\(\int \frac {15+60 x-146 x^2+355 x^4+(30 x^2-146 x^4) \log (3)+15 x^4 \log ^2(3)+(6 x^2-10 x^4+2 x^4 \log (3)) \log (x^2)}{15-150 x^2+375 x^4+(30 x^2-150 x^4) \log (3)+15 x^4 \log ^2(3)} \, dx\) [1176]

Optimal result

Integrand size = 99, antiderivative size = 23 \[ \int \frac {15+60 x-146 x^2+355 x^4+\left (30 x^2-146 x^4\right ) \log (3)+15 x^4 \log ^2(3)+\left (6 x^2-10 x^4+2 x^4 \log (3)\right ) \log \left (x^2\right )}{15-150 x^2+375 x^4+\left (30 x^2-150 x^4\right ) \log (3)+15 x^4 \log ^2(3)} \, dx=x+\frac {2+\frac {2}{15} x \log \left (x^2\right )}{-5+\frac {1}{x^2}+\log (3)} \]

[Out]

x+(2/15*x*ln(x^2)+2)/(1/x^2-5+ln(3))

Rubi [C] (verified)

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

Time = 0.53 (sec) , antiderivative size = 545, normalized size of antiderivative = 23.70, number of steps used = 25, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.141, Rules used = {6, 6820, 12, 6874, 205, 212, 267, 294, 327, 2404, 2361, 6031, 2332, 2360} \[ \int \frac {15+60 x-146 x^2+355 x^4+\left (30 x^2-146 x^4\right ) \log (3)+15 x^4 \log ^2(3)+\left (6 x^2-10 x^4+2 x^4 \log (3)\right ) \log \left (x^2\right )}{15-150 x^2+375 x^4+\left (30 x^2-150 x^4\right ) \log (3)+15 x^4 \log ^2(3)} \, dx=\frac {\log (81) \log \left (x^2\right ) \text {arctanh}\left (x \sqrt {5-\log (3)}\right )}{30 (5-\log (3))^{3/2} \log (3)}-\frac {\log (9) \log \left (x^2\right ) \text {arctanh}\left (x \sqrt {5-\log (3)}\right )}{15 (5-\log (3))^{3/2} \log (3)}-\frac {\log (81) \text {arctanh}\left (x \sqrt {5-\log (3)}\right )}{15 (5-\log (3))^{3/2} \log (3)}+\frac {\text {arctanh}\left (x \sqrt {5-\log (3)}\right )}{2 \sqrt {5-\log (3)}}+\frac {(73-15 \log (3)) \text {arctanh}\left (x \sqrt {5-\log (3)}\right )}{15 (5-\log (3))^{3/2}}-\frac {(71-15 \log (3)) \text {arctanh}\left (x \sqrt {5-\log (3)}\right )}{10 (5-\log (3))^{3/2}}+\frac {\log (81) \operatorname {PolyLog}\left (2,-x \sqrt {5-\log (3)}\right )}{30 (5-\log (3))^{3/2} \log (3)}-\frac {\log (9) \operatorname {PolyLog}\left (2,-x \sqrt {5-\log (3)}\right )}{15 (5-\log (3))^{3/2} \log (3)}-\frac {\log (81) \operatorname {PolyLog}\left (2,x \sqrt {5-\log (3)}\right )}{30 (5-\log (3))^{3/2} \log (3)}+\frac {\log (9) \operatorname {PolyLog}\left (2,x \sqrt {5-\log (3)}\right )}{15 (5-\log (3))^{3/2} \log (3)}+\frac {x (20-\log (81)) \log \left (x^2\right )}{30 (5-\log (3))^2 \left (1-x^2 (5-\log (3))\right )}-\frac {x (10-\log (9)) \log \left (x^2\right )}{15 (5-\log (3))^2}+\frac {x}{2 \left (1-x^2 (5-\log (3))\right )}-\frac {x (73-15 \log (3))}{15 (5-\log (3)) \left (1-x^2 (5-\log (3))\right )}+\frac {2}{(5-\log (3)) \left (1-x^2 (5-\log (3))\right )}+\frac {x^3 (71-15 \log (3))}{30 \left (1-x^2 (5-\log (3))\right )}+\frac {2 x (10-\log (9))}{15 (5-\log (3))^2}+\frac {x (71-15 \log (3))}{10 (5-\log (3))} \]

[In]

Int[(15 + 60*x - 146*x^2 + 355*x^4 + (30*x^2 - 146*x^4)*Log[3] + 15*x^4*Log[3]^2 + (6*x^2 - 10*x^4 + 2*x^4*Log
[3])*Log[x^2])/(15 - 150*x^2 + 375*x^4 + (30*x^2 - 150*x^4)*Log[3] + 15*x^4*Log[3]^2),x]

[Out]

x/(2*(1 - x^2*(5 - Log[3]))) + (x^3*(71 - 15*Log[3]))/(30*(1 - x^2*(5 - Log[3]))) - (ArcTanh[x*Sqrt[5 - Log[3]
]]*(71 - 15*Log[3]))/(10*(5 - Log[3])^(3/2)) + (ArcTanh[x*Sqrt[5 - Log[3]]]*(73 - 15*Log[3]))/(15*(5 - Log[3])
^(3/2)) + 2/((1 - x^2*(5 - Log[3]))*(5 - Log[3])) + (x*(71 - 15*Log[3]))/(10*(5 - Log[3])) - (x*(73 - 15*Log[3
]))/(15*(1 - x^2*(5 - Log[3]))*(5 - Log[3])) + ArcTanh[x*Sqrt[5 - Log[3]]]/(2*Sqrt[5 - Log[3]]) + (2*x*(10 - L
og[9]))/(15*(5 - Log[3])^2) - (ArcTanh[x*Sqrt[5 - Log[3]]]*Log[81])/(15*(5 - Log[3])^(3/2)*Log[3]) - (x*(10 -
Log[9])*Log[x^2])/(15*(5 - Log[3])^2) - (ArcTanh[x*Sqrt[5 - Log[3]]]*Log[9]*Log[x^2])/(15*(5 - Log[3])^(3/2)*L
og[3]) + (x*(20 - Log[81])*Log[x^2])/(30*(1 - x^2*(5 - Log[3]))*(5 - Log[3])^2) + (ArcTanh[x*Sqrt[5 - Log[3]]]
*Log[81]*Log[x^2])/(30*(5 - Log[3])^(3/2)*Log[3]) - (Log[9]*PolyLog[2, -(x*Sqrt[5 - Log[3]])])/(15*(5 - Log[3]
)^(3/2)*Log[3]) + (Log[81]*PolyLog[2, -(x*Sqrt[5 - Log[3]])])/(30*(5 - Log[3])^(3/2)*Log[3]) + (Log[9]*PolyLog
[2, x*Sqrt[5 - Log[3]]])/(15*(5 - Log[3])^(3/2)*Log[3]) - (Log[81]*PolyLog[2, x*Sqrt[5 - Log[3]]])/(30*(5 - Lo
g[3])^(3/2)*Log[3])

Rule 6

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

Rule 12

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

Rule 205

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^n)^(p + 1)/(a*n*(p + 1))), x] + Dist[(n*(p
+ 1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[p, -1] && (
IntegerQ[2*p] || (n == 2 && IntegerQ[4*p]) || (n == 2 && IntegerQ[3*p]) || Denominator[p + 1/n] < Denominator[
p])

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 267

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

Rule 294

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^(n - 1)*(c*x)^(m - n + 1)*((a + b*x^
n)^(p + 1)/(b*n*(p + 1))), x] - Dist[c^n*((m - n + 1)/(b*n*(p + 1))), Int[(c*x)^(m - n)*(a + b*x^n)^(p + 1), x
], x] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m + 1, n] &&  !ILtQ[(m + n*(p + 1) + 1)/n, 0]
&& IntBinomialQ[a, b, c, n, m, p, x]

Rule 327

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^(n - 1)*(c*x)^(m - n + 1)*((a + b*x^n
)^(p + 1)/(b*(m + n*p + 1))), x] - Dist[a*c^n*((m - n + 1)/(b*(m + n*p + 1))), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

Rule 2332

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{c, n}, x]

Rule 2360

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_) + (e_.)*(x_)^2)^(q_), x_Symbol] :> Simp[(-x)*(d + e*x^2)^(q +
1)*((a + b*Log[c*x^n])/(2*d*(q + 1))), x] + (Dist[(2*q + 3)/(2*d*(q + 1)), Int[(d + e*x^2)^(q + 1)*(a + b*Log[
c*x^n]), x], x] + Dist[b*(n/(2*d*(q + 1))), Int[(d + e*x^2)^(q + 1), x], x]) /; FreeQ[{a, b, c, d, e, n}, x] &
& LtQ[q, -1]

Rule 2361

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

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 6031

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))/(x_), x_Symbol] :> Simp[a*Log[x], x] + (-Simp[(b/2)*PolyLog[2, (-c)*x]
, x] + Simp[(b/2)*PolyLog[2, c*x], x]) /; FreeQ[{a, b, c}, x]

Rule 6820

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

Rule 6874

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

Rubi steps \begin{align*} \text {integral}& = \int \frac {15+60 x-146 x^2+355 x^4+\left (30 x^2-146 x^4\right ) \log (3)+15 x^4 \log ^2(3)+\left (6 x^2-10 x^4+2 x^4 \log (3)\right ) \log \left (x^2\right )}{15-150 x^2+\left (30 x^2-150 x^4\right ) \log (3)+x^4 \left (375+15 \log ^2(3)\right )} \, dx \\ & = \int \frac {15+60 x-146 x^2+\left (30 x^2-146 x^4\right ) \log (3)+x^4 \left (355+15 \log ^2(3)\right )+\left (6 x^2-10 x^4+2 x^4 \log (3)\right ) \log \left (x^2\right )}{15-150 x^2+\left (30 x^2-150 x^4\right ) \log (3)+x^4 \left (375+15 \log ^2(3)\right )} \, dx \\ & = \int \frac {15+60 x+2 x^2 (-73+15 \log (3))+x^4 \left (355-146 \log (3)+15 \log ^2(3)\right )+x^2 \left (6+x^2 (-10+\log (9))\right ) \log \left (x^2\right )}{15 \left (1+x^2 (-5+\log (3))\right )^2} \, dx \\ & = \frac {1}{15} \int \frac {15+60 x+2 x^2 (-73+15 \log (3))+x^4 \left (355-146 \log (3)+15 \log ^2(3)\right )+x^2 \left (6+x^2 (-10+\log (9))\right ) \log \left (x^2\right )}{\left (1+x^2 (-5+\log (3))\right )^2} \, dx \\ & = \frac {1}{15} \int \left (\frac {15}{\left (1-x^2 (5-\log (3))\right )^2}+\frac {60 x}{\left (1-x^2 (5-\log (3))\right )^2}+\frac {x^4 (71-15 \log (3)) (5-\log (3))}{\left (1-x^2 (5-\log (3))\right )^2}+\frac {2 x^2 (-73+15 \log (3))}{\left (1-x^2 (5-\log (3))\right )^2}+\frac {x^2 \left (6-x^2 (10-\log (9))\right ) \log \left (x^2\right )}{\left (1-x^2 (5-\log (3))\right )^2}\right ) \, dx \\ & = \frac {1}{15} \int \frac {x^2 \left (6-x^2 (10-\log (9))\right ) \log \left (x^2\right )}{\left (1-x^2 (5-\log (3))\right )^2} \, dx+4 \int \frac {x}{\left (1-x^2 (5-\log (3))\right )^2} \, dx-\frac {1}{15} (2 (73-15 \log (3))) \int \frac {x^2}{\left (1-x^2 (5-\log (3))\right )^2} \, dx+\frac {1}{15} ((71-15 \log (3)) (5-\log (3))) \int \frac {x^4}{\left (1-x^2 (5-\log (3))\right )^2} \, dx+\int \frac {1}{\left (1-x^2 (5-\log (3))\right )^2} \, dx \\ & = \frac {x}{2 \left (1-x^2 (5-\log (3))\right )}+\frac {x^3 (71-15 \log (3))}{30 \left (1-x^2 (5-\log (3))\right )}+\frac {2}{\left (1-x^2 (5-\log (3))\right ) (5-\log (3))}-\frac {x (73-15 \log (3))}{15 \left (1-x^2 (5-\log (3))\right ) (5-\log (3))}+\frac {1}{15} \int \left (\frac {(-10+\log (9)) \log \left (x^2\right )}{\left (1-x^2 (5-\log (3))\right ) (5-\log (3))^2}+\frac {(-10+\log (9)) \log \left (x^2\right )}{(-5+\log (3))^2}+\frac {(20-\log (81)) \log \left (x^2\right )}{\left (1-x^2 (5-\log (3))\right )^2 (5-\log (3))^2}\right ) \, dx+\frac {1}{2} \int \frac {1}{1+x^2 (-5+\log (3))} \, dx+\frac {(73-15 \log (3)) \int \frac {1}{1+x^2 (-5+\log (3))} \, dx}{15 (5-\log (3))}+\frac {1}{10} (-71+15 \log (3)) \int \frac {x^2}{1+x^2 (-5+\log (3))} \, dx \\ & = \frac {x}{2 \left (1-x^2 (5-\log (3))\right )}+\frac {x^3 (71-15 \log (3))}{30 \left (1-x^2 (5-\log (3))\right )}+\frac {\text {arctanh}\left (x \sqrt {5-\log (3)}\right ) (73-15 \log (3))}{15 (5-\log (3))^{3/2}}+\frac {2}{\left (1-x^2 (5-\log (3))\right ) (5-\log (3))}+\frac {x (71-15 \log (3))}{10 (5-\log (3))}-\frac {x (73-15 \log (3))}{15 \left (1-x^2 (5-\log (3))\right ) (5-\log (3))}+\frac {\text {arctanh}\left (x \sqrt {5-\log (3)}\right )}{2 \sqrt {5-\log (3)}}-\frac {(71-15 \log (3)) \int \frac {1}{1+x^2 (-5+\log (3))} \, dx}{10 (5-\log (3))}-\frac {(10-\log (9)) \int \log \left (x^2\right ) \, dx}{15 (5-\log (3))^2}-\frac {(10-\log (9)) \int \frac {\log \left (x^2\right )}{1+x^2 (-5+\log (3))} \, dx}{15 (5-\log (3))^2}+\frac {(20-\log (81)) \int \frac {\log \left (x^2\right )}{\left (1+x^2 (-5+\log (3))\right )^2} \, dx}{15 (5-\log (3))^2} \\ & = \frac {x}{2 \left (1-x^2 (5-\log (3))\right )}+\frac {x^3 (71-15 \log (3))}{30 \left (1-x^2 (5-\log (3))\right )}-\frac {\text {arctanh}\left (x \sqrt {5-\log (3)}\right ) (71-15 \log (3))}{10 (5-\log (3))^{3/2}}+\frac {\text {arctanh}\left (x \sqrt {5-\log (3)}\right ) (73-15 \log (3))}{15 (5-\log (3))^{3/2}}+\frac {2}{\left (1-x^2 (5-\log (3))\right ) (5-\log (3))}+\frac {x (71-15 \log (3))}{10 (5-\log (3))}-\frac {x (73-15 \log (3))}{15 \left (1-x^2 (5-\log (3))\right ) (5-\log (3))}+\frac {\text {arctanh}\left (x \sqrt {5-\log (3)}\right )}{2 \sqrt {5-\log (3)}}+\frac {2 x (10-\log (9))}{15 (5-\log (3))^2}-\frac {x (10-\log (9)) \log \left (x^2\right )}{15 (5-\log (3))^2}-\frac {\text {arctanh}\left (x \sqrt {5-\log (3)}\right ) \log (9) \log \left (x^2\right )}{15 (5-\log (3))^{3/2} \log (3)}+\frac {x (20-\log (81)) \log \left (x^2\right )}{30 \left (1-x^2 (5-\log (3))\right ) (5-\log (3))^2}+\frac {(2 (10-\log (9))) \int \frac {\text {arctanh}\left (x \sqrt {5-\log (3)}\right )}{x \sqrt {5-\log (3)}} \, dx}{15 (5-\log (3))^2}+\frac {(20-\log (81)) \int \frac {\log \left (x^2\right )}{1+x^2 (-5+\log (3))} \, dx}{30 (5-\log (3))^2}-\frac {(20-\log (81)) \int \frac {1}{1+x^2 (-5+\log (3))} \, dx}{15 (5-\log (3))^2} \\ & = \frac {x}{2 \left (1-x^2 (5-\log (3))\right )}+\frac {x^3 (71-15 \log (3))}{30 \left (1-x^2 (5-\log (3))\right )}-\frac {\text {arctanh}\left (x \sqrt {5-\log (3)}\right ) (71-15 \log (3))}{10 (5-\log (3))^{3/2}}+\frac {\text {arctanh}\left (x \sqrt {5-\log (3)}\right ) (73-15 \log (3))}{15 (5-\log (3))^{3/2}}+\frac {2}{\left (1-x^2 (5-\log (3))\right ) (5-\log (3))}+\frac {x (71-15 \log (3))}{10 (5-\log (3))}-\frac {x (73-15 \log (3))}{15 \left (1-x^2 (5-\log (3))\right ) (5-\log (3))}+\frac {\text {arctanh}\left (x \sqrt {5-\log (3)}\right )}{2 \sqrt {5-\log (3)}}+\frac {2 x (10-\log (9))}{15 (5-\log (3))^2}-\frac {\text {arctanh}\left (x \sqrt {5-\log (3)}\right ) \log (81)}{15 (5-\log (3))^{3/2} \log (3)}-\frac {x (10-\log (9)) \log \left (x^2\right )}{15 (5-\log (3))^2}-\frac {\text {arctanh}\left (x \sqrt {5-\log (3)}\right ) \log (9) \log \left (x^2\right )}{15 (5-\log (3))^{3/2} \log (3)}+\frac {x (20-\log (81)) \log \left (x^2\right )}{30 \left (1-x^2 (5-\log (3))\right ) (5-\log (3))^2}+\frac {\text {arctanh}\left (x \sqrt {5-\log (3)}\right ) \log (81) \log \left (x^2\right )}{30 (5-\log (3))^{3/2} \log (3)}+\frac {(2 \log (9)) \int \frac {\text {arctanh}\left (x \sqrt {5-\log (3)}\right )}{x} \, dx}{15 (5-\log (3))^{3/2} \log (3)}-\frac {(20-\log (81)) \int \frac {\text {arctanh}\left (x \sqrt {5-\log (3)}\right )}{x \sqrt {5-\log (3)}} \, dx}{15 (5-\log (3))^2} \\ & = \frac {x}{2 \left (1-x^2 (5-\log (3))\right )}+\frac {x^3 (71-15 \log (3))}{30 \left (1-x^2 (5-\log (3))\right )}-\frac {\text {arctanh}\left (x \sqrt {5-\log (3)}\right ) (71-15 \log (3))}{10 (5-\log (3))^{3/2}}+\frac {\text {arctanh}\left (x \sqrt {5-\log (3)}\right ) (73-15 \log (3))}{15 (5-\log (3))^{3/2}}+\frac {2}{\left (1-x^2 (5-\log (3))\right ) (5-\log (3))}+\frac {x (71-15 \log (3))}{10 (5-\log (3))}-\frac {x (73-15 \log (3))}{15 \left (1-x^2 (5-\log (3))\right ) (5-\log (3))}+\frac {\text {arctanh}\left (x \sqrt {5-\log (3)}\right )}{2 \sqrt {5-\log (3)}}+\frac {2 x (10-\log (9))}{15 (5-\log (3))^2}-\frac {\text {arctanh}\left (x \sqrt {5-\log (3)}\right ) \log (81)}{15 (5-\log (3))^{3/2} \log (3)}-\frac {x (10-\log (9)) \log \left (x^2\right )}{15 (5-\log (3))^2}-\frac {\text {arctanh}\left (x \sqrt {5-\log (3)}\right ) \log (9) \log \left (x^2\right )}{15 (5-\log (3))^{3/2} \log (3)}+\frac {x (20-\log (81)) \log \left (x^2\right )}{30 \left (1-x^2 (5-\log (3))\right ) (5-\log (3))^2}+\frac {\text {arctanh}\left (x \sqrt {5-\log (3)}\right ) \log (81) \log \left (x^2\right )}{30 (5-\log (3))^{3/2} \log (3)}-\frac {\log (9) \operatorname {PolyLog}\left (2,-x \sqrt {5-\log (3)}\right )}{15 (5-\log (3))^{3/2} \log (3)}+\frac {\log (9) \operatorname {PolyLog}\left (2,x \sqrt {5-\log (3)}\right )}{15 (5-\log (3))^{3/2} \log (3)}-\frac {\log (81) \int \frac {\text {arctanh}\left (x \sqrt {5-\log (3)}\right )}{x} \, dx}{15 (5-\log (3))^{3/2} \log (3)} \\ & = \frac {x}{2 \left (1-x^2 (5-\log (3))\right )}+\frac {x^3 (71-15 \log (3))}{30 \left (1-x^2 (5-\log (3))\right )}-\frac {\text {arctanh}\left (x \sqrt {5-\log (3)}\right ) (71-15 \log (3))}{10 (5-\log (3))^{3/2}}+\frac {\text {arctanh}\left (x \sqrt {5-\log (3)}\right ) (73-15 \log (3))}{15 (5-\log (3))^{3/2}}+\frac {2}{\left (1-x^2 (5-\log (3))\right ) (5-\log (3))}+\frac {x (71-15 \log (3))}{10 (5-\log (3))}-\frac {x (73-15 \log (3))}{15 \left (1-x^2 (5-\log (3))\right ) (5-\log (3))}+\frac {\text {arctanh}\left (x \sqrt {5-\log (3)}\right )}{2 \sqrt {5-\log (3)}}+\frac {2 x (10-\log (9))}{15 (5-\log (3))^2}-\frac {\text {arctanh}\left (x \sqrt {5-\log (3)}\right ) \log (81)}{15 (5-\log (3))^{3/2} \log (3)}-\frac {x (10-\log (9)) \log \left (x^2\right )}{15 (5-\log (3))^2}-\frac {\text {arctanh}\left (x \sqrt {5-\log (3)}\right ) \log (9) \log \left (x^2\right )}{15 (5-\log (3))^{3/2} \log (3)}+\frac {x (20-\log (81)) \log \left (x^2\right )}{30 \left (1-x^2 (5-\log (3))\right ) (5-\log (3))^2}+\frac {\text {arctanh}\left (x \sqrt {5-\log (3)}\right ) \log (81) \log \left (x^2\right )}{30 (5-\log (3))^{3/2} \log (3)}-\frac {\log (9) \operatorname {PolyLog}\left (2,-x \sqrt {5-\log (3)}\right )}{15 (5-\log (3))^{3/2} \log (3)}+\frac {\log (81) \operatorname {PolyLog}\left (2,-x \sqrt {5-\log (3)}\right )}{30 (5-\log (3))^{3/2} \log (3)}+\frac {\log (9) \operatorname {PolyLog}\left (2,x \sqrt {5-\log (3)}\right )}{15 (5-\log (3))^{3/2} \log (3)}-\frac {\log (81) \operatorname {PolyLog}\left (2,x \sqrt {5-\log (3)}\right )}{30 (5-\log (3))^{3/2} \log (3)} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(250\) vs. \(2(23)=46\).

Time = 0.64 (sec) , antiderivative size = 250, normalized size of antiderivative = 10.87 \[ \int \frac {15+60 x-146 x^2+355 x^4+\left (30 x^2-146 x^4\right ) \log (3)+15 x^4 \log ^2(3)+\left (6 x^2-10 x^4+2 x^4 \log (3)\right ) \log \left (x^2\right )}{15-150 x^2+375 x^4+\left (30 x^2-150 x^4\right ) \log (3)+15 x^4 \log ^2(3)} \, dx=-\frac {-150-375 x+1875 x^3+30 \log (3)+150 x \log (3)-1125 x^3 \log (3)-15 x \log ^2(3)+225 x^3 \log ^2(3)-15 x^3 \log ^3(3)-2 x^3 (-5+\log (3))^2 \log \left (x^2\right )+\sqrt {5-\log (3)} \left (2+x^2 (-10+\log (9))\right ) \log \left (1-x \sqrt {5-\log (3)}\right )-2 \sqrt {5-\log (3)} \log \left (\sqrt {5-\log (3)}+x (-5+\log (3))\right )+10 x^2 \sqrt {5-\log (3)} \log \left (\sqrt {5-\log (3)}+x (-5+\log (3))\right )-2 x^2 \sqrt {5-\log (3)} \log (3) \log \left (\sqrt {5-\log (3)}+x (-5+\log (3))\right )}{15 \left (1+x \sqrt {5-\log (3)}\right ) \left (\sqrt {5-\log (3)}+x (-5+\log (3))\right ) (5-\log (3))^{3/2}} \]

[In]

Integrate[(15 + 60*x - 146*x^2 + 355*x^4 + (30*x^2 - 146*x^4)*Log[3] + 15*x^4*Log[3]^2 + (6*x^2 - 10*x^4 + 2*x
^4*Log[3])*Log[x^2])/(15 - 150*x^2 + 375*x^4 + (30*x^2 - 150*x^4)*Log[3] + 15*x^4*Log[3]^2),x]

[Out]

-1/15*(-150 - 375*x + 1875*x^3 + 30*Log[3] + 150*x*Log[3] - 1125*x^3*Log[3] - 15*x*Log[3]^2 + 225*x^3*Log[3]^2
- 15*x^3*Log[3]^3 - 2*x^3*(-5 + Log[3])^2*Log[x^2] + Sqrt[5 - Log[3]]*(2 + x^2*(-10 + Log[9]))*Log[1 - x*Sqrt
[5 - Log[3]]] - 2*Sqrt[5 - Log[3]]*Log[Sqrt[5 - Log[3]] + x*(-5 + Log[3])] + 10*x^2*Sqrt[5 - Log[3]]*Log[Sqrt[
5 - Log[3]] + x*(-5 + Log[3])] - 2*x^2*Sqrt[5 - Log[3]]*Log[3]*Log[Sqrt[5 - Log[3]] + x*(-5 + Log[3])])/((1 +
x*Sqrt[5 - Log[3]])*(Sqrt[5 - Log[3]] + x*(-5 + Log[3]))*(5 - Log[3])^(3/2))

Maple [A] (verified)

Time = 0.24 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.78

method	result	size
norman	\(\frac {x +\left (\ln \left (3\right )-5\right ) x^{3}+2 x^{2}+\frac {2 x^{3} \ln \left (x^{2}\right )}{15}}{x^{2} \ln \left (3\right )-5 x^{2}+1}\)	\(41\)
parallelrisch	\(\frac {30 x^{3} \ln \left (3\right )+4 x^{3} \ln \left (x^{2}\right )-150 x^{3}+60 x^{2}+30 x}{30 x^{2} \ln \left (3\right )-150 x^{2}+30}\)	\(48\)
risch	\(\frac {2 x^{3} \ln \left (x^{2}\right )}{15 \left (x^{2} \ln \left (3\right )-5 x^{2}+1\right )}+\frac {x^{3} \ln \left (3\right )^{2}-10 x^{3} \ln \left (3\right )+25 x^{3}+x \ln \left (3\right )-5 x -2}{\left (x^{2} \ln \left (3\right )-5 x^{2}+1\right ) \left (\ln \left (3\right )-5\right )}\)	\(77\)

[In]

int(((2*x^4*ln(3)-10*x^4+6*x^2)*ln(x^2)+15*x^4*ln(3)^2+(-146*x^4+30*x^2)*ln(3)+355*x^4-146*x^2+60*x+15)/(15*x^
4*ln(3)^2+(-150*x^4+30*x^2)*ln(3)+375*x^4-150*x^2+15),x,method=_RETURNVERBOSE)

[Out]

(x+(ln(3)-5)*x^3+2*x^2+2/15*x^3*ln(x^2))/(x^2*ln(3)-5*x^2+1)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 80 vs. \(2 (21) = 42\).

Time = 0.25 (sec) , antiderivative size = 80, normalized size of antiderivative = 3.48 \[ \int \frac {15+60 x-146 x^2+355 x^4+\left (30 x^2-146 x^4\right ) \log (3)+15 x^4 \log ^2(3)+\left (6 x^2-10 x^4+2 x^4 \log (3)\right ) \log \left (x^2\right )}{15-150 x^2+375 x^4+\left (30 x^2-150 x^4\right ) \log (3)+15 x^4 \log ^2(3)} \, dx=\frac {15 \, x^{3} \log \left (3\right )^{2} + 375 \, x^{3} - 15 \, {\left (10 \, x^{3} - x\right )} \log \left (3\right ) + 2 \, {\left (x^{3} \log \left (3\right ) - 5 \, x^{3}\right )} \log \left (x^{2}\right ) - 75 \, x - 30}{15 \, {\left (x^{2} \log \left (3\right )^{2} + 25 \, x^{2} - {\left (10 \, x^{2} - 1\right )} \log \left (3\right ) - 5\right )}} \]

[In]

integrate(((2*x^4*log(3)-10*x^4+6*x^2)*log(x^2)+15*x^4*log(3)^2+(-146*x^4+30*x^2)*log(3)+355*x^4-146*x^2+60*x+
15)/(15*x^4*log(3)^2+(-150*x^4+30*x^2)*log(3)+375*x^4-150*x^2+15),x, algorithm="fricas")

[Out]

1/15*(15*x^3*log(3)^2 + 375*x^3 - 15*(10*x^3 - x)*log(3) + 2*(x^3*log(3) - 5*x^3)*log(x^2) - 75*x - 30)/(x^2*l
og(3)^2 + 25*x^2 - (10*x^2 - 1)*log(3) - 5)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 48 vs. \(2 (22) = 44\).

Time = 0.30 (sec) , antiderivative size = 48, normalized size of antiderivative = 2.09 \[ \int \frac {15+60 x-146 x^2+355 x^4+\left (30 x^2-146 x^4\right ) \log (3)+15 x^4 \log ^2(3)+\left (6 x^2-10 x^4+2 x^4 \log (3)\right ) \log \left (x^2\right )}{15-150 x^2+375 x^4+\left (30 x^2-150 x^4\right ) \log (3)+15 x^4 \log ^2(3)} \, dx=\frac {2 x^{3} \log {\left (x^{2} \right )}}{- 75 x^{2} + 15 x^{2} \log {\left (3 \right )} + 15} + x - \frac {2}{x^{2} \left (- 10 \log {\left (3 \right )} + \log {\left (3 \right )}^{2} + 25\right ) - 5 + \log {\left (3 \right )}} \]

[In]

integrate(((2*x**4*ln(3)-10*x**4+6*x**2)*ln(x**2)+15*x**4*ln(3)**2+(-146*x**4+30*x**2)*ln(3)+355*x**4-146*x**2
+60*x+15)/(15*x**4*ln(3)**2+(-150*x**4+30*x**2)*ln(3)+375*x**4-150*x**2+15),x)

[Out]

2*x**3*log(x**2)/(-75*x**2 + 15*x**2*log(3) + 15) + x - 2/(x**2*(-10*log(3) + log(3)**2 + 25) - 5 + log(3))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 620 vs. \(2 (21) = 42\).

Time = 0.33 (sec) , antiderivative size = 620, normalized size of antiderivative = 26.96 \[ \int \frac {15+60 x-146 x^2+355 x^4+\left (30 x^2-146 x^4\right ) \log (3)+15 x^4 \log ^2(3)+\left (6 x^2-10 x^4+2 x^4 \log (3)\right ) \log \left (x^2\right )}{15-150 x^2+375 x^4+\left (30 x^2-150 x^4\right ) \log (3)+15 x^4 \log ^2(3)} \, dx=\text {Too large to display} \]

[In]

integrate(((2*x^4*log(3)-10*x^4+6*x^2)*log(x^2)+15*x^4*log(3)^2+(-146*x^4+30*x^2)*log(3)+355*x^4-146*x^2+60*x+
15)/(15*x^4*log(3)^2+(-150*x^4+30*x^2)*log(3)+375*x^4-150*x^2+15),x, algorithm="maxima")

[Out]

1/4*(2*x/((log(3)^3 - 15*log(3)^2 + 75*log(3) - 125)*x^2 + log(3)^2 - 10*log(3) + 25) + 4*x/(log(3)^2 - 10*log
(3) + 25) - 3*log((x*(log(3) - 5) - sqrt(-log(3) + 5))/(x*(log(3) - 5) + sqrt(-log(3) + 5)))/((log(3)^2 - 10*l
og(3) + 25)*sqrt(-log(3) + 5)))*log(3)^2 - 73/30*(2*x/((log(3)^3 - 15*log(3)^2 + 75*log(3) - 125)*x^2 + log(3)
^2 - 10*log(3) + 25) + 4*x/(log(3)^2 - 10*log(3) + 25) - 3*log((x*(log(3) - 5) - sqrt(-log(3) + 5))/(x*(log(3)
- 5) + sqrt(-log(3) + 5)))/((log(3)^2 - 10*log(3) + 25)*sqrt(-log(3) + 5)))*log(3) - 1/2*(2*x/((log(3)^2 - 10
*log(3) + 25)*x^2 + log(3) - 5) - log((x*(log(3) - 5) - sqrt(-log(3) + 5))/(x*(log(3) - 5) + sqrt(-log(3) + 5)
))/((log(3) - 5)*sqrt(-log(3) + 5)))*log(3) + 1/4*log((x*(log(3) - 5) - sqrt(-log(3) + 5))/(x*(log(3) - 5) + s
qrt(-log(3) + 5)))/sqrt(-log(3) + 5) + 4/15*(x^3*(log(3) - 5)*log(x) - x^3*(log(3) - 5) - x)/((log(3)^2 - 10*l
og(3) + 25)*x^2 + log(3) - 5) + 71/6*x/((log(3)^3 - 15*log(3)^2 + 75*log(3) - 125)*x^2 + log(3)^2 - 10*log(3)
+ 25) + 73/15*x/((log(3)^2 - 10*log(3) + 25)*x^2 + log(3) - 5) + 1/2*x/(x^2*(log(3) - 5) + 1) + 71/3*x/(log(3)
^2 - 10*log(3) + 25) - 71/4*log((x*(log(3) - 5) - sqrt(-log(3) + 5))/(x*(log(3) - 5) + sqrt(-log(3) + 5)))/((l
og(3)^2 - 10*log(3) + 25)*sqrt(-log(3) + 5)) - 23/10*log((x*(log(3) - 5) - sqrt(-log(3) + 5))/(x*(log(3) - 5)
+ sqrt(-log(3) + 5)))/((log(3) - 5)*sqrt(-log(3) + 5)) - 2/((log(3)^2 - 10*log(3) + 25)*x^2 + log(3) - 5)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 74 vs. \(2 (21) = 42\).

Time = 0.28 (sec) , antiderivative size = 74, normalized size of antiderivative = 3.22 \[ \int \frac {15+60 x-146 x^2+355 x^4+\left (30 x^2-146 x^4\right ) \log (3)+15 x^4 \log ^2(3)+\left (6 x^2-10 x^4+2 x^4 \log (3)\right ) \log \left (x^2\right )}{15-150 x^2+375 x^4+\left (30 x^2-150 x^4\right ) \log (3)+15 x^4 \log ^2(3)} \, dx=-\frac {2}{15} \, {\left (\frac {x}{x^{2} \log \left (3\right )^{2} - 10 \, x^{2} \log \left (3\right ) + 25 \, x^{2} + \log \left (3\right ) - 5} - \frac {x}{\log \left (3\right ) - 5}\right )} \log \left (x^{2}\right ) + x - \frac {2}{x^{2} \log \left (3\right )^{2} - 10 \, x^{2} \log \left (3\right ) + 25 \, x^{2} + \log \left (3\right ) - 5} \]

[In]

integrate(((2*x^4*log(3)-10*x^4+6*x^2)*log(x^2)+15*x^4*log(3)^2+(-146*x^4+30*x^2)*log(3)+355*x^4-146*x^2+60*x+
15)/(15*x^4*log(3)^2+(-150*x^4+30*x^2)*log(3)+375*x^4-150*x^2+15),x, algorithm="giac")

[Out]

-2/15*(x/(x^2*log(3)^2 - 10*x^2*log(3) + 25*x^2 + log(3) - 5) - x/(log(3) - 5))*log(x^2) + x - 2/(x^2*log(3)^2
- 10*x^2*log(3) + 25*x^2 + log(3) - 5)

Mupad [F(-1)]

Timed out. \[ \int \frac {15+60 x-146 x^2+355 x^4+\left (30 x^2-146 x^4\right ) \log (3)+15 x^4 \log ^2(3)+\left (6 x^2-10 x^4+2 x^4 \log (3)\right ) \log \left (x^2\right )}{15-150 x^2+375 x^4+\left (30 x^2-150 x^4\right ) \log (3)+15 x^4 \log ^2(3)} \, dx=\int \frac {60\,x+15\,x^4\,{\ln \left (3\right )}^2+\ln \left (x^2\right )\,\left (2\,x^4\,\ln \left (3\right )+6\,x^2-10\,x^4\right )+\ln \left (3\right )\,\left (30\,x^2-146\,x^4\right )-146\,x^2+355\,x^4+15}{15\,x^4\,{\ln \left (3\right )}^2+\ln \left (3\right )\,\left (30\,x^2-150\,x^4\right )-150\,x^2+375\,x^4+15} \,d x \]

[In]

int((60*x + 15*x^4*log(3)^2 + log(x^2)*(2*x^4*log(3) + 6*x^2 - 10*x^4) + log(3)*(30*x^2 - 146*x^4) - 146*x^2 +
355*x^4 + 15)/(15*x^4*log(3)^2 + log(3)*(30*x^2 - 150*x^4) - 150*x^2 + 375*x^4 + 15),x)

[Out]

int((60*x + 15*x^4*log(3)^2 + log(x^2)*(2*x^4*log(3) + 6*x^2 - 10*x^4) + log(3)*(30*x^2 - 146*x^4) - 146*x^2 +
355*x^4 + 15)/(15*x^4*log(3)^2 + log(3)*(30*x^2 - 150*x^4) - 150*x^2 + 375*x^4 + 15), x)