∫ 36−e3xx+12 log(2)+log2(2)+e2x(1−12x−2x log(2))+ex(12−32x+(2−12x) log(2)−x log2(2))+(72+2e2x+24 log(2)+2 log2(2)+ex(24+4 log(2))) log(x)_________________________________________________________________________________________________________ 36x+e2xx+12x log(2)+x log2(2)+ex(12x+2x log(2)) dx

Optimal. Leaf size=25 \[ -e^x-\frac {4}{6+e^x+\log (2)}+\log ^2(x)+\log (2 x) \]

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Rubi [A] time = 1.58, antiderivative size = 29, normalized size of antiderivative = 1.16, number of steps used = 14, number of rules used = 10, integrand size = 128, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.078, Rules used = {6, 6741, 6742, 2194, 2282, 36, 29, 31, 44, 2301} \begin {gather*} -e^x+\frac {1}{4} (2 \log (x)+1)^2-\frac {4}{e^x+6+\log (2)} \end {gather*}

Int[(36 - E^(3*x)*x + 12*Log[2] + Log[2]^2 + E^(2*x)*(1 - 12*x - 2*x*Log[2]) + E^x*(12 - 32*x + (2 - 12*x)*Log

[2] - x*Log[2]^2) + (72 + 2*E^(2*x) + 24*Log[2] + 2*Log[2]^2 + E^x*(24 + 4*Log[2]))*Log[x])/(36*x + E^(2*x)*x

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

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Dist[b/(b*c - a*d), Int[1/(a + b*x), x], x] -

Int[((a_) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*

x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] && !(IGtQ[n, 0] && L

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

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

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/(x_), x_Symbol] :> Simp[(a + b*Log[c*x^n])^2/(2*b*n), x] /; FreeQ[{a

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {36-e^{3 x} x+12 \log (2)+\log ^2(2)+e^{2 x} (1-12 x-2 x \log (2))+e^x \left (12-32 x+(2-12 x) \log (2)-x \log ^2(2)\right )+\left (72+2 e^{2 x}+24 \log (2)+2 \log ^2(2)+e^x (24+4 \log (2))\right ) \log (x)}{e^{2 x} x+x \log ^2(2)+x (36+12 \log (2))+e^x (12 x+2 x \log (2))} \, dx\\ &=\int \frac {36-e^{3 x} x+12 \log (2)+\log ^2(2)+e^{2 x} (1-12 x-2 x \log (2))+e^x \left (12-32 x+(2-12 x) \log (2)-x \log ^2(2)\right )+\left (72+2 e^{2 x}+24 \log (2)+2 \log ^2(2)+e^x (24+4 \log (2))\right ) \log (x)}{e^{2 x} x+e^x (12 x+2 x \log (2))+x \left (36+12 \log (2)+\log ^2(2)\right )} \, dx\\ &=\int \frac {-e^{3 x} x+e^{2 x} (1-12 x-2 x \log (2))+e^x \left (12-32 x+(2-12 x) \log (2)-x \log ^2(2)\right )+36 \left (1+\frac {1}{36} \log (2) (12+\log (2))\right )+\left (72+2 e^{2 x}+24 \log (2)+2 \log ^2(2)+e^x (24+4 \log (2))\right ) \log (x)}{x \left (e^x+6 \left (1+\frac {\log (2)}{6}\right )\right )^2} \, dx\\ &=\int \left (-e^x+\frac {4}{e^x+6 \left (1+\frac {\log (2)}{6}\right )}+\frac {4 (-6-\log (2))}{\left (e^x+6 \left (1+\frac {\log (2)}{6}\right )\right )^2}+\frac {1+2 \log (x)}{x}\right ) \, dx\\ &=4 \int \frac {1}{e^x+6 \left (1+\frac {\log (2)}{6}\right )} \, dx-(4 (6+\log (2))) \int \frac {1}{\left (e^x+6 \left (1+\frac {\log (2)}{6}\right )\right )^2} \, dx-\int e^x \, dx+\int \frac {1+2 \log (x)}{x} \, dx\\ &=-e^x+\frac {1}{4} (1+2 \log (x))^2+4 \operatorname {Subst}\left (\int \frac {1}{x (6+x+\log (2))} \, dx,x,e^x\right )-(4 (6+\log (2))) \operatorname {Subst}\left (\int \frac {1}{x (6+x+\log (2))^2} \, dx,x,e^x\right )\\ &=-e^x+\frac {1}{4} (1+2 \log (x))^2+\frac {4 \operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,e^x\right )}{6+\log (2)}-\frac {4 \operatorname {Subst}\left (\int \frac {1}{6+x+\log (2)} \, dx,x,e^x\right )}{6+\log (2)}-(4 (6+\log (2))) \operatorname {Subst}\left (\int \left (\frac {1}{x (6+\log (2))^2}-\frac {1}{(6+\log (2)) (6+x+\log (2))^2}-\frac {1}{(6+\log (2))^2 (6+x+\log (2))}\right ) \, dx,x,e^x\right )\\ &=-e^x-\frac {4}{6+e^x+\log (2)}+\frac {1}{4} (1+2 \log (x))^2\\ \end {aligned} \end {gather*}

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Mathematica [B] time = 0.36, size = 59, normalized size = 2.36 \begin {gather*} -e^x-\frac {24-2 \log ^3(2)+\log ^2(2) (-12+\log (4))+\log (16)-\log (2) \log (4096)+\log (4) \log (4096)}{(6+\log (2)) \left (6+e^x+\log (2)\right )}+\log (x)+\log ^2(x) \end {gather*}

Integrate[(36 - E^(3*x)*x + 12*Log[2] + Log[2]^2 + E^(2*x)*(1 - 12*x - 2*x*Log[2]) + E^x*(12 - 32*x + (2 - 12*

x)*Log[2] - x*Log[2]^2) + (72 + 2*E^(2*x) + 24*Log[2] + 2*Log[2]^2 + E^x*(24 + 4*Log[2]))*Log[x])/(36*x + E^(2

-E^x - (24 - 2*Log[2]^3 + Log[2]^2*(-12 + Log[4]) + Log[16] - Log[2]*Log[4096] + Log[4]*Log[4096])/((6 + Log[2

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fricas [A] time = 0.45, size = 45, normalized size = 1.80 \begin {gather*} \frac {{\left (e^{x} + \log \relax (2) + 6\right )} \log \relax (x)^{2} - {\left (\log \relax (2) + 6\right )} e^{x} + {\left (e^{x} + \log \relax (2) + 6\right )} \log \relax (x) - e^{\left (2 \, x\right )} - 4}{e^{x} + \log \relax (2) + 6} \end {gather*}

integrate(((2*exp(x)^2+(4*log(2)+24)*exp(x)+2*log(2)^2+24*log(2)+72)*log(x)-x*exp(x)^3+(-2*x*log(2)-12*x+1)*ex

p(x)^2+(-x*log(2)^2+(-12*x+2)*log(2)-32*x+12)*exp(x)+log(2)^2+12*log(2)+36)/(x*exp(x)^2+(2*x*log(2)+12*x)*exp(

((e^x + log(2) + 6)*log(x)^2 - (log(2) + 6)*e^x + (e^x + log(2) + 6)*log(x) - e^(2*x) - 4)/(e^x + log(2) + 6)

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giac [B] time = 0.23, size = 61, normalized size = 2.44 \begin {gather*} \frac {e^{x} \log \relax (x)^{2} + \log \relax (2) \log \relax (x)^{2} - e^{x} \log \relax (2) + e^{x} \log \relax (x) + \log \relax (2) \log \relax (x) + 6 \, \log \relax (x)^{2} - e^{\left (2 \, x\right )} - 6 \, e^{x} + 6 \, \log \relax (x) - 4}{e^{x} + \log \relax (2) + 6} \end {gather*}

integrate(((2*exp(x)^2+(4*log(2)+24)*exp(x)+2*log(2)^2+24*log(2)+72)*log(x)-x*exp(x)^3+(-2*x*log(2)-12*x+1)*ex

p(x)^2+(-x*log(2)^2+(-12*x+2)*log(2)-32*x+12)*exp(x)+log(2)^2+12*log(2)+36)/(x*exp(x)^2+(2*x*log(2)+12*x)*exp(

(e^x*log(x)^2 + log(2)*log(x)^2 - e^x*log(2) + e^x*log(x) + log(2)*log(x) + 6*log(x)^2 - e^(2*x) - 6*e^x + 6*l

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method	result	size

risch	\(\ln \relax (x )^{2}+\frac {\ln \relax (2) \ln \relax (x )+{\mathrm e}^{x} \ln \relax (x )-{\mathrm e}^{x} \ln \relax (2)-{\mathrm e}^{2 x}+6 \ln \relax (x )-6 \,{\mathrm e}^{x}-4}{6+\ln \relax (2)+{\mathrm e}^{x}}\)	\(47\)
norman	\(\frac {\left (\ln \relax (2)+6\right ) \ln \relax (x )^{2}+\left (\ln \relax (2)+6\right ) \ln \relax (x )+{\mathrm e}^{x} \ln \relax (x )^{2}+{\mathrm e}^{x} \ln \relax (x )-{\mathrm e}^{2 x}+\ln \relax (2)^{2}+12 \ln \relax (2)+32}{6+\ln \relax (2)+{\mathrm e}^{x}}\)	\(54\)

int(((2*exp(x)^2+(4*ln(2)+24)*exp(x)+2*ln(2)^2+24*ln(2)+72)*ln(x)-x*exp(x)^3+(-2*x*ln(2)-12*x+1)*exp(x)^2+(-x*

ln(2)^2+(-12*x+2)*ln(2)-32*x+12)*exp(x)+ln(2)^2+12*ln(2)+36)/(x*exp(x)^2+(2*x*ln(2)+12*x)*exp(x)+x*ln(2)^2+12*

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maxima [B] time = 0.50, size = 48, normalized size = 1.92 \begin {gather*} \frac {{\left (\log \relax (2) + 6\right )} \log \relax (x)^{2} + {\left (\log \relax (x)^{2} - \log \relax (2) + \log \relax (x) - 6\right )} e^{x} + {\left (\log \relax (2) + 6\right )} \log \relax (x) - e^{\left (2 \, x\right )} - 4}{e^{x} + \log \relax (2) + 6} \end {gather*}

integrate(((2*exp(x)^2+(4*log(2)+24)*exp(x)+2*log(2)^2+24*log(2)+72)*log(x)-x*exp(x)^3+(-2*x*log(2)-12*x+1)*ex

p(x)^2+(-x*log(2)^2+(-12*x+2)*log(2)-32*x+12)*exp(x)+log(2)^2+12*log(2)+36)/(x*exp(x)^2+(2*x*log(2)+12*x)*exp(

((log(2) + 6)*log(x)^2 + (log(x)^2 - log(2) + log(x) - 6)*e^x + (log(2) + 6)*log(x) - e^(2*x) - 4)/(e^x + log(

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mupad [F] time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {12\,\ln \relax (2)-x\,{\mathrm {e}}^{3\,x}-{\mathrm {e}}^{2\,x}\,\left (12\,x+2\,x\,\ln \relax (2)-1\right )-{\mathrm {e}}^x\,\left (32\,x+\ln \relax (2)\,\left (12\,x-2\right )+x\,{\ln \relax (2)}^2-12\right )+\ln \relax (x)\,\left (2\,{\mathrm {e}}^{2\,x}+24\,\ln \relax (2)+{\mathrm {e}}^x\,\left (4\,\ln \relax (2)+24\right )+2\,{\ln \relax (2)}^2+72\right )+{\ln \relax (2)}^2+36}{36\,x+x\,{\mathrm {e}}^{2\,x}+12\,x\,\ln \relax (2)+x\,{\ln \relax (2)}^2+{\mathrm {e}}^x\,\left (12\,x+2\,x\,\ln \relax (2)\right )} \,d x \end {gather*}

int((12*log(2) - x*exp(3*x) - exp(2*x)*(12*x + 2*x*log(2) - 1) - exp(x)*(32*x + log(2)*(12*x - 2) + x*log(2)^2

- 12) + log(x)*(2*exp(2*x) + 24*log(2) + exp(x)*(4*log(2) + 24) + 2*log(2)^2 + 72) + log(2)^2 + 36)/(36*x + x

int((12*log(2) - x*exp(3*x) - exp(2*x)*(12*x + 2*x*log(2) - 1) - exp(x)*(32*x + log(2)*(12*x - 2) + x*log(2)^2

- 12) + log(x)*(2*exp(2*x) + 24*log(2) + exp(x)*(4*log(2) + 24) + 2*log(2)^2 + 72) + log(2)^2 + 36)/(36*x + x

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sympy [A] time = 0.41, size = 20, normalized size = 0.80 \begin {gather*} - e^{x} + \log {\relax (x )}^{2} + \log {\relax (x )} - \frac {4}{e^{x} + \log {\relax (2 )} + 6} \end {gather*}

integrate(((2*exp(x)**2+(4*ln(2)+24)*exp(x)+2*ln(2)**2+24*ln(2)+72)*ln(x)-x*exp(x)**3+(-2*x*ln(2)-12*x+1)*exp(

x)**2+(-x*ln(2)**2+(-12*x+2)*ln(2)-32*x+12)*exp(x)+ln(2)**2+12*ln(2)+36)/(x*exp(x)**2+(2*x*ln(2)+12*x)*exp(x)+

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3.53.65 \(\int \frac {36-e^{3 x} x+12 \log (2)+\log ^2(2)+e^{2 x} (1-12 x-2 x \log (2))+e^x (12-32 x+(2-12 x) \log (2)-x \log ^2(2))+(72+2 e^{2 x}+24 \log (2)+2 \log ^2(2)+e^x (24+4 \log (2))) \log (x)}{36 x+e^{2 x} x+12 x \log (2)+x \log ^2(2)+e^x (12 x+2 x \log (2))} \, dx\)