3.70.16 \(\int \frac {100-25 x^2+7 x^4+(10 x^6-2 x^8) \log (x)+(25 x^6-3 x^8) \log ^2(x)}{25 x^2-10 x^4+x^6} \, dx\) [6916]

Optimal. Leaf size=28 \[ -\frac {4}{x}+\frac {3+x^4 \log ^2(x)}{\frac {5}{x}-x} \]

[Out]

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

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Rubi [C] Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
time = 0.63, antiderivative size = 174, normalized size of antiderivative = 6.21, number of steps used = 44, number of rules used = 25, integrand size = 59, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.424, Rules used = {1608, 28, 6874, 1273, 21, 30, 308, 213, 2393, 2332, 2341, 2361, 12, 6031, 2404, 2333, 2342, 2367, 2355, 2353, 2352, 2354, 2438, 2421, 6724} \begin {gather*} -5 \sqrt {5} \text {PolyLog}\left (2,\frac {x}{\sqrt {5}}\right )-5 \sqrt {5} \text {PolyLog}\left (2,1-\frac {x}{\sqrt {5}}\right )+x^3 \left (-\log ^2(x)\right )+\frac {3 x}{5-x^2}-\frac {4}{x}+\frac {5 \sqrt {5} x \log ^2(x)}{2 \left (\sqrt {5}-x\right )}+\frac {5 \sqrt {5} x \log ^2(x)}{2 \left (x+\sqrt {5}\right )}-5 x \log ^2(x)+\frac {5}{2} \sqrt {5} \log (5) \log \left (\sqrt {5}-x\right )-5 \sqrt {5} \log (x) \log \left (\frac {x}{\sqrt {5}}+1\right )+10 \sqrt {5} \log (x) \tanh ^{-1}\left (\frac {x}{\sqrt {5}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(100 - 25*x^2 + 7*x^4 + (10*x^6 - 2*x^8)*Log[x] + (25*x^6 - 3*x^8)*Log[x]^2)/(25*x^2 - 10*x^4 + x^6),x]

[Out]

-4/x + (3*x)/(5 - x^2) + (5*Sqrt[5]*Log[5]*Log[Sqrt[5] - x])/2 + 10*Sqrt[5]*ArcTanh[x/Sqrt[5]]*Log[x] - 5*x*Lo
g[x]^2 + (5*Sqrt[5]*x*Log[x]^2)/(2*(Sqrt[5] - x)) - x^3*Log[x]^2 + (5*Sqrt[5]*x*Log[x]^2)/(2*(Sqrt[5] + x)) -
5*Sqrt[5]*Log[x]*Log[1 + x/Sqrt[5]] - 5*Sqrt[5]*PolyLog[2, x/Sqrt[5]] - 5*Sqrt[5]*PolyLog[2, 1 - x/Sqrt[5]]

Rule 12

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

Rule 21

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + b*x])

Rule 28

Int[(u_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Dist[1/c^p, Int[u*(b/2 + c*x^n)^(2*
p), x], x] /; FreeQ[{a, b, c, n}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 213

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

Rule 308

Int[(x_)^(m_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Int[PolynomialDivide[x^m, a + b*x^n, x], x] /; FreeQ[{a,
b}, x] && IGtQ[m, 0] && IGtQ[n, 0] && GtQ[m, 2*n - 1]

Rule 1273

Int[(x_)^(m_)*((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Simp[(-d)^(m
/2 - 1)*(c*d^2 - b*d*e + a*e^2)^p*x*((d + e*x^2)^(q + 1)/(2*e^(2*p + m/2)*(q + 1))), x] + Dist[(-d)^(m/2 - 1)/
(2*e^(2*p)*(q + 1)), Int[x^m*(d + e*x^2)^(q + 1)*ExpandToSum[Together[(1/(d + e*x^2))*(2*(-d)^(-m/2 + 1)*e^(2*
p)*(q + 1)*(a + b*x^2 + c*x^4)^p - ((c*d^2 - b*d*e + a*e^2)^p/(e^(m/2)*x^m))*(d + e*(2*q + 3)*x^2))], x], x],
x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && IGtQ[p, 0] && ILtQ[q, -1] && ILtQ[m/2, 0]

Rule 1608

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.) + (c_.)*(x_)^(r_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^
(q - p) + c*x^(r - p))^n, x] /; FreeQ[{a, b, c, p, q, r}, x] && IntegerQ[n] && PosQ[q - p] && PosQ[r - p]

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 2333

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

Rule 2341

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*Log[c*x^
n])/(d*(m + 1))), x] - Simp[b*n*((d*x)^(m + 1)/(d*(m + 1)^2)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1
]

Rule 2342

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

Rule 2352

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-e^(-1))*PolyLog[2, 1 - c*x], x] /; FreeQ[{c, d, e
}, x] && EqQ[e + c*d, 0]

Rule 2353

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

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 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 2367

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol] :> With[{u = Expand
Integrand[(a + b*Log[c*x^n])^p, (d + e*x^r)^q, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, d, e, n, p, q, r}
, x] && IntegerQ[q] && (GtQ[q, 0] || (IGtQ[p, 0] && IntegerQ[r]))

Rule 2393

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol] :> Wit
h[{u = ExpandIntegrand[a + b*Log[c*x^n], (f*x)^m*(d + e*x^r)^q, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c,
d, e, f, m, n, q, r}, x] && IntegerQ[q] && (GtQ[q, 0] || (IntegerQ[m] && IntegerQ[r]))

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 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 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 6874

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {100-25 x^2+7 x^4+\left (10 x^6-2 x^8\right ) \log (x)+\left (25 x^6-3 x^8\right ) \log ^2(x)}{x^2 \left (25-10 x^2+x^4\right )} \, dx\\ &=\int \frac {100-25 x^2+7 x^4+\left (10 x^6-2 x^8\right ) \log (x)+\left (25 x^6-3 x^8\right ) \log ^2(x)}{x^2 \left (-5+x^2\right )^2} \, dx\\ &=\int \left (\frac {100-25 x^2+7 x^4}{x^2 \left (-5+x^2\right )^2}-\frac {2 x^4 \log (x)}{-5+x^2}-\frac {x^4 \left (-25+3 x^2\right ) \log ^2(x)}{\left (-5+x^2\right )^2}\right ) \, dx\\ &=-\left (2 \int \frac {x^4 \log (x)}{-5+x^2} \, dx\right )+\int \frac {100-25 x^2+7 x^4}{x^2 \left (-5+x^2\right )^2} \, dx-\int \frac {x^4 \left (-25+3 x^2\right ) \log ^2(x)}{\left (-5+x^2\right )^2} \, dx\\ &=\frac {3 x}{5-x^2}-\frac {1}{50} \int \frac {1000-200 x^2}{x^2 \left (-5+x^2\right )} \, dx-2 \int \left (5 \log (x)+x^2 \log (x)+\frac {25 \log (x)}{-5+x^2}\right ) \, dx-\int \left (5 \log ^2(x)+3 x^2 \log ^2(x)-\frac {250 \log ^2(x)}{\left (-5+x^2\right )^2}-\frac {25 \log ^2(x)}{-5+x^2}\right ) \, dx\\ &=\frac {3 x}{5-x^2}-2 \int x^2 \log (x) \, dx-3 \int x^2 \log ^2(x) \, dx+4 \int \frac {1}{x^2} \, dx-5 \int \log ^2(x) \, dx-10 \int \log (x) \, dx+25 \int \frac {\log ^2(x)}{-5+x^2} \, dx-50 \int \frac {\log (x)}{-5+x^2} \, dx+250 \int \frac {\log ^2(x)}{\left (-5+x^2\right )^2} \, dx\\ &=-\frac {4}{x}+10 x+\frac {2 x^3}{9}+\frac {3 x}{5-x^2}-10 x \log (x)-\frac {2}{3} x^3 \log (x)+10 \sqrt {5} \tanh ^{-1}\left (\frac {x}{\sqrt {5}}\right ) \log (x)-5 x \log ^2(x)-x^3 \log ^2(x)+2 \int x^2 \log (x) \, dx+10 \int \log (x) \, dx+25 \int \left (-\frac {\log ^2(x)}{2 \sqrt {5} \left (\sqrt {5}-x\right )}-\frac {\log ^2(x)}{2 \sqrt {5} \left (\sqrt {5}+x\right )}\right ) \, dx-50 \int \frac {\tanh ^{-1}\left (\frac {x}{\sqrt {5}}\right )}{\sqrt {5} x} \, dx+250 \int \left (\frac {\log ^2(x)}{20 \left (\sqrt {5}-x\right )^2}+\frac {\log ^2(x)}{20 \left (\sqrt {5}+x\right )^2}+\frac {\log ^2(x)}{10 \left (5-x^2\right )}\right ) \, dx\\ &=-\frac {4}{x}+\frac {3 x}{5-x^2}+10 \sqrt {5} \tanh ^{-1}\left (\frac {x}{\sqrt {5}}\right ) \log (x)-5 x \log ^2(x)-x^3 \log ^2(x)+\frac {25}{2} \int \frac {\log ^2(x)}{\left (\sqrt {5}-x\right )^2} \, dx+\frac {25}{2} \int \frac {\log ^2(x)}{\left (\sqrt {5}+x\right )^2} \, dx+25 \int \frac {\log ^2(x)}{5-x^2} \, dx-\frac {1}{2} \left (5 \sqrt {5}\right ) \int \frac {\log ^2(x)}{\sqrt {5}-x} \, dx-\frac {1}{2} \left (5 \sqrt {5}\right ) \int \frac {\log ^2(x)}{\sqrt {5}+x} \, dx-\left (10 \sqrt {5}\right ) \int \frac {\tanh ^{-1}\left (\frac {x}{\sqrt {5}}\right )}{x} \, dx\\ &=-\frac {4}{x}+\frac {3 x}{5-x^2}+10 \sqrt {5} \tanh ^{-1}\left (\frac {x}{\sqrt {5}}\right ) \log (x)-5 x \log ^2(x)+\frac {5 \sqrt {5} x \log ^2(x)}{2 \left (\sqrt {5}-x\right )}-x^3 \log ^2(x)+\frac {5 \sqrt {5} x \log ^2(x)}{2 \left (\sqrt {5}+x\right )}+\frac {5}{2} \sqrt {5} \log ^2(x) \log \left (1-\frac {x}{\sqrt {5}}\right )-\frac {5}{2} \sqrt {5} \log ^2(x) \log \left (1+\frac {x}{\sqrt {5}}\right )+5 \sqrt {5} \text {Li}_2\left (-\frac {x}{\sqrt {5}}\right )-5 \sqrt {5} \text {Li}_2\left (\frac {x}{\sqrt {5}}\right )+25 \int \left (\frac {\log ^2(x)}{2 \sqrt {5} \left (\sqrt {5}-x\right )}+\frac {\log ^2(x)}{2 \sqrt {5} \left (\sqrt {5}+x\right )}\right ) \, dx-\left (5 \sqrt {5}\right ) \int \frac {\log (x)}{\sqrt {5}-x} \, dx-\left (5 \sqrt {5}\right ) \int \frac {\log (x)}{\sqrt {5}+x} \, dx-\left (5 \sqrt {5}\right ) \int \frac {\log (x) \log \left (1-\frac {x}{\sqrt {5}}\right )}{x} \, dx+\left (5 \sqrt {5}\right ) \int \frac {\log (x) \log \left (1+\frac {x}{\sqrt {5}}\right )}{x} \, dx\\ &=-\frac {4}{x}+\frac {3 x}{5-x^2}+\frac {5}{2} \sqrt {5} \log (5) \log \left (\sqrt {5}-x\right )+10 \sqrt {5} \tanh ^{-1}\left (\frac {x}{\sqrt {5}}\right ) \log (x)-5 x \log ^2(x)+\frac {5 \sqrt {5} x \log ^2(x)}{2 \left (\sqrt {5}-x\right )}-x^3 \log ^2(x)+\frac {5 \sqrt {5} x \log ^2(x)}{2 \left (\sqrt {5}+x\right )}+\frac {5}{2} \sqrt {5} \log ^2(x) \log \left (1-\frac {x}{\sqrt {5}}\right )-5 \sqrt {5} \log (x) \log \left (1+\frac {x}{\sqrt {5}}\right )-\frac {5}{2} \sqrt {5} \log ^2(x) \log \left (1+\frac {x}{\sqrt {5}}\right )+5 \sqrt {5} \text {Li}_2\left (-\frac {x}{\sqrt {5}}\right )-5 \sqrt {5} \log (x) \text {Li}_2\left (-\frac {x}{\sqrt {5}}\right )-5 \sqrt {5} \text {Li}_2\left (\frac {x}{\sqrt {5}}\right )+5 \sqrt {5} \log (x) \text {Li}_2\left (\frac {x}{\sqrt {5}}\right )+\frac {1}{2} \left (5 \sqrt {5}\right ) \int \frac {\log ^2(x)}{\sqrt {5}-x} \, dx+\frac {1}{2} \left (5 \sqrt {5}\right ) \int \frac {\log ^2(x)}{\sqrt {5}+x} \, dx-\left (5 \sqrt {5}\right ) \int \frac {\log \left (\frac {x}{\sqrt {5}}\right )}{\sqrt {5}-x} \, dx+\left (5 \sqrt {5}\right ) \int \frac {\log \left (1+\frac {x}{\sqrt {5}}\right )}{x} \, dx+\left (5 \sqrt {5}\right ) \int \frac {\text {Li}_2\left (-\frac {x}{\sqrt {5}}\right )}{x} \, dx-\left (5 \sqrt {5}\right ) \int \frac {\text {Li}_2\left (\frac {x}{\sqrt {5}}\right )}{x} \, dx\\ &=-\frac {4}{x}+\frac {3 x}{5-x^2}+\frac {5}{2} \sqrt {5} \log (5) \log \left (\sqrt {5}-x\right )+10 \sqrt {5} \tanh ^{-1}\left (\frac {x}{\sqrt {5}}\right ) \log (x)-5 x \log ^2(x)+\frac {5 \sqrt {5} x \log ^2(x)}{2 \left (\sqrt {5}-x\right )}-x^3 \log ^2(x)+\frac {5 \sqrt {5} x \log ^2(x)}{2 \left (\sqrt {5}+x\right )}-5 \sqrt {5} \log (x) \log \left (1+\frac {x}{\sqrt {5}}\right )-5 \sqrt {5} \log (x) \text {Li}_2\left (-\frac {x}{\sqrt {5}}\right )-5 \sqrt {5} \text {Li}_2\left (\frac {x}{\sqrt {5}}\right )+5 \sqrt {5} \log (x) \text {Li}_2\left (\frac {x}{\sqrt {5}}\right )-5 \sqrt {5} \text {Li}_2\left (1-\frac {x}{\sqrt {5}}\right )+5 \sqrt {5} \text {Li}_3\left (-\frac {x}{\sqrt {5}}\right )-5 \sqrt {5} \text {Li}_3\left (\frac {x}{\sqrt {5}}\right )+\left (5 \sqrt {5}\right ) \int \frac {\log (x) \log \left (1-\frac {x}{\sqrt {5}}\right )}{x} \, dx-\left (5 \sqrt {5}\right ) \int \frac {\log (x) \log \left (1+\frac {x}{\sqrt {5}}\right )}{x} \, dx\\ &=-\frac {4}{x}+\frac {3 x}{5-x^2}+\frac {5}{2} \sqrt {5} \log (5) \log \left (\sqrt {5}-x\right )+10 \sqrt {5} \tanh ^{-1}\left (\frac {x}{\sqrt {5}}\right ) \log (x)-5 x \log ^2(x)+\frac {5 \sqrt {5} x \log ^2(x)}{2 \left (\sqrt {5}-x\right )}-x^3 \log ^2(x)+\frac {5 \sqrt {5} x \log ^2(x)}{2 \left (\sqrt {5}+x\right )}-5 \sqrt {5} \log (x) \log \left (1+\frac {x}{\sqrt {5}}\right )-5 \sqrt {5} \text {Li}_2\left (\frac {x}{\sqrt {5}}\right )-5 \sqrt {5} \text {Li}_2\left (1-\frac {x}{\sqrt {5}}\right )+5 \sqrt {5} \text {Li}_3\left (-\frac {x}{\sqrt {5}}\right )-5 \sqrt {5} \text {Li}_3\left (\frac {x}{\sqrt {5}}\right )-\left (5 \sqrt {5}\right ) \int \frac {\text {Li}_2\left (-\frac {x}{\sqrt {5}}\right )}{x} \, dx+\left (5 \sqrt {5}\right ) \int \frac {\text {Li}_2\left (\frac {x}{\sqrt {5}}\right )}{x} \, dx\\ &=-\frac {4}{x}+\frac {3 x}{5-x^2}+\frac {5}{2} \sqrt {5} \log (5) \log \left (\sqrt {5}-x\right )+10 \sqrt {5} \tanh ^{-1}\left (\frac {x}{\sqrt {5}}\right ) \log (x)-5 x \log ^2(x)+\frac {5 \sqrt {5} x \log ^2(x)}{2 \left (\sqrt {5}-x\right )}-x^3 \log ^2(x)+\frac {5 \sqrt {5} x \log ^2(x)}{2 \left (\sqrt {5}+x\right )}-5 \sqrt {5} \log (x) \log \left (1+\frac {x}{\sqrt {5}}\right )-5 \sqrt {5} \text {Li}_2\left (\frac {x}{\sqrt {5}}\right )-5 \sqrt {5} \text {Li}_2\left (1-\frac {x}{\sqrt {5}}\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]
time = 0.08, size = 27, normalized size = 0.96 \begin {gather*} \frac {20-7 x^2-x^6 \log ^2(x)}{x \left (-5+x^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(100 - 25*x^2 + 7*x^4 + (10*x^6 - 2*x^8)*Log[x] + (25*x^6 - 3*x^8)*Log[x]^2)/(25*x^2 - 10*x^4 + x^6)
,x]

[Out]

(20 - 7*x^2 - x^6*Log[x]^2)/(x*(-5 + x^2))

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Maple [A]
time = 0.22, size = 37, normalized size = 1.32

method	result	size
risch	\(-\frac {x^{5} \ln \left (x \right )^{2}}{x^{2}-5}-\frac {7 x^{2}-20}{x \left (x^{2}-5\right )}\)	\(37\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-3*x^8+25*x^6)*ln(x)^2+(-2*x^8+10*x^6)*ln(x)+7*x^4-25*x^2+100)/(x^6-10*x^4+25*x^2),x,method=_RETURNVERBO
SE)

[Out]

-x^5/(x^2-5)*ln(x)^2-(7*x^2-20)/x/(x^2-5)

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Maxima [A]
time = 0.54, size = 45, normalized size = 1.61 \begin {gather*} -\frac {x^{5} \log \left (x\right )^{2}}{x^{2} - 5} - \frac {2 \, {\left (3 \, x^{2} - 10\right )}}{x^{3} - 5 \, x} - \frac {x}{x^{2} - 5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-3*x^8+25*x^6)*log(x)^2+(-2*x^8+10*x^6)*log(x)+7*x^4-25*x^2+100)/(x^6-10*x^4+25*x^2),x, algorithm=
"maxima")

[Out]

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

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Fricas [A]
time = 0.37, size = 26, normalized size = 0.93 \begin {gather*} -\frac {x^{6} \log \left (x\right )^{2} + 7 \, x^{2} - 20}{x^{3} - 5 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-3*x^8+25*x^6)*log(x)^2+(-2*x^8+10*x^6)*log(x)+7*x^4-25*x^2+100)/(x^6-10*x^4+25*x^2),x, algorithm=
"fricas")

[Out]

-(x^6*log(x)^2 + 7*x^2 - 20)/(x^3 - 5*x)

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Sympy [A]
time = 0.09, size = 26, normalized size = 0.93 \begin {gather*} - \frac {x^{5} \log {\left (x \right )}^{2}}{x^{2} - 5} + \frac {20 - 7 x^{2}}{x^{3} - 5 x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-3*x**8+25*x**6)*ln(x)**2+(-2*x**8+10*x**6)*ln(x)+7*x**4-25*x**2+100)/(x**6-10*x**4+25*x**2),x)

[Out]

-x**5*log(x)**2/(x**2 - 5) + (20 - 7*x**2)/(x**3 - 5*x)

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Giac [A]
time = 0.40, size = 39, normalized size = 1.39 \begin {gather*} -{\left (x^{3} + 5 \, x + \frac {25 \, x}{x^{2} - 5}\right )} \log \left (x\right )^{2} - \frac {3 \, x}{x^{2} - 5} - \frac {4}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-3*x^8+25*x^6)*log(x)^2+(-2*x^8+10*x^6)*log(x)+7*x^4-25*x^2+100)/(x^6-10*x^4+25*x^2),x, algorithm=
"giac")

[Out]

-(x^3 + 5*x + 25*x/(x^2 - 5))*log(x)^2 - 3*x/(x^2 - 5) - 4/x

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Mupad [B]
time = 4.33, size = 27, normalized size = 0.96 \begin {gather*} -\frac {x^6\,{\ln \left (x\right )}^2+7\,x^2-20}{x\,\left (x^2-5\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(x)*(10*x^6 - 2*x^8) + log(x)^2*(25*x^6 - 3*x^8) - 25*x^2 + 7*x^4 + 100)/(25*x^2 - 10*x^4 + x^6),x)

[Out]

-(x^6*log(x)^2 + 7*x^2 - 20)/(x*(x^2 - 5))

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