\(\int \frac {-62208+373392 x^2-1728 x^4+2 x^6+(62208-432 x^2) \log (x)}{186624 x^2-864 x^4+x^6} \, dx\) [9674]

Optimal result

Integrand size = 44, antiderivative size = 23 \[ \int \frac {-62208+373392 x^2-1728 x^4+2 x^6+\left (62208-432 x^2\right ) \log (x)}{186624 x^2-864 x^4+x^6} \, dx=-6+2 x-\frac {\log (x)}{x \left (3-\frac {x^2}{144}\right )} \]

[Out]

2*x-6-ln(x)/x/(3-1/144*x^2)

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(77\) vs. \(2(23)=46\).

Time = 0.43 (sec) , antiderivative size = 77, normalized size of antiderivative = 3.35, number of steps used = 27, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {1608, 28, 6873, 12, 6874, 205, 213, 296, 331, 294, 327, 2404, 2341, 2360, 2361, 6031} \[ \int \frac {-62208+373392 x^2-1728 x^4+2 x^6+\left (62208-432 x^2\right ) \log (x)}{186624 x^2-864 x^4+x^6} \, dx=-\frac {2591 x}{6 \left (432-x^2\right )}-\frac {72}{\left (432-x^2\right ) x}-\frac {x \log (x)}{3 \left (432-x^2\right )}+\frac {x^3}{432-x^2}+3 x+\frac {1}{6 x}-\frac {\log (x)}{3 x} \]

[In]

Int[(-62208 + 373392*x^2 - 1728*x^4 + 2*x^6 + (62208 - 432*x^2)*Log[x])/(186624*x^2 - 864*x^4 + x^6),x]

[Out]

1/(6*x) + 3*x - 72/(x*(432 - x^2)) - (2591*x)/(6*(432 - x^2)) + x^3/(432 - x^2) - Log[x]/(3*x) - (x*Log[x])/(3
*(432 - x^2))

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[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 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 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 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 296

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(-(c*x)^(m + 1))*((a + b*x^n)^(p + 1)/
(a*c*n*(p + 1))), x] + Dist[(m + n*(p + 1) + 1)/(a*n*(p + 1)), Int[(c*x)^m*(a + b*x^n)^(p + 1), x], x] /; Free
Q[{a, b, c, m}, x] && IGtQ[n, 0] && LtQ[p, -1] && 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 331

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

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 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 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 6873

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

Rule 6874

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

Rubi steps \begin{align*} \text {integral}& = \int \frac {-62208+373392 x^2-1728 x^4+2 x^6+\left (62208-432 x^2\right ) \log (x)}{x^2 \left (186624-864 x^2+x^4\right )} \, dx \\ & = \int \frac {-62208+373392 x^2-1728 x^4+2 x^6+\left (62208-432 x^2\right ) \log (x)}{x^2 \left (-432+x^2\right )^2} \, dx \\ & = \int \frac {2 \left (-31104+186696 x^2-864 x^4+x^6+31104 \log (x)-216 x^2 \log (x)\right )}{x^2 \left (432-x^2\right )^2} \, dx \\ & = 2 \int \frac {-31104+186696 x^2-864 x^4+x^6+31104 \log (x)-216 x^2 \log (x)}{x^2 \left (432-x^2\right )^2} \, dx \\ & = 2 \int \left (\frac {186696}{\left (-432+x^2\right )^2}-\frac {31104}{x^2 \left (-432+x^2\right )^2}-\frac {864 x^2}{\left (-432+x^2\right )^2}+\frac {x^4}{\left (-432+x^2\right )^2}-\frac {216 \left (-144+x^2\right ) \log (x)}{x^2 \left (-432+x^2\right )^2}\right ) \, dx \\ & = 2 \int \frac {x^4}{\left (-432+x^2\right )^2} \, dx-432 \int \frac {\left (-144+x^2\right ) \log (x)}{x^2 \left (-432+x^2\right )^2} \, dx-1728 \int \frac {x^2}{\left (-432+x^2\right )^2} \, dx-62208 \int \frac {1}{x^2 \left (-432+x^2\right )^2} \, dx+373392 \int \frac {1}{\left (-432+x^2\right )^2} \, dx \\ & = -\frac {72}{x \left (432-x^2\right )}-\frac {2591 x}{6 \left (432-x^2\right )}+\frac {x^3}{432-x^2}+3 \int \frac {x^2}{-432+x^2} \, dx+216 \int \frac {1}{x^2 \left (-432+x^2\right )} \, dx-432 \int \left (-\frac {\log (x)}{1296 x^2}+\frac {2 \log (x)}{3 \left (-432+x^2\right )^2}+\frac {\log (x)}{1296 \left (-432+x^2\right )}\right ) \, dx-\frac {2593}{6} \int \frac {1}{-432+x^2} \, dx-864 \int \frac {1}{-432+x^2} \, dx \\ & = \frac {1}{2 x}+3 x-\frac {72}{x \left (432-x^2\right )}-\frac {2591 x}{6 \left (432-x^2\right )}+\frac {x^3}{432-x^2}+\frac {2593 \text {arctanh}\left (\frac {x}{12 \sqrt {3}}\right )}{72 \sqrt {3}}+24 \sqrt {3} \text {arctanh}\left (\frac {x}{12 \sqrt {3}}\right )+\frac {1}{3} \int \frac {\log (x)}{x^2} \, dx-\frac {1}{3} \int \frac {\log (x)}{-432+x^2} \, dx+\frac {1}{2} \int \frac {1}{-432+x^2} \, dx-288 \int \frac {\log (x)}{\left (-432+x^2\right )^2} \, dx+1296 \int \frac {1}{-432+x^2} \, dx \\ & = \frac {1}{6 x}+3 x-\frac {72}{x \left (432-x^2\right )}-\frac {2591 x}{6 \left (432-x^2\right )}+\frac {x^3}{432-x^2}+\frac {1295 \text {arctanh}\left (\frac {x}{12 \sqrt {3}}\right )}{36 \sqrt {3}}-12 \sqrt {3} \text {arctanh}\left (\frac {x}{12 \sqrt {3}}\right )-\frac {\log (x)}{3 x}-\frac {x \log (x)}{3 \left (432-x^2\right )}+\frac {\text {arctanh}\left (\frac {x}{12 \sqrt {3}}\right ) \log (x)}{36 \sqrt {3}}-\frac {1}{3} \int \frac {1}{-432+x^2} \, dx+\frac {1}{3} \int -\frac {\text {arctanh}\left (\frac {x}{12 \sqrt {3}}\right )}{12 \sqrt {3} x} \, dx+\frac {1}{3} \int \frac {\log (x)}{-432+x^2} \, dx \\ & = \frac {1}{6 x}+3 x-\frac {72}{x \left (432-x^2\right )}-\frac {2591 x}{6 \left (432-x^2\right )}+\frac {x^3}{432-x^2}-\frac {\log (x)}{3 x}-\frac {x \log (x)}{3 \left (432-x^2\right )}-\frac {1}{3} \int -\frac {\text {arctanh}\left (\frac {x}{12 \sqrt {3}}\right )}{12 \sqrt {3} x} \, dx-\frac {\int \frac {\text {arctanh}\left (\frac {x}{12 \sqrt {3}}\right )}{x} \, dx}{36 \sqrt {3}} \\ & = \frac {1}{6 x}+3 x-\frac {72}{x \left (432-x^2\right )}-\frac {2591 x}{6 \left (432-x^2\right )}+\frac {x^3}{432-x^2}-\frac {\log (x)}{3 x}-\frac {x \log (x)}{3 \left (432-x^2\right )}+\frac {\operatorname {PolyLog}\left (2,-\frac {x}{12 \sqrt {3}}\right )}{72 \sqrt {3}}-\frac {\operatorname {PolyLog}\left (2,\frac {x}{12 \sqrt {3}}\right )}{72 \sqrt {3}}+\frac {\int \frac {\text {arctanh}\left (\frac {x}{12 \sqrt {3}}\right )}{x} \, dx}{36 \sqrt {3}} \\ & = \frac {1}{6 x}+3 x-\frac {72}{x \left (432-x^2\right )}-\frac {2591 x}{6 \left (432-x^2\right )}+\frac {x^3}{432-x^2}-\frac {\log (x)}{3 x}-\frac {x \log (x)}{3 \left (432-x^2\right )} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(61\) vs. \(2(23)=46\).

Time = 0.20 (sec) , antiderivative size = 61, normalized size of antiderivative = 2.65 \[ \int \frac {-62208+373392 x^2-1728 x^4+2 x^6+\left (62208-432 x^2\right ) \log (x)}{186624 x^2-864 x^4+x^6} \, dx=2 \left (\frac {72 \log (x)}{x \left (-432+x^2\right )}+\frac {1}{432} \left (432 x+\sqrt {3} \log \left (12 \sqrt {3}+x\right )-\sqrt {3} \log \left (1+\frac {x}{12 \sqrt {3}}\right )\right )\right ) \]

[In]

Integrate[(-62208 + 373392*x^2 - 1728*x^4 + 2*x^6 + (62208 - 432*x^2)*Log[x])/(186624*x^2 - 864*x^4 + x^6),x]

[Out]

2*((72*Log[x])/(x*(-432 + x^2)) + (432*x + Sqrt[3]*Log[12*Sqrt[3] + x] - Sqrt[3]*Log[1 + x/(12*Sqrt[3])])/432)

Maple [A] (verified)

Time = 0.22 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.83

method	result	size
risch	\(\frac {144 \ln \left (x \right )}{x \left (x^{2}-432\right )}+2 x\)	\(19\)
norman	\(\frac {-864 x^{2}+2 x^{4}+144 \ln \left (x \right )}{x \left (x^{2}-432\right )}\)	\(27\)
parallelrisch	\(\frac {-864 x^{2}+2 x^{4}+144 \ln \left (x \right )}{x \left (x^{2}-432\right )}\)	\(27\)
default	\(2 x +\frac {\ln \left (x \right ) \left (\sqrt {3}\, \ln \left (1-\frac {x \sqrt {3}}{36}\right ) x^{2}-\sqrt {3}\, \ln \left (1+\frac {x \sqrt {3}}{36}\right ) x^{2}-432 \sqrt {3}\, \ln \left (1-\frac {x \sqrt {3}}{36}\right )+432 \sqrt {3}\, \ln \left (1+\frac {x \sqrt {3}}{36}\right )+72 x \right )}{216 x^{2}-93312}-\frac {\sqrt {3}\, \ln \left (x \right ) \ln \left (1-\frac {x \sqrt {3}}{36}\right )}{216}+\frac {\sqrt {3}\, \ln \left (x \right ) \ln \left (1+\frac {x \sqrt {3}}{36}\right )}{216}-\frac {\ln \left (x \right )}{3 x}\)	\(120\)
parts	\(2 x +\frac {\ln \left (x \right ) \left (\sqrt {3}\, \ln \left (1-\frac {x \sqrt {3}}{36}\right ) x^{2}-\sqrt {3}\, \ln \left (1+\frac {x \sqrt {3}}{36}\right ) x^{2}-432 \sqrt {3}\, \ln \left (1-\frac {x \sqrt {3}}{36}\right )+432 \sqrt {3}\, \ln \left (1+\frac {x \sqrt {3}}{36}\right )+72 x \right )}{216 x^{2}-93312}-\frac {\sqrt {3}\, \ln \left (x \right ) \ln \left (1-\frac {x \sqrt {3}}{36}\right )}{216}+\frac {\sqrt {3}\, \ln \left (x \right ) \ln \left (1+\frac {x \sqrt {3}}{36}\right )}{216}-\frac {\ln \left (x \right )}{3 x}\)	\(120\)

[In]

int(((-432*x^2+62208)*ln(x)+2*x^6-1728*x^4+373392*x^2-62208)/(x^6-864*x^4+186624*x^2),x,method=_RETURNVERBOSE)

[Out]

144/x/(x^2-432)*ln(x)+2*x

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.04 \[ \int \frac {-62208+373392 x^2-1728 x^4+2 x^6+\left (62208-432 x^2\right ) \log (x)}{186624 x^2-864 x^4+x^6} \, dx=\frac {2 \, {\left (x^{4} - 432 \, x^{2} + 72 \, \log \left (x\right )\right )}}{x^{3} - 432 \, x} \]

[In]

integrate(((-432*x^2+62208)*log(x)+2*x^6-1728*x^4+373392*x^2-62208)/(x^6-864*x^4+186624*x^2),x, algorithm="fri
cas")

[Out]

2*(x^4 - 432*x^2 + 72*log(x))/(x^3 - 432*x)

Sympy [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.61 \[ \int \frac {-62208+373392 x^2-1728 x^4+2 x^6+\left (62208-432 x^2\right ) \log (x)}{186624 x^2-864 x^4+x^6} \, dx=2 x + \frac {144 \log {\left (x \right )}}{x^{3} - 432 x} \]

[In]

integrate(((-432*x**2+62208)*ln(x)+2*x**6-1728*x**4+373392*x**2-62208)/(x**6-864*x**4+186624*x**2),x)

[Out]

2*x + 144*log(x)/(x**3 - 432*x)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 50 vs. \(2 (19) = 38\).

Time = 0.34 (sec) , antiderivative size = 50, normalized size of antiderivative = 2.17 \[ \int \frac {-62208+373392 x^2-1728 x^4+2 x^6+\left (62208-432 x^2\right ) \log (x)}{186624 x^2-864 x^4+x^6} \, dx=2 \, x - \frac {x^{2} - 432 \, \log \left (x\right ) - 432}{3 \, {\left (x^{3} - 432 \, x\right )}} + \frac {x^{2} - 288}{2 \, {\left (x^{3} - 432 \, x\right )}} - \frac {x}{6 \, {\left (x^{2} - 432\right )}} \]

[In]

integrate(((-432*x^2+62208)*log(x)+2*x^6-1728*x^4+373392*x^2-62208)/(x^6-864*x^4+186624*x^2),x, algorithm="max
ima")

[Out]

2*x - 1/3*(x^2 - 432*log(x) - 432)/(x^3 - 432*x) + 1/2*(x^2 - 288)/(x^3 - 432*x) - 1/6*x/(x^2 - 432)

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int \frac {-62208+373392 x^2-1728 x^4+2 x^6+\left (62208-432 x^2\right ) \log (x)}{186624 x^2-864 x^4+x^6} \, dx=\frac {1}{3} \, {\left (\frac {x}{x^{2} - 432} - \frac {1}{x}\right )} \log \left (x\right ) + 2 \, x \]

[In]

integrate(((-432*x^2+62208)*log(x)+2*x^6-1728*x^4+373392*x^2-62208)/(x^6-864*x^4+186624*x^2),x, algorithm="gia
c")

[Out]

1/3*(x/(x^2 - 432) - 1/x)*log(x) + 2*x

Mupad [B] (verification not implemented)

Time = 14.21 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.78 \[ \int \frac {-62208+373392 x^2-1728 x^4+2 x^6+\left (62208-432 x^2\right ) \log (x)}{186624 x^2-864 x^4+x^6} \, dx=2\,x+\frac {144\,\ln \left (x\right )}{x\,\left (x^2-432\right )} \]

[In]

int(-(1728*x^4 - 373392*x^2 - 2*x^6 + log(x)*(432*x^2 - 62208) + 62208)/(186624*x^2 - 864*x^4 + x^6),x)

[Out]

2*x + (144*log(x))/(x*(x^2 - 432))