\(\int \frac {-16-32 x-175 x^2-158 x^3-389 x^4+10 x^5+25 x^6+(-8 x^2-8 x^3-40 x^4) \log (x)+(-4 x^2+20 x^4) \log ^2(x)}{x^2+2 x^3+11 x^4+10 x^5+25 x^6} \, dx\) [3637]

Optimal result

Integrand size = 92, antiderivative size = 31 \[ \int \frac {-16-32 x-175 x^2-158 x^3-389 x^4+10 x^5+25 x^6+\left (-8 x^2-8 x^3-40 x^4\right ) \log (x)+\left (-4 x^2+20 x^4\right ) \log ^2(x)}{x^2+2 x^3+11 x^4+10 x^5+25 x^6} \, dx=\frac {(-4+x)^2+x}{x}-\frac {4 \log ^2(x)}{5 x+\frac {1+x}{x}} \]

[Out]

(x+(x-4)^2)/x-4*ln(x)^2/(5*x+(1+x)/x)

Rubi [C] (verified)

Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.

Time = 0.79 (sec) , antiderivative size = 487, normalized size of antiderivative = 15.71, number of steps used = 48, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.076, Rules used = {6820, 2404, 2354, 2438, 2355, 2421, 6724} \[ \int \frac {-16-32 x-175 x^2-158 x^3-389 x^4+10 x^5+25 x^6+\left (-8 x^2-8 x^3-40 x^4\right ) \log (x)+\left (-4 x^2+20 x^4\right ) \log ^2(x)}{x^2+2 x^3+11 x^4+10 x^5+25 x^6} \, dx=-\frac {160 \operatorname {PolyLog}\left (2,-\frac {10 x}{1-i \sqrt {19}}\right )}{19 \left (1-i \sqrt {19}\right )}+\frac {8 i \operatorname {PolyLog}\left (2,-\frac {10 x}{1-i \sqrt {19}}\right )}{\sqrt {19}}+\frac {8}{19} \operatorname {PolyLog}\left (2,-\frac {10 x}{1-i \sqrt {19}}\right )-\frac {160 \operatorname {PolyLog}\left (2,-\frac {10 x}{1+i \sqrt {19}}\right )}{19 \left (1+i \sqrt {19}\right )}-\frac {8 i \operatorname {PolyLog}\left (2,-\frac {10 x}{1+i \sqrt {19}}\right )}{\sqrt {19}}+\frac {8}{19} \operatorname {PolyLog}\left (2,-\frac {10 x}{1+i \sqrt {19}}\right )+x+\frac {16}{x}+\frac {800 x \log ^2(x)}{19 \left (1-i \sqrt {19}\right ) \left (10 x-i \sqrt {19}+1\right )}-\frac {40 x \log ^2(x)}{19 \left (10 x-i \sqrt {19}+1\right )}+\frac {800 x \log ^2(x)}{19 \left (1+i \sqrt {19}\right ) \left (10 x+i \sqrt {19}+1\right )}-\frac {40 x \log ^2(x)}{19 \left (10 x+i \sqrt {19}+1\right )}-\frac {160 \log \left (1+\frac {10 x}{1-i \sqrt {19}}\right ) \log (x)}{19 \left (1-i \sqrt {19}\right )}+\frac {8 i \log \left (1+\frac {10 x}{1-i \sqrt {19}}\right ) \log (x)}{\sqrt {19}}+\frac {8}{19} \log \left (1+\frac {10 x}{1-i \sqrt {19}}\right ) \log (x)-\frac {160 \log \left (1+\frac {10 x}{1+i \sqrt {19}}\right ) \log (x)}{19 \left (1+i \sqrt {19}\right )}-\frac {8 i \log \left (1+\frac {10 x}{1+i \sqrt {19}}\right ) \log (x)}{\sqrt {19}}+\frac {8}{19} \log \left (1+\frac {10 x}{1+i \sqrt {19}}\right ) \log (x) \]

[In]

Int[(-16 - 32*x - 175*x^2 - 158*x^3 - 389*x^4 + 10*x^5 + 25*x^6 + (-8*x^2 - 8*x^3 - 40*x^4)*Log[x] + (-4*x^2 +
20*x^4)*Log[x]^2)/(x^2 + 2*x^3 + 11*x^4 + 10*x^5 + 25*x^6),x]

[Out]

16/x + x - (40*x*Log[x]^2)/(19*(1 - I*Sqrt[19] + 10*x)) + (800*x*Log[x]^2)/(19*(1 - I*Sqrt[19])*(1 - I*Sqrt[19
] + 10*x)) - (40*x*Log[x]^2)/(19*(1 + I*Sqrt[19] + 10*x)) + (800*x*Log[x]^2)/(19*(1 + I*Sqrt[19])*(1 + I*Sqrt[
19] + 10*x)) + (8*Log[x]*Log[1 + (10*x)/(1 - I*Sqrt[19])])/19 + ((8*I)*Log[x]*Log[1 + (10*x)/(1 - I*Sqrt[19])]
)/Sqrt[19] - (160*Log[x]*Log[1 + (10*x)/(1 - I*Sqrt[19])])/(19*(1 - I*Sqrt[19])) + (8*Log[x]*Log[1 + (10*x)/(1
+ I*Sqrt[19])])/19 - ((8*I)*Log[x]*Log[1 + (10*x)/(1 + I*Sqrt[19])])/Sqrt[19] - (160*Log[x]*Log[1 + (10*x)/(1
+ I*Sqrt[19])])/(19*(1 + I*Sqrt[19])) + (8*PolyLog[2, (-10*x)/(1 - I*Sqrt[19])])/19 + ((8*I)*PolyLog[2, (-10*
x)/(1 - I*Sqrt[19])])/Sqrt[19] - (160*PolyLog[2, (-10*x)/(1 - I*Sqrt[19])])/(19*(1 - I*Sqrt[19])) + (8*PolyLog
[2, (-10*x)/(1 + I*Sqrt[19])])/19 - ((8*I)*PolyLog[2, (-10*x)/(1 + I*Sqrt[19])])/Sqrt[19] - (160*PolyLog[2, (-
10*x)/(1 + I*Sqrt[19])])/(19*(1 + I*Sqrt[19]))

Rule 2354

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[
{a, b, c, d, e, n}, x] && IGtQ[p, 0]

Rule 2355

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((d_) + (e_.)*(x_))^2, x_Symbol] :> Simp[x*((a + b*Log[c*x^n])
^p/(d*(d + e*x))), x] - Dist[b*n*(p/d), Int[(a + b*Log[c*x^n])^(p - 1)/(d + e*x), x], x] /; FreeQ[{a, b, c, d,
e, n, p}, x] && GtQ[p, 0]

Rule 2404

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(RFx_), x_Symbol] :> With[{u = ExpandIntegrand[(a + b*Log[c*x^
n])^p, RFx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, n}, x] && RationalFunctionQ[RFx, x] && IGtQ[p, 0]

Rule 2421

Int[(Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.))/(x_), x_Symbol] :> Simp
[(-PolyLog[2, (-d)*f*x^m])*((a + b*Log[c*x^n])^p/m), x] + Dist[b*n*(p/m), Int[PolyLog[2, (-d)*f*x^m]*((a + b*L
og[c*x^n])^(p - 1)/x), x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && IGtQ[p, 0] && EqQ[d*e, 1]

Rule 2438

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2, (-c)*e*x^n]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 6724

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[PolyLog[n + 1, c*(a
+ b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*d, a*e]

Rule 6820

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

Rubi steps \begin{align*} \text {integral}& = \int \left (1-\frac {16}{x^2}-\frac {8 \log (x)}{1+x+5 x^2}+\frac {4 \left (-1+5 x^2\right ) \log ^2(x)}{\left (1+x+5 x^2\right )^2}\right ) \, dx \\ & = \frac {16}{x}+x+4 \int \frac {\left (-1+5 x^2\right ) \log ^2(x)}{\left (1+x+5 x^2\right )^2} \, dx-8 \int \frac {\log (x)}{1+x+5 x^2} \, dx \\ & = \frac {16}{x}+x+4 \int \left (\frac {(-2-x) \log ^2(x)}{\left (1+x+5 x^2\right )^2}+\frac {\log ^2(x)}{1+x+5 x^2}\right ) \, dx-8 \int \left (\frac {10 i \log (x)}{\sqrt {19} \left (-1+i \sqrt {19}-10 x\right )}+\frac {10 i \log (x)}{\sqrt {19} \left (1+i \sqrt {19}+10 x\right )}\right ) \, dx \\ & = \frac {16}{x}+x+4 \int \frac {(-2-x) \log ^2(x)}{\left (1+x+5 x^2\right )^2} \, dx+4 \int \frac {\log ^2(x)}{1+x+5 x^2} \, dx-\frac {(80 i) \int \frac {\log (x)}{-1+i \sqrt {19}-10 x} \, dx}{\sqrt {19}}-\frac {(80 i) \int \frac {\log (x)}{1+i \sqrt {19}+10 x} \, dx}{\sqrt {19}} \\ & = \frac {16}{x}+x+\frac {8 i \log (x) \log \left (1+\frac {10 x}{1-i \sqrt {19}}\right )}{\sqrt {19}}-\frac {8 i \log (x) \log \left (1+\frac {10 x}{1+i \sqrt {19}}\right )}{\sqrt {19}}+4 \int \left (\frac {10 i \log ^2(x)}{\sqrt {19} \left (-1+i \sqrt {19}-10 x\right )}+\frac {10 i \log ^2(x)}{\sqrt {19} \left (1+i \sqrt {19}+10 x\right )}\right ) \, dx+4 \int \left (-\frac {2 \log ^2(x)}{\left (1+x+5 x^2\right )^2}-\frac {x \log ^2(x)}{\left (1+x+5 x^2\right )^2}\right ) \, dx-\frac {(8 i) \int \frac {\log \left (1-\frac {10 x}{-1+i \sqrt {19}}\right )}{x} \, dx}{\sqrt {19}}+\frac {(8 i) \int \frac {\log \left (1+\frac {10 x}{1+i \sqrt {19}}\right )}{x} \, dx}{\sqrt {19}} \\ & = \frac {16}{x}+x+\frac {8 i \log (x) \log \left (1+\frac {10 x}{1-i \sqrt {19}}\right )}{\sqrt {19}}-\frac {8 i \log (x) \log \left (1+\frac {10 x}{1+i \sqrt {19}}\right )}{\sqrt {19}}+\frac {8 i \text {Li}_2\left (-\frac {10 x}{1-i \sqrt {19}}\right )}{\sqrt {19}}-\frac {8 i \text {Li}_2\left (-\frac {10 x}{1+i \sqrt {19}}\right )}{\sqrt {19}}-4 \int \frac {x \log ^2(x)}{\left (1+x+5 x^2\right )^2} \, dx-8 \int \frac {\log ^2(x)}{\left (1+x+5 x^2\right )^2} \, dx+\frac {(40 i) \int \frac {\log ^2(x)}{-1+i \sqrt {19}-10 x} \, dx}{\sqrt {19}}+\frac {(40 i) \int \frac {\log ^2(x)}{1+i \sqrt {19}+10 x} \, dx}{\sqrt {19}} \\ & = \frac {16}{x}+x+\frac {8 i \log (x) \log \left (1+\frac {10 x}{1-i \sqrt {19}}\right )}{\sqrt {19}}-\frac {4 i \log ^2(x) \log \left (1+\frac {10 x}{1-i \sqrt {19}}\right )}{\sqrt {19}}-\frac {8 i \log (x) \log \left (1+\frac {10 x}{1+i \sqrt {19}}\right )}{\sqrt {19}}+\frac {4 i \log ^2(x) \log \left (1+\frac {10 x}{1+i \sqrt {19}}\right )}{\sqrt {19}}+\frac {8 i \text {Li}_2\left (-\frac {10 x}{1-i \sqrt {19}}\right )}{\sqrt {19}}-\frac {8 i \text {Li}_2\left (-\frac {10 x}{1+i \sqrt {19}}\right )}{\sqrt {19}}-4 \int \left (-\frac {10 \left (-1+i \sqrt {19}\right ) \log ^2(x)}{19 \left (-1+i \sqrt {19}-10 x\right )^2}-\frac {10 i \log ^2(x)}{19 \sqrt {19} \left (-1+i \sqrt {19}-10 x\right )}-\frac {10 \left (-1-i \sqrt {19}\right ) \log ^2(x)}{19 \left (1+i \sqrt {19}+10 x\right )^2}-\frac {10 i \log ^2(x)}{19 \sqrt {19} \left (1+i \sqrt {19}+10 x\right )}\right ) \, dx-8 \int \left (-\frac {100 \log ^2(x)}{19 \left (-1+i \sqrt {19}-10 x\right )^2}+\frac {100 i \log ^2(x)}{19 \sqrt {19} \left (-1+i \sqrt {19}-10 x\right )}-\frac {100 \log ^2(x)}{19 \left (1+i \sqrt {19}+10 x\right )^2}+\frac {100 i \log ^2(x)}{19 \sqrt {19} \left (1+i \sqrt {19}+10 x\right )}\right ) \, dx+\frac {(8 i) \int \frac {\log (x) \log \left (1-\frac {10 x}{-1+i \sqrt {19}}\right )}{x} \, dx}{\sqrt {19}}-\frac {(8 i) \int \frac {\log (x) \log \left (1+\frac {10 x}{1+i \sqrt {19}}\right )}{x} \, dx}{\sqrt {19}} \\ & = \frac {16}{x}+x+\frac {8 i \log (x) \log \left (1+\frac {10 x}{1-i \sqrt {19}}\right )}{\sqrt {19}}-\frac {4 i \log ^2(x) \log \left (1+\frac {10 x}{1-i \sqrt {19}}\right )}{\sqrt {19}}-\frac {8 i \log (x) \log \left (1+\frac {10 x}{1+i \sqrt {19}}\right )}{\sqrt {19}}+\frac {4 i \log ^2(x) \log \left (1+\frac {10 x}{1+i \sqrt {19}}\right )}{\sqrt {19}}+\frac {8 i \text {Li}_2\left (-\frac {10 x}{1-i \sqrt {19}}\right )}{\sqrt {19}}-\frac {8 i \log (x) \text {Li}_2\left (-\frac {10 x}{1-i \sqrt {19}}\right )}{\sqrt {19}}-\frac {8 i \text {Li}_2\left (-\frac {10 x}{1+i \sqrt {19}}\right )}{\sqrt {19}}+\frac {8 i \log (x) \text {Li}_2\left (-\frac {10 x}{1+i \sqrt {19}}\right )}{\sqrt {19}}+\frac {800}{19} \int \frac {\log ^2(x)}{\left (-1+i \sqrt {19}-10 x\right )^2} \, dx+\frac {800}{19} \int \frac {\log ^2(x)}{\left (1+i \sqrt {19}+10 x\right )^2} \, dx+\frac {(40 i) \int \frac {\log ^2(x)}{-1+i \sqrt {19}-10 x} \, dx}{19 \sqrt {19}}+\frac {(40 i) \int \frac {\log ^2(x)}{1+i \sqrt {19}+10 x} \, dx}{19 \sqrt {19}}+\frac {(8 i) \int \frac {\text {Li}_2\left (\frac {10 x}{-1+i \sqrt {19}}\right )}{x} \, dx}{\sqrt {19}}-\frac {(8 i) \int \frac {\text {Li}_2\left (-\frac {10 x}{1+i \sqrt {19}}\right )}{x} \, dx}{\sqrt {19}}-\frac {(800 i) \int \frac {\log ^2(x)}{-1+i \sqrt {19}-10 x} \, dx}{19 \sqrt {19}}-\frac {(800 i) \int \frac {\log ^2(x)}{1+i \sqrt {19}+10 x} \, dx}{19 \sqrt {19}}-\frac {1}{19} \left (40 \left (1-i \sqrt {19}\right )\right ) \int \frac {\log ^2(x)}{\left (-1+i \sqrt {19}-10 x\right )^2} \, dx-\frac {1}{19} \left (40 \left (1+i \sqrt {19}\right )\right ) \int \frac {\log ^2(x)}{\left (1+i \sqrt {19}+10 x\right )^2} \, dx \\ & = \frac {16}{x}+x-\frac {40 x \log ^2(x)}{19 \left (1-i \sqrt {19}+10 x\right )}+\frac {800 x \log ^2(x)}{19 \left (1-i \sqrt {19}\right ) \left (1-i \sqrt {19}+10 x\right )}-\frac {40 x \log ^2(x)}{19 \left (1+i \sqrt {19}+10 x\right )}+\frac {800 x \log ^2(x)}{19 \left (1+i \sqrt {19}\right ) \left (1+i \sqrt {19}+10 x\right )}+\frac {8 i \log (x) \log \left (1+\frac {10 x}{1-i \sqrt {19}}\right )}{\sqrt {19}}-\frac {8 i \log (x) \log \left (1+\frac {10 x}{1+i \sqrt {19}}\right )}{\sqrt {19}}+\frac {8 i \text {Li}_2\left (-\frac {10 x}{1-i \sqrt {19}}\right )}{\sqrt {19}}-\frac {8 i \log (x) \text {Li}_2\left (-\frac {10 x}{1-i \sqrt {19}}\right )}{\sqrt {19}}-\frac {8 i \text {Li}_2\left (-\frac {10 x}{1+i \sqrt {19}}\right )}{\sqrt {19}}+\frac {8 i \log (x) \text {Li}_2\left (-\frac {10 x}{1+i \sqrt {19}}\right )}{\sqrt {19}}+\frac {8 i \text {Li}_3\left (-\frac {10 x}{1-i \sqrt {19}}\right )}{\sqrt {19}}-\frac {8 i \text {Li}_3\left (-\frac {10 x}{1+i \sqrt {19}}\right )}{\sqrt {19}}-\frac {80}{19} \int \frac {\log (x)}{-1+i \sqrt {19}-10 x} \, dx+\frac {80}{19} \int \frac {\log (x)}{1+i \sqrt {19}+10 x} \, dx+\frac {(8 i) \int \frac {\log (x) \log \left (1-\frac {10 x}{-1+i \sqrt {19}}\right )}{x} \, dx}{19 \sqrt {19}}-\frac {(8 i) \int \frac {\log (x) \log \left (1+\frac {10 x}{1+i \sqrt {19}}\right )}{x} \, dx}{19 \sqrt {19}}-\frac {(160 i) \int \frac {\log (x) \log \left (1-\frac {10 x}{-1+i \sqrt {19}}\right )}{x} \, dx}{19 \sqrt {19}}+\frac {(160 i) \int \frac {\log (x) \log \left (1+\frac {10 x}{1+i \sqrt {19}}\right )}{x} \, dx}{19 \sqrt {19}}+\frac {1600 \int \frac {\log (x)}{-1+i \sqrt {19}-10 x} \, dx}{19 \left (1-i \sqrt {19}\right )}-\frac {1600 \int \frac {\log (x)}{1+i \sqrt {19}+10 x} \, dx}{19 \left (1+i \sqrt {19}\right )} \\ & = \frac {16}{x}+x-\frac {40 x \log ^2(x)}{19 \left (1-i \sqrt {19}+10 x\right )}+\frac {800 x \log ^2(x)}{19 \left (1-i \sqrt {19}\right ) \left (1-i \sqrt {19}+10 x\right )}-\frac {40 x \log ^2(x)}{19 \left (1+i \sqrt {19}+10 x\right )}+\frac {800 x \log ^2(x)}{19 \left (1+i \sqrt {19}\right ) \left (1+i \sqrt {19}+10 x\right )}+\frac {8}{19} \log (x) \log \left (1+\frac {10 x}{1-i \sqrt {19}}\right )+\frac {8 i \log (x) \log \left (1+\frac {10 x}{1-i \sqrt {19}}\right )}{\sqrt {19}}-\frac {160 \log (x) \log \left (1+\frac {10 x}{1-i \sqrt {19}}\right )}{19 \left (1-i \sqrt {19}\right )}+\frac {8}{19} \log (x) \log \left (1+\frac {10 x}{1+i \sqrt {19}}\right )-\frac {8 i \log (x) \log \left (1+\frac {10 x}{1+i \sqrt {19}}\right )}{\sqrt {19}}-\frac {160 \log (x) \log \left (1+\frac {10 x}{1+i \sqrt {19}}\right )}{19 \left (1+i \sqrt {19}\right )}+\frac {8 i \text {Li}_2\left (-\frac {10 x}{1-i \sqrt {19}}\right )}{\sqrt {19}}-\frac {8 i \text {Li}_2\left (-\frac {10 x}{1+i \sqrt {19}}\right )}{\sqrt {19}}+\frac {8 i \text {Li}_3\left (-\frac {10 x}{1-i \sqrt {19}}\right )}{\sqrt {19}}-\frac {8 i \text {Li}_3\left (-\frac {10 x}{1+i \sqrt {19}}\right )}{\sqrt {19}}-\frac {8}{19} \int \frac {\log \left (1-\frac {10 x}{-1+i \sqrt {19}}\right )}{x} \, dx-\frac {8}{19} \int \frac {\log \left (1+\frac {10 x}{1+i \sqrt {19}}\right )}{x} \, dx+\frac {(8 i) \int \frac {\text {Li}_2\left (\frac {10 x}{-1+i \sqrt {19}}\right )}{x} \, dx}{19 \sqrt {19}}-\frac {(8 i) \int \frac {\text {Li}_2\left (-\frac {10 x}{1+i \sqrt {19}}\right )}{x} \, dx}{19 \sqrt {19}}-\frac {(160 i) \int \frac {\text {Li}_2\left (\frac {10 x}{-1+i \sqrt {19}}\right )}{x} \, dx}{19 \sqrt {19}}+\frac {(160 i) \int \frac {\text {Li}_2\left (-\frac {10 x}{1+i \sqrt {19}}\right )}{x} \, dx}{19 \sqrt {19}}+\frac {160 \int \frac {\log \left (1-\frac {10 x}{-1+i \sqrt {19}}\right )}{x} \, dx}{19 \left (1-i \sqrt {19}\right )}+\frac {160 \int \frac {\log \left (1+\frac {10 x}{1+i \sqrt {19}}\right )}{x} \, dx}{19 \left (1+i \sqrt {19}\right )} \\ & = \frac {16}{x}+x-\frac {40 x \log ^2(x)}{19 \left (1-i \sqrt {19}+10 x\right )}+\frac {800 x \log ^2(x)}{19 \left (1-i \sqrt {19}\right ) \left (1-i \sqrt {19}+10 x\right )}-\frac {40 x \log ^2(x)}{19 \left (1+i \sqrt {19}+10 x\right )}+\frac {800 x \log ^2(x)}{19 \left (1+i \sqrt {19}\right ) \left (1+i \sqrt {19}+10 x\right )}+\frac {8}{19} \log (x) \log \left (1+\frac {10 x}{1-i \sqrt {19}}\right )+\frac {8 i \log (x) \log \left (1+\frac {10 x}{1-i \sqrt {19}}\right )}{\sqrt {19}}-\frac {160 \log (x) \log \left (1+\frac {10 x}{1-i \sqrt {19}}\right )}{19 \left (1-i \sqrt {19}\right )}+\frac {8}{19} \log (x) \log \left (1+\frac {10 x}{1+i \sqrt {19}}\right )-\frac {8 i \log (x) \log \left (1+\frac {10 x}{1+i \sqrt {19}}\right )}{\sqrt {19}}-\frac {160 \log (x) \log \left (1+\frac {10 x}{1+i \sqrt {19}}\right )}{19 \left (1+i \sqrt {19}\right )}+\frac {8}{19} \text {Li}_2\left (-\frac {10 x}{1-i \sqrt {19}}\right )+\frac {8 i \text {Li}_2\left (-\frac {10 x}{1-i \sqrt {19}}\right )}{\sqrt {19}}-\frac {160 \text {Li}_2\left (-\frac {10 x}{1-i \sqrt {19}}\right )}{19 \left (1-i \sqrt {19}\right )}+\frac {8}{19} \text {Li}_2\left (-\frac {10 x}{1+i \sqrt {19}}\right )-\frac {8 i \text {Li}_2\left (-\frac {10 x}{1+i \sqrt {19}}\right )}{\sqrt {19}}-\frac {160 \text {Li}_2\left (-\frac {10 x}{1+i \sqrt {19}}\right )}{19 \left (1+i \sqrt {19}\right )} \\ \end{align*}

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.

Time = 1.08 (sec) , antiderivative size = 380, normalized size of antiderivative = 12.26 \[ \int \frac {-16-32 x-175 x^2-158 x^3-389 x^4+10 x^5+25 x^6+\left (-8 x^2-8 x^3-40 x^4\right ) \log (x)+\left (-4 x^2+20 x^4\right ) \log ^2(x)}{x^2+2 x^3+11 x^4+10 x^5+25 x^6} \, dx=\frac {16 \sqrt {19}+16 \sqrt {19} x+81 \sqrt {19} x^2+\sqrt {19} x^3+5 \sqrt {19} x^4-8 i x \log \left (\frac {-i+\sqrt {19}-10 i x}{-i+\sqrt {19}}\right ) \log (x)-8 i x^2 \log \left (\frac {-i+\sqrt {19}-10 i x}{-i+\sqrt {19}}\right ) \log (x)-40 i x^3 \log \left (\frac {-i+\sqrt {19}-10 i x}{-i+\sqrt {19}}\right ) \log (x)-4 \sqrt {19} x^2 \log ^2(x)+8 i x \log (x) \log \left (\frac {9 i+\sqrt {19}+5 \left (-i+\sqrt {19}\right ) x}{9 i+\sqrt {19}}\right )+8 i x^2 \log (x) \log \left (\frac {9 i+\sqrt {19}+5 \left (-i+\sqrt {19}\right ) x}{9 i+\sqrt {19}}\right )+40 i x^3 \log (x) \log \left (\frac {9 i+\sqrt {19}+5 \left (-i+\sqrt {19}\right ) x}{9 i+\sqrt {19}}\right )-8 i x \left (1+x+5 x^2\right ) \operatorname {PolyLog}\left (2,\frac {10 i x}{-i+\sqrt {19}}\right )+8 i x \left (1+x+5 x^2\right ) \operatorname {PolyLog}\left (2,-\frac {5 \left (-i+\sqrt {19}\right ) x}{9 i+\sqrt {19}}\right )}{\sqrt {19} x \left (1+x+5 x^2\right )} \]

[In]

Integrate[(-16 - 32*x - 175*x^2 - 158*x^3 - 389*x^4 + 10*x^5 + 25*x^6 + (-8*x^2 - 8*x^3 - 40*x^4)*Log[x] + (-4
*x^2 + 20*x^4)*Log[x]^2)/(x^2 + 2*x^3 + 11*x^4 + 10*x^5 + 25*x^6),x]

[Out]

(16*Sqrt[19] + 16*Sqrt[19]*x + 81*Sqrt[19]*x^2 + Sqrt[19]*x^3 + 5*Sqrt[19]*x^4 - (8*I)*x*Log[(-I + Sqrt[19] -
(10*I)*x)/(-I + Sqrt[19])]*Log[x] - (8*I)*x^2*Log[(-I + Sqrt[19] - (10*I)*x)/(-I + Sqrt[19])]*Log[x] - (40*I)*
x^3*Log[(-I + Sqrt[19] - (10*I)*x)/(-I + Sqrt[19])]*Log[x] - 4*Sqrt[19]*x^2*Log[x]^2 + (8*I)*x*Log[x]*Log[(9*I
+ Sqrt[19] + 5*(-I + Sqrt[19])*x)/(9*I + Sqrt[19])] + (8*I)*x^2*Log[x]*Log[(9*I + Sqrt[19] + 5*(-I + Sqrt[19]
)*x)/(9*I + Sqrt[19])] + (40*I)*x^3*Log[x]*Log[(9*I + Sqrt[19] + 5*(-I + Sqrt[19])*x)/(9*I + Sqrt[19])] - (8*I
)*x*(1 + x + 5*x^2)*PolyLog[2, ((10*I)*x)/(-I + Sqrt[19])] + (8*I)*x*(1 + x + 5*x^2)*PolyLog[2, (-5*(-I + Sqrt
[19])*x)/(9*I + Sqrt[19])])/(Sqrt[19]*x*(1 + x + 5*x^2))

Maple [A] (verified)

Time = 0.37 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.90

method	result	size
risch	\(-\frac {4 x \ln \left (x \right )^{2}}{5 x^{2}+x +1}+\frac {x^{2}+16}{x}\)	\(28\)
norman	\(\frac {16+\frac {79 x}{5}+\frac {404 x^{2}}{5}+5 x^{4}-4 x^{2} \ln \left (x \right )^{2}}{x \left (5 x^{2}+x +1\right )}\)	\(39\)
parallelrisch	\(\frac {25 x^{4}+80-20 x^{2} \ln \left (x \right )^{2}+404 x^{2}+79 x}{5 x \left (5 x^{2}+x +1\right )}\)	\(40\)

[In]

int(((20*x^4-4*x^2)*ln(x)^2+(-40*x^4-8*x^3-8*x^2)*ln(x)+25*x^6+10*x^5-389*x^4-158*x^3-175*x^2-32*x-16)/(25*x^6
+10*x^5+11*x^4+2*x^3+x^2),x,method=_RETURNVERBOSE)

[Out]

-4*x/(5*x^2+x+1)*ln(x)^2+(x^2+16)/x

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.29 \[ \int \frac {-16-32 x-175 x^2-158 x^3-389 x^4+10 x^5+25 x^6+\left (-8 x^2-8 x^3-40 x^4\right ) \log (x)+\left (-4 x^2+20 x^4\right ) \log ^2(x)}{x^2+2 x^3+11 x^4+10 x^5+25 x^6} \, dx=\frac {5 \, x^{4} - 4 \, x^{2} \log \left (x\right )^{2} + x^{3} + 81 \, x^{2} + 16 \, x + 16}{5 \, x^{3} + x^{2} + x} \]

[In]

integrate(((20*x^4-4*x^2)*log(x)^2+(-40*x^4-8*x^3-8*x^2)*log(x)+25*x^6+10*x^5-389*x^4-158*x^3-175*x^2-32*x-16)
/(25*x^6+10*x^5+11*x^4+2*x^3+x^2),x, algorithm="fricas")

[Out]

(5*x^4 - 4*x^2*log(x)^2 + x^3 + 81*x^2 + 16*x + 16)/(5*x^3 + x^2 + x)

Sympy [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.65 \[ \int \frac {-16-32 x-175 x^2-158 x^3-389 x^4+10 x^5+25 x^6+\left (-8 x^2-8 x^3-40 x^4\right ) \log (x)+\left (-4 x^2+20 x^4\right ) \log ^2(x)}{x^2+2 x^3+11 x^4+10 x^5+25 x^6} \, dx=x - \frac {4 x \log {\left (x \right )}^{2}}{5 x^{2} + x + 1} + \frac {16}{x} \]

[In]

integrate(((20*x**4-4*x**2)*ln(x)**2+(-40*x**4-8*x**3-8*x**2)*ln(x)+25*x**6+10*x**5-389*x**4-158*x**3-175*x**2
-32*x-16)/(25*x**6+10*x**5+11*x**4+2*x**3+x**2),x)

[Out]

x - 4*x*log(x)**2/(5*x**2 + x + 1) + 16/x

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 143 vs. \(2 (31) = 62\).

Time = 0.47 (sec) , antiderivative size = 143, normalized size of antiderivative = 4.61 \[ \int \frac {-16-32 x-175 x^2-158 x^3-389 x^4+10 x^5+25 x^6+\left (-8 x^2-8 x^3-40 x^4\right ) \log (x)+\left (-4 x^2+20 x^4\right ) \log ^2(x)}{x^2+2 x^3+11 x^4+10 x^5+25 x^6} \, dx=-\frac {4 \, x \log \left (x\right )^{2}}{5 \, x^{2} + x + 1} + x + \frac {16 \, {\left (140 \, x^{2} + 33 \, x + 19\right )}}{19 \, {\left (5 \, x^{3} + x^{2} + x\right )}} + \frac {31 \, x - 14}{95 \, {\left (5 \, x^{2} + x + 1\right )}} + \frac {2 \, {\left (14 \, x + 9\right )}}{95 \, {\left (5 \, x^{2} + x + 1\right )}} - \frac {175 \, {\left (10 \, x + 1\right )}}{19 \, {\left (5 \, x^{2} + x + 1\right )}} + \frac {389 \, {\left (9 \, x - 1\right )}}{95 \, {\left (5 \, x^{2} + x + 1\right )}} + \frac {32 \, {\left (5 \, x - 9\right )}}{19 \, {\left (5 \, x^{2} + x + 1\right )}} + \frac {158 \, {\left (x + 2\right )}}{19 \, {\left (5 \, x^{2} + x + 1\right )}} \]

[In]

integrate(((20*x^4-4*x^2)*log(x)^2+(-40*x^4-8*x^3-8*x^2)*log(x)+25*x^6+10*x^5-389*x^4-158*x^3-175*x^2-32*x-16)
/(25*x^6+10*x^5+11*x^4+2*x^3+x^2),x, algorithm="maxima")

[Out]

-4*x*log(x)^2/(5*x^2 + x + 1) + x + 16/19*(140*x^2 + 33*x + 19)/(5*x^3 + x^2 + x) + 1/95*(31*x - 14)/(5*x^2 +
x + 1) + 2/95*(14*x + 9)/(5*x^2 + x + 1) - 175/19*(10*x + 1)/(5*x^2 + x + 1) + 389/95*(9*x - 1)/(5*x^2 + x + 1
) + 32/19*(5*x - 9)/(5*x^2 + x + 1) + 158/19*(x + 2)/(5*x^2 + x + 1)

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.77 \[ \int \frac {-16-32 x-175 x^2-158 x^3-389 x^4+10 x^5+25 x^6+\left (-8 x^2-8 x^3-40 x^4\right ) \log (x)+\left (-4 x^2+20 x^4\right ) \log ^2(x)}{x^2+2 x^3+11 x^4+10 x^5+25 x^6} \, dx=-\frac {4 \, x \log \left (x\right )^{2}}{5 \, x^{2} + x + 1} + x + \frac {16}{x} \]

[In]

integrate(((20*x^4-4*x^2)*log(x)^2+(-40*x^4-8*x^3-8*x^2)*log(x)+25*x^6+10*x^5-389*x^4-158*x^3-175*x^2-32*x-16)
/(25*x^6+10*x^5+11*x^4+2*x^3+x^2),x, algorithm="giac")

[Out]

-4*x*log(x)^2/(5*x^2 + x + 1) + x + 16/x

Mupad [B] (verification not implemented)

Time = 9.24 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.10 \[ \int \frac {-16-32 x-175 x^2-158 x^3-389 x^4+10 x^5+25 x^6+\left (-8 x^2-8 x^3-40 x^4\right ) \log (x)+\left (-4 x^2+20 x^4\right ) \log ^2(x)}{x^2+2 x^3+11 x^4+10 x^5+25 x^6} \, dx=x+\frac {16\,x-x^2\,\left (4\,{\ln \left (x\right )}^2-80\right )+16}{x\,\left (5\,x^2+x+1\right )} \]

[In]

int(-(32*x + log(x)*(8*x^2 + 8*x^3 + 40*x^4) + log(x)^2*(4*x^2 - 20*x^4) + 175*x^2 + 158*x^3 + 389*x^4 - 10*x^
5 - 25*x^6 + 16)/(x^2 + 2*x^3 + 11*x^4 + 10*x^5 + 25*x^6),x)

[Out]

x + (16*x - x^2*(4*log(x)^2 - 80) + 16)/(x*(x + 5*x^2 + 1))