∫ −1458−162x+e2x(−18+18x)+ex(−324+144x+18x2)+e2 log2(x) (−162+324 log(x))+elog2 (x)(−972−54x+ex(−108+54x)+(972+108ex+108x) log(x))________________________________________________________________________________________________________

Optimal. Leaf size=24 \[ -5+\frac {9 \left (e^x+3 \left (3+e^{\log ^2(x)}\right )+x\right )^2}{x^2} \]

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Rubi [B] time = 0.72, antiderivative size = 80, normalized size of antiderivative = 3.33, number of steps used = 16, number of rules used = 7, integrand size = 83, $\frac {\text {number of rules}}{\text {integrand size}}$ = 0.084, Rules used = {14, 2197, 37, 2199, 2177, 2178, 2288} \begin {gather*} \frac {9 (x+9)^2}{x^2}+\frac {162 e^x}{x^2}+\frac {9 e^{2 x}}{x^2}+\frac {54 e^{\log ^2(x)} \left (e^x \log (x)+x \log (x)+9 \log (x)\right )}{x^2 \log (x)}+\frac {81 e^{2 \log ^2(x)}}{x^2}+\frac {18 e^x}{x} \end {gather*}

Int[(-1458 - 162*x + E^(2*x)*(-18 + 18*x) + E^x*(-324 + 144*x + 18*x^2) + E^(2*Log[x]^2)*(-162 + 324*Log[x]) +

E^Log[x]^2*(-972 - 54*x + E^x*(-108 + 54*x) + (972 + 108*E^x + 108*x)*Log[x]))/x^3,x]

(162*E^x)/x^2 + (9*E^(2*x))/x^2 + (81*E^(2*Log[x]^2))/x^2 + (18*E^x)/x + (9*(9 + x)^2)/x^2 + (54*E^Log[x]^2*(9

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]

&& !LinearQ[u, x] && !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n +

1))/((b*c - a*d)*(m + 1)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ

Int[((b_.)*(F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_), x_Symbol] :> Simp[((c + d*x)^(m

+ 1)*(b*F^(g*(e + f*x)))^n)/(d*(m + 1)), x] - Dist[(f*g*n*Log[F])/(d*(m + 1)), 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] && LtQ[m, -1] && IntegerQ[2*m] && !$UseGamma ===

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[(F^(g*(e - (c*f)/d))*ExpIntegral

Ei[(f*g*(c + d*x)*Log[F])/d])/d, x] /; FreeQ[{F, c, d, e, f, g}, x] && !$UseGamma === True

Int[(F_)^((c_.)*(v_))*(u_)^(m_.)*(w_), x_Symbol] :> With[{b = Coefficient[v, x, 1], d = Coefficient[u, x, 0],

e = Coefficient[u, x, 1], f = Coefficient[w, x, 0], g = Coefficient[w, x, 1]}, Simp[(g*u^(m + 1)*F^(c*v))/(b*c

*e*Log[F]), x] /; EqQ[e*g*(m + 1) - b*c*(e*f - d*g)*Log[F], 0]] /; FreeQ[{F, c, m}, x] && LinearQ[{u, v, w}, x

Int[(F_)^((c_.)*(v_))*(u_)^(m_.)*(w_), x_Symbol] :> Int[ExpandIntegrand[F^(c*ExpandToSum[v, x]), w*NormalizePo

werOfLinear[u, x]^m, x], x] /; FreeQ[{F, c}, x] && PolynomialQ[w, x] && LinearQ[v, x] && PowerOfLinearQ[u, x]

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,

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {18 \left (-81-18 e^x-e^{2 x}-9 x+8 e^x x+e^{2 x} x+e^x x^2\right )}{x^3}+\frac {162 e^{2 \log ^2(x)} (-1+2 \log (x))}{x^3}+\frac {54 e^{\log ^2(x)} \left (-18-2 e^x-x+e^x x+18 \log (x)+2 e^x \log (x)+2 x \log (x)\right )}{x^3}\right ) \, dx\\ &=18 \int \frac {-81-18 e^x-e^{2 x}-9 x+8 e^x x+e^{2 x} x+e^x x^2}{x^3} \, dx+54 \int \frac {e^{\log ^2(x)} \left (-18-2 e^x-x+e^x x+18 \log (x)+2 e^x \log (x)+2 x \log (x)\right )}{x^3} \, dx+162 \int \frac {e^{2 \log ^2(x)} (-1+2 \log (x))}{x^3} \, dx\\ &=\frac {81 e^{2 \log ^2(x)}}{x^2}+\frac {54 e^{\log ^2(x)} \left (9 \log (x)+e^x \log (x)+x \log (x)\right )}{x^2 \log (x)}+18 \int \left (\frac {e^{2 x} (-1+x)}{x^3}-\frac {9 (9+x)}{x^3}+\frac {e^x \left (-18+8 x+x^2\right )}{x^3}\right ) \, dx\\ &=\frac {81 e^{2 \log ^2(x)}}{x^2}+\frac {54 e^{\log ^2(x)} \left (9 \log (x)+e^x \log (x)+x \log (x)\right )}{x^2 \log (x)}+18 \int \frac {e^{2 x} (-1+x)}{x^3} \, dx+18 \int \frac {e^x \left (-18+8 x+x^2\right )}{x^3} \, dx-162 \int \frac {9+x}{x^3} \, dx\\ &=\frac {9 e^{2 x}}{x^2}+\frac {81 e^{2 \log ^2(x)}}{x^2}+\frac {9 (9+x)^2}{x^2}+\frac {54 e^{\log ^2(x)} \left (9 \log (x)+e^x \log (x)+x \log (x)\right )}{x^2 \log (x)}+18 \int \left (-\frac {18 e^x}{x^3}+\frac {8 e^x}{x^2}+\frac {e^x}{x}\right ) \, dx\\ &=\frac {9 e^{2 x}}{x^2}+\frac {81 e^{2 \log ^2(x)}}{x^2}+\frac {9 (9+x)^2}{x^2}+\frac {54 e^{\log ^2(x)} \left (9 \log (x)+e^x \log (x)+x \log (x)\right )}{x^2 \log (x)}+18 \int \frac {e^x}{x} \, dx+144 \int \frac {e^x}{x^2} \, dx-324 \int \frac {e^x}{x^3} \, dx\\ &=\frac {162 e^x}{x^2}+\frac {9 e^{2 x}}{x^2}+\frac {81 e^{2 \log ^2(x)}}{x^2}-\frac {144 e^x}{x}+\frac {9 (9+x)^2}{x^2}+18 \text {Ei}(x)+\frac {54 e^{\log ^2(x)} \left (9 \log (x)+e^x \log (x)+x \log (x)\right )}{x^2 \log (x)}+144 \int \frac {e^x}{x} \, dx-162 \int \frac {e^x}{x^2} \, dx\\ &=\frac {162 e^x}{x^2}+\frac {9 e^{2 x}}{x^2}+\frac {81 e^{2 \log ^2(x)}}{x^2}+\frac {18 e^x}{x}+\frac {9 (9+x)^2}{x^2}+162 \text {Ei}(x)+\frac {54 e^{\log ^2(x)} \left (9 \log (x)+e^x \log (x)+x \log (x)\right )}{x^2 \log (x)}-162 \int \frac {e^x}{x} \, dx\\ &=\frac {162 e^x}{x^2}+\frac {9 e^{2 x}}{x^2}+\frac {81 e^{2 \log ^2(x)}}{x^2}+\frac {18 e^x}{x}+\frac {9 (9+x)^2}{x^2}+\frac {54 e^{\log ^2(x)} \left (9 \log (x)+e^x \log (x)+x \log (x)\right )}{x^2 \log (x)}\\ \end {aligned} \end {gather*}

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Mathematica [A] time = 0.79, size = 34, normalized size = 1.42 \begin {gather*} \frac {9 \left (9+e^x+3 e^{\log ^2(x)}\right ) \left (9+e^x+3 e^{\log ^2(x)}+2 x\right )}{x^2} \end {gather*}

Integrate[(-1458 - 162*x + E^(2*x)*(-18 + 18*x) + E^x*(-324 + 144*x + 18*x^2) + E^(2*Log[x]^2)*(-162 + 324*Log

[x]) + E^Log[x]^2*(-972 - 54*x + E^x*(-108 + 54*x) + (972 + 108*E^x + 108*x)*Log[x]))/x^3,x]

(9*(9 + E^x + 3*E^Log[x]^2)*(9 + E^x + 3*E^Log[x]^2 + 2*x))/x^2

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fricas [A] time = 0.62, size = 42, normalized size = 1.75 \begin {gather*} \frac {9 \, {\left (6 \, {\left (x + e^{x} + 9\right )} e^{\left (\log \relax (x)^{2}\right )} + 2 \, {\left (x + 9\right )} e^{x} + 18 \, x + 9 \, e^{\left (2 \, \log \relax (x)^{2}\right )} + e^{\left (2 \, x\right )} + 81\right )}}{x^{2}} \end {gather*}

integrate(((324*log(x)-162)*exp(log(x)^2)^2+((108*exp(x)+108*x+972)*log(x)+(54*x-108)*exp(x)-54*x-972)*exp(log

(x)^2)+(18*x-18)*exp(x)^2+(18*x^2+144*x-324)*exp(x)-162*x-1458)/x^3,x, algorithm="fricas")

9*(6*(x + e^x + 9)*e^(log(x)^2) + 2*(x + 9)*e^x + 18*x + 9*e^(2*log(x)^2) + e^(2*x) + 81)/x^2

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giac [B] time = 0.20, size = 56, normalized size = 2.33 \begin {gather*} \frac {9 \, {\left (6 \, x e^{\left (\log \relax (x)^{2}\right )} + 2 \, x e^{x} + 18 \, x + 9 \, e^{\left (2 \, \log \relax (x)^{2}\right )} + 6 \, e^{\left (\log \relax (x)^{2} + x\right )} + 54 \, e^{\left (\log \relax (x)^{2}\right )} + e^{\left (2 \, x\right )} + 18 \, e^{x} + 81\right )}}{x^{2}} \end {gather*}

integrate(((324*log(x)-162)*exp(log(x)^2)^2+((108*exp(x)+108*x+972)*log(x)+(54*x-108)*exp(x)-54*x-972)*exp(log

(x)^2)+(18*x-18)*exp(x)^2+(18*x^2+144*x-324)*exp(x)-162*x-1458)/x^3,x, algorithm="giac")

9*(6*x*e^(log(x)^2) + 2*x*e^x + 18*x + 9*e^(2*log(x)^2) + 6*e^(log(x)^2 + x) + 54*e^(log(x)^2) + e^(2*x) + 18*

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method	result	size

risch	$\frac {18 \,{\mathrm e}^{x} x +9 \,{\mathrm e}^{2 x}+162 x +162 \,{\mathrm e}^{x}+729}{x^{2}}+\frac {81 \,{\mathrm e}^{2 \ln \relax (x )^{2}}}{x^{2}}+\frac {54 \left (x +{\mathrm e}^{x}+9\right ) {\mathrm e}^{\ln \relax (x )^{2}}}{x^{2}}$	$52$

int(((324*ln(x)-162)*exp(ln(x)^2)^2+((108*exp(x)+108*x+972)*ln(x)+(54*x-108)*exp(x)-54*x-972)*exp(ln(x)^2)+(18

*x-18)*exp(x)^2+(18*x^2+144*x-324)*exp(x)-162*x-1458)/x^3,x,method=_RETURNVERBOSE)

9*(2*exp(x)*x+exp(2*x)+18*x+18*exp(x)+81)/x^2+81/x^2*exp(2*ln(x)^2)+54*(x+exp(x)+9)/x^2*exp(ln(x)^2)

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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {81}{2} i \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (i \, \sqrt {2} \log \relax (x) - \frac {1}{2} i \, \sqrt {2}\right ) e^{\left (-\frac {1}{2}\right )} + 27 i \, \sqrt {\pi } \operatorname {erf}\left (i \, \log \relax (x) - \frac {1}{2} i\right ) e^{\left (-\frac {1}{4}\right )} + 486 i \, \sqrt {\pi } \operatorname {erf}\left (i \, \log \relax (x) - i\right ) e^{\left (-1\right )} + \frac {162}{x} + \frac {54 \, e^{\left (\log \relax (x)^{2} + x\right )}}{x^{2}} + \frac {729}{x^{2}} + 18 \, {\rm Ei}\relax (x) + 144 \, \Gamma \left (-1, -x\right ) + 36 \, \Gamma \left (-1, -2 \, x\right ) + 324 \, \Gamma \left (-2, -x\right ) + 72 \, \Gamma \left (-2, -2 \, x\right ) + 18 \, \int \frac {6 \, {\left (x + 9\right )} e^{\left (\log \relax (x)^{2}\right )} \log \relax (x)}{x^{3}}\,{d x} + 324 \, \int \frac {e^{\left (2 \, \log \relax (x)^{2}\right )} \log \relax (x)}{x^{3}}\,{d x} \end {gather*}

integrate(((324*log(x)-162)*exp(log(x)^2)^2+((108*exp(x)+108*x+972)*log(x)+(54*x-108)*exp(x)-54*x-972)*exp(log

(x)^2)+(18*x-18)*exp(x)^2+(18*x^2+144*x-324)*exp(x)-162*x-1458)/x^3,x, algorithm="maxima")

81/2*I*sqrt(2)*sqrt(pi)*erf(I*sqrt(2)*log(x) - 1/2*I*sqrt(2))*e^(-1/2) + 27*I*sqrt(pi)*erf(I*log(x) - 1/2*I)*e

^(-1/4) + 486*I*sqrt(pi)*erf(I*log(x) - I)*e^(-1) + 162/x + 54*e^(log(x)^2 + x)/x^2 + 729/x^2 + 18*Ei(x) + 144

*gamma(-1, -x) + 36*gamma(-1, -2*x) + 324*gamma(-2, -x) + 72*gamma(-2, -2*x) + 18*integrate(6*(x + 9)*e^(log(x

)^2)*log(x)/x^3, x) + 324*integrate(e^(2*log(x)^2)*log(x)/x^3, x)

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mupad [B] time = 6.78, size = 30, normalized size = 1.25 \begin {gather*} \frac {9\,\left (3\,{\mathrm {e}}^{{\ln \relax (x)}^2}+{\mathrm {e}}^x+9\right )\,\left (2\,x+3\,{\mathrm {e}}^{{\ln \relax (x)}^2}+{\mathrm {e}}^x+9\right )}{x^2} \end {gather*}

int(-(162*x - exp(2*log(x)^2)*(324*log(x) - 162) + exp(log(x)^2)*(54*x - exp(x)*(54*x - 108) - log(x)*(108*x +

108*exp(x) + 972) + 972) - exp(x)*(144*x + 18*x^2 - 324) - exp(2*x)*(18*x - 18) + 1458)/x^3,x)

(9*(3*exp(log(x)^2) + exp(x) + 9)*(2*x + 3*exp(log(x)^2) + exp(x) + 9))/x^2

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sympy [B] time = 0.63, size = 82, normalized size = 3.42 \begin {gather*} - \frac {- 162 x - 729}{x^{2}} + \frac {9 x^{2} e^{2 x} + \left (18 x^{3} + 54 x^{2} e^{\log {\relax (x )}^{2}} + 162 x^{2}\right ) e^{x}}{x^{4}} + \frac {81 x^{2} e^{2 \log {\relax (x )}^{2}} + \left (54 x^{3} + 486 x^{2}\right ) e^{\log {\relax (x )}^{2}}}{x^{4}} \end {gather*}

integrate(((324*ln(x)-162)*exp(ln(x)**2)**2+((108*exp(x)+108*x+972)*ln(x)+(54*x-108)*exp(x)-54*x-972)*exp(ln(x

)**2)+(18*x-18)*exp(x)**2+(18*x**2+144*x-324)*exp(x)-162*x-1458)/x**3,x)

-(-162*x - 729)/x**2 + (9*x**2*exp(2*x) + (18*x**3 + 54*x**2*exp(log(x)**2) + 162*x**2)*exp(x))/x**4 + (81*x**

2*exp(2*log(x)**2) + (54*x**3 + 486*x**2)*exp(log(x)**2))/x**4

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3.93.83 \(\int \frac {-1458-162 x+e^{2 x} (-18+18 x)+e^x (-324+144 x+18 x^2)+e^{2 \log ^2(x)} (-162+324 \log (x))+e^{\log ^2(x)} (-972-54 x+e^x (-108+54 x)+(972+108 e^x+108 x) \log (x))}{x^3} \, dx\)