Integrand size = 237, antiderivative size = 27 \[ \int \frac {e^{1-e^{\frac {x+5 e x+e x \log (3+x)}{5 e+e \log (3+x)}}+x} \left (e (75+25 x)+e (30+10 x) \log (3+x)+e (3+x) \log ^2(3+x)+e^{\frac {x+5 e x+e x \log (3+x)}{5 e+e \log (3+x)}} \left (-15+e (-75-25 x)-4 x+(-3+e (-30-10 x)-x) \log (3+x)+e (-3-x) \log ^2(3+x)\right )\right )}{e (150+50 x)+e (60+20 x) \log (3+x)+e (6+2 x) \log ^2(3+x)+e^{1-e^{\frac {x+5 e x+e x \log (3+x)}{5 e+e \log (3+x)}}+x} \left (e (75+25 x)+e (30+10 x) \log (3+x)+e (3+x) \log ^2(3+x)\right )} \, dx=\log \left (2+e^{1-e^{x+\frac {x}{e (5+\log (3+x))}}+x}\right ) \]
Time = 1.15 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.78 \[ \int \frac {e^{1-e^{\frac {x+5 e x+e x \log (3+x)}{5 e+e \log (3+x)}}+x} \left (e (75+25 x)+e (30+10 x) \log (3+x)+e (3+x) \log ^2(3+x)+e^{\frac {x+5 e x+e x \log (3+x)}{5 e+e \log (3+x)}} \left (-15+e (-75-25 x)-4 x+(-3+e (-30-10 x)-x) \log (3+x)+e (-3-x) \log ^2(3+x)\right )\right )}{e (150+50 x)+e (60+20 x) \log (3+x)+e (6+2 x) \log ^2(3+x)+e^{1-e^{\frac {x+5 e x+e x \log (3+x)}{5 e+e \log (3+x)}}+x} \left (e (75+25 x)+e (30+10 x) \log (3+x)+e (3+x) \log ^2(3+x)\right )} \, dx=-e^{x+\frac {x}{e (5+\log (3+x))}}+\log \left (2 e^{e^{x+\frac {x}{e (5+\log (3+x))}}}+e^{1+x}\right ) \]
Integrate[(E^(1 - E^((x + 5*E*x + E*x*Log[3 + x])/(5*E + E*Log[3 + x])) + x)*(E*(75 + 25*x) + E*(30 + 10*x)*Log[3 + x] + E*(3 + x)*Log[3 + x]^2 + E^ ((x + 5*E*x + E*x*Log[3 + x])/(5*E + E*Log[3 + x]))*(-15 + E*(-75 - 25*x) - 4*x + (-3 + E*(-30 - 10*x) - x)*Log[3 + x] + E*(-3 - x)*Log[3 + x]^2)))/ (E*(150 + 50*x) + E*(60 + 20*x)*Log[3 + x] + E*(6 + 2*x)*Log[3 + x]^2 + E^ (1 - E^((x + 5*E*x + E*x*Log[3 + x])/(5*E + E*Log[3 + x])) + x)*(E*(75 + 2 5*x) + E*(30 + 10*x)*Log[3 + x] + E*(3 + x)*Log[3 + x]^2)),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.
| \(\displaystyle \int \frac {\exp \left (-\exp \left (\frac {5 e x+x+e x \log (x+3)}{e \log (x+3)+5 e}\right )+x+1\right ) \left (\left (e (-25 x-75)-4 x+e (-x-3) \log ^2(x+3)+(e (-10 x-30)-x-3) \log (x+3)-15\right ) \exp \left (\frac {5 e x+x+e x \log (x+3)}{e \log (x+3)+5 e}\right )+e (25 x+75)+e (x+3) \log ^2(x+3)+e (10 x+30) \log (x+3)\right )}{\left (e (25 x+75)+e (x+3) \log ^2(x+3)+e (10 x+30) \log (x+3)\right ) \exp \left (-\exp \left (\frac {5 e x+x+e x \log (x+3)}{e \log (x+3)+5 e}\right )+x+1\right )+e (50 x+150)+e (2 x+6) \log ^2(x+3)+e (20 x+60) \log (x+3)} \, dx\) |
| \(\Big \downarrow \) 2007 |
| \(\displaystyle \int \frac {459 x^4-1188 x^3-144 x^2+e^{4 x} \left (108 x^5-378 x^4+360 x^3+32 x^2-128 x\right )+e^{2 x} \left (324 x^5-540 x^4-504 x^3+864 x^2\right )+64 x}{(3 x-4)^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {459 x^4}{(3 x-4)^3}-\frac {1188 x^3}{(3 x-4)^3}+\frac {36 e^{2 x} \left (3 x^2-x-6\right ) x^2}{(3 x-4)^2}-\frac {144 x^2}{(3 x-4)^3}+2 e^{4 x} (2 x+1) x+\frac {64 x}{(3 x-4)^3}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 6 e^{2 x} x^2+e^{4 x} x^2-\frac {8 x^2}{(4-3 x)^2}+\frac {17 x^2}{2}+8 e^{2 x} x+24 x+\frac {32 e^{2 x}}{3}-\frac {128 e^{2 x}}{3 (4-3 x)}-\frac {2368}{9 (4-3 x)}+\frac {2432}{9 (4-3 x)^2}\) |
Int[(E^(1 - E^((x + 5*E*x + E*x*Log[3 + x])/(5*E + E*Log[3 + x])) + x)*(E* (75 + 25*x) + E*(30 + 10*x)*Log[3 + x] + E*(3 + x)*Log[3 + x]^2 + E^((x + 5*E*x + E*x*Log[3 + x])/(5*E + E*Log[3 + x]))*(-15 + E*(-75 - 25*x) - 4*x + (-3 + E*(-30 - 10*x) - x)*Log[3 + x] + E*(-3 - x)*Log[3 + x]^2)))/(E*(15 0 + 50*x) + E*(60 + 20*x)*Log[3 + x] + E*(6 + 2*x)*Log[3 + x]^2 + E^(1 - E ^((x + 5*E*x + E*x*Log[3 + x])/(5*E + E*Log[3 + x])) + x)*(E*(75 + 25*x) + E*(30 + 10*x)*Log[3 + x] + E*(3 + x)*Log[3 + x]^2)),x]
3.4.62.3.1 Defintions of rubi rules used
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]
Time = 26.28 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.41
| method | result | size |
| risch | \(-1+\ln \left ({\mathrm e}^{-{\mathrm e}^{\frac {x \left ({\mathrm e} \ln \left (3+x \right )+5 \,{\mathrm e}+1\right ) {\mathrm e}^{-1}}{\ln \left (3+x \right )+5}}+x +1}+2\right )\) | \(38\) |
| parallelrisch | \(\ln \left ({\mathrm e}^{-{\mathrm e}^{\frac {x \left ({\mathrm e} \ln \left (3+x \right )+5 \,{\mathrm e}+1\right ) {\mathrm e}^{-1}}{\ln \left (3+x \right )+5}}+x +1}+2\right )\) | \(38\) |
int((((-3-x)*exp(1)*ln(3+x)^2+((-10*x-30)*exp(1)-3-x)*ln(3+x)+(-25*x-75)*e xp(1)-4*x-15)*exp((x*exp(1)*ln(3+x)+5*x*exp(1)+x)/(exp(1)*ln(3+x)+5*exp(1) ))+(3+x)*exp(1)*ln(3+x)^2+(10*x+30)*exp(1)*ln(3+x)+(25*x+75)*exp(1))*exp(- exp((x*exp(1)*ln(3+x)+5*x*exp(1)+x)/(exp(1)*ln(3+x)+5*exp(1)))+x+1)/(((3+x )*exp(1)*ln(3+x)^2+(10*x+30)*exp(1)*ln(3+x)+(25*x+75)*exp(1))*exp(-exp((x* exp(1)*ln(3+x)+5*x*exp(1)+x)/(exp(1)*ln(3+x)+5*exp(1)))+x+1)+(2*x+6)*exp(1 )*ln(3+x)^2+(20*x+60)*exp(1)*ln(3+x)+(50*x+150)*exp(1)),x,method=_RETURNVE RBOSE)
Time = 0.24 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.48 \[ \int \frac {e^{1-e^{\frac {x+5 e x+e x \log (3+x)}{5 e+e \log (3+x)}}+x} \left (e (75+25 x)+e (30+10 x) \log (3+x)+e (3+x) \log ^2(3+x)+e^{\frac {x+5 e x+e x \log (3+x)}{5 e+e \log (3+x)}} \left (-15+e (-75-25 x)-4 x+(-3+e (-30-10 x)-x) \log (3+x)+e (-3-x) \log ^2(3+x)\right )\right )}{e (150+50 x)+e (60+20 x) \log (3+x)+e (6+2 x) \log ^2(3+x)+e^{1-e^{\frac {x+5 e x+e x \log (3+x)}{5 e+e \log (3+x)}}+x} \left (e (75+25 x)+e (30+10 x) \log (3+x)+e (3+x) \log ^2(3+x)\right )} \, dx=\log \left (e^{\left (x - e^{\left (\frac {x e \log \left (x + 3\right ) + 5 \, x e + x}{e \log \left (x + 3\right ) + 5 \, e}\right )} + 1\right )} + 2\right ) \]
integrate((((-3-x)*exp(1)*log(3+x)^2+((-10*x-30)*exp(1)-3-x)*log(3+x)+(-25 *x-75)*exp(1)-4*x-15)*exp((x*exp(1)*log(3+x)+5*x*exp(1)+x)/(exp(1)*log(3+x )+5*exp(1)))+(3+x)*exp(1)*log(3+x)^2+(10*x+30)*exp(1)*log(3+x)+(25*x+75)*e xp(1))*exp(-exp((x*exp(1)*log(3+x)+5*x*exp(1)+x)/(exp(1)*log(3+x)+5*exp(1) ))+x+1)/(((3+x)*exp(1)*log(3+x)^2+(10*x+30)*exp(1)*log(3+x)+(25*x+75)*exp( 1))*exp(-exp((x*exp(1)*log(3+x)+5*x*exp(1)+x)/(exp(1)*log(3+x)+5*exp(1)))+ x+1)+(2*x+6)*exp(1)*log(3+x)^2+(20*x+60)*exp(1)*log(3+x)+(50*x+150)*exp(1) ),x, algorithm=\
Time = 15.55 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.52 \[ \int \frac {e^{1-e^{\frac {x+5 e x+e x \log (3+x)}{5 e+e \log (3+x)}}+x} \left (e (75+25 x)+e (30+10 x) \log (3+x)+e (3+x) \log ^2(3+x)+e^{\frac {x+5 e x+e x \log (3+x)}{5 e+e \log (3+x)}} \left (-15+e (-75-25 x)-4 x+(-3+e (-30-10 x)-x) \log (3+x)+e (-3-x) \log ^2(3+x)\right )\right )}{e (150+50 x)+e (60+20 x) \log (3+x)+e (6+2 x) \log ^2(3+x)+e^{1-e^{\frac {x+5 e x+e x \log (3+x)}{5 e+e \log (3+x)}}+x} \left (e (75+25 x)+e (30+10 x) \log (3+x)+e (3+x) \log ^2(3+x)\right )} \, dx=\log {\left (e^{x - e^{\frac {e x \log {\left (x + 3 \right )} + x + 5 e x}{e \log {\left (x + 3 \right )} + 5 e}} + 1} + 2 \right )} \]
integrate((((-3-x)*exp(1)*ln(3+x)**2+((-10*x-30)*exp(1)-3-x)*ln(3+x)+(-25* x-75)*exp(1)-4*x-15)*exp((x*exp(1)*ln(3+x)+5*x*exp(1)+x)/(exp(1)*ln(3+x)+5 *exp(1)))+(3+x)*exp(1)*ln(3+x)**2+(10*x+30)*exp(1)*ln(3+x)+(25*x+75)*exp(1 ))*exp(-exp((x*exp(1)*ln(3+x)+5*x*exp(1)+x)/(exp(1)*ln(3+x)+5*exp(1)))+x+1 )/(((3+x)*exp(1)*ln(3+x)**2+(10*x+30)*exp(1)*ln(3+x)+(25*x+75)*exp(1))*exp (-exp((x*exp(1)*ln(3+x)+5*x*exp(1)+x)/(exp(1)*ln(3+x)+5*exp(1)))+x+1)+(2*x +6)*exp(1)*ln(3+x)**2+(20*x+60)*exp(1)*ln(3+x)+(50*x+150)*exp(1)),x)
Leaf count of result is larger than twice the leaf count of optimal. 98 vs. \(2 (24) = 48\).
Time = 0.33 (sec) , antiderivative size = 98, normalized size of antiderivative = 3.63 \[ \int \frac {e^{1-e^{\frac {x+5 e x+e x \log (3+x)}{5 e+e \log (3+x)}}+x} \left (e (75+25 x)+e (30+10 x) \log (3+x)+e (3+x) \log ^2(3+x)+e^{\frac {x+5 e x+e x \log (3+x)}{5 e+e \log (3+x)}} \left (-15+e (-75-25 x)-4 x+(-3+e (-30-10 x)-x) \log (3+x)+e (-3-x) \log ^2(3+x)\right )\right )}{e (150+50 x)+e (60+20 x) \log (3+x)+e (6+2 x) \log ^2(3+x)+e^{1-e^{\frac {x+5 e x+e x \log (3+x)}{5 e+e \log (3+x)}}+x} \left (e (75+25 x)+e (30+10 x) \log (3+x)+e (3+x) \log ^2(3+x)\right )} \, dx=-e^{\left (\frac {x \log \left (x + 3\right )}{\log \left (x + 3\right ) + 5} + \frac {x}{e \log \left (x + 3\right ) + 5 \, e} + \frac {5 \, x}{\log \left (x + 3\right ) + 5}\right )} + \log \left (\frac {1}{2} \, e^{\left (x + 1\right )} + e^{\left (e^{\left (\frac {x \log \left (x + 3\right )}{\log \left (x + 3\right ) + 5} + \frac {x}{e \log \left (x + 3\right ) + 5 \, e} + \frac {5 \, x}{\log \left (x + 3\right ) + 5}\right )}\right )}\right ) \]
integrate((((-3-x)*exp(1)*log(3+x)^2+((-10*x-30)*exp(1)-3-x)*log(3+x)+(-25 *x-75)*exp(1)-4*x-15)*exp((x*exp(1)*log(3+x)+5*x*exp(1)+x)/(exp(1)*log(3+x )+5*exp(1)))+(3+x)*exp(1)*log(3+x)^2+(10*x+30)*exp(1)*log(3+x)+(25*x+75)*e xp(1))*exp(-exp((x*exp(1)*log(3+x)+5*x*exp(1)+x)/(exp(1)*log(3+x)+5*exp(1) ))+x+1)/(((3+x)*exp(1)*log(3+x)^2+(10*x+30)*exp(1)*log(3+x)+(25*x+75)*exp( 1))*exp(-exp((x*exp(1)*log(3+x)+5*x*exp(1)+x)/(exp(1)*log(3+x)+5*exp(1)))+ x+1)+(2*x+6)*exp(1)*log(3+x)^2+(20*x+60)*exp(1)*log(3+x)+(50*x+150)*exp(1) ),x, algorithm=\
-e^(x*log(x + 3)/(log(x + 3) + 5) + x/(e*log(x + 3) + 5*e) + 5*x/(log(x + 3) + 5)) + log(1/2*e^(x + 1) + e^(e^(x*log(x + 3)/(log(x + 3) + 5) + x/(e* log(x + 3) + 5*e) + 5*x/(log(x + 3) + 5))))
Leaf count of result is larger than twice the leaf count of optimal. 329 vs. \(2 (24) = 48\).
Time = 11.64 (sec) , antiderivative size = 329, normalized size of antiderivative = 12.19 \[ \int \frac {e^{1-e^{\frac {x+5 e x+e x \log (3+x)}{5 e+e \log (3+x)}}+x} \left (e (75+25 x)+e (30+10 x) \log (3+x)+e (3+x) \log ^2(3+x)+e^{\frac {x+5 e x+e x \log (3+x)}{5 e+e \log (3+x)}} \left (-15+e (-75-25 x)-4 x+(-3+e (-30-10 x)-x) \log (3+x)+e (-3-x) \log ^2(3+x)\right )\right )}{e (150+50 x)+e (60+20 x) \log (3+x)+e (6+2 x) \log ^2(3+x)+e^{1-e^{\frac {x+5 e x+e x \log (3+x)}{5 e+e \log (3+x)}}+x} \left (e (75+25 x)+e (30+10 x) \log (3+x)+e (3+x) \log ^2(3+x)\right )} \, dx=-\frac {x e \log \left (x + 3\right ) - e \log \left (x + 3\right ) \log \left (e^{\left (\frac {2 \, x e \log \left (x + 3\right ) + 10 \, x e - e^{\left (\frac {x e \log \left (x + 3\right ) + 5 \, x e + x}{e \log \left (x + 3\right ) + 5 \, e} + 1\right )} \log \left (x + 3\right ) + x - 5 \, e^{\left (\frac {x e \log \left (x + 3\right ) + 5 \, x e + x}{e \log \left (x + 3\right ) + 5 \, e} + 1\right )}}{e \log \left (x + 3\right ) + 5 \, e} + 1\right )} + 2 \, e^{\left (\frac {x e \log \left (x + 3\right ) + 5 \, x e + x}{e \log \left (x + 3\right ) + 5 \, e}\right )}\right ) + 5 \, x e - 5 \, e \log \left (e^{\left (\frac {2 \, x e \log \left (x + 3\right ) + 10 \, x e - e^{\left (\frac {x e \log \left (x + 3\right ) + 5 \, x e + x}{e \log \left (x + 3\right ) + 5 \, e} + 1\right )} \log \left (x + 3\right ) + x - 5 \, e^{\left (\frac {x e \log \left (x + 3\right ) + 5 \, x e + x}{e \log \left (x + 3\right ) + 5 \, e} + 1\right )}}{e \log \left (x + 3\right ) + 5 \, e} + 1\right )} + 2 \, e^{\left (\frac {x e \log \left (x + 3\right ) + 5 \, x e + x}{e \log \left (x + 3\right ) + 5 \, e}\right )}\right ) + x}{e \log \left (x + 3\right ) + 5 \, e} \]
integrate((((-3-x)*exp(1)*log(3+x)^2+((-10*x-30)*exp(1)-3-x)*log(3+x)+(-25 *x-75)*exp(1)-4*x-15)*exp((x*exp(1)*log(3+x)+5*x*exp(1)+x)/(exp(1)*log(3+x )+5*exp(1)))+(3+x)*exp(1)*log(3+x)^2+(10*x+30)*exp(1)*log(3+x)+(25*x+75)*e xp(1))*exp(-exp((x*exp(1)*log(3+x)+5*x*exp(1)+x)/(exp(1)*log(3+x)+5*exp(1) ))+x+1)/(((3+x)*exp(1)*log(3+x)^2+(10*x+30)*exp(1)*log(3+x)+(25*x+75)*exp( 1))*exp(-exp((x*exp(1)*log(3+x)+5*x*exp(1)+x)/(exp(1)*log(3+x)+5*exp(1)))+ x+1)+(2*x+6)*exp(1)*log(3+x)^2+(20*x+60)*exp(1)*log(3+x)+(50*x+150)*exp(1) ),x, algorithm=\
-(x*e*log(x + 3) - e*log(x + 3)*log(e^((2*x*e*log(x + 3) + 10*x*e - e^((x* e*log(x + 3) + 5*x*e + x)/(e*log(x + 3) + 5*e) + 1)*log(x + 3) + x - 5*e^( (x*e*log(x + 3) + 5*x*e + x)/(e*log(x + 3) + 5*e) + 1))/(e*log(x + 3) + 5* e) + 1) + 2*e^((x*e*log(x + 3) + 5*x*e + x)/(e*log(x + 3) + 5*e))) + 5*x*e - 5*e*log(e^((2*x*e*log(x + 3) + 10*x*e - e^((x*e*log(x + 3) + 5*x*e + x) /(e*log(x + 3) + 5*e) + 1)*log(x + 3) + x - 5*e^((x*e*log(x + 3) + 5*x*e + x)/(e*log(x + 3) + 5*e) + 1))/(e*log(x + 3) + 5*e) + 1) + 2*e^((x*e*log(x + 3) + 5*x*e + x)/(e*log(x + 3) + 5*e))) + x)/(e*log(x + 3) + 5*e)
Time = 1.38 (sec) , antiderivative size = 54, normalized size of antiderivative = 2.00 \[ \int \frac {e^{1-e^{\frac {x+5 e x+e x \log (3+x)}{5 e+e \log (3+x)}}+x} \left (e (75+25 x)+e (30+10 x) \log (3+x)+e (3+x) \log ^2(3+x)+e^{\frac {x+5 e x+e x \log (3+x)}{5 e+e \log (3+x)}} \left (-15+e (-75-25 x)-4 x+(-3+e (-30-10 x)-x) \log (3+x)+e (-3-x) \log ^2(3+x)\right )\right )}{e (150+50 x)+e (60+20 x) \log (3+x)+e (6+2 x) \log ^2(3+x)+e^{1-e^{\frac {x+5 e x+e x \log (3+x)}{5 e+e \log (3+x)}}+x} \left (e (75+25 x)+e (30+10 x) \log (3+x)+e (3+x) \log ^2(3+x)\right )} \, dx=\ln \left (\mathrm {e}\,{\mathrm {e}}^x\,{\mathrm {e}}^{-{\mathrm {e}}^{\frac {5\,x}{\ln \left (x+3\right )+5}}\,{\mathrm {e}}^{\frac {x}{5\,\mathrm {e}+\ln \left (x+3\right )\,\mathrm {e}}}\,{\left (x+3\right )}^{\frac {x}{\ln \left (x+3\right )+5}}}+2\right ) \]
int((exp(x - exp((x + 5*x*exp(1) + x*log(x + 3)*exp(1))/(5*exp(1) + log(x + 3)*exp(1))) + 1)*(exp(1)*(25*x + 75) - exp((x + 5*x*exp(1) + x*log(x + 3 )*exp(1))/(5*exp(1) + log(x + 3)*exp(1)))*(4*x + log(x + 3)*(x + exp(1)*(1 0*x + 30) + 3) + exp(1)*(25*x + 75) + log(x + 3)^2*exp(1)*(x + 3) + 15) + log(x + 3)*exp(1)*(10*x + 30) + log(x + 3)^2*exp(1)*(x + 3)))/(exp(x - exp ((x + 5*x*exp(1) + x*log(x + 3)*exp(1))/(5*exp(1) + log(x + 3)*exp(1))) + 1)*(exp(1)*(25*x + 75) + log(x + 3)*exp(1)*(10*x + 30) + log(x + 3)^2*exp( 1)*(x + 3)) + exp(1)*(50*x + 150) + log(x + 3)*exp(1)*(20*x + 60) + log(x + 3)^2*exp(1)*(2*x + 6)),x)