\(\int \frac {10 x+7 x^2+x^3+e^4 (2 x+x^2)+(4-6 x+x^3) \log (\frac {-2+x}{e^2})+(-2 x-x^2) \log (x)+(2 x+x^2) \log (4+2 x)}{-4 x+x^3} \, dx\) [7921]

Optimal result

Integrand size = 76, antiderivative size = 25 \[ \int \frac {10 x+7 x^2+x^3+e^4 \left (2 x+x^2\right )+\left (4-6 x+x^3\right ) \log \left (\frac {-2+x}{e^2}\right )+\left (-2 x-x^2\right ) \log (x)+\left (2 x+x^2\right ) \log (4+2 x)}{-4 x+x^3} \, dx=\log \left (\frac {-2+x}{e^2}\right ) \left (5+e^4+x-\log (x)+\log (4+2 x)\right ) \]

[Out]

ln((-2+x)/exp(2))*(exp(4)+x-ln(x)+5+ln(4+2*x))

Rubi [C] (verified)

Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.

Time = 0.33 (sec) , antiderivative size = 123, normalized size of antiderivative = 4.92, number of steps used = 21, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.171, Rules used = {1607, 6857, 213, 266, 327, 2465, 2441, 2352, 2440, 2438, 2436, 2332, 2353} \[ \int \frac {10 x+7 x^2+x^3+e^4 \left (2 x+x^2\right )+\left (4-6 x+x^3\right ) \log \left (\frac {-2+x}{e^2}\right )+\left (-2 x-x^2\right ) \log (x)+\left (2 x+x^2\right ) \log (4+2 x)}{-4 x+x^3} \, dx=-7 \text {arctanh}\left (\frac {x}{2}\right )+\operatorname {PolyLog}\left (2,\frac {2-x}{4}\right )+\operatorname {PolyLog}\left (2,\frac {x+2}{4}\right )+\frac {7}{2} \log \left (4-x^2\right )-2 x+e^4 \log (2-x)-(2-x) \log (x-2)-\log (2) \log (x-2)+(2-\log (x-2)) \log \left (\frac {x}{2}\right )-(2-\log (x-2)) \log \left (\frac {x+2}{4}\right )+\log \left (\frac {2-x}{4}\right ) \log (2 x+4) \]

[In]

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

[Out]

-2*x - 7*ArcTanh[x/2] + E^4*Log[2 - x] - (2 - x)*Log[-2 + x] - Log[2]*Log[-2 + x] + (2 - Log[-2 + x])*Log[x/2]
- (2 - Log[-2 + x])*Log[(2 + x)/4] + Log[(2 - x)/4]*Log[4 + 2*x] + (7*Log[4 - x^2])/2 + PolyLog[2, (2 - x)/4]
+ PolyLog[2, (2 + x)/4]

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 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 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 1607

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^(q - p))^n, x] /; F
reeQ[{a, b, p, q}, x] && IntegerQ[n] && PosQ[q - 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 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 2436

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.), x_Symbol] :> Dist[1/e, Subst[Int[(a + b*Log[c*
x^n])^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, n, p}, x]

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 2440

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))]*(b_.))/((f_.) + (g_.)*(x_)), x_Symbol] :> Dist[1/g, Subst[Int[(a +
b*Log[1 + c*e*(x/g)])/x, x], x, f + g*x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && EqQ[g
+ c*(e*f - d*g), 0]

Rule 2441

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

Rule 2465

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*(RFx_), x_Symbol] :> With[{u = ExpandIntegrand[
(a + b*Log[c*(d + e*x)^n])^p, RFx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, d, e, n}, x] && RationalFunct
ionQ[RFx, x] && IntegerQ[p]

Rule 6857

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a + b*x^n), x]}, Int[v, x]
/; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ[n, 0]

Rubi steps \begin{align*} \text {integral}& = \int \frac {10 x+7 x^2+x^3+e^4 \left (2 x+x^2\right )+\left (4-6 x+x^3\right ) \log \left (\frac {-2+x}{e^2}\right )+\left (-2 x-x^2\right ) \log (x)+\left (2 x+x^2\right ) \log (4+2 x)}{x \left (-4+x^2\right )} \, dx \\ & = \int \left (\frac {e^4}{-2+x}+\frac {10}{-4+x^2}+\frac {7 x}{-4+x^2}+\frac {x^2}{-4+x^2}+\frac {\left (-2+2 x+x^2\right ) (-2+\log (-2+x))}{x (2+x)}-\frac {\log (x)}{-2+x}+\frac {\log (4+2 x)}{-2+x}\right ) \, dx \\ & = e^4 \log (2-x)+7 \int \frac {x}{-4+x^2} \, dx+10 \int \frac {1}{-4+x^2} \, dx+\int \frac {x^2}{-4+x^2} \, dx+\int \frac {\left (-2+2 x+x^2\right ) (-2+\log (-2+x))}{x (2+x)} \, dx-\int \frac {\log (x)}{-2+x} \, dx+\int \frac {\log (4+2 x)}{-2+x} \, dx \\ & = x-5 \text {arctanh}\left (\frac {x}{2}\right )+e^4 \log (2-x)-\log (2) \log (-2+x)+\log \left (\frac {2-x}{4}\right ) \log (4+2 x)+\frac {7}{2} \log \left (4-x^2\right )-2 \int \frac {\log \left (\frac {2-x}{4}\right )}{4+2 x} \, dx+4 \int \frac {1}{-4+x^2} \, dx+\int \left (-2-\frac {-2+\log (-2+x)}{x}+\frac {-2+\log (-2+x)}{2+x}+\log (-2+x)\right ) \, dx-\int \frac {\log \left (\frac {x}{2}\right )}{-2+x} \, dx \\ & = -x-7 \text {arctanh}\left (\frac {x}{2}\right )+e^4 \log (2-x)-\log (2) \log (-2+x)+\log \left (\frac {2-x}{4}\right ) \log (4+2 x)+\frac {7}{2} \log \left (4-x^2\right )+\operatorname {PolyLog}\left (2,1-\frac {x}{2}\right )-\int \frac {-2+\log (-2+x)}{x} \, dx+\int \frac {-2+\log (-2+x)}{2+x} \, dx+\int \log (-2+x) \, dx-\text {Subst}\left (\int \frac {\log \left (1-\frac {x}{8}\right )}{x} \, dx,x,4+2 x\right ) \\ & = -x-7 \text {arctanh}\left (\frac {x}{2}\right )+e^4 \log (2-x)-\log (2) \log (-2+x)+(2-\log (-2+x)) \log \left (\frac {x}{2}\right )-(2-\log (-2+x)) \log \left (\frac {2+x}{4}\right )+\log \left (\frac {2-x}{4}\right ) \log (4+2 x)+\frac {7}{2} \log \left (4-x^2\right )+\operatorname {PolyLog}\left (2,1-\frac {x}{2}\right )+\operatorname {PolyLog}\left (2,\frac {2+x}{4}\right )+\int \frac {\log \left (\frac {x}{2}\right )}{-2+x} \, dx-\int \frac {\log \left (\frac {2+x}{4}\right )}{-2+x} \, dx+\text {Subst}(\int \log (x) \, dx,x,-2+x) \\ & = -2 x-7 \text {arctanh}\left (\frac {x}{2}\right )+e^4 \log (2-x)+(-2+x) \log (-2+x)-\log (2) \log (-2+x)+(2-\log (-2+x)) \log \left (\frac {x}{2}\right )-(2-\log (-2+x)) \log \left (\frac {2+x}{4}\right )+\log \left (\frac {2-x}{4}\right ) \log (4+2 x)+\frac {7}{2} \log \left (4-x^2\right )+\operatorname {PolyLog}\left (2,\frac {2+x}{4}\right )-\text {Subst}\left (\int \frac {\log \left (1+\frac {x}{4}\right )}{x} \, dx,x,-2+x\right ) \\ & = -2 x-7 \text {arctanh}\left (\frac {x}{2}\right )+e^4 \log (2-x)+(-2+x) \log (-2+x)-\log (2) \log (-2+x)+(2-\log (-2+x)) \log \left (\frac {x}{2}\right )-(2-\log (-2+x)) \log \left (\frac {2+x}{4}\right )+\log \left (\frac {2-x}{4}\right ) \log (4+2 x)+\frac {7}{2} \log \left (4-x^2\right )+\operatorname {PolyLog}\left (2,\frac {2-x}{4}\right )+\operatorname {PolyLog}\left (2,\frac {2+x}{4}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.12 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.84 \[ \int \frac {10 x+7 x^2+x^3+e^4 \left (2 x+x^2\right )+\left (4-6 x+x^3\right ) \log \left (\frac {-2+x}{e^2}\right )+\left (-2 x-x^2\right ) \log (x)+\left (2 x+x^2\right ) \log (4+2 x)}{-4 x+x^3} \, dx=-2 x+\log (4)+\left (7+e^4\right ) \log (2-x)+2 \log (x)-2 \log (2+x)+\log (-2+x) (-2+x-\log (x)+\log (2 (2+x))) \]

[In]

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

[Out]

-2*x + Log[4] + (7 + E^4)*Log[2 - x] + 2*Log[x] - 2*Log[2 + x] + Log[-2 + x]*(-2 + x - Log[x] + Log[2*(2 + x)]
)

Maple [A] (verified)

Time = 3.91 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.40

method	result	size
risch	\(\left (x -\ln \left (x \right )+\ln \left (4+2 x \right )\right ) \ln \left (\left (-2+x \right ) {\mathrm e}^{-2}\right )+\ln \left (-2+x \right ) {\mathrm e}^{4}+5 \ln \left (-2+x \right )\)	\(35\)
parallelrisch	\(\ln \left (\left (-2+x \right ) {\mathrm e}^{-2}\right ) {\mathrm e}^{4}+\ln \left (\left (-2+x \right ) {\mathrm e}^{-2}\right ) x -\ln \left (\left (-2+x \right ) {\mathrm e}^{-2}\right ) \ln \left (x \right )+\ln \left (\left (-2+x \right ) {\mathrm e}^{-2}\right ) \ln \left (4+2 x \right )+5 \ln \left (\left (-2+x \right ) {\mathrm e}^{-2}\right )\)	\(65\)
default	\(\ln \left (2\right ) \ln \left (-2+x \right )+\left (\ln \left (2+x \right )-\ln \left (\frac {1}{2}+\frac {x}{4}\right )\right ) \ln \left (\frac {1}{2}-\frac {x}{4}\right )-2 x +2 \ln \left (x \right )-2 \ln \left (2+x \right )+\left ({\mathrm e}^{4}+7\right ) \ln \left (-2+x \right )+\left (-2+x \right ) \ln \left (-2+x \right )+2-\ln \left (-2+x \right ) \ln \left (\frac {x}{2}\right )+\ln \left (-2+x \right ) \ln \left (\frac {1}{2}+\frac {x}{4}\right )-\left (\ln \left (x \right )-\ln \left (\frac {x}{2}\right )\right ) \ln \left (1-\frac {x}{2}\right )\)	\(98\)
parts	\(\left (\ln \left (4+2 x \right )-\ln \left (\frac {1}{2}+\frac {x}{4}\right )\right ) \ln \left (\frac {1}{2}-\frac {x}{4}\right )-\operatorname {dilog}\left (\frac {1}{2}+\frac {x}{4}\right )+x +\left ({\mathrm e}^{4}+7\right ) \ln \left (-2+x \right )+{\mathrm e}^{-4} {\mathrm e}^{4} \ln \left (x \,{\mathrm e}^{-2}-2 \,{\mathrm e}^{-2}\right ) x -2 \,{\mathrm e}^{-4} {\mathrm e}^{4} \ln \left (x \,{\mathrm e}^{-2}-2 \,{\mathrm e}^{-2}\right )-{\mathrm e}^{-4} {\mathrm e}^{4} x +2 \,{\mathrm e}^{-4} {\mathrm e}^{4}+\ln \left (x \,{\mathrm e}^{-2}-2 \,{\mathrm e}^{-2}\right ) \ln \left (\frac {{\mathrm e}^{2} \left (x \,{\mathrm e}^{-2}-2 \,{\mathrm e}^{-2}\right )}{4}+1\right )+\operatorname {dilog}\left (\frac {{\mathrm e}^{2} \left (x \,{\mathrm e}^{-2}-2 \,{\mathrm e}^{-2}\right )}{4}+1\right )-\ln \left (x \,{\mathrm e}^{-2}-2 \,{\mathrm e}^{-2}\right ) \ln \left (\frac {{\mathrm e}^{2} \left (x \,{\mathrm e}^{-2}-2 \,{\mathrm e}^{-2}\right )}{2}+1\right )-\operatorname {dilog}\left (\frac {{\mathrm e}^{2} \left (x \,{\mathrm e}^{-2}-2 \,{\mathrm e}^{-2}\right )}{2}+1\right )-\left (\ln \left (x \right )-\ln \left (\frac {x}{2}\right )\right ) \ln \left (1-\frac {x}{2}\right )+\operatorname {dilog}\left (\frac {x}{2}\right )\)	\(237\)

[In]

int(((x^2+2*x)*ln(4+2*x)+(-x^2-2*x)*ln(x)+(x^3-6*x+4)*ln((-2+x)/exp(2))+(x^2+2*x)*exp(4)+x^3+7*x^2+10*x)/(x^3-
4*x),x,method=_RETURNVERBOSE)

[Out]

(x-ln(x)+ln(4+2*x))*ln((-2+x)*exp(-2))+ln(-2+x)*exp(4)+5*ln(-2+x)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.56 \[ \int \frac {10 x+7 x^2+x^3+e^4 \left (2 x+x^2\right )+\left (4-6 x+x^3\right ) \log \left (\frac {-2+x}{e^2}\right )+\left (-2 x-x^2\right ) \log (x)+\left (2 x+x^2\right ) \log (4+2 x)}{-4 x+x^3} \, dx={\left (x + e^{4} + 5\right )} \log \left ({\left (x - 2\right )} e^{\left (-2\right )}\right ) + \log \left ({\left (x - 2\right )} e^{\left (-2\right )}\right ) \log \left (2 \, x + 4\right ) - \log \left ({\left (x - 2\right )} e^{\left (-2\right )}\right ) \log \left (x\right ) \]

[In]

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

[Out]

(x + e^4 + 5)*log((x - 2)*e^(-2)) + log((x - 2)*e^(-2))*log(2*x + 4) - log((x - 2)*e^(-2))*log(x)

Sympy [A] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.16 \[ \int \frac {10 x+7 x^2+x^3+e^4 \left (2 x+x^2\right )+\left (4-6 x+x^3\right ) \log \left (\frac {-2+x}{e^2}\right )+\left (-2 x-x^2\right ) \log (x)+\left (2 x+x^2\right ) \log (4+2 x)}{-4 x+x^3} \, dx=\left (x - \log {\left (x \right )} + \log {\left (2 x + 4 \right )}\right ) \log {\left (\frac {x - 2}{e^{2}} \right )} + \left (5 + e^{4}\right ) \log {\left (x - 2 \right )} \]

[In]

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

[Out]

(x - log(x) + log(2*x + 4))*log((x - 2)*exp(-2)) + (5 + exp(4))*log(x - 2)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 67 vs. \(2 (23) = 46\).

Time = 0.32 (sec) , antiderivative size = 67, normalized size of antiderivative = 2.68 \[ \int \frac {10 x+7 x^2+x^3+e^4 \left (2 x+x^2\right )+\left (4-6 x+x^3\right ) \log \left (\frac {-2+x}{e^2}\right )+\left (-2 x-x^2\right ) \log (x)+\left (2 x+x^2\right ) \log (4+2 x)}{-4 x+x^3} \, dx=\frac {1}{2} \, {\left (\log \left (x + 2\right ) + \log \left (x - 2\right )\right )} e^{4} - \frac {1}{2} \, {\left (\log \left (x + 2\right ) - \log \left (x - 2\right )\right )} e^{4} + {\left (\log \left (x - 2\right ) - 2\right )} \log \left (x + 2\right ) + {\left (x + \log \left (2\right ) - \log \left (x\right ) - 2\right )} \log \left (x - 2\right ) - 2 \, x + 7 \, \log \left (x - 2\right ) + 2 \, \log \left (x\right ) \]

[In]

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

[Out]

1/2*(log(x + 2) + log(x - 2))*e^4 - 1/2*(log(x + 2) - log(x - 2))*e^4 + (log(x - 2) - 2)*log(x + 2) + (x + log
(2) - log(x) - 2)*log(x - 2) - 2*x + 7*log(x - 2) + 2*log(x)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 52 vs. \(2 (23) = 46\).

Time = 0.29 (sec) , antiderivative size = 52, normalized size of antiderivative = 2.08 \[ \int \frac {10 x+7 x^2+x^3+e^4 \left (2 x+x^2\right )+\left (4-6 x+x^3\right ) \log \left (\frac {-2+x}{e^2}\right )+\left (-2 x-x^2\right ) \log (x)+\left (2 x+x^2\right ) \log (4+2 x)}{-4 x+x^3} \, dx=x \log \left (x - 2\right ) + e^{4} \log \left (x - 2\right ) + \log \left (2 \, x + 4\right ) \log \left (x - 2\right ) - \log \left (x - 2\right ) \log \left (x\right ) - 2 \, x - 2 \, \log \left (x + 2\right ) + 5 \, \log \left (x - 2\right ) + 2 \, \log \left (x\right ) \]

[In]

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

[Out]

x*log(x - 2) + e^4*log(x - 2) + log(2*x + 4)*log(x - 2) - log(x - 2)*log(x) - 2*x - 2*log(x + 2) + 5*log(x - 2
) + 2*log(x)

Mupad [B] (verification not implemented)

Time = 12.80 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.92 \[ \int \frac {10 x+7 x^2+x^3+e^4 \left (2 x+x^2\right )+\left (4-6 x+x^3\right ) \log \left (\frac {-2+x}{e^2}\right )+\left (-2 x-x^2\right ) \log (x)+\left (2 x+x^2\right ) \log (4+2 x)}{-4 x+x^3} \, dx=5\,\ln \left (x-2\right )+\ln \left (x-2\right )\,{\mathrm {e}}^4+\ln \left ({\mathrm {e}}^{-2}\,\left (x-2\right )\right )\,\ln \left (2\,x+4\right )-\ln \left ({\mathrm {e}}^{-2}\,\left (x-2\right )\right )\,\ln \left (x\right )+x\,\ln \left ({\mathrm {e}}^{-2}\,\left (x-2\right )\right ) \]

[In]

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

[Out]

5*log(x - 2) + log(x - 2)*exp(4) + log(exp(-2)*(x - 2))*log(2*x + 4) - log(exp(-2)*(x - 2))*log(x) + x*log(exp
(-2)*(x - 2))