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

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

[Out]

-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*a*e*x^5-2/5*b*e*x*arccoth(c*x)/c^4-2/1
5*b*e*x^3*arccoth(c*x)/c^2-2/25*b*e*x^5*arccoth(c*x)+1/5*b*e*arccoth(c*x)^2/c^5-1/20*(4*a+3*b)*e*ln(-c*x+1)/c^
5+1/20*(4*a-3*b)*e*ln(c*x+1)/c^5-23/75*b*e*ln(-c^2*x^2+1)/c^5-1/20*b*e*ln(-c^2*x^2+1)^2/c^5+1/10*b*x^2*(d+e*ln
(-c^2*x^2+1))/c^3+1/20*b*x^4*(d+e*ln(-c^2*x^2+1))/c+1/5*x^5*(a+b*arccoth(c*x))*(d+e*ln(-c^2*x^2+1))+1/10*b*ln(
-c^2*x^2+1)*(d+e*ln(-c^2*x^2+1))/c^5

________________________________________________________________________________________

Rubi [A]  time = 0.75, antiderivative size = 315, normalized size of antiderivative = 1.00, 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 31

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

Rule 43

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

Rule 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 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 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 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 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 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 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 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 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 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 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]

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*}

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Mathematica [A]  time = 0.16, size = 236, normalized size = 0.75 \[ \frac {30 c^2 e x^2 \log \left (1-c^2 x^2\right ) \left (4 a c^3 x^3+4 b c^3 x^3 \coth ^{-1}(c x)+b \left (c^2 x^2+2\right )\right )+2 \log (1-c x) (-60 a e+30 b d-137 b e)+2 \log (c x+1) (60 a e+30 b d-137 b e)+24 a c^5 x^5 (5 d-2 e)-80 a c^3 e x^3-240 a c e x+3 b c^4 x^4 (10 d-9 e)+2 b c^2 x^2 (30 d-77 e)+30 b e \log ^2\left (1-c^2 x^2\right )-8 b c x \coth ^{-1}(c x) \left (2 e \left (3 c^4 x^4+5 c^2 x^2+15\right )-15 c^4 d x^4\right )+120 b e \coth ^{-1}(c x)^2}{600 c^5} \]

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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fricas [A]  time = 0.61, size = 249, normalized size = 0.79 \[ -\frac {80 \, a c^{3} e x^{3} - 24 \, {\left (5 \, a c^{5} d - 2 \, a c^{5} e\right )} x^{5} - 3 \, {\left (10 \, b c^{4} d - 9 \, b c^{4} e\right )} x^{4} + 240 \, a c e x - 30 \, b e \log \left (-c^{2} x^{2} + 1\right )^{2} - 30 \, b e \log \left (\frac {c x + 1}{c x - 1}\right )^{2} - 2 \, {\left (30 \, b c^{2} d - 77 \, b c^{2} e\right )} x^{2} - 2 \, {\left (60 \, a c^{5} e x^{5} + 15 \, b c^{4} e x^{4} + 30 \, b c^{2} e x^{2} + 30 \, b d - 137 \, b e\right )} \log \left (-c^{2} x^{2} + 1\right ) - 4 \, {\left (15 \, b c^{5} e x^{5} \log \left (-c^{2} x^{2} + 1\right ) - 10 \, b c^{3} e x^{3} + 3 \, {\left (5 \, b c^{5} d - 2 \, b c^{5} e\right )} x^{5} - 30 \, b c e x + 30 \, a e\right )} \log \left (\frac {c x + 1}{c x - 1}\right )}{600 \, c^{5}} \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b \operatorname {arcoth}\left (c x\right ) + a\right )} {\left (e \log \left (-c^{2} x^{2} + 1\right ) + d\right )} x^{4}\,{d x} \]

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)

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maple [C]  time = 4.75, size = 4194, normalized size = 13.31 \[ \text {output too large to display} \]

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]

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

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maxima [C]  time = 0.34, size = 314, normalized size = 1.00 \[ \frac {1}{5} \, a d x^{5} + \frac {1}{75} \, {\left (15 \, x^{5} \log \left (-c^{2} x^{2} + 1\right ) - c^{2} {\left (\frac {2 \, {\left (3 \, c^{4} x^{5} + 5 \, c^{2} x^{3} + 15 \, x\right )}}{c^{6}} - \frac {15 \, \log \left (c x + 1\right )}{c^{7}} + \frac {15 \, \log \left (c x - 1\right )}{c^{7}}\right )}\right )} b e \operatorname {arcoth}\left (c x\right ) + \frac {1}{20} \, {\left (4 \, x^{5} \operatorname {arcoth}\left (c x\right ) + c {\left (\frac {c^{2} x^{4} + 2 \, x^{2}}{c^{4}} + \frac {2 \, \log \left (c^{2} x^{2} - 1\right )}{c^{6}}\right )}\right )} b d + \frac {1}{75} \, {\left (15 \, x^{5} \log \left (-c^{2} x^{2} + 1\right ) - c^{2} {\left (\frac {2 \, {\left (3 \, c^{4} x^{5} + 5 \, c^{2} x^{3} + 15 \, x\right )}}{c^{6}} - \frac {15 \, \log \left (c x + 1\right )}{c^{7}} + \frac {15 \, \log \left (c x - 1\right )}{c^{7}}\right )}\right )} a e + \frac {{\left ({\left (30 i \, \pi c^{4} - 27 \, c^{4}\right )} x^{4} + {\left (60 i \, \pi c^{2} - 154 \, c^{2}\right )} x^{2} + {\left (60 i \, \pi + 30 \, c^{4} x^{4} + 60 \, c^{2} x^{2} + 120 \, \log \left (c x - 1\right ) - 274\right )} \log \left (c x + 1\right ) - 2 \, {\left (-30 i \, \pi - 15 \, c^{4} x^{4} - 30 \, c^{2} x^{2} + 137\right )} \log \left (c x - 1\right )\right )} b e}{600 \, c^{5}} \]

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*((30*I*pi*c^4 - 27*c^4)*x^4 + (60*I*pi*c^2 - 154*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) - 2*(-30*I*pi - 15*c^4*x^4 - 30*c^2*x^2 + 13
7)*log(c*x - 1))*b*e/c^5

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mupad [B]  time = 2.34, size = 497, normalized size = 1.58 \[ \ln \left (\frac {1}{c\,x}+1\right )\,\left (\frac {b\,d\,x^5}{10}-\frac {\frac {2\,b\,e\,c^5\,x^5}{5}+\frac {2\,b\,e\,c^3\,x^3}{3}+2\,b\,e\,c\,x}{10\,c^5}+\frac {b\,e\,x^5\,\ln \left (1-c^2\,x^2\right )}{10}\right )+\ln \left (1-\frac {1}{c\,x}\right )\,\left (\frac {\frac {b\,d\,x^6}{5}-\frac {b\,c^2\,d\,x^8}{5}}{2\,\left (c\,x^2+x\right )\,\left (c\,x-1\right )}+\frac {\frac {4\,b\,e\,x^6}{75}+\frac {4\,b\,e\,x^4}{15\,c^2}-\frac {2\,b\,e\,x^2}{5\,c^4}+\frac {2\,b\,c^2\,e\,x^8}{25}}{2\,\left (c\,x^2+x\right )\,\left (c\,x-1\right )}+\frac {\ln \left (1-c^2\,x^2\right )\,\left (\frac {b\,e\,x^6}{5}-\frac {b\,c^2\,e\,x^8}{5}\right )}{2\,\left (c\,x^2+x\right )\,\left (c\,x-1\right )}-\frac {b\,e\,\ln \left (\frac {1}{c\,x}+1\right )}{10\,c^5}\right )+x^3\,\left (\frac {a\,\left (5\,d-2\,e\right )}{15\,c^2}-\frac {a\,d}{3\,c^2}\right )+x^2\,\left (\frac {b\,\left (10\,d-9\,e\right )}{100\,c^3}-\frac {b\,e}{6\,c^3}\right )+\frac {x\,\left (\frac {a\,\left (5\,d-2\,e\right )}{5\,c^2}-\frac {a\,d}{c^2}\right )}{c^2}+\frac {a\,x^5\,\left (5\,d-2\,e\right )}{25}+c^2\,\ln \left (1-c^2\,x^2\right )\,\left (\frac {a\,e\,x^5}{5\,c^2}+\frac {b\,e\,x^4}{20\,c^3}+\frac {b\,e\,x^2}{10\,c^5}\right )-\frac {\ln \left (c\,x-1\right )\,\left (60\,a\,e-30\,b\,d+137\,b\,e\right )}{300\,c^5}+\frac {\ln \left (c\,x+1\right )\,\left (60\,a\,e+30\,b\,d-137\,b\,e\right )}{300\,c^5}+\frac {b\,e\,{\ln \left (\frac {1}{c\,x}+1\right )}^2}{20\,c^5}+\frac {b\,e\,{\ln \left (1-\frac {1}{c\,x}\right )}^2}{20\,c^5}+\frac {b\,e\,{\ln \left (1-c^2\,x^2\right )}^2}{20\,c^5}+\frac {b\,x^4\,\left (10\,d-9\,e\right )}{200\,c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(1/(c*x) + 1)*((b*d*x^5)/10 - (2*b*c*e*x + (2*b*c^3*e*x^3)/3 + (2*b*c^5*e*x^5)/5)/(10*c^5) + (b*e*x^5*log(1
- c^2*x^2))/10) + log(1 - 1/(c*x))*(((b*d*x^6)/5 - (b*c^2*d*x^8)/5)/(2*(x + c*x^2)*(c*x - 1)) + ((4*b*e*x^6)/
75 + (4*b*e*x^4)/(15*c^2) - (2*b*e*x^2)/(5*c^4) + (2*b*c^2*e*x^8)/25)/(2*(x + c*x^2)*(c*x - 1)) + (log(1 - c^2
*x^2)*((b*e*x^6)/5 - (b*c^2*e*x^8)/5))/(2*(x + c*x^2)*(c*x - 1)) - (b*e*log(1/(c*x) + 1))/(10*c^5)) + x^3*((a*
(5*d - 2*e))/(15*c^2) - (a*d)/(3*c^2)) + x^2*((b*(10*d - 9*e))/(100*c^3) - (b*e)/(6*c^3)) + (x*((a*(5*d - 2*e)
)/(5*c^2) - (a*d)/c^2))/c^2 + (a*x^5*(5*d - 2*e))/25 + c^2*log(1 - c^2*x^2)*((a*e*x^5)/(5*c^2) + (b*e*x^4)/(20
*c^3) + (b*e*x^2)/(10*c^5)) - (log(c*x - 1)*(60*a*e - 30*b*d + 137*b*e))/(300*c^5) + (log(c*x + 1)*(60*a*e + 3
0*b*d - 137*b*e))/(300*c^5) + (b*e*log(1/(c*x) + 1)^2)/(20*c^5) + (b*e*log(1 - 1/(c*x))^2)/(20*c^5) + (b*e*log
(1 - c^2*x^2)^2)/(20*c^5) + (b*x^4*(10*d - 9*e))/(200*c)

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sympy [A]  time = 15.30, size = 345, normalized size = 1.10 \[ \begin {cases} \frac {a d x^{5}}{5} + \frac {a e x^{5} \log {\left (- c^{2} x^{2} + 1 \right )}}{5} - \frac {2 a e x^{5}}{25} - \frac {2 a e x^{3}}{15 c^{2}} - \frac {2 a e x}{5 c^{4}} + \frac {2 a e \operatorname {acoth}{\left (c x \right )}}{5 c^{5}} + \frac {b d x^{5} \operatorname {acoth}{\left (c x \right )}}{5} + \frac {b e x^{5} \log {\left (- c^{2} x^{2} + 1 \right )} \operatorname {acoth}{\left (c x \right )}}{5} - \frac {2 b e x^{5} \operatorname {acoth}{\left (c x \right )}}{25} + \frac {b d x^{4}}{20 c} + \frac {b e x^{4} \log {\left (- c^{2} x^{2} + 1 \right )}}{20 c} - \frac {9 b e x^{4}}{200 c} - \frac {2 b e x^{3} \operatorname {acoth}{\left (c x \right )}}{15 c^{2}} + \frac {b d x^{2}}{10 c^{3}} + \frac {b e x^{2} \log {\left (- c^{2} x^{2} + 1 \right )}}{10 c^{3}} - \frac {77 b e x^{2}}{300 c^{3}} - \frac {2 b e x \operatorname {acoth}{\left (c x \right )}}{5 c^{4}} + \frac {b d \log {\left (- c^{2} x^{2} + 1 \right )}}{10 c^{5}} + \frac {b e \log {\left (- c^{2} x^{2} + 1 \right )}^{2}}{20 c^{5}} - \frac {137 b e \log {\left (- c^{2} x^{2} + 1 \right )}}{300 c^{5}} + \frac {b e \operatorname {acoth}^{2}{\left (c x \right )}}{5 c^{5}} & \text {for}\: c \neq 0 \\\frac {d x^{5} \left (a + \frac {i \pi b}{2}\right )}{5} & \text {otherwise} \end {cases} \]

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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