\(\int \frac {-12 x+4 x^3+4 x^5+(-24 x-8 x^5) \log (x)}{9-6 x^2-5 x^4+2 x^6+x^8} \, dx\) [7695]

Optimal result

Integrand size = 49, antiderivative size = 23 \[ \int \frac {-12 x+4 x^3+4 x^5+\left (-24 x-8 x^5\right ) \log (x)}{9-6 x^2-5 x^4+2 x^6+x^8} \, dx=4-\frac {4 x^2 \log (x)}{3-x^2-x^4} \]

[Out]

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

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(230\) vs. \(2(23)=46\).

Time = 0.67 (sec) , antiderivative size = 230, normalized size of antiderivative = 10.00, number of steps used = 37, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.224, Rules used = {6820, 12, 6874, 1121, 632, 212, 2404, 2373, 266, 2375, 2438} \[ \int \frac {-12 x+4 x^3+4 x^5+\left (-24 x-8 x^5\right ) \log (x)}{9-6 x^2-5 x^4+2 x^6+x^8} \, dx=-\frac {4 \text {arctanh}\left (\frac {2 x^2+1}{\sqrt {13}}\right )}{\sqrt {13}}-\frac {96 x^2 \log (x)}{13 \left (1-\sqrt {13}\right ) \left (2 x^2-\sqrt {13}+1\right )}-\frac {8 x^2 \log (x)}{13 \left (2 x^2-\sqrt {13}+1\right )}-\frac {96 x^2 \log (x)}{13 \left (1+\sqrt {13}\right ) \left (2 x^2+\sqrt {13}+1\right )}-\frac {8 x^2 \log (x)}{13 \left (2 x^2+\sqrt {13}+1\right )}+\frac {24 \log \left (2 x^2-\sqrt {13}+1\right )}{13 \left (1-\sqrt {13}\right )}+\frac {2}{13} \log \left (2 x^2-\sqrt {13}+1\right )+\frac {24 \log \left (2 x^2+\sqrt {13}+1\right )}{13 \left (1+\sqrt {13}\right )}+\frac {2}{13} \log \left (2 x^2+\sqrt {13}+1\right ) \]

[In]

Int[(-12*x + 4*x^3 + 4*x^5 + (-24*x - 8*x^5)*Log[x])/(9 - 6*x^2 - 5*x^4 + 2*x^6 + x^8),x]

[Out]

(-4*ArcTanh[(1 + 2*x^2)/Sqrt[13]])/Sqrt[13] - (8*x^2*Log[x])/(13*(1 - Sqrt[13] + 2*x^2)) - (96*x^2*Log[x])/(13
*(1 - Sqrt[13])*(1 - Sqrt[13] + 2*x^2)) - (8*x^2*Log[x])/(13*(1 + Sqrt[13] + 2*x^2)) - (96*x^2*Log[x])/(13*(1
+ Sqrt[13])*(1 + Sqrt[13] + 2*x^2)) + (2*Log[1 - Sqrt[13] + 2*x^2])/13 + (24*Log[1 - Sqrt[13] + 2*x^2])/(13*(1
- Sqrt[13])) + (2*Log[1 + Sqrt[13] + 2*x^2])/13 + (24*Log[1 + Sqrt[13] + 2*x^2])/(13*(1 + Sqrt[13]))

Rule 12

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

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 266

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

Rule 632

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 1121

Int[(x_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2, Subst[Int[(a + b*x + c*x^2)^p, x],
x, x^2], x] /; FreeQ[{a, b, c, p}, x]

Rule 2373

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(r_.))^(q_), x_Symbol] :> Simp
[(f*x)^(m + 1)*(d + e*x^r)^(q + 1)*((a + b*Log[c*x^n])/(d*f*(m + 1))), x] - Dist[b*(n/(d*(m + 1))), Int[(f*x)^
m*(d + e*x^r)^(q + 1), x], x] /; FreeQ[{a, b, c, d, e, f, m, n, q, r}, x] && EqQ[m + r*(q + 1) + 1, 0] && NeQ[
m, -1]

Rule 2375

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.))/((d_) + (e_.)*(x_)^(r_)), x_Symbol] :> Si
mp[f^m*Log[1 + e*(x^r/d)]*((a + b*Log[c*x^n])^p/(e*r)), x] - Dist[b*f^m*n*(p/(e*r)), Int[Log[1 + e*(x^r/d)]*((
a + b*Log[c*x^n])^(p - 1)/x), x], x] /; FreeQ[{a, b, c, d, e, f, m, n, r}, x] && EqQ[m, r - 1] && IGtQ[p, 0] &
& (IntegerQ[m] || GtQ[f, 0]) && NeQ[r, n]

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 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 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 {4 x \left (-3+x^2+x^4-2 \left (3+x^4\right ) \log (x)\right )}{\left (3-x^2-x^4\right )^2} \, dx \\ & = 4 \int \frac {x \left (-3+x^2+x^4-2 \left (3+x^4\right ) \log (x)\right )}{\left (3-x^2-x^4\right )^2} \, dx \\ & = 4 \int \left (\frac {x}{-3+x^2+x^4}-\frac {2 x \left (3+x^4\right ) \log (x)}{\left (-3+x^2+x^4\right )^2}\right ) \, dx \\ & = 4 \int \frac {x}{-3+x^2+x^4} \, dx-8 \int \frac {x \left (3+x^4\right ) \log (x)}{\left (-3+x^2+x^4\right )^2} \, dx \\ & = 2 \text {Subst}\left (\int \frac {1}{-3+x+x^2} \, dx,x,x^2\right )-8 \int \left (-\frac {x \left (-6+x^2\right ) \log (x)}{\left (-3+x^2+x^4\right )^2}+\frac {x \log (x)}{-3+x^2+x^4}\right ) \, dx \\ & = -\left (4 \text {Subst}\left (\int \frac {1}{13-x^2} \, dx,x,1+2 x^2\right )\right )+8 \int \frac {x \left (-6+x^2\right ) \log (x)}{\left (-3+x^2+x^4\right )^2} \, dx-8 \int \frac {x \log (x)}{-3+x^2+x^4} \, dx \\ & = -\frac {4 \text {arctanh}\left (\frac {1+2 x^2}{\sqrt {13}}\right )}{\sqrt {13}}-8 \int \left (-\frac {2 x \log (x)}{\sqrt {13} \left (-1+\sqrt {13}-2 x^2\right )}-\frac {2 x \log (x)}{\sqrt {13} \left (1+\sqrt {13}+2 x^2\right )}\right ) \, dx+8 \int \left (-\frac {6 x \log (x)}{\left (-3+x^2+x^4\right )^2}+\frac {x^3 \log (x)}{\left (-3+x^2+x^4\right )^2}\right ) \, dx \\ & = -\frac {4 \text {arctanh}\left (\frac {1+2 x^2}{\sqrt {13}}\right )}{\sqrt {13}}+8 \int \frac {x^3 \log (x)}{\left (-3+x^2+x^4\right )^2} \, dx-48 \int \frac {x \log (x)}{\left (-3+x^2+x^4\right )^2} \, dx+\frac {16 \int \frac {x \log (x)}{-1+\sqrt {13}-2 x^2} \, dx}{\sqrt {13}}+\frac {16 \int \frac {x \log (x)}{1+\sqrt {13}+2 x^2} \, dx}{\sqrt {13}} \\ & = -\frac {4 \text {arctanh}\left (\frac {1+2 x^2}{\sqrt {13}}\right )}{\sqrt {13}}-\frac {4 \log (x) \log \left (1+\frac {2 x^2}{1-\sqrt {13}}\right )}{\sqrt {13}}+\frac {4 \log (x) \log \left (1+\frac {2 x^2}{1+\sqrt {13}}\right )}{\sqrt {13}}+8 \int \left (\frac {2 \left (-1+\sqrt {13}\right ) x \log (x)}{13 \left (-1+\sqrt {13}-2 x^2\right )^2}-\frac {2 x \log (x)}{13 \sqrt {13} \left (-1+\sqrt {13}-2 x^2\right )}-\frac {2 \left (1+\sqrt {13}\right ) x \log (x)}{13 \left (1+\sqrt {13}+2 x^2\right )^2}-\frac {2 x \log (x)}{13 \sqrt {13} \left (1+\sqrt {13}+2 x^2\right )}\right ) \, dx-48 \int \left (\frac {4 x \log (x)}{13 \left (-1+\sqrt {13}-2 x^2\right )^2}+\frac {4 x \log (x)}{13 \sqrt {13} \left (-1+\sqrt {13}-2 x^2\right )}+\frac {4 x \log (x)}{13 \left (1+\sqrt {13}+2 x^2\right )^2}+\frac {4 x \log (x)}{13 \sqrt {13} \left (1+\sqrt {13}+2 x^2\right )}\right ) \, dx+\frac {4 \int \frac {\log \left (1-\frac {2 x^2}{-1+\sqrt {13}}\right )}{x} \, dx}{\sqrt {13}}-\frac {4 \int \frac {\log \left (1+\frac {2 x^2}{1+\sqrt {13}}\right )}{x} \, dx}{\sqrt {13}} \\ & = -\frac {4 \text {arctanh}\left (\frac {1+2 x^2}{\sqrt {13}}\right )}{\sqrt {13}}-\frac {4 \log (x) \log \left (1+\frac {2 x^2}{1-\sqrt {13}}\right )}{\sqrt {13}}+\frac {4 \log (x) \log \left (1+\frac {2 x^2}{1+\sqrt {13}}\right )}{\sqrt {13}}-\frac {2 \operatorname {PolyLog}\left (2,-\frac {2 x^2}{1-\sqrt {13}}\right )}{\sqrt {13}}+\frac {2 \operatorname {PolyLog}\left (2,-\frac {2 x^2}{1+\sqrt {13}}\right )}{\sqrt {13}}-\frac {192}{13} \int \frac {x \log (x)}{\left (-1+\sqrt {13}-2 x^2\right )^2} \, dx-\frac {192}{13} \int \frac {x \log (x)}{\left (1+\sqrt {13}+2 x^2\right )^2} \, dx-\frac {16 \int \frac {x \log (x)}{-1+\sqrt {13}-2 x^2} \, dx}{13 \sqrt {13}}-\frac {16 \int \frac {x \log (x)}{1+\sqrt {13}+2 x^2} \, dx}{13 \sqrt {13}}-\frac {192 \int \frac {x \log (x)}{-1+\sqrt {13}-2 x^2} \, dx}{13 \sqrt {13}}-\frac {192 \int \frac {x \log (x)}{1+\sqrt {13}+2 x^2} \, dx}{13 \sqrt {13}}-\frac {1}{13} \left (16 \left (1-\sqrt {13}\right )\right ) \int \frac {x \log (x)}{\left (-1+\sqrt {13}-2 x^2\right )^2} \, dx-\frac {1}{13} \left (16 \left (1+\sqrt {13}\right )\right ) \int \frac {x \log (x)}{\left (1+\sqrt {13}+2 x^2\right )^2} \, dx \\ & = -\frac {4 \text {arctanh}\left (\frac {1+2 x^2}{\sqrt {13}}\right )}{\sqrt {13}}-\frac {8 x^2 \log (x)}{13 \left (1-\sqrt {13}+2 x^2\right )}-\frac {96 x^2 \log (x)}{13 \left (1-\sqrt {13}\right ) \left (1-\sqrt {13}+2 x^2\right )}-\frac {8 x^2 \log (x)}{13 \left (1+\sqrt {13}+2 x^2\right )}-\frac {96 x^2 \log (x)}{13 \left (1+\sqrt {13}\right ) \left (1+\sqrt {13}+2 x^2\right )}-\frac {2 \operatorname {PolyLog}\left (2,-\frac {2 x^2}{1-\sqrt {13}}\right )}{\sqrt {13}}+\frac {2 \operatorname {PolyLog}\left (2,-\frac {2 x^2}{1+\sqrt {13}}\right )}{\sqrt {13}}-\frac {8}{13} \int \frac {x}{-1+\sqrt {13}-2 x^2} \, dx+\frac {8}{13} \int \frac {x}{1+\sqrt {13}+2 x^2} \, dx-\frac {4 \int \frac {\log \left (1-\frac {2 x^2}{-1+\sqrt {13}}\right )}{x} \, dx}{13 \sqrt {13}}+\frac {4 \int \frac {\log \left (1+\frac {2 x^2}{1+\sqrt {13}}\right )}{x} \, dx}{13 \sqrt {13}}-\frac {48 \int \frac {\log \left (1-\frac {2 x^2}{-1+\sqrt {13}}\right )}{x} \, dx}{13 \sqrt {13}}+\frac {48 \int \frac {\log \left (1+\frac {2 x^2}{1+\sqrt {13}}\right )}{x} \, dx}{13 \sqrt {13}}+\frac {96 \int \frac {x}{-1+\sqrt {13}-2 x^2} \, dx}{13 \left (-1+\sqrt {13}\right )}+\frac {96 \int \frac {x}{1+\sqrt {13}+2 x^2} \, dx}{13 \left (1+\sqrt {13}\right )} \\ & = -\frac {4 \text {arctanh}\left (\frac {1+2 x^2}{\sqrt {13}}\right )}{\sqrt {13}}-\frac {8 x^2 \log (x)}{13 \left (1-\sqrt {13}+2 x^2\right )}-\frac {96 x^2 \log (x)}{13 \left (1-\sqrt {13}\right ) \left (1-\sqrt {13}+2 x^2\right )}-\frac {8 x^2 \log (x)}{13 \left (1+\sqrt {13}+2 x^2\right )}-\frac {96 x^2 \log (x)}{13 \left (1+\sqrt {13}\right ) \left (1+\sqrt {13}+2 x^2\right )}+\frac {2}{13} \log \left (1-\sqrt {13}+2 x^2\right )+\frac {24 \log \left (1-\sqrt {13}+2 x^2\right )}{13 \left (1-\sqrt {13}\right )}+\frac {2}{13} \log \left (1+\sqrt {13}+2 x^2\right )+\frac {24 \log \left (1+\sqrt {13}+2 x^2\right )}{13 \left (1+\sqrt {13}\right )} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.13 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.74 \[ \int \frac {-12 x+4 x^3+4 x^5+\left (-24 x-8 x^5\right ) \log (x)}{9-6 x^2-5 x^4+2 x^6+x^8} \, dx=\frac {4 x^2 \log (x)}{-3+x^2+x^4} \]

[In]

Integrate[(-12*x + 4*x^3 + 4*x^5 + (-24*x - 8*x^5)*Log[x])/(9 - 6*x^2 - 5*x^4 + 2*x^6 + x^8),x]

[Out]

(4*x^2*Log[x])/(-3 + x^2 + x^4)

Maple [A] (verified)

Time = 0.16 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.78

method	result	size
default	\(\frac {4 \ln \left (x \right ) x^{2}}{x^{4}+x^{2}-3}\)	\(18\)
norman	\(\frac {4 \ln \left (x \right ) x^{2}}{x^{4}+x^{2}-3}\)	\(18\)
risch	\(\frac {4 \ln \left (x \right ) x^{2}}{x^{4}+x^{2}-3}\)	\(18\)
parallelrisch	\(\frac {4 \ln \left (x \right ) x^{2}}{x^{4}+x^{2}-3}\)	\(18\)
parts	\(\frac {4 \ln \left (x \right ) x^{2}}{x^{4}+x^{2}-3}\)	\(18\)

[In]

int(((-8*x^5-24*x)*ln(x)+4*x^5+4*x^3-12*x)/(x^8+2*x^6-5*x^4-6*x^2+9),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.33 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.74 \[ \int \frac {-12 x+4 x^3+4 x^5+\left (-24 x-8 x^5\right ) \log (x)}{9-6 x^2-5 x^4+2 x^6+x^8} \, dx=\frac {4 \, x^{2} \log \left (x\right )}{x^{4} + x^{2} - 3} \]

[In]

integrate(((-8*x^5-24*x)*log(x)+4*x^5+4*x^3-12*x)/(x^8+2*x^6-5*x^4-6*x^2+9),x, algorithm="fricas")

[Out]

4*x^2*log(x)/(x^4 + x^2 - 3)

Sympy [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.65 \[ \int \frac {-12 x+4 x^3+4 x^5+\left (-24 x-8 x^5\right ) \log (x)}{9-6 x^2-5 x^4+2 x^6+x^8} \, dx=\frac {4 x^{2} \log {\left (x \right )}}{x^{4} + x^{2} - 3} \]

[In]

integrate(((-8*x**5-24*x)*ln(x)+4*x**5+4*x**3-12*x)/(x**8+2*x**6-5*x**4-6*x**2+9),x)

[Out]

4*x**2*log(x)/(x**4 + x**2 - 3)

Maxima [F]

\[ \int \frac {-12 x+4 x^3+4 x^5+\left (-24 x-8 x^5\right ) \log (x)}{9-6 x^2-5 x^4+2 x^6+x^8} \, dx=\int { \frac {4 \, {\left (x^{5} + x^{3} - 2 \, {\left (x^{5} + 3 \, x\right )} \log \left (x\right ) - 3 \, x\right )}}{x^{8} + 2 \, x^{6} - 5 \, x^{4} - 6 \, x^{2} + 9} \,d x } \]

[In]

integrate(((-8*x^5-24*x)*log(x)+4*x^5+4*x^3-12*x)/(x^8+2*x^6-5*x^4-6*x^2+9),x, algorithm="maxima")

[Out]

4*x^2*log(x)/(x^4 + x^2 - 3) + 2/13*sqrt(13)*log((2*x^2 - sqrt(13) + 1)/(2*x^2 + sqrt(13) + 1)) - 2/13*(7*x^2
- 3)/(x^4 + x^2 - 3) + 6/13*(2*x^2 + 1)/(x^4 + x^2 - 3) + 2/13*(x^2 - 6)/(x^4 + x^2 - 3) - 4*integrate(x/(x^4
+ x^2 - 3), x)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.74 \[ \int \frac {-12 x+4 x^3+4 x^5+\left (-24 x-8 x^5\right ) \log (x)}{9-6 x^2-5 x^4+2 x^6+x^8} \, dx=\frac {4 \, x^{2} \log \left (x\right )}{x^{4} + x^{2} - 3} \]

[In]

integrate(((-8*x^5-24*x)*log(x)+4*x^5+4*x^3-12*x)/(x^8+2*x^6-5*x^4-6*x^2+9),x, algorithm="giac")

[Out]

4*x^2*log(x)/(x^4 + x^2 - 3)

Mupad [B] (verification not implemented)

Time = 13.30 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.74 \[ \int \frac {-12 x+4 x^3+4 x^5+\left (-24 x-8 x^5\right ) \log (x)}{9-6 x^2-5 x^4+2 x^6+x^8} \, dx=\frac {4\,x^2\,\ln \left (x\right )}{x^4+x^2-3} \]

[In]

int(-(12*x + log(x)*(24*x + 8*x^5) - 4*x^3 - 4*x^5)/(2*x^6 - 5*x^4 - 6*x^2 + x^8 + 9),x)

[Out]

(4*x^2*log(x))/(x^2 + x^4 - 3)