3.63.14 \(\int \frac {(36 x^2+48 x^3-44 x^4-4 e^x x^4-16 x^5+4 x^6) \log (5+x^2)+(-90-60 x-18 x^2-32 x^3+10 x^4-4 x^5+2 x^6+e^x (-5 x^3-x^5)) \log ^2(5+x^2)}{5 x^3+x^5} \, dx\)

Optimal. Leaf size=30 \[ \left (-9-e^x+\frac {\left (-3-2 x+x^2\right )^2}{x^2}\right ) \log ^2\left (5+x^2\right ) \]

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Rubi [C]  time = 5.81, antiderivative size = 620, normalized size of antiderivative = 20.67, number of steps used = 66, number of rules used = 43, integrand size = 108, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.398, Rules used = {1593, 6688, 14, 6742, 6725, 2178, 2194, 2554, 12, 1802, 635, 203, 260, 2528, 2448, 321, 2454, 2395, 36, 29, 31, 2455, 2389, 2295, 2288, 2392, 2391, 2462, 2416, 2390, 2301, 2394, 2393, 2450, 2476, 2470, 4920, 4854, 2402, 2315, 2397, 2457, 2296} \begin {gather*} -16 i \sqrt {5} \text {Li}_2\left (1-\frac {2 \sqrt {5}}{i x+\sqrt {5}}\right )-\frac {48 i \text {Li}_2\left (1-\frac {2 \sqrt {5}}{i x+\sqrt {5}}\right )}{\sqrt {5}}+\frac {2}{5} \left (89-32 i \sqrt {5}\right ) \text {Li}_2\left (\frac {\sqrt {5}-i x}{2 \sqrt {5}}\right )+\frac {2}{5} \left (89+32 i \sqrt {5}\right ) \text {Li}_2\left (\frac {i x+\sqrt {5}}{2 \sqrt {5}}\right )-4 x \log ^2\left (x^2+5\right )+\frac {9 \left (x^2+5\right ) \log ^2\left (x^2+5\right )}{5 x^2}+\left (x^2+5\right ) \log ^2\left (x^2+5\right )+\frac {12 \log ^2\left (x^2+5\right )}{x}-\frac {2}{5} \left (89+32 i \sqrt {5}\right ) \log \left (-x+i \sqrt {5}\right ) \log \left (x^2+5\right )-\frac {2}{5} \left (89-32 i \sqrt {5}\right ) \log \left (x+i \sqrt {5}\right ) \log \left (x^2+5\right )-\frac {e^x \log \left (x^2+5\right ) \left (x^2 \log \left (x^2+5\right )+5 \log \left (x^2+5\right )\right )}{x^2+5}-16 \sqrt {5} \log \left (x^2+5\right ) \tan ^{-1}\left (\frac {x}{\sqrt {5}}\right )-\frac {48 \log \left (x^2+5\right ) \tan ^{-1}\left (\frac {x}{\sqrt {5}}\right )}{\sqrt {5}}+\frac {1}{5} \left (89+32 i \sqrt {5}\right ) \log ^2\left (-x+i \sqrt {5}\right )+\frac {1}{5} \left (89-32 i \sqrt {5}\right ) \log ^2\left (x+i \sqrt {5}\right )+\frac {2}{5} \left (89-32 i \sqrt {5}\right ) \log \left (-\frac {i \left (-x+i \sqrt {5}\right )}{2 \sqrt {5}}\right ) \log \left (x+i \sqrt {5}\right )+\frac {2}{5} \left (89+32 i \sqrt {5}\right ) \log \left (-x+i \sqrt {5}\right ) \log \left (-\frac {i \left (x+i \sqrt {5}\right )}{2 \sqrt {5}}\right )-16 i \sqrt {5} \tan ^{-1}\left (\frac {x}{\sqrt {5}}\right )^2-\frac {48 i \tan ^{-1}\left (\frac {x}{\sqrt {5}}\right )^2}{\sqrt {5}}-32 \sqrt {5} \log \left (\frac {2 \sqrt {5}}{\sqrt {5}+i x}\right ) \tan ^{-1}\left (\frac {x}{\sqrt {5}}\right )-\frac {96 \log \left (\frac {2 \sqrt {5}}{\sqrt {5}+i x}\right ) \tan ^{-1}\left (\frac {x}{\sqrt {5}}\right )}{\sqrt {5}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((36*x^2 + 48*x^3 - 44*x^4 - 4*E^x*x^4 - 16*x^5 + 4*x^6)*Log[5 + x^2] + (-90 - 60*x - 18*x^2 - 32*x^3 + 10
*x^4 - 4*x^5 + 2*x^6 + E^x*(-5*x^3 - x^5))*Log[5 + x^2]^2)/(5*x^3 + x^5),x]

[Out]

((-48*I)*ArcTan[x/Sqrt[5]]^2)/Sqrt[5] - (16*I)*Sqrt[5]*ArcTan[x/Sqrt[5]]^2 + ((89 + (32*I)*Sqrt[5])*Log[I*Sqrt
[5] - x]^2)/5 - (96*ArcTan[x/Sqrt[5]]*Log[(2*Sqrt[5])/(Sqrt[5] + I*x)])/Sqrt[5] - 32*Sqrt[5]*ArcTan[x/Sqrt[5]]
*Log[(2*Sqrt[5])/(Sqrt[5] + I*x)] + (2*(89 - (32*I)*Sqrt[5])*Log[((-1/2*I)*(I*Sqrt[5] - x))/Sqrt[5]]*Log[I*Sqr
t[5] + x])/5 + ((89 - (32*I)*Sqrt[5])*Log[I*Sqrt[5] + x]^2)/5 + (2*(89 + (32*I)*Sqrt[5])*Log[I*Sqrt[5] - x]*Lo
g[((-1/2*I)*(I*Sqrt[5] + x))/Sqrt[5]])/5 - (48*ArcTan[x/Sqrt[5]]*Log[5 + x^2])/Sqrt[5] - 16*Sqrt[5]*ArcTan[x/S
qrt[5]]*Log[5 + x^2] - (2*(89 + (32*I)*Sqrt[5])*Log[I*Sqrt[5] - x]*Log[5 + x^2])/5 - (2*(89 - (32*I)*Sqrt[5])*
Log[I*Sqrt[5] + x]*Log[5 + x^2])/5 + (12*Log[5 + x^2]^2)/x - 4*x*Log[5 + x^2]^2 + (5 + x^2)*Log[5 + x^2]^2 + (
9*(5 + x^2)*Log[5 + x^2]^2)/(5*x^2) - (E^x*Log[5 + x^2]*(5*Log[5 + x^2] + x^2*Log[5 + x^2]))/(5 + x^2) - ((48*
I)*PolyLog[2, 1 - (2*Sqrt[5])/(Sqrt[5] + I*x)])/Sqrt[5] - (16*I)*Sqrt[5]*PolyLog[2, 1 - (2*Sqrt[5])/(Sqrt[5] +
I*x)] + (2*(89 - (32*I)*Sqrt[5])*PolyLog[2, (Sqrt[5] - I*x)/(2*Sqrt[5])])/5 + (2*(89 + (32*I)*Sqrt[5])*PolyLo
g[2, (Sqrt[5] + I*x)/(2*Sqrt[5])])/5

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 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 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[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 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 635

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

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 1802

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

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 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 2295

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

Rule 2296

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a + b*Log[c*x^n])^p, x] - Dist[b*n*p, In
t[(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{a, b, c, n}, x] && GtQ[p, 0] && IntegerQ[2*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 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 2389

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

Rule 2390

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_) + (g_.)*(x_))^(q_.), x_Symbol] :> Dist[1/
e, Subst[Int[((f*x)/d)^q*(a + b*Log[c*x^n])^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p, q}, x]
&& EqQ[e*f - d*g, 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 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 2393

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

Rule 2394

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))/((f_.) + (g_.)*(x_)), x_Symbol] :> Simp[(Log[(e*(f +
g*x))/(e*f - d*g)]*(a + b*Log[c*(d + e*x)^n]))/g, x] - Dist[(b*e*n)/g, Int[Log[(e*(f + g*x))/(e*f - d*g)]/(d +
e*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && NeQ[e*f - d*g, 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 2397

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_)/((f_.) + (g_.)*(x_))^2, x_Symbol] :> Simp[((d +
e*x)*(a + b*Log[c*(d + e*x)^n])^p)/((e*f - d*g)*(f + g*x)), x] - Dist[(b*e*n*p)/(e*f - d*g), Int[(a + b*Log[c*
(d + e*x)^n])^(p - 1)/(f + g*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && NeQ[e*f - d*g, 0] && GtQ[p, 0
]

Rule 2402

Int[Log[(c_.)/((d_) + (e_.)*(x_))]/((f_) + (g_.)*(x_)^2), x_Symbol] :> -Dist[e/g, Subst[Int[Log[2*d*x]/(1 - 2*
d*x), x], x, 1/(d + e*x)], x] /; FreeQ[{c, d, e, f, g}, x] && EqQ[c, 2*d] && EqQ[e^2*f + d^2*g, 0]

Rule 2416

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((h_.)*(x_))^(m_.)*((f_) + (g_.)*(x_)^(r_.))^(q
_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*Log[c*(d + e*x)^n])^p, (h*x)^m*(f + g*x^r)^q, x], x] /; FreeQ[{a,
b, c, d, e, f, g, h, m, n, p, q, r}, x] && IntegerQ[m] && IntegerQ[q]

Rule 2448

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)], x_Symbol] :> Simp[x*Log[c*(d + e*x^n)^p], x] - Dist[e*n*p, Int[
x^n/(d + e*x^n), x], x] /; FreeQ[{c, d, e, n, p}, x]

Rule 2450

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

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 2455

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

Rule 2457

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

Rule 2462

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

Rule 2470

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))/((f_) + (g_.)*(x_)^2), x_Symbol] :> With[{u = In
tHide[1/(f + g*x^2), x]}, Simp[u*(a + b*Log[c*(d + e*x^n)^p]), x] - Dist[b*e*n*p, Int[(u*x^(n - 1))/(d + e*x^n
), x], x]] /; FreeQ[{a, b, c, d, e, f, g, n, p}, x] && IntegerQ[n]

Rule 2476

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

Rule 2528

Int[((a_.) + Log[(c_.)*(RFx_)^(p_.)]*(b_.))^(n_.)*(RGx_), x_Symbol] :> With[{u = ExpandIntegrand[(a + b*Log[c*
RFx^p])^n, RGx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, p}, x] && RationalFunctionQ[RFx, x] && RationalF
unctionQ[RGx, x] && IGtQ[n, 0]

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 4854

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

Rule 4920

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

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 (36 x^2+48 x^3-44 x^4-4 e^x x^4-16 x^5+4 x^6\right ) \log \left (5+x^2\right )+\left (-90-60 x-18 x^2-32 x^3+10 x^4-4 x^5+2 x^6+e^x \left (-5 x^3-x^5\right )\right ) \log ^2\left (5+x^2\right )}{x^3 \left (5+x^2\right )} \, dx\\ &=\int \frac {\log \left (5+x^2\right ) \left (\frac {4 x^2 \left (9+12 x-\left (11+e^x\right ) x^2-4 x^3+x^4\right )}{5+x^2}+\left (-18-12 x-\left (4+e^x\right ) x^3+2 x^4\right ) \log \left (5+x^2\right )\right )}{x^3} \, dx\\ &=\int \left (-\frac {e^x \log \left (5+x^2\right ) \left (4 x+5 \log \left (5+x^2\right )+x^2 \log \left (5+x^2\right )\right )}{5+x^2}+\frac {2 \log \left (5+x^2\right ) \left (18 x^2+24 x^3-22 x^4-8 x^5+2 x^6-45 \log \left (5+x^2\right )-30 x \log \left (5+x^2\right )-9 x^2 \log \left (5+x^2\right )-16 x^3 \log \left (5+x^2\right )+5 x^4 \log \left (5+x^2\right )-2 x^5 \log \left (5+x^2\right )+x^6 \log \left (5+x^2\right )\right )}{x^3 \left (5+x^2\right )}\right ) \, dx\\ &=2 \int \frac {\log \left (5+x^2\right ) \left (18 x^2+24 x^3-22 x^4-8 x^5+2 x^6-45 \log \left (5+x^2\right )-30 x \log \left (5+x^2\right )-9 x^2 \log \left (5+x^2\right )-16 x^3 \log \left (5+x^2\right )+5 x^4 \log \left (5+x^2\right )-2 x^5 \log \left (5+x^2\right )+x^6 \log \left (5+x^2\right )\right )}{x^3 \left (5+x^2\right )} \, dx-\int \frac {e^x \log \left (5+x^2\right ) \left (4 x+5 \log \left (5+x^2\right )+x^2 \log \left (5+x^2\right )\right )}{5+x^2} \, dx\\ &=-\frac {e^x \log \left (5+x^2\right ) \left (5 \log \left (5+x^2\right )+x^2 \log \left (5+x^2\right )\right )}{5+x^2}+2 \int \frac {\log \left (5+x^2\right ) \left (2 x^2 \left (9+12 x-11 x^2-4 x^3+x^4\right )+\left (-45-30 x-9 x^2-16 x^3+5 x^4-2 x^5+x^6\right ) \log \left (5+x^2\right )\right )}{x^3 \left (5+x^2\right )} \, dx\\ &=-\frac {e^x \log \left (5+x^2\right ) \left (5 \log \left (5+x^2\right )+x^2 \log \left (5+x^2\right )\right )}{5+x^2}+2 \int \left (\frac {2 \left (-3-5 x+x^2\right ) \left (-3+x+x^2\right ) \log \left (5+x^2\right )}{x \left (5+x^2\right )}+\frac {(-3+x) (1+x) \left (3+x^2\right ) \log ^2\left (5+x^2\right )}{x^3}\right ) \, dx\\ &=-\frac {e^x \log \left (5+x^2\right ) \left (5 \log \left (5+x^2\right )+x^2 \log \left (5+x^2\right )\right )}{5+x^2}+2 \int \frac {(-3+x) (1+x) \left (3+x^2\right ) \log ^2\left (5+x^2\right )}{x^3} \, dx+4 \int \frac {\left (-3-5 x+x^2\right ) \left (-3+x+x^2\right ) \log \left (5+x^2\right )}{x \left (5+x^2\right )} \, dx\\ &=-\frac {e^x \log \left (5+x^2\right ) \left (5 \log \left (5+x^2\right )+x^2 \log \left (5+x^2\right )\right )}{5+x^2}+2 \int \left (-2 \log ^2\left (5+x^2\right )-\frac {9 \log ^2\left (5+x^2\right )}{x^3}-\frac {6 \log ^2\left (5+x^2\right )}{x^2}+x \log ^2\left (5+x^2\right )\right ) \, dx+4 \int \left (-4 \log \left (5+x^2\right )+\frac {9 \log \left (5+x^2\right )}{5 x}+x \log \left (5+x^2\right )+\frac {(160-89 x) \log \left (5+x^2\right )}{5 \left (5+x^2\right )}\right ) \, dx\\ &=-\frac {e^x \log \left (5+x^2\right ) \left (5 \log \left (5+x^2\right )+x^2 \log \left (5+x^2\right )\right )}{5+x^2}+\frac {4}{5} \int \frac {(160-89 x) \log \left (5+x^2\right )}{5+x^2} \, dx+2 \int x \log ^2\left (5+x^2\right ) \, dx+4 \int x \log \left (5+x^2\right ) \, dx-4 \int \log ^2\left (5+x^2\right ) \, dx+\frac {36}{5} \int \frac {\log \left (5+x^2\right )}{x} \, dx-12 \int \frac {\log ^2\left (5+x^2\right )}{x^2} \, dx-16 \int \log \left (5+x^2\right ) \, dx-18 \int \frac {\log ^2\left (5+x^2\right )}{x^3} \, dx\\ &=-16 x \log \left (5+x^2\right )+\frac {12 \log ^2\left (5+x^2\right )}{x}-4 x \log ^2\left (5+x^2\right )-\frac {e^x \log \left (5+x^2\right ) \left (5 \log \left (5+x^2\right )+x^2 \log \left (5+x^2\right )\right )}{5+x^2}+\frac {4}{5} \int \left (\frac {\left (445+160 i \sqrt {5}\right ) \log \left (5+x^2\right )}{10 \left (i \sqrt {5}-x\right )}+\frac {\left (-445+160 i \sqrt {5}\right ) \log \left (5+x^2\right )}{10 \left (i \sqrt {5}+x\right )}\right ) \, dx+2 \operatorname {Subst}\left (\int \log (5+x) \, dx,x,x^2\right )+\frac {18}{5} \operatorname {Subst}\left (\int \frac {\log (5+x)}{x} \, dx,x,x^2\right )-9 \operatorname {Subst}\left (\int \frac {\log ^2(5+x)}{x^2} \, dx,x,x^2\right )+16 \int \frac {x^2 \log \left (5+x^2\right )}{5+x^2} \, dx+32 \int \frac {x^2}{5+x^2} \, dx-48 \int \frac {\log \left (5+x^2\right )}{5+x^2} \, dx+\operatorname {Subst}\left (\int \log ^2(5+x) \, dx,x,x^2\right )\\ &=32 x+\frac {36}{5} \log (5) \log (x)-16 x \log \left (5+x^2\right )-\frac {48 \tan ^{-1}\left (\frac {x}{\sqrt {5}}\right ) \log \left (5+x^2\right )}{\sqrt {5}}+\frac {12 \log ^2\left (5+x^2\right )}{x}-4 x \log ^2\left (5+x^2\right )+\frac {9 \left (5+x^2\right ) \log ^2\left (5+x^2\right )}{5 x^2}-\frac {e^x \log \left (5+x^2\right ) \left (5 \log \left (5+x^2\right )+x^2 \log \left (5+x^2\right )\right )}{5+x^2}+2 \operatorname {Subst}\left (\int \log (x) \, dx,x,5+x^2\right )+\frac {18}{5} \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {x}{5}\right )}{x} \, dx,x,x^2\right )-\frac {18}{5} \operatorname {Subst}\left (\int \frac {\log (5+x)}{x} \, dx,x,x^2\right )+16 \int \left (\log \left (5+x^2\right )-\frac {5 \log \left (5+x^2\right )}{5+x^2}\right ) \, dx+96 \int \frac {x \tan ^{-1}\left (\frac {x}{\sqrt {5}}\right )}{\sqrt {5} \left (5+x^2\right )} \, dx-160 \int \frac {1}{5+x^2} \, dx-\frac {1}{5} \left (2 \left (89-32 i \sqrt {5}\right )\right ) \int \frac {\log \left (5+x^2\right )}{i \sqrt {5}+x} \, dx+\frac {1}{5} \left (2 \left (89+32 i \sqrt {5}\right )\right ) \int \frac {\log \left (5+x^2\right )}{i \sqrt {5}-x} \, dx+\operatorname {Subst}\left (\int \log ^2(x) \, dx,x,5+x^2\right )\\ &=32 x-2 x^2-32 \sqrt {5} \tan ^{-1}\left (\frac {x}{\sqrt {5}}\right )-16 x \log \left (5+x^2\right )+2 \left (5+x^2\right ) \log \left (5+x^2\right )-\frac {48 \tan ^{-1}\left (\frac {x}{\sqrt {5}}\right ) \log \left (5+x^2\right )}{\sqrt {5}}-\frac {2}{5} \left (89+32 i \sqrt {5}\right ) \log \left (i \sqrt {5}-x\right ) \log \left (5+x^2\right )-\frac {2}{5} \left (89-32 i \sqrt {5}\right ) \log \left (i \sqrt {5}+x\right ) \log \left (5+x^2\right )+\frac {12 \log ^2\left (5+x^2\right )}{x}-4 x \log ^2\left (5+x^2\right )+\left (5+x^2\right ) \log ^2\left (5+x^2\right )+\frac {9 \left (5+x^2\right ) \log ^2\left (5+x^2\right )}{5 x^2}-\frac {e^x \log \left (5+x^2\right ) \left (5 \log \left (5+x^2\right )+x^2 \log \left (5+x^2\right )\right )}{5+x^2}-\frac {18}{5} \text {Li}_2\left (-\frac {x^2}{5}\right )-2 \operatorname {Subst}\left (\int \log (x) \, dx,x,5+x^2\right )-\frac {18}{5} \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {x}{5}\right )}{x} \, dx,x,x^2\right )+16 \int \log \left (5+x^2\right ) \, dx-80 \int \frac {\log \left (5+x^2\right )}{5+x^2} \, dx+\frac {96 \int \frac {x \tan ^{-1}\left (\frac {x}{\sqrt {5}}\right )}{5+x^2} \, dx}{\sqrt {5}}+\frac {1}{5} \left (4 \left (89-32 i \sqrt {5}\right )\right ) \int \frac {x \log \left (i \sqrt {5}+x\right )}{5+x^2} \, dx+\frac {1}{5} \left (4 \left (89+32 i \sqrt {5}\right )\right ) \int \frac {x \log \left (i \sqrt {5}-x\right )}{5+x^2} \, dx\\ &=32 x-32 \sqrt {5} \tan ^{-1}\left (\frac {x}{\sqrt {5}}\right )-\frac {48 i \tan ^{-1}\left (\frac {x}{\sqrt {5}}\right )^2}{\sqrt {5}}-\frac {48 \tan ^{-1}\left (\frac {x}{\sqrt {5}}\right ) \log \left (5+x^2\right )}{\sqrt {5}}-16 \sqrt {5} \tan ^{-1}\left (\frac {x}{\sqrt {5}}\right ) \log \left (5+x^2\right )-\frac {2}{5} \left (89+32 i \sqrt {5}\right ) \log \left (i \sqrt {5}-x\right ) \log \left (5+x^2\right )-\frac {2}{5} \left (89-32 i \sqrt {5}\right ) \log \left (i \sqrt {5}+x\right ) \log \left (5+x^2\right )+\frac {12 \log ^2\left (5+x^2\right )}{x}-4 x \log ^2\left (5+x^2\right )+\left (5+x^2\right ) \log ^2\left (5+x^2\right )+\frac {9 \left (5+x^2\right ) \log ^2\left (5+x^2\right )}{5 x^2}-\frac {e^x \log \left (5+x^2\right ) \left (5 \log \left (5+x^2\right )+x^2 \log \left (5+x^2\right )\right )}{5+x^2}-\frac {96}{5} \int \frac {\tan ^{-1}\left (\frac {x}{\sqrt {5}}\right )}{i-\frac {x}{\sqrt {5}}} \, dx-32 \int \frac {x^2}{5+x^2} \, dx+160 \int \frac {x \tan ^{-1}\left (\frac {x}{\sqrt {5}}\right )}{\sqrt {5} \left (5+x^2\right )} \, dx+\frac {1}{5} \left (4 \left (89-32 i \sqrt {5}\right )\right ) \int \left (-\frac {\log \left (i \sqrt {5}+x\right )}{2 \left (i \sqrt {5}-x\right )}+\frac {\log \left (i \sqrt {5}+x\right )}{2 \left (i \sqrt {5}+x\right )}\right ) \, dx+\frac {1}{5} \left (4 \left (89+32 i \sqrt {5}\right )\right ) \int \left (-\frac {\log \left (i \sqrt {5}-x\right )}{2 \left (i \sqrt {5}-x\right )}+\frac {\log \left (i \sqrt {5}-x\right )}{2 \left (i \sqrt {5}+x\right )}\right ) \, dx\\ &=-32 \sqrt {5} \tan ^{-1}\left (\frac {x}{\sqrt {5}}\right )-\frac {48 i \tan ^{-1}\left (\frac {x}{\sqrt {5}}\right )^2}{\sqrt {5}}-\frac {96 \tan ^{-1}\left (\frac {x}{\sqrt {5}}\right ) \log \left (\frac {2 \sqrt {5}}{\sqrt {5}+i x}\right )}{\sqrt {5}}-\frac {48 \tan ^{-1}\left (\frac {x}{\sqrt {5}}\right ) \log \left (5+x^2\right )}{\sqrt {5}}-16 \sqrt {5} \tan ^{-1}\left (\frac {x}{\sqrt {5}}\right ) \log \left (5+x^2\right )-\frac {2}{5} \left (89+32 i \sqrt {5}\right ) \log \left (i \sqrt {5}-x\right ) \log \left (5+x^2\right )-\frac {2}{5} \left (89-32 i \sqrt {5}\right ) \log \left (i \sqrt {5}+x\right ) \log \left (5+x^2\right )+\frac {12 \log ^2\left (5+x^2\right )}{x}-4 x \log ^2\left (5+x^2\right )+\left (5+x^2\right ) \log ^2\left (5+x^2\right )+\frac {9 \left (5+x^2\right ) \log ^2\left (5+x^2\right )}{5 x^2}-\frac {e^x \log \left (5+x^2\right ) \left (5 \log \left (5+x^2\right )+x^2 \log \left (5+x^2\right )\right )}{5+x^2}+\frac {96}{5} \int \frac {\log \left (\frac {2}{1+\frac {i x}{\sqrt {5}}}\right )}{1+\frac {x^2}{5}} \, dx+160 \int \frac {1}{5+x^2} \, dx+\left (32 \sqrt {5}\right ) \int \frac {x \tan ^{-1}\left (\frac {x}{\sqrt {5}}\right )}{5+x^2} \, dx-\frac {1}{5} \left (2 \left (89-32 i \sqrt {5}\right )\right ) \int \frac {\log \left (i \sqrt {5}+x\right )}{i \sqrt {5}-x} \, dx+\frac {1}{5} \left (2 \left (89-32 i \sqrt {5}\right )\right ) \int \frac {\log \left (i \sqrt {5}+x\right )}{i \sqrt {5}+x} \, dx-\frac {1}{5} \left (2 \left (89+32 i \sqrt {5}\right )\right ) \int \frac {\log \left (i \sqrt {5}-x\right )}{i \sqrt {5}-x} \, dx+\frac {1}{5} \left (2 \left (89+32 i \sqrt {5}\right )\right ) \int \frac {\log \left (i \sqrt {5}-x\right )}{i \sqrt {5}+x} \, dx\\ &=-\frac {48 i \tan ^{-1}\left (\frac {x}{\sqrt {5}}\right )^2}{\sqrt {5}}-16 i \sqrt {5} \tan ^{-1}\left (\frac {x}{\sqrt {5}}\right )^2-\frac {96 \tan ^{-1}\left (\frac {x}{\sqrt {5}}\right ) \log \left (\frac {2 \sqrt {5}}{\sqrt {5}+i x}\right )}{\sqrt {5}}+\frac {2}{5} \left (89-32 i \sqrt {5}\right ) \log \left (-\frac {i \left (i \sqrt {5}-x\right )}{2 \sqrt {5}}\right ) \log \left (i \sqrt {5}+x\right )+\frac {2}{5} \left (89+32 i \sqrt {5}\right ) \log \left (i \sqrt {5}-x\right ) \log \left (-\frac {i \left (i \sqrt {5}+x\right )}{2 \sqrt {5}}\right )-\frac {48 \tan ^{-1}\left (\frac {x}{\sqrt {5}}\right ) \log \left (5+x^2\right )}{\sqrt {5}}-16 \sqrt {5} \tan ^{-1}\left (\frac {x}{\sqrt {5}}\right ) \log \left (5+x^2\right )-\frac {2}{5} \left (89+32 i \sqrt {5}\right ) \log \left (i \sqrt {5}-x\right ) \log \left (5+x^2\right )-\frac {2}{5} \left (89-32 i \sqrt {5}\right ) \log \left (i \sqrt {5}+x\right ) \log \left (5+x^2\right )+\frac {12 \log ^2\left (5+x^2\right )}{x}-4 x \log ^2\left (5+x^2\right )+\left (5+x^2\right ) \log ^2\left (5+x^2\right )+\frac {9 \left (5+x^2\right ) \log ^2\left (5+x^2\right )}{5 x^2}-\frac {e^x \log \left (5+x^2\right ) \left (5 \log \left (5+x^2\right )+x^2 \log \left (5+x^2\right )\right )}{5+x^2}-32 \int \frac {\tan ^{-1}\left (\frac {x}{\sqrt {5}}\right )}{i-\frac {x}{\sqrt {5}}} \, dx-\frac {(96 i) \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+\frac {i x}{\sqrt {5}}}\right )}{\sqrt {5}}-\frac {1}{5} \left (2 \left (89-32 i \sqrt {5}\right )\right ) \int \frac {\log \left (-\frac {i \left (i \sqrt {5}-x\right )}{2 \sqrt {5}}\right )}{i \sqrt {5}+x} \, dx+\frac {1}{5} \left (2 \left (89-32 i \sqrt {5}\right )\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,i \sqrt {5}+x\right )+\frac {1}{5} \left (2 \left (89+32 i \sqrt {5}\right )\right ) \int \frac {\log \left (\frac {i \left (-i \sqrt {5}-x\right )}{2 \sqrt {5}}\right )}{i \sqrt {5}-x} \, dx+\frac {1}{5} \left (2 \left (89+32 i \sqrt {5}\right )\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,i \sqrt {5}-x\right )\\ &=-\frac {48 i \tan ^{-1}\left (\frac {x}{\sqrt {5}}\right )^2}{\sqrt {5}}-16 i \sqrt {5} \tan ^{-1}\left (\frac {x}{\sqrt {5}}\right )^2+\frac {1}{5} \left (89+32 i \sqrt {5}\right ) \log ^2\left (i \sqrt {5}-x\right )-\frac {96 \tan ^{-1}\left (\frac {x}{\sqrt {5}}\right ) \log \left (\frac {2 \sqrt {5}}{\sqrt {5}+i x}\right )}{\sqrt {5}}-32 \sqrt {5} \tan ^{-1}\left (\frac {x}{\sqrt {5}}\right ) \log \left (\frac {2 \sqrt {5}}{\sqrt {5}+i x}\right )+\frac {2}{5} \left (89-32 i \sqrt {5}\right ) \log \left (-\frac {i \left (i \sqrt {5}-x\right )}{2 \sqrt {5}}\right ) \log \left (i \sqrt {5}+x\right )+\frac {1}{5} \left (89-32 i \sqrt {5}\right ) \log ^2\left (i \sqrt {5}+x\right )+\frac {2}{5} \left (89+32 i \sqrt {5}\right ) \log \left (i \sqrt {5}-x\right ) \log \left (-\frac {i \left (i \sqrt {5}+x\right )}{2 \sqrt {5}}\right )-\frac {48 \tan ^{-1}\left (\frac {x}{\sqrt {5}}\right ) \log \left (5+x^2\right )}{\sqrt {5}}-16 \sqrt {5} \tan ^{-1}\left (\frac {x}{\sqrt {5}}\right ) \log \left (5+x^2\right )-\frac {2}{5} \left (89+32 i \sqrt {5}\right ) \log \left (i \sqrt {5}-x\right ) \log \left (5+x^2\right )-\frac {2}{5} \left (89-32 i \sqrt {5}\right ) \log \left (i \sqrt {5}+x\right ) \log \left (5+x^2\right )+\frac {12 \log ^2\left (5+x^2\right )}{x}-4 x \log ^2\left (5+x^2\right )+\left (5+x^2\right ) \log ^2\left (5+x^2\right )+\frac {9 \left (5+x^2\right ) \log ^2\left (5+x^2\right )}{5 x^2}-\frac {e^x \log \left (5+x^2\right ) \left (5 \log \left (5+x^2\right )+x^2 \log \left (5+x^2\right )\right )}{5+x^2}-\frac {48 i \text {Li}_2\left (1-\frac {2 \sqrt {5}}{\sqrt {5}+i x}\right )}{\sqrt {5}}+32 \int \frac {\log \left (\frac {2}{1+\frac {i x}{\sqrt {5}}}\right )}{1+\frac {x^2}{5}} \, dx-\frac {1}{5} \left (2 \left (89-32 i \sqrt {5}\right )\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {i x}{2 \sqrt {5}}\right )}{x} \, dx,x,i \sqrt {5}+x\right )-\frac {1}{5} \left (2 \left (89+32 i \sqrt {5}\right )\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {i x}{2 \sqrt {5}}\right )}{x} \, dx,x,i \sqrt {5}-x\right )\\ &=-\frac {48 i \tan ^{-1}\left (\frac {x}{\sqrt {5}}\right )^2}{\sqrt {5}}-16 i \sqrt {5} \tan ^{-1}\left (\frac {x}{\sqrt {5}}\right )^2+\frac {1}{5} \left (89+32 i \sqrt {5}\right ) \log ^2\left (i \sqrt {5}-x\right )-\frac {96 \tan ^{-1}\left (\frac {x}{\sqrt {5}}\right ) \log \left (\frac {2 \sqrt {5}}{\sqrt {5}+i x}\right )}{\sqrt {5}}-32 \sqrt {5} \tan ^{-1}\left (\frac {x}{\sqrt {5}}\right ) \log \left (\frac {2 \sqrt {5}}{\sqrt {5}+i x}\right )+\frac {2}{5} \left (89-32 i \sqrt {5}\right ) \log \left (-\frac {i \left (i \sqrt {5}-x\right )}{2 \sqrt {5}}\right ) \log \left (i \sqrt {5}+x\right )+\frac {1}{5} \left (89-32 i \sqrt {5}\right ) \log ^2\left (i \sqrt {5}+x\right )+\frac {2}{5} \left (89+32 i \sqrt {5}\right ) \log \left (i \sqrt {5}-x\right ) \log \left (-\frac {i \left (i \sqrt {5}+x\right )}{2 \sqrt {5}}\right )-\frac {48 \tan ^{-1}\left (\frac {x}{\sqrt {5}}\right ) \log \left (5+x^2\right )}{\sqrt {5}}-16 \sqrt {5} \tan ^{-1}\left (\frac {x}{\sqrt {5}}\right ) \log \left (5+x^2\right )-\frac {2}{5} \left (89+32 i \sqrt {5}\right ) \log \left (i \sqrt {5}-x\right ) \log \left (5+x^2\right )-\frac {2}{5} \left (89-32 i \sqrt {5}\right ) \log \left (i \sqrt {5}+x\right ) \log \left (5+x^2\right )+\frac {12 \log ^2\left (5+x^2\right )}{x}-4 x \log ^2\left (5+x^2\right )+\left (5+x^2\right ) \log ^2\left (5+x^2\right )+\frac {9 \left (5+x^2\right ) \log ^2\left (5+x^2\right )}{5 x^2}-\frac {e^x \log \left (5+x^2\right ) \left (5 \log \left (5+x^2\right )+x^2 \log \left (5+x^2\right )\right )}{5+x^2}-\frac {48 i \text {Li}_2\left (1-\frac {2 \sqrt {5}}{\sqrt {5}+i x}\right )}{\sqrt {5}}+\frac {2}{5} \left (89-32 i \sqrt {5}\right ) \text {Li}_2\left (\frac {\sqrt {5}-i x}{2 \sqrt {5}}\right )+\frac {2}{5} \left (89+32 i \sqrt {5}\right ) \text {Li}_2\left (\frac {\sqrt {5}+i x}{2 \sqrt {5}}\right )-\left (32 i \sqrt {5}\right ) \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+\frac {i x}{\sqrt {5}}}\right )\\ &=-\frac {48 i \tan ^{-1}\left (\frac {x}{\sqrt {5}}\right )^2}{\sqrt {5}}-16 i \sqrt {5} \tan ^{-1}\left (\frac {x}{\sqrt {5}}\right )^2+\frac {1}{5} \left (89+32 i \sqrt {5}\right ) \log ^2\left (i \sqrt {5}-x\right )-\frac {96 \tan ^{-1}\left (\frac {x}{\sqrt {5}}\right ) \log \left (\frac {2 \sqrt {5}}{\sqrt {5}+i x}\right )}{\sqrt {5}}-32 \sqrt {5} \tan ^{-1}\left (\frac {x}{\sqrt {5}}\right ) \log \left (\frac {2 \sqrt {5}}{\sqrt {5}+i x}\right )+\frac {2}{5} \left (89-32 i \sqrt {5}\right ) \log \left (-\frac {i \left (i \sqrt {5}-x\right )}{2 \sqrt {5}}\right ) \log \left (i \sqrt {5}+x\right )+\frac {1}{5} \left (89-32 i \sqrt {5}\right ) \log ^2\left (i \sqrt {5}+x\right )+\frac {2}{5} \left (89+32 i \sqrt {5}\right ) \log \left (i \sqrt {5}-x\right ) \log \left (-\frac {i \left (i \sqrt {5}+x\right )}{2 \sqrt {5}}\right )-\frac {48 \tan ^{-1}\left (\frac {x}{\sqrt {5}}\right ) \log \left (5+x^2\right )}{\sqrt {5}}-16 \sqrt {5} \tan ^{-1}\left (\frac {x}{\sqrt {5}}\right ) \log \left (5+x^2\right )-\frac {2}{5} \left (89+32 i \sqrt {5}\right ) \log \left (i \sqrt {5}-x\right ) \log \left (5+x^2\right )-\frac {2}{5} \left (89-32 i \sqrt {5}\right ) \log \left (i \sqrt {5}+x\right ) \log \left (5+x^2\right )+\frac {12 \log ^2\left (5+x^2\right )}{x}-4 x \log ^2\left (5+x^2\right )+\left (5+x^2\right ) \log ^2\left (5+x^2\right )+\frac {9 \left (5+x^2\right ) \log ^2\left (5+x^2\right )}{5 x^2}-\frac {e^x \log \left (5+x^2\right ) \left (5 \log \left (5+x^2\right )+x^2 \log \left (5+x^2\right )\right )}{5+x^2}-\frac {48 i \text {Li}_2\left (1-\frac {2 \sqrt {5}}{\sqrt {5}+i x}\right )}{\sqrt {5}}-16 i \sqrt {5} \text {Li}_2\left (1-\frac {2 \sqrt {5}}{\sqrt {5}+i x}\right )+\frac {2}{5} \left (89-32 i \sqrt {5}\right ) \text {Li}_2\left (\frac {\sqrt {5}-i x}{2 \sqrt {5}}\right )+\frac {2}{5} \left (89+32 i \sqrt {5}\right ) \text {Li}_2\left (\frac {\sqrt {5}+i x}{2 \sqrt {5}}\right )\\ \end {aligned} \end {gather*}

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Mathematica [C]  time = 1.06, size = 572, normalized size = 19.07 \begin {gather*} \frac {1}{5} \left (-128 i \sqrt {5} \tan ^{-1}\left (\frac {x}{\sqrt {5}}\right )^2+89 \log ^2\left (i \sqrt {5}-x\right )+32 i \sqrt {5} \log ^2\left (i \sqrt {5}-x\right )+89 \log ^2\left (i \sqrt {5}+x\right )-32 i \sqrt {5} \log ^2\left (i \sqrt {5}+x\right )-256 \sqrt {5} \tan ^{-1}\left (\frac {x}{\sqrt {5}}\right ) \log \left (\frac {10 i}{5 i-\sqrt {5} x}\right )+178 \log \left (i \sqrt {5}-x\right ) \log \left (\frac {1}{10} \left (5-i \sqrt {5} x\right )\right )+64 i \sqrt {5} \log \left (i \sqrt {5}-x\right ) \log \left (\frac {1}{10} \left (5-i \sqrt {5} x\right )\right )+178 \log \left (i \sqrt {5}+x\right ) \log \left (\frac {1}{10} \left (5+i \sqrt {5} x\right )\right )-64 i \sqrt {5} \log \left (i \sqrt {5}+x\right ) \log \left (\frac {1}{10} \left (5+i \sqrt {5} x\right )\right )-128 \sqrt {5} \tan ^{-1}\left (\frac {x}{\sqrt {5}}\right ) \log \left (5+x^2\right )-178 \log \left (i \sqrt {5}-x\right ) \log \left (5+x^2\right )-64 i \sqrt {5} \log \left (i \sqrt {5}-x\right ) \log \left (5+x^2\right )-178 \log \left (i \sqrt {5}+x\right ) \log \left (5+x^2\right )+64 i \sqrt {5} \log \left (i \sqrt {5}+x\right ) \log \left (5+x^2\right )+34 \log ^2\left (5+x^2\right )-5 e^x \log ^2\left (5+x^2\right )+\frac {45 \log ^2\left (5+x^2\right )}{x^2}+\frac {60 \log ^2\left (5+x^2\right )}{x}-20 x \log ^2\left (5+x^2\right )+5 x^2 \log ^2\left (5+x^2\right )+2 \left (89-32 i \sqrt {5}\right ) \text {Li}_2\left (\frac {1}{2}-\frac {i x}{2 \sqrt {5}}\right )+2 \left (89+32 i \sqrt {5}\right ) \text {Li}_2\left (\frac {1}{2}+\frac {i x}{2 \sqrt {5}}\right )-128 i \sqrt {5} \text {Li}_2\left (\frac {5 i+\sqrt {5} x}{-5 i+\sqrt {5} x}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((36*x^2 + 48*x^3 - 44*x^4 - 4*E^x*x^4 - 16*x^5 + 4*x^6)*Log[5 + x^2] + (-90 - 60*x - 18*x^2 - 32*x^
3 + 10*x^4 - 4*x^5 + 2*x^6 + E^x*(-5*x^3 - x^5))*Log[5 + x^2]^2)/(5*x^3 + x^5),x]

[Out]

((-128*I)*Sqrt[5]*ArcTan[x/Sqrt[5]]^2 + 89*Log[I*Sqrt[5] - x]^2 + (32*I)*Sqrt[5]*Log[I*Sqrt[5] - x]^2 + 89*Log
[I*Sqrt[5] + x]^2 - (32*I)*Sqrt[5]*Log[I*Sqrt[5] + x]^2 - 256*Sqrt[5]*ArcTan[x/Sqrt[5]]*Log[(10*I)/(5*I - Sqrt
[5]*x)] + 178*Log[I*Sqrt[5] - x]*Log[(5 - I*Sqrt[5]*x)/10] + (64*I)*Sqrt[5]*Log[I*Sqrt[5] - x]*Log[(5 - I*Sqrt
[5]*x)/10] + 178*Log[I*Sqrt[5] + x]*Log[(5 + I*Sqrt[5]*x)/10] - (64*I)*Sqrt[5]*Log[I*Sqrt[5] + x]*Log[(5 + I*S
qrt[5]*x)/10] - 128*Sqrt[5]*ArcTan[x/Sqrt[5]]*Log[5 + x^2] - 178*Log[I*Sqrt[5] - x]*Log[5 + x^2] - (64*I)*Sqrt
[5]*Log[I*Sqrt[5] - x]*Log[5 + x^2] - 178*Log[I*Sqrt[5] + x]*Log[5 + x^2] + (64*I)*Sqrt[5]*Log[I*Sqrt[5] + x]*
Log[5 + x^2] + 34*Log[5 + x^2]^2 - 5*E^x*Log[5 + x^2]^2 + (45*Log[5 + x^2]^2)/x^2 + (60*Log[5 + x^2]^2)/x - 20
*x*Log[5 + x^2]^2 + 5*x^2*Log[5 + x^2]^2 + 2*(89 - (32*I)*Sqrt[5])*PolyLog[2, 1/2 - ((I/2)*x)/Sqrt[5]] + 2*(89
+ (32*I)*Sqrt[5])*PolyLog[2, 1/2 + ((I/2)*x)/Sqrt[5]] - (128*I)*Sqrt[5]*PolyLog[2, (5*I + Sqrt[5]*x)/(-5*I +
Sqrt[5]*x)])/5

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fricas [A]  time = 0.86, size = 37, normalized size = 1.23 \begin {gather*} \frac {{\left (x^{4} - 4 \, x^{3} - x^{2} e^{x} - 11 \, x^{2} + 12 \, x + 9\right )} \log \left (x^{2} + 5\right )^{2}}{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-x^5-5*x^3)*exp(x)+2*x^6-4*x^5+10*x^4-32*x^3-18*x^2-60*x-90)*log(x^2+5)^2+(-4*exp(x)*x^4+4*x^6-16
*x^5-44*x^4+48*x^3+36*x^2)*log(x^2+5))/(x^5+5*x^3),x, algorithm="fricas")

[Out]

(x^4 - 4*x^3 - x^2*e^x - 11*x^2 + 12*x + 9)*log(x^2 + 5)^2/x^2

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giac [B]  time = 0.34, size = 79, normalized size = 2.63 \begin {gather*} \frac {x^{4} \log \left (x^{2} + 5\right )^{2} - 4 \, x^{3} \log \left (x^{2} + 5\right )^{2} - x^{2} e^{x} \log \left (x^{2} + 5\right )^{2} - 11 \, x^{2} \log \left (x^{2} + 5\right )^{2} + 12 \, x \log \left (x^{2} + 5\right )^{2} + 9 \, \log \left (x^{2} + 5\right )^{2}}{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-x^5-5*x^3)*exp(x)+2*x^6-4*x^5+10*x^4-32*x^3-18*x^2-60*x-90)*log(x^2+5)^2+(-4*exp(x)*x^4+4*x^6-16
*x^5-44*x^4+48*x^3+36*x^2)*log(x^2+5))/(x^5+5*x^3),x, algorithm="giac")

[Out]

(x^4*log(x^2 + 5)^2 - 4*x^3*log(x^2 + 5)^2 - x^2*e^x*log(x^2 + 5)^2 - 11*x^2*log(x^2 + 5)^2 + 12*x*log(x^2 + 5
)^2 + 9*log(x^2 + 5)^2)/x^2

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maple [A]  time = 0.36, size = 38, normalized size = 1.27

method	result	size
risch	\(\frac {\left (x^{4}-4 x^{3}-{\mathrm e}^{x} x^{2}-11 x^{2}+12 x +9\right ) \ln \left (x^{2}+5\right )^{2}}{x^{2}}\)	\(38\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((-x^5-5*x^3)*exp(x)+2*x^6-4*x^5+10*x^4-32*x^3-18*x^2-60*x-90)*ln(x^2+5)^2+(-4*exp(x)*x^4+4*x^6-16*x^5-44
*x^4+48*x^3+36*x^2)*ln(x^2+5))/(x^5+5*x^3),x,method=_RETURNVERBOSE)

[Out]

(x^4-4*x^3-exp(x)*x^2-11*x^2+12*x+9)/x^2*ln(x^2+5)^2

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maxima [A]  time = 0.40, size = 37, normalized size = 1.23 \begin {gather*} \frac {{\left (x^{4} - 4 \, x^{3} - x^{2} e^{x} - 11 \, x^{2} + 12 \, x + 9\right )} \log \left (x^{2} + 5\right )^{2}}{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-x^5-5*x^3)*exp(x)+2*x^6-4*x^5+10*x^4-32*x^3-18*x^2-60*x-90)*log(x^2+5)^2+(-4*exp(x)*x^4+4*x^6-16
*x^5-44*x^4+48*x^3+36*x^2)*log(x^2+5))/(x^5+5*x^3),x, algorithm="maxima")

[Out]

(x^4 - 4*x^3 - x^2*e^x - 11*x^2 + 12*x + 9)*log(x^2 + 5)^2/x^2

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mupad [B]  time = 0.18, size = 49, normalized size = 1.63 \begin {gather*} -{\ln \left (x^2+5\right )}^2\,\left ({\mathrm {e}}^x+\frac {12\,x^3-6\,x^4}{x^2}-\frac {-5\,x^4+8\,x^3+12\,x+9}{x^2}+11\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(log(x^2 + 5)^2*(60*x + exp(x)*(5*x^3 + x^5) + 18*x^2 + 32*x^3 - 10*x^4 + 4*x^5 - 2*x^6 + 90) + log(x^2 +
5)*(4*x^4*exp(x) - 36*x^2 - 48*x^3 + 44*x^4 + 16*x^5 - 4*x^6))/(5*x^3 + x^5),x)

[Out]

-log(x^2 + 5)^2*(exp(x) + (12*x^3 - 6*x^4)/x^2 - (12*x + 8*x^3 - 5*x^4 + 9)/x^2 + 11)

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sympy [A]  time = 9.60, size = 41, normalized size = 1.37 \begin {gather*} - e^{x} \log {\left (x^{2} + 5 \right )}^{2} + \frac {\left (x^{4} - 4 x^{3} - 11 x^{2} + 12 x + 9\right ) \log {\left (x^{2} + 5 \right )}^{2}}{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-x**5-5*x**3)*exp(x)+2*x**6-4*x**5+10*x**4-32*x**3-18*x**2-60*x-90)*ln(x**2+5)**2+(-4*exp(x)*x**4
+4*x**6-16*x**5-44*x**4+48*x**3+36*x**2)*ln(x**2+5))/(x**5+5*x**3),x)

[Out]

-exp(x)*log(x**2 + 5)**2 + (x**4 - 4*x**3 - 11*x**2 + 12*x + 9)*log(x**2 + 5)**2/x**2

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