\(\int \frac {1}{4} (20-e^{x/4}+e^{4 x} (4+16 x)+e^{3 x} (-144-432 x) \log (3)+1296 \log ^2(3)+26244 \log ^4(3)+e^{2 x} (16+32 x+(1944+3888 x) \log ^2(3))+e^x ((-288-288 x) \log (3)+(-11664-11664 x) \log ^3(3))) \, dx\) [10026]

Optimal result

Integrand size = 95, antiderivative size = 27 \[ \int \frac {1}{4} \left (20-e^{x/4}+e^{4 x} (4+16 x)+e^{3 x} (-144-432 x) \log (3)+1296 \log ^2(3)+26244 \log ^4(3)+e^{2 x} \left (16+32 x+(1944+3888 x) \log ^2(3)\right )+e^x \left ((-288-288 x) \log (3)+(-11664-11664 x) \log ^3(3)\right )\right ) \, dx=-e^{x/4}+x+x \left (2+\left (e^x-9 \log (3)\right )^2\right )^2 \]

[Out]

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

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(152\) vs. \(2(27)=54\).

Time = 0.08 (sec) , antiderivative size = 152, normalized size of antiderivative = 5.63, number of steps used = 13, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {12, 2225, 2207, 2218} \[ \int \frac {1}{4} \left (20-e^{x/4}+e^{4 x} (4+16 x)+e^{3 x} (-144-432 x) \log (3)+1296 \log ^2(3)+26244 \log ^4(3)+e^{2 x} \left (16+32 x+(1944+3888 x) \log ^2(3)\right )+e^x \left ((-288-288 x) \log (3)+(-11664-11664 x) \log ^3(3)\right )\right ) \, dx=-e^{x/4}-\frac {e^{4 x}}{4}+\frac {1}{4} e^{4 x} (4 x+1)-36 e^x \log (3) \left (x \left (2+81 \log ^2(3)\right )+2+81 \log ^2(3)\right )+e^{2 x} \left (2 x \left (2+243 \log ^2(3)\right )+2+243 \log ^2(3)\right )-e^{2 x} \left (2+243 \log ^2(3)\right )+36 e^x \log (3) \left (2+81 \log ^2(3)\right )+x \left (5+6561 \log ^4(3)+324 \log ^2(3)\right )+12 e^{3 x} \log (3)-12 e^{3 x} (3 x+1) \log (3) \]

[In]

Int[(20 - E^(x/4) + E^(4*x)*(4 + 16*x) + E^(3*x)*(-144 - 432*x)*Log[3] + 1296*Log[3]^2 + 26244*Log[3]^4 + E^(2
*x)*(16 + 32*x + (1944 + 3888*x)*Log[3]^2) + E^x*((-288 - 288*x)*Log[3] + (-11664 - 11664*x)*Log[3]^3))/4,x]

[Out]

-E^(x/4) - E^(4*x)/4 + (E^(4*x)*(1 + 4*x))/4 + 12*E^(3*x)*Log[3] - 12*E^(3*x)*(1 + 3*x)*Log[3] + 36*E^x*Log[3]
*(2 + 81*Log[3]^2) - E^(2*x)*(2 + 243*Log[3]^2) + x*(5 + 324*Log[3]^2 + 6561*Log[3]^4) - 36*E^x*Log[3]*(2 + 81
*Log[3]^2 + x*(2 + 81*Log[3]^2)) + E^(2*x)*(2 + 243*Log[3]^2 + 2*x*(2 + 243*Log[3]^2))

Rule 12

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

Rule 2207

Int[((b_.)*(F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[(c + d*x)^m*
((b*F^(g*(e + f*x)))^n/(f*g*n*Log[F])), x] - Dist[d*(m/(f*g*n*Log[F])), Int[(c + d*x)^(m - 1)*(b*F^(g*(e + f*x
)))^n, x], x] /; FreeQ[{F, b, c, d, e, f, g, n}, x] && GtQ[m, 0] && IntegerQ[2*m] &&  !TrueQ[$UseGamma]

Rule 2218

Int[((a_.) + (b_.)*((F_)^((g_.)*(v_)))^(n_.))^(p_.)*(u_)^(m_.), x_Symbol] :> Int[NormalizePowerOfLinear[u, x]^
m*(a + b*(F^(g*ExpandToSum[v, x]))^n)^p, x] /; FreeQ[{F, a, b, g, n, p}, x] && LinearQ[v, x] && PowerOfLinearQ
[u, x] &&  !(LinearMatchQ[v, x] && PowerOfLinearMatchQ[u, x]) && IntegerQ[m]

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]

Rubi steps \begin{align*} \text {integral}& = \frac {1}{4} \int \left (20-e^{x/4}+e^{4 x} (4+16 x)+e^{3 x} (-144-432 x) \log (3)+1296 \log ^2(3)+26244 \log ^4(3)+e^{2 x} \left (16+32 x+(1944+3888 x) \log ^2(3)\right )+e^x \left ((-288-288 x) \log (3)+(-11664-11664 x) \log ^3(3)\right )\right ) \, dx \\ & = x \left (5+324 \log ^2(3)+6561 \log ^4(3)\right )-\frac {1}{4} \int e^{x/4} \, dx+\frac {1}{4} \int e^{4 x} (4+16 x) \, dx+\frac {1}{4} \int e^{2 x} \left (16+32 x+(1944+3888 x) \log ^2(3)\right ) \, dx+\frac {1}{4} \int e^x \left ((-288-288 x) \log (3)+(-11664-11664 x) \log ^3(3)\right ) \, dx+\frac {1}{4} \log (3) \int e^{3 x} (-144-432 x) \, dx \\ & = -e^{x/4}+\frac {1}{4} e^{4 x} (1+4 x)-12 e^{3 x} (1+3 x) \log (3)+x \left (5+324 \log ^2(3)+6561 \log ^4(3)\right )+\frac {1}{4} \int e^x \left (-144 \log (3) \left (2+81 \log ^2(3)\right )-144 x \log (3) \left (2+81 \log ^2(3)\right )\right ) \, dx+\frac {1}{4} \int e^{2 x} \left (8 \left (2+243 \log ^2(3)\right )+16 x \left (2+243 \log ^2(3)\right )\right ) \, dx+(36 \log (3)) \int e^{3 x} \, dx-\int e^{4 x} \, dx \\ & = -e^{x/4}-\frac {e^{4 x}}{4}+\frac {1}{4} e^{4 x} (1+4 x)+12 e^{3 x} \log (3)-12 e^{3 x} (1+3 x) \log (3)+x \left (5+324 \log ^2(3)+6561 \log ^4(3)\right )-36 e^x \log (3) \left (2+81 \log ^2(3)+x \left (2+81 \log ^2(3)\right )\right )+e^{2 x} \left (2+243 \log ^2(3)+2 x \left (2+243 \log ^2(3)\right )\right )+\left (36 \log (3) \left (2+81 \log ^2(3)\right )\right ) \int e^x \, dx-\left (2 \left (2+243 \log ^2(3)\right )\right ) \int e^{2 x} \, dx \\ & = -e^{x/4}-\frac {e^{4 x}}{4}+\frac {1}{4} e^{4 x} (1+4 x)+12 e^{3 x} \log (3)-12 e^{3 x} (1+3 x) \log (3)+36 e^x \log (3) \left (2+81 \log ^2(3)\right )-e^{2 x} \left (2+243 \log ^2(3)\right )+x \left (5+324 \log ^2(3)+6561 \log ^4(3)\right )-36 e^x \log (3) \left (2+81 \log ^2(3)+x \left (2+81 \log ^2(3)\right )\right )+e^{2 x} \left (2+243 \log ^2(3)+2 x \left (2+243 \log ^2(3)\right )\right ) \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(76\) vs. \(2(27)=54\).

Time = 0.07 (sec) , antiderivative size = 76, normalized size of antiderivative = 2.81 \[ \int \frac {1}{4} \left (20-e^{x/4}+e^{4 x} (4+16 x)+e^{3 x} (-144-432 x) \log (3)+1296 \log ^2(3)+26244 \log ^4(3)+e^{2 x} \left (16+32 x+(1944+3888 x) \log ^2(3)\right )+e^x \left ((-288-288 x) \log (3)+(-11664-11664 x) \log ^3(3)\right )\right ) \, dx=-e^{x/4}+5 x+e^{4 x} x-36 e^{3 x} x \log (3)+324 x \log ^2(3)+6561 x \log ^4(3)-36 e^x x \log (3) \left (2+81 \log ^2(3)\right )+2 e^{2 x} x \left (2+243 \log ^2(3)\right ) \]

[In]

Integrate[(20 - E^(x/4) + E^(4*x)*(4 + 16*x) + E^(3*x)*(-144 - 432*x)*Log[3] + 1296*Log[3]^2 + 26244*Log[3]^4
+ E^(2*x)*(16 + 32*x + (1944 + 3888*x)*Log[3]^2) + E^x*((-288 - 288*x)*Log[3] + (-11664 - 11664*x)*Log[3]^3))/
4,x]

[Out]

-E^(x/4) + 5*x + E^(4*x)*x - 36*E^(3*x)*x*Log[3] + 324*x*Log[3]^2 + 6561*x*Log[3]^4 - 36*E^x*x*Log[3]*(2 + 81*
Log[3]^2) + 2*E^(2*x)*x*(2 + 243*Log[3]^2)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(69\) vs. \(2(23)=46\).

Time = 0.27 (sec) , antiderivative size = 70, normalized size of antiderivative = 2.59

method	result	size
risch	\(x \,{\mathrm e}^{4 x}-36 \ln \left (3\right ) {\mathrm e}^{3 x} x +2 \left (243 \ln \left (3\right )^{2}+2\right ) x \,{\mathrm e}^{2 x}-36 \ln \left (3\right ) \left (81 \ln \left (3\right )^{2}+2\right ) x \,{\mathrm e}^{x}-{\mathrm e}^{\frac {x}{4}}+6561 x \ln \left (3\right )^{4}+324 x \ln \left (3\right )^{2}+5 x\)	\(70\)
parallelrisch	\(x \,{\mathrm e}^{4 x}-36 \ln \left (3\right ) {\mathrm e}^{3 x} x +486 \ln \left (3\right )^{2} {\mathrm e}^{2 x} x +4 x \,{\mathrm e}^{2 x}-2916 \,{\mathrm e}^{x} \ln \left (3\right )^{3} x -72 x \ln \left (3\right ) {\mathrm e}^{x}-{\mathrm e}^{\frac {x}{4}}+\left (6561 \ln \left (3\right )^{4}+324 \ln \left (3\right )^{2}+5\right ) x\)	\(73\)
default	\(-2916 \,{\mathrm e}^{x} \ln \left (3\right )^{3} x +6561 x \ln \left (3\right )^{4}+486 \ln \left (3\right )^{2} {\mathrm e}^{2 x} x -72 x \ln \left (3\right ) {\mathrm e}^{x}+324 x \ln \left (3\right )^{2}-36 \ln \left (3\right ) {\mathrm e}^{3 x} x +x \,{\mathrm e}^{4 x}+4 x \,{\mathrm e}^{2 x}-{\mathrm e}^{\frac {x}{4}}+5 x\)	\(74\)
parts	\(-2916 \,{\mathrm e}^{x} \ln \left (3\right )^{3} x +6561 x \ln \left (3\right )^{4}+486 \ln \left (3\right )^{2} {\mathrm e}^{2 x} x -72 x \ln \left (3\right ) {\mathrm e}^{x}+324 x \ln \left (3\right )^{2}-36 \ln \left (3\right ) {\mathrm e}^{3 x} x +x \,{\mathrm e}^{4 x}+4 x \,{\mathrm e}^{2 x}-{\mathrm e}^{\frac {x}{4}}+5 x\)	\(74\)

[In]

int(1/4*(16*x+4)*exp(x)^4+1/4*(-432*x-144)*ln(3)*exp(x)^3+1/4*((3888*x+1944)*ln(3)^2+32*x+16)*exp(x)^2+1/4*((-
11664*x-11664)*ln(3)^3+(-288*x-288)*ln(3))*exp(x)-1/4*exp(1/4*x)+6561*ln(3)^4+324*ln(3)^2+5,x,method=_RETURNVE
RBOSE)

[Out]

x*exp(x)^4-36*ln(3)*exp(x)^3*x+2*(243*ln(3)^2+2)*x*exp(x)^2-36*ln(3)*(81*ln(3)^2+2)*x*exp(x)-exp(1/4*x)+6561*x
*ln(3)^4+324*x*ln(3)^2+5*x

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 73 vs. \(2 (23) = 46\).

Time = 0.27 (sec) , antiderivative size = 73, normalized size of antiderivative = 2.70 \[ \int \frac {1}{4} \left (20-e^{x/4}+e^{4 x} (4+16 x)+e^{3 x} (-144-432 x) \log (3)+1296 \log ^2(3)+26244 \log ^4(3)+e^{2 x} \left (16+32 x+(1944+3888 x) \log ^2(3)\right )+e^x \left ((-288-288 x) \log (3)+(-11664-11664 x) \log ^3(3)\right )\right ) \, dx=6561 \, x \log \left (3\right )^{4} - 36 \, x e^{\left (3 \, x\right )} \log \left (3\right ) + 324 \, x \log \left (3\right )^{2} + x e^{\left (4 \, x\right )} + 2 \, {\left (243 \, x \log \left (3\right )^{2} + 2 \, x\right )} e^{\left (2 \, x\right )} - 36 \, {\left (81 \, x \log \left (3\right )^{3} + 2 \, x \log \left (3\right )\right )} e^{x} + 5 \, x - e^{\left (\frac {1}{4} \, x\right )} \]

[In]

integrate(1/4*(16*x+4)*exp(x)^4+1/4*(-432*x-144)*log(3)*exp(x)^3+1/4*((3888*x+1944)*log(3)^2+32*x+16)*exp(x)^2
+1/4*((-11664*x-11664)*log(3)^3+(-288*x-288)*log(3))*exp(x)-1/4*exp(1/4*x)+6561*log(3)^4+324*log(3)^2+5,x, alg
orithm="fricas")

[Out]

6561*x*log(3)^4 - 36*x*e^(3*x)*log(3) + 324*x*log(3)^2 + x*e^(4*x) + 2*(243*x*log(3)^2 + 2*x)*e^(2*x) - 36*(81
*x*log(3)^3 + 2*x*log(3))*e^x + 5*x - e^(1/4*x)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 76 vs. \(2 (20) = 40\).

Time = 0.20 (sec) , antiderivative size = 76, normalized size of antiderivative = 2.81 \[ \int \frac {1}{4} \left (20-e^{x/4}+e^{4 x} (4+16 x)+e^{3 x} (-144-432 x) \log (3)+1296 \log ^2(3)+26244 \log ^4(3)+e^{2 x} \left (16+32 x+(1944+3888 x) \log ^2(3)\right )+e^x \left ((-288-288 x) \log (3)+(-11664-11664 x) \log ^3(3)\right )\right ) \, dx=x e^{4 x} - 36 x e^{3 x} \log {\left (3 \right )} + x \left (5 + 324 \log {\left (3 \right )}^{2} + 6561 \log {\left (3 \right )}^{4}\right ) + \left (4 x + 486 x \log {\left (3 \right )}^{2}\right ) e^{2 x} + \left (- 2916 x \log {\left (3 \right )}^{3} - 72 x \log {\left (3 \right )}\right ) e^{x} - e^{\frac {x}{4}} \]

[In]

integrate(1/4*(16*x+4)*exp(x)**4+1/4*(-432*x-144)*ln(3)*exp(x)**3+1/4*((3888*x+1944)*ln(3)**2+32*x+16)*exp(x)*
*2+1/4*((-11664*x-11664)*ln(3)**3+(-288*x-288)*ln(3))*exp(x)-1/4*exp(1/4*x)+6561*ln(3)**4+324*ln(3)**2+5,x)

[Out]

x*exp(4*x) - 36*x*exp(3*x)*log(3) + x*(5 + 324*log(3)**2 + 6561*log(3)**4) + (4*x + 486*x*log(3)**2)*exp(2*x)
+ (-2916*x*log(3)**3 - 72*x*log(3))*exp(x) - exp(x/4)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 70 vs. \(2 (23) = 46\).

Time = 0.28 (sec) , antiderivative size = 70, normalized size of antiderivative = 2.59 \[ \int \frac {1}{4} \left (20-e^{x/4}+e^{4 x} (4+16 x)+e^{3 x} (-144-432 x) \log (3)+1296 \log ^2(3)+26244 \log ^4(3)+e^{2 x} \left (16+32 x+(1944+3888 x) \log ^2(3)\right )+e^x \left ((-288-288 x) \log (3)+(-11664-11664 x) \log ^3(3)\right )\right ) \, dx=6561 \, x \log \left (3\right )^{4} + 2 \, {\left (243 \, \log \left (3\right )^{2} + 2\right )} x e^{\left (2 \, x\right )} - 36 \, {\left (81 \, \log \left (3\right )^{3} + 2 \, \log \left (3\right )\right )} x e^{x} - 36 \, x e^{\left (3 \, x\right )} \log \left (3\right ) + 324 \, x \log \left (3\right )^{2} + x e^{\left (4 \, x\right )} + 5 \, x - e^{\left (\frac {1}{4} \, x\right )} \]

[In]

integrate(1/4*(16*x+4)*exp(x)^4+1/4*(-432*x-144)*log(3)*exp(x)^3+1/4*((3888*x+1944)*log(3)^2+32*x+16)*exp(x)^2
+1/4*((-11664*x-11664)*log(3)^3+(-288*x-288)*log(3))*exp(x)-1/4*exp(1/4*x)+6561*log(3)^4+324*log(3)^2+5,x, alg
orithm="maxima")

[Out]

6561*x*log(3)^4 + 2*(243*log(3)^2 + 2)*x*e^(2*x) - 36*(81*log(3)^3 + 2*log(3))*x*e^x - 36*x*e^(3*x)*log(3) + 3
24*x*log(3)^2 + x*e^(4*x) + 5*x - e^(1/4*x)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 73 vs. \(2 (23) = 46\).

Time = 0.27 (sec) , antiderivative size = 73, normalized size of antiderivative = 2.70 \[ \int \frac {1}{4} \left (20-e^{x/4}+e^{4 x} (4+16 x)+e^{3 x} (-144-432 x) \log (3)+1296 \log ^2(3)+26244 \log ^4(3)+e^{2 x} \left (16+32 x+(1944+3888 x) \log ^2(3)\right )+e^x \left ((-288-288 x) \log (3)+(-11664-11664 x) \log ^3(3)\right )\right ) \, dx=6561 \, x \log \left (3\right )^{4} - 36 \, x e^{\left (3 \, x\right )} \log \left (3\right ) + 324 \, x \log \left (3\right )^{2} + x e^{\left (4 \, x\right )} + 2 \, {\left (243 \, x \log \left (3\right )^{2} + 2 \, x\right )} e^{\left (2 \, x\right )} - 36 \, {\left (81 \, x \log \left (3\right )^{3} + 2 \, x \log \left (3\right )\right )} e^{x} + 5 \, x - e^{\left (\frac {1}{4} \, x\right )} \]

[In]

integrate(1/4*(16*x+4)*exp(x)^4+1/4*(-432*x-144)*log(3)*exp(x)^3+1/4*((3888*x+1944)*log(3)^2+32*x+16)*exp(x)^2
+1/4*((-11664*x-11664)*log(3)^3+(-288*x-288)*log(3))*exp(x)-1/4*exp(1/4*x)+6561*log(3)^4+324*log(3)^2+5,x, alg
orithm="giac")

[Out]

6561*x*log(3)^4 - 36*x*e^(3*x)*log(3) + 324*x*log(3)^2 + x*e^(4*x) + 2*(243*x*log(3)^2 + 2*x)*e^(2*x) - 36*(81
*x*log(3)^3 + 2*x*log(3))*e^x + 5*x - e^(1/4*x)

Mupad [B] (verification not implemented)

Time = 18.98 (sec) , antiderivative size = 68, normalized size of antiderivative = 2.52 \[ \int \frac {1}{4} \left (20-e^{x/4}+e^{4 x} (4+16 x)+e^{3 x} (-144-432 x) \log (3)+1296 \log ^2(3)+26244 \log ^4(3)+e^{2 x} \left (16+32 x+(1944+3888 x) \log ^2(3)\right )+e^x \left ((-288-288 x) \log (3)+(-11664-11664 x) \log ^3(3)\right )\right ) \, dx=x\,{\mathrm {e}}^{4\,x}-{\mathrm {e}}^{x/4}+x\,\left (324\,{\ln \left (3\right )}^2+6561\,{\ln \left (3\right )}^4+5\right )-36\,x\,{\mathrm {e}}^{3\,x}\,\ln \left (3\right )+\frac {x\,{\mathrm {e}}^{2\,x}\,\left (1944\,{\ln \left (3\right )}^2+16\right )}{4}-36\,x\,{\mathrm {e}}^x\,\ln \left (3\right )\,\left (81\,{\ln \left (3\right )}^2+2\right ) \]

[In]

int((exp(2*x)*(32*x + log(3)^2*(3888*x + 1944) + 16))/4 - (exp(x)*(log(3)*(288*x + 288) + log(3)^3*(11664*x +
11664)))/4 - exp(x/4)/4 + 324*log(3)^2 + 6561*log(3)^4 + (exp(4*x)*(16*x + 4))/4 - (exp(3*x)*log(3)*(432*x + 1
44))/4 + 5,x)

[Out]

x*exp(4*x) - exp(x/4) + x*(324*log(3)^2 + 6561*log(3)^4 + 5) - 36*x*exp(3*x)*log(3) + (x*exp(2*x)*(1944*log(3)
^2 + 16))/4 - 36*x*exp(x)*log(3)*(81*log(3)^2 + 2)