3.13.25 \(\int \frac {(648 x^4+162 x^6+e^x (864 x^5+216 x^7)+e^{2 x} (432 x^6+108 x^8)+e^{3 x} (96 x^7+24 x^9)+e^{4 x} (8 x^8+2 x^{10})) \log (x)+(-216 x^2+1242 x^4+162 x^6+e^x (-144 x^3+2124 x^5+432 x^6+324 x^7+108 x^8)+e^{2 x} (-24 x^4+1290 x^6+432 x^7+216 x^8+108 x^9)+e^{3 x} (336 x^7+144 x^8+60 x^9+36 x^{10})+e^{4 x} (32 x^8+16 x^9+6 x^{10}+4 x^{11})) \log ^2(x)+(16-140 x^2+e^x (-144 x^3-48 x^4-12 x^5-12 x^6)+e^{2 x} (-32 x^4-16 x^5-4 x^6-4 x^7)) \log ^3(x)-2 x^2 \log ^4(x)}{144 x+72 x^3+9 x^5} \, dx\)

Optimal. Leaf size=34 \[ \frac {\log ^2(x) \left (-x^2 \left (3+e^x x\right )^2+\log (x)\right )^2}{9 \left (4+x^2\right )} \]

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Rubi [C]  time = 30.92, antiderivative size = 343, normalized size of antiderivative = 10.09, number of steps used = 54, number of rules used = 32, integrand size = 308, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.104, Rules used = {1594, 28, 6688, 12, 6742, 266, 43, 2335, 260, 2351, 2337, 2391, 2304, 2349, 2345, 2374, 6589, 2338, 6725, 2176, 2194, 2178, 2288, 2554, 2271, 2269, 14, 2199, 2177, 2353, 2305, 2383} \begin {gather*} -18 \text {Li}_2\left (-\frac {4}{x^2}\right )-18 \text {Li}_2\left (-\frac {x^2}{4}\right )-\frac {3}{2} \text {Li}_3\left (-\frac {4}{x^2}\right )+\frac {3}{2} \text {Li}_3\left (-\frac {x^2}{4}\right )-3 \text {Li}_2\left (-\frac {4}{x^2}\right ) \log (x)-3 \text {Li}_2\left (-\frac {x^2}{4}\right ) \log (x)+\frac {\log ^4(x)}{9 \left (x^2+4\right )}+\frac {8 \log ^3(x)}{x^2+4}+9 x^2 \log ^2(x)+3 \log \left (\frac {4}{x^2}+1\right ) \log ^2(x)+\frac {144 \log ^2(x)}{x^2+4}-3 \log ^2(x) \log \left (\frac {x^2}{4}+1\right )+36 \log \left (\frac {4}{x^2}+1\right ) \log (x)-36 \log (x) \log \left (\frac {x^2}{4}+1\right )+\frac {e^{4 x} x^7 \log (x) \left (x^3 \log (x)+4 x \log (x)\right )}{9 \left (x^2+4\right )^2}+\frac {4 e^{3 x} x^6 \log (x) \left (x^3 \log (x)+4 x \log (x)\right )}{3 \left (x^2+4\right )^2}+\frac {2 e^{2 x} x^3 \log (x) \left (27 x^5 \log (x)-x^3 \log ^2(x)+108 x^3 \log (x)-4 x \log ^2(x)\right )}{9 \left (x^2+4\right )^2}+\frac {4 e^x x^2 \log (x) \left (9 x^5 \log (x)-x^3 \log ^2(x)+36 x^3 \log (x)-4 x \log ^2(x)\right )}{3 \left (x^2+4\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((648*x^4 + 162*x^6 + E^x*(864*x^5 + 216*x^7) + E^(2*x)*(432*x^6 + 108*x^8) + E^(3*x)*(96*x^7 + 24*x^9) +
E^(4*x)*(8*x^8 + 2*x^10))*Log[x] + (-216*x^2 + 1242*x^4 + 162*x^6 + E^x*(-144*x^3 + 2124*x^5 + 432*x^6 + 324*x
^7 + 108*x^8) + E^(2*x)*(-24*x^4 + 1290*x^6 + 432*x^7 + 216*x^8 + 108*x^9) + E^(3*x)*(336*x^7 + 144*x^8 + 60*x
^9 + 36*x^10) + E^(4*x)*(32*x^8 + 16*x^9 + 6*x^10 + 4*x^11))*Log[x]^2 + (16 - 140*x^2 + E^x*(-144*x^3 - 48*x^4
- 12*x^5 - 12*x^6) + E^(2*x)*(-32*x^4 - 16*x^5 - 4*x^6 - 4*x^7))*Log[x]^3 - 2*x^2*Log[x]^4)/(144*x + 72*x^3 +
9*x^5),x]

[Out]

36*Log[1 + 4/x^2]*Log[x] + 9*x^2*Log[x]^2 + (144*Log[x]^2)/(4 + x^2) + 3*Log[1 + 4/x^2]*Log[x]^2 + (8*Log[x]^3
)/(4 + x^2) + Log[x]^4/(9*(4 + x^2)) + (4*E^(3*x)*x^6*Log[x]*(4*x*Log[x] + x^3*Log[x]))/(3*(4 + x^2)^2) + (E^(
4*x)*x^7*Log[x]*(4*x*Log[x] + x^3*Log[x]))/(9*(4 + x^2)^2) + (4*E^x*x^2*Log[x]*(36*x^3*Log[x] + 9*x^5*Log[x] -
4*x*Log[x]^2 - x^3*Log[x]^2))/(3*(4 + x^2)^2) + (2*E^(2*x)*x^3*Log[x]*(108*x^3*Log[x] + 27*x^5*Log[x] - 4*x*L
og[x]^2 - x^3*Log[x]^2))/(9*(4 + x^2)^2) - 36*Log[x]*Log[1 + x^2/4] - 3*Log[x]^2*Log[1 + x^2/4] - 18*PolyLog[2
, -4/x^2] - 3*Log[x]*PolyLog[2, -4/x^2] - 18*PolyLog[2, -1/4*x^2] - 3*Log[x]*PolyLog[2, -1/4*x^2] - (3*PolyLog
[3, -4/x^2])/2 + (3*PolyLog[3, -1/4*x^2])/2

Rule 12

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

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 28

Int[(u_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Dist[1/c^p, Int[u*(b/2 + c*x^n)^(2*
p), x], x] /; FreeQ[{a, b, c, n}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

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 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 1594

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

Rule 2176

Int[((b_.)*(F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^m
*(b*F^(g*(e + f*x)))^n)/(f*g*n*Log[F]), x] - Dist[(d*m)/(f*g*n*Log[F]), Int[(c + d*x)^(m - 1)*(b*F^(g*(e + f*x
)))^n, x], x] /; FreeQ[{F, b, c, d, e, f, g, n}, x] && GtQ[m, 0] && IntegerQ[2*m] &&  !$UseGamma === True

Rule 2177

Int[((b_.)*(F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_), x_Symbol] :> Simp[((c + d*x)^(m
+ 1)*(b*F^(g*(e + f*x)))^n)/(d*(m + 1)), x] - Dist[(f*g*n*Log[F])/(d*(m + 1)), Int[(c + d*x)^(m + 1)*(b*F^(g*
(e + f*x)))^n, x], x] /; FreeQ[{F, b, c, d, e, f, g, n}, x] && LtQ[m, -1] && IntegerQ[2*m] &&  !$UseGamma ===
True

Rule 2178

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[(F^(g*(e - (c*f)/d))*ExpIntegral
Ei[(f*g*(c + d*x)*Log[F])/d])/d, x] /; FreeQ[{F, c, d, e, f, g}, x] &&  !$UseGamma === True

Rule 2194

Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre
eQ[{F, a, b, c, n}, x]

Rule 2199

Int[(F_)^((c_.)*(v_))*(u_)^(m_.)*(w_), x_Symbol] :> Int[ExpandIntegrand[F^(c*ExpandToSum[v, x]), w*NormalizePo
werOfLinear[u, x]^m, x], x] /; FreeQ[{F, c}, x] && PolynomialQ[w, x] && LinearQ[v, x] && PowerOfLinearQ[u, x]
&& IntegerQ[m] &&  !$UseGamma === True

Rule 2269

Int[(F_)^((g_.)*((d_.) + (e_.)*(x_))^(n_.))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegrand[F^(g*(d +
e*x)^n), 1/(a + c*x^2), x], x] /; FreeQ[{F, a, c, d, e, g, n}, x]

Rule 2271

Int[((F_)^((g_.)*((d_.) + (e_.)*(x_))^(n_.))*(u_)^(m_.))/((a_) + (c_)*(x_)^2), x_Symbol] :> Int[ExpandIntegran
d[F^(g*(d + e*x)^n), u^m/(a + c*x^2), x], x] /; FreeQ[{F, a, c, d, e, g, n}, x] && PolynomialQ[u, x] && Intege
rQ[m]

Rule 2288

Int[(y_.)*(F_)^(u_)*((v_) + (w_)), x_Symbol] :> With[{z = (v*y)/(Log[F]*D[u, x])}, Simp[F^u*z, x] /; EqQ[D[z,
x], w*y]] /; FreeQ[F, x]

Rule 2304

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*Log[c*x^
n]))/(d*(m + 1)), x] - Simp[(b*n*(d*x)^(m + 1))/(d*(m + 1)^2), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1
]

Rule 2305

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

Rule 2335

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

Rule 2337

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.))/((d_) + (e_.)*(x_)^(r_)), x_Symbol] :> Si
mp[(f^m*Log[1 + (e*x^r)/d]*(a + b*Log[c*x^n])^p)/(e*r), x] - Dist[(b*f^m*n*p)/(e*r), Int[(Log[1 + (e*x^r)/d]*(
a + b*Log[c*x^n])^(p - 1))/x, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, r}, x] && EqQ[m, r - 1] && IGtQ[p, 0] &
& (IntegerQ[m] || GtQ[f, 0]) && NeQ[r, n]

Rule 2338

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

Rule 2345

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

Rule 2349

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

Rule 2351

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol] :> Wit
h[{u = ExpandIntegrand[a + b*Log[c*x^n], (f*x)^m*(d + e*x^r)^q, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c,
d, e, f, m, n, q, r}, x] && IntegerQ[q] && (GtQ[q, 0] || (IntegerQ[m] && IntegerQ[r]))

Rule 2353

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol]
:> With[{u = ExpandIntegrand[(a + b*Log[c*x^n])^p, (f*x)^m*(d + e*x^r)^q, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[
{a, b, c, d, e, f, m, n, p, q, r}, x] && IntegerQ[q] && (GtQ[q, 0] || (IGtQ[p, 0] && IntegerQ[m] && IntegerQ[r
]))

Rule 2374

Int[(Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.))/(x_), x_Symbol] :> -Sim
p[(PolyLog[2, -(d*f*x^m)]*(a + b*Log[c*x^n])^p)/m, x] + Dist[(b*n*p)/m, Int[(PolyLog[2, -(d*f*x^m)]*(a + b*Log
[c*x^n])^(p - 1))/x, x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && IGtQ[p, 0] && EqQ[d*e, 1]

Rule 2383

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*PolyLog[k_, (e_.)*(x_)^(q_.)])/(x_), x_Symbol] :> Simp[(PolyL
og[k + 1, e*x^q]*(a + b*Log[c*x^n])^p)/q, x] - Dist[(b*n*p)/q, Int[(PolyLog[k + 1, e*x^q]*(a + b*Log[c*x^n])^(
p - 1))/x, x], x] /; FreeQ[{a, b, c, e, k, n, q}, x] && GtQ[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 2554

Int[Log[u_]*(v_), x_Symbol] :> With[{w = IntHide[v, x]}, Dist[Log[u], w, x] - Int[SimplifyIntegrand[(w*D[u, x]
)/u, x], x] /; InverseFunctionFreeQ[w, x]] /; InverseFunctionFreeQ[u, x]

Rule 6589

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

Rule 6688

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

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]

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 \frac {\left (648 x^4+162 x^6+e^x \left (864 x^5+216 x^7\right )+e^{2 x} \left (432 x^6+108 x^8\right )+e^{3 x} \left (96 x^7+24 x^9\right )+e^{4 x} \left (8 x^8+2 x^{10}\right )\right ) \log (x)+\left (-216 x^2+1242 x^4+162 x^6+e^x \left (-144 x^3+2124 x^5+432 x^6+324 x^7+108 x^8\right )+e^{2 x} \left (-24 x^4+1290 x^6+432 x^7+216 x^8+108 x^9\right )+e^{3 x} \left (336 x^7+144 x^8+60 x^9+36 x^{10}\right )+e^{4 x} \left (32 x^8+16 x^9+6 x^{10}+4 x^{11}\right )\right ) \log ^2(x)+\left (16-140 x^2+e^x \left (-144 x^3-48 x^4-12 x^5-12 x^6\right )+e^{2 x} \left (-32 x^4-16 x^5-4 x^6-4 x^7\right )\right ) \log ^3(x)-2 x^2 \log ^4(x)}{x \left (144+72 x^2+9 x^4\right )} \, dx\\ &=9 \int \frac {\left (648 x^4+162 x^6+e^x \left (864 x^5+216 x^7\right )+e^{2 x} \left (432 x^6+108 x^8\right )+e^{3 x} \left (96 x^7+24 x^9\right )+e^{4 x} \left (8 x^8+2 x^{10}\right )\right ) \log (x)+\left (-216 x^2+1242 x^4+162 x^6+e^x \left (-144 x^3+2124 x^5+432 x^6+324 x^7+108 x^8\right )+e^{2 x} \left (-24 x^4+1290 x^6+432 x^7+216 x^8+108 x^9\right )+e^{3 x} \left (336 x^7+144 x^8+60 x^9+36 x^{10}\right )+e^{4 x} \left (32 x^8+16 x^9+6 x^{10}+4 x^{11}\right )\right ) \log ^2(x)+\left (16-140 x^2+e^x \left (-144 x^3-48 x^4-12 x^5-12 x^6\right )+e^{2 x} \left (-32 x^4-16 x^5-4 x^6-4 x^7\right )\right ) \log ^3(x)-2 x^2 \log ^4(x)}{x \left (36+9 x^2\right )^2} \, dx\\ &=9 \int \frac {2 \log (x) \left (x^4 \left (3+e^x x\right )^4 \left (4+x^2\right )+x^2 \left (3+e^x x\right )^2 \left (-12+69 x^2+72 e^x x^3+\left (3+4 e^x\right )^2 x^4+4 e^x \left (3+2 e^x\right ) x^5+3 e^x \left (2+e^x\right ) x^6+2 e^{2 x} x^7\right ) \log (x)-2 \left (-4+35 x^2+36 e^x x^3+4 e^x \left (3+2 e^x\right ) x^4+e^x \left (3+4 e^x\right ) x^5+e^x \left (3+e^x\right ) x^6+e^{2 x} x^7\right ) \log ^2(x)-x^2 \log ^3(x)\right )}{81 x \left (4+x^2\right )^2} \, dx\\ &=\frac {2}{9} \int \frac {\log (x) \left (x^4 \left (3+e^x x\right )^4 \left (4+x^2\right )+x^2 \left (3+e^x x\right )^2 \left (-12+69 x^2+72 e^x x^3+\left (3+4 e^x\right )^2 x^4+4 e^x \left (3+2 e^x\right ) x^5+3 e^x \left (2+e^x\right ) x^6+2 e^{2 x} x^7\right ) \log (x)-2 \left (-4+35 x^2+36 e^x x^3+4 e^x \left (3+2 e^x\right ) x^4+e^x \left (3+4 e^x\right ) x^5+e^x \left (3+e^x\right ) x^6+e^{2 x} x^7\right ) \log ^2(x)-x^2 \log ^3(x)\right )}{x \left (4+x^2\right )^2} \, dx\\ &=\frac {2}{9} \int \left (\frac {81 x^3 \log (x)}{4+x^2}-\frac {108 x \log ^2(x)}{\left (4+x^2\right )^2}+\frac {621 x^3 \log ^2(x)}{\left (4+x^2\right )^2}+\frac {81 x^5 \log ^2(x)}{\left (4+x^2\right )^2}+\frac {8 \log ^3(x)}{x \left (4+x^2\right )^2}-\frac {70 x \log ^3(x)}{\left (4+x^2\right )^2}-\frac {x \log ^4(x)}{\left (4+x^2\right )^2}+\frac {e^{4 x} x^7 \log (x) \left (4+x^2+16 \log (x)+8 x \log (x)+3 x^2 \log (x)+2 x^3 \log (x)\right )}{\left (4+x^2\right )^2}+\frac {6 e^{3 x} x^6 \log (x) \left (8+2 x^2+28 \log (x)+12 x \log (x)+5 x^2 \log (x)+3 x^3 \log (x)\right )}{\left (4+x^2\right )^2}+\frac {e^{2 x} x^3 \log (x) \left (216 x^2+54 x^4-12 \log (x)+645 x^2 \log (x)+216 x^3 \log (x)+108 x^4 \log (x)+54 x^5 \log (x)-16 \log ^2(x)-8 x \log ^2(x)-2 x^2 \log ^2(x)-2 x^3 \log ^2(x)\right )}{\left (4+x^2\right )^2}+\frac {6 e^x x^2 \log (x) \left (72 x^2+18 x^4-12 \log (x)+177 x^2 \log (x)+36 x^3 \log (x)+27 x^4 \log (x)+9 x^5 \log (x)-12 \log ^2(x)-4 x \log ^2(x)-x^2 \log ^2(x)-x^3 \log ^2(x)\right )}{\left (4+x^2\right )^2}\right ) \, dx\\ &=-\left (\frac {2}{9} \int \frac {x \log ^4(x)}{\left (4+x^2\right )^2} \, dx\right )+\frac {2}{9} \int \frac {e^{4 x} x^7 \log (x) \left (4+x^2+16 \log (x)+8 x \log (x)+3 x^2 \log (x)+2 x^3 \log (x)\right )}{\left (4+x^2\right )^2} \, dx+\frac {2}{9} \int \frac {e^{2 x} x^3 \log (x) \left (216 x^2+54 x^4-12 \log (x)+645 x^2 \log (x)+216 x^3 \log (x)+108 x^4 \log (x)+54 x^5 \log (x)-16 \log ^2(x)-8 x \log ^2(x)-2 x^2 \log ^2(x)-2 x^3 \log ^2(x)\right )}{\left (4+x^2\right )^2} \, dx+\frac {4}{3} \int \frac {e^{3 x} x^6 \log (x) \left (8+2 x^2+28 \log (x)+12 x \log (x)+5 x^2 \log (x)+3 x^3 \log (x)\right )}{\left (4+x^2\right )^2} \, dx+\frac {4}{3} \int \frac {e^x x^2 \log (x) \left (72 x^2+18 x^4-12 \log (x)+177 x^2 \log (x)+36 x^3 \log (x)+27 x^4 \log (x)+9 x^5 \log (x)-12 \log ^2(x)-4 x \log ^2(x)-x^2 \log ^2(x)-x^3 \log ^2(x)\right )}{\left (4+x^2\right )^2} \, dx+\frac {16}{9} \int \frac {\log ^3(x)}{x \left (4+x^2\right )^2} \, dx-\frac {140}{9} \int \frac {x \log ^3(x)}{\left (4+x^2\right )^2} \, dx+18 \int \frac {x^3 \log (x)}{4+x^2} \, dx+18 \int \frac {x^5 \log ^2(x)}{\left (4+x^2\right )^2} \, dx-24 \int \frac {x \log ^2(x)}{\left (4+x^2\right )^2} \, dx+138 \int \frac {x^3 \log ^2(x)}{\left (4+x^2\right )^2} \, dx\\ &=\frac {12 \log ^2(x)}{4+x^2}+\frac {70 \log ^3(x)}{9 \left (4+x^2\right )}+\frac {\log ^4(x)}{9 \left (4+x^2\right )}+\frac {4 e^{3 x} x^6 \log (x) \left (4 x \log (x)+x^3 \log (x)\right )}{3 \left (4+x^2\right )^2}+\frac {e^{4 x} x^7 \log (x) \left (4 x \log (x)+x^3 \log (x)\right )}{9 \left (4+x^2\right )^2}+\frac {4 e^x x^2 \log (x) \left (36 x^3 \log (x)+9 x^5 \log (x)-4 x \log ^2(x)-x^3 \log ^2(x)\right )}{3 \left (4+x^2\right )^2}+\frac {2 e^{2 x} x^3 \log (x) \left (108 x^3 \log (x)+27 x^5 \log (x)-4 x \log ^2(x)-x^3 \log ^2(x)\right )}{9 \left (4+x^2\right )^2}-\frac {4}{9} \int \frac {x \log ^3(x)}{\left (4+x^2\right )^2} \, dx+18 \int \left (x \log (x)-\frac {4 x \log (x)}{4+x^2}\right ) \, dx+18 \int \left (x \log ^2(x)+\frac {16 x \log ^2(x)}{\left (4+x^2\right )^2}-\frac {8 x \log ^2(x)}{4+x^2}\right ) \, dx-\frac {70}{3} \int \frac {\log ^2(x)}{x \left (4+x^2\right )} \, dx-24 \int \frac {\log (x)}{x \left (4+x^2\right )} \, dx+138 \int \left (-\frac {4 x \log ^2(x)}{\left (4+x^2\right )^2}+\frac {x \log ^2(x)}{4+x^2}\right ) \, dx\\ &=3 \log \left (1+\frac {4}{x^2}\right ) \log (x)+\frac {12 \log ^2(x)}{4+x^2}+\frac {35}{12} \log \left (1+\frac {4}{x^2}\right ) \log ^2(x)+\frac {8 \log ^3(x)}{4+x^2}+\frac {\log ^4(x)}{9 \left (4+x^2\right )}+\frac {4 e^{3 x} x^6 \log (x) \left (4 x \log (x)+x^3 \log (x)\right )}{3 \left (4+x^2\right )^2}+\frac {e^{4 x} x^7 \log (x) \left (4 x \log (x)+x^3 \log (x)\right )}{9 \left (4+x^2\right )^2}+\frac {4 e^x x^2 \log (x) \left (36 x^3 \log (x)+9 x^5 \log (x)-4 x \log ^2(x)-x^3 \log ^2(x)\right )}{3 \left (4+x^2\right )^2}+\frac {2 e^{2 x} x^3 \log (x) \left (108 x^3 \log (x)+27 x^5 \log (x)-4 x \log ^2(x)-x^3 \log ^2(x)\right )}{9 \left (4+x^2\right )^2}-\frac {2}{3} \int \frac {\log ^2(x)}{x \left (4+x^2\right )} \, dx-3 \int \frac {\log \left (1+\frac {4}{x^2}\right )}{x} \, dx-\frac {35}{6} \int \frac {\log \left (1+\frac {4}{x^2}\right ) \log (x)}{x} \, dx+18 \int x \log (x) \, dx+18 \int x \log ^2(x) \, dx-72 \int \frac {x \log (x)}{4+x^2} \, dx+138 \int \frac {x \log ^2(x)}{4+x^2} \, dx-144 \int \frac {x \log ^2(x)}{4+x^2} \, dx+288 \int \frac {x \log ^2(x)}{\left (4+x^2\right )^2} \, dx-552 \int \frac {x \log ^2(x)}{\left (4+x^2\right )^2} \, dx\\ &=-\frac {9 x^2}{2}+9 x^2 \log (x)+3 \log \left (1+\frac {4}{x^2}\right ) \log (x)+9 x^2 \log ^2(x)+\frac {144 \log ^2(x)}{4+x^2}+3 \log \left (1+\frac {4}{x^2}\right ) \log ^2(x)+\frac {8 \log ^3(x)}{4+x^2}+\frac {\log ^4(x)}{9 \left (4+x^2\right )}+\frac {4 e^{3 x} x^6 \log (x) \left (4 x \log (x)+x^3 \log (x)\right )}{3 \left (4+x^2\right )^2}+\frac {e^{4 x} x^7 \log (x) \left (4 x \log (x)+x^3 \log (x)\right )}{9 \left (4+x^2\right )^2}+\frac {4 e^x x^2 \log (x) \left (36 x^3 \log (x)+9 x^5 \log (x)-4 x \log ^2(x)-x^3 \log ^2(x)\right )}{3 \left (4+x^2\right )^2}+\frac {2 e^{2 x} x^3 \log (x) \left (108 x^3 \log (x)+27 x^5 \log (x)-4 x \log ^2(x)-x^3 \log ^2(x)\right )}{9 \left (4+x^2\right )^2}-36 \log (x) \log \left (1+\frac {x^2}{4}\right )-3 \log ^2(x) \log \left (1+\frac {x^2}{4}\right )-\frac {3}{2} \text {Li}_2\left (-\frac {4}{x^2}\right )-\frac {35}{12} \log (x) \text {Li}_2\left (-\frac {4}{x^2}\right )-\frac {1}{6} \int \frac {\log \left (1+\frac {4}{x^2}\right ) \log (x)}{x} \, dx+\frac {35}{12} \int \frac {\text {Li}_2\left (-\frac {4}{x^2}\right )}{x} \, dx-18 \int x \log (x) \, dx+36 \int \frac {\log \left (1+\frac {x^2}{4}\right )}{x} \, dx-138 \int \frac {\log (x) \log \left (1+\frac {x^2}{4}\right )}{x} \, dx+144 \int \frac {\log (x) \log \left (1+\frac {x^2}{4}\right )}{x} \, dx+288 \int \frac {\log (x)}{x \left (4+x^2\right )} \, dx-552 \int \frac {\log (x)}{x \left (4+x^2\right )} \, dx\\ &=36 \log \left (1+\frac {4}{x^2}\right ) \log (x)+9 x^2 \log ^2(x)+\frac {144 \log ^2(x)}{4+x^2}+3 \log \left (1+\frac {4}{x^2}\right ) \log ^2(x)+\frac {8 \log ^3(x)}{4+x^2}+\frac {\log ^4(x)}{9 \left (4+x^2\right )}+\frac {4 e^{3 x} x^6 \log (x) \left (4 x \log (x)+x^3 \log (x)\right )}{3 \left (4+x^2\right )^2}+\frac {e^{4 x} x^7 \log (x) \left (4 x \log (x)+x^3 \log (x)\right )}{9 \left (4+x^2\right )^2}+\frac {4 e^x x^2 \log (x) \left (36 x^3 \log (x)+9 x^5 \log (x)-4 x \log ^2(x)-x^3 \log ^2(x)\right )}{3 \left (4+x^2\right )^2}+\frac {2 e^{2 x} x^3 \log (x) \left (108 x^3 \log (x)+27 x^5 \log (x)-4 x \log ^2(x)-x^3 \log ^2(x)\right )}{9 \left (4+x^2\right )^2}-36 \log (x) \log \left (1+\frac {x^2}{4}\right )-3 \log ^2(x) \log \left (1+\frac {x^2}{4}\right )-\frac {3}{2} \text {Li}_2\left (-\frac {4}{x^2}\right )-3 \log (x) \text {Li}_2\left (-\frac {4}{x^2}\right )-18 \text {Li}_2\left (-\frac {x^2}{4}\right )-3 \log (x) \text {Li}_2\left (-\frac {x^2}{4}\right )-\frac {35}{24} \text {Li}_3\left (-\frac {4}{x^2}\right )+\frac {1}{12} \int \frac {\text {Li}_2\left (-\frac {4}{x^2}\right )}{x} \, dx+36 \int \frac {\log \left (1+\frac {4}{x^2}\right )}{x} \, dx-69 \int \frac {\log \left (1+\frac {4}{x^2}\right )}{x} \, dx-69 \int \frac {\text {Li}_2\left (-\frac {x^2}{4}\right )}{x} \, dx+72 \int \frac {\text {Li}_2\left (-\frac {x^2}{4}\right )}{x} \, dx\\ &=36 \log \left (1+\frac {4}{x^2}\right ) \log (x)+9 x^2 \log ^2(x)+\frac {144 \log ^2(x)}{4+x^2}+3 \log \left (1+\frac {4}{x^2}\right ) \log ^2(x)+\frac {8 \log ^3(x)}{4+x^2}+\frac {\log ^4(x)}{9 \left (4+x^2\right )}+\frac {4 e^{3 x} x^6 \log (x) \left (4 x \log (x)+x^3 \log (x)\right )}{3 \left (4+x^2\right )^2}+\frac {e^{4 x} x^7 \log (x) \left (4 x \log (x)+x^3 \log (x)\right )}{9 \left (4+x^2\right )^2}+\frac {4 e^x x^2 \log (x) \left (36 x^3 \log (x)+9 x^5 \log (x)-4 x \log ^2(x)-x^3 \log ^2(x)\right )}{3 \left (4+x^2\right )^2}+\frac {2 e^{2 x} x^3 \log (x) \left (108 x^3 \log (x)+27 x^5 \log (x)-4 x \log ^2(x)-x^3 \log ^2(x)\right )}{9 \left (4+x^2\right )^2}-36 \log (x) \log \left (1+\frac {x^2}{4}\right )-3 \log ^2(x) \log \left (1+\frac {x^2}{4}\right )-18 \text {Li}_2\left (-\frac {4}{x^2}\right )-3 \log (x) \text {Li}_2\left (-\frac {4}{x^2}\right )-18 \text {Li}_2\left (-\frac {x^2}{4}\right )-3 \log (x) \text {Li}_2\left (-\frac {x^2}{4}\right )-\frac {3}{2} \text {Li}_3\left (-\frac {4}{x^2}\right )+\frac {3}{2} \text {Li}_3\left (-\frac {x^2}{4}\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.26, size = 34, normalized size = 1.00 \begin {gather*} \frac {\log ^2(x) \left (-x^2 \left (3+e^x x\right )^2+\log (x)\right )^2}{9 \left (4+x^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((648*x^4 + 162*x^6 + E^x*(864*x^5 + 216*x^7) + E^(2*x)*(432*x^6 + 108*x^8) + E^(3*x)*(96*x^7 + 24*x
^9) + E^(4*x)*(8*x^8 + 2*x^10))*Log[x] + (-216*x^2 + 1242*x^4 + 162*x^6 + E^x*(-144*x^3 + 2124*x^5 + 432*x^6 +
324*x^7 + 108*x^8) + E^(2*x)*(-24*x^4 + 1290*x^6 + 432*x^7 + 216*x^8 + 108*x^9) + E^(3*x)*(336*x^7 + 144*x^8
+ 60*x^9 + 36*x^10) + E^(4*x)*(32*x^8 + 16*x^9 + 6*x^10 + 4*x^11))*Log[x]^2 + (16 - 140*x^2 + E^x*(-144*x^3 -
48*x^4 - 12*x^5 - 12*x^6) + E^(2*x)*(-32*x^4 - 16*x^5 - 4*x^6 - 4*x^7))*Log[x]^3 - 2*x^2*Log[x]^4)/(144*x + 72
*x^3 + 9*x^5),x]

[Out]

(Log[x]^2*(-(x^2*(3 + E^x*x)^2) + Log[x])^2)/(9*(4 + x^2))

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fricas [B]  time = 0.88, size = 88, normalized size = 2.59 \begin {gather*} -\frac {2 \, {\left (x^{4} e^{\left (2 \, x\right )} + 6 \, x^{3} e^{x} + 9 \, x^{2}\right )} \log \relax (x)^{3} - \log \relax (x)^{4} - {\left (x^{8} e^{\left (4 \, x\right )} + 12 \, x^{7} e^{\left (3 \, x\right )} + 54 \, x^{6} e^{\left (2 \, x\right )} + 108 \, x^{5} e^{x} + 81 \, x^{4}\right )} \log \relax (x)^{2}}{9 \, {\left (x^{2} + 4\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x^2*log(x)^4+((-4*x^7-4*x^6-16*x^5-32*x^4)*exp(x)^2+(-12*x^6-12*x^5-48*x^4-144*x^3)*exp(x)-140*x
^2+16)*log(x)^3+((4*x^11+6*x^10+16*x^9+32*x^8)*exp(x)^4+(36*x^10+60*x^9+144*x^8+336*x^7)*exp(x)^3+(108*x^9+216
*x^8+432*x^7+1290*x^6-24*x^4)*exp(x)^2+(108*x^8+324*x^7+432*x^6+2124*x^5-144*x^3)*exp(x)+162*x^6+1242*x^4-216*
x^2)*log(x)^2+((2*x^10+8*x^8)*exp(x)^4+(24*x^9+96*x^7)*exp(x)^3+(108*x^8+432*x^6)*exp(x)^2+(216*x^7+864*x^5)*e
xp(x)+162*x^6+648*x^4)*log(x))/(9*x^5+72*x^3+144*x),x, algorithm="fricas")

[Out]

-1/9*(2*(x^4*e^(2*x) + 6*x^3*e^x + 9*x^2)*log(x)^3 - log(x)^4 - (x^8*e^(4*x) + 12*x^7*e^(3*x) + 54*x^6*e^(2*x)
+ 108*x^5*e^x + 81*x^4)*log(x)^2)/(x^2 + 4)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int -\frac {2 \, {\left (x^{2} \log \relax (x)^{4} + 2 \, {\left (35 \, x^{2} + {\left (x^{7} + x^{6} + 4 \, x^{5} + 8 \, x^{4}\right )} e^{\left (2 \, x\right )} + 3 \, {\left (x^{6} + x^{5} + 4 \, x^{4} + 12 \, x^{3}\right )} e^{x} - 4\right )} \log \relax (x)^{3} - {\left (81 \, x^{6} + 621 \, x^{4} - 108 \, x^{2} + {\left (2 \, x^{11} + 3 \, x^{10} + 8 \, x^{9} + 16 \, x^{8}\right )} e^{\left (4 \, x\right )} + 6 \, {\left (3 \, x^{10} + 5 \, x^{9} + 12 \, x^{8} + 28 \, x^{7}\right )} e^{\left (3 \, x\right )} + 3 \, {\left (18 \, x^{9} + 36 \, x^{8} + 72 \, x^{7} + 215 \, x^{6} - 4 \, x^{4}\right )} e^{\left (2 \, x\right )} + 18 \, {\left (3 \, x^{8} + 9 \, x^{7} + 12 \, x^{6} + 59 \, x^{5} - 4 \, x^{3}\right )} e^{x}\right )} \log \relax (x)^{2} - {\left (81 \, x^{6} + 324 \, x^{4} + {\left (x^{10} + 4 \, x^{8}\right )} e^{\left (4 \, x\right )} + 12 \, {\left (x^{9} + 4 \, x^{7}\right )} e^{\left (3 \, x\right )} + 54 \, {\left (x^{8} + 4 \, x^{6}\right )} e^{\left (2 \, x\right )} + 108 \, {\left (x^{7} + 4 \, x^{5}\right )} e^{x}\right )} \log \relax (x)\right )}}{9 \, {\left (x^{5} + 8 \, x^{3} + 16 \, x\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x^2*log(x)^4+((-4*x^7-4*x^6-16*x^5-32*x^4)*exp(x)^2+(-12*x^6-12*x^5-48*x^4-144*x^3)*exp(x)-140*x
^2+16)*log(x)^3+((4*x^11+6*x^10+16*x^9+32*x^8)*exp(x)^4+(36*x^10+60*x^9+144*x^8+336*x^7)*exp(x)^3+(108*x^9+216
*x^8+432*x^7+1290*x^6-24*x^4)*exp(x)^2+(108*x^8+324*x^7+432*x^6+2124*x^5-144*x^3)*exp(x)+162*x^6+1242*x^4-216*
x^2)*log(x)^2+((2*x^10+8*x^8)*exp(x)^4+(24*x^9+96*x^7)*exp(x)^3+(108*x^8+432*x^6)*exp(x)^2+(216*x^7+864*x^5)*e
xp(x)+162*x^6+648*x^4)*log(x))/(9*x^5+72*x^3+144*x),x, algorithm="giac")

[Out]

integrate(-2/9*(x^2*log(x)^4 + 2*(35*x^2 + (x^7 + x^6 + 4*x^5 + 8*x^4)*e^(2*x) + 3*(x^6 + x^5 + 4*x^4 + 12*x^3
)*e^x - 4)*log(x)^3 - (81*x^6 + 621*x^4 - 108*x^2 + (2*x^11 + 3*x^10 + 8*x^9 + 16*x^8)*e^(4*x) + 6*(3*x^10 + 5
*x^9 + 12*x^8 + 28*x^7)*e^(3*x) + 3*(18*x^9 + 36*x^8 + 72*x^7 + 215*x^6 - 4*x^4)*e^(2*x) + 18*(3*x^8 + 9*x^7 +
12*x^6 + 59*x^5 - 4*x^3)*e^x)*log(x)^2 - (81*x^6 + 324*x^4 + (x^10 + 4*x^8)*e^(4*x) + 12*(x^9 + 4*x^7)*e^(3*x
) + 54*(x^8 + 4*x^6)*e^(2*x) + 108*(x^7 + 4*x^5)*e^x)*log(x))/(x^5 + 8*x^3 + 16*x), x)

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maple [B]  time = 0.10, size = 95, normalized size = 2.79

method	result	size
risch	\(\frac {\ln \relax (x )^{4}}{9 x^{2}+36}-\frac {2 x^{2} \left ({\mathrm e}^{2 x} x^{2}+6 \,{\mathrm e}^{x} x +9\right ) \ln \relax (x )^{3}}{9 \left (x^{2}+4\right )}+\frac {\left (x^{4} {\mathrm e}^{4 x}+12 x^{3} {\mathrm e}^{3 x}+54 \,{\mathrm e}^{2 x} x^{2}+108 \,{\mathrm e}^{x} x +81\right ) x^{4} \ln \relax (x )^{2}}{9 x^{2}+36}\)	\(95\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-2*x^2*ln(x)^4+((-4*x^7-4*x^6-16*x^5-32*x^4)*exp(x)^2+(-12*x^6-12*x^5-48*x^4-144*x^3)*exp(x)-140*x^2+16)*
ln(x)^3+((4*x^11+6*x^10+16*x^9+32*x^8)*exp(x)^4+(36*x^10+60*x^9+144*x^8+336*x^7)*exp(x)^3+(108*x^9+216*x^8+432
*x^7+1290*x^6-24*x^4)*exp(x)^2+(108*x^8+324*x^7+432*x^6+2124*x^5-144*x^3)*exp(x)+162*x^6+1242*x^4-216*x^2)*ln(
x)^2+((2*x^10+8*x^8)*exp(x)^4+(24*x^9+96*x^7)*exp(x)^3+(108*x^8+432*x^6)*exp(x)^2+(216*x^7+864*x^5)*exp(x)+162
*x^6+648*x^4)*ln(x))/(9*x^5+72*x^3+144*x),x,method=_RETURNVERBOSE)

[Out]

1/9/(x^2+4)*ln(x)^4-2/9*x^2*(exp(2*x)*x^2+6*exp(x)*x+9)/(x^2+4)*ln(x)^3+1/9*(x^4*exp(4*x)+12*x^3*exp(3*x)+54*e
xp(2*x)*x^2+108*exp(x)*x+81)*x^4/(x^2+4)*ln(x)^2

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maxima [B]  time = 0.56, size = 105, normalized size = 3.09 \begin {gather*} \frac {x^{8} e^{\left (4 \, x\right )} \log \relax (x)^{2} + 12 \, x^{7} e^{\left (3 \, x\right )} \log \relax (x)^{2} + 81 \, x^{4} \log \relax (x)^{2} - 18 \, x^{2} \log \relax (x)^{3} + \log \relax (x)^{4} + 2 \, {\left (27 \, x^{6} \log \relax (x)^{2} - x^{4} \log \relax (x)^{3}\right )} e^{\left (2 \, x\right )} + 12 \, {\left (9 \, x^{5} \log \relax (x)^{2} - x^{3} \log \relax (x)^{3}\right )} e^{x}}{9 \, {\left (x^{2} + 4\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x^2*log(x)^4+((-4*x^7-4*x^6-16*x^5-32*x^4)*exp(x)^2+(-12*x^6-12*x^5-48*x^4-144*x^3)*exp(x)-140*x
^2+16)*log(x)^3+((4*x^11+6*x^10+16*x^9+32*x^8)*exp(x)^4+(36*x^10+60*x^9+144*x^8+336*x^7)*exp(x)^3+(108*x^9+216
*x^8+432*x^7+1290*x^6-24*x^4)*exp(x)^2+(108*x^8+324*x^7+432*x^6+2124*x^5-144*x^3)*exp(x)+162*x^6+1242*x^4-216*
x^2)*log(x)^2+((2*x^10+8*x^8)*exp(x)^4+(24*x^9+96*x^7)*exp(x)^3+(108*x^8+432*x^6)*exp(x)^2+(216*x^7+864*x^5)*e
xp(x)+162*x^6+648*x^4)*log(x))/(9*x^5+72*x^3+144*x),x, algorithm="maxima")

[Out]

1/9*(x^8*e^(4*x)*log(x)^2 + 12*x^7*e^(3*x)*log(x)^2 + 81*x^4*log(x)^2 - 18*x^2*log(x)^3 + log(x)^4 + 2*(27*x^6
*log(x)^2 - x^4*log(x)^3)*e^(2*x) + 12*(9*x^5*log(x)^2 - x^3*log(x)^3)*e^x)/(x^2 + 4)

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mupad [B]  time = 1.44, size = 173, normalized size = 5.09 \begin {gather*} \frac {8\,{\ln \relax (x)}^3}{x^2+4}+\frac {{\ln \relax (x)}^4}{9\,\left (x^2+4\right )}-2\,{\ln \relax (x)}^3+\frac {9\,x^4\,{\ln \relax (x)}^2}{x^2+4}-\frac {2\,x^4\,{\mathrm {e}}^{2\,x}\,{\ln \relax (x)}^3}{9\,\left (x^2+4\right )}+\frac {6\,x^6\,{\mathrm {e}}^{2\,x}\,{\ln \relax (x)}^2}{x^2+4}+\frac {4\,x^7\,{\mathrm {e}}^{3\,x}\,{\ln \relax (x)}^2}{3\,\left (x^2+4\right )}+\frac {x^8\,{\mathrm {e}}^{4\,x}\,{\ln \relax (x)}^2}{9\,\left (x^2+4\right )}-\frac {4\,x^3\,{\mathrm {e}}^x\,{\ln \relax (x)}^3}{3\,\left (x^2+4\right )}+\frac {12\,x^5\,{\mathrm {e}}^x\,{\ln \relax (x)}^2}{x^2+4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(x)^2*(exp(2*x)*(1290*x^6 - 24*x^4 + 432*x^7 + 216*x^8 + 108*x^9) + exp(x)*(2124*x^5 - 144*x^3 + 432*x
^6 + 324*x^7 + 108*x^8) + exp(4*x)*(32*x^8 + 16*x^9 + 6*x^10 + 4*x^11) + exp(3*x)*(336*x^7 + 144*x^8 + 60*x^9
+ 36*x^10) - 216*x^2 + 1242*x^4 + 162*x^6) + log(x)*(exp(x)*(864*x^5 + 216*x^7) + exp(4*x)*(8*x^8 + 2*x^10) +
exp(3*x)*(96*x^7 + 24*x^9) + exp(2*x)*(432*x^6 + 108*x^8) + 648*x^4 + 162*x^6) - 2*x^2*log(x)^4 - log(x)^3*(ex
p(x)*(144*x^3 + 48*x^4 + 12*x^5 + 12*x^6) + exp(2*x)*(32*x^4 + 16*x^5 + 4*x^6 + 4*x^7) + 140*x^2 - 16))/(144*x
+ 72*x^3 + 9*x^5),x)

[Out]

(8*log(x)^3)/(x^2 + 4) + log(x)^4/(9*(x^2 + 4)) - 2*log(x)^3 + (9*x^4*log(x)^2)/(x^2 + 4) - (2*x^4*exp(2*x)*lo
g(x)^3)/(9*(x^2 + 4)) + (6*x^6*exp(2*x)*log(x)^2)/(x^2 + 4) + (4*x^7*exp(3*x)*log(x)^2)/(3*(x^2 + 4)) + (x^8*e
xp(4*x)*log(x)^2)/(9*(x^2 + 4)) - (4*x^3*exp(x)*log(x)^3)/(3*(x^2 + 4)) + (12*x^5*exp(x)*log(x)^2)/(x^2 + 4)

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sympy [B]  time = 0.96, size = 326, normalized size = 9.59 \begin {gather*} \frac {9 x^{4} \log {\relax (x )}^{2}}{x^{2} + 4} - \frac {2 x^{2} \log {\relax (x )}^{3}}{x^{2} + 4} + \frac {\left (972 x^{13} \log {\relax (x )}^{2} + 11664 x^{11} \log {\relax (x )}^{2} + 46656 x^{9} \log {\relax (x )}^{2} + 62208 x^{7} \log {\relax (x )}^{2}\right ) e^{3 x} + \left (81 x^{14} \log {\relax (x )}^{2} + 972 x^{12} \log {\relax (x )}^{2} + 3888 x^{10} \log {\relax (x )}^{2} + 5184 x^{8} \log {\relax (x )}^{2}\right ) e^{4 x} + \left (8748 x^{11} \log {\relax (x )}^{2} - 972 x^{9} \log {\relax (x )}^{3} + 104976 x^{9} \log {\relax (x )}^{2} - 11664 x^{7} \log {\relax (x )}^{3} + 419904 x^{7} \log {\relax (x )}^{2} - 46656 x^{5} \log {\relax (x )}^{3} + 559872 x^{5} \log {\relax (x )}^{2} - 62208 x^{3} \log {\relax (x )}^{3}\right ) e^{x} + \left (4374 x^{12} \log {\relax (x )}^{2} - 162 x^{10} \log {\relax (x )}^{3} + 52488 x^{10} \log {\relax (x )}^{2} - 1944 x^{8} \log {\relax (x )}^{3} + 209952 x^{8} \log {\relax (x )}^{2} - 7776 x^{6} \log {\relax (x )}^{3} + 279936 x^{6} \log {\relax (x )}^{2} - 10368 x^{4} \log {\relax (x )}^{3}\right ) e^{2 x}}{729 x^{8} + 11664 x^{6} + 69984 x^{4} + 186624 x^{2} + 186624} + \frac {\log {\relax (x )}^{4}}{9 x^{2} + 36} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x**2*ln(x)**4+((-4*x**7-4*x**6-16*x**5-32*x**4)*exp(x)**2+(-12*x**6-12*x**5-48*x**4-144*x**3)*ex
p(x)-140*x**2+16)*ln(x)**3+((4*x**11+6*x**10+16*x**9+32*x**8)*exp(x)**4+(36*x**10+60*x**9+144*x**8+336*x**7)*e
xp(x)**3+(108*x**9+216*x**8+432*x**7+1290*x**6-24*x**4)*exp(x)**2+(108*x**8+324*x**7+432*x**6+2124*x**5-144*x*
*3)*exp(x)+162*x**6+1242*x**4-216*x**2)*ln(x)**2+((2*x**10+8*x**8)*exp(x)**4+(24*x**9+96*x**7)*exp(x)**3+(108*
x**8+432*x**6)*exp(x)**2+(216*x**7+864*x**5)*exp(x)+162*x**6+648*x**4)*ln(x))/(9*x**5+72*x**3+144*x),x)

[Out]

9*x**4*log(x)**2/(x**2 + 4) - 2*x**2*log(x)**3/(x**2 + 4) + ((972*x**13*log(x)**2 + 11664*x**11*log(x)**2 + 46
656*x**9*log(x)**2 + 62208*x**7*log(x)**2)*exp(3*x) + (81*x**14*log(x)**2 + 972*x**12*log(x)**2 + 3888*x**10*l
og(x)**2 + 5184*x**8*log(x)**2)*exp(4*x) + (8748*x**11*log(x)**2 - 972*x**9*log(x)**3 + 104976*x**9*log(x)**2
- 11664*x**7*log(x)**3 + 419904*x**7*log(x)**2 - 46656*x**5*log(x)**3 + 559872*x**5*log(x)**2 - 62208*x**3*log
(x)**3)*exp(x) + (4374*x**12*log(x)**2 - 162*x**10*log(x)**3 + 52488*x**10*log(x)**2 - 1944*x**8*log(x)**3 + 2
09952*x**8*log(x)**2 - 7776*x**6*log(x)**3 + 279936*x**6*log(x)**2 - 10368*x**4*log(x)**3)*exp(2*x))/(729*x**8
+ 11664*x**6 + 69984*x**4 + 186624*x**2 + 186624) + log(x)**4/(9*x**2 + 36)

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