2} + 56 \, x \log \left (3\right ) + 49\right )} \log \left (x\right ) + 196\right )} \log \left (e^{12} - e^{x}\right ) - \log \left (-e^{12} + e^{x}\right ) \end {dmath*}
integrate((((4*x^4*log(3)^4+24*x^3*log(3)^3+60*x^2*log(3)^2+56*x*log(3))*e xp(4+x)-4*x^4*exp(16)*log(3)^4-24*x^3*exp(16)*log(3)^3-60*x^2*exp(16)*log( 3)^2-56*x*exp(16)*log(3))*log(-x*exp(4+x)+x*exp(16))^2+(((2*x^5+2*x^4)*log (3)^4+(16*x^4+16*x^3)*log(3)^3+(60*x^3+60*x^2)*log(3)^2+(112*x^2+112*x)*lo g(3)+98*x+98)*exp(4+x)-2*x^4*exp(16)*log(3)^4-16*x^3*exp(16)*log(3)^3-60*x ^2*exp(16)*log(3)^2-112*x*exp(16)*log(3)-98*exp(16))*log(-x*exp(4+x)+x*exp (16))-x*exp(4+x)+x*exp(16))/(x*exp(4+x)-x*exp(16)),x, algorithm=\
16*x^4*log(3)^4 + 128*x^3*log(3)^3 + 480*x^2*log(3)^2 + (x^4*log(3)^4 + 8* x^3*log(3)^3 + 30*x^2*log(3)^2 + 56*x*log(3) + 49)*log(x)^2 + (x^4*log(3)^ 4 + 8*x^3*log(3)^3 + 30*x^2*log(3)^2 + 56*x*log(3) + 49)*log(e^12 - e^x)^2 - (x*e^(-16) - e^(-16)*log(-e^12 + e^x))*e^16 + 896*x*log(3) + 8*(x^4*log (3)^4 + 8*x^3*log(3)^3 + 30*x^2*log(3)^2 + 56*x*log(3) + 49)*log(x) + 2*(4 *x^4*log(3)^4 + 32*x^3*log(3)^3 + 120*x^2*log(3)^2 + 224*x*log(3) + (x^4*l og(3)^4 + 8*x^3*log(3)^3 + 30*x^2*log(3)^2 + 56*x*log(3) + 49)*log(x) + 19 6)*log(e^12 - e^x) - log(-e^12 + e^x)
\begin {dmath*} \int \frac {e^{16} x-e^{4+x} x+\left (-98 e^{16}-112 e^{16} x \log (3)-60 e^{16} x^2 \log ^2(3)-16 e^{16} x^3 \log ^3(3)-2 e^{16} x^4 \log ^4(3)+e^{4+x} \left (98+98 x+\left (112 x+112 x^2\right ) \log (3)+\left (60 x^2+60 x^3\right ) \log ^2(3)+\left (16 x^3+16 x^4\right ) \log ^3(3)+\left (2 x^4+2 x^5\right ) \log ^4(3)\right )\right ) \log \left (e^{16} x-e^{4+x} x\right )+\left (-56 e^{16} x \log (3)-60 e^{16} x^2 \log ^2(3)-24 e^{16} x^3 \log ^3(3)-4 e^{16} x^4 \log ^4(3)+e^{4+x} \left (56 x \log (3)+60 x^2 \log ^2(3)+24 x^3 \log ^3(3)+4 x^4 \log ^4(3)\right )\right ) \log ^2\left (e^{16} x-e^{4+x} x\right )}{-e^{16} x+e^{4+x} x} \, dx=\int { \frac {4 \, {\left (x^{4} e^{16} \log \left (3\right )^{4} + 6 \, x^{3} e^{16} \log \left (3\right )^{3} + 15 \, x^{2} e^{16} \log \left (3\right )^{2} + 14 \, x e^{16} \log \left (3\right ) - {\left (x^{4} \log \left (3\right )^{4} + 6 \, x^{3} \log \left (3\right )^{3} + 15 \, x^{2} \log \left (3\right )^{2} + 14 \, x \log \left (3\right )\right )} e^{\left (x + 4\right )}\right )} \log \left (x e^{16} - x e^{\left (x + 4\right )}\right )^{2} - x e^{16} + x e^{\left (x + 4\right )} + 2 \, {\left (x^{4} e^{16} \log \left (3\right )^{4} + 8 \, x^{3} e^{16} \log \left (3\right )^{3} + 30 \, x^{2} e^{16} \log \left (3\right )^{2} + 56 \, x e^{16} \log \left (3\right ) - {\left ({\left (x^{5} + x^{4}\right )} \log \left (3\right )^{4} + 8 \, {\left (x^{4} + x^{3}\right )} \log \left (3\right )^{3} + 30 \, {\left (x^{3} + x^{2}\right )} \log \left (3\right )^{2} + 56 \, {\left (x^{2} + x\right )} \log \left (3\right ) + 49 \, x + 49\right )} e^{\left (x + 4\right )} + 49 \, e^{16}\right )} \log \left (x e^{16} - x e^{\left (x + 4\right )}\right )}{x e^{16} - x e^{\left (x + 4\right )}} \,d x } \end {dmath*}
integrate((((4*x^4*log(3)^4+24*x^3*log(3)^3+60*x^2*log(3)^2+56*x*log(3))*e xp(4+x)-4*x^4*exp(16)*log(3)^4-24*x^3*exp(16)*log(3)^3-60*x^2*exp(16)*log( 3)^2-56*x*exp(16)*log(3))*log(-x*exp(4+x)+x*exp(16))^2+(((2*x^5+2*x^4)*log (3)^4+(16*x^4+16*x^3)*log(3)^3+(60*x^3+60*x^2)*log(3)^2+(112*x^2+112*x)*lo g(3)+98*x+98)*exp(4+x)-2*x^4*exp(16)*log(3)^4-16*x^3*exp(16)*log(3)^3-60*x ^2*exp(16)*log(3)^2-112*x*exp(16)*log(3)-98*exp(16))*log(-x*exp(4+x)+x*exp (16))-x*exp(4+x)+x*exp(16))/(x*exp(4+x)-x*exp(16)),x, algorithm=\
integrate((4*(x^4*e^16*log(3)^4 + 6*x^3*e^16*log(3)^3 + 15*x^2*e^16*log(3) ^2 + 14*x*e^16*log(3) - (x^4*log(3)^4 + 6*x^3*log(3)^3 + 15*x^2*log(3)^2 + 14*x*log(3))*e^(x + 4))*log(x*e^16 - x*e^(x + 4))^2 - x*e^16 + x*e^(x + 4 ) + 2*(x^4*e^16*log(3)^4 + 8*x^3*e^16*log(3)^3 + 30*x^2*e^16*log(3)^2 + 56 *x*e^16*log(3) - ((x^5 + x^4)*log(3)^4 + 8*(x^4 + x^3)*log(3)^3 + 30*(x^3 + x^2)*log(3)^2 + 56*(x^2 + x)*log(3) + 49*x + 49)*e^(x + 4) + 49*e^16)*lo g(x*e^16 - x*e^(x + 4)))/(x*e^16 - x*e^(x + 4)), x)
Time = 13.84 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.61 \begin {dmath*} \int \frac {e^{16} x-e^{4+x} x+\left (-98 e^{16}-112 e^{16} x \log (3)-60 e^{16} x^2 \log ^2(3)-16 e^{16} x^3 \log ^3(3)-2 e^{16} x^4 \log ^4(3)+e^{4+x} \left (98+98 x+\left (112 x+112 x^2\right ) \log (3)+\left (60 x^2+60 x^3\right ) \log ^2(3)+\left (16 x^3+16 x^4\right ) \log ^3(3)+\left (2 x^4+2 x^5\right ) \log ^4(3)\right )\right ) \log \left (e^{16} x-e^{4+x} x\right )+\left (-56 e^{16} x \log (3)-60 e^{16} x^2 \log ^2(3)-24 e^{16} x^3 \log ^3(3)-4 e^{16} x^4 \log ^4(3)+e^{4+x} \left (56 x \log (3)+60 x^2 \log ^2(3)+24 x^3 \log ^3(3)+4 x^4 \log ^4(3)\right )\right ) \log ^2\left (e^{16} x-e^{4+x} x\right )}{-e^{16} x+e^{4+x} x} \, dx={\ln \left (x\,{\mathrm {e}}^{16}-x\,{\mathrm {e}}^4\,{\mathrm {e}}^x\right )}^2\,\left ({\ln \left (3\right )}^4\,x^4+8\,{\ln \left (3\right )}^3\,x^3+30\,{\ln \left (3\right )}^2\,x^2+56\,\ln \left (3\right )\,x+49\right )-x \end {dmath*}
int(-(x*exp(x + 4) - x*exp(16) + log(x*exp(16) - x*exp(x + 4))^2*(60*x^2*e xp(16)*log(3)^2 - exp(x + 4)*(60*x^2*log(3)^2 + 24*x^3*log(3)^3 + 4*x^4*lo g(3)^4 + 56*x*log(3)) + 24*x^3*exp(16)*log(3)^3 + 4*x^4*exp(16)*log(3)^4 + 56*x*exp(16)*log(3)) + log(x*exp(16) - x*exp(x + 4))*(98*exp(16) - exp(x + 4)*(98*x + log(3)*(112*x + 112*x^2) + log(3)^4*(2*x^4 + 2*x^5) + log(3)^ 3*(16*x^3 + 16*x^4) + log(3)^2*(60*x^2 + 60*x^3) + 98) + 60*x^2*exp(16)*lo g(3)^2 + 16*x^3*exp(16)*log(3)^3 + 2*x^4*exp(16)*log(3)^4 + 112*x*exp(16)* log(3)))/(x*exp(x + 4) - x*exp(16)),x)