\(\int \frac {201326592 e^{-1+4 e^{-1+3 x}+3 x}+64 x^3-256 e^3 x^3+384 e^6 x^3-256 e^9 x^3+64 e^{12} x^3+e^{3 e^{-1+3 x}} (2097152-2097152 e^3+e^{-1+3 x} (18874368 x-18874368 e^3 x))+e^{2 e^{-1+3 x}} (196608 x-393216 e^3 x+196608 e^6 x+e^{-1+3 x} (589824 x^2-1179648 e^3 x^2+589824 e^6 x^2))+e^{e^{-1+3 x}} (6144 x^2-18432 e^3 x^2+18432 e^6 x^2-6144 e^9 x^2+e^{-1+3 x} (6144 x^3-18432 e^3 x^3+18432 e^6 x^3-6144 e^9 x^3))}{1-4 e^3+6 e^6-4 e^9+e^{12}} \, dx\) [6737]

Optimal result

Integrand size = 252, antiderivative size = 26 \[ \int \frac {201326592 e^{-1+4 e^{-1+3 x}+3 x}+64 x^3-256 e^3 x^3+384 e^6 x^3-256 e^9 x^3+64 e^{12} x^3+e^{3 e^{-1+3 x}} \left (2097152-2097152 e^3+e^{-1+3 x} \left (18874368 x-18874368 e^3 x\right )\right )+e^{2 e^{-1+3 x}} \left (196608 x-393216 e^3 x+196608 e^6 x+e^{-1+3 x} \left (589824 x^2-1179648 e^3 x^2+589824 e^6 x^2\right )\right )+e^{e^{-1+3 x}} \left (6144 x^2-18432 e^3 x^2+18432 e^6 x^2-6144 e^9 x^2+e^{-1+3 x} \left (6144 x^3-18432 e^3 x^3+18432 e^6 x^3-6144 e^9 x^3\right )\right )}{1-4 e^3+6 e^6-4 e^9+e^{12}} \, dx=\left (\frac {64 e^{e^{-1+3 x}}}{1-e^3}+2 x\right )^4 \]

[Out]

(2*x+64*exp(exp(-1+3*x))/(-exp(3)+1))^4

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(140\) vs. \(2(26)=52\).

Time = 0.10 (sec) , antiderivative size = 140, normalized size of antiderivative = 5.38, number of steps used = 16, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.020, Rules used = {6, 12, 2320, 2225, 2326} \[ \int \frac {201326592 e^{-1+4 e^{-1+3 x}+3 x}+64 x^3-256 e^3 x^3+384 e^6 x^3-256 e^9 x^3+64 e^{12} x^3+e^{3 e^{-1+3 x}} \left (2097152-2097152 e^3+e^{-1+3 x} \left (18874368 x-18874368 e^3 x\right )\right )+e^{2 e^{-1+3 x}} \left (196608 x-393216 e^3 x+196608 e^6 x+e^{-1+3 x} \left (589824 x^2-1179648 e^3 x^2+589824 e^6 x^2\right )\right )+e^{e^{-1+3 x}} \left (6144 x^2-18432 e^3 x^2+18432 e^6 x^2-6144 e^9 x^2+e^{-1+3 x} \left (6144 x^3-18432 e^3 x^3+18432 e^6 x^3-6144 e^9 x^3\right )\right )}{1-4 e^3+6 e^6-4 e^9+e^{12}} \, dx=16 x^4+\frac {2048 e^{e^{3 x-1}} \left (-e^9 x^3+3 e^6 x^3-3 e^3 x^3+x^3\right )}{\left (1-e^3\right )^4}+\frac {98304 e^{2 e^{3 x-1}} \left (e^6 x^2-2 e^3 x^2+x^2\right )}{\left (1-e^3\right )^4}+\frac {2097152 e^{3 e^{3 x-1}} x}{\left (1-e^3\right )^3}+\frac {16777216 e^{4 e^{3 x-1}}}{\left (1-e^3\right )^4} \]

[In]

Int[(201326592*E^(-1 + 4*E^(-1 + 3*x) + 3*x) + 64*x^3 - 256*E^3*x^3 + 384*E^6*x^3 - 256*E^9*x^3 + 64*E^12*x^3
+ E^(3*E^(-1 + 3*x))*(2097152 - 2097152*E^3 + E^(-1 + 3*x)*(18874368*x - 18874368*E^3*x)) + E^(2*E^(-1 + 3*x))
*(196608*x - 393216*E^3*x + 196608*E^6*x + E^(-1 + 3*x)*(589824*x^2 - 1179648*E^3*x^2 + 589824*E^6*x^2)) + E^E
^(-1 + 3*x)*(6144*x^2 - 18432*E^3*x^2 + 18432*E^6*x^2 - 6144*E^9*x^2 + E^(-1 + 3*x)*(6144*x^3 - 18432*E^3*x^3
+ 18432*E^6*x^3 - 6144*E^9*x^3)))/(1 - 4*E^3 + 6*E^6 - 4*E^9 + E^12),x]

[Out]

(16777216*E^(4*E^(-1 + 3*x)))/(1 - E^3)^4 + (2097152*E^(3*E^(-1 + 3*x))*x)/(1 - E^3)^3 + 16*x^4 + (98304*E^(2*
E^(-1 + 3*x))*(x^2 - 2*E^3*x^2 + E^6*x^2))/(1 - E^3)^4 + (2048*E^E^(-1 + 3*x)*(x^3 - 3*E^3*x^3 + 3*E^6*x^3 - E
^9*x^3))/(1 - E^3)^4

Rule 6

Int[(u_.)*((w_.) + (a_.)*(v_) + (b_.)*(v_))^(p_.), x_Symbol] :> Int[u*((a + b)*v + w)^p, x] /; FreeQ[{a, b}, x
] &&  !FreeQ[v, x]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 2225

Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre
eQ[{F, a, b, c, n}, x]

Rule 2320

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rule 2326

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]

Rubi steps \begin{align*} \text {integral}& = \int \frac {201326592 e^{-1+4 e^{-1+3 x}+3 x}+384 e^6 x^3-256 e^9 x^3+64 e^{12} x^3+\left (64-256 e^3\right ) x^3+e^{3 e^{-1+3 x}} \left (2097152-2097152 e^3+e^{-1+3 x} \left (18874368 x-18874368 e^3 x\right )\right )+e^{2 e^{-1+3 x}} \left (196608 x-393216 e^3 x+196608 e^6 x+e^{-1+3 x} \left (589824 x^2-1179648 e^3 x^2+589824 e^6 x^2\right )\right )+e^{e^{-1+3 x}} \left (6144 x^2-18432 e^3 x^2+18432 e^6 x^2-6144 e^9 x^2+e^{-1+3 x} \left (6144 x^3-18432 e^3 x^3+18432 e^6 x^3-6144 e^9 x^3\right )\right )}{1-4 e^3+6 e^6-4 e^9+e^{12}} \, dx \\ & = \int \frac {201326592 e^{-1+4 e^{-1+3 x}+3 x}+64 e^{12} x^3+\left (64-256 e^3\right ) x^3+\left (384 e^6-256 e^9\right ) x^3+e^{3 e^{-1+3 x}} \left (2097152-2097152 e^3+e^{-1+3 x} \left (18874368 x-18874368 e^3 x\right )\right )+e^{2 e^{-1+3 x}} \left (196608 x-393216 e^3 x+196608 e^6 x+e^{-1+3 x} \left (589824 x^2-1179648 e^3 x^2+589824 e^6 x^2\right )\right )+e^{e^{-1+3 x}} \left (6144 x^2-18432 e^3 x^2+18432 e^6 x^2-6144 e^9 x^2+e^{-1+3 x} \left (6144 x^3-18432 e^3 x^3+18432 e^6 x^3-6144 e^9 x^3\right )\right )}{1-4 e^3+6 e^6-4 e^9+e^{12}} \, dx \\ & = \int \frac {201326592 e^{-1+4 e^{-1+3 x}+3 x}+\left (384 e^6-256 e^9\right ) x^3+\left (64-256 e^3+64 e^{12}\right ) x^3+e^{3 e^{-1+3 x}} \left (2097152-2097152 e^3+e^{-1+3 x} \left (18874368 x-18874368 e^3 x\right )\right )+e^{2 e^{-1+3 x}} \left (196608 x-393216 e^3 x+196608 e^6 x+e^{-1+3 x} \left (589824 x^2-1179648 e^3 x^2+589824 e^6 x^2\right )\right )+e^{e^{-1+3 x}} \left (6144 x^2-18432 e^3 x^2+18432 e^6 x^2-6144 e^9 x^2+e^{-1+3 x} \left (6144 x^3-18432 e^3 x^3+18432 e^6 x^3-6144 e^9 x^3\right )\right )}{1-4 e^3+6 e^6-4 e^9+e^{12}} \, dx \\ & = \int \frac {201326592 e^{-1+4 e^{-1+3 x}+3 x}+\left (64-256 e^3+384 e^6-256 e^9+64 e^{12}\right ) x^3+e^{3 e^{-1+3 x}} \left (2097152-2097152 e^3+e^{-1+3 x} \left (18874368 x-18874368 e^3 x\right )\right )+e^{2 e^{-1+3 x}} \left (196608 x-393216 e^3 x+196608 e^6 x+e^{-1+3 x} \left (589824 x^2-1179648 e^3 x^2+589824 e^6 x^2\right )\right )+e^{e^{-1+3 x}} \left (6144 x^2-18432 e^3 x^2+18432 e^6 x^2-6144 e^9 x^2+e^{-1+3 x} \left (6144 x^3-18432 e^3 x^3+18432 e^6 x^3-6144 e^9 x^3\right )\right )}{1-4 e^3+6 e^6-4 e^9+e^{12}} \, dx \\ & = \frac {\int \left (201326592 e^{-1+4 e^{-1+3 x}+3 x}+\left (64-256 e^3+384 e^6-256 e^9+64 e^{12}\right ) x^3+e^{3 e^{-1+3 x}} \left (2097152-2097152 e^3+e^{-1+3 x} \left (18874368 x-18874368 e^3 x\right )\right )+e^{2 e^{-1+3 x}} \left (196608 x-393216 e^3 x+196608 e^6 x+e^{-1+3 x} \left (589824 x^2-1179648 e^3 x^2+589824 e^6 x^2\right )\right )+e^{e^{-1+3 x}} \left (6144 x^2-18432 e^3 x^2+18432 e^6 x^2-6144 e^9 x^2+e^{-1+3 x} \left (6144 x^3-18432 e^3 x^3+18432 e^6 x^3-6144 e^9 x^3\right )\right )\right ) \, dx}{1-4 e^3+6 e^6-4 e^9+e^{12}} \\ & = 16 x^4+\frac {201326592 \int e^{-1+4 e^{-1+3 x}+3 x} \, dx}{\left (1-e^3\right )^4}+\frac {\int e^{3 e^{-1+3 x}} \left (2097152-2097152 e^3+e^{-1+3 x} \left (18874368 x-18874368 e^3 x\right )\right ) \, dx}{1-4 e^3+6 e^6-4 e^9+e^{12}}+\frac {\int e^{2 e^{-1+3 x}} \left (196608 x-393216 e^3 x+196608 e^6 x+e^{-1+3 x} \left (589824 x^2-1179648 e^3 x^2+589824 e^6 x^2\right )\right ) \, dx}{1-4 e^3+6 e^6-4 e^9+e^{12}}+\frac {\int e^{e^{-1+3 x}} \left (6144 x^2-18432 e^3 x^2+18432 e^6 x^2-6144 e^9 x^2+e^{-1+3 x} \left (6144 x^3-18432 e^3 x^3+18432 e^6 x^3-6144 e^9 x^3\right )\right ) \, dx}{1-4 e^3+6 e^6-4 e^9+e^{12}} \\ & = \frac {2097152 e^{3 e^{-1+3 x}} x}{\left (1-e^3\right )^3}+16 x^4+\frac {67108864 \text {Subst}\left (\int e^{-1+\frac {4 x}{e}} \, dx,x,e^{3 x}\right )}{\left (1-e^3\right )^4}+\frac {\int e^{2 e^{-1+3 x}} \left (196608 e^6 x+\left (196608-393216 e^3\right ) x+e^{-1+3 x} \left (589824 x^2-1179648 e^3 x^2+589824 e^6 x^2\right )\right ) \, dx}{1-4 e^3+6 e^6-4 e^9+e^{12}}+\frac {\int e^{e^{-1+3 x}} \left (18432 e^6 x^2-6144 e^9 x^2+\left (6144-18432 e^3\right ) x^2+e^{-1+3 x} \left (6144 x^3-18432 e^3 x^3+18432 e^6 x^3-6144 e^9 x^3\right )\right ) \, dx}{1-4 e^3+6 e^6-4 e^9+e^{12}} \\ & = \frac {16777216 e^{4 e^{-1+3 x}}}{\left (1-e^3\right )^4}+\frac {2097152 e^{3 e^{-1+3 x}} x}{\left (1-e^3\right )^3}+16 x^4+\frac {\int e^{2 e^{-1+3 x}} \left (\left (196608-393216 e^3+196608 e^6\right ) x+e^{-1+3 x} \left (589824 x^2-1179648 e^3 x^2+589824 e^6 x^2\right )\right ) \, dx}{1-4 e^3+6 e^6-4 e^9+e^{12}}+\frac {\int e^{e^{-1+3 x}} \left (\left (6144-18432 e^3\right ) x^2+\left (18432 e^6-6144 e^9\right ) x^2+e^{-1+3 x} \left (6144 x^3-18432 e^3 x^3+18432 e^6 x^3-6144 e^9 x^3\right )\right ) \, dx}{1-4 e^3+6 e^6-4 e^9+e^{12}} \\ & = \frac {16777216 e^{4 e^{-1+3 x}}}{\left (1-e^3\right )^4}+\frac {2097152 e^{3 e^{-1+3 x}} x}{\left (1-e^3\right )^3}+16 x^4+\frac {98304 e^{2 e^{-1+3 x}} \left (x^2-2 e^3 x^2+e^6 x^2\right )}{\left (1-e^3\right )^4}+\frac {\int e^{e^{-1+3 x}} \left (\left (6144-18432 e^3+18432 e^6-6144 e^9\right ) x^2+e^{-1+3 x} \left (6144 x^3-18432 e^3 x^3+18432 e^6 x^3-6144 e^9 x^3\right )\right ) \, dx}{1-4 e^3+6 e^6-4 e^9+e^{12}} \\ & = \frac {16777216 e^{4 e^{-1+3 x}}}{\left (1-e^3\right )^4}+\frac {2097152 e^{3 e^{-1+3 x}} x}{\left (1-e^3\right )^3}+16 x^4+\frac {98304 e^{2 e^{-1+3 x}} \left (x^2-2 e^3 x^2+e^6 x^2\right )}{\left (1-e^3\right )^4}+\frac {2048 e^{e^{-1+3 x}} \left (x^3-3 e^3 x^3+3 e^6 x^3-e^9 x^3\right )}{\left (1-e^3\right )^4} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.15 \[ \int \frac {201326592 e^{-1+4 e^{-1+3 x}+3 x}+64 x^3-256 e^3 x^3+384 e^6 x^3-256 e^9 x^3+64 e^{12} x^3+e^{3 e^{-1+3 x}} \left (2097152-2097152 e^3+e^{-1+3 x} \left (18874368 x-18874368 e^3 x\right )\right )+e^{2 e^{-1+3 x}} \left (196608 x-393216 e^3 x+196608 e^6 x+e^{-1+3 x} \left (589824 x^2-1179648 e^3 x^2+589824 e^6 x^2\right )\right )+e^{e^{-1+3 x}} \left (6144 x^2-18432 e^3 x^2+18432 e^6 x^2-6144 e^9 x^2+e^{-1+3 x} \left (6144 x^3-18432 e^3 x^3+18432 e^6 x^3-6144 e^9 x^3\right )\right )}{1-4 e^3+6 e^6-4 e^9+e^{12}} \, dx=\frac {16 \left (32 e^{e^{-1+3 x}}+x-e^3 x\right )^4}{\left (-1+e^3\right )^4} \]

[In]

Integrate[(201326592*E^(-1 + 4*E^(-1 + 3*x) + 3*x) + 64*x^3 - 256*E^3*x^3 + 384*E^6*x^3 - 256*E^9*x^3 + 64*E^1
2*x^3 + E^(3*E^(-1 + 3*x))*(2097152 - 2097152*E^3 + E^(-1 + 3*x)*(18874368*x - 18874368*E^3*x)) + E^(2*E^(-1 +
3*x))*(196608*x - 393216*E^3*x + 196608*E^6*x + E^(-1 + 3*x)*(589824*x^2 - 1179648*E^3*x^2 + 589824*E^6*x^2))
+ E^E^(-1 + 3*x)*(6144*x^2 - 18432*E^3*x^2 + 18432*E^6*x^2 - 6144*E^9*x^2 + E^(-1 + 3*x)*(6144*x^3 - 18432*E^
3*x^3 + 18432*E^6*x^3 - 6144*E^9*x^3)))/(1 - 4*E^3 + 6*E^6 - 4*E^9 + E^12),x]

[Out]

(16*(32*E^E^(-1 + 3*x) + x - E^3*x)^4)/(-1 + E^3)^4

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(147\) vs. \(2(23)=46\).

Time = 0.55 (sec) , antiderivative size = 148, normalized size of antiderivative = 5.69

method	result	size
default	\(\frac {\left (-2097152 \,{\mathrm e}^{3}+2097152\right ) x \,{\mathrm e}^{3 \,{\mathrm e}^{-1+3 x}}+\left (98304 \,{\mathrm e}^{6}-196608 \,{\mathrm e}^{3}+98304\right ) x^{2} {\mathrm e}^{2 \,{\mathrm e}^{-1+3 x}}+\left (-2048 \,{\mathrm e}^{9}+6144 \,{\mathrm e}^{6}-6144 \,{\mathrm e}^{3}+2048\right ) x^{3} {\mathrm e}^{{\mathrm e}^{-1+3 x}}+16 x^{4}-64 x^{4} {\mathrm e}^{3}+96 \,{\mathrm e}^{6} x^{4}-64 x^{4} {\mathrm e}^{9}+16 x^{4} {\mathrm e}^{12}[phone] \,{\mathrm e}^{4 \,{\mathrm e}^{-1+3 x}}}{{\mathrm e}^{12}-4 \,{\mathrm e}^{9}+6 \,{\mathrm e}^{6}-4 \,{\mathrm e}^{3}+1}\)	\(148\)
risch	\(16 x^{4}+\frac {16777216 \,{\mathrm e}^{4 \,{\mathrm e}^{-1+3 x}}}{{\mathrm e}^{12}-4 \,{\mathrm e}^{9}+6 \,{\mathrm e}^{6}-4 \,{\mathrm e}^{3}+1}+\frac {\left (-2097152 x \,{\mathrm e}^{3}+2097152 x \right ) {\mathrm e}^{3 \,{\mathrm e}^{-1+3 x}}}{{\mathrm e}^{12}-4 \,{\mathrm e}^{9}+6 \,{\mathrm e}^{6}-4 \,{\mathrm e}^{3}+1}+\frac {\left (98304 x^{2} {\mathrm e}^{6}-196608 x^{2} {\mathrm e}^{3}+98304 x^{2}\right ) {\mathrm e}^{2 \,{\mathrm e}^{-1+3 x}}}{{\mathrm e}^{12}-4 \,{\mathrm e}^{9}+6 \,{\mathrm e}^{6}-4 \,{\mathrm e}^{3}+1}+\frac {\left (-2048 x^{3} {\mathrm e}^{9}+6144 x^{3} {\mathrm e}^{6}-6144 x^{3} {\mathrm e}^{3}+2048 x^{3}\right ) {\mathrm e}^{{\mathrm e}^{-1+3 x}}}{{\mathrm e}^{12}-4 \,{\mathrm e}^{9}+6 \,{\mathrm e}^{6}-4 \,{\mathrm e}^{3}+1}\)	\(174\)
parallelrisch	\(\frac {16 x^{4} {\mathrm e}^{12}-64 x^{4} {\mathrm e}^{9}-2048 x^{3} {\mathrm e}^{{\mathrm e}^{-1+3 x}} {\mathrm e}^{9}+96 \,{\mathrm e}^{6} x^{4}+6144 x^{3} {\mathrm e}^{{\mathrm e}^{-1+3 x}} {\mathrm e}^{6}+98304 x^{2} {\mathrm e}^{2 \,{\mathrm e}^{-1+3 x}} {\mathrm e}^{6}-64 x^{4} {\mathrm e}^{3}-6144 x^{3} {\mathrm e}^{{\mathrm e}^{-1+3 x}} {\mathrm e}^{3}-196608 x^{2} {\mathrm e}^{2 \,{\mathrm e}^{-1+3 x}} {\mathrm e}^{3}-2097152 x \,{\mathrm e}^{3 \,{\mathrm e}^{-1+3 x}} {\mathrm e}^{3}+16 x^{4}+2048 x^{3} {\mathrm e}^{{\mathrm e}^{-1+3 x}}+98304 x^{2} {\mathrm e}^{2 \,{\mathrm e}^{-1+3 x}}+2097152 x \,{\mathrm e}^{3 \,{\mathrm e}^{-1+3 x}}[phone] \,{\mathrm e}^{4 \,{\mathrm e}^{-1+3 x}}}{{\mathrm e}^{12}-4 \,{\mathrm e}^{9}+6 \,{\mathrm e}^{6}-4 \,{\mathrm e}^{3}+1}\)	\(209\)

[In]

int((201326592*exp(-1+3*x)*exp(exp(-1+3*x))^4+((-18874368*x*exp(3)[phone]*x)*exp(-1+3*x)-2097152*exp(3)+2097
152)*exp(exp(-1+3*x))^3+((589824*x^2*exp(3)^2-1179648*x^2*exp(3)+589824*x^2)*exp(-1+3*x)+196608*x*exp(3)^2-393
216*x*exp(3)+196608*x)*exp(exp(-1+3*x))^2+((-6144*x^3*exp(3)^3+18432*x^3*exp(3)^2-18432*x^3*exp(3)+6144*x^3)*e
xp(-1+3*x)-6144*x^2*exp(3)^3+18432*x^2*exp(3)^2-18432*x^2*exp(3)+6144*x^2)*exp(exp(-1+3*x))+64*x^3*exp(3)^4-25
6*x^3*exp(3)^3+384*x^3*exp(3)^2-256*x^3*exp(3)+64*x^3)/(exp(3)^4-4*exp(3)^3+6*exp(3)^2-4*exp(3)+1),x,method=_R
ETURNVERBOSE)

[Out]

1/(exp(3)^4-4*exp(3)^3+6*exp(3)^2-4*exp(3)+1)*((-2097152*exp(3)+2097152)*x*exp(exp(-1+3*x))^3+(98304*exp(3)^2-
196608*exp(3)+98304)*x^2*exp(exp(-1+3*x))^2+(-2048*exp(3)^3+6144*exp(3)^2-6144*exp(3)+2048)*x^3*exp(exp(-1+3*x
))+16*x^4-64*x^4*exp(3)+96*x^4*exp(3)^2-64*x^4*exp(3)^3+16*x^4*exp(3)^4+16777216*exp(exp(-1+3*x))^4)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 144 vs. \(2 (21) = 42\).

Time = 0.40 (sec) , antiderivative size = 144, normalized size of antiderivative = 5.54 \[ \int \frac {201326592 e^{-1+4 e^{-1+3 x}+3 x}+64 x^3-256 e^3 x^3+384 e^6 x^3-256 e^9 x^3+64 e^{12} x^3+e^{3 e^{-1+3 x}} \left (2097152-2097152 e^3+e^{-1+3 x} \left (18874368 x-18874368 e^3 x\right )\right )+e^{2 e^{-1+3 x}} \left (196608 x-393216 e^3 x+196608 e^6 x+e^{-1+3 x} \left (589824 x^2-1179648 e^3 x^2+589824 e^6 x^2\right )\right )+e^{e^{-1+3 x}} \left (6144 x^2-18432 e^3 x^2+18432 e^6 x^2-6144 e^9 x^2+e^{-1+3 x} \left (6144 x^3-18432 e^3 x^3+18432 e^6 x^3-6144 e^9 x^3\right )\right )}{1-4 e^3+6 e^6-4 e^9+e^{12}} \, dx=\frac {16 \, {\left (x^{4} e^{12} - 4 \, x^{4} e^{9} + 6 \, x^{4} e^{6} - 4 \, x^{4} e^{3} + x^{4} - 131072 \, {\left (x e^{3} - x\right )} e^{\left (3 \, e^{\left (3 \, x - 1\right )}\right )} + 6144 \, {\left (x^{2} e^{6} - 2 \, x^{2} e^{3} + x^{2}\right )} e^{\left (2 \, e^{\left (3 \, x - 1\right )}\right )} - 128 \, {\left (x^{3} e^{9} - 3 \, x^{3} e^{6} + 3 \, x^{3} e^{3} - x^{3}\right )} e^{\left (e^{\left (3 \, x - 1\right )}\right )} + 1048576 \, e^{\left (4 \, e^{\left (3 \, x - 1\right )}\right )}\right )}}{e^{12} - 4 \, e^{9} + 6 \, e^{6} - 4 \, e^{3} + 1} \]

[In]

integrate((201326592*exp(-1+3*x)*exp(exp(-1+3*x))^4+((-18874368*x*exp(3)[phone]*x)*exp(-1+3*x)-2097152*exp(3
)+2097152)*exp(exp(-1+3*x))^3+((589824*x^2*exp(3)^2-1179648*x^2*exp(3)+589824*x^2)*exp(-1+3*x)+196608*x*exp(3)
^2-393216*x*exp(3)+196608*x)*exp(exp(-1+3*x))^2+((-6144*x^3*exp(3)^3+18432*x^3*exp(3)^2-18432*x^3*exp(3)+6144*
x^3)*exp(-1+3*x)-6144*x^2*exp(3)^3+18432*x^2*exp(3)^2-18432*x^2*exp(3)+6144*x^2)*exp(exp(-1+3*x))+64*x^3*exp(3
)^4-256*x^3*exp(3)^3+384*x^3*exp(3)^2-256*x^3*exp(3)+64*x^3)/(exp(3)^4-4*exp(3)^3+6*exp(3)^2-4*exp(3)+1),x, al
gorithm="fricas")

[Out]

16*(x^4*e^12 - 4*x^4*e^9 + 6*x^4*e^6 - 4*x^4*e^3 + x^4 - 131072*(x*e^3 - x)*e^(3*e^(3*x - 1)) + 6144*(x^2*e^6
- 2*x^2*e^3 + x^2)*e^(2*e^(3*x - 1)) - 128*(x^3*e^9 - 3*x^3*e^6 + 3*x^3*e^3 - x^3)*e^(e^(3*x - 1)) + 1048576*e
^(4*e^(3*x - 1)))/(e^12 - 4*e^9 + 6*e^6 - 4*e^3 + 1)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 332 vs. \(2 (19) = 38\).

Time = 0.47 (sec) , antiderivative size = 332, normalized size of antiderivative = 12.77 \[ \int \frac {201326592 e^{-1+4 e^{-1+3 x}+3 x}+64 x^3-256 e^3 x^3+384 e^6 x^3-256 e^9 x^3+64 e^{12} x^3+e^{3 e^{-1+3 x}} \left (2097152-2097152 e^3+e^{-1+3 x} \left (18874368 x-18874368 e^3 x\right )\right )+e^{2 e^{-1+3 x}} \left (196608 x-393216 e^3 x+196608 e^6 x+e^{-1+3 x} \left (589824 x^2-1179648 e^3 x^2+589824 e^6 x^2\right )\right )+e^{e^{-1+3 x}} \left (6144 x^2-18432 e^3 x^2+18432 e^6 x^2-6144 e^9 x^2+e^{-1+3 x} \left (6144 x^3-18432 e^3 x^3+18432 e^6 x^3-6144 e^9 x^3\right )\right )}{1-4 e^3+6 e^6-4 e^9+e^{12}} \, dx=16 x^{4} + \frac {\left (- 2097152 x e^{21} - 44040192 x e^{15} - 73400320 x e^{9} - 14680064 x e^{3} + 2097152 x + 44040192 x e^{6} + 73400320 x e^{12} + 14680064 x e^{18}\right ) e^{3 e^{3 x - 1}} + \left (- 786432 x^{2} e^{21} - 5505024 x^{2} e^{15} - 5505024 x^{2} e^{9} - 786432 x^{2} e^{3} + 98304 x^{2} + 2752512 x^{2} e^{6} + 6881280 x^{2} e^{12} + 2752512 x^{2} e^{18} + 98304 x^{2} e^{24}\right ) e^{2 e^{3 x - 1}} + \left (- 2048 x^{3} e^{27} - 73728 x^{3} e^{21} - 258048 x^{3} e^{15} - 172032 x^{3} e^{9} - 18432 x^{3} e^{3} + 2048 x^{3} + 73728 x^{3} e^{6} + 258048 x^{3} e^{12} + 172032 x^{3} e^{18} + 18432 x^{3} e^{24}\right ) e^{e^{3 x - 1}} + \left (- 100663296 e^{15} - 335544320 e^{9} - 100663296 e^{3} + 16777216 + 251658240 e^{6} + 251658240 e^{12} + 16777216 e^{18}\right ) e^{4 e^{3 x - 1}}}{- 10 e^{27} - 120 e^{21} - 252 e^{15} - 120 e^{9} - 10 e^{3} + 1 + 45 e^{6} + 210 e^{12} + 210 e^{18} + 45 e^{24} + e^{30}} \]

[In]

integrate((201326592*exp(-1+3*x)*exp(exp(-1+3*x))**4+((-18874368*x*exp(3)[phone]*x)*exp(-1+3*x)-2097152*exp(
3)+2097152)*exp(exp(-1+3*x))**3+((589824*x**2*exp(3)**2-1179648*x**2*exp(3)+589824*x**2)*exp(-1+3*x)+196608*x*
exp(3)**2-393216*x*exp(3)+196608*x)*exp(exp(-1+3*x))**2+((-6144*x**3*exp(3)**3+18432*x**3*exp(3)**2-18432*x**3
*exp(3)+6144*x**3)*exp(-1+3*x)-6144*x**2*exp(3)**3+18432*x**2*exp(3)**2-18432*x**2*exp(3)+6144*x**2)*exp(exp(-
1+3*x))+64*x**3*exp(3)**4-256*x**3*exp(3)**3+384*x**3*exp(3)**2-256*x**3*exp(3)+64*x**3)/(exp(3)**4-4*exp(3)**
3+6*exp(3)**2-4*exp(3)+1),x)

[Out]

16*x**4 + ((-2097152*x*exp(21) - 44040192*x*exp(15) - 73400320*x*exp(9) - 14680064*x*exp(3) + 2097152*x + 4404
0192*x*exp(6) + 73400320*x*exp(12) + 14680064*x*exp(18))*exp(3*exp(3*x - 1)) + (-786432*x**2*exp(21) - 5505024
*x**2*exp(15) - 5505024*x**2*exp(9) - 786432*x**2*exp(3) + 98304*x**2 + 2752512*x**2*exp(6) + 6881280*x**2*exp
(12) + 2752512*x**2*exp(18) + 98304*x**2*exp(24))*exp(2*exp(3*x - 1)) + (-2048*x**3*exp(27) - 73728*x**3*exp(2
1) - 258048*x**3*exp(15) - 172032*x**3*exp(9) - 18432*x**3*exp(3) + 2048*x**3 + 73728*x**3*exp(6) + 258048*x**
3*exp(12) + 172032*x**3*exp(18) + 18432*x**3*exp(24))*exp(exp(3*x - 1)) + (-100663296*exp(15) - 335544320*exp(
9) - 100663296*exp(3) + 16777216 + 251658240*exp(6) + 251658240*exp(12) + 16777216*exp(18))*exp(4*exp(3*x - 1)
))/(-10*exp(27) - 120*exp(21) - 252*exp(15) - 120*exp(9) - 10*exp(3) + 1 + 45*exp(6) + 210*exp(12) + 210*exp(1
8) + 45*exp(24) + exp(30))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 124 vs. \(2 (21) = 42\).

Time = 0.20 (sec) , antiderivative size = 124, normalized size of antiderivative = 4.77 \[ \int \frac {201326592 e^{-1+4 e^{-1+3 x}+3 x}+64 x^3-256 e^3 x^3+384 e^6 x^3-256 e^9 x^3+64 e^{12} x^3+e^{3 e^{-1+3 x}} \left (2097152-2097152 e^3+e^{-1+3 x} \left (18874368 x-18874368 e^3 x\right )\right )+e^{2 e^{-1+3 x}} \left (196608 x-393216 e^3 x+196608 e^6 x+e^{-1+3 x} \left (589824 x^2-1179648 e^3 x^2+589824 e^6 x^2\right )\right )+e^{e^{-1+3 x}} \left (6144 x^2-18432 e^3 x^2+18432 e^6 x^2-6144 e^9 x^2+e^{-1+3 x} \left (6144 x^3-18432 e^3 x^3+18432 e^6 x^3-6144 e^9 x^3\right )\right )}{1-4 e^3+6 e^6-4 e^9+e^{12}} \, dx=\frac {16 \, {\left (x^{4} e^{12} - 4 \, x^{4} e^{9} + 6 \, x^{4} e^{6} - 4 \, x^{4} e^{3} - 128 \, x^{3} {\left (e^{9} - 3 \, e^{6} + 3 \, e^{3} - 1\right )} e^{\left (e^{\left (3 \, x - 1\right )}\right )} + x^{4} + 6144 \, x^{2} {\left (e^{6} - 2 \, e^{3} + 1\right )} e^{\left (2 \, e^{\left (3 \, x - 1\right )}\right )} - 131072 \, x {\left (e^{3} - 1\right )} e^{\left (3 \, e^{\left (3 \, x - 1\right )}\right )} + 1048576 \, e^{\left (4 \, e^{\left (3 \, x - 1\right )}\right )}\right )}}{e^{12} - 4 \, e^{9} + 6 \, e^{6} - 4 \, e^{3} + 1} \]

[In]

integrate((201326592*exp(-1+3*x)*exp(exp(-1+3*x))^4+((-18874368*x*exp(3)[phone]*x)*exp(-1+3*x)-2097152*exp(3
)+2097152)*exp(exp(-1+3*x))^3+((589824*x^2*exp(3)^2-1179648*x^2*exp(3)+589824*x^2)*exp(-1+3*x)+196608*x*exp(3)
^2-393216*x*exp(3)+196608*x)*exp(exp(-1+3*x))^2+((-6144*x^3*exp(3)^3+18432*x^3*exp(3)^2-18432*x^3*exp(3)+6144*
x^3)*exp(-1+3*x)-6144*x^2*exp(3)^3+18432*x^2*exp(3)^2-18432*x^2*exp(3)+6144*x^2)*exp(exp(-1+3*x))+64*x^3*exp(3
)^4-256*x^3*exp(3)^3+384*x^3*exp(3)^2-256*x^3*exp(3)+64*x^3)/(exp(3)^4-4*exp(3)^3+6*exp(3)^2-4*exp(3)+1),x, al
gorithm="maxima")

[Out]

16*(x^4*e^12 - 4*x^4*e^9 + 6*x^4*e^6 - 4*x^4*e^3 - 128*x^3*(e^9 - 3*e^6 + 3*e^3 - 1)*e^(e^(3*x - 1)) + x^4 + 6
144*x^2*(e^6 - 2*e^3 + 1)*e^(2*e^(3*x - 1)) - 131072*x*(e^3 - 1)*e^(3*e^(3*x - 1)) + 1048576*e^(4*e^(3*x - 1))
)/(e^12 - 4*e^9 + 6*e^6 - 4*e^3 + 1)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 207 vs. \(2 (21) = 42\).

Time = 0.29 (sec) , antiderivative size = 207, normalized size of antiderivative = 7.96 \[ \int \frac {201326592 e^{-1+4 e^{-1+3 x}+3 x}+64 x^3-256 e^3 x^3+384 e^6 x^3-256 e^9 x^3+64 e^{12} x^3+e^{3 e^{-1+3 x}} \left (2097152-2097152 e^3+e^{-1+3 x} \left (18874368 x-18874368 e^3 x\right )\right )+e^{2 e^{-1+3 x}} \left (196608 x-393216 e^3 x+196608 e^6 x+e^{-1+3 x} \left (589824 x^2-1179648 e^3 x^2+589824 e^6 x^2\right )\right )+e^{e^{-1+3 x}} \left (6144 x^2-18432 e^3 x^2+18432 e^6 x^2-6144 e^9 x^2+e^{-1+3 x} \left (6144 x^3-18432 e^3 x^3+18432 e^6 x^3-6144 e^9 x^3\right )\right )}{1-4 e^3+6 e^6-4 e^9+e^{12}} \, dx=\frac {16 \, {\left (x^{4} e^{12} - 4 \, x^{4} e^{9} + 6 \, x^{4} e^{6} - 4 \, x^{4} e^{3} + x^{4} + 6144 \, x^{2} e^{\left (2 \, e^{\left (3 \, x - 1\right )}\right )} + 6144 \, x^{2} e^{\left (2 \, e^{\left (3 \, x - 1\right )} + 6\right )} - 12288 \, x^{2} e^{\left (2 \, e^{\left (3 \, x - 1\right )} + 3\right )} - 128 \, {\left (x^{3} e^{\left (3 \, x + e^{\left (3 \, x - 1\right )} + 9\right )} - 3 \, x^{3} e^{\left (3 \, x + e^{\left (3 \, x - 1\right )} + 6\right )} + 3 \, x^{3} e^{\left (3 \, x + e^{\left (3 \, x - 1\right )} + 3\right )} - x^{3} e^{\left (3 \, x + e^{\left (3 \, x - 1\right )}\right )}\right )} e^{\left (-3 \, x\right )} + 131072 \, x e^{\left (3 \, e^{\left (3 \, x - 1\right )}\right )} - 131072 \, x e^{\left (3 \, e^{\left (3 \, x - 1\right )} + 3\right )} + 1048576 \, e^{\left (4 \, e^{\left (3 \, x - 1\right )}\right )}\right )}}{e^{12} - 4 \, e^{9} + 6 \, e^{6} - 4 \, e^{3} + 1} \]

[In]

integrate((201326592*exp(-1+3*x)*exp(exp(-1+3*x))^4+((-18874368*x*exp(3)[phone]*x)*exp(-1+3*x)-2097152*exp(3
)+2097152)*exp(exp(-1+3*x))^3+((589824*x^2*exp(3)^2-1179648*x^2*exp(3)+589824*x^2)*exp(-1+3*x)+196608*x*exp(3)
^2-393216*x*exp(3)+196608*x)*exp(exp(-1+3*x))^2+((-6144*x^3*exp(3)^3+18432*x^3*exp(3)^2-18432*x^3*exp(3)+6144*
x^3)*exp(-1+3*x)-6144*x^2*exp(3)^3+18432*x^2*exp(3)^2-18432*x^2*exp(3)+6144*x^2)*exp(exp(-1+3*x))+64*x^3*exp(3
)^4-256*x^3*exp(3)^3+384*x^3*exp(3)^2-256*x^3*exp(3)+64*x^3)/(exp(3)^4-4*exp(3)^3+6*exp(3)^2-4*exp(3)+1),x, al
gorithm="giac")

[Out]

16*(x^4*e^12 - 4*x^4*e^9 + 6*x^4*e^6 - 4*x^4*e^3 + x^4 + 6144*x^2*e^(2*e^(3*x - 1)) + 6144*x^2*e^(2*e^(3*x - 1
) + 6) - 12288*x^2*e^(2*e^(3*x - 1) + 3) - 128*(x^3*e^(3*x + e^(3*x - 1) + 9) - 3*x^3*e^(3*x + e^(3*x - 1) + 6
) + 3*x^3*e^(3*x + e^(3*x - 1) + 3) - x^3*e^(3*x + e^(3*x - 1)))*e^(-3*x) + 131072*x*e^(3*e^(3*x - 1)) - 13107
2*x*e^(3*e^(3*x - 1) + 3) + 1048576*e^(4*e^(3*x - 1)))/(e^12 - 4*e^9 + 6*e^6 - 4*e^3 + 1)

Mupad [B] (verification not implemented)

Time = 18.98 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.04 \[ \int \frac {201326592 e^{-1+4 e^{-1+3 x}+3 x}+64 x^3-256 e^3 x^3+384 e^6 x^3-256 e^9 x^3+64 e^{12} x^3+e^{3 e^{-1+3 x}} \left (2097152-2097152 e^3+e^{-1+3 x} \left (18874368 x-18874368 e^3 x\right )\right )+e^{2 e^{-1+3 x}} \left (196608 x-393216 e^3 x+196608 e^6 x+e^{-1+3 x} \left (589824 x^2-1179648 e^3 x^2+589824 e^6 x^2\right )\right )+e^{e^{-1+3 x}} \left (6144 x^2-18432 e^3 x^2+18432 e^6 x^2-6144 e^9 x^2+e^{-1+3 x} \left (6144 x^3-18432 e^3 x^3+18432 e^6 x^3-6144 e^9 x^3\right )\right )}{1-4 e^3+6 e^6-4 e^9+e^{12}} \, dx=\frac {16\,{\left (x+32\,{\mathrm {e}}^{{\mathrm {e}}^{3\,x}\,{\mathrm {e}}^{-1}}-x\,{\mathrm {e}}^3\right )}^4}{{\left ({\mathrm {e}}^3-1\right )}^4} \]

[In]

int((exp(2*exp(3*x - 1))*(196608*x - 393216*x*exp(3) + 196608*x*exp(6) + exp(3*x - 1)*(589824*x^2*exp(6) - 117
9648*x^2*exp(3) + 589824*x^2)) - exp(exp(3*x - 1))*(exp(3*x - 1)*(18432*x^3*exp(3) - 18432*x^3*exp(6) + 6144*x
^3*exp(9) - 6144*x^3) + 18432*x^2*exp(3) - 18432*x^2*exp(6) + 6144*x^2*exp(9) - 6144*x^2) + exp(3*exp(3*x - 1)
)*(exp(3*x - 1)*(18874368*x - 18874368*x*exp(3)) - 2097152*exp(3) + 2097152) + 201326592*exp(4*exp(3*x - 1))*e
xp(3*x - 1) - 256*x^3*exp(3) + 384*x^3*exp(6) - 256*x^3*exp(9) + 64*x^3*exp(12) + 64*x^3)/(6*exp(6) - 4*exp(3)
- 4*exp(9) + exp(12) + 1),x)

[Out]

(16*(x + 32*exp(exp(3*x)*exp(-1)) - x*exp(3))^4)/(exp(3) - 1)^4