3.272 \(\int x^4 (a+b \coth ^{-1}(c x)) (d+e \log (1-c^2 x^2)) \, dx\)

Optimal. Leaf size=315 \[ \frac{1}{5} x^5 \left (a+b \coth ^{-1}(c x)\right ) \left (e \log \left (1-c^2 x^2\right )+d\right )-\frac{e (4 a+3 b) \log (1-c x)}{20 c^5}+\frac{e (4 a-3 b) \log (c x+1)}{20 c^5}-\frac{2 a e x^3}{15 c^2}-\frac{2 a e x}{5 c^4}-\frac{2}{25} a e x^5+\frac{b x^4 \left (e \log \left (1-c^2 x^2\right )+d\right )}{20 c}+\frac{b x^2 \left (e \log \left (1-c^2 x^2\right )+d\right )}{10 c^3}+\frac{b \log \left (1-c^2 x^2\right ) \left (e \log \left (1-c^2 x^2\right )+d\right )}{10 c^5}-\frac{77 b e x^2}{300 c^3}-\frac{b e \log ^2\left (1-c^2 x^2\right )}{20 c^5}-\frac{23 b e \log \left (1-c^2 x^2\right )}{75 c^5}-\frac{2 b e x^3 \coth ^{-1}(c x)}{15 c^2}-\frac{2 b e x \coth ^{-1}(c x)}{5 c^4}+\frac{b e \coth ^{-1}(c x)^2}{5 c^5}-\frac{9 b e x^4}{200 c}-\frac{2}{25} b e x^5 \coth ^{-1}(c x) \]

[Out]

(-2*a*e*x)/(5*c^4) - (77*b*e*x^2)/(300*c^3) - (2*a*e*x^3)/(15*c^2) - (9*b*e*x^4)/(200*c) - (2*a*e*x^5)/25 - (2
*b*e*x*ArcCoth[c*x])/(5*c^4) - (2*b*e*x^3*ArcCoth[c*x])/(15*c^2) - (2*b*e*x^5*ArcCoth[c*x])/25 + (b*e*ArcCoth[
c*x]^2)/(5*c^5) - ((4*a + 3*b)*e*Log[1 - c*x])/(20*c^5) + ((4*a - 3*b)*e*Log[1 + c*x])/(20*c^5) - (23*b*e*Log[
1 - c^2*x^2])/(75*c^5) - (b*e*Log[1 - c^2*x^2]^2)/(20*c^5) + (b*x^2*(d + e*Log[1 - c^2*x^2]))/(10*c^3) + (b*x^
4*(d + e*Log[1 - c^2*x^2]))/(20*c) + (x^5*(a + b*ArcCoth[c*x])*(d + e*Log[1 - c^2*x^2]))/5 + (b*Log[1 - c^2*x^
2]*(d + e*Log[1 - c^2*x^2]))/(10*c^5)

________________________________________________________________________________________

Rubi [A]  time = 0.749021, antiderivative size = 315, normalized size of antiderivative = 1., number of steps used = 26, number of rules used = 15, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.556, Rules used = {5917, 266, 43, 6086, 6725, 1802, 633, 31, 5981, 5911, 260, 5949, 2475, 2390, 2301} \[ \frac{1}{5} x^5 \left (a+b \coth ^{-1}(c x)\right ) \left (e \log \left (1-c^2 x^2\right )+d\right )-\frac{e (4 a+3 b) \log (1-c x)}{20 c^5}+\frac{e (4 a-3 b) \log (c x+1)}{20 c^5}-\frac{2 a e x^3}{15 c^2}-\frac{2 a e x}{5 c^4}-\frac{2}{25} a e x^5+\frac{b x^4 \left (e \log \left (1-c^2 x^2\right )+d\right )}{20 c}+\frac{b x^2 \left (e \log \left (1-c^2 x^2\right )+d\right )}{10 c^3}+\frac{b \log \left (1-c^2 x^2\right ) \left (e \log \left (1-c^2 x^2\right )+d\right )}{10 c^5}-\frac{77 b e x^2}{300 c^3}-\frac{b e \log ^2\left (1-c^2 x^2\right )}{20 c^5}-\frac{23 b e \log \left (1-c^2 x^2\right )}{75 c^5}-\frac{2 b e x^3 \coth ^{-1}(c x)}{15 c^2}-\frac{2 b e x \coth ^{-1}(c x)}{5 c^4}+\frac{b e \coth ^{-1}(c x)^2}{5 c^5}-\frac{9 b e x^4}{200 c}-\frac{2}{25} b e x^5 \coth ^{-1}(c x) \]

Antiderivative was successfully verified.

[In]

Int[x^4*(a + b*ArcCoth[c*x])*(d + e*Log[1 - c^2*x^2]),x]

[Out]

(-2*a*e*x)/(5*c^4) - (77*b*e*x^2)/(300*c^3) - (2*a*e*x^3)/(15*c^2) - (9*b*e*x^4)/(200*c) - (2*a*e*x^5)/25 - (2
*b*e*x*ArcCoth[c*x])/(5*c^4) - (2*b*e*x^3*ArcCoth[c*x])/(15*c^2) - (2*b*e*x^5*ArcCoth[c*x])/25 + (b*e*ArcCoth[
c*x]^2)/(5*c^5) - ((4*a + 3*b)*e*Log[1 - c*x])/(20*c^5) + ((4*a - 3*b)*e*Log[1 + c*x])/(20*c^5) - (23*b*e*Log[
1 - c^2*x^2])/(75*c^5) - (b*e*Log[1 - c^2*x^2]^2)/(20*c^5) + (b*x^2*(d + e*Log[1 - c^2*x^2]))/(10*c^3) + (b*x^
4*(d + e*Log[1 - c^2*x^2]))/(20*c) + (x^5*(a + b*ArcCoth[c*x])*(d + e*Log[1 - c^2*x^2]))/5 + (b*Log[1 - c^2*x^
2]*(d + e*Log[1 - c^2*x^2]))/(10*c^5)

Rule 5917

Int[((a_.) + ArcCoth[(c_.)*(x_)]*(b_.))^(p_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*ArcC
oth[c*x])^p)/(d*(m + 1)), x] - Dist[(b*c*p)/(d*(m + 1)), Int[((d*x)^(m + 1)*(a + b*ArcCoth[c*x])^(p - 1))/(1 -
c^2*x^2), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[p, 0] && (EqQ[p, 1] || IntegerQ[m]) && NeQ[m, -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 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 6086

Int[((a_.) + ArcCoth[(c_.)*(x_)]*(b_.))*((d_.) + Log[(f_.) + (g_.)*(x_)^2]*(e_.))*(x_)^(m_.), x_Symbol] :> Wit
h[{u = IntHide[x^m*(a + b*ArcCoth[c*x]), x]}, Dist[d + e*Log[f + g*x^2], u, x] - Dist[2*e*g, Int[ExpandIntegra
nd[(x*u)/(f + g*x^2), x], x], x]] /; FreeQ[{a, b, c, d, e, f, g}, x] && IntegerQ[m] && NeQ[m, -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]

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 633

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

Rule 31

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

Rule 5981

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

Rule 5911

Int[((a_.) + ArcCoth[(c_.)*(x_)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a + b*ArcCoth[c*x])^p, x] - Dist[b*c*p, In
t[(x*(a + b*ArcCoth[c*x])^(p - 1))/(1 - c^2*x^2), x], x] /; FreeQ[{a, b, c}, x] && IGtQ[p, 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 5949

Int[((a_.) + ArcCoth[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[(a + b*ArcCoth[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 2475

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

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 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]

Rubi steps

\begin{align*} \int x^4 \left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right ) \, dx &=\frac{b x^2 \left (d+e \log \left (1-c^2 x^2\right )\right )}{10 c^3}+\frac{b x^4 \left (d+e \log \left (1-c^2 x^2\right )\right )}{20 c}+\frac{1}{5} x^5 \left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )+\frac{b \log \left (1-c^2 x^2\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )}{10 c^5}+\left (2 c^2 e\right ) \int \left (\frac{-2 b x^3-b c^2 x^5-4 a c^3 x^6-4 b c^3 x^6 \coth ^{-1}(c x)}{20 c^3 \left (-1+c^2 x^2\right )}-\frac{b x \log \left (1-c^2 x^2\right )}{10 c^5 \left (-1+c^2 x^2\right )}\right ) \, dx\\ &=\frac{b x^2 \left (d+e \log \left (1-c^2 x^2\right )\right )}{10 c^3}+\frac{b x^4 \left (d+e \log \left (1-c^2 x^2\right )\right )}{20 c}+\frac{1}{5} x^5 \left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )+\frac{b \log \left (1-c^2 x^2\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )}{10 c^5}-\frac{(b e) \int \frac{x \log \left (1-c^2 x^2\right )}{-1+c^2 x^2} \, dx}{5 c^3}+\frac{e \int \frac{-2 b x^3-b c^2 x^5-4 a c^3 x^6-4 b c^3 x^6 \coth ^{-1}(c x)}{-1+c^2 x^2} \, dx}{10 c}\\ &=\frac{b x^2 \left (d+e \log \left (1-c^2 x^2\right )\right )}{10 c^3}+\frac{b x^4 \left (d+e \log \left (1-c^2 x^2\right )\right )}{20 c}+\frac{1}{5} x^5 \left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )+\frac{b \log \left (1-c^2 x^2\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )}{10 c^5}-\frac{(b e) \operatorname{Subst}\left (\int \frac{\log \left (1-c^2 x\right )}{-1+c^2 x} \, dx,x,x^2\right )}{10 c^3}+\frac{e \int \left (\frac{x^3 \left (2 b+b c^2 x^2+4 a c^3 x^3\right )}{1-c^2 x^2}-\frac{4 b c^3 x^6 \coth ^{-1}(c x)}{-1+c^2 x^2}\right ) \, dx}{10 c}\\ &=\frac{b x^2 \left (d+e \log \left (1-c^2 x^2\right )\right )}{10 c^3}+\frac{b x^4 \left (d+e \log \left (1-c^2 x^2\right )\right )}{20 c}+\frac{1}{5} x^5 \left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )+\frac{b \log \left (1-c^2 x^2\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )}{10 c^5}-\frac{(b e) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,1-c^2 x^2\right )}{10 c^5}+\frac{e \int \frac{x^3 \left (2 b+b c^2 x^2+4 a c^3 x^3\right )}{1-c^2 x^2} \, dx}{10 c}-\frac{1}{5} \left (2 b c^2 e\right ) \int \frac{x^6 \coth ^{-1}(c x)}{-1+c^2 x^2} \, dx\\ &=-\frac{b e \log ^2\left (1-c^2 x^2\right )}{20 c^5}+\frac{b x^2 \left (d+e \log \left (1-c^2 x^2\right )\right )}{10 c^3}+\frac{b x^4 \left (d+e \log \left (1-c^2 x^2\right )\right )}{20 c}+\frac{1}{5} x^5 \left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )+\frac{b \log \left (1-c^2 x^2\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )}{10 c^5}-\frac{1}{5} (2 b e) \int x^4 \coth ^{-1}(c x) \, dx-\frac{1}{5} (2 b e) \int \frac{x^4 \coth ^{-1}(c x)}{-1+c^2 x^2} \, dx+\frac{e \int \left (-\frac{4 a}{c^3}-\frac{3 b x}{c^2}-\frac{4 a x^2}{c}-b x^3-4 a c x^4+\frac{4 a+3 b c x}{c^3 \left (1-c^2 x^2\right )}\right ) \, dx}{10 c}\\ &=-\frac{2 a e x}{5 c^4}-\frac{3 b e x^2}{20 c^3}-\frac{2 a e x^3}{15 c^2}-\frac{b e x^4}{40 c}-\frac{2}{25} a e x^5-\frac{2}{25} b e x^5 \coth ^{-1}(c x)-\frac{b e \log ^2\left (1-c^2 x^2\right )}{20 c^5}+\frac{b x^2 \left (d+e \log \left (1-c^2 x^2\right )\right )}{10 c^3}+\frac{b x^4 \left (d+e \log \left (1-c^2 x^2\right )\right )}{20 c}+\frac{1}{5} x^5 \left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )+\frac{b \log \left (1-c^2 x^2\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )}{10 c^5}+\frac{e \int \frac{4 a+3 b c x}{1-c^2 x^2} \, dx}{10 c^4}-\frac{(2 b e) \int x^2 \coth ^{-1}(c x) \, dx}{5 c^2}-\frac{(2 b e) \int \frac{x^2 \coth ^{-1}(c x)}{-1+c^2 x^2} \, dx}{5 c^2}+\frac{1}{25} (2 b c e) \int \frac{x^5}{1-c^2 x^2} \, dx\\ &=-\frac{2 a e x}{5 c^4}-\frac{3 b e x^2}{20 c^3}-\frac{2 a e x^3}{15 c^2}-\frac{b e x^4}{40 c}-\frac{2}{25} a e x^5-\frac{2 b e x^3 \coth ^{-1}(c x)}{15 c^2}-\frac{2}{25} b e x^5 \coth ^{-1}(c x)-\frac{b e \log ^2\left (1-c^2 x^2\right )}{20 c^5}+\frac{b x^2 \left (d+e \log \left (1-c^2 x^2\right )\right )}{10 c^3}+\frac{b x^4 \left (d+e \log \left (1-c^2 x^2\right )\right )}{20 c}+\frac{1}{5} x^5 \left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )+\frac{b \log \left (1-c^2 x^2\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )}{10 c^5}-\frac{(2 b e) \int \coth ^{-1}(c x) \, dx}{5 c^4}-\frac{(2 b e) \int \frac{\coth ^{-1}(c x)}{-1+c^2 x^2} \, dx}{5 c^4}-\frac{((4 a-3 b) e) \int \frac{1}{-c-c^2 x} \, dx}{20 c^3}+\frac{((4 a+3 b) e) \int \frac{1}{c-c^2 x} \, dx}{20 c^3}+\frac{(2 b e) \int \frac{x^3}{1-c^2 x^2} \, dx}{15 c}+\frac{1}{25} (b c e) \operatorname{Subst}\left (\int \frac{x^2}{1-c^2 x} \, dx,x,x^2\right )\\ &=-\frac{2 a e x}{5 c^4}-\frac{3 b e x^2}{20 c^3}-\frac{2 a e x^3}{15 c^2}-\frac{b e x^4}{40 c}-\frac{2}{25} a e x^5-\frac{2 b e x \coth ^{-1}(c x)}{5 c^4}-\frac{2 b e x^3 \coth ^{-1}(c x)}{15 c^2}-\frac{2}{25} b e x^5 \coth ^{-1}(c x)+\frac{b e \coth ^{-1}(c x)^2}{5 c^5}-\frac{(4 a+3 b) e \log (1-c x)}{20 c^5}+\frac{(4 a-3 b) e \log (1+c x)}{20 c^5}-\frac{b e \log ^2\left (1-c^2 x^2\right )}{20 c^5}+\frac{b x^2 \left (d+e \log \left (1-c^2 x^2\right )\right )}{10 c^3}+\frac{b x^4 \left (d+e \log \left (1-c^2 x^2\right )\right )}{20 c}+\frac{1}{5} x^5 \left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )+\frac{b \log \left (1-c^2 x^2\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )}{10 c^5}+\frac{(2 b e) \int \frac{x}{1-c^2 x^2} \, dx}{5 c^3}+\frac{(b e) \operatorname{Subst}\left (\int \frac{x}{1-c^2 x} \, dx,x,x^2\right )}{15 c}+\frac{1}{25} (b c e) \operatorname{Subst}\left (\int \left (-\frac{1}{c^4}-\frac{x}{c^2}-\frac{1}{c^4 \left (-1+c^2 x\right )}\right ) \, dx,x,x^2\right )\\ &=-\frac{2 a e x}{5 c^4}-\frac{19 b e x^2}{100 c^3}-\frac{2 a e x^3}{15 c^2}-\frac{9 b e x^4}{200 c}-\frac{2}{25} a e x^5-\frac{2 b e x \coth ^{-1}(c x)}{5 c^4}-\frac{2 b e x^3 \coth ^{-1}(c x)}{15 c^2}-\frac{2}{25} b e x^5 \coth ^{-1}(c x)+\frac{b e \coth ^{-1}(c x)^2}{5 c^5}-\frac{(4 a+3 b) e \log (1-c x)}{20 c^5}+\frac{(4 a-3 b) e \log (1+c x)}{20 c^5}-\frac{6 b e \log \left (1-c^2 x^2\right )}{25 c^5}-\frac{b e \log ^2\left (1-c^2 x^2\right )}{20 c^5}+\frac{b x^2 \left (d+e \log \left (1-c^2 x^2\right )\right )}{10 c^3}+\frac{b x^4 \left (d+e \log \left (1-c^2 x^2\right )\right )}{20 c}+\frac{1}{5} x^5 \left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )+\frac{b \log \left (1-c^2 x^2\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )}{10 c^5}+\frac{(b e) \operatorname{Subst}\left (\int \left (-\frac{1}{c^2}-\frac{1}{c^2 \left (-1+c^2 x\right )}\right ) \, dx,x,x^2\right )}{15 c}\\ &=-\frac{2 a e x}{5 c^4}-\frac{77 b e x^2}{300 c^3}-\frac{2 a e x^3}{15 c^2}-\frac{9 b e x^4}{200 c}-\frac{2}{25} a e x^5-\frac{2 b e x \coth ^{-1}(c x)}{5 c^4}-\frac{2 b e x^3 \coth ^{-1}(c x)}{15 c^2}-\frac{2}{25} b e x^5 \coth ^{-1}(c x)+\frac{b e \coth ^{-1}(c x)^2}{5 c^5}-\frac{(4 a+3 b) e \log (1-c x)}{20 c^5}+\frac{(4 a-3 b) e \log (1+c x)}{20 c^5}-\frac{23 b e \log \left (1-c^2 x^2\right )}{75 c^5}-\frac{b e \log ^2\left (1-c^2 x^2\right )}{20 c^5}+\frac{b x^2 \left (d+e \log \left (1-c^2 x^2\right )\right )}{10 c^3}+\frac{b x^4 \left (d+e \log \left (1-c^2 x^2\right )\right )}{20 c}+\frac{1}{5} x^5 \left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )+\frac{b \log \left (1-c^2 x^2\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )}{10 c^5}\\ \end{align*}

Mathematica [A]  time = 0.148721, size = 236, normalized size = 0.75 \[ \frac{30 c^2 e x^2 \log \left (1-c^2 x^2\right ) \left (4 a c^3 x^3+b \left (c^2 x^2+2\right )+4 b c^3 x^3 \coth ^{-1}(c x)\right )+2 \log (1-c x) (-60 a e+30 b d-137 b e)+2 \log (c x+1) (60 a e+30 b d-137 b e)+24 a c^5 x^5 (5 d-2 e)-80 a c^3 e x^3-240 a c e x+3 b c^4 x^4 (10 d-9 e)+2 b c^2 x^2 (30 d-77 e)-8 b c x \coth ^{-1}(c x) \left (2 e \left (3 c^4 x^4+5 c^2 x^2+15\right )-15 c^4 d x^4\right )+30 b e \log ^2\left (1-c^2 x^2\right )+120 b e \coth ^{-1}(c x)^2}{600 c^5} \]

Antiderivative was successfully verified.

[In]

Integrate[x^4*(a + b*ArcCoth[c*x])*(d + e*Log[1 - c^2*x^2]),x]

[Out]

(-240*a*c*e*x + 2*b*c^2*(30*d - 77*e)*x^2 - 80*a*c^3*e*x^3 + 3*b*c^4*(10*d - 9*e)*x^4 + 24*a*c^5*(5*d - 2*e)*x
^5 - 8*b*c*x*(-15*c^4*d*x^4 + 2*e*(15 + 5*c^2*x^2 + 3*c^4*x^4))*ArcCoth[c*x] + 120*b*e*ArcCoth[c*x]^2 + 2*(30*
b*d - 60*a*e - 137*b*e)*Log[1 - c*x] + 2*(30*b*d + 60*a*e - 137*b*e)*Log[1 + c*x] + 30*c^2*e*x^2*(4*a*c^3*x^3
+ b*(2 + c^2*x^2) + 4*b*c^3*x^3*ArcCoth[c*x])*Log[1 - c^2*x^2] + 30*b*e*Log[1 - c^2*x^2]^2)/(600*c^5)

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Maple [C]  time = 1.72, size = 4194, normalized size = 13.3 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(a+b*arccoth(c*x))*(d+e*ln(-c^2*x^2+1)),x)

[Out]

1/10/c^3*b*x^2*d+1/20/c*b*x^4*d+1/5*b*arccoth(c*x)*x^5*d-46/75/c^5*b*arccoth(c*x)*e-1/5/c^5*b*ln((c*x+1)/(c*x-
1)-1)*d+1/5/c^5*b*e*ln((c*x+1)/(c*x-1)-1)^2+137/150/c^5*b*ln((c*x+1)/(c*x-1)-1)*e+1/5/c^5*b*arccoth(c*x)*d-2/5
*b*e*x*arccoth(c*x)/c^4-2/15*b*e*x^3*arccoth(c*x)/c^2+181/600/c^5*b*e-2/5*a*e*x/c^4-77/300*b*e*x^2/c^3-2/15*a*
e*x^3/c^2-9/200*b*e*x^4/c-2/25*b*e*x^5*arccoth(c*x)+1/20*I/c^3*b*csgn(I*(c*x+1)/(c*x-1))*csgn(I*(c*x+1)/(c*x-1
)/((c*x+1)/(c*x-1)-1)^2)^2*Pi*x^2*e+1/40*I/c*b*csgn(I*((c*x+1)/(c*x-1)-1)^2)*csgn(I*((c*x+1)/(c*x-1)-1))^2*Pi*
x^4*e+1/20*I/c^3*b*csgn(I*((c*x+1)/(c*x-1)-1)^2)*csgn(I*((c*x+1)/(c*x-1)-1))^2*Pi*x^2*e+1/20*I/c*b*csgn(I/((c*
x-1)/(c*x+1))^(1/2))*csgn(I*(c*x+1)/(c*x-1))^2*Pi*x^4*e+1/10*I/c^3*b*csgn(I/((c*x-1)/(c*x+1))^(1/2))*csgn(I*(c
*x+1)/(c*x-1))^2*Pi*x^2*e-1/40*I/c*b*csgn(I/((c*x-1)/(c*x+1))^(1/2))^2*csgn(I*(c*x+1)/(c*x-1))*Pi*x^4*e-2/25*a
*e*x^5+1/10*I/c^5*b*Pi*ln((c*x+1)/(c*x-1)-1)*e*csgn(I/((c*x+1)/(c*x-1)-1)^2)*csgn(I*(c*x+1)/(c*x-1)/((c*x+1)/(
c*x-1)-1)^2)*csgn(I*(c*x+1)/(c*x-1))-1/10*I*b*arccoth(c*x)*csgn(I*(c*x+1)/(c*x-1))*csgn(I*(c*x+1)/(c*x-1)/((c*
x+1)/(c*x-1)-1)^2)*csgn(I/((c*x+1)/(c*x-1)-1)^2)*Pi*x^5*e-3/20/c^5*b*d-1/10*I/c^5*b*arccoth(c*x)*e*Pi*csgn(I/(
(c*x+1)/(c*x-1)-1)^2)*csgn(I*(c*x+1)/(c*x-1)/((c*x+1)/(c*x-1)-1)^2)*csgn(I*(c*x+1)/(c*x-1))-1/40*I/c*b*csgn(I*
(c*x+1)/(c*x-1))*csgn(I*(c*x+1)/(c*x-1)/((c*x+1)/(c*x-1)-1)^2)*csgn(I/((c*x+1)/(c*x-1)-1)^2)*Pi*x^4*e-1/20*I/c
^3*b*csgn(I*(c*x+1)/(c*x-1))*csgn(I*(c*x+1)/(c*x-1)/((c*x+1)/(c*x-1)-1)^2)*csgn(I/((c*x+1)/(c*x-1)-1)^2)*Pi*x^
2*e+1/5*x^5*a*d-3/40*I/c^5*b*Pi*e*csgn(I/((c*x+1)/(c*x-1)-1)^2)*csgn(I*(c*x+1)/(c*x-1)/((c*x+1)/(c*x-1)-1)^2)^
2-3/40*I/c^5*b*Pi*e*csgn(I*(c*x+1)/(c*x-1)/((c*x+1)/(c*x-1)-1)^2)^2*csgn(I*(c*x+1)/(c*x-1))-3/40*I/c^5*b*Pi*e*
csgn(I*((c*x+1)/(c*x-1)-1)^2)*csgn(I*((c*x+1)/(c*x-1)-1))^2-1/10*I/c^5*b*arccoth(c*x)*e*Pi*csgn(I*(c*x+1)/(c*x
-1))^3-3/20*I/c^5*b*e*Pi*csgn(I/((c*x-1)/(c*x+1))^(1/2))*csgn(I*(c*x+1)/(c*x-1))^2-1/5*I/c^5*b*csgn(I*(c*x+1)/
(c*x-1)/((c*x+1)/(c*x-1)-1)^2)^2*Pi*e*arccoth(c*x)+1/5*I/c^5*b*Pi*ln((c*x+1)/(c*x-1)-1)*e*csgn(I*(c*x+1)/(c*x-
1)/((c*x+1)/(c*x-1)-1)^2)^2-1/40*I/c*b*csgn(I*(c*x+1)/(c*x-1))^3*Pi*x^4*e-1/20*I/c^3*b*csgn(I*(c*x+1)/(c*x-1))
^3*Pi*x^2*e+1/10*I/c^5*b*Pi*ln((c*x+1)/(c*x-1)-1)*e*csgn(I*(c*x+1)/(c*x-1))^3+1/10*I/c^5*b*arccoth(c*x)*Pi*e*c
sgn(I*(c*x+1)/(c*x-1)/((c*x+1)/(c*x-1)-1)^2)^3-1/20*I/c*b*csgn(I*(c*x+1)/(c*x-1)/((c*x+1)/(c*x-1)-1)^2)^2*Pi*x
^4*e-1/10*I/c^3*b*csgn(I*(c*x+1)/(c*x-1)/((c*x+1)/(c*x-1)-1)^2)^2*Pi*x^2*e+1/40*I/c*b*csgn(I*(c*x+1)/(c*x-1)/(
(c*x+1)/(c*x-1)-1)^2)^3*Pi*x^4*e+1/20*I/c^3*b*csgn(I*(c*x+1)/(c*x-1)/((c*x+1)/(c*x-1)-1)^2)^3*Pi*x^2*e-1/5*I*b
*arccoth(c*x)*csgn(I*(c*x+1)/(c*x-1)/((c*x+1)/(c*x-1)-1)^2)^2*Pi*x^5*e+1/10*I*b*arccoth(c*x)*csgn(I*(c*x+1)/(c
*x-1)/((c*x+1)/(c*x-1)-1)^2)^3*Pi*x^5*e+1/10*I*b*arccoth(c*x)*csgn(I*((c*x+1)/(c*x-1)-1)^2)^3*Pi*x^5*e+1/10*I/
c^5*b*arccoth(c*x)*e*Pi*csgn(I*((c*x+1)/(c*x-1)-1)^2)^3+3/20*I/c^5*b*Pi*e*csgn(I*((c*x+1)/(c*x-1)-1)^2)^2*csgn
(I*((c*x+1)/(c*x-1)-1))+1/5*a*e*x^5*ln(-c^2*x^2+1)-1/5*a*e/c^5*ln(c*x-1)+1/5*a*e/c^5*ln(c*x+1)-1/10*I*b*arccot
h(c*x)*csgn(I*(c*x+1)/(c*x-1))^3*Pi*x^5*e-1/10/c*b*ln((c*x+1)/(c*x-1)-1)*x^4*e-1/5/c^3*b*ln((c*x+1)/(c*x-1)-1)
*x^2*e+1/10/c*b*ln(2)*x^4*e+1/5/c^3*b*ln(2)*x^2*e+2/5/c^5*b*arccoth(c*x)*ln(2)*e-2/5/c^5*b*ln(2)*ln((c*x+1)/(c
*x-1)-1)*e-2/5*b*arccoth(c*x)*ln((c*x+1)/(c*x-1)-1)*x^5*e+2/5*b*arccoth(c*x)*ln(2)*x^5*e-3/20*I/c^5*b*Pi*e-1/2
0*I/c^3*b*csgn(I/((c*x-1)/(c*x+1))^(1/2))^2*csgn(I*(c*x+1)/(c*x-1))*Pi*x^2*e+1/10*I/c^5*b*arccoth(c*x)*e*Pi*cs
gn(I*((c*x+1)/(c*x-1)-1)^2)*csgn(I*((c*x+1)/(c*x-1)-1))^2-1/10*I/c^5*b*Pi*ln((c*x+1)/(c*x-1)-1)*e*csgn(I*((c*x
+1)/(c*x-1)-1)^2)*csgn(I*((c*x+1)/(c*x-1)-1))^2+1/10*I/c^5*b*arccoth(c*x)*e*Pi*csgn(I/((c*x+1)/(c*x-1)-1)^2)*c
sgn(I*(c*x+1)/(c*x-1)/((c*x+1)/(c*x-1)-1)^2)^2+3/40*I/c^5*b*e*Pi*csgn(I/((c*x+1)/(c*x-1)-1)^2)*csgn(I*(c*x+1)/
(c*x-1)/((c*x+1)/(c*x-1)-1)^2)*csgn(I*(c*x+1)/(c*x-1))+1/5*I/c^5*b*arccoth(c*x)*Pi*e*csgn(I/((c*x-1)/(c*x+1))^
(1/2))*csgn(I*(c*x+1)/(c*x-1))^2-1/10*I/c^5*b*arccoth(c*x)*e*Pi*csgn(I/((c*x-1)/(c*x+1))^(1/2))^2*csgn(I*(c*x+
1)/(c*x-1))-1/10*I/c^5*b*Pi*ln((c*x+1)/(c*x-1)-1)*e*csgn(I/((c*x+1)/(c*x-1)-1)^2)*csgn(I*(c*x+1)/(c*x-1)/((c*x
+1)/(c*x-1)-1)^2)^2-1/5*I/c^5*b*Pi*ln((c*x+1)/(c*x-1)-1)*e*csgn(I/((c*x-1)/(c*x+1))^(1/2))*csgn(I*(c*x+1)/(c*x
-1))^2+1/5*I/c^5*b*Pi*ln((c*x+1)/(c*x-1)-1)*e*csgn(I*((c*x+1)/(c*x-1)-1)^2)^2*csgn(I*((c*x+1)/(c*x-1)-1))-1/20
*I/c*b*csgn(I*((c*x+1)/(c*x-1)-1)^2)^2*csgn(I*((c*x+1)/(c*x-1)-1))*Pi*x^4*e-1/10*I/c^3*b*csgn(I*((c*x+1)/(c*x-
1)-1)^2)^2*csgn(I*((c*x+1)/(c*x-1)-1))*Pi*x^2*e-1/10*I/c^5*b*Pi*ln((c*x+1)/(c*x-1)-1)*e*csgn(I*(c*x+1)/(c*x-1)
/((c*x+1)/(c*x-1)-1)^2)^2*csgn(I*(c*x+1)/(c*x-1))+1/10*I/c^5*b*Pi*ln((c*x+1)/(c*x-1)-1)*e*csgn(I/((c*x-1)/(c*x
+1))^(1/2))^2*csgn(I*(c*x+1)/(c*x-1))+1/10*I*b*arccoth(c*x)*csgn(I*(c*x+1)/(c*x-1)/((c*x+1)/(c*x-1)-1)^2)^2*cs
gn(I/((c*x+1)/(c*x-1)-1)^2)*Pi*x^5*e+1/10*I*b*arccoth(c*x)*csgn(I*((c*x+1)/(c*x-1)-1)^2)*csgn(I*((c*x+1)/(c*x-
1)-1))^2*Pi*x^5*e+1/5*I*b*arccoth(c*x)*csgn(I/((c*x-1)/(c*x+1))^(1/2))*csgn(I*(c*x+1)/(c*x-1))^2*Pi*x^5*e-1/10
*I*b*arccoth(c*x)*csgn(I/((c*x-1)/(c*x+1))^(1/2))^2*csgn(I*(c*x+1)/(c*x-1))*Pi*x^5*e+1/10*I*b*arccoth(c*x)*csg
n(I*(c*x+1)/(c*x-1))*csgn(I*(c*x+1)/(c*x-1)/((c*x+1)/(c*x-1)-1)^2)^2*Pi*x^5*e-1/5*I*b*arccoth(c*x)*csgn(I*((c*
x+1)/(c*x-1)-1)^2)^2*csgn(I*((c*x+1)/(c*x-1)-1))*Pi*x^5*e-1/5*I/c^5*b*arccoth(c*x)*Pi*e*csgn(I*((c*x+1)/(c*x-1
)-1)^2)^2*csgn(I*((c*x+1)/(c*x-1)-1))+1/10*I/c^5*b*arccoth(c*x)*Pi*e*csgn(I*(c*x+1)/(c*x-1)/((c*x+1)/(c*x-1)-1
)^2)^2*csgn(I*(c*x+1)/(c*x-1))+1/40*I/c*b*csgn(I*(c*x+1)/(c*x-1)/((c*x+1)/(c*x-1)-1)^2)^2*csgn(I/((c*x+1)/(c*x
-1)-1)^2)*Pi*x^4*e+1/20*I/c^3*b*csgn(I*(c*x+1)/(c*x-1)/((c*x+1)/(c*x-1)-1)^2)^2*csgn(I/((c*x+1)/(c*x-1)-1)^2)*
Pi*x^2*e+1/40*I/c*b*csgn(I*(c*x+1)/(c*x-1))*csgn(I*(c*x+1)/(c*x-1)/((c*x+1)/(c*x-1)-1)^2)^2*Pi*x^4*e+1/40*I/c*
b*csgn(I*((c*x+1)/(c*x-1)-1)^2)^3*Pi*x^4*e+1/20*I/c^3*b*csgn(I*((c*x+1)/(c*x-1)-1)^2)^3*Pi*x^2*e-1/10*I/c^5*b*
Pi*ln((c*x+1)/(c*x-1)-1)*e*csgn(I*((c*x+1)/(c*x-1)-1)^2)^3-1/10*I/c^5*b*Pi*ln((c*x+1)/(c*x-1)-1)*e*csgn(I*(c*x
+1)/(c*x-1)/((c*x+1)/(c*x-1)-1)^2)^3+3/40*I/c^5*b*e*Pi*csgn(I/((c*x-1)/(c*x+1))^(1/2))^2*csgn(I*(c*x+1)/(c*x-1
))-3/10/c^5*b*e*ln(2)-1/20/c^5*b*e*(4*arccoth(c*x)*x^5*c^5+c^4*x^4+2*c^2*x^2+4*arccoth(c*x)-4*ln((c*x+1)/(c*x-
1)-1)-3)*ln((c*x-1)/(c*x+1))+1/5*I*b*arccoth(c*x)*Pi*x^5*e+1/5*I/c^5*b*arccoth(c*x)*Pi*e-1/5*I/c^5*b*Pi*ln((c*
x+1)/(c*x-1)-1)*e-3/40*I/c^5*b*Pi*e*csgn(I*((c*x+1)/(c*x-1)-1)^2)^3+3/40*I/c^5*b*e*Pi*csgn(I*(c*x+1)/(c*x-1))^
3+3/20*I/c^5*b*Pi*e*csgn(I*(c*x+1)/(c*x-1)/((c*x+1)/(c*x-1)-1)^2)^2-3/40*I/c^5*b*Pi*e*csgn(I*(c*x+1)/(c*x-1)/(
(c*x+1)/(c*x-1)-1)^2)^3+1/20*I/c*b*Pi*x^4*e+1/10*I/c^3*b*Pi*x^2*e

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Maxima [C]  time = 1.10989, size = 428, normalized size = 1.36 \begin{align*} \frac{1}{5} \, a d x^{5} + \frac{1}{75} \,{\left (15 \, x^{5} \log \left (-c^{2} x^{2} + 1\right ) - c^{2}{\left (\frac{2 \,{\left (3 \, c^{4} x^{5} + 5 \, c^{2} x^{3} + 15 \, x\right )}}{c^{6}} - \frac{15 \, \log \left (c x + 1\right )}{c^{7}} + \frac{15 \, \log \left (c x - 1\right )}{c^{7}}\right )}\right )} b e \operatorname{arcoth}\left (c x\right ) + \frac{1}{20} \,{\left (4 \, x^{5} \operatorname{arcoth}\left (c x\right ) + c{\left (\frac{c^{2} x^{4} + 2 \, x^{2}}{c^{4}} + \frac{2 \, \log \left (c^{2} x^{2} - 1\right )}{c^{6}}\right )}\right )} b d + \frac{1}{75} \,{\left (15 \, x^{5} \log \left (-c^{2} x^{2} + 1\right ) - c^{2}{\left (\frac{2 \,{\left (3 \, c^{4} x^{5} + 5 \, c^{2} x^{3} + 15 \, x\right )}}{c^{6}} - \frac{15 \, \log \left (c x + 1\right )}{c^{7}} + \frac{15 \, \log \left (c x - 1\right )}{c^{7}}\right )}\right )} a e - \frac{{\left (3 \,{\left (-10 i \, \pi c^{4} + 9 \, c^{4}\right )} x^{4} + 2 \,{\left (-30 i \, \pi c^{2} + 77 \, c^{2}\right )} x^{2} -{\left (60 i \, \pi + 30 \, c^{4} x^{4} + 60 \, c^{2} x^{2} + 120 \, \log \left (c x - 1\right ) - 274\right )} \log \left (c x + 1\right ) -{\left (60 i \, \pi + 30 \, c^{4} x^{4} + 60 \, c^{2} x^{2} - 274\right )} \log \left (c x - 1\right )\right )} b e}{600 \, c^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(a+b*arccoth(c*x))*(d+e*log(-c^2*x^2+1)),x, algorithm="maxima")

[Out]

1/5*a*d*x^5 + 1/75*(15*x^5*log(-c^2*x^2 + 1) - c^2*(2*(3*c^4*x^5 + 5*c^2*x^3 + 15*x)/c^6 - 15*log(c*x + 1)/c^7
+ 15*log(c*x - 1)/c^7))*b*e*arccoth(c*x) + 1/20*(4*x^5*arccoth(c*x) + c*((c^2*x^4 + 2*x^2)/c^4 + 2*log(c^2*x^
2 - 1)/c^6))*b*d + 1/75*(15*x^5*log(-c^2*x^2 + 1) - c^2*(2*(3*c^4*x^5 + 5*c^2*x^3 + 15*x)/c^6 - 15*log(c*x + 1
)/c^7 + 15*log(c*x - 1)/c^7))*a*e - 1/600*(3*(-10*I*pi*c^4 + 9*c^4)*x^4 + 2*(-30*I*pi*c^2 + 77*c^2)*x^2 - (60*
I*pi + 30*c^4*x^4 + 60*c^2*x^2 + 120*log(c*x - 1) - 274)*log(c*x + 1) - (60*I*pi + 30*c^4*x^4 + 60*c^2*x^2 - 2
74)*log(c*x - 1))*b*e/c^5

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Fricas [A]  time = 1.67589, size = 587, normalized size = 1.86 \begin{align*} -\frac{80 \, a c^{3} e x^{3} - 24 \,{\left (5 \, a c^{5} d - 2 \, a c^{5} e\right )} x^{5} - 3 \,{\left (10 \, b c^{4} d - 9 \, b c^{4} e\right )} x^{4} + 240 \, a c e x - 30 \, b e \log \left (-c^{2} x^{2} + 1\right )^{2} - 30 \, b e \log \left (\frac{c x + 1}{c x - 1}\right )^{2} - 2 \,{\left (30 \, b c^{2} d - 77 \, b c^{2} e\right )} x^{2} - 2 \,{\left (60 \, a c^{5} e x^{5} + 15 \, b c^{4} e x^{4} + 30 \, b c^{2} e x^{2} + 30 \, b d - 137 \, b e\right )} \log \left (-c^{2} x^{2} + 1\right ) - 4 \,{\left (15 \, b c^{5} e x^{5} \log \left (-c^{2} x^{2} + 1\right ) - 10 \, b c^{3} e x^{3} + 3 \,{\left (5 \, b c^{5} d - 2 \, b c^{5} e\right )} x^{5} - 30 \, b c e x + 30 \, a e\right )} \log \left (\frac{c x + 1}{c x - 1}\right )}{600 \, c^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(a+b*arccoth(c*x))*(d+e*log(-c^2*x^2+1)),x, algorithm="fricas")

[Out]

-1/600*(80*a*c^3*e*x^3 - 24*(5*a*c^5*d - 2*a*c^5*e)*x^5 - 3*(10*b*c^4*d - 9*b*c^4*e)*x^4 + 240*a*c*e*x - 30*b*
e*log(-c^2*x^2 + 1)^2 - 30*b*e*log((c*x + 1)/(c*x - 1))^2 - 2*(30*b*c^2*d - 77*b*c^2*e)*x^2 - 2*(60*a*c^5*e*x^
5 + 15*b*c^4*e*x^4 + 30*b*c^2*e*x^2 + 30*b*d - 137*b*e)*log(-c^2*x^2 + 1) - 4*(15*b*c^5*e*x^5*log(-c^2*x^2 + 1
) - 10*b*c^3*e*x^3 + 3*(5*b*c^5*d - 2*b*c^5*e)*x^5 - 30*b*c*e*x + 30*a*e)*log((c*x + 1)/(c*x - 1)))/c^5

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Sympy [A]  time = 43.4042, size = 345, normalized size = 1.1 \begin{align*} \begin{cases} \frac{a d x^{5}}{5} + \frac{a e x^{5} \log{\left (- c^{2} x^{2} + 1 \right )}}{5} - \frac{2 a e x^{5}}{25} - \frac{2 a e x^{3}}{15 c^{2}} - \frac{2 a e x}{5 c^{4}} + \frac{2 a e \operatorname{acoth}{\left (c x \right )}}{5 c^{5}} + \frac{b d x^{5} \operatorname{acoth}{\left (c x \right )}}{5} + \frac{b e x^{5} \log{\left (- c^{2} x^{2} + 1 \right )} \operatorname{acoth}{\left (c x \right )}}{5} - \frac{2 b e x^{5} \operatorname{acoth}{\left (c x \right )}}{25} + \frac{b d x^{4}}{20 c} + \frac{b e x^{4} \log{\left (- c^{2} x^{2} + 1 \right )}}{20 c} - \frac{9 b e x^{4}}{200 c} - \frac{2 b e x^{3} \operatorname{acoth}{\left (c x \right )}}{15 c^{2}} + \frac{b d x^{2}}{10 c^{3}} + \frac{b e x^{2} \log{\left (- c^{2} x^{2} + 1 \right )}}{10 c^{3}} - \frac{77 b e x^{2}}{300 c^{3}} - \frac{2 b e x \operatorname{acoth}{\left (c x \right )}}{5 c^{4}} + \frac{b d \log{\left (- c^{2} x^{2} + 1 \right )}}{10 c^{5}} + \frac{b e \log{\left (- c^{2} x^{2} + 1 \right )}^{2}}{20 c^{5}} - \frac{137 b e \log{\left (- c^{2} x^{2} + 1 \right )}}{300 c^{5}} + \frac{b e \operatorname{acoth}^{2}{\left (c x \right )}}{5 c^{5}} & \text{for}\: c \neq 0 \\\frac{d x^{5} \left (a + \frac{i \pi b}{2}\right )}{5} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4*(a+b*acoth(c*x))*(d+e*ln(-c**2*x**2+1)),x)

[Out]

Piecewise((a*d*x**5/5 + a*e*x**5*log(-c**2*x**2 + 1)/5 - 2*a*e*x**5/25 - 2*a*e*x**3/(15*c**2) - 2*a*e*x/(5*c**
4) + 2*a*e*acoth(c*x)/(5*c**5) + b*d*x**5*acoth(c*x)/5 + b*e*x**5*log(-c**2*x**2 + 1)*acoth(c*x)/5 - 2*b*e*x**
5*acoth(c*x)/25 + b*d*x**4/(20*c) + b*e*x**4*log(-c**2*x**2 + 1)/(20*c) - 9*b*e*x**4/(200*c) - 2*b*e*x**3*acot
h(c*x)/(15*c**2) + b*d*x**2/(10*c**3) + b*e*x**2*log(-c**2*x**2 + 1)/(10*c**3) - 77*b*e*x**2/(300*c**3) - 2*b*
e*x*acoth(c*x)/(5*c**4) + b*d*log(-c**2*x**2 + 1)/(10*c**5) + b*e*log(-c**2*x**2 + 1)**2/(20*c**5) - 137*b*e*l
og(-c**2*x**2 + 1)/(300*c**5) + b*e*acoth(c*x)**2/(5*c**5), Ne(c, 0)), (d*x**5*(a + I*pi*b/2)/5, True))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \operatorname{arcoth}\left (c x\right ) + a\right )}{\left (e \log \left (-c^{2} x^{2} + 1\right ) + d\right )} x^{4}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(a+b*arccoth(c*x))*(d+e*log(-c^2*x^2+1)),x, algorithm="giac")

[Out]

integrate((b*arccoth(c*x) + a)*(e*log(-c^2*x^2 + 1) + d)*x^4, x)