∫ 6x−e5x+6x2−5x3−2x4−x log( 1_ x2 )+(6+2x+e5(3+x)+(3+x) log( 1_ x2 )) log(3+x 2 ) ______________________________________________________________________________________________________

Optimal. Leaf size=34 \[ 1-x^2+\frac {\left (-e^5+x-\log \left (\frac {1}{x^2}\right )\right ) \left (x+\log \left (\frac {3+x}{2}\right )\right )}{x} \]

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Rubi [B] time = 0.70, antiderivative size = 160, normalized size of antiderivative = 4.71, number of steps used = 21, number of rules used = 14, integrand size = 73, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {6, 1593, 6742, 1620, 2344, 2301, 2317, 2391, 2395, 36, 31, 29, 2376, 2392} \begin {gather*} -x^2+\frac {1}{12} \log ^2\left (\frac {1}{x^2}\right )+\frac {1}{3} \log \left (\frac {x}{3}+1\right ) \log \left (\frac {1}{x^2}\right )+\frac {1}{3} \left (\log \left (\frac {1}{x^2}\right )+e^5+2\right ) \log (x)-\frac {1}{3} \left (\log \left (\frac {1}{x^2}\right )+e^5+2\right ) \log (x+3)-\frac {\log \left (\frac {x}{2}+\frac {3}{2}\right ) \left (\log \left (\frac {1}{x^2}\right )+e^5+2\right )}{x}+x+\frac {\log ^2(x)}{3}-\frac {2}{3} \log (3) \log (x)+\frac {1}{3} \left (6-e^5\right ) \log (x)-\frac {2 \log (x)}{3}+\frac {1}{3} \left (3+e^5\right ) \log (x+3)+\frac {2}{3} \log (x+3)+\frac {2 \log \left (\frac {x}{2}+\frac {3}{2}\right )}{x} \end {gather*}

Int[(6*x - E^5*x + 6*x^2 - 5*x^3 - 2*x^4 - x*Log[x^(-2)] + (6 + 2*x + E^5*(3 + x) + (3 + x)*Log[x^(-2)])*Log[(

x - x^2 + (2*Log[3/2 + x/2])/x + (Log[1 + x/3]*Log[x^(-2)])/3 + Log[x^(-2)]^2/12 - (Log[3/2 + x/2]*(2 + E^5 +

Log[x^(-2)]))/x - (2*Log[x])/3 + ((6 - E^5)*Log[x])/3 - (2*Log[3]*Log[x])/3 + ((2 + E^5 + Log[x^(-2)])*Log[x])

/3 + Log[x]^2/3 + (2*Log[3 + x])/3 + ((3 + E^5)*Log[3 + x])/3 - ((2 + E^5 + Log[x^(-2)])*Log[3 + x])/3

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

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, 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] -

Dist[d/(b*c - a*d), Int[1/(c + d*x), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^(q - p))^n, x] /; F

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

^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && PolyQ[Px, x] && (IntegersQ[m, n] || IGtQ[m, -2]) &&

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

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(Log[1 + (e*x)/d]*(a +

b*Log[c*x^n])^p)/e, x] - Dist[(b*n*p)/e, Int[(Log[1 + (e*x)/d]*(a + b*Log[c*x^n])^(p - 1))/x, x], x] /; FreeQ[

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((x_)*((d_) + (e_.)*(x_))), x_Symbol] :> Dist[1/d, Int[(a + b*

Log[c*x^n])^p/x, x], x] - Dist[e/d, Int[(a + b*Log[c*x^n])^p/(d + e*x), x], x] /; FreeQ[{a, b, c, d, e, n}, x]

Int[Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))^(r_.)]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((g_.)*(x_))^(q_.), x_Sym

bol] :> With[{u = IntHide[(g*x)^q*Log[d*(e + f*x^m)^r], x]}, Dist[a + b*Log[c*x^n], u, x] - Dist[b*n, Int[Dist

[1/x, u, x], x], x]] /; FreeQ[{a, b, c, d, e, f, g, r, m, n, q}, x] && (IntegerQ[(q + 1)/m] || (RationalQ[m] &

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> -Simp[PolyLog[2, -(c*e*x^n)]/n, x] /; FreeQ[{c, d,

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

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))*((f_.) + (g_.)*(x_))^(q_.), x_Symbol] :> Simp[((f + g

*x)^(q + 1)*(a + b*Log[c*(d + e*x)^n]))/(g*(q + 1)), x] - Dist[(b*e*n)/(g*(q + 1)), Int[(f + g*x)^(q + 1)/(d +

e*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, n, q}, x] && NeQ[e*f - d*g, 0] && NeQ[q, -1]

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {\left (6-e^5\right ) x+6 x^2-5 x^3-2 x^4-x \log \left (\frac {1}{x^2}\right )+\left (6+2 x+e^5 (3+x)+(3+x) \log \left (\frac {1}{x^2}\right )\right ) \log \left (\frac {3+x}{2}\right )}{3 x^2+x^3} \, dx\\ &=\int \frac {\left (6-e^5\right ) x+6 x^2-5 x^3-2 x^4-x \log \left (\frac {1}{x^2}\right )+\left (6+2 x+e^5 (3+x)+(3+x) \log \left (\frac {1}{x^2}\right )\right ) \log \left (\frac {3+x}{2}\right )}{x^2 (3+x)} \, dx\\ &=\int \left (\frac {6 \left (1-\frac {e^5}{6}\right )+6 x-5 x^2-2 x^3-\log \left (\frac {1}{x^2}\right )}{x (3+x)}+\frac {\log \left (\frac {3}{2}+\frac {x}{2}\right ) \left (2 \left (1+\frac {e^5}{2}\right )+\log \left (\frac {1}{x^2}\right )\right )}{x^2}\right ) \, dx\\ &=\int \frac {6 \left (1-\frac {e^5}{6}\right )+6 x-5 x^2-2 x^3-\log \left (\frac {1}{x^2}\right )}{x (3+x)} \, dx+\int \frac {\log \left (\frac {3}{2}+\frac {x}{2}\right ) \left (2 \left (1+\frac {e^5}{2}\right )+\log \left (\frac {1}{x^2}\right )\right )}{x^2} \, dx\\ &=-\frac {\log \left (\frac {3}{2}+\frac {x}{2}\right ) \left (2+e^5+\log \left (\frac {1}{x^2}\right )\right )}{x}+\frac {1}{3} \left (2+e^5+\log \left (\frac {1}{x^2}\right )\right ) \log (x)-\frac {1}{3} \left (2+e^5+\log \left (\frac {1}{x^2}\right )\right ) \log (3+x)+2 \int \left (-\frac {\log \left (\frac {3}{2}+\frac {x}{2}\right )}{x^2}+\frac {\log (x)}{3 x}-\frac {\log (3+x)}{3 x}\right ) \, dx+\int \left (\frac {6-e^5+6 x-5 x^2-2 x^3}{x (3+x)}-\frac {\log \left (\frac {1}{x^2}\right )}{x (3+x)}\right ) \, dx\\ &=-\frac {\log \left (\frac {3}{2}+\frac {x}{2}\right ) \left (2+e^5+\log \left (\frac {1}{x^2}\right )\right )}{x}+\frac {1}{3} \left (2+e^5+\log \left (\frac {1}{x^2}\right )\right ) \log (x)-\frac {1}{3} \left (2+e^5+\log \left (\frac {1}{x^2}\right )\right ) \log (3+x)+\frac {2}{3} \int \frac {\log (x)}{x} \, dx-\frac {2}{3} \int \frac {\log (3+x)}{x} \, dx-2 \int \frac {\log \left (\frac {3}{2}+\frac {x}{2}\right )}{x^2} \, dx+\int \frac {6-e^5+6 x-5 x^2-2 x^3}{x (3+x)} \, dx-\int \frac {\log \left (\frac {1}{x^2}\right )}{x (3+x)} \, dx\\ &=\frac {2 \log \left (\frac {3}{2}+\frac {x}{2}\right )}{x}-\frac {\log \left (\frac {3}{2}+\frac {x}{2}\right ) \left (2+e^5+\log \left (\frac {1}{x^2}\right )\right )}{x}-\frac {2}{3} \log (3) \log (x)+\frac {1}{3} \left (2+e^5+\log \left (\frac {1}{x^2}\right )\right ) \log (x)+\frac {\log ^2(x)}{3}-\frac {1}{3} \left (2+e^5+\log \left (\frac {1}{x^2}\right )\right ) \log (3+x)-\frac {1}{3} \int \frac {\log \left (\frac {1}{x^2}\right )}{x} \, dx+\frac {1}{3} \int \frac {\log \left (\frac {1}{x^2}\right )}{3+x} \, dx-\frac {2}{3} \int \frac {\log \left (1+\frac {x}{3}\right )}{x} \, dx-\int \frac {1}{\left (\frac {3}{2}+\frac {x}{2}\right ) x} \, dx+\int \left (1+\frac {6-e^5}{3 x}-2 x+\frac {3+e^5}{3 (3+x)}\right ) \, dx\\ &=x-x^2+\frac {2 \log \left (\frac {3}{2}+\frac {x}{2}\right )}{x}+\frac {1}{3} \log \left (1+\frac {x}{3}\right ) \log \left (\frac {1}{x^2}\right )+\frac {1}{12} \log ^2\left (\frac {1}{x^2}\right )-\frac {\log \left (\frac {3}{2}+\frac {x}{2}\right ) \left (2+e^5+\log \left (\frac {1}{x^2}\right )\right )}{x}+\frac {1}{3} \left (6-e^5\right ) \log (x)-\frac {2}{3} \log (3) \log (x)+\frac {1}{3} \left (2+e^5+\log \left (\frac {1}{x^2}\right )\right ) \log (x)+\frac {\log ^2(x)}{3}+\frac {1}{3} \left (3+e^5\right ) \log (3+x)-\frac {1}{3} \left (2+e^5+\log \left (\frac {1}{x^2}\right )\right ) \log (3+x)+\frac {2 \text {Li}_2\left (-\frac {x}{3}\right )}{3}+\frac {1}{3} \int \frac {1}{\frac {3}{2}+\frac {x}{2}} \, dx-\frac {2}{3} \int \frac {1}{x} \, dx+\frac {2}{3} \int \frac {\log \left (1+\frac {x}{3}\right )}{x} \, dx\\ &=x-x^2+\frac {2 \log \left (\frac {3}{2}+\frac {x}{2}\right )}{x}+\frac {1}{3} \log \left (1+\frac {x}{3}\right ) \log \left (\frac {1}{x^2}\right )+\frac {1}{12} \log ^2\left (\frac {1}{x^2}\right )-\frac {\log \left (\frac {3}{2}+\frac {x}{2}\right ) \left (2+e^5+\log \left (\frac {1}{x^2}\right )\right )}{x}-\frac {2 \log (x)}{3}+\frac {1}{3} \left (6-e^5\right ) \log (x)-\frac {2}{3} \log (3) \log (x)+\frac {1}{3} \left (2+e^5+\log \left (\frac {1}{x^2}\right )\right ) \log (x)+\frac {\log ^2(x)}{3}+\frac {2}{3} \log (3+x)+\frac {1}{3} \left (3+e^5\right ) \log (3+x)-\frac {1}{3} \left (2+e^5+\log \left (\frac {1}{x^2}\right )\right ) \log (3+x)\\ \end {aligned} \end {gather*}

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Mathematica [A] time = 0.17, size = 36, normalized size = 1.06 \begin {gather*} x-x^2+2 \log (x)-\frac {\left (e^5+\log \left (\frac {1}{x^2}\right )\right ) \log \left (\frac {3+x}{2}\right )}{x}+\log (3+x) \end {gather*}

Integrate[(6*x - E^5*x + 6*x^2 - 5*x^3 - 2*x^4 - x*Log[x^(-2)] + (6 + 2*x + E^5*(3 + x) + (3 + x)*Log[x^(-2)])

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fricas [A] time = 0.56, size = 40, normalized size = 1.18 \begin {gather*} -\frac {x^{3} - x^{2} - {\left (x - e^{5} - \log \left (\frac {1}{x^{2}}\right )\right )} \log \left (\frac {1}{2} \, x + \frac {3}{2}\right ) + x \log \left (\frac {1}{x^{2}}\right )}{x} \end {gather*}

integrate((((3+x)*log(1/x^2)+(3+x)*exp(5)+2*x+6)*log(3/2+1/2*x)-x*log(1/x^2)-x*exp(5)-2*x^4-5*x^3+6*x^2+6*x)/(

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giac [A] time = 0.23, size = 52, normalized size = 1.53 \begin {gather*} -\frac {x^{3} - x^{2} + \log \relax (2) \log \left (x^{2}\right ) - x \log \left (x + 3\right ) - \log \left (x^{2}\right ) \log \left (x + 3\right ) - 2 \, x \log \relax (x) + e^{5} \log \left (\frac {1}{2} \, x + \frac {3}{2}\right )}{x} \end {gather*}

integrate((((3+x)*log(1/x^2)+(3+x)*exp(5)+2*x+6)*log(3/2+1/2*x)-x*log(1/x^2)-x*exp(5)-2*x^4-5*x^3+6*x^2+6*x)/(

-(x^3 - x^2 + log(2)*log(x^2) - x*log(x + 3) - log(x^2)*log(x + 3) - 2*x*log(x) + e^5*log(1/2*x + 3/2))/x

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method	result	size

risch	\(-\frac {\left (i \pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-2 i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}+i \pi \mathrm {csgn}\left (i x^{2}\right )^{3}+2 \,{\mathrm e}^{5}-4 \ln \relax (x )\right ) \ln \left (\frac {3}{2}+\frac {x}{2}\right )}{2 x}-x^{2}+x +\ln \left (3+x \right )+2 \ln \relax (x )\)	\(85\)

int((((3+x)*ln(1/x^2)+(3+x)*exp(5)+2*x+6)*ln(3/2+1/2*x)-x*ln(1/x^2)-x*exp(5)-2*x^4-5*x^3+6*x^2+6*x)/(x^3+3*x^2

-1/2*(I*Pi*csgn(I*x)^2*csgn(I*x^2)-2*I*Pi*csgn(I*x)*csgn(I*x^2)^2+I*Pi*csgn(I*x^2)^3+2*exp(5)-4*ln(x))/x*ln(3/

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maxima [A] time = 0.50, size = 46, normalized size = 1.35 \begin {gather*} -\frac {x^{3} - x^{2} - e^{5} \log \relax (2) - {\left (x - e^{5} + 2 \, \log \relax (x)\right )} \log \left (x + 3\right ) - 2 \, {\left (x - \log \relax (2)\right )} \log \relax (x)}{x} \end {gather*}

integrate((((3+x)*log(1/x^2)+(3+x)*exp(5)+2*x+6)*log(3/2+1/2*x)-x*log(1/x^2)-x*exp(5)-2*x^4-5*x^3+6*x^2+6*x)/(

-(x^3 - x^2 - e^5*log(2) - (x - e^5 + 2*log(x))*log(x + 3) - 2*(x - log(2))*log(x))/x

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mupad [B] time = 6.32, size = 43, normalized size = 1.26 \begin {gather*} x+\ln \left (x+3\right )+2\,\ln \relax (x)-x^2-\frac {\ln \left (\frac {1}{x^2}\right )\,\ln \left (\frac {x}{2}+\frac {3}{2}\right )}{x}-\frac {{\mathrm {e}}^5\,\ln \left (\frac {x}{2}+\frac {3}{2}\right )}{x} \end {gather*}

int(-(x*log(1/x^2) - 6*x + x*exp(5) - log(x/2 + 3/2)*(2*x + log(1/x^2)*(x + 3) + exp(5)*(x + 3) + 6) - 6*x^2 +

x + log(x + 3) + 2*log(x) - x^2 - (log(1/x^2)*log(x/2 + 3/2))/x - (exp(5)*log(x/2 + 3/2))/x

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sympy [A] time = 0.68, size = 36, normalized size = 1.06 \begin {gather*} - x^{2} + x + 2 \log {\relax (x )} + \log {\left (x + 3 \right )} + \frac {\left (- \log {\left (\frac {1}{x^{2}} \right )} - e^{5}\right ) \log {\left (\frac {x}{2} + \frac {3}{2} \right )}}{x} \end {gather*}

integrate((((3+x)*ln(1/x**2)+(3+x)*exp(5)+2*x+6)*ln(3/2+1/2*x)-x*ln(1/x**2)-x*exp(5)-2*x**4-5*x**3+6*x**2+6*x)

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3.97.93 \(\int \frac {6 x-e^5 x+6 x^2-5 x^3-2 x^4-x \log (\frac {1}{x^2})+(6+2 x+e^5 (3+x)+(3+x) \log (\frac {1}{x^2})) \log (\frac {3+x}{2})}{3 x^2+x^3} \, dx\)