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

Optimal. Leaf size=27 \[ -e^{x/4}+x+x \left (2+\left (e^x-9 \log (3)\right )^2\right )^2 \]

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Rubi [B]  time = 0.13, antiderivative size = 153, normalized size of antiderivative = 5.67, number of steps used = 13, number of rules used = 4, integrand size = 95, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {12, 2194, 2176, 2187} \begin {gather*} -e^{x/4}-\frac {e^{4 x}}{4}+\frac {1}{4} e^{4 x} (4 x+1)+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 )-36 e^x \left (x \log (3) \left (2+81 \log ^2(3)\right )+81 \log ^3(3)+\log (9)\right )+12 e^{3 x} \log (3)-12 e^{3 x} (3 x+1) \log (3) \end {gather*}

Antiderivative was successfully verified.

[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) + E^(2*x)*(2 + 243*Log[
3]^2 + 2*x*(2 + 243*Log[3]^2)) - 36*E^x*(81*Log[3]^3 + x*Log[3]*(2 + 81*Log[3]^2) + Log[9])

Rule 12

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

Rule 2176

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] &&  !$UseGamma === True

Rule 2187

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 2194

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 {gather*} \begin {aligned} \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^{2 x} \left (8 \left (2+243 \log ^2(3)\right )+16 x \left (2+243 \log ^2(3)\right )\right ) \, dx+\frac {1}{4} \int e^x \left (-144 x \log (3) \left (2+81 \log ^2(3)\right )-144 \left (81 \log ^3(3)+\log (9)\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 )+e^{2 x} \left (2+243 \log ^2(3)+2 x \left (2+243 \log ^2(3)\right )\right )-36 e^x \left (81 \log ^3(3)+x \log (3) \left (2+81 \log ^2(3)\right )+\log (9)\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 )+e^{2 x} \left (2+243 \log ^2(3)+2 x \left (2+243 \log ^2(3)\right )\right )-36 e^x \left (81 \log ^3(3)+x \log (3) \left (2+81 \log ^2(3)\right )+\log (9)\right )\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.05, size = 76, normalized size = 2.81 \begin {gather*} -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 ) \end {gather*}

Antiderivative was successfully verified.

[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)

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fricas [B]  time = 0.99, size = 73, normalized size = 2.70 \begin {gather*} 6561 \, x \log \relax (3)^{4} - 36 \, x e^{\left (3 \, x\right )} \log \relax (3) + 324 \, x \log \relax (3)^{2} + x e^{\left (4 \, x\right )} + 2 \, {\left (243 \, x \log \relax (3)^{2} + 2 \, x\right )} e^{\left (2 \, x\right )} - 36 \, {\left (81 \, x \log \relax (3)^{3} + 2 \, x \log \relax (3)\right )} e^{x} + 5 \, x - e^{\left (\frac {1}{4} \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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giac [B]  time = 0.14, size = 73, normalized size = 2.70 \begin {gather*} 6561 \, x \log \relax (3)^{4} - 36 \, x e^{\left (3 \, x\right )} \log \relax (3) + 324 \, x \log \relax (3)^{2} + x e^{\left (4 \, x\right )} + 2 \, {\left (243 \, x \log \relax (3)^{2} + 2 \, x\right )} e^{\left (2 \, x\right )} - 36 \, {\left (81 \, x \log \relax (3)^{3} + 2 \, x \log \relax (3)\right )} e^{x} + 5 \, x - e^{\left (\frac {1}{4} \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maple [B]  time = 0.04, size = 70, normalized size = 2.59

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

Verification of antiderivative is not currently implemented for this CAS.

[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(4*x)-36*ln(3)*exp(3*x)*x+2*(243*ln(3)^2+2)*x*exp(2*x)-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

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maxima [B]  time = 0.51, size = 70, normalized size = 2.59 \begin {gather*} 6561 \, x \log \relax (3)^{4} + 2 \, {\left (243 \, \log \relax (3)^{2} + 2\right )} x e^{\left (2 \, x\right )} - 36 \, {\left (81 \, \log \relax (3)^{3} + 2 \, \log \relax (3)\right )} x e^{x} - 36 \, x e^{\left (3 \, x\right )} \log \relax (3) + 324 \, x \log \relax (3)^{2} + x e^{\left (4 \, x\right )} + 5 \, x - e^{\left (\frac {1}{4} \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [B]  time = 8.07, size = 68, normalized size = 2.52 \begin {gather*} x\,{\mathrm {e}}^{4\,x}-{\mathrm {e}}^{x/4}+x\,\left (324\,{\ln \relax (3)}^2+6561\,{\ln \relax (3)}^4+5\right )-36\,x\,{\mathrm {e}}^{3\,x}\,\ln \relax (3)+\frac {x\,{\mathrm {e}}^{2\,x}\,\left (1944\,{\ln \relax (3)}^2+16\right )}{4}-36\,x\,{\mathrm {e}}^x\,\ln \relax (3)\,\left (81\,{\ln \relax (3)}^2+2\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [B]  time = 0.32, size = 76, normalized size = 2.81 \begin {gather*} x e^{4 x} - 36 x e^{3 x} \log {\relax (3 )} + x \left (5 + 324 \log {\relax (3 )}^{2} + 6561 \log {\relax (3 )}^{4}\right ) + \left (4 x + 486 x \log {\relax (3 )}^{2}\right ) e^{2 x} + \left (- 2916 x \log {\relax (3 )}^{3} - 72 x \log {\relax (3 )}\right ) e^{x} - e^{\frac {x}{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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