3.60.90 \(\int \frac {5022+2832 x-48 x^2-292 x^3-62 x^4-4 x^5+(186+36 x-22 x^2-4 x^3) \log (\frac {e^x (3-x)}{3+x})}{-9+x^2} \, dx\)

Optimal. Leaf size=28 \[ 1-\left (2+(5+x)^2+\log \left (\frac {e^x (3-x)}{3+x}\right )\right )^2 \]

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Rubi [B]  time = 0.55, antiderivative size = 107, normalized size of antiderivative = 3.82, number of steps used = 32, number of rules used = 18, integrand size = 64, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.281, Rules used = {6725, 207, 260, 321, 266, 43, 302, 1810, 2548, 444, 2551, 459, 206, 2552, 12, 388, 5910, 5948} \begin {gather*} -x^4-20 x^3-153 x^2-2 x^2 \log \left (\frac {e^x (3-x)}{x+3}\right )-594 x-22 x \log \left (\frac {e^x (3-x)}{x+3}\right )-4 x \tanh ^{-1}\left (\frac {x}{3}\right )+4 \tanh ^{-1}\left (\frac {x}{3}\right )^2+108 \tanh ^{-1}\left (\frac {x}{3}\right )+4 \log \left (\frac {e^x (3-x)}{x+3}\right ) \tanh ^{-1}\left (\frac {x}{3}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(5022 + 2832*x - 48*x^2 - 292*x^3 - 62*x^4 - 4*x^5 + (186 + 36*x - 22*x^2 - 4*x^3)*Log[(E^x*(3 - x))/(3 +
x)])/(-9 + x^2),x]

[Out]

-594*x - 153*x^2 - 20*x^3 - x^4 + 108*ArcTanh[x/3] - 4*x*ArcTanh[x/3] + 4*ArcTanh[x/3]^2 - 22*x*Log[(E^x*(3 -
x))/(3 + x)] - 2*x^2*Log[(E^x*(3 - x))/(3 + x)] + 4*ArcTanh[x/3]*Log[(E^x*(3 - x))/(3 + x)]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

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 207

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTanh[(Rt[b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[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 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 302

Int[(x_)^(m_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Int[PolynomialDivide[x^m, a + b*x^n, x], x] /; FreeQ[{a,
b}, x] && IGtQ[m, 0] && IGtQ[n, 0] && GtQ[m, 2*n - 1]

Rule 321

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 388

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[(d*x*(a + b*x^n)^(p + 1))/(b*(n*
(p + 1) + 1)), x] - Dist[(a*d - b*c*(n*(p + 1) + 1))/(b*(n*(p + 1) + 1)), Int[(a + b*x^n)^p, x], x] /; FreeQ[{
a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && NeQ[n*(p + 1) + 1, 0]

Rule 444

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && EqQ[m
- n + 1, 0]

Rule 459

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[(d*(e*x)^(m
+ 1)*(a + b*x^n)^(p + 1))/(b*e*(m + n*(p + 1) + 1)), x] - Dist[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(b*(m +
n*(p + 1) + 1)), Int[(e*x)^m*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && NeQ[b*c - a*d, 0]
&& NeQ[m + n*(p + 1) + 1, 0]

Rule 1810

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

Rule 2548

Int[Log[u_], x_Symbol] :> Simp[x*Log[u], x] - Int[SimplifyIntegrand[(x*D[u, x])/u, x], x] /; InverseFunctionFr
eeQ[u, x]

Rule 2551

Int[Log[u_]*((a_.) + (b_.)*(x_))^(m_.), x_Symbol] :> Simp[((a + b*x)^(m + 1)*Log[u])/(b*(m + 1)), x] - Dist[1/
(b*(m + 1)), Int[SimplifyIntegrand[((a + b*x)^(m + 1)*D[u, x])/u, x], x], x] /; FreeQ[{a, b, m}, x] && Inverse
FunctionFreeQ[u, x] && NeQ[m, -1]

Rule 2552

Int[Log[u_]/(Qx_), x_Symbol] :> With[{v = IntHide[1/Qx, x]}, Simp[v*Log[u], x] - Int[SimplifyIntegrand[(v*D[u,
x])/u, x], x]] /; QuadraticQ[Qx, x] && InverseFunctionFreeQ[u, x]

Rule 5910

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a + b*ArcTanh[c*x])^p, x] - Dist[b*c*p, In
t[(x*(a + b*ArcTanh[c*x])^(p - 1))/(1 - c^2*x^2), x], x] /; FreeQ[{a, b, c}, x] && IGtQ[p, 0]

Rule 5948

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

Rule 6725

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 {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {5022}{-9+x^2}+\frac {2832 x}{-9+x^2}-\frac {48 x^2}{-9+x^2}-\frac {292 x^3}{-9+x^2}-\frac {62 x^4}{-9+x^2}-\frac {4 x^5}{-9+x^2}-\frac {2 \left (-93-18 x+11 x^2+2 x^3\right ) \log \left (-\frac {e^x (-3+x)}{3+x}\right )}{-9+x^2}\right ) \, dx\\ &=-\left (2 \int \frac {\left (-93-18 x+11 x^2+2 x^3\right ) \log \left (-\frac {e^x (-3+x)}{3+x}\right )}{-9+x^2} \, dx\right )-4 \int \frac {x^5}{-9+x^2} \, dx-48 \int \frac {x^2}{-9+x^2} \, dx-62 \int \frac {x^4}{-9+x^2} \, dx-292 \int \frac {x^3}{-9+x^2} \, dx+2832 \int \frac {x}{-9+x^2} \, dx+5022 \int \frac {1}{-9+x^2} \, dx\\ &=-48 x-1674 \tanh ^{-1}\left (\frac {x}{3}\right )+1416 \log \left (9-x^2\right )-2 \int \left (11 \log \left (-\frac {e^x (-3+x)}{3+x}\right )+2 x \log \left (-\frac {e^x (-3+x)}{3+x}\right )+\frac {6 \log \left (-\frac {e^x (-3+x)}{3+x}\right )}{-9+x^2}\right ) \, dx-2 \operatorname {Subst}\left (\int \frac {x^2}{-9+x} \, dx,x,x^2\right )-62 \int \left (9+x^2+\frac {81}{-9+x^2}\right ) \, dx-146 \operatorname {Subst}\left (\int \frac {x}{-9+x} \, dx,x,x^2\right )-432 \int \frac {1}{-9+x^2} \, dx\\ &=-606 x-\frac {62 x^3}{3}-1530 \tanh ^{-1}\left (\frac {x}{3}\right )+1416 \log \left (9-x^2\right )-2 \operatorname {Subst}\left (\int \left (9+\frac {81}{-9+x}+x\right ) \, dx,x,x^2\right )-4 \int x \log \left (-\frac {e^x (-3+x)}{3+x}\right ) \, dx-12 \int \frac {\log \left (-\frac {e^x (-3+x)}{3+x}\right )}{-9+x^2} \, dx-22 \int \log \left (-\frac {e^x (-3+x)}{3+x}\right ) \, dx-146 \operatorname {Subst}\left (\int \left (1+\frac {9}{-9+x}\right ) \, dx,x,x^2\right )-5022 \int \frac {1}{-9+x^2} \, dx\\ &=-606 x-164 x^2-\frac {62 x^3}{3}-x^4+144 \tanh ^{-1}\left (\frac {x}{3}\right )-22 x \log \left (\frac {e^x (3-x)}{3+x}\right )-2 x^2 \log \left (\frac {e^x (3-x)}{3+x}\right )+4 \tanh ^{-1}\left (\frac {x}{3}\right ) \log \left (\frac {e^x (3-x)}{3+x}\right )-60 \log \left (9-x^2\right )+2 \int \frac {x^2 \left (3-x^2\right )}{9-x^2} \, dx+12 \int \frac {\left (-3+x^2\right ) \tanh ^{-1}\left (\frac {x}{3}\right )}{3 \left (9-x^2\right )} \, dx+22 \int \frac {x \left (3-x^2\right )}{9-x^2} \, dx\\ &=-606 x-164 x^2-20 x^3-x^4+144 \tanh ^{-1}\left (\frac {x}{3}\right )-22 x \log \left (\frac {e^x (3-x)}{3+x}\right )-2 x^2 \log \left (\frac {e^x (3-x)}{3+x}\right )+4 \tanh ^{-1}\left (\frac {x}{3}\right ) \log \left (\frac {e^x (3-x)}{3+x}\right )-60 \log \left (9-x^2\right )+4 \int \frac {\left (-3+x^2\right ) \tanh ^{-1}\left (\frac {x}{3}\right )}{9-x^2} \, dx+11 \operatorname {Subst}\left (\int \frac {3-x}{9-x} \, dx,x,x^2\right )-12 \int \frac {x^2}{9-x^2} \, dx\\ &=-594 x-164 x^2-20 x^3-x^4+144 \tanh ^{-1}\left (\frac {x}{3}\right )-22 x \log \left (\frac {e^x (3-x)}{3+x}\right )-2 x^2 \log \left (\frac {e^x (3-x)}{3+x}\right )+4 \tanh ^{-1}\left (\frac {x}{3}\right ) \log \left (\frac {e^x (3-x)}{3+x}\right )-60 \log \left (9-x^2\right )+4 \int \left (-\tanh ^{-1}\left (\frac {x}{3}\right )-\frac {6 \tanh ^{-1}\left (\frac {x}{3}\right )}{-9+x^2}\right ) \, dx+11 \operatorname {Subst}\left (\int \left (1+\frac {6}{-9+x}\right ) \, dx,x,x^2\right )-108 \int \frac {1}{9-x^2} \, dx\\ &=-594 x-153 x^2-20 x^3-x^4+108 \tanh ^{-1}\left (\frac {x}{3}\right )-22 x \log \left (\frac {e^x (3-x)}{3+x}\right )-2 x^2 \log \left (\frac {e^x (3-x)}{3+x}\right )+4 \tanh ^{-1}\left (\frac {x}{3}\right ) \log \left (\frac {e^x (3-x)}{3+x}\right )+6 \log \left (9-x^2\right )-4 \int \tanh ^{-1}\left (\frac {x}{3}\right ) \, dx-24 \int \frac {\tanh ^{-1}\left (\frac {x}{3}\right )}{-9+x^2} \, dx\\ &=-594 x-153 x^2-20 x^3-x^4+108 \tanh ^{-1}\left (\frac {x}{3}\right )-4 x \tanh ^{-1}\left (\frac {x}{3}\right )+4 \tanh ^{-1}\left (\frac {x}{3}\right )^2-22 x \log \left (\frac {e^x (3-x)}{3+x}\right )-2 x^2 \log \left (\frac {e^x (3-x)}{3+x}\right )+4 \tanh ^{-1}\left (\frac {x}{3}\right ) \log \left (\frac {e^x (3-x)}{3+x}\right )+6 \log \left (9-x^2\right )+\frac {4}{3} \int \frac {x}{1-\frac {x^2}{9}} \, dx\\ &=-594 x-153 x^2-20 x^3-x^4+108 \tanh ^{-1}\left (\frac {x}{3}\right )-4 x \tanh ^{-1}\left (\frac {x}{3}\right )+4 \tanh ^{-1}\left (\frac {x}{3}\right )^2-22 x \log \left (\frac {e^x (3-x)}{3+x}\right )-2 x^2 \log \left (\frac {e^x (3-x)}{3+x}\right )+4 \tanh ^{-1}\left (\frac {x}{3}\right ) \log \left (\frac {e^x (3-x)}{3+x}\right )\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.16, size = 130, normalized size = 4.64 \begin {gather*} -594 x-153 x^2-20 x^3-x^4+84 \log (3-x)-\log ^2\left (\frac {-3+x}{3+x}\right )-4 \tanh ^{-1}\left (\frac {x}{3}\right ) \left (-36+x+\log \left (\frac {-3+x}{3+x}\right )-\log \left (-\frac {e^x (-3+x)}{3+x}\right )\right )-22 x \log \left (-\frac {e^x (-3+x)}{3+x}\right )-2 x^2 \log \left (-\frac {e^x (-3+x)}{3+x}\right )+48 \log (3+x)-66 \log \left (9-x^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(5022 + 2832*x - 48*x^2 - 292*x^3 - 62*x^4 - 4*x^5 + (186 + 36*x - 22*x^2 - 4*x^3)*Log[(E^x*(3 - x))
/(3 + x)])/(-9 + x^2),x]

[Out]

-594*x - 153*x^2 - 20*x^3 - x^4 + 84*Log[3 - x] - Log[(-3 + x)/(3 + x)]^2 - 4*ArcTanh[x/3]*(-36 + x + Log[(-3
+ x)/(3 + x)] - Log[-((E^x*(-3 + x))/(3 + x))]) - 22*x*Log[-((E^x*(-3 + x))/(3 + x))] - 2*x^2*Log[-((E^x*(-3 +
x))/(3 + x))] + 48*Log[3 + x] - 66*Log[9 - x^2]

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fricas [B]  time = 0.81, size = 59, normalized size = 2.11 \begin {gather*} -x^{4} - 20 \, x^{3} - 154 \, x^{2} - 2 \, {\left (x^{2} + 10 \, x + 27\right )} \log \left (-\frac {{\left (x - 3\right )} e^{x}}{x + 3}\right ) - \log \left (-\frac {{\left (x - 3\right )} e^{x}}{x + 3}\right )^{2} - 540 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*x^3-22*x^2+36*x+186)*log((3-x)*exp(x)/(3+x))-4*x^5-62*x^4-292*x^3-48*x^2+2832*x+5022)/(x^2-9),x
, algorithm="fricas")

[Out]

-x^4 - 20*x^3 - 154*x^2 - 2*(x^2 + 10*x + 27)*log(-(x - 3)*e^x/(x + 3)) - log(-(x - 3)*e^x/(x + 3))^2 - 540*x

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giac [B]  time = 0.17, size = 66, normalized size = 2.36 \begin {gather*} -x^{4} - 22 \, x^{3} - 175 \, x^{2} - 2 \, {\left (x^{2} + 11 \, x\right )} \log \left (-\frac {x - 3}{x + 3}\right ) - \log \left (-\frac {x - 3}{x + 3}\right )^{2} - 594 \, x + 54 \, \log \left (x + 3\right ) - 54 \, \log \left (x - 3\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*x^3-22*x^2+36*x+186)*log((3-x)*exp(x)/(3+x))-4*x^5-62*x^4-292*x^3-48*x^2+2832*x+5022)/(x^2-9),x
, algorithm="giac")

[Out]

-x^4 - 22*x^3 - 175*x^2 - 2*(x^2 + 11*x)*log(-(x - 3)/(x + 3)) - log(-(x - 3)/(x + 3))^2 - 594*x + 54*log(x +
3) - 54*log(x - 3)

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maple [B]  time = 0.86, size = 172, normalized size = 6.14

method	result	size
default	\(-x^{4}-20 x^{3}-153 x^{2}-594 x +60 \ln \left (3+x \right )-48 \ln \left (x -3\right )-2 \ln \left (\frac {\left (3-x \right ) {\mathrm e}^{x}}{3+x}\right ) x^{2}-2 \ln \left (x -3\right ) \ln \left (\frac {\left (3-x \right ) {\mathrm e}^{x}}{3+x}\right )+2 \ln \left (\frac {\left (3-x \right ) {\mathrm e}^{x}}{3+x}\right ) \ln \left (3+x \right )-22 \ln \left (\frac {\left (3-x \right ) {\mathrm e}^{x}}{3+x}\right ) x +2 \left (x -3\right ) \ln \left (x -3\right )+12-2 \ln \left (x -3\right ) \ln \left (\frac {x}{6}+\frac {1}{2}\right )+\ln \left (x -3\right )^{2}-2 \left (3+x \right ) \ln \left (3+x \right )-2 \left (\ln \left (3+x \right )-\ln \left (\frac {x}{6}+\frac {1}{2}\right )\right ) \ln \left (\frac {1}{2}-\frac {x}{6}\right )+\ln \left (3+x \right )^{2}\)	\(172\)
risch	\(-594 x -54 \ln \left (x -3\right )+20 x \ln \left (3+x \right )+54 \ln \left (3+x \right )-x^{4}-20 x^{3}-153 x^{2}-20 \ln \left (x -3\right ) x -2 i \pi \,x^{2}-2 i \ln \left (x -3\right ) \pi +2 i \ln \left (3+x \right ) \pi -i \ln \left (x -3\right ) \pi \,\mathrm {csgn}\left (i \left (x -3\right )\right ) \mathrm {csgn}\left (\frac {i \left (x -3\right )}{3+x}\right )^{2}-i \ln \left (x -3\right ) \pi \,\mathrm {csgn}\left (\frac {i \left (x -3\right )}{3+x}\right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{x} \left (x -3\right )}{3+x}\right )^{2}-i \pi \,x^{2} \mathrm {csgn}\left (\frac {i}{3+x}\right ) \mathrm {csgn}\left (\frac {i \left (x -3\right )}{3+x}\right )^{2}+2 \ln \left (3+x \right ) x^{2}-\ln \left (x -3\right )^{2}-\ln \left (3+x \right )^{2}-22 i x \pi +i \ln \left (3+x \right ) \pi \mathrm {csgn}\left (\frac {i {\mathrm e}^{x} \left (x -3\right )}{3+x}\right )^{2} \mathrm {csgn}\left (i {\mathrm e}^{x}\right )+i \ln \left (3+x \right ) \pi \,\mathrm {csgn}\left (\frac {i \left (x -3\right )}{3+x}\right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{x} \left (x -3\right )}{3+x}\right )^{2}-11 i \pi x \,\mathrm {csgn}\left (\frac {i \left (x -3\right )}{3+x}\right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{x} \left (x -3\right )}{3+x}\right )^{2}-11 i \pi x \mathrm {csgn}\left (\frac {i {\mathrm e}^{x} \left (x -3\right )}{3+x}\right )^{2} \mathrm {csgn}\left (i {\mathrm e}^{x}\right )-i \pi \,x^{2} \mathrm {csgn}\left (\frac {i {\mathrm e}^{x} \left (x -3\right )}{3+x}\right )^{2} \mathrm {csgn}\left (i {\mathrm e}^{x}\right )-11 i \pi x \,\mathrm {csgn}\left (\frac {i}{3+x}\right ) \mathrm {csgn}\left (\frac {i \left (x -3\right )}{3+x}\right )^{2}-11 i \pi x \,\mathrm {csgn}\left (i \left (x -3\right )\right ) \mathrm {csgn}\left (\frac {i \left (x -3\right )}{3+x}\right )^{2}+2 i \ln \left (x -3\right ) \pi \mathrm {csgn}\left (\frac {i {\mathrm e}^{x} \left (x -3\right )}{3+x}\right )^{2}+22 i \pi x \mathrm {csgn}\left (\frac {i {\mathrm e}^{x} \left (x -3\right )}{3+x}\right )^{2}-2 i \ln \left (3+x \right ) \pi \mathrm {csgn}\left (\frac {i {\mathrm e}^{x} \left (x -3\right )}{3+x}\right )^{2}+2 \ln \left (3+x \right ) \ln \left (x -3\right )-2 x^{2} \ln \left (x -3\right )+\left (-2 x^{2}-22 x +2 \ln \left (3+x \right )-2 \ln \left (x -3\right )\right ) \ln \left ({\mathrm e}^{x}\right )+11 i \pi x \mathrm {csgn}\left (\frac {i \left (x -3\right )}{3+x}\right )^{3}-i \ln \left (3+x \right ) \pi \mathrm {csgn}\left (\frac {i \left (x -3\right )}{3+x}\right )^{3}+i \ln \left (x -3\right ) \pi \mathrm {csgn}\left (\frac {i \left (x -3\right )}{3+x}\right )^{3}+i \ln \left (x -3\right ) \pi \,\mathrm {csgn}\left (\frac {i}{3+x}\right ) \mathrm {csgn}\left (i \left (x -3\right )\right ) \mathrm {csgn}\left (\frac {i \left (x -3\right )}{3+x}\right )+i \ln \left (x -3\right ) \pi \,\mathrm {csgn}\left (\frac {i \left (x -3\right )}{3+x}\right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{x} \left (x -3\right )}{3+x}\right ) \mathrm {csgn}\left (i {\mathrm e}^{x}\right )+11 i \pi x \,\mathrm {csgn}\left (\frac {i}{3+x}\right ) \mathrm {csgn}\left (i \left (x -3\right )\right ) \mathrm {csgn}\left (\frac {i \left (x -3\right )}{3+x}\right )+11 i \pi x \,\mathrm {csgn}\left (\frac {i \left (x -3\right )}{3+x}\right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{x} \left (x -3\right )}{3+x}\right ) \mathrm {csgn}\left (i {\mathrm e}^{x}\right )-i \pi \,x^{2} \mathrm {csgn}\left (\frac {i {\mathrm e}^{x} \left (x -3\right )}{3+x}\right )^{3}-i \ln \left (x -3\right ) \pi \mathrm {csgn}\left (\frac {i {\mathrm e}^{x} \left (x -3\right )}{3+x}\right )^{3}+i \pi \,x^{2} \mathrm {csgn}\left (\frac {i \left (x -3\right )}{3+x}\right )^{3}+i \ln \left (3+x \right ) \pi \mathrm {csgn}\left (\frac {i {\mathrm e}^{x} \left (x -3\right )}{3+x}\right )^{3}-i \pi \,x^{2} \mathrm {csgn}\left (i \left (x -3\right )\right ) \mathrm {csgn}\left (\frac {i \left (x -3\right )}{3+x}\right )^{2}-i \pi \,x^{2} \mathrm {csgn}\left (\frac {i \left (x -3\right )}{3+x}\right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{x} \left (x -3\right )}{3+x}\right )^{2}-i \ln \left (x -3\right ) \pi \,\mathrm {csgn}\left (\frac {i}{3+x}\right ) \mathrm {csgn}\left (\frac {i \left (x -3\right )}{3+x}\right )^{2}-i \ln \left (x -3\right ) \pi \mathrm {csgn}\left (\frac {i {\mathrm e}^{x} \left (x -3\right )}{3+x}\right )^{2} \mathrm {csgn}\left (i {\mathrm e}^{x}\right )+i \ln \left (3+x \right ) \pi \,\mathrm {csgn}\left (\frac {i}{3+x}\right ) \mathrm {csgn}\left (\frac {i \left (x -3\right )}{3+x}\right )^{2}+i \ln \left (3+x \right ) \pi \,\mathrm {csgn}\left (i \left (x -3\right )\right ) \mathrm {csgn}\left (\frac {i \left (x -3\right )}{3+x}\right )^{2}-i \ln \left (3+x \right ) \pi \,\mathrm {csgn}\left (\frac {i}{3+x}\right ) \mathrm {csgn}\left (i \left (x -3\right )\right ) \mathrm {csgn}\left (\frac {i \left (x -3\right )}{3+x}\right )-i \ln \left (3+x \right ) \pi \,\mathrm {csgn}\left (\frac {i \left (x -3\right )}{3+x}\right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{x} \left (x -3\right )}{3+x}\right ) \mathrm {csgn}\left (i {\mathrm e}^{x}\right )+i \pi \,x^{2} \mathrm {csgn}\left (\frac {i}{3+x}\right ) \mathrm {csgn}\left (i \left (x -3\right )\right ) \mathrm {csgn}\left (\frac {i \left (x -3\right )}{3+x}\right )+i \pi \,x^{2} \mathrm {csgn}\left (\frac {i \left (x -3\right )}{3+x}\right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{x} \left (x -3\right )}{3+x}\right ) \mathrm {csgn}\left (i {\mathrm e}^{x}\right )-11 i \pi x \mathrm {csgn}\left (\frac {i {\mathrm e}^{x} \left (x -3\right )}{3+x}\right )^{3}+2 i \pi \,x^{2} \mathrm {csgn}\left (\frac {i {\mathrm e}^{x} \left (x -3\right )}{3+x}\right )^{2}\)	\(1195\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-4*x^3-22*x^2+36*x+186)*ln((3-x)*exp(x)/(3+x))-4*x^5-62*x^4-292*x^3-48*x^2+2832*x+5022)/(x^2-9),x,method
=_RETURNVERBOSE)

[Out]

-x^4-20*x^3-153*x^2-594*x+60*ln(3+x)-48*ln(x-3)-2*ln((3-x)*exp(x)/(3+x))*x^2-2*ln(x-3)*ln((3-x)*exp(x)/(3+x))+
2*ln((3-x)*exp(x)/(3+x))*ln(3+x)-22*ln((3-x)*exp(x)/(3+x))*x+2*(x-3)*ln(x-3)+12-2*ln(x-3)*ln(1/6*x+1/2)+ln(x-3
)^2-2*(3+x)*ln(3+x)-2*(ln(3+x)-ln(1/6*x+1/2))*ln(1/2-1/6*x)+ln(3+x)^2

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maxima [B]  time = 0.45, size = 160, normalized size = 5.71 \begin {gather*} -x^{4} - 22 \, x^{3} - 175 \, x^{2} + {\left (2 \, x^{2} + 22 \, x - 51\right )} \log \left (x + 3\right ) + 31 \, {\left (x + \log \left (x - 3\right ) + 3\right )} \log \left (x + 3\right ) - 32 \, \log \left (x + 3\right )^{2} - 31 \, {\left (x - 3\right )} \log \left (x - 3\right ) - \frac {31}{2} \, \log \left (x - 3\right )^{2} - {\left (2 \, x^{2} + 22 \, x - 33 \, \log \left (x + 3\right ) + 15\right )} \log \left (-x + 3\right ) - \frac {33}{2} \, \log \left (-x + 3\right )^{2} - 31 \, {\left (\log \left (x + 3\right ) - \log \left (x - 3\right )\right )} \log \left (-\frac {x e^{x}}{x + 3} + \frac {3 \, e^{x}}{x + 3}\right ) - 594 \, x - 60 \, \log \left (x^{2} - 9\right ) + 72 \, \log \left (x + 3\right ) - 72 \, \log \left (x - 3\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*x^3-22*x^2+36*x+186)*log((3-x)*exp(x)/(3+x))-4*x^5-62*x^4-292*x^3-48*x^2+2832*x+5022)/(x^2-9),x
, algorithm="maxima")

[Out]

-x^4 - 22*x^3 - 175*x^2 + (2*x^2 + 22*x - 51)*log(x + 3) + 31*(x + log(x - 3) + 3)*log(x + 3) - 32*log(x + 3)^
2 - 31*(x - 3)*log(x - 3) - 31/2*log(x - 3)^2 - (2*x^2 + 22*x - 33*log(x + 3) + 15)*log(-x + 3) - 33/2*log(-x
+ 3)^2 - 31*(log(x + 3) - log(x - 3))*log(-x*e^x/(x + 3) + 3*e^x/(x + 3)) - 594*x - 60*log(x^2 - 9) + 72*log(x
+ 3) - 72*log(x - 3)

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mupad [B]  time = 5.41, size = 72, normalized size = 2.57 \begin {gather*} -594\,x-2\,x^2\,\ln \left (-\frac {x-3}{x+3}\right )-{\ln \left (-\frac {x-3}{x+3}\right )}^2-22\,x\,\ln \left (-\frac {x-3}{x+3}\right )-175\,x^2-22\,x^3-x^4-\mathrm {atan}\left (\frac {x\,1{}\mathrm {i}}{3}\right )\,108{}\mathrm {i} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(48*x^2 - log(-(exp(x)*(x - 3))/(x + 3))*(36*x - 22*x^2 - 4*x^3 + 186) - 2832*x + 292*x^3 + 62*x^4 + 4*x^
5 - 5022)/(x^2 - 9),x)

[Out]

- 594*x - atan((x*1i)/3)*108i - 2*x^2*log(-(x - 3)/(x + 3)) - log(-(x - 3)/(x + 3))^2 - 22*x*log(-(x - 3)/(x +
3)) - 175*x^2 - 22*x^3 - x^4

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sympy [B]  time = 0.24, size = 65, normalized size = 2.32 \begin {gather*} - x^{4} - 20 x^{3} - 154 x^{2} - 594 x + \left (- 2 x^{2} - 20 x\right ) \log {\left (\frac {\left (3 - x\right ) e^{x}}{x + 3} \right )} - \log {\left (\frac {\left (3 - x\right ) e^{x}}{x + 3} \right )}^{2} - 54 \log {\left (x - 3 \right )} + 54 \log {\left (x + 3 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*x**3-22*x**2+36*x+186)*ln((3-x)*exp(x)/(3+x))-4*x**5-62*x**4-292*x**3-48*x**2+2832*x+5022)/(x**
2-9),x)

[Out]

-x**4 - 20*x**3 - 154*x**2 - 594*x + (-2*x**2 - 20*x)*log((3 - x)*exp(x)/(x + 3)) - log((3 - x)*exp(x)/(x + 3)
)**2 - 54*log(x - 3) + 54*log(x + 3)

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