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

Optimal. Leaf size=23 \[ 4-\frac {4 x^2 \log (x)}{3-x^2-x^4} \]

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Rubi [B]  time = 1.07, antiderivative size = 230, normalized size of antiderivative = 10.00, number of steps used = 37, number of rules used = 11, integrand size = 49, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.224, Rules used = {6688, 12, 6742, 1107, 618, 206, 2357, 2335, 260, 2337, 2391} \begin {gather*} -\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 )-\frac {4 \tanh ^{-1}\left (\frac {2 x^2+1}{\sqrt {13}}\right )}{\sqrt {13}} \end {gather*}

Antiderivative was successfully verified.

[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 206

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

Rule 260

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 618

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 1107

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 2335

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 2337

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 2357

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 2391

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 6688

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

Rule 6742

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

Rubi steps

\begin {gather*} \begin {aligned} \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 \operatorname {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 \operatorname {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 \tanh ^{-1}\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 \tanh ^{-1}\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 \tanh ^{-1}\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 \tanh ^{-1}\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 \text {Li}_2\left (-\frac {2 x^2}{1-\sqrt {13}}\right )}{\sqrt {13}}+\frac {2 \text {Li}_2\left (-\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 \tanh ^{-1}\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 \text {Li}_2\left (-\frac {2 x^2}{1-\sqrt {13}}\right )}{\sqrt {13}}+\frac {2 \text {Li}_2\left (-\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 \tanh ^{-1}\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 {aligned} \end {gather*}

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Mathematica [A]  time = 0.12, size = 17, normalized size = 0.74 \begin {gather*} \frac {4 x^2 \log (x)}{-3+x^2+x^4} \end {gather*}

Antiderivative was successfully verified.

[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)

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fricas [A]  time = 0.58, size = 17, normalized size = 0.74 \begin {gather*} \frac {4 \, x^{2} \log \relax (x)}{x^{4} + x^{2} - 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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giac [A]  time = 0.14, size = 17, normalized size = 0.74 \begin {gather*} \frac {4 \, x^{2} \log \relax (x)}{x^{4} + x^{2} - 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maple [A]  time = 0.10, size = 18, normalized size = 0.78

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

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {4 \, x^{2} \log \relax (x)}{x^{4} + x^{2} - 3} + \frac {2}{13} \, \sqrt {13} \log \left (\frac {2 \, x^{2} - \sqrt {13} + 1}{2 \, x^{2} + \sqrt {13} + 1}\right ) - \frac {2 \, {\left (7 \, x^{2} - 3\right )}}{13 \, {\left (x^{4} + x^{2} - 3\right )}} + \frac {6 \, {\left (2 \, x^{2} + 1\right )}}{13 \, {\left (x^{4} + x^{2} - 3\right )}} + \frac {2 \, {\left (x^{2} - 6\right )}}{13 \, {\left (x^{4} + x^{2} - 3\right )}} - 4 \, \int \frac {x}{x^{4} + x^{2} - 3}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [B]  time = 5.39, size = 17, normalized size = 0.74 \begin {gather*} \frac {4\,x^2\,\ln \relax (x)}{x^4+x^2-3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [A]  time = 0.14, size = 15, normalized size = 0.65 \begin {gather*} \frac {4 x^{2} \log {\relax (x )}}{x^{4} + x^{2} - 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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