3.42.27 \(\int \frac {2 x^2 \log ^2(x)+((96+2 x^2) \log (x)+(-96-2 x^2) \log ^2(x)) \log (\frac {1}{16} e^4 (48+x^2))}{48 x^3+x^5} \, dx\)

Optimal. Leaf size=22 \[ \frac {\log ^2(x) \log \left (e^4 \left (3+\frac {x^2}{16}\right )\right )}{x^2} \]

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Rubi [C]  time = 2.08, antiderivative size = 188, normalized size of antiderivative = 8.55, number of steps used = 42, number of rules used = 27, integrand size = 59, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.458, Rules used = {1593, 6688, 12, 14, 2357, 2304, 2301, 2337, 2391, 2454, 2395, 36, 31, 29, 2376, 2392, 6742, 2305, 2302, 30, 2374, 6589, 2366, 2303, 2378, 266, 2345} \begin {gather*} -\frac {1}{96} \text {Li}_2\left (-\frac {48}{x^2}\right )-\frac {1}{96} \text {Li}_2\left (-\frac {x^2}{48}\right )-\frac {1}{96} \text {Li}_3\left (-\frac {48}{x^2}\right )+\frac {1}{96} \text {Li}_3\left (-\frac {x^2}{48}\right )-\frac {1}{48} \text {Li}_2\left (-\frac {48}{x^2}\right ) \log (x)-\frac {1}{48} \text {Li}_2\left (-\frac {x^2}{48}\right ) \log (x)+\frac {1}{48} \log \left (\frac {48}{x^2}+1\right ) \log ^2(x)-\frac {1}{48} \log \left (\frac {x^2}{48}+1\right ) \log ^2(x)+\frac {\log \left (\frac {x^2}{16}+3\right ) \log ^2(x)}{x^2}+\frac {4 \log ^2(x)}{x^2}+\frac {1}{48} \log \left (\frac {48}{x^2}+1\right ) \log (x)-\frac {1}{48} \log \left (x^2+48\right ) \log (x)+\frac {\log ^3(x)}{72}+\frac {\log ^2(x)}{48}+\frac {1}{48} \log (48) \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(2*x^2*Log[x]^2 + ((96 + 2*x^2)*Log[x] + (-96 - 2*x^2)*Log[x]^2)*Log[(E^4*(48 + x^2))/16])/(48*x^3 + x^5),
x]

[Out]

(Log[48]*Log[x])/48 + (Log[1 + 48/x^2]*Log[x])/48 + Log[x]^2/48 + (4*Log[x]^2)/x^2 + (Log[1 + 48/x^2]*Log[x]^2
)/48 + Log[x]^3/72 - (Log[x]^2*Log[1 + x^2/48])/48 + (Log[x]^2*Log[3 + x^2/16])/x^2 - (Log[x]*Log[48 + x^2])/4
8 - PolyLog[2, -48/x^2]/96 - (Log[x]*PolyLog[2, -48/x^2])/48 - PolyLog[2, -1/48*x^2]/96 - (Log[x]*PolyLog[2, -
1/48*x^2])/48 - PolyLog[3, -48/x^2]/96 + PolyLog[3, -1/48*x^2]/96

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 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 36

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

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 1593

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 2301

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/(x_), x_Symbol] :> Simp[(a + b*Log[c*x^n])^2/(2*b*n), x] /; FreeQ[{a
, b, c, n}, x]

Rule 2302

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

Rule 2303

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

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 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 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 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 2366

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_.) + Log[(f_.)*(x_)^(r_.)]*(e_.))*((g_.)*(x_))^(m_.), x_Sy
mbol] :> With[{u = IntHide[(g*x)^m*(a + b*Log[c*x^n])^p, x]}, Dist[d + e*Log[f*x^r], u, x] - Dist[e*r, Int[Sim
plifyIntegrand[u/x, x], x], x]] /; FreeQ[{a, b, c, d, e, f, g, m, n, p, r}, x] &&  !(EqQ[p, 1] && EqQ[a, 0] &&
NeQ[d, 0])

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 2376

Int[Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))^(r_.)]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((g_.)*(x_))^(q_.), x_Sym
bol] :> With[{u = IntHide[(g*x)^q*Log[d*(e + f*x^m)^r], x]}, Dist[a + b*Log[c*x^n], u, x] - Dist[b*n, Int[Dist
[1/x, u, x], x], x]] /; FreeQ[{a, b, c, d, e, f, g, r, m, n, q}, x] && (IntegerQ[(q + 1)/m] || (RationalQ[m] &
& RationalQ[q])) && NeQ[q, -1]

Rule 2378

Int[Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))^(r_.)]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((g_.)*(x_))^(q_.),
x_Symbol] :> With[{u = IntHide[(g*x)^q*(a + b*Log[c*x^n])^p, x]}, Dist[Log[d*(e + f*x^m)^r], u, x] - Dist[f*m
*r, Int[Dist[x^(m - 1)/(e + f*x^m), u, x], x], x]] /; FreeQ[{a, b, c, d, e, f, g, r, m, n, q}, x] && IGtQ[p, 0
] && RationalQ[m] && RationalQ[q]

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 2392

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))]*(b_.))/(x_), x_Symbol] :> Simp[(a + b*Log[c*d])*Log[x], x] + Dist[
b, Int[Log[1 + (e*x)/d]/x, x], x] /; FreeQ[{a, b, c, d, e}, x] && GtQ[c*d, 0]

Rule 2395

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

Rule 2454

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

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 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 {2 x^2 \log ^2(x)+\left (\left (96+2 x^2\right ) \log (x)+\left (-96-2 x^2\right ) \log ^2(x)\right ) \log \left (\frac {1}{16} e^4 \left (48+x^2\right )\right )}{x^3 \left (48+x^2\right )} \, dx\\ &=\int \frac {2 \log (x) \left (\frac {x^2 \log (x)}{48+x^2}-(-1+\log (x)) \left (4+\log \left (3+\frac {x^2}{16}\right )\right )\right )}{x^3} \, dx\\ &=2 \int \frac {\log (x) \left (\frac {x^2 \log (x)}{48+x^2}-(-1+\log (x)) \left (4+\log \left (3+\frac {x^2}{16}\right )\right )\right )}{x^3} \, dx\\ &=2 \int \left (-\frac {\log (x) \left (-192-4 x^2+192 \log (x)+3 x^2 \log (x)\right )}{x^3 \left (48+x^2\right )}-\frac {(-1+\log (x)) \log (x) \log \left (3+\frac {x^2}{16}\right )}{x^3}\right ) \, dx\\ &=-\left (2 \int \frac {\log (x) \left (-192-4 x^2+192 \log (x)+3 x^2 \log (x)\right )}{x^3 \left (48+x^2\right )} \, dx\right )-2 \int \frac {(-1+\log (x)) \log (x) \log \left (3+\frac {x^2}{16}\right )}{x^3} \, dx\\ &=-\left (2 \int \frac {\log (x) \left (-4+\frac {3 \left (64+x^2\right ) \log (x)}{48+x^2}\right )}{x^3} \, dx\right )-2 \int \left (-\frac {\log (x) \log \left (3+\frac {x^2}{16}\right )}{x^3}+\frac {\log ^2(x) \log \left (3+\frac {x^2}{16}\right )}{x^3}\right ) \, dx\\ &=-\left (2 \int \left (-\frac {4 \log (x)}{x^3}+\frac {3 \left (64+x^2\right ) \log ^2(x)}{x^3 \left (48+x^2\right )}\right ) \, dx\right )+2 \int \frac {\log (x) \log \left (3+\frac {x^2}{16}\right )}{x^3} \, dx-2 \int \frac {\log ^2(x) \log \left (3+\frac {x^2}{16}\right )}{x^3} \, dx\\ &=\frac {\log ^2(x)}{24}+\frac {\log \left (3+\frac {x^2}{16}\right )}{2 x^2}+\frac {\log ^2(x) \log \left (3+\frac {x^2}{16}\right )}{x^2}-\frac {1}{48} \log (x) \log \left (48+x^2\right )+\frac {1}{4} \int \left (-\frac {4}{x \left (48+x^2\right )}-\frac {8 \log (x)}{x \left (48+x^2\right )}-\frac {8 \log ^2(x)}{x \left (48+x^2\right )}\right ) \, dx-2 \int \left (\frac {\log (x)}{48 x}-\frac {\log \left (3+\frac {x^2}{16}\right )}{2 x^3}-\frac {\log \left (48+x^2\right )}{96 x}\right ) \, dx-6 \int \frac {\left (64+x^2\right ) \log ^2(x)}{x^3 \left (48+x^2\right )} \, dx+8 \int \frac {\log (x)}{x^3} \, dx\\ &=-\frac {2}{x^2}-\frac {4 \log (x)}{x^2}+\frac {\log ^2(x)}{24}+\frac {\log \left (3+\frac {x^2}{16}\right )}{2 x^2}+\frac {\log ^2(x) \log \left (3+\frac {x^2}{16}\right )}{x^2}-\frac {1}{48} \log (x) \log \left (48+x^2\right )+\frac {1}{48} \int \frac {\log \left (48+x^2\right )}{x} \, dx-\frac {1}{24} \int \frac {\log (x)}{x} \, dx-2 \int \frac {\log (x)}{x \left (48+x^2\right )} \, dx-2 \int \frac {\log ^2(x)}{x \left (48+x^2\right )} \, dx-6 \int \left (\frac {4 \log ^2(x)}{3 x^3}-\frac {\log ^2(x)}{144 x}+\frac {x \log ^2(x)}{144 \left (48+x^2\right )}\right ) \, dx-\int \frac {1}{x \left (48+x^2\right )} \, dx+\int \frac {\log \left (3+\frac {x^2}{16}\right )}{x^3} \, dx\\ &=-\frac {2}{x^2}-\frac {4 \log (x)}{x^2}+\frac {1}{48} \log \left (1+\frac {48}{x^2}\right ) \log (x)+\frac {\log ^2(x)}{48}+\frac {1}{48} \log \left (1+\frac {48}{x^2}\right ) \log ^2(x)+\frac {\log \left (3+\frac {x^2}{16}\right )}{2 x^2}+\frac {\log ^2(x) \log \left (3+\frac {x^2}{16}\right )}{x^2}-\frac {1}{48} \log (x) \log \left (48+x^2\right )+\frac {1}{96} \operatorname {Subst}\left (\int \frac {\log (48+x)}{x} \, dx,x,x^2\right )-\frac {1}{48} \int \frac {\log \left (1+\frac {48}{x^2}\right )}{x} \, dx-\frac {1}{24} \int \frac {\log \left (1+\frac {48}{x^2}\right ) \log (x)}{x} \, dx+\frac {1}{24} \int \frac {\log ^2(x)}{x} \, dx-\frac {1}{24} \int \frac {x \log ^2(x)}{48+x^2} \, dx-\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{x (48+x)} \, dx,x,x^2\right )+\frac {1}{2} \operatorname {Subst}\left (\int \frac {\log \left (3+\frac {x}{16}\right )}{x^2} \, dx,x,x^2\right )-8 \int \frac {\log ^2(x)}{x^3} \, dx\\ &=-\frac {2}{x^2}-\frac {4 \log (x)}{x^2}+\frac {1}{48} \log (48) \log (x)+\frac {1}{48} \log \left (1+\frac {48}{x^2}\right ) \log (x)+\frac {\log ^2(x)}{48}+\frac {4 \log ^2(x)}{x^2}+\frac {1}{48} \log \left (1+\frac {48}{x^2}\right ) \log ^2(x)-\frac {1}{48} \log ^2(x) \log \left (1+\frac {x^2}{48}\right )+\frac {\log ^2(x) \log \left (3+\frac {x^2}{16}\right )}{x^2}-\frac {1}{48} \log (x) \log \left (48+x^2\right )-\frac {1}{96} \text {Li}_2\left (-\frac {48}{x^2}\right )-\frac {1}{48} \log (x) \text {Li}_2\left (-\frac {48}{x^2}\right )-\frac {1}{96} \operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,x^2\right )+\frac {1}{96} \operatorname {Subst}\left (\int \frac {1}{48+x} \, dx,x,x^2\right )+\frac {1}{96} \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {x}{48}\right )}{x} \, dx,x,x^2\right )+\frac {1}{48} \int \frac {\text {Li}_2\left (-\frac {48}{x^2}\right )}{x} \, dx+\frac {1}{32} \operatorname {Subst}\left (\int \frac {1}{\left (3+\frac {x}{16}\right ) x} \, dx,x,x^2\right )+\frac {1}{24} \int \frac {\log (x) \log \left (1+\frac {x^2}{48}\right )}{x} \, dx+\frac {1}{24} \operatorname {Subst}\left (\int x^2 \, dx,x,\log (x)\right )-8 \int \frac {\log (x)}{x^3} \, dx\\ &=-\frac {\log (x)}{48}+\frac {1}{48} \log (48) \log (x)+\frac {1}{48} \log \left (1+\frac {48}{x^2}\right ) \log (x)+\frac {\log ^2(x)}{48}+\frac {4 \log ^2(x)}{x^2}+\frac {1}{48} \log \left (1+\frac {48}{x^2}\right ) \log ^2(x)+\frac {\log ^3(x)}{72}-\frac {1}{48} \log ^2(x) \log \left (1+\frac {x^2}{48}\right )+\frac {\log ^2(x) \log \left (3+\frac {x^2}{16}\right )}{x^2}+\frac {1}{96} \log \left (48+x^2\right )-\frac {1}{48} \log (x) \log \left (48+x^2\right )-\frac {1}{96} \text {Li}_2\left (-\frac {48}{x^2}\right )-\frac {1}{48} \log (x) \text {Li}_2\left (-\frac {48}{x^2}\right )-\frac {1}{96} \text {Li}_2\left (-\frac {x^2}{48}\right )-\frac {1}{48} \log (x) \text {Li}_2\left (-\frac {x^2}{48}\right )-\frac {1}{96} \text {Li}_3\left (-\frac {48}{x^2}\right )-\frac {\operatorname {Subst}\left (\int \frac {1}{3+\frac {x}{16}} \, dx,x,x^2\right )}{1536}+\frac {1}{96} \operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,x^2\right )+\frac {1}{48} \int \frac {\text {Li}_2\left (-\frac {x^2}{48}\right )}{x} \, dx\\ &=\frac {1}{48} \log (48) \log (x)+\frac {1}{48} \log \left (1+\frac {48}{x^2}\right ) \log (x)+\frac {\log ^2(x)}{48}+\frac {4 \log ^2(x)}{x^2}+\frac {1}{48} \log \left (1+\frac {48}{x^2}\right ) \log ^2(x)+\frac {\log ^3(x)}{72}-\frac {1}{48} \log ^2(x) \log \left (1+\frac {x^2}{48}\right )+\frac {\log ^2(x) \log \left (3+\frac {x^2}{16}\right )}{x^2}-\frac {1}{48} \log (x) \log \left (48+x^2\right )-\frac {1}{96} \text {Li}_2\left (-\frac {48}{x^2}\right )-\frac {1}{48} \log (x) \text {Li}_2\left (-\frac {48}{x^2}\right )-\frac {1}{96} \text {Li}_2\left (-\frac {x^2}{48}\right )-\frac {1}{48} \log (x) \text {Li}_2\left (-\frac {x^2}{48}\right )-\frac {1}{96} \text {Li}_3\left (-\frac {48}{x^2}\right )+\frac {1}{96} \text {Li}_3\left (-\frac {x^2}{48}\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.20, size = 33, normalized size = 1.50 \begin {gather*} 2 \left (\frac {2 \log ^2(x)}{x^2}+\frac {\log ^2(x) \log \left (3+\frac {x^2}{16}\right )}{2 x^2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(2*x^2*Log[x]^2 + ((96 + 2*x^2)*Log[x] + (-96 - 2*x^2)*Log[x]^2)*Log[(E^4*(48 + x^2))/16])/(48*x^3 +
x^5),x]

[Out]

2*((2*Log[x]^2)/x^2 + (Log[x]^2*Log[3 + x^2/16])/(2*x^2))

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fricas [A]  time = 0.78, size = 18, normalized size = 0.82 \begin {gather*} \frac {\log \left (\frac {1}{16} \, {\left (x^{2} + 48\right )} e^{4}\right ) \log \relax (x)^{2}}{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-2*x^2-96)*log(x)^2+(2*x^2+96)*log(x))*log(1/16*(x^2+48)*exp(4))+2*x^2*log(x)^2)/(x^5+48*x^3),x,
algorithm="fricas")

[Out]

log(1/16*(x^2 + 48)*e^4)*log(x)^2/x^2

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giac [A]  time = 0.22, size = 28, normalized size = 1.27 \begin {gather*} -\frac {4 \, {\left (\log \relax (2) - 1\right )} \log \relax (x)^{2}}{x^{2}} + \frac {\log \left (x^{2} + 48\right ) \log \relax (x)^{2}}{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-2*x^2-96)*log(x)^2+(2*x^2+96)*log(x))*log(1/16*(x^2+48)*exp(4))+2*x^2*log(x)^2)/(x^5+48*x^3),x,
algorithm="giac")

[Out]

-4*(log(2) - 1)*log(x)^2/x^2 + log(x^2 + 48)*log(x)^2/x^2

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maple [A]  time = 0.07, size = 19, normalized size = 0.86

method	result	size
risch	\(\frac {\ln \relax (x )^{2} \ln \left (\frac {\left (x^{2}+48\right ) {\mathrm e}^{4}}{16}\right )}{x^{2}}\)	\(19\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((-2*x^2-96)*ln(x)^2+(2*x^2+96)*ln(x))*ln(1/16*(x^2+48)*exp(4))+2*x^2*ln(x)^2)/(x^5+48*x^3),x,method=_RET
URNVERBOSE)

[Out]

1/x^2*ln(x)^2*ln(1/16*(x^2+48)*exp(4))

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maxima [A]  time = 0.51, size = 28, normalized size = 1.27 \begin {gather*} -\frac {4 \, {\left (\log \relax (2) - 1\right )} \log \relax (x)^{2} - \log \left (x^{2} + 48\right ) \log \relax (x)^{2}}{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-2*x^2-96)*log(x)^2+(2*x^2+96)*log(x))*log(1/16*(x^2+48)*exp(4))+2*x^2*log(x)^2)/(x^5+48*x^3),x,
algorithm="maxima")

[Out]

-(4*(log(2) - 1)*log(x)^2 - log(x^2 + 48)*log(x)^2)/x^2

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mupad [B]  time = 4.31, size = 20, normalized size = 0.91 \begin {gather*} \frac {{\ln \relax (x)}^2\,\left (\ln \left (x^2+48\right )-\ln \left (16\right )+4\right )}{x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(log((exp(4)*(x^2 + 48))/16)*(log(x)^2*(2*x^2 + 96) - log(x)*(2*x^2 + 96)) - 2*x^2*log(x)^2)/(48*x^3 + x^
5),x)

[Out]

(log(x)^2*(log(x^2 + 48) - log(16) + 4))/x^2

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sympy [A]  time = 2.81, size = 19, normalized size = 0.86 \begin {gather*} \frac {\log {\relax (x )}^{2} \log {\left (\left (\frac {x^{2}}{16} + 3\right ) e^{4} \right )}}{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-2*x**2-96)*ln(x)**2+(2*x**2+96)*ln(x))*ln(1/16*(x**2+48)*exp(4))+2*x**2*ln(x)**2)/(x**5+48*x**3)
,x)

[Out]

log(x)**2*log((x**2/16 + 3)*exp(4))/x**2

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