Integrand size = 85, antiderivative size = 24 \[ \int \frac {e^{\frac {e}{\log \left (\frac {3+e^{8+x}}{e^8}\right )}} \left (e^{8+x} \left (-24 e-e^3 x\right )+\left (3 e^2+e^{10+x}\right ) \log ^2\left (\frac {3+e^{8+x}}{e^8}\right )\right )}{\left (3+e^{8+x}\right ) \log ^2\left (\frac {3+e^{8+x}}{e^8}\right )} \, dx=e^{\frac {e}{\log \left (\frac {3}{e^8}+e^x\right )}} \left (24+e^2 x\right ) \] Output:
(24+exp(2)*x)*exp(exp(1)/ln(exp(x)+3/exp(4)^2))
Time = 0.10 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {e^{\frac {e}{\log \left (\frac {3+e^{8+x}}{e^8}\right )}} \left (e^{8+x} \left (-24 e-e^3 x\right )+\left (3 e^2+e^{10+x}\right ) \log ^2\left (\frac {3+e^{8+x}}{e^8}\right )\right )}{\left (3+e^{8+x}\right ) \log ^2\left (\frac {3+e^{8+x}}{e^8}\right )} \, dx=e^{\frac {e}{-8+\log \left (3+e^{8+x}\right )}} \left (24+e^2 x\right ) \] Input:
Integrate[(E^(E/Log[(3 + E^(8 + x))/E^8])*(E^(8 + x)*(-24*E - E^3*x) + (3* E^2 + E^(10 + x))*Log[(3 + E^(8 + x))/E^8]^2))/((3 + E^(8 + x))*Log[(3 + E ^(8 + x))/E^8]^2),x]
Output:
E^(E/(-8 + Log[3 + E^(8 + x)]))*(24 + E^2*x)
Time = 0.37 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.012, Rules used = {2726}
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.
\(\displaystyle \int \frac {e^{\frac {e}{\log \left (\frac {e^{x+8}+3}{e^8}\right )}} \left (e^{x+8} \left (-e^3 x-24 e\right )+\left (e^{x+10}+3 e^2\right ) \log ^2\left (\frac {e^{x+8}+3}{e^8}\right )\right )}{\left (e^{x+8}+3\right ) \log ^2\left (\frac {e^{x+8}+3}{e^8}\right )} \, dx\) |
\(\Big \downarrow \) 2726 |
\(\displaystyle \left (e^2 x+24\right ) e^{\frac {e}{\log \left (\frac {e^{x+8}+3}{e^8}\right )}}\) |
Input:
Int[(E^(E/Log[(3 + E^(8 + x))/E^8])*(E^(8 + x)*(-24*E - E^3*x) + (3*E^2 + E^(10 + x))*Log[(3 + E^(8 + x))/E^8]^2))/((3 + E^(8 + x))*Log[(3 + E^(8 + x))/E^8]^2),x]
Output:
E^(E/Log[(3 + E^(8 + x))/E^8])*(24 + E^2*x)
Int[(y_.)*(F_)^(u_)*((v_) + (w_)), x_Symbol] :> With[{z = v*(y/(Log[F]*D[u, x]))}, Simp[F^u*z, x] /; EqQ[D[z, x], w*y]] /; FreeQ[F, x]
Time = 3.43 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00
method | result | size |
risch | \(\left (24+{\mathrm e}^{2} x \right ) {\mathrm e}^{\frac {{\mathrm e}}{\ln \left (\left ({\mathrm e}^{x +8}+3\right ) {\mathrm e}^{-8}\right )}}\) | \(24\) |
parallelrisch | \({\mathrm e}^{2} {\mathrm e}^{\frac {{\mathrm e}}{\ln \left (\left ({\mathrm e}^{8} {\mathrm e}^{x}+3\right ) {\mathrm e}^{-8}\right )}} x +24 \,{\mathrm e}^{\frac {{\mathrm e}}{\ln \left (\left ({\mathrm e}^{8} {\mathrm e}^{x}+3\right ) {\mathrm e}^{-8}\right )}}\) | \(50\) |
Input:
int(((exp(2)*exp(4)^2*exp(x)+3*exp(2))*ln((exp(4)^2*exp(x)+3)/exp(4)^2)^2+ (-x*exp(1)*exp(2)-24*exp(1))*exp(4)^2*exp(x))*exp(exp(1)/ln((exp(4)^2*exp( x)+3)/exp(4)^2))/(exp(4)^2*exp(x)+3)/ln((exp(4)^2*exp(x)+3)/exp(4)^2)^2,x, method=_RETURNVERBOSE)
Output:
(24+exp(2)*x)*exp(exp(1)/ln((exp(x+8)+3)*exp(-8)))
Time = 0.10 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {e^{\frac {e}{\log \left (\frac {3+e^{8+x}}{e^8}\right )}} \left (e^{8+x} \left (-24 e-e^3 x\right )+\left (3 e^2+e^{10+x}\right ) \log ^2\left (\frac {3+e^{8+x}}{e^8}\right )\right )}{\left (3+e^{8+x}\right ) \log ^2\left (\frac {3+e^{8+x}}{e^8}\right )} \, dx={\left (x e^{2} + 24\right )} e^{\left (\frac {e}{\log \left ({\left (3 \, e^{2} + e^{\left (x + 10\right )}\right )} e^{\left (-10\right )}\right )}\right )} \] Input:
integrate(((exp(2)*exp(4)^2*exp(x)+3*exp(2))*log((exp(4)^2*exp(x)+3)/exp(4 )^2)^2+(-x*exp(1)*exp(2)-24*exp(1))*exp(4)^2*exp(x))*exp(exp(1)/log((exp(4 )^2*exp(x)+3)/exp(4)^2))/(exp(4)^2*exp(x)+3)/log((exp(4)^2*exp(x)+3)/exp(4 )^2)^2,x, algorithm="fricas")
Output:
(x*e^2 + 24)*e^(e/log((3*e^2 + e^(x + 10))*e^(-10)))
Timed out. \[ \int \frac {e^{\frac {e}{\log \left (\frac {3+e^{8+x}}{e^8}\right )}} \left (e^{8+x} \left (-24 e-e^3 x\right )+\left (3 e^2+e^{10+x}\right ) \log ^2\left (\frac {3+e^{8+x}}{e^8}\right )\right )}{\left (3+e^{8+x}\right ) \log ^2\left (\frac {3+e^{8+x}}{e^8}\right )} \, dx=\text {Timed out} \] Input:
integrate(((exp(2)*exp(4)**2*exp(x)+3*exp(2))*ln((exp(4)**2*exp(x)+3)/exp( 4)**2)**2+(-x*exp(1)*exp(2)-24*exp(1))*exp(4)**2*exp(x))*exp(exp(1)/ln((ex p(4)**2*exp(x)+3)/exp(4)**2))/(exp(4)**2*exp(x)+3)/ln((exp(4)**2*exp(x)+3) /exp(4)**2)**2,x)
Output:
Timed out
\[ \int \frac {e^{\frac {e}{\log \left (\frac {3+e^{8+x}}{e^8}\right )}} \left (e^{8+x} \left (-24 e-e^3 x\right )+\left (3 e^2+e^{10+x}\right ) \log ^2\left (\frac {3+e^{8+x}}{e^8}\right )\right )}{\left (3+e^{8+x}\right ) \log ^2\left (\frac {3+e^{8+x}}{e^8}\right )} \, dx=\int { \frac {{\left ({\left (3 \, e^{2} + e^{\left (x + 10\right )}\right )} \log \left ({\left (e^{\left (x + 8\right )} + 3\right )} e^{\left (-8\right )}\right )^{2} - {\left (x e^{3} + 24 \, e\right )} e^{\left (x + 8\right )}\right )} e^{\left (\frac {e}{\log \left ({\left (e^{\left (x + 8\right )} + 3\right )} e^{\left (-8\right )}\right )}\right )}}{{\left (e^{\left (x + 8\right )} + 3\right )} \log \left ({\left (e^{\left (x + 8\right )} + 3\right )} e^{\left (-8\right )}\right )^{2}} \,d x } \] Input:
integrate(((exp(2)*exp(4)^2*exp(x)+3*exp(2))*log((exp(4)^2*exp(x)+3)/exp(4 )^2)^2+(-x*exp(1)*exp(2)-24*exp(1))*exp(4)^2*exp(x))*exp(exp(1)/log((exp(4 )^2*exp(x)+3)/exp(4)^2))/(exp(4)^2*exp(x)+3)/log((exp(4)^2*exp(x)+3)/exp(4 )^2)^2,x, algorithm="maxima")
Output:
x*e^(e/(log(e^(x + 8) + 3) - 8) + 2) + 24*e^(e/(log(e^(x + 8) + 3) - 8)) + integrate(e^(e/(log(e^(x + 8) + 3) - 8) + 2), x)
Exception generated. \[ \int \frac {e^{\frac {e}{\log \left (\frac {3+e^{8+x}}{e^8}\right )}} \left (e^{8+x} \left (-24 e-e^3 x\right )+\left (3 e^2+e^{10+x}\right ) \log ^2\left (\frac {3+e^{8+x}}{e^8}\right )\right )}{\left (3+e^{8+x}\right ) \log ^2\left (\frac {3+e^{8+x}}{e^8}\right )} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(((exp(2)*exp(4)^2*exp(x)+3*exp(2))*log((exp(4)^2*exp(x)+3)/exp(4 )^2)^2+(-x*exp(1)*exp(2)-24*exp(1))*exp(4)^2*exp(x))*exp(exp(1)/log((exp(4 )^2*exp(x)+3)/exp(4)^2))/(exp(4)^2*exp(x)+3)/log((exp(4)^2*exp(x)+3)/exp(4 )^2)^2,x, algorithm="giac")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Unable to divide, perhaps due to ro unding error%%%{1,[1,17,9,10,0,80,1]%%%}+%%%{21,[1,17,9,9,0,72,1]%%%}+%%%{ 189,[1,17
Timed out. \[ \int \frac {e^{\frac {e}{\log \left (\frac {3+e^{8+x}}{e^8}\right )}} \left (e^{8+x} \left (-24 e-e^3 x\right )+\left (3 e^2+e^{10+x}\right ) \log ^2\left (\frac {3+e^{8+x}}{e^8}\right )\right )}{\left (3+e^{8+x}\right ) \log ^2\left (\frac {3+e^{8+x}}{e^8}\right )} \, dx=\int \frac {{\mathrm {e}}^{\frac {\mathrm {e}}{\ln \left ({\mathrm {e}}^{-8}\,\left ({\mathrm {e}}^8\,{\mathrm {e}}^x+3\right )\right )}}\,\left ({\ln \left ({\mathrm {e}}^{-8}\,\left ({\mathrm {e}}^8\,{\mathrm {e}}^x+3\right )\right )}^2\,\left ({\mathrm {e}}^{x+10}+3\,{\mathrm {e}}^2\right )-{\mathrm {e}}^{x+8}\,\left (24\,\mathrm {e}+x\,{\mathrm {e}}^3\right )\right )}{{\ln \left ({\mathrm {e}}^{-8}\,\left ({\mathrm {e}}^8\,{\mathrm {e}}^x+3\right )\right )}^2\,\left ({\mathrm {e}}^{x+8}+3\right )} \,d x \] Input:
int((exp(exp(1)/log(exp(-8)*(exp(8)*exp(x) + 3)))*(log(exp(-8)*(exp(8)*exp (x) + 3))^2*(3*exp(2) + exp(10)*exp(x)) - exp(8)*exp(x)*(24*exp(1) + x*exp (3))))/(log(exp(-8)*(exp(8)*exp(x) + 3))^2*(exp(8)*exp(x) + 3)),x)
Output:
int((exp(exp(1)/log(exp(-8)*(exp(8)*exp(x) + 3)))*(log(exp(-8)*(exp(8)*exp (x) + 3))^2*(exp(x + 10) + 3*exp(2)) - exp(x + 8)*(24*exp(1) + x*exp(3)))) /(log(exp(-8)*(exp(8)*exp(x) + 3))^2*(exp(x + 8) + 3)), x)
Time = 0.16 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.17 \[ \int \frac {e^{\frac {e}{\log \left (\frac {3+e^{8+x}}{e^8}\right )}} \left (e^{8+x} \left (-24 e-e^3 x\right )+\left (3 e^2+e^{10+x}\right ) \log ^2\left (\frac {3+e^{8+x}}{e^8}\right )\right )}{\left (3+e^{8+x}\right ) \log ^2\left (\frac {3+e^{8+x}}{e^8}\right )} \, dx=e^{\frac {e}{\mathrm {log}\left (\frac {e^{x} e^{8}+3}{e^{8}}\right )}} \left (e^{2} x +24\right ) \] Input:
int(((exp(2)*exp(4)^2*exp(x)+3*exp(2))*log((exp(4)^2*exp(x)+3)/exp(4)^2)^2 +(-x*exp(1)*exp(2)-24*exp(1))*exp(4)^2*exp(x))*exp(exp(1)/log((exp(4)^2*ex p(x)+3)/exp(4)^2))/(exp(4)^2*exp(x)+3)/log((exp(4)^2*exp(x)+3)/exp(4)^2)^2 ,x)
Output:
e**(e/log((e**x*e**8 + 3)/e**8))*(e**2*x + 24)