3.69.70 $\int \frac{e^2(-1-2x)+6x+14x^2+4x^3+(e^2-6x-2x^2)\log (\frac{-e^2+6x+2x^2}{2x})+\log (x)(-3e^2+12x+2x^2+(e^2-6x-2x^2)\log (\frac{-e^2+6x+2x^2}{2x}))}{e^2-6x-2x^2}dx$

Optimal. Leaf size=28 $$-1+x-x^2+x\log (x)(-2+\log (3-\frac{e^2}{2x}+x))$$

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Rubi [A]  time = 1.00, antiderivative size = 30, normalized size of antiderivative = 1.07, number of steps used = 37, number of rules used = 15, integrand size = 125, $\frac{\text{number of rules}}{\text{integrand size}}$ = 0.120, Rules used = {6688, 2523, 1657, 634, 618, 206, 628, 6728, 2357, 2295, 2317, 2391, 2316, 2315, 2556} $$\begin{array}{c}-x^2+x-2x\log (x)+x\log (x)\log (x-\frac{e^2}{2x}+3)\end{array}$$

Antiderivative was successfully verified.

[In]

Int[(E^2*(-1 - 2*x) + 6*x + 14*x^2 + 4*x^3 + (E^2 - 6*x - 2*x^2)*Log[(-E^2 + 6*x + 2*x^2)/(2*x)] + Log[x]*(-3*
E^2 + 12*x + 2*x^2 + (E^2 - 6*x - 2*x^2)*Log[(-E^2 + 6*x + 2*x^2)/(2*x)]))/(E^2 - 6*x - 2*x^2),x]

[Out]

x - x^2 - 2*x*Log[x] + x*Log[x]*Log[3 - E^2/(2*x) + 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 628

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

Rule 634

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

Rule 1657

Int[(Pq_)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[Pq*(a + b*x + c*x^2)^p, x
], x] /; FreeQ[{a, b, c}, x] && PolyQ[Pq, x] && IGtQ[p, -2]

Rule 2295

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{c, n}, x]

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 2523

Int[((a_.) + Log[(c_.)*(RFx_)^(p_.)]*(b_.))^(n_.), x_Symbol] :> Simp[x*(a + b*Log[c*RFx^p])^n, x] - Dist[b*n*p
, Int[SimplifyIntegrand[(x*(a + b*Log[c*RFx^p])^(n - 1)*D[RFx, x])/RFx, x], x], x] /; FreeQ[{a, b, c, p}, x] &
& RationalFunctionQ[RFx, x] && IGtQ[n, 0]

Rule 2556

Int[Log[v_]*Log[w_], x_Symbol] :> Simp[x*Log[v]*Log[w], x] + (-Int[SimplifyIntegrand[(x*Log[w]*D[v, x])/v, x],
x] - Int[SimplifyIntegrand[(x*Log[v]*D[w, x])/w, x], x]) /; InverseFunctionFreeQ[v, x] && InverseFunctionFree
Q[w, x]

Rule 6688

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

Rule 6728

Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a +
b*x^n + c*x^(2*n)), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]

Rubi steps

$$\begin{array}{c}\begin{array}{rl}\text{integral}& =\int (-1-2x+\log (3-\frac{e^2}{2x}+x)+\frac{\log (x)(-3e^2+2x(6+x)+(e^2-2x(3+x))\log (3-\frac{e^2}{2x}+x))}{e^2-6x-2x^2})dx\\ & =-x-x^2+\int \log (3-\frac{e^2}{2x}+x)dx+\int \frac{\log (x)(-3e^2+2x(6+x)+(e^2-2x(3+x))\log (3-\frac{e^2}{2x}+x))}{e^2-6x-2x^2}dx\\ & =-x-x^2+x\log (3-\frac{e^2}{2x}+x)-\int \frac{-e^2-2x^2}{e^2-6x-2x^2}dx+\int (-\frac{(3e^2-12x-2x^2)\log (x)}{e^2-6x-2x^2}+\log (x)\log (3-\frac{e^2}{2x}+x))dx\\ & =-x-x^2+x\log (3-\frac{e^2}{2x}+x)-\int (1-\frac{2(e^2-3x)}{e^2-6x-2x^2})dx-\int \frac{(3e^2-12x-2x^2)\log (x)}{e^2-6x-2x^2}dx+\int \log (x)\log (3-\frac{e^2}{2x}+x)dx\\ & =-2x-x^2+x\log (3-\frac{e^2}{2x}+x)+x\log (x)\log (3-\frac{e^2}{2x}+x)+2\int \frac{e^2-3x}{e^2-6x-2x^2}dx-\int \frac{(-e^2-2x^2)\log (x)}{e^2-6x-2x^2}dx-\int (\log (x)+\frac{2(e^2-3x)\log (x)}{e^2-6x-2x^2})dx-\int \log (3-\frac{e^2}{2x}+x)dx\\ & =-2x-x^2+x\log (x)\log (3-\frac{e^2}{2x}+x)+\frac{3}{2}\int \frac{-6-4x}{e^2-6x-2x^2}dx-2\int \frac{(e^2-3x)\log (x)}{e^2-6x-2x^2}dx+(9+2e^2)\int \frac{1}{e^2-6x-2x^2}dx+\int \frac{-e^2-2x^2}{e^2-6x-2x^2}dx-\int \log (x)dx-\int (\log (x)-\frac{2(e^2-3x)\log (x)}{e^2-6x-2x^2})dx\\ & =-x-x^2-x\log (x)+x\log (x)\log (3-\frac{e^2}{2x}+x)+\frac{3}{2}\log (e^2-6x-2x^2)+2\int \frac{(e^2-3x)\log (x)}{e^2-6x-2x^2}dx-2\int (\frac{(-3-\sqrt{9+2e^2})\log (x)}{-6-2\sqrt{9+2e^2}-4x}+\frac{(-3+\sqrt{9+2e^2})\log (x)}{-6+2\sqrt{9+2e^2}-4x})dx-\left(2(9+2e^2)\right)\mathrm{Subst}(\int \frac{1}{4(9+2e^2)-x^2}dx,x,-6-4x)+\int (1-\frac{2(e^2-3x)}{e^2-6x-2x^2})dx-\int \log (x)dx\\ & =x-x^2+\sqrt{9+2e^2}{\tanh }^{-1}\left(\frac{3+2x}{\sqrt{9+2e^2}}\right)-2x\log (x)+x\log (x)\log (3-\frac{e^2}{2x}+x)+\frac{3}{2}\log (e^2-6x-2x^2)-2\int \frac{e^2-3x}{e^2-6x-2x^2}dx+2\int (\frac{(-3-\sqrt{9+2e^2})\log (x)}{-6-2\sqrt{9+2e^2}-4x}+\frac{(-3+\sqrt{9+2e^2})\log (x)}{-6+2\sqrt{9+2e^2}-4x})dx+\left(2(3-\sqrt{9+2e^2})\right)\int \frac{\log (x)}{-6+2\sqrt{9+2e^2}-4x}dx+\left(2(3+\sqrt{9+2e^2})\right)\int \frac{\log (x)}{-6-2\sqrt{9+2e^2}-4x}dx\\ & =x-x^2+\sqrt{9+2e^2}{\tanh }^{-1}\left(\frac{3+2x}{\sqrt{9+2e^2}}\right)-\frac{1}{2}(3-\sqrt{9+2e^2})\log \left(\frac{1}{2}(-3+\sqrt{9+2e^2})\right)\log (-2(3-\sqrt{9+2e^2})-4x)-2x\log (x)+x\log (x)\log (3-\frac{e^2}{2x}+x)-\frac{1}{2}(3+\sqrt{9+2e^2})\log (x)\log (1+\frac{2x}{3+\sqrt{9+2e^2}})+\frac{3}{2}\log (e^2-6x-2x^2)-\frac{3}{2}\int \frac{-6-4x}{e^2-6x-2x^2}dx-(9+2e^2)\int \frac{1}{e^2-6x-2x^2}dx-\left(2(3-\sqrt{9+2e^2})\right)\int \frac{\log (x)}{-6+2\sqrt{9+2e^2}-4x}dx+\left(2(3-\sqrt{9+2e^2})\right)\int \frac{\log \left(\frac{4x}{-6+2\sqrt{9+2e^2}}\right)}{-6+2\sqrt{9+2e^2}-4x}dx+\frac{1}{2}(3+\sqrt{9+2e^2})\int \frac{\log (1-\frac{4x}{-6-2\sqrt{9+2e^2}})}{x}dx-\left(2(3+\sqrt{9+2e^2})\right)\int \frac{\log (x)}{-6-2\sqrt{9+2e^2}-4x}dx\\ & =x-x^2+\sqrt{9+2e^2}{\tanh }^{-1}\left(\frac{3+2x}{\sqrt{9+2e^2}}\right)-2x\log (x)+x\log (x)\log (3-\frac{e^2}{2x}+x)-\frac{1}{2}(3+\sqrt{9+2e^2}){\text{Li}}_2(-\frac{2x}{3+\sqrt{9+2e^2}})+\frac{1}{2}(3-\sqrt{9+2e^2}){\text{Li}}_2(1+\frac{2x}{3-\sqrt{9+2e^2}})+\left(2(9+2e^2)\right)\mathrm{Subst}(\int \frac{1}{4(9+2e^2)-x^2}dx,x,-6-4x)-\left(2(3-\sqrt{9+2e^2})\right)\int \frac{\log \left(\frac{4x}{-6+2\sqrt{9+2e^2}}\right)}{-6+2\sqrt{9+2e^2}-4x}dx-\frac{1}{2}(3+\sqrt{9+2e^2})\int \frac{\log (1-\frac{4x}{-6-2\sqrt{9+2e^2}})}{x}dx\\ & =x-x^2-2x\log (x)+x\log (x)\log (3-\frac{e^2}{2x}+x)\end{array}\end{array}$$

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Mathematica [A]  time = 0.23, size = 26, normalized size = 0.93 $$\begin{array}{c}x(1-x+\log (x)(-2+\log (3-\frac{e^2}{2x}+x)))\end{array}$$

Antiderivative was successfully verified.

[In]

Integrate[(E^2*(-1 - 2*x) + 6*x + 14*x^2 + 4*x^3 + (E^2 - 6*x - 2*x^2)*Log[(-E^2 + 6*x + 2*x^2)/(2*x)] + Log[x
]*(-3*E^2 + 12*x + 2*x^2 + (E^2 - 6*x - 2*x^2)*Log[(-E^2 + 6*x + 2*x^2)/(2*x)]))/(E^2 - 6*x - 2*x^2),x]

[Out]

x*(1 - x + Log[x]*(-2 + Log[3 - E^2/(2*x) + x]))

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fricas [A]  time = 1.15, size = 35, normalized size = 1.25 $$\begin{array}{c}-x^2+(x\log \left(\frac{2x^2+6x-e^2}{2x}\right)-2x)\log (x)+x\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((exp(2)-2*x^2-6*x)*log(1/2*(-exp(2)+2*x^2+6*x)/x)-3*exp(2)+2*x^2+12*x)*log(x)+(exp(2)-2*x^2-6*x)*l
og(1/2*(-exp(2)+2*x^2+6*x)/x)+(-2*x-1)*exp(2)+4*x^3+14*x^2+6*x)/(exp(2)-2*x^2-6*x),x, algorithm="fricas")

[Out]

-x^2 + (x*log(1/2*(2*x^2 + 6*x - e^2)/x) - 2*x)*log(x) + x

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giac [A]  time = 0.28, size = 44, normalized size = 1.57 $$\begin{array}{c}-x\log (2)\log (x)+x\log (2x^2+6x-e^2)\log (x)-x\log (x{)}^2-x^2-2x\log (x)+x\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((exp(2)-2*x^2-6*x)*log(1/2*(-exp(2)+2*x^2+6*x)/x)-3*exp(2)+2*x^2+12*x)*log(x)+(exp(2)-2*x^2-6*x)*l
og(1/2*(-exp(2)+2*x^2+6*x)/x)+(-2*x-1)*exp(2)+4*x^3+14*x^2+6*x)/(exp(2)-2*x^2-6*x),x, algorithm="giac")

[Out]

-x*log(2)*log(x) + x*log(2*x^2 + 6*x - e^2)*log(x) - x*log(x)^2 - x^2 - 2*x*log(x) + x

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maple [C]  time = 0.31, size = 227, normalized size = 8.11

method	result	size
risch	$\ln (x)x\ln ({\mathrm{e}}^2-2x^2-6x)-x\ln (x{)}^2-\frac{i\ln (x)\pi x\mathrm{csgn}\left(\frac{i}{x}\right)\mathrm{csgn}\left(i({\mathrm{e}}^2-2x^2-6x)\right)\mathrm{csgn}\left(\frac{i({\mathrm{e}}^2-2x^2-6x)}{x}\right)}{2}+\frac{i\ln (x)\pi x\mathrm{csgn}\left(\frac{i}{x}\right)\mathrm{csgn}{\left(\frac{i({\mathrm{e}}^2-2x^2-6x)}{x}\right)}^2}{2}-i\ln (x)\pi x\mathrm{csgn}{\left(\frac{i({\mathrm{e}}^2-2x^2-6x)}{x}\right)}^2+\frac{i\ln (x)\pi x\mathrm{csgn}\left(i({\mathrm{e}}^2-2x^2-6x)\right)\mathrm{csgn}{\left(\frac{i({\mathrm{e}}^2-2x^2-6x)}{x}\right)}^2}{2}+\frac{i\ln (x)\pi x\mathrm{csgn}{\left(\frac{i({\mathrm{e}}^2-2x^2-6x)}{x}\right)}^3}{2}+i\pi x\ln (x)-x\ln (2)\ln (x)-2x\ln (x)-x^2+x$	$227$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((exp(2)-2*x^2-6*x)*ln(1/2*(-exp(2)+2*x^2+6*x)/x)-3*exp(2)+2*x^2+12*x)*ln(x)+(exp(2)-2*x^2-6*x)*ln(1/2*(-
exp(2)+2*x^2+6*x)/x)+(-2*x-1)*exp(2)+4*x^3+14*x^2+6*x)/(exp(2)-2*x^2-6*x),x,method=_RETURNVERBOSE)

[Out]

ln(x)*x*ln(exp(2)-2*x^2-6*x)-x*ln(x)^2-1/2*I*ln(x)*Pi*x*csgn(I/x)*csgn(I*(exp(2)-2*x^2-6*x))*csgn(I/x*(exp(2)-
2*x^2-6*x))+1/2*I*ln(x)*Pi*x*csgn(I/x)*csgn(I/x*(exp(2)-2*x^2-6*x))^2-I*ln(x)*Pi*x*csgn(I/x*(exp(2)-2*x^2-6*x)
)^2+1/2*I*ln(x)*Pi*x*csgn(I*(exp(2)-2*x^2-6*x))*csgn(I/x*(exp(2)-2*x^2-6*x))^2+1/2*I*ln(x)*Pi*x*csgn(I/x*(exp(
2)-2*x^2-6*x))^3+I*Pi*x*ln(x)-x*ln(2)*ln(x)-2*x*ln(x)-x^2+x

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maxima [B]  time = 0.52, size = 318, normalized size = 11.36 $$\begin{array}{c}-x(\log (2)+2)\log (x)+x\log (2x^2+6x-e^2)\log (x)-x\log (x{)}^2-x^2-\frac{1}{2}(\frac{3\log \left(\frac{2x-\sqrt{2e^2+9}+3}{2x+\sqrt{2e^2+9}+3}\right)}{\sqrt{2e^2+9}}-\log (2x^2+6x-e^2))e^2-\frac{1}{2}(e^2+18)\log (2x^2+6x-e^2)-\frac{7(e^2+9)\log \left(\frac{2x-\sqrt{2e^2+9}+3}{2x+\sqrt{2e^2+9}+3}\right)}{2\sqrt{2e^2+9}}+\frac{9(e^2+6)\log \left(\frac{2x-\sqrt{2e^2+9}+3}{2x+\sqrt{2e^2+9}+3}\right)}{2\sqrt{2e^2+9}}+\frac{e^2\log \left(\frac{2x-\sqrt{2e^2+9}+3}{2x+\sqrt{2e^2+9}+3}\right)}{2\sqrt{2e^2+9}}+x+\frac{9\log \left(\frac{2x-\sqrt{2e^2+9}+3}{2x+\sqrt{2e^2+9}+3}\right)}{2\sqrt{2e^2+9}}+9\log (2x^2+6x-e^2)\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((exp(2)-2*x^2-6*x)*log(1/2*(-exp(2)+2*x^2+6*x)/x)-3*exp(2)+2*x^2+12*x)*log(x)+(exp(2)-2*x^2-6*x)*l
og(1/2*(-exp(2)+2*x^2+6*x)/x)+(-2*x-1)*exp(2)+4*x^3+14*x^2+6*x)/(exp(2)-2*x^2-6*x),x, algorithm="maxima")

[Out]

-x*(log(2) + 2)*log(x) + x*log(2*x^2 + 6*x - e^2)*log(x) - x*log(x)^2 - x^2 - 1/2*(3*log((2*x - sqrt(2*e^2 + 9
) + 3)/(2*x + sqrt(2*e^2 + 9) + 3))/sqrt(2*e^2 + 9) - log(2*x^2 + 6*x - e^2))*e^2 - 1/2*(e^2 + 18)*log(2*x^2 +
6*x - e^2) - 7/2*(e^2 + 9)*log((2*x - sqrt(2*e^2 + 9) + 3)/(2*x + sqrt(2*e^2 + 9) + 3))/sqrt(2*e^2 + 9) + 9/2
*(e^2 + 6)*log((2*x - sqrt(2*e^2 + 9) + 3)/(2*x + sqrt(2*e^2 + 9) + 3))/sqrt(2*e^2 + 9) + 1/2*e^2*log((2*x - s
qrt(2*e^2 + 9) + 3)/(2*x + sqrt(2*e^2 + 9) + 3))/sqrt(2*e^2 + 9) + x + 9/2*log((2*x - sqrt(2*e^2 + 9) + 3)/(2*
x + sqrt(2*e^2 + 9) + 3))/sqrt(2*e^2 + 9) + 9*log(2*x^2 + 6*x - e^2)

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mupad [B]  time = 4.45, size = 32, normalized size = 1.14 $$\begin{array}{c}x-2x\ln (x)-x^2+x\ln \left(\frac{x^2+3x-\frac{{\mathrm{e}}^2}{2}}{x}\right)\ln (x)\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(6*x - log((3*x - exp(2)/2 + x^2)/x)*(6*x - exp(2) + 2*x^2) + log(x)*(12*x - 3*exp(2) - log((3*x - exp(2)
/2 + x^2)/x)*(6*x - exp(2) + 2*x^2) + 2*x^2) + 14*x^2 + 4*x^3 - exp(2)*(2*x + 1))/(6*x - exp(2) + 2*x^2),x)

[Out]

x - 2*x*log(x) - x^2 + x*log((3*x - exp(2)/2 + x^2)/x)*log(x)

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sympy [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 $$\begin{array}{c}\text{Exception raised: CoercionFailed}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((exp(2)-2*x**2-6*x)*ln(1/2*(-exp(2)+2*x**2+6*x)/x)-3*exp(2)+2*x**2+12*x)*ln(x)+(exp(2)-2*x**2-6*x)
*ln(1/2*(-exp(2)+2*x**2+6*x)/x)+(-2*x-1)*exp(2)+4*x**3+14*x**2+6*x)/(exp(2)-2*x**2-6*x),x)

[Out]

Exception raised: CoercionFailed

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