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

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

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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 {gather*} -x^2+x-2 x \log (x)+x \log (x) \log \left (x-\frac {e^2}{2 x}+3\right ) \end {gather*}

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 {gather*} \begin {aligned} \text {integral} &=\int \left (-1-2 x+\log \left (3-\frac {e^2}{2 x}+x\right )+\frac {\log (x) \left (-3 e^2+2 x (6+x)+\left (e^2-2 x (3+x)\right ) \log \left (3-\frac {e^2}{2 x}+x\right )\right )}{e^2-6 x-2 x^2}\right ) \, dx\\ &=-x-x^2+\int \log \left (3-\frac {e^2}{2 x}+x\right ) \, dx+\int \frac {\log (x) \left (-3 e^2+2 x (6+x)+\left (e^2-2 x (3+x)\right ) \log \left (3-\frac {e^2}{2 x}+x\right )\right )}{e^2-6 x-2 x^2} \, dx\\ &=-x-x^2+x \log \left (3-\frac {e^2}{2 x}+x\right )-\int \frac {-e^2-2 x^2}{e^2-6 x-2 x^2} \, dx+\int \left (-\frac {\left (3 e^2-12 x-2 x^2\right ) \log (x)}{e^2-6 x-2 x^2}+\log (x) \log \left (3-\frac {e^2}{2 x}+x\right )\right ) \, dx\\ &=-x-x^2+x \log \left (3-\frac {e^2}{2 x}+x\right )-\int \left (1-\frac {2 \left (e^2-3 x\right )}{e^2-6 x-2 x^2}\right ) \, dx-\int \frac {\left (3 e^2-12 x-2 x^2\right ) \log (x)}{e^2-6 x-2 x^2} \, dx+\int \log (x) \log \left (3-\frac {e^2}{2 x}+x\right ) \, dx\\ &=-2 x-x^2+x \log \left (3-\frac {e^2}{2 x}+x\right )+x \log (x) \log \left (3-\frac {e^2}{2 x}+x\right )+2 \int \frac {e^2-3 x}{e^2-6 x-2 x^2} \, dx-\int \frac {\left (-e^2-2 x^2\right ) \log (x)}{e^2-6 x-2 x^2} \, dx-\int \left (\log (x)+\frac {2 \left (e^2-3 x\right ) \log (x)}{e^2-6 x-2 x^2}\right ) \, dx-\int \log \left (3-\frac {e^2}{2 x}+x\right ) \, dx\\ &=-2 x-x^2+x \log (x) \log \left (3-\frac {e^2}{2 x}+x\right )+\frac {3}{2} \int \frac {-6-4 x}{e^2-6 x-2 x^2} \, dx-2 \int \frac {\left (e^2-3 x\right ) \log (x)}{e^2-6 x-2 x^2} \, dx+\left (9+2 e^2\right ) \int \frac {1}{e^2-6 x-2 x^2} \, dx+\int \frac {-e^2-2 x^2}{e^2-6 x-2 x^2} \, dx-\int \log (x) \, dx-\int \left (\log (x)-\frac {2 \left (e^2-3 x\right ) \log (x)}{e^2-6 x-2 x^2}\right ) \, dx\\ &=-x-x^2-x \log (x)+x \log (x) \log \left (3-\frac {e^2}{2 x}+x\right )+\frac {3}{2} \log \left (e^2-6 x-2 x^2\right )+2 \int \frac {\left (e^2-3 x\right ) \log (x)}{e^2-6 x-2 x^2} \, dx-2 \int \left (\frac {\left (-3-\sqrt {9+2 e^2}\right ) \log (x)}{-6-2 \sqrt {9+2 e^2}-4 x}+\frac {\left (-3+\sqrt {9+2 e^2}\right ) \log (x)}{-6+2 \sqrt {9+2 e^2}-4 x}\right ) \, dx-\left (2 \left (9+2 e^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{4 \left (9+2 e^2\right )-x^2} \, dx,x,-6-4 x\right )+\int \left (1-\frac {2 \left (e^2-3 x\right )}{e^2-6 x-2 x^2}\right ) \, dx-\int \log (x) \, dx\\ &=x-x^2+\sqrt {9+2 e^2} \tanh ^{-1}\left (\frac {3+2 x}{\sqrt {9+2 e^2}}\right )-2 x \log (x)+x \log (x) \log \left (3-\frac {e^2}{2 x}+x\right )+\frac {3}{2} \log \left (e^2-6 x-2 x^2\right )-2 \int \frac {e^2-3 x}{e^2-6 x-2 x^2} \, dx+2 \int \left (\frac {\left (-3-\sqrt {9+2 e^2}\right ) \log (x)}{-6-2 \sqrt {9+2 e^2}-4 x}+\frac {\left (-3+\sqrt {9+2 e^2}\right ) \log (x)}{-6+2 \sqrt {9+2 e^2}-4 x}\right ) \, dx+\left (2 \left (3-\sqrt {9+2 e^2}\right )\right ) \int \frac {\log (x)}{-6+2 \sqrt {9+2 e^2}-4 x} \, dx+\left (2 \left (3+\sqrt {9+2 e^2}\right )\right ) \int \frac {\log (x)}{-6-2 \sqrt {9+2 e^2}-4 x} \, dx\\ &=x-x^2+\sqrt {9+2 e^2} \tanh ^{-1}\left (\frac {3+2 x}{\sqrt {9+2 e^2}}\right )-\frac {1}{2} \left (3-\sqrt {9+2 e^2}\right ) \log \left (\frac {1}{2} \left (-3+\sqrt {9+2 e^2}\right )\right ) \log \left (-2 \left (3-\sqrt {9+2 e^2}\right )-4 x\right )-2 x \log (x)+x \log (x) \log \left (3-\frac {e^2}{2 x}+x\right )-\frac {1}{2} \left (3+\sqrt {9+2 e^2}\right ) \log (x) \log \left (1+\frac {2 x}{3+\sqrt {9+2 e^2}}\right )+\frac {3}{2} \log \left (e^2-6 x-2 x^2\right )-\frac {3}{2} \int \frac {-6-4 x}{e^2-6 x-2 x^2} \, dx-\left (9+2 e^2\right ) \int \frac {1}{e^2-6 x-2 x^2} \, dx-\left (2 \left (3-\sqrt {9+2 e^2}\right )\right ) \int \frac {\log (x)}{-6+2 \sqrt {9+2 e^2}-4 x} \, dx+\left (2 \left (3-\sqrt {9+2 e^2}\right )\right ) \int \frac {\log \left (\frac {4 x}{-6+2 \sqrt {9+2 e^2}}\right )}{-6+2 \sqrt {9+2 e^2}-4 x} \, dx+\frac {1}{2} \left (3+\sqrt {9+2 e^2}\right ) \int \frac {\log \left (1-\frac {4 x}{-6-2 \sqrt {9+2 e^2}}\right )}{x} \, dx-\left (2 \left (3+\sqrt {9+2 e^2}\right )\right ) \int \frac {\log (x)}{-6-2 \sqrt {9+2 e^2}-4 x} \, dx\\ &=x-x^2+\sqrt {9+2 e^2} \tanh ^{-1}\left (\frac {3+2 x}{\sqrt {9+2 e^2}}\right )-2 x \log (x)+x \log (x) \log \left (3-\frac {e^2}{2 x}+x\right )-\frac {1}{2} \left (3+\sqrt {9+2 e^2}\right ) \text {Li}_2\left (-\frac {2 x}{3+\sqrt {9+2 e^2}}\right )+\frac {1}{2} \left (3-\sqrt {9+2 e^2}\right ) \text {Li}_2\left (1+\frac {2 x}{3-\sqrt {9+2 e^2}}\right )+\left (2 \left (9+2 e^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{4 \left (9+2 e^2\right )-x^2} \, dx,x,-6-4 x\right )-\left (2 \left (3-\sqrt {9+2 e^2}\right )\right ) \int \frac {\log \left (\frac {4 x}{-6+2 \sqrt {9+2 e^2}}\right )}{-6+2 \sqrt {9+2 e^2}-4 x} \, dx-\frac {1}{2} \left (3+\sqrt {9+2 e^2}\right ) \int \frac {\log \left (1-\frac {4 x}{-6-2 \sqrt {9+2 e^2}}\right )}{x} \, dx\\ &=x-x^2-2 x \log (x)+x \log (x) \log \left (3-\frac {e^2}{2 x}+x\right )\\ \end {aligned} \end {gather*}

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

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 {gather*} -x^{2} + {\left (x \log \left (\frac {2 \, x^{2} + 6 \, x - e^{2}}{2 \, x}\right ) - 2 \, x\right )} \log \relax (x) + x \end {gather*}

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 {gather*} -x \log \relax (2) \log \relax (x) + x \log \left (2 \, x^{2} + 6 \, x - e^{2}\right ) \log \relax (x) - x \log \relax (x)^{2} - x^{2} - 2 \, x \log \relax (x) + x \end {gather*}

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	\(\ln \relax (x ) x \ln \left ({\mathrm e}^{2}-2 x^{2}-6 x \right )-x \ln \relax (x )^{2}-\frac {i \ln \relax (x ) \pi x \,\mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (i \left ({\mathrm e}^{2}-2 x^{2}-6 x \right )\right ) \mathrm {csgn}\left (\frac {i \left ({\mathrm e}^{2}-2 x^{2}-6 x \right )}{x}\right )}{2}+\frac {i \ln \relax (x ) \pi x \,\mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (\frac {i \left ({\mathrm e}^{2}-2 x^{2}-6 x \right )}{x}\right )^{2}}{2}-i \ln \relax (x ) \pi x \mathrm {csgn}\left (\frac {i \left ({\mathrm e}^{2}-2 x^{2}-6 x \right )}{x}\right )^{2}+\frac {i \ln \relax (x ) \pi x \,\mathrm {csgn}\left (i \left ({\mathrm e}^{2}-2 x^{2}-6 x \right )\right ) \mathrm {csgn}\left (\frac {i \left ({\mathrm e}^{2}-2 x^{2}-6 x \right )}{x}\right )^{2}}{2}+\frac {i \ln \relax (x ) \pi x \mathrm {csgn}\left (\frac {i \left ({\mathrm e}^{2}-2 x^{2}-6 x \right )}{x}\right )^{3}}{2}+i \pi x \ln \relax (x )-x \ln \relax (2) \ln \relax (x )-2 x \ln \relax (x )-x^{2}+x\)	\(227\)

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 {gather*} -x {\left (\log \relax (2) + 2\right )} \log \relax (x) + x \log \left (2 \, x^{2} + 6 \, x - e^{2}\right ) \log \relax (x) - x \log \relax (x)^{2} - x^{2} - \frac {1}{2} \, {\left (\frac {3 \, \log \left (\frac {2 \, x - \sqrt {2 \, e^{2} + 9} + 3}{2 \, x + \sqrt {2 \, e^{2} + 9} + 3}\right )}{\sqrt {2 \, e^{2} + 9}} - \log \left (2 \, x^{2} + 6 \, x - e^{2}\right )\right )} e^{2} - \frac {1}{2} \, {\left (e^{2} + 18\right )} \log \left (2 \, x^{2} + 6 \, x - e^{2}\right ) - \frac {7 \, {\left (e^{2} + 9\right )} \log \left (\frac {2 \, x - \sqrt {2 \, e^{2} + 9} + 3}{2 \, x + \sqrt {2 \, e^{2} + 9} + 3}\right )}{2 \, \sqrt {2 \, e^{2} + 9}} + \frac {9 \, {\left (e^{2} + 6\right )} \log \left (\frac {2 \, x - \sqrt {2 \, e^{2} + 9} + 3}{2 \, x + \sqrt {2 \, e^{2} + 9} + 3}\right )}{2 \, \sqrt {2 \, e^{2} + 9}} + \frac {e^{2} \log \left (\frac {2 \, x - \sqrt {2 \, e^{2} + 9} + 3}{2 \, x + \sqrt {2 \, e^{2} + 9} + 3}\right )}{2 \, \sqrt {2 \, e^{2} + 9}} + x + \frac {9 \, \log \left (\frac {2 \, x - \sqrt {2 \, e^{2} + 9} + 3}{2 \, x + \sqrt {2 \, e^{2} + 9} + 3}\right )}{2 \, \sqrt {2 \, e^{2} + 9}} + 9 \, \log \left (2 \, x^{2} + 6 \, x - e^{2}\right ) \end {gather*}

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 {gather*} x-2\,x\,\ln \relax (x)-x^2+x\,\ln \left (\frac {x^2+3\,x-\frac {{\mathrm {e}}^2}{2}}{x}\right )\,\ln \relax (x) \end {gather*}

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 {gather*} \text {Exception raised: CoercionFailed} \end {gather*}

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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