3.46.78 \(\int \frac {80-16 x-4 x^2+(80+4 x^2) \log (-\frac {x}{\log (5)})}{400-160 x-24 x^2+8 x^3+x^4} \, dx\)

Optimal. Leaf size=24 \[ \frac {x \log \left (-\frac {x}{\log (5)}\right )}{5-\left (1+\frac {x}{4}\right ) x} \]

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Rubi [B]  time = 0.83, antiderivative size = 238, normalized size of antiderivative = 9.92, number of steps used = 34, number of rules used = 10, integrand size = 47, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.213, Rules used = {6742, 618, 206, 2357, 2314, 31, 2317, 2391, 2316, 2315} \begin {gather*} -\frac {5 \log \left (x+2 \left (1-\sqrt {6}\right )\right )}{6 \left (1-\sqrt {6}\right )}-\frac {1}{6} \log \left (x+2 \left (1-\sqrt {6}\right )\right )-\frac {5 \log \left (x+2 \left (1+\sqrt {6}\right )\right )}{6 \left (1+\sqrt {6}\right )}-\frac {1}{6} \log \left (x+2 \left (1+\sqrt {6}\right )\right )+\frac {5 x \log \left (-\frac {x}{\log (5)}\right )}{6 \left (1-\sqrt {6}\right ) \left (x+2 \left (1-\sqrt {6}\right )\right )}+\frac {x \log \left (-\frac {x}{\log (5)}\right )}{6 \left (x+2 \left (1-\sqrt {6}\right )\right )}+\frac {5 x \log \left (-\frac {x}{\log (5)}\right )}{6 \left (1+\sqrt {6}\right ) \left (x+2 \left (1+\sqrt {6}\right )\right )}+\frac {x \log \left (-\frac {x}{\log (5)}\right )}{6 \left (x+2 \left (1+\sqrt {6}\right )\right )}+\sqrt {\frac {2}{3}} \tanh ^{-1}\left (\frac {x+2}{2 \sqrt {6}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(80 - 16*x - 4*x^2 + (80 + 4*x^2)*Log[-(x/Log[5])])/(400 - 160*x - 24*x^2 + 8*x^3 + x^4),x]

[Out]

Sqrt[2/3]*ArcTanh[(2 + x)/(2*Sqrt[6])] - Log[2*(1 - Sqrt[6]) + x]/6 - (5*Log[2*(1 - Sqrt[6]) + x])/(6*(1 - Sqr
t[6])) - Log[2*(1 + Sqrt[6]) + x]/6 - (5*Log[2*(1 + Sqrt[6]) + x])/(6*(1 + Sqrt[6])) + (x*Log[-(x/Log[5])])/(6
*(2*(1 - Sqrt[6]) + x)) + (5*x*Log[-(x/Log[5])])/(6*(1 - Sqrt[6])*(2*(1 - Sqrt[6]) + x)) + (x*Log[-(x/Log[5])]
)/(6*(2*(1 + Sqrt[6]) + x)) + (5*x*Log[-(x/Log[5])])/(6*(1 + Sqrt[6])*(2*(1 + Sqrt[6]) + x))

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, 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 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 2314

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

Rule 2315

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

Rule 2316

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 2317

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 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 6742

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-\frac {4}{-20+4 x+x^2}+\frac {4 \left (20+x^2\right ) \log \left (-\frac {x}{\log (5)}\right )}{\left (-20+4 x+x^2\right )^2}\right ) \, dx\\ &=-\left (4 \int \frac {1}{-20+4 x+x^2} \, dx\right )+4 \int \frac {\left (20+x^2\right ) \log \left (-\frac {x}{\log (5)}\right )}{\left (-20+4 x+x^2\right )^2} \, dx\\ &=4 \int \left (-\frac {4 (-10+x) \log \left (-\frac {x}{\log (5)}\right )}{\left (-20+4 x+x^2\right )^2}+\frac {\log \left (-\frac {x}{\log (5)}\right )}{-20+4 x+x^2}\right ) \, dx+8 \operatorname {Subst}\left (\int \frac {1}{96-x^2} \, dx,x,4+2 x\right )\\ &=\sqrt {\frac {2}{3}} \tanh ^{-1}\left (\frac {2+x}{2 \sqrt {6}}\right )+4 \int \frac {\log \left (-\frac {x}{\log (5)}\right )}{-20+4 x+x^2} \, dx-16 \int \frac {(-10+x) \log \left (-\frac {x}{\log (5)}\right )}{\left (-20+4 x+x^2\right )^2} \, dx\\ &=\sqrt {\frac {2}{3}} \tanh ^{-1}\left (\frac {2+x}{2 \sqrt {6}}\right )+4 \int \left (-\frac {\log \left (-\frac {x}{\log (5)}\right )}{2 \sqrt {6} \left (-4+4 \sqrt {6}-2 x\right )}-\frac {\log \left (-\frac {x}{\log (5)}\right )}{2 \sqrt {6} \left (4+4 \sqrt {6}+2 x\right )}\right ) \, dx-16 \int \left (-\frac {10 \log \left (-\frac {x}{\log (5)}\right )}{\left (-20+4 x+x^2\right )^2}+\frac {x \log \left (-\frac {x}{\log (5)}\right )}{\left (-20+4 x+x^2\right )^2}\right ) \, dx\\ &=\sqrt {\frac {2}{3}} \tanh ^{-1}\left (\frac {2+x}{2 \sqrt {6}}\right )-16 \int \frac {x \log \left (-\frac {x}{\log (5)}\right )}{\left (-20+4 x+x^2\right )^2} \, dx+160 \int \frac {\log \left (-\frac {x}{\log (5)}\right )}{\left (-20+4 x+x^2\right )^2} \, dx-\sqrt {\frac {2}{3}} \int \frac {\log \left (-\frac {x}{\log (5)}\right )}{-4+4 \sqrt {6}-2 x} \, dx-\sqrt {\frac {2}{3}} \int \frac {\log \left (-\frac {x}{\log (5)}\right )}{4+4 \sqrt {6}+2 x} \, dx\\ &=\sqrt {\frac {2}{3}} \tanh ^{-1}\left (\frac {2+x}{2 \sqrt {6}}\right )-\frac {\log \left (4 \left (1+\sqrt {6}\right )+2 x\right ) \log \left (\frac {2 \left (1+\sqrt {6}\right )}{\log (5)}\right )}{\sqrt {6}}+\frac {\log \left (1+\frac {x}{2 \left (1-\sqrt {6}\right )}\right ) \log \left (-\frac {x}{\log (5)}\right )}{\sqrt {6}}-16 \int \left (\frac {\left (-4+4 \sqrt {6}\right ) \log \left (-\frac {x}{\log (5)}\right )}{48 \left (-4+4 \sqrt {6}-2 x\right )^2}-\frac {\log \left (-\frac {x}{\log (5)}\right )}{48 \sqrt {6} \left (-4+4 \sqrt {6}-2 x\right )}+\frac {\left (-4-4 \sqrt {6}\right ) \log \left (-\frac {x}{\log (5)}\right )}{48 \left (4+4 \sqrt {6}+2 x\right )^2}-\frac {\log \left (-\frac {x}{\log (5)}\right )}{48 \sqrt {6} \left (4+4 \sqrt {6}+2 x\right )}\right ) \, dx+160 \int \left (\frac {\log \left (-\frac {x}{\log (5)}\right )}{24 \left (-4+4 \sqrt {6}-2 x\right )^2}+\frac {\log \left (-\frac {x}{\log (5)}\right )}{96 \sqrt {6} \left (-4+4 \sqrt {6}-2 x\right )}+\frac {\log \left (-\frac {x}{\log (5)}\right )}{24 \left (4+4 \sqrt {6}+2 x\right )^2}+\frac {\log \left (-\frac {x}{\log (5)}\right )}{96 \sqrt {6} \left (4+4 \sqrt {6}+2 x\right )}\right ) \, dx-\sqrt {\frac {2}{3}} \int \frac {\log \left (-\frac {2 x}{4+4 \sqrt {6}}\right )}{4+4 \sqrt {6}+2 x} \, dx-\frac {\int \frac {\log \left (1-\frac {2 x}{-4+4 \sqrt {6}}\right )}{x} \, dx}{\sqrt {6}}\\ &=\sqrt {\frac {2}{3}} \tanh ^{-1}\left (\frac {2+x}{2 \sqrt {6}}\right )-\frac {\log \left (4 \left (1+\sqrt {6}\right )+2 x\right ) \log \left (\frac {2 \left (1+\sqrt {6}\right )}{\log (5)}\right )}{\sqrt {6}}+\frac {\log \left (1+\frac {x}{2 \left (1-\sqrt {6}\right )}\right ) \log \left (-\frac {x}{\log (5)}\right )}{\sqrt {6}}+\frac {\text {Li}_2\left (-\frac {x}{2 \left (1-\sqrt {6}\right )}\right )}{\sqrt {6}}+\frac {\text {Li}_2\left (1+\frac {x}{2 \left (1+\sqrt {6}\right )}\right )}{\sqrt {6}}+\frac {20}{3} \int \frac {\log \left (-\frac {x}{\log (5)}\right )}{\left (-4+4 \sqrt {6}-2 x\right )^2} \, dx+\frac {20}{3} \int \frac {\log \left (-\frac {x}{\log (5)}\right )}{\left (4+4 \sqrt {6}+2 x\right )^2} \, dx+\frac {\int \frac {\log \left (-\frac {x}{\log (5)}\right )}{-4+4 \sqrt {6}-2 x} \, dx}{3 \sqrt {6}}+\frac {\int \frac {\log \left (-\frac {x}{\log (5)}\right )}{4+4 \sqrt {6}+2 x} \, dx}{3 \sqrt {6}}+\frac {5 \int \frac {\log \left (-\frac {x}{\log (5)}\right )}{-4+4 \sqrt {6}-2 x} \, dx}{3 \sqrt {6}}+\frac {5 \int \frac {\log \left (-\frac {x}{\log (5)}\right )}{4+4 \sqrt {6}+2 x} \, dx}{3 \sqrt {6}}+\frac {1}{3} \left (4 \left (1-\sqrt {6}\right )\right ) \int \frac {\log \left (-\frac {x}{\log (5)}\right )}{\left (-4+4 \sqrt {6}-2 x\right )^2} \, dx+\frac {1}{3} \left (4 \left (1+\sqrt {6}\right )\right ) \int \frac {\log \left (-\frac {x}{\log (5)}\right )}{\left (4+4 \sqrt {6}+2 x\right )^2} \, dx\\ &=\sqrt {\frac {2}{3}} \tanh ^{-1}\left (\frac {2+x}{2 \sqrt {6}}\right )+\frac {x \log \left (-\frac {x}{\log (5)}\right )}{6 \left (2 \left (1-\sqrt {6}\right )+x\right )}+\frac {5 x \log \left (-\frac {x}{\log (5)}\right )}{6 \left (1-\sqrt {6}\right ) \left (2 \left (1-\sqrt {6}\right )+x\right )}+\frac {x \log \left (-\frac {x}{\log (5)}\right )}{6 \left (2 \left (1+\sqrt {6}\right )+x\right )}+\frac {5 x \log \left (-\frac {x}{\log (5)}\right )}{6 \left (1+\sqrt {6}\right ) \left (2 \left (1+\sqrt {6}\right )+x\right )}+\frac {\text {Li}_2\left (-\frac {x}{2 \left (1-\sqrt {6}\right )}\right )}{\sqrt {6}}+\frac {\text {Li}_2\left (1+\frac {x}{2 \left (1+\sqrt {6}\right )}\right )}{\sqrt {6}}+\frac {1}{3} \int \frac {1}{-4+4 \sqrt {6}-2 x} \, dx-\frac {1}{3} \int \frac {1}{4+4 \sqrt {6}+2 x} \, dx+\frac {\int \frac {\log \left (1-\frac {2 x}{-4+4 \sqrt {6}}\right )}{x} \, dx}{6 \sqrt {6}}+\frac {\int \frac {\log \left (-\frac {2 x}{4+4 \sqrt {6}}\right )}{4+4 \sqrt {6}+2 x} \, dx}{3 \sqrt {6}}+\frac {5 \int \frac {\log \left (1-\frac {2 x}{-4+4 \sqrt {6}}\right )}{x} \, dx}{6 \sqrt {6}}+\frac {5 \int \frac {\log \left (-\frac {2 x}{4+4 \sqrt {6}}\right )}{4+4 \sqrt {6}+2 x} \, dx}{3 \sqrt {6}}-\frac {5 \int \frac {1}{4+4 \sqrt {6}+2 x} \, dx}{3 \left (1+\sqrt {6}\right )}-\frac {20 \int \frac {1}{-4+4 \sqrt {6}-2 x} \, dx}{3 \left (-4+4 \sqrt {6}\right )}\\ &=\sqrt {\frac {2}{3}} \tanh ^{-1}\left (\frac {2+x}{2 \sqrt {6}}\right )-\frac {1}{6} \log \left (2 \left (1-\sqrt {6}\right )+x\right )-\frac {5 \log \left (2 \left (1-\sqrt {6}\right )+x\right )}{6 \left (1-\sqrt {6}\right )}-\frac {1}{6} \log \left (2 \left (1+\sqrt {6}\right )+x\right )-\frac {5 \log \left (2 \left (1+\sqrt {6}\right )+x\right )}{6 \left (1+\sqrt {6}\right )}+\frac {x \log \left (-\frac {x}{\log (5)}\right )}{6 \left (2 \left (1-\sqrt {6}\right )+x\right )}+\frac {5 x \log \left (-\frac {x}{\log (5)}\right )}{6 \left (1-\sqrt {6}\right ) \left (2 \left (1-\sqrt {6}\right )+x\right )}+\frac {x \log \left (-\frac {x}{\log (5)}\right )}{6 \left (2 \left (1+\sqrt {6}\right )+x\right )}+\frac {5 x \log \left (-\frac {x}{\log (5)}\right )}{6 \left (1+\sqrt {6}\right ) \left (2 \left (1+\sqrt {6}\right )+x\right )}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.34, size = 21, normalized size = 0.88 \begin {gather*} -\frac {4 x \log \left (-\frac {x}{\log (5)}\right )}{-20+4 x+x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(80 - 16*x - 4*x^2 + (80 + 4*x^2)*Log[-(x/Log[5])])/(400 - 160*x - 24*x^2 + 8*x^3 + x^4),x]

[Out]

(-4*x*Log[-(x/Log[5])])/(-20 + 4*x + x^2)

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fricas [A]  time = 0.74, size = 21, normalized size = 0.88 \begin {gather*} -\frac {4 \, x \log \left (-\frac {x}{\log \relax (5)}\right )}{x^{2} + 4 \, x - 20} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x^2+80)*log(-x/log(5))-4*x^2-16*x+80)/(x^4+8*x^3-24*x^2-160*x+400),x, algorithm="fricas")

[Out]

-4*x*log(-x/log(5))/(x^2 + 4*x - 20)

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giac [A]  time = 0.14, size = 34, normalized size = 1.42 \begin {gather*} -\frac {4 \, x \log \left (-x\right )}{x^{2} + 4 \, x - 20} + \frac {4 \, x \log \left (\log \relax (5)\right )}{x^{2} + 4 \, x - 20} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x^2+80)*log(-x/log(5))-4*x^2-16*x+80)/(x^4+8*x^3-24*x^2-160*x+400),x, algorithm="giac")

[Out]

-4*x*log(-x)/(x^2 + 4*x - 20) + 4*x*log(log(5))/(x^2 + 4*x - 20)

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maple [A]  time = 0.07, size = 22, normalized size = 0.92

method	result	size
norman	\(-\frac {4 x \ln \left (-\frac {x}{\ln \relax (5)}\right )}{x^{2}+4 x -20}\)	\(22\)
risch	\(-\frac {4 x \ln \left (-\frac {x}{\ln \relax (5)}\right )}{x^{2}+4 x -20}\)	\(22\)
derivativedivides	\(-\ln \relax (5) \left (-\frac {\ln \left (-\frac {x}{\ln \relax (5)}\right ) \left (\ln \left (-\frac {-x -2 \sqrt {6}-2}{2 \left (\sqrt {6}+1\right )}\right ) \sqrt {6}\, x^{2}-\ln \left (\frac {-x +2 \sqrt {6}-2}{2 \sqrt {6}-2}\right ) \sqrt {6}\, x^{2}+4 \ln \left (-\frac {-x -2 \sqrt {6}-2}{2 \left (\sqrt {6}+1\right )}\right ) \sqrt {6}\, x -4 \ln \left (\frac {-x +2 \sqrt {6}-2}{2 \sqrt {6}-2}\right ) \sqrt {6}\, x -24 x -20 \sqrt {6}\, \ln \left (-\frac {-x -2 \sqrt {6}-2}{2 \left (\sqrt {6}+1\right )}\right )+20 \sqrt {6}\, \ln \left (\frac {-x +2 \sqrt {6}-2}{2 \sqrt {6}-2}\right )\right )}{6 \left (x^{2}+4 x -20\right ) \ln \relax (5)}+\frac {\sqrt {6}\, \ln \left (-\frac {x}{\ln \relax (5)}\right ) \left (\ln \left (-\frac {-x -2 \sqrt {6}-2}{2 \left (\sqrt {6}+1\right )}\right )-\ln \left (\frac {-x +2 \sqrt {6}-2}{2 \sqrt {6}-2}\right )\right )}{6 \ln \relax (5)}\right )\)	\(251\)
default	\(-\ln \relax (5) \left (-\frac {\ln \left (-\frac {x}{\ln \relax (5)}\right ) \left (\ln \left (-\frac {-x -2 \sqrt {6}-2}{2 \left (\sqrt {6}+1\right )}\right ) \sqrt {6}\, x^{2}-\ln \left (\frac {-x +2 \sqrt {6}-2}{2 \sqrt {6}-2}\right ) \sqrt {6}\, x^{2}+4 \ln \left (-\frac {-x -2 \sqrt {6}-2}{2 \left (\sqrt {6}+1\right )}\right ) \sqrt {6}\, x -4 \ln \left (\frac {-x +2 \sqrt {6}-2}{2 \sqrt {6}-2}\right ) \sqrt {6}\, x -24 x -20 \sqrt {6}\, \ln \left (-\frac {-x -2 \sqrt {6}-2}{2 \left (\sqrt {6}+1\right )}\right )+20 \sqrt {6}\, \ln \left (\frac {-x +2 \sqrt {6}-2}{2 \sqrt {6}-2}\right )\right )}{6 \left (x^{2}+4 x -20\right ) \ln \relax (5)}+\frac {\sqrt {6}\, \ln \left (-\frac {x}{\ln \relax (5)}\right ) \left (\ln \left (-\frac {-x -2 \sqrt {6}-2}{2 \left (\sqrt {6}+1\right )}\right )-\ln \left (\frac {-x +2 \sqrt {6}-2}{2 \sqrt {6}-2}\right )\right )}{6 \ln \relax (5)}\right )\)	\(251\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((4*x^2+80)*ln(-x/ln(5))-4*x^2-16*x+80)/(x^4+8*x^3-24*x^2-160*x+400),x,method=_RETURNVERBOSE)

[Out]

-4*x*ln(-x/ln(5))/(x^2+4*x-20)

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maxima [B]  time = 0.49, size = 73, normalized size = 3.04 \begin {gather*} -\frac {4 \, {\left (x \log \left (-x\right ) - x \log \left (\log \relax (5)\right )\right )}}{x^{2} + 4 \, x - 20} + \frac {7 \, x - 10}{3 \, {\left (x^{2} + 4 \, x - 20\right )}} - \frac {5 \, {\left (x + 2\right )}}{3 \, {\left (x^{2} + 4 \, x - 20\right )}} - \frac {2 \, {\left (x - 10\right )}}{3 \, {\left (x^{2} + 4 \, x - 20\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x^2+80)*log(-x/log(5))-4*x^2-16*x+80)/(x^4+8*x^3-24*x^2-160*x+400),x, algorithm="maxima")

[Out]

-4*(x*log(-x) - x*log(log(5)))/(x^2 + 4*x - 20) + 1/3*(7*x - 10)/(x^2 + 4*x - 20) - 5/3*(x + 2)/(x^2 + 4*x - 2
0) - 2/3*(x - 10)/(x^2 + 4*x - 20)

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mupad [B]  time = 3.46, size = 23, normalized size = 0.96 \begin {gather*} -\frac {4\,x\,\left (\ln \left (-x\right )-\ln \left (\ln \relax (5)\right )\right )}{x^2+4\,x-20} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(16*x - log(-x/log(5))*(4*x^2 + 80) + 4*x^2 - 80)/(8*x^3 - 24*x^2 - 160*x + x^4 + 400),x)

[Out]

-(4*x*(log(-x) - log(log(5))))/(4*x + x^2 - 20)

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sympy [A]  time = 0.14, size = 20, normalized size = 0.83 \begin {gather*} - \frac {4 x \log {\left (- \frac {x}{\log {\relax (5 )}} \right )}}{x^{2} + 4 x - 20} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x**2+80)*ln(-x/ln(5))-4*x**2-16*x+80)/(x**4+8*x**3-24*x**2-160*x+400),x)

[Out]

-4*x*log(-x/log(5))/(x**2 + 4*x - 20)

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