\(\int \frac {e^{\frac {81 x^2}{625-500 x+150 x^2-20 x^3+x^4+(50 x-20 x^2+2 x^3) \log (2+4 x)+x^2 \log ^2(2+4 x)}} (4050 x+8100 x^2-486 x^3-324 x^4)}{46875+37500 x-84375 x^2+48750 x^3-13875 x^4+2160 x^5-177 x^6+6 x^7+(5625 x+6750 x^2-7650 x^3+2520 x^4-351 x^5+18 x^6) \log (2+4 x)+(225 x^2+360 x^3-171 x^4+18 x^5) \log ^2(2+4 x)+(3 x^3+6 x^4) \log ^3(2+4 x)+e^{\frac {81 x^2}{625-500 x+150 x^2-20 x^3+x^4+(50 x-20 x^2+2 x^3) \log (2+4 x)+x^2 \log ^2(2+4 x)}} (15625+12500 x-28125 x^2+16250 x^3-4625 x^4+720 x^5-59 x^6+2 x^7+(1875 x+2250 x^2-2550 x^3+840 x^4-117 x^5+6 x^6) \log (2+4 x)+(75 x^2+120 x^3-57 x^4+6 x^5) \log ^2(2+4 x)+(x^3+2 x^4) \log ^3(2+4 x))} \, dx\) [1040]

Optimal result

Integrand size = 383, antiderivative size = 28 \[ \int \frac {e^{\frac {81 x^2}{625-500 x+150 x^2-20 x^3+x^4+\left (50 x-20 x^2+2 x^3\right ) \log (2+4 x)+x^2 \log ^2(2+4 x)}} \left (4050 x+8100 x^2-486 x^3-324 x^4\right )}{46875+37500 x-84375 x^2+48750 x^3-13875 x^4+2160 x^5-177 x^6+6 x^7+\left (5625 x+6750 x^2-7650 x^3+2520 x^4-351 x^5+18 x^6\right ) \log (2+4 x)+\left (225 x^2+360 x^3-171 x^4+18 x^5\right ) \log ^2(2+4 x)+\left (3 x^3+6 x^4\right ) \log ^3(2+4 x)+e^{\frac {81 x^2}{625-500 x+150 x^2-20 x^3+x^4+\left (50 x-20 x^2+2 x^3\right ) \log (2+4 x)+x^2 \log ^2(2+4 x)}} \left (15625+12500 x-28125 x^2+16250 x^3-4625 x^4+720 x^5-59 x^6+2 x^7+\left (1875 x+2250 x^2-2550 x^3+840 x^4-117 x^5+6 x^6\right ) \log (2+4 x)+\left (75 x^2+120 x^3-57 x^4+6 x^5\right ) \log ^2(2+4 x)+\left (x^3+2 x^4\right ) \log ^3(2+4 x)\right )} \, dx=\log \left (3+e^{\frac {81 x^2}{\left ((5-x)^2+x \log (2+4 x)\right )^2}}\right ) \] Output:

ln(3+exp(x^2/(1/9*ln(4*x+2)*x+1/9*(5-x)^2)^2))
 

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.93 \[ \int \frac {e^{\frac {81 x^2}{625-500 x+150 x^2-20 x^3+x^4+\left (50 x-20 x^2+2 x^3\right ) \log (2+4 x)+x^2 \log ^2(2+4 x)}} \left (4050 x+8100 x^2-486 x^3-324 x^4\right )}{46875+37500 x-84375 x^2+48750 x^3-13875 x^4+2160 x^5-177 x^6+6 x^7+\left (5625 x+6750 x^2-7650 x^3+2520 x^4-351 x^5+18 x^6\right ) \log (2+4 x)+\left (225 x^2+360 x^3-171 x^4+18 x^5\right ) \log ^2(2+4 x)+\left (3 x^3+6 x^4\right ) \log ^3(2+4 x)+e^{\frac {81 x^2}{625-500 x+150 x^2-20 x^3+x^4+\left (50 x-20 x^2+2 x^3\right ) \log (2+4 x)+x^2 \log ^2(2+4 x)}} \left (15625+12500 x-28125 x^2+16250 x^3-4625 x^4+720 x^5-59 x^6+2 x^7+\left (1875 x+2250 x^2-2550 x^3+840 x^4-117 x^5+6 x^6\right ) \log (2+4 x)+\left (75 x^2+120 x^3-57 x^4+6 x^5\right ) \log ^2(2+4 x)+\left (x^3+2 x^4\right ) \log ^3(2+4 x)\right )} \, dx=\log \left (3+e^{\frac {81 x^2}{\left ((-5+x)^2+x \log (2+4 x)\right )^2}}\right ) \] Input:

Integrate[(E^((81*x^2)/(625 - 500*x + 150*x^2 - 20*x^3 + x^4 + (50*x - 20* 
x^2 + 2*x^3)*Log[2 + 4*x] + x^2*Log[2 + 4*x]^2))*(4050*x + 8100*x^2 - 486* 
x^3 - 324*x^4))/(46875 + 37500*x - 84375*x^2 + 48750*x^3 - 13875*x^4 + 216 
0*x^5 - 177*x^6 + 6*x^7 + (5625*x + 6750*x^2 - 7650*x^3 + 2520*x^4 - 351*x 
^5 + 18*x^6)*Log[2 + 4*x] + (225*x^2 + 360*x^3 - 171*x^4 + 18*x^5)*Log[2 + 
 4*x]^2 + (3*x^3 + 6*x^4)*Log[2 + 4*x]^3 + E^((81*x^2)/(625 - 500*x + 150* 
x^2 - 20*x^3 + x^4 + (50*x - 20*x^2 + 2*x^3)*Log[2 + 4*x] + x^2*Log[2 + 4* 
x]^2))*(15625 + 12500*x - 28125*x^2 + 16250*x^3 - 4625*x^4 + 720*x^5 - 59* 
x^6 + 2*x^7 + (1875*x + 2250*x^2 - 2550*x^3 + 840*x^4 - 117*x^5 + 6*x^6)*L 
og[2 + 4*x] + (75*x^2 + 120*x^3 - 57*x^4 + 6*x^5)*Log[2 + 4*x]^2 + (x^3 + 
2*x^4)*Log[2 + 4*x]^3)),x]
 

Output:

Log[3 + E^((81*x^2)/((-5 + x)^2 + x*Log[2 + 4*x])^2)]
 

Rubi [A] (verified)

Time = 5.07 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.93, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.010, Rules used = {2029, 7239, 27, 7235}

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.

\(\displaystyle \int \frac {\left (-324 x^4-486 x^3+8100 x^2+4050 x\right ) \exp \left (\frac {81 x^2}{x^4-20 x^3+150 x^2+x^2 \log ^2(4 x+2)+\left (2 x^3-20 x^2+50 x\right ) \log (4 x+2)-500 x+625}\right )}{\left (2 x^7-59 x^6+720 x^5-4625 x^4+16250 x^3-28125 x^2+\left (2 x^4+x^3\right ) \log ^3(4 x+2)+\left (6 x^5-57 x^4+120 x^3+75 x^2\right ) \log ^2(4 x+2)+\left (6 x^6-117 x^5+840 x^4-2550 x^3+2250 x^2+1875 x\right ) \log (4 x+2)+12500 x+15625\right ) \exp \left (\frac {81 x^2}{x^4-20 x^3+150 x^2+x^2 \log ^2(4 x+2)+\left (2 x^3-20 x^2+50 x\right ) \log (4 x+2)-500 x+625}\right )+6 x^7-177 x^6+2160 x^5-13875 x^4+48750 x^3-84375 x^2+\left (6 x^4+3 x^3\right ) \log ^3(4 x+2)+\left (18 x^5-171 x^4+360 x^3+225 x^2\right ) \log ^2(4 x+2)+\left (18 x^6-351 x^5+2520 x^4-7650 x^3+6750 x^2+5625 x\right ) \log (4 x+2)+37500 x+46875} \, dx\)

\(\Big \downarrow \) 2029

\(\displaystyle \int \frac {x \left (-324 x^3-486 x^2+8100 x+4050\right ) \exp \left (\frac {81 x^2}{x^4-20 x^3+150 x^2+x^2 \log ^2(4 x+2)+\left (2 x^3-20 x^2+50 x\right ) \log (4 x+2)-500 x+625}\right )}{\left (2 x^7-59 x^6+720 x^5-4625 x^4+16250 x^3-28125 x^2+\left (2 x^4+x^3\right ) \log ^3(4 x+2)+\left (6 x^5-57 x^4+120 x^3+75 x^2\right ) \log ^2(4 x+2)+\left (6 x^6-117 x^5+840 x^4-2550 x^3+2250 x^2+1875 x\right ) \log (4 x+2)+12500 x+15625\right ) \exp \left (\frac {81 x^2}{x^4-20 x^3+150 x^2+x^2 \log ^2(4 x+2)+\left (2 x^3-20 x^2+50 x\right ) \log (4 x+2)-500 x+625}\right )+6 x^7-177 x^6+2160 x^5-13875 x^4+48750 x^3-84375 x^2+\left (6 x^4+3 x^3\right ) \log ^3(4 x+2)+\left (18 x^5-171 x^4+360 x^3+225 x^2\right ) \log ^2(4 x+2)+\left (18 x^6-351 x^5+2520 x^4-7650 x^3+6750 x^2+5625 x\right ) \log (4 x+2)+37500 x+46875}dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {162 x \left (-2 x^3-3 x^2+50 x+25\right ) e^{\frac {81 x^2}{\left ((x-5)^2+x \log (4 x+2)\right )^2}}}{(2 x+1) \left (e^{\frac {81 x^2}{\left ((x-5)^2+x \log (4 x+2)\right )^2}}+3\right ) \left ((x-5)^2+x \log (4 x+2)\right )^3}dx\)

\(\Big \downarrow \) 27

\(\displaystyle 162 \int \frac {e^{\frac {81 x^2}{\left ((x-5)^2+x \log (4 x+2)\right )^2}} x \left (-2 x^3-3 x^2+50 x+25\right )}{\left (3+e^{\frac {81 x^2}{\left ((x-5)^2+x \log (4 x+2)\right )^2}}\right ) (2 x+1) \left ((x-5)^2+x \log (4 x+2)\right )^3}dx\)

\(\Big \downarrow \) 7235

\(\displaystyle \log \left (e^{\frac {81 x^2}{\left ((x-5)^2+x \log (4 x+2)\right )^2}}+3\right )\)

Input:

Int[(E^((81*x^2)/(625 - 500*x + 150*x^2 - 20*x^3 + x^4 + (50*x - 20*x^2 + 
2*x^3)*Log[2 + 4*x] + x^2*Log[2 + 4*x]^2))*(4050*x + 8100*x^2 - 486*x^3 - 
324*x^4))/(46875 + 37500*x - 84375*x^2 + 48750*x^3 - 13875*x^4 + 2160*x^5 
- 177*x^6 + 6*x^7 + (5625*x + 6750*x^2 - 7650*x^3 + 2520*x^4 - 351*x^5 + 1 
8*x^6)*Log[2 + 4*x] + (225*x^2 + 360*x^3 - 171*x^4 + 18*x^5)*Log[2 + 4*x]^ 
2 + (3*x^3 + 6*x^4)*Log[2 + 4*x]^3 + E^((81*x^2)/(625 - 500*x + 150*x^2 - 
20*x^3 + x^4 + (50*x - 20*x^2 + 2*x^3)*Log[2 + 4*x] + x^2*Log[2 + 4*x]^2)) 
*(15625 + 12500*x - 28125*x^2 + 16250*x^3 - 4625*x^4 + 720*x^5 - 59*x^6 + 
2*x^7 + (1875*x + 2250*x^2 - 2550*x^3 + 840*x^4 - 117*x^5 + 6*x^6)*Log[2 + 
 4*x] + (75*x^2 + 120*x^3 - 57*x^4 + 6*x^5)*Log[2 + 4*x]^2 + (x^3 + 2*x^4) 
*Log[2 + 4*x]^3)),x]
 

Output:

Log[3 + E^((81*x^2)/((-5 + x)^2 + x*Log[2 + 4*x])^2)]
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 2029

Int[(Fx_.)*((d_.)*(x_)^(q_.) + (a_.)*(x_)^(r_.) + (b_.)*(x_)^(s_.) + (c_.)* 
(x_)^(t_.))^(p_.), x_Symbol] :> Int[x^(p*r)*(a + b*x^(s - r) + c*x^(t - r) 
+ d*x^(q - r))^p*Fx, x] /; FreeQ[{a, b, c, d, r, s, t, q}, x] && IntegerQ[p 
] && PosQ[s - r] && PosQ[t - r] && PosQ[q - r] &&  !(EqQ[p, 1] && EqQ[u, 1] 
)
 

rule 7235

Int[(u_)/(y_), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[q*L 
og[RemoveContent[y, x]], x] /;  !FalseQ[q]]
 

rule 7239

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl 
erIntegrandQ[v, u, x]]
 

Maple [A] (verified)

Time = 31.92 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00

method	result	size
risch	\(\ln \left ({\mathrm e}^{\frac {81 x^{2}}{\left (\ln \left (4 x +2\right ) x +x^{2}-10 x +25\right )^{2}}}+3\right )\)	\(28\)
parallelrisch	\(\ln \left ({\mathrm e}^{\frac {81 x^{2}}{x^{2} \ln \left (4 x +2\right )^{2}+2 \ln \left (4 x +2\right ) x^{3}+x^{4}-20 \ln \left (4 x +2\right ) x^{2}-20 x^{3}+50 \ln \left (4 x +2\right ) x +150 x^{2}-500 x +625}}+3\right )\)	\(73\)

Input:

int((-324*x^4-486*x^3+8100*x^2+4050*x)*exp(81*x^2/(x^2*ln(4*x+2)^2+(2*x^3- 
20*x^2+50*x)*ln(4*x+2)+x^4-20*x^3+150*x^2-500*x+625))/(((2*x^4+x^3)*ln(4*x 
+2)^3+(6*x^5-57*x^4+120*x^3+75*x^2)*ln(4*x+2)^2+(6*x^6-117*x^5+840*x^4-255 
0*x^3+2250*x^2+1875*x)*ln(4*x+2)+2*x^7-59*x^6+720*x^5-4625*x^4+16250*x^3-2 
8125*x^2+12500*x+15625)*exp(81*x^2/(x^2*ln(4*x+2)^2+(2*x^3-20*x^2+50*x)*ln 
(4*x+2)+x^4-20*x^3+150*x^2-500*x+625))+(6*x^4+3*x^3)*ln(4*x+2)^3+(18*x^5-1 
71*x^4+360*x^3+225*x^2)*ln(4*x+2)^2+(18*x^6-351*x^5+2520*x^4-7650*x^3+6750 
*x^2+5625*x)*ln(4*x+2)+6*x^7-177*x^6+2160*x^5-13875*x^4+48750*x^3-84375*x^ 
2+37500*x+46875),x,method=_RETURNVERBOSE)
 

Output:

ln(exp(81*x^2/(ln(4*x+2)*x+x^2-10*x+25)^2)+3)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 61 vs. \(2 (25) = 50\).

Time = 0.13 (sec) , antiderivative size = 61, normalized size of antiderivative = 2.18 \[ \int \frac {e^{\frac {81 x^2}{625-500 x+150 x^2-20 x^3+x^4+\left (50 x-20 x^2+2 x^3\right ) \log (2+4 x)+x^2 \log ^2(2+4 x)}} \left (4050 x+8100 x^2-486 x^3-324 x^4\right )}{46875+37500 x-84375 x^2+48750 x^3-13875 x^4+2160 x^5-177 x^6+6 x^7+\left (5625 x+6750 x^2-7650 x^3+2520 x^4-351 x^5+18 x^6\right ) \log (2+4 x)+\left (225 x^2+360 x^3-171 x^4+18 x^5\right ) \log ^2(2+4 x)+\left (3 x^3+6 x^4\right ) \log ^3(2+4 x)+e^{\frac {81 x^2}{625-500 x+150 x^2-20 x^3+x^4+\left (50 x-20 x^2+2 x^3\right ) \log (2+4 x)+x^2 \log ^2(2+4 x)}} \left (15625+12500 x-28125 x^2+16250 x^3-4625 x^4+720 x^5-59 x^6+2 x^7+\left (1875 x+2250 x^2-2550 x^3+840 x^4-117 x^5+6 x^6\right ) \log (2+4 x)+\left (75 x^2+120 x^3-57 x^4+6 x^5\right ) \log ^2(2+4 x)+\left (x^3+2 x^4\right ) \log ^3(2+4 x)\right )} \, dx=\log \left (e^{\left (\frac {81 \, x^{2}}{x^{4} + x^{2} \log \left (4 \, x + 2\right )^{2} - 20 \, x^{3} + 150 \, x^{2} + 2 \, {\left (x^{3} - 10 \, x^{2} + 25 \, x\right )} \log \left (4 \, x + 2\right ) - 500 \, x + 625}\right )} + 3\right ) \] Input:

integrate((-324*x^4-486*x^3+8100*x^2+4050*x)*exp(81*x^2/(x^2*log(2+4*x)^2+ 
(2*x^3-20*x^2+50*x)*log(2+4*x)+x^4-20*x^3+150*x^2-500*x+625))/(((2*x^4+x^3 
)*log(2+4*x)^3+(6*x^5-57*x^4+120*x^3+75*x^2)*log(2+4*x)^2+(6*x^6-117*x^5+8 
40*x^4-2550*x^3+2250*x^2+1875*x)*log(2+4*x)+2*x^7-59*x^6+720*x^5-4625*x^4+ 
16250*x^3-28125*x^2+12500*x+15625)*exp(81*x^2/(x^2*log(2+4*x)^2+(2*x^3-20* 
x^2+50*x)*log(2+4*x)+x^4-20*x^3+150*x^2-500*x+625))+(6*x^4+3*x^3)*log(2+4* 
x)^3+(18*x^5-171*x^4+360*x^3+225*x^2)*log(2+4*x)^2+(18*x^6-351*x^5+2520*x^ 
4-7650*x^3+6750*x^2+5625*x)*log(2+4*x)+6*x^7-177*x^6+2160*x^5-13875*x^4+48 
750*x^3-84375*x^2+37500*x+46875),x, algorithm="fricas")
 

Output:

log(e^(81*x^2/(x^4 + x^2*log(4*x + 2)^2 - 20*x^3 + 150*x^2 + 2*(x^3 - 10*x 
^2 + 25*x)*log(4*x + 2) - 500*x + 625)) + 3)
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 60 vs. \(2 (26) = 52\).

Time = 1.78 (sec) , antiderivative size = 60, normalized size of antiderivative = 2.14 \[ \int \frac {e^{\frac {81 x^2}{625-500 x+150 x^2-20 x^3+x^4+\left (50 x-20 x^2+2 x^3\right ) \log (2+4 x)+x^2 \log ^2(2+4 x)}} \left (4050 x+8100 x^2-486 x^3-324 x^4\right )}{46875+37500 x-84375 x^2+48750 x^3-13875 x^4+2160 x^5-177 x^6+6 x^7+\left (5625 x+6750 x^2-7650 x^3+2520 x^4-351 x^5+18 x^6\right ) \log (2+4 x)+\left (225 x^2+360 x^3-171 x^4+18 x^5\right ) \log ^2(2+4 x)+\left (3 x^3+6 x^4\right ) \log ^3(2+4 x)+e^{\frac {81 x^2}{625-500 x+150 x^2-20 x^3+x^4+\left (50 x-20 x^2+2 x^3\right ) \log (2+4 x)+x^2 \log ^2(2+4 x)}} \left (15625+12500 x-28125 x^2+16250 x^3-4625 x^4+720 x^5-59 x^6+2 x^7+\left (1875 x+2250 x^2-2550 x^3+840 x^4-117 x^5+6 x^6\right ) \log (2+4 x)+\left (75 x^2+120 x^3-57 x^4+6 x^5\right ) \log ^2(2+4 x)+\left (x^3+2 x^4\right ) \log ^3(2+4 x)\right )} \, dx=\log {\left (e^{\frac {81 x^{2}}{x^{4} - 20 x^{3} + x^{2} \log {\left (4 x + 2 \right )}^{2} + 150 x^{2} - 500 x + \left (2 x^{3} - 20 x^{2} + 50 x\right ) \log {\left (4 x + 2 \right )} + 625}} + 3 \right )} \] Input:

integrate((-324*x**4-486*x**3+8100*x**2+4050*x)*exp(81*x**2/(x**2*ln(2+4*x 
)**2+(2*x**3-20*x**2+50*x)*ln(2+4*x)+x**4-20*x**3+150*x**2-500*x+625))/((( 
2*x**4+x**3)*ln(2+4*x)**3+(6*x**5-57*x**4+120*x**3+75*x**2)*ln(2+4*x)**2+( 
6*x**6-117*x**5+840*x**4-2550*x**3+2250*x**2+1875*x)*ln(2+4*x)+2*x**7-59*x 
**6+720*x**5-4625*x**4+16250*x**3-28125*x**2+12500*x+15625)*exp(81*x**2/(x 
**2*ln(2+4*x)**2+(2*x**3-20*x**2+50*x)*ln(2+4*x)+x**4-20*x**3+150*x**2-500 
*x+625))+(6*x**4+3*x**3)*ln(2+4*x)**3+(18*x**5-171*x**4+360*x**3+225*x**2) 
*ln(2+4*x)**2+(18*x**6-351*x**5+2520*x**4-7650*x**3+6750*x**2+5625*x)*ln(2 
+4*x)+6*x**7-177*x**6+2160*x**5-13875*x**4+48750*x**3-84375*x**2+37500*x+4 
6875),x)
 

Output:

log(exp(81*x**2/(x**4 - 20*x**3 + x**2*log(4*x + 2)**2 + 150*x**2 - 500*x 
+ (2*x**3 - 20*x**2 + 50*x)*log(4*x + 2) + 625)) + 3)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 555 vs. \(2 (25) = 50\).

Time = 0.48 (sec) , antiderivative size = 555, normalized size of antiderivative = 19.82 \[ \int \frac {e^{\frac {81 x^2}{625-500 x+150 x^2-20 x^3+x^4+\left (50 x-20 x^2+2 x^3\right ) \log (2+4 x)+x^2 \log ^2(2+4 x)}} \left (4050 x+8100 x^2-486 x^3-324 x^4\right )}{46875+37500 x-84375 x^2+48750 x^3-13875 x^4+2160 x^5-177 x^6+6 x^7+\left (5625 x+6750 x^2-7650 x^3+2520 x^4-351 x^5+18 x^6\right ) \log (2+4 x)+\left (225 x^2+360 x^3-171 x^4+18 x^5\right ) \log ^2(2+4 x)+\left (3 x^3+6 x^4\right ) \log ^3(2+4 x)+e^{\frac {81 x^2}{625-500 x+150 x^2-20 x^3+x^4+\left (50 x-20 x^2+2 x^3\right ) \log (2+4 x)+x^2 \log ^2(2+4 x)}} \left (15625+12500 x-28125 x^2+16250 x^3-4625 x^4+720 x^5-59 x^6+2 x^7+\left (1875 x+2250 x^2-2550 x^3+840 x^4-117 x^5+6 x^6\right ) \log (2+4 x)+\left (75 x^2+120 x^3-57 x^4+6 x^5\right ) \log ^2(2+4 x)+\left (x^3+2 x^4\right ) \log ^3(2+4 x)\right )} \, dx =\text {Too large to display} \] Input:

integrate((-324*x^4-486*x^3+8100*x^2+4050*x)*exp(81*x^2/(x^2*log(2+4*x)^2+ 
(2*x^3-20*x^2+50*x)*log(2+4*x)+x^4-20*x^3+150*x^2-500*x+625))/(((2*x^4+x^3 
)*log(2+4*x)^3+(6*x^5-57*x^4+120*x^3+75*x^2)*log(2+4*x)^2+(6*x^6-117*x^5+8 
40*x^4-2550*x^3+2250*x^2+1875*x)*log(2+4*x)+2*x^7-59*x^6+720*x^5-4625*x^4+ 
16250*x^3-28125*x^2+12500*x+15625)*exp(81*x^2/(x^2*log(2+4*x)^2+(2*x^3-20* 
x^2+50*x)*log(2+4*x)+x^4-20*x^3+150*x^2-500*x+625))+(6*x^4+3*x^3)*log(2+4* 
x)^3+(18*x^5-171*x^4+360*x^3+225*x^2)*log(2+4*x)^2+(18*x^6-351*x^5+2520*x^ 
4-7650*x^3+6750*x^2+5625*x)*log(2+4*x)+6*x^7-177*x^6+2160*x^5-13875*x^4+48 
750*x^3-84375*x^2+37500*x+46875),x, algorithm="maxima")
 

Output:

-81*x*log(2*x + 1)/(x^4 + 2*x^3*(log(2) - 10) + x^2*log(2*x + 1)^2 + (log( 
2)^2 - 20*log(2) + 150)*x^2 + 50*x*(log(2) - 10) + 2*(x^3 + x^2*(log(2) - 
10) + 25*x)*log(2*x + 1) + 625) + log(1/3*(3*e^(81*x*log(2)/(x^4 + 2*x^3*( 
log(2) + log(2*x + 1) - 10) + (log(2)^2 + 2*(log(2) - 10)*log(2*x + 1) + l 
og(2*x + 1)^2 - 20*log(2) + 150)*x^2 + 50*x*(log(2) + log(2*x + 1) - 10) + 
 625) + 81*x*log(2*x + 1)/(x^4 + 2*x^3*(log(2) + log(2*x + 1) - 10) + (log 
(2)^2 + 2*(log(2) - 10)*log(2*x + 1) + log(2*x + 1)^2 - 20*log(2) + 150)*x 
^2 + 50*x*(log(2) + log(2*x + 1) - 10) + 625) + 2025/(x^4 + 2*x^3*(log(2) 
+ log(2*x + 1) - 10) + (log(2)^2 + 2*(log(2) - 10)*log(2*x + 1) + log(2*x 
+ 1)^2 - 20*log(2) + 150)*x^2 + 50*x*(log(2) + log(2*x + 1) - 10) + 625)) 
+ e^(810*x/(x^4 + 2*x^3*(log(2) + log(2*x + 1) - 10) + (log(2)^2 + 2*(log( 
2) - 10)*log(2*x + 1) + log(2*x + 1)^2 - 20*log(2) + 150)*x^2 + 50*x*(log( 
2) + log(2*x + 1) - 10) + 625) + 81/(x^2 + x*(log(2) + log(2*x + 1) - 10) 
+ 25)))*e^(-81*x*log(2)/(x^4 + 2*x^3*(log(2) + log(2*x + 1) - 10) + (log(2 
)^2 + 2*(log(2) - 10)*log(2*x + 1) + log(2*x + 1)^2 - 20*log(2) + 150)*x^2 
 + 50*x*(log(2) + log(2*x + 1) - 10) + 625) - 2025/(x^4 + 2*x^3*(log(2) + 
log(2*x + 1) - 10) + (log(2)^2 + 2*(log(2) - 10)*log(2*x + 1) + log(2*x + 
1)^2 - 20*log(2) + 150)*x^2 + 50*x*(log(2) + log(2*x + 1) - 10) + 625)))
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 72 vs. \(2 (25) = 50\).

Time = 0.37 (sec) , antiderivative size = 72, normalized size of antiderivative = 2.57 \[ \int \frac {e^{\frac {81 x^2}{625-500 x+150 x^2-20 x^3+x^4+\left (50 x-20 x^2+2 x^3\right ) \log (2+4 x)+x^2 \log ^2(2+4 x)}} \left (4050 x+8100 x^2-486 x^3-324 x^4\right )}{46875+37500 x-84375 x^2+48750 x^3-13875 x^4+2160 x^5-177 x^6+6 x^7+\left (5625 x+6750 x^2-7650 x^3+2520 x^4-351 x^5+18 x^6\right ) \log (2+4 x)+\left (225 x^2+360 x^3-171 x^4+18 x^5\right ) \log ^2(2+4 x)+\left (3 x^3+6 x^4\right ) \log ^3(2+4 x)+e^{\frac {81 x^2}{625-500 x+150 x^2-20 x^3+x^4+\left (50 x-20 x^2+2 x^3\right ) \log (2+4 x)+x^2 \log ^2(2+4 x)}} \left (15625+12500 x-28125 x^2+16250 x^3-4625 x^4+720 x^5-59 x^6+2 x^7+\left (1875 x+2250 x^2-2550 x^3+840 x^4-117 x^5+6 x^6\right ) \log (2+4 x)+\left (75 x^2+120 x^3-57 x^4+6 x^5\right ) \log ^2(2+4 x)+\left (x^3+2 x^4\right ) \log ^3(2+4 x)\right )} \, dx=\log \left (e^{\left (\frac {81 \, x^{2}}{x^{4} + 2 \, x^{3} \log \left (4 \, x + 2\right ) + x^{2} \log \left (4 \, x + 2\right )^{2} - 20 \, x^{3} - 20 \, x^{2} \log \left (4 \, x + 2\right ) + 150 \, x^{2} + 50 \, x \log \left (4 \, x + 2\right ) - 500 \, x + 625}\right )} + 3\right ) \] Input:

integrate((-324*x^4-486*x^3+8100*x^2+4050*x)*exp(81*x^2/(x^2*log(2+4*x)^2+ 
(2*x^3-20*x^2+50*x)*log(2+4*x)+x^4-20*x^3+150*x^2-500*x+625))/(((2*x^4+x^3 
)*log(2+4*x)^3+(6*x^5-57*x^4+120*x^3+75*x^2)*log(2+4*x)^2+(6*x^6-117*x^5+8 
40*x^4-2550*x^3+2250*x^2+1875*x)*log(2+4*x)+2*x^7-59*x^6+720*x^5-4625*x^4+ 
16250*x^3-28125*x^2+12500*x+15625)*exp(81*x^2/(x^2*log(2+4*x)^2+(2*x^3-20* 
x^2+50*x)*log(2+4*x)+x^4-20*x^3+150*x^2-500*x+625))+(6*x^4+3*x^3)*log(2+4* 
x)^3+(18*x^5-171*x^4+360*x^3+225*x^2)*log(2+4*x)^2+(18*x^6-351*x^5+2520*x^ 
4-7650*x^3+6750*x^2+5625*x)*log(2+4*x)+6*x^7-177*x^6+2160*x^5-13875*x^4+48 
750*x^3-84375*x^2+37500*x+46875),x, algorithm="giac")
 

Output:

log(e^(81*x^2/(x^4 + 2*x^3*log(4*x + 2) + x^2*log(4*x + 2)^2 - 20*x^3 - 20 
*x^2*log(4*x + 2) + 150*x^2 + 50*x*log(4*x + 2) - 500*x + 625)) + 3)
 

Mupad [B] (verification not implemented)

Time = 7.40 (sec) , antiderivative size = 72, normalized size of antiderivative = 2.57 \[ \int \frac {e^{\frac {81 x^2}{625-500 x+150 x^2-20 x^3+x^4+\left (50 x-20 x^2+2 x^3\right ) \log (2+4 x)+x^2 \log ^2(2+4 x)}} \left (4050 x+8100 x^2-486 x^3-324 x^4\right )}{46875+37500 x-84375 x^2+48750 x^3-13875 x^4+2160 x^5-177 x^6+6 x^7+\left (5625 x+6750 x^2-7650 x^3+2520 x^4-351 x^5+18 x^6\right ) \log (2+4 x)+\left (225 x^2+360 x^3-171 x^4+18 x^5\right ) \log ^2(2+4 x)+\left (3 x^3+6 x^4\right ) \log ^3(2+4 x)+e^{\frac {81 x^2}{625-500 x+150 x^2-20 x^3+x^4+\left (50 x-20 x^2+2 x^3\right ) \log (2+4 x)+x^2 \log ^2(2+4 x)}} \left (15625+12500 x-28125 x^2+16250 x^3-4625 x^4+720 x^5-59 x^6+2 x^7+\left (1875 x+2250 x^2-2550 x^3+840 x^4-117 x^5+6 x^6\right ) \log (2+4 x)+\left (75 x^2+120 x^3-57 x^4+6 x^5\right ) \log ^2(2+4 x)+\left (x^3+2 x^4\right ) \log ^3(2+4 x)\right )} \, dx=\ln \left ({\mathrm {e}}^{\frac {81\,x^2}{x^4+2\,x^3\,\ln \left (4\,x+2\right )-20\,x^3+x^2\,{\ln \left (4\,x+2\right )}^2-20\,x^2\,\ln \left (4\,x+2\right )+150\,x^2+50\,x\,\ln \left (4\,x+2\right )-500\,x+625}}+3\right ) \] Input:

int((exp((81*x^2)/(x^2*log(4*x + 2)^2 - 500*x + log(4*x + 2)*(50*x - 20*x^ 
2 + 2*x^3) + 150*x^2 - 20*x^3 + x^4 + 625))*(4050*x + 8100*x^2 - 486*x^3 - 
 324*x^4))/(37500*x + log(4*x + 2)^3*(3*x^3 + 6*x^4) + log(4*x + 2)*(5625* 
x + 6750*x^2 - 7650*x^3 + 2520*x^4 - 351*x^5 + 18*x^6) + log(4*x + 2)^2*(2 
25*x^2 + 360*x^3 - 171*x^4 + 18*x^5) - 84375*x^2 + 48750*x^3 - 13875*x^4 + 
 2160*x^5 - 177*x^6 + 6*x^7 + exp((81*x^2)/(x^2*log(4*x + 2)^2 - 500*x + l 
og(4*x + 2)*(50*x - 20*x^2 + 2*x^3) + 150*x^2 - 20*x^3 + x^4 + 625))*(1250 
0*x + log(4*x + 2)*(1875*x + 2250*x^2 - 2550*x^3 + 840*x^4 - 117*x^5 + 6*x 
^6) + log(4*x + 2)^2*(75*x^2 + 120*x^3 - 57*x^4 + 6*x^5) + log(4*x + 2)^3* 
(x^3 + 2*x^4) - 28125*x^2 + 16250*x^3 - 4625*x^4 + 720*x^5 - 59*x^6 + 2*x^ 
7 + 15625) + 46875),x)
 

Output:

log(exp((81*x^2)/(x^2*log(4*x + 2)^2 - 500*x + 50*x*log(4*x + 2) + 150*x^2 
 - 20*x^3 + x^4 - 20*x^2*log(4*x + 2) + 2*x^3*log(4*x + 2) + 625)) + 3)
 

Reduce [B] (verification not implemented)

Time = 0.94 (sec) , antiderivative size = 376, normalized size of antiderivative = 13.43 \[ \int \frac {e^{\frac {81 x^2}{625-500 x+150 x^2-20 x^3+x^4+\left (50 x-20 x^2+2 x^3\right ) \log (2+4 x)+x^2 \log ^2(2+4 x)}} \left (4050 x+8100 x^2-486 x^3-324 x^4\right )}{46875+37500 x-84375 x^2+48750 x^3-13875 x^4+2160 x^5-177 x^6+6 x^7+\left (5625 x+6750 x^2-7650 x^3+2520 x^4-351 x^5+18 x^6\right ) \log (2+4 x)+\left (225 x^2+360 x^3-171 x^4+18 x^5\right ) \log ^2(2+4 x)+\left (3 x^3+6 x^4\right ) \log ^3(2+4 x)+e^{\frac {81 x^2}{625-500 x+150 x^2-20 x^3+x^4+\left (50 x-20 x^2+2 x^3\right ) \log (2+4 x)+x^2 \log ^2(2+4 x)}} \left (15625+12500 x-28125 x^2+16250 x^3-4625 x^4+720 x^5-59 x^6+2 x^7+\left (1875 x+2250 x^2-2550 x^3+840 x^4-117 x^5+6 x^6\right ) \log (2+4 x)+\left (75 x^2+120 x^3-57 x^4+6 x^5\right ) \log ^2(2+4 x)+\left (x^3+2 x^4\right ) \log ^3(2+4 x)\right )} \, dx=\mathrm {log}\left (e^{\frac {54 x^{2}}{\mathrm {log}\left (4 x +2\right )^{2} x^{2}+2 \,\mathrm {log}\left (4 x +2\right ) x^{3}-20 \,\mathrm {log}\left (4 x +2\right ) x^{2}+50 \,\mathrm {log}\left (4 x +2\right ) x +x^{4}-20 x^{3}+150 x^{2}-500 x +625}}-e^{\frac {27 x^{2}}{\mathrm {log}\left (4 x +2\right )^{2} x^{2}+2 \,\mathrm {log}\left (4 x +2\right ) x^{3}-20 \,\mathrm {log}\left (4 x +2\right ) x^{2}+50 \,\mathrm {log}\left (4 x +2\right ) x +x^{4}-20 x^{3}+150 x^{2}-500 x +625}} 3^{\frac {1}{3}}+3^{\frac {2}{3}}\right )+\mathrm {log}\left (e^{\frac {18 x^{2}}{\mathrm {log}\left (4 x +2\right )^{2} x^{2}+2 \,\mathrm {log}\left (4 x +2\right ) x^{3}-20 \,\mathrm {log}\left (4 x +2\right ) x^{2}+50 \,\mathrm {log}\left (4 x +2\right ) x +x^{4}-20 x^{3}+150 x^{2}-500 x +625}}-e^{\frac {9 x^{2}}{\mathrm {log}\left (4 x +2\right )^{2} x^{2}+2 \,\mathrm {log}\left (4 x +2\right ) x^{3}-20 \,\mathrm {log}\left (4 x +2\right ) x^{2}+50 \,\mathrm {log}\left (4 x +2\right ) x +x^{4}-20 x^{3}+150 x^{2}-500 x +625}} 3^{\frac {1}{9}}+3^{\frac {2}{9}}\right )+\mathrm {log}\left (e^{\frac {9 x^{2}}{\mathrm {log}\left (4 x +2\right )^{2} x^{2}+2 \,\mathrm {log}\left (4 x +2\right ) x^{3}-20 \,\mathrm {log}\left (4 x +2\right ) x^{2}+50 \,\mathrm {log}\left (4 x +2\right ) x +x^{4}-20 x^{3}+150 x^{2}-500 x +625}}+3^{\frac {1}{9}}\right ) \] Input:

int((-324*x^4-486*x^3+8100*x^2+4050*x)*exp(81*x^2/(x^2*log(2+4*x)^2+(2*x^3 
-20*x^2+50*x)*log(2+4*x)+x^4-20*x^3+150*x^2-500*x+625))/(((2*x^4+x^3)*log( 
2+4*x)^3+(6*x^5-57*x^4+120*x^3+75*x^2)*log(2+4*x)^2+(6*x^6-117*x^5+840*x^4 
-2550*x^3+2250*x^2+1875*x)*log(2+4*x)+2*x^7-59*x^6+720*x^5-4625*x^4+16250* 
x^3-28125*x^2+12500*x+15625)*exp(81*x^2/(x^2*log(2+4*x)^2+(2*x^3-20*x^2+50 
*x)*log(2+4*x)+x^4-20*x^3+150*x^2-500*x+625))+(6*x^4+3*x^3)*log(2+4*x)^3+( 
18*x^5-171*x^4+360*x^3+225*x^2)*log(2+4*x)^2+(18*x^6-351*x^5+2520*x^4-7650 
*x^3+6750*x^2+5625*x)*log(2+4*x)+6*x^7-177*x^6+2160*x^5-13875*x^4+48750*x^ 
3-84375*x^2+37500*x+46875),x)
 

Output:

log(e**((54*x**2)/(log(4*x + 2)**2*x**2 + 2*log(4*x + 2)*x**3 - 20*log(4*x 
 + 2)*x**2 + 50*log(4*x + 2)*x + x**4 - 20*x**3 + 150*x**2 - 500*x + 625)) 
 - e**((27*x**2)/(log(4*x + 2)**2*x**2 + 2*log(4*x + 2)*x**3 - 20*log(4*x 
+ 2)*x**2 + 50*log(4*x + 2)*x + x**4 - 20*x**3 + 150*x**2 - 500*x + 625))* 
3**(1/3) + 3**(2/3)) + log(e**((18*x**2)/(log(4*x + 2)**2*x**2 + 2*log(4*x 
 + 2)*x**3 - 20*log(4*x + 2)*x**2 + 50*log(4*x + 2)*x + x**4 - 20*x**3 + 1 
50*x**2 - 500*x + 625)) - e**((9*x**2)/(log(4*x + 2)**2*x**2 + 2*log(4*x + 
 2)*x**3 - 20*log(4*x + 2)*x**2 + 50*log(4*x + 2)*x + x**4 - 20*x**3 + 150 
*x**2 - 500*x + 625))*3**(1/9) + 3**(2/9)) + log(e**((9*x**2)/(log(4*x + 2 
)**2*x**2 + 2*log(4*x + 2)*x**3 - 20*log(4*x + 2)*x**2 + 50*log(4*x + 2)*x 
 + x**4 - 20*x**3 + 150*x**2 - 500*x + 625)) + 3**(1/9))