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

Optimal. Leaf size=247 \[ \frac{1}{3} x^3 \left (a+b \coth ^{-1}(c x)\right ) \left (e \log \left (1-c^2 x^2\right )+d\right )-\frac{e (2 a+b) \log (1-c x)}{6 c^3}+\frac{e (2 a-b) \log (c x+1)}{6 c^3}-\frac{2 a e x}{3 c^2}-\frac{2}{9} a e x^3+\frac{b x^2 \left (e \log \left (1-c^2 x^2\right )+d\right )}{6 c}+\frac{b \log \left (1-c^2 x^2\right ) \left (e \log \left (1-c^2 x^2\right )+d\right )}{6 c^3}-\frac{b e \log ^2\left (1-c^2 x^2\right )}{12 c^3}-\frac{4 b e \log \left (1-c^2 x^2\right )}{9 c^3}-\frac{2 b e x \coth ^{-1}(c x)}{3 c^2}+\frac{b e \coth ^{-1}(c x)^2}{3 c^3}-\frac{5 b e x^2}{18 c}-\frac{2}{9} b e x^3 \coth ^{-1}(c x) \]

[Out]

(-2*a*e*x)/(3*c^2) - (5*b*e*x^2)/(18*c) - (2*a*e*x^3)/9 - (2*b*e*x*ArcCoth[c*x])/(3*c^2) - (2*b*e*x^3*ArcCoth[
c*x])/9 + (b*e*ArcCoth[c*x]^2)/(3*c^3) - ((2*a + b)*e*Log[1 - c*x])/(6*c^3) + ((2*a - b)*e*Log[1 + c*x])/(6*c^
3) - (4*b*e*Log[1 - c^2*x^2])/(9*c^3) - (b*e*Log[1 - c^2*x^2]^2)/(12*c^3) + (b*x^2*(d + e*Log[1 - c^2*x^2]))/(
6*c) + (x^3*(a + b*ArcCoth[c*x])*(d + e*Log[1 - c^2*x^2]))/3 + (b*Log[1 - c^2*x^2]*(d + e*Log[1 - c^2*x^2]))/(
6*c^3)

________________________________________________________________________________________

Rubi [A]  time = 0.620957, antiderivative size = 247, normalized size of antiderivative = 1., number of steps used = 21, 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, 801, 633, 31, 5981, 5911, 260, 5949, 2475, 2390, 2301} \[ \frac{1}{3} x^3 \left (a+b \coth ^{-1}(c x)\right ) \left (e \log \left (1-c^2 x^2\right )+d\right )-\frac{e (2 a+b) \log (1-c x)}{6 c^3}+\frac{e (2 a-b) \log (c x+1)}{6 c^3}-\frac{2 a e x}{3 c^2}-\frac{2}{9} a e x^3+\frac{b x^2 \left (e \log \left (1-c^2 x^2\right )+d\right )}{6 c}+\frac{b \log \left (1-c^2 x^2\right ) \left (e \log \left (1-c^2 x^2\right )+d\right )}{6 c^3}-\frac{b e \log ^2\left (1-c^2 x^2\right )}{12 c^3}-\frac{4 b e \log \left (1-c^2 x^2\right )}{9 c^3}-\frac{2 b e x \coth ^{-1}(c x)}{3 c^2}+\frac{b e \coth ^{-1}(c x)^2}{3 c^3}-\frac{5 b e x^2}{18 c}-\frac{2}{9} b e x^3 \coth ^{-1}(c x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*a*e*x)/(3*c^2) - (5*b*e*x^2)/(18*c) - (2*a*e*x^3)/9 - (2*b*e*x*ArcCoth[c*x])/(3*c^2) - (2*b*e*x^3*ArcCoth[
c*x])/9 + (b*e*ArcCoth[c*x]^2)/(3*c^3) - ((2*a + b)*e*Log[1 - c*x])/(6*c^3) + ((2*a - b)*e*Log[1 + c*x])/(6*c^
3) - (4*b*e*Log[1 - c^2*x^2])/(9*c^3) - (b*e*Log[1 - c^2*x^2]^2)/(12*c^3) + (b*x^2*(d + e*Log[1 - c^2*x^2]))/(
6*c) + (x^3*(a + b*ArcCoth[c*x])*(d + e*Log[1 - c^2*x^2]))/3 + (b*Log[1 - c^2*x^2]*(d + e*Log[1 - c^2*x^2]))/(
6*c^3)

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 801

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegrand[(
(d + e*x)^m*(f + g*x))/(a + c*x^2), x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[c*d^2 + a*e^2, 0] && Integer
Q[m]

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^2 \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 )}{6 c}+\frac{1}{3} x^3 \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 )}{6 c^3}+\left (2 c^2 e\right ) \int \left (-\frac{x^3 \left (b+2 a c x+2 b c x \coth ^{-1}(c x)\right )}{6 c \left (-1+c^2 x^2\right )}-\frac{b x \log \left (1-c^2 x^2\right )}{6 c^3 \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 )}{6 c}+\frac{1}{3} x^3 \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 )}{6 c^3}-\frac{(b e) \int \frac{x \log \left (1-c^2 x^2\right )}{-1+c^2 x^2} \, dx}{3 c}-\frac{1}{3} (c e) \int \frac{x^3 \left (b+2 a c x+2 b c x \coth ^{-1}(c x)\right )}{-1+c^2 x^2} \, dx\\ &=\frac{b x^2 \left (d+e \log \left (1-c^2 x^2\right )\right )}{6 c}+\frac{1}{3} x^3 \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 )}{6 c^3}-\frac{(b e) \operatorname{Subst}\left (\int \frac{\log \left (1-c^2 x\right )}{-1+c^2 x} \, dx,x,x^2\right )}{6 c}-\frac{1}{3} (c e) \int \left (\frac{x^3 (b+2 a c x)}{-1+c^2 x^2}+\frac{2 b c x^4 \coth ^{-1}(c x)}{-1+c^2 x^2}\right ) \, dx\\ &=\frac{b x^2 \left (d+e \log \left (1-c^2 x^2\right )\right )}{6 c}+\frac{1}{3} x^3 \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 )}{6 c^3}-\frac{(b e) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,1-c^2 x^2\right )}{6 c^3}-\frac{1}{3} (c e) \int \frac{x^3 (b+2 a c x)}{-1+c^2 x^2} \, dx-\frac{1}{3} \left (2 b c^2 e\right ) \int \frac{x^4 \coth ^{-1}(c x)}{-1+c^2 x^2} \, dx\\ &=-\frac{b e \log ^2\left (1-c^2 x^2\right )}{12 c^3}+\frac{b x^2 \left (d+e \log \left (1-c^2 x^2\right )\right )}{6 c}+\frac{1}{3} x^3 \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 )}{6 c^3}-\frac{1}{3} (2 b e) \int x^2 \coth ^{-1}(c x) \, dx-\frac{1}{3} (2 b e) \int \frac{x^2 \coth ^{-1}(c x)}{-1+c^2 x^2} \, dx-\frac{1}{3} (c e) \int \left (\frac{2 a}{c^3}+\frac{b x}{c^2}+\frac{2 a x^2}{c}+\frac{2 a+b c x}{c^3 \left (-1+c^2 x^2\right )}\right ) \, dx\\ &=-\frac{2 a e x}{3 c^2}-\frac{b e x^2}{6 c}-\frac{2}{9} a e x^3-\frac{2}{9} b e x^3 \coth ^{-1}(c x)-\frac{b e \log ^2\left (1-c^2 x^2\right )}{12 c^3}+\frac{b x^2 \left (d+e \log \left (1-c^2 x^2\right )\right )}{6 c}+\frac{1}{3} x^3 \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 )}{6 c^3}-\frac{e \int \frac{2 a+b c x}{-1+c^2 x^2} \, dx}{3 c^2}-\frac{(2 b e) \int \coth ^{-1}(c x) \, dx}{3 c^2}-\frac{(2 b e) \int \frac{\coth ^{-1}(c x)}{-1+c^2 x^2} \, dx}{3 c^2}+\frac{1}{9} (2 b c e) \int \frac{x^3}{1-c^2 x^2} \, dx\\ &=-\frac{2 a e x}{3 c^2}-\frac{b e x^2}{6 c}-\frac{2}{9} a e x^3-\frac{2 b e x \coth ^{-1}(c x)}{3 c^2}-\frac{2}{9} b e x^3 \coth ^{-1}(c x)+\frac{b e \coth ^{-1}(c x)^2}{3 c^3}-\frac{b e \log ^2\left (1-c^2 x^2\right )}{12 c^3}+\frac{b x^2 \left (d+e \log \left (1-c^2 x^2\right )\right )}{6 c}+\frac{1}{3} x^3 \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 )}{6 c^3}+\frac{((2 a-b) e) \int \frac{1}{c+c^2 x} \, dx}{6 c}+\frac{(2 b e) \int \frac{x}{1-c^2 x^2} \, dx}{3 c}-\frac{((2 a+b) e) \int \frac{1}{-c+c^2 x} \, dx}{6 c}+\frac{1}{9} (b c e) \operatorname{Subst}\left (\int \frac{x}{1-c^2 x} \, dx,x,x^2\right )\\ &=-\frac{2 a e x}{3 c^2}-\frac{b e x^2}{6 c}-\frac{2}{9} a e x^3-\frac{2 b e x \coth ^{-1}(c x)}{3 c^2}-\frac{2}{9} b e x^3 \coth ^{-1}(c x)+\frac{b e \coth ^{-1}(c x)^2}{3 c^3}-\frac{(2 a+b) e \log (1-c x)}{6 c^3}+\frac{(2 a-b) e \log (1+c x)}{6 c^3}-\frac{b e \log \left (1-c^2 x^2\right )}{3 c^3}-\frac{b e \log ^2\left (1-c^2 x^2\right )}{12 c^3}+\frac{b x^2 \left (d+e \log \left (1-c^2 x^2\right )\right )}{6 c}+\frac{1}{3} x^3 \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 )}{6 c^3}+\frac{1}{9} (b c 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 )\\ &=-\frac{2 a e x}{3 c^2}-\frac{5 b e x^2}{18 c}-\frac{2}{9} a e x^3-\frac{2 b e x \coth ^{-1}(c x)}{3 c^2}-\frac{2}{9} b e x^3 \coth ^{-1}(c x)+\frac{b e \coth ^{-1}(c x)^2}{3 c^3}-\frac{(2 a+b) e \log (1-c x)}{6 c^3}+\frac{(2 a-b) e \log (1+c x)}{6 c^3}-\frac{4 b e \log \left (1-c^2 x^2\right )}{9 c^3}-\frac{b e \log ^2\left (1-c^2 x^2\right )}{12 c^3}+\frac{b x^2 \left (d+e \log \left (1-c^2 x^2\right )\right )}{6 c}+\frac{1}{3} x^3 \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 )}{6 c^3}\\ \end{align*}

Mathematica [A]  time = 0.123003, size = 183, normalized size = 0.74 \[ \frac{6 c^2 e x^2 \log \left (1-c^2 x^2\right ) \left (2 a c x+2 b c x \coth ^{-1}(c x)+b\right )+2 \log (1-c x) (-6 a e+3 b d-11 b e)+2 \log (c x+1) (6 a e+3 b d-11 b e)+4 a c^3 x^3 (3 d-2 e)-24 a c e x+2 b c^2 x^2 (3 d-5 e)+4 b c x \coth ^{-1}(c x) \left (3 c^2 d x^2-2 e \left (c^2 x^2+3\right )\right )+3 b e \log ^2\left (1-c^2 x^2\right )+12 b e \coth ^{-1}(c x)^2}{36 c^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-24*a*c*e*x + 2*b*c^2*(3*d - 5*e)*x^2 + 4*a*c^3*(3*d - 2*e)*x^3 + 4*b*c*x*(3*c^2*d*x^2 - 2*e*(3 + c^2*x^2))*A
rcCoth[c*x] + 12*b*e*ArcCoth[c*x]^2 + 2*(3*b*d - 6*a*e - 11*b*e)*Log[1 - c*x] + 2*(3*b*d + 6*a*e - 11*b*e)*Log
[1 + c*x] + 6*c^2*e*x^2*(b + 2*a*c*x + 2*b*c*x*ArcCoth[c*x])*Log[1 - c^2*x^2] + 3*b*e*Log[1 - c^2*x^2]^2)/(36*
c^3)

________________________________________________________________________________________

Maple [C]  time = 1.083, size = 3515, normalized size = 14.2 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3*b*e*x*arccoth(c*x)/c^2-2/3*a*e*x/c^2-5/18*b*e*x^2/c-2/9*b*e*x^3*arccoth(c*x)+1/3*a*e*x^3*ln(-c^2*x^2+1)-1
/3*a*e/c^3*ln(c*x-1)+1/3*a*e/c^3*ln(c*x+1)+1/6/c*b*x^2*d+1/3/c^3*b*e*ln((c*x+1)/(c*x-1)-1)^2-1/3/c^3*b*ln((c*x
+1)/(c*x-1)-1)*d+11/9/c^3*b*ln((c*x+1)/(c*x-1)-1)*e+1/3/c^3*b*arccoth(c*x)*d-8/9/c^3*b*e*arccoth(c*x)+1/3*b*ar
ccoth(c*x)*x^3*d-1/6/c^3*b*d+5/18/c^3*b*e-2/9*a*e*x^3+1/3*x^3*a*d-1/3/c^3*b*e*ln(2)+1/3*I/c^3*b*e*ln((c*x+1)/(
c*x-1)-1)*Pi*csgn(I*((c*x+1)/(c*x-1)-1)^2)^2*csgn(I*((c*x+1)/(c*x-1)-1))-1/6*I/c^3*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))^2-1/6*I/c^3*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))+1/3*I/c^3*b*arccoth(c*x)*e*Pi*csgn(I/((c*x-1)/(c*x+1))^(1/2))*csgn(I*(
c*x+1)/(c*x-1))^2+1/6*I/c^3*b*arccoth(c*x)*e*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))+1/6*I/c^3*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)^2-1/3*I/c^3*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))+1/6*I/c^3*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+1/12*I/c^3*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))+1/6*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^3*e-1/12*I/c*b*Pi*csgn(I/((c*x-1)/(c*x+1))^(1/2)
)^2*csgn(I*(c*x+1)/(c*x-1))*x^2*e-1/6*I/c^3*b*Pi*e-1/6*I*b*arccoth(c*x)*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)*x^3*e-1/12*I/c*b*Pi*csgn(I*(c*x+1)/(c*x-1))*c
sgn(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)*x^2*e+1/6*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)*e*ln((c*x+1)/(c*x-1)-1)*Pi-
1/6*I/c^3*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))-1/3/c*b*ln((c*x+1)/(c*x-1)-1)*x^2*e+1/3/c*b*ln(2)*x^2*e-2/3/c^3*b*ln(2)*ln((c*x+1)/(c*x-1)
-1)*e+2/3/c^3*b*arccoth(c*x)*ln(2)*e-2/3*b*ln((c*x+1)/(c*x-1)-1)*arccoth(c*x)*x^3*e+2/3*b*arccoth(c*x)*ln(2)*x
^3*e+1/6/c^3*b*e*(-2*arccoth(c*x)*x^3*c^3-c^2*x^2+2*ln((c*x+1)/(c*x-1)-1)-2*arccoth(c*x)+1)*ln((c*x-1)/(c*x+1)
)+1/6*I/c*b*Pi*csgn(I/((c*x-1)/(c*x+1))^(1/2))*csgn(I*(c*x+1)/(c*x-1))^2*x^2*e+1/12*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^2*e+1/12*I/c*b*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)^2*x^2*e-1/6*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^2*e+1/12*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^2*e-1/6*I*b*arcc
oth(c*x)*Pi*csgn(I/((c*x-1)/(c*x+1))^(1/2))^2*csgn(I*(c*x+1)/(c*x-1))*x^3*e+1/3*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^3*e+1/6*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^3*e+1/6*I*b*arccoth(c*x)*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)^2*x^3*e-1/3*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^3*e+1/6*I/c^3*b*e*ln((c*x+1)/(c*x-1)-1)*Pi*csgn(I/((c*x-1)/(c*x+1))^(1/2))^2*csgn(I*(c*x+1
)/(c*x-1))-1/3*I/c^3*b*e*ln((c*x+1)/(c*x-1)-1)*Pi*csgn(I/((c*x-1)/(c*x+1))^(1/2))*csgn(I*(c*x+1)/(c*x-1))^2-1/
6*I/c^3*b*e*ln((c*x+1)/(c*x-1)-1)*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
/6*I/c^3*b*csgn(I/((c*x+1)/(c*x-1)-1)^2)*e*ln((c*x+1)/(c*x-1)-1)*Pi*csgn(I*(c*x+1)/(c*x-1)/((c*x+1)/(c*x-1)-1)
^2)^2+1/6*I/c^3*b*arccoth(c*x)*e*Pi*csgn(I*(c*x+1)/(c*x-1)/((c*x+1)/(c*x-1)-1)^2)^3+1/6*I/c^3*b*arccoth(c*x)*P
i*e*csgn(I*((c*x+1)/(c*x-1)-1)^2)^3+1/3*I/c^3*b*e*ln((c*x+1)/(c*x-1)-1)*Pi*csgn(I*(c*x+1)/(c*x-1)/((c*x+1)/(c*
x-1)-1)^2)^2-1/3*I/c^3*b*arccoth(c*x)*e*Pi*csgn(I*(c*x+1)/(c*x-1)/((c*x+1)/(c*x-1)-1)^2)^2+1/12*I/c^3*b*Pi*e*c
sgn(I/((c*x-1)/(c*x+1))^(1/2))^2*csgn(I*(c*x+1)/(c*x-1))-1/6*I/c^3*b*Pi*e*csgn(I/((c*x-1)/(c*x+1))^(1/2))*csgn
(I*(c*x+1)/(c*x-1))^2-1/12*I/c^3*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)
)-1/6*I*b*arccoth(c*x)*Pi*csgn(I*(c*x+1)/(c*x-1))^3*x^3*e+1/6*I*b*arccoth(c*x)*Pi*csgn(I*(c*x+1)/(c*x-1)/((c*x
+1)/(c*x-1)-1)^2)^3*x^3*e+1/6*I*b*arccoth(c*x)*Pi*csgn(I*((c*x+1)/(c*x-1)-1)^2)^3*x^3*e-1/3*I*b*arccoth(c*x)*P
i*csgn(I*(c*x+1)/(c*x-1)/((c*x+1)/(c*x-1)-1)^2)^2*x^3*e-1/12*I/c^3*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)^2+1/6*I/c^3*b*Pi*e*csgn(I*((c*x+1)/(c*x-1)-1)^2)^2*csgn(I*((c*x+1)/(c*
x-1)-1))-1/12*I/c^3*b*Pi*e*csgn(I*((c*x+1)/(c*x-1)-1)^2)*csgn(I*((c*x+1)/(c*x-1)-1))^2-1/12*I/c*b*Pi*csgn(I*(c
*x+1)/(c*x-1))^3*x^2*e+1/12*I/c*b*Pi*csgn(I*(c*x+1)/(c*x-1)/((c*x+1)/(c*x-1)-1)^2)^3*x^2*e+1/12*I/c*b*Pi*csgn(
I*((c*x+1)/(c*x-1)-1)^2)^3*x^2*e-1/6*I/c*b*Pi*csgn(I*(c*x+1)/(c*x-1)/((c*x+1)/(c*x-1)-1)^2)^2*x^2*e+1/6*I/c^3*
b*e*ln((c*x+1)/(c*x-1)-1)*Pi*csgn(I*(c*x+1)/(c*x-1))^3-1/6*I/c^3*b*e*ln((c*x+1)/(c*x-1)-1)*Pi*csgn(I*(c*x+1)/(
c*x-1)/((c*x+1)/(c*x-1)-1)^2)^3-1/6*I/c^3*b*e*ln((c*x+1)/(c*x-1)-1)*Pi*csgn(I*((c*x+1)/(c*x-1)-1)^2)^3-1/6*I/c
^3*b*arccoth(c*x)*e*Pi*csgn(I*(c*x+1)/(c*x-1))^3+1/3*I*b*arccoth(c*x)*Pi*x^3*e+1/6*I/c*b*Pi*x^2*e+1/12*I/c^3*b
*Pi*e*csgn(I*(c*x+1)/(c*x-1))^3-1/12*I/c^3*b*Pi*e*csgn(I*(c*x+1)/(c*x-1)/((c*x+1)/(c*x-1)-1)^2)^3-1/12*I/c^3*b
*Pi*e*csgn(I*((c*x+1)/(c*x-1)-1)^2)^3+1/6*I/c^3*b*Pi*e*csgn(I*(c*x+1)/(c*x-1)/((c*x+1)/(c*x-1)-1)^2)^2-1/3*I/c
^3*b*e*ln((c*x+1)/(c*x-1)-1)*Pi+1/3*I/c^3*b*arccoth(c*x)*Pi*e

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Maxima [C]  time = 1.19273, size = 340, normalized size = 1.38 \begin{align*} \frac{1}{3} \, a d x^{3} + \frac{1}{9} \,{\left (3 \, x^{3} \log \left (-c^{2} x^{2} + 1\right ) - c^{2}{\left (\frac{2 \,{\left (c^{2} x^{3} + 3 \, x\right )}}{c^{4}} - \frac{3 \, \log \left (c x + 1\right )}{c^{5}} + \frac{3 \, \log \left (c x - 1\right )}{c^{5}}\right )}\right )} b e \operatorname{arcoth}\left (c x\right ) + \frac{1}{6} \,{\left (2 \, x^{3} \operatorname{arcoth}\left (c x\right ) + c{\left (\frac{x^{2}}{c^{2}} + \frac{\log \left (c^{2} x^{2} - 1\right )}{c^{4}}\right )}\right )} b d + \frac{1}{9} \,{\left (3 \, x^{3} \log \left (-c^{2} x^{2} + 1\right ) - c^{2}{\left (\frac{2 \,{\left (c^{2} x^{3} + 3 \, x\right )}}{c^{4}} - \frac{3 \, \log \left (c x + 1\right )}{c^{5}} + \frac{3 \, \log \left (c x - 1\right )}{c^{5}}\right )}\right )} a e + \frac{{\left ({\left (3 i \, \pi c^{2} - 5 \, c^{2}\right )} x^{2} +{\left (3 i \, \pi + 3 \, c^{2} x^{2} + 6 \, \log \left (c x - 1\right ) - 11\right )} \log \left (c x + 1\right ) +{\left (3 i \, \pi + 3 \, c^{2} x^{2} - 11\right )} \log \left (c x - 1\right )\right )} b e}{18 \, c^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*a*d*x^3 + 1/9*(3*x^3*log(-c^2*x^2 + 1) - c^2*(2*(c^2*x^3 + 3*x)/c^4 - 3*log(c*x + 1)/c^5 + 3*log(c*x - 1)/
c^5))*b*e*arccoth(c*x) + 1/6*(2*x^3*arccoth(c*x) + c*(x^2/c^2 + log(c^2*x^2 - 1)/c^4))*b*d + 1/9*(3*x^3*log(-c
^2*x^2 + 1) - c^2*(2*(c^2*x^3 + 3*x)/c^4 - 3*log(c*x + 1)/c^5 + 3*log(c*x - 1)/c^5))*a*e + 1/18*((3*I*pi*c^2 -
5*c^2)*x^2 + (3*I*pi + 3*c^2*x^2 + 6*log(c*x - 1) - 11)*log(c*x + 1) + (3*I*pi + 3*c^2*x^2 - 11)*log(c*x - 1)
)*b*e/c^3

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Fricas [A]  time = 1.72752, size = 452, normalized size = 1.83 \begin{align*} -\frac{24 \, a c e x - 4 \,{\left (3 \, a c^{3} d - 2 \, a c^{3} e\right )} x^{3} - 3 \, b e \log \left (-c^{2} x^{2} + 1\right )^{2} - 3 \, b e \log \left (\frac{c x + 1}{c x - 1}\right )^{2} - 2 \,{\left (3 \, b c^{2} d - 5 \, b c^{2} e\right )} x^{2} - 2 \,{\left (6 \, a c^{3} e x^{3} + 3 \, b c^{2} e x^{2} + 3 \, b d - 11 \, b e\right )} \log \left (-c^{2} x^{2} + 1\right ) - 2 \,{\left (3 \, b c^{3} e x^{3} \log \left (-c^{2} x^{2} + 1\right ) - 6 \, b c e x +{\left (3 \, b c^{3} d - 2 \, b c^{3} e\right )} x^{3} + 6 \, a e\right )} \log \left (\frac{c x + 1}{c x - 1}\right )}{36 \, c^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/36*(24*a*c*e*x - 4*(3*a*c^3*d - 2*a*c^3*e)*x^3 - 3*b*e*log(-c^2*x^2 + 1)^2 - 3*b*e*log((c*x + 1)/(c*x - 1))
^2 - 2*(3*b*c^2*d - 5*b*c^2*e)*x^2 - 2*(6*a*c^3*e*x^3 + 3*b*c^2*e*x^2 + 3*b*d - 11*b*e)*log(-c^2*x^2 + 1) - 2*
(3*b*c^3*e*x^3*log(-c^2*x^2 + 1) - 6*b*c*e*x + (3*b*c^3*d - 2*b*c^3*e)*x^3 + 6*a*e)*log((c*x + 1)/(c*x - 1)))/
c^3

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Sympy [A]  time = 15.7038, size = 265, normalized size = 1.07 \begin{align*} \begin{cases} \frac{a d x^{3}}{3} + \frac{a e x^{3} \log{\left (- c^{2} x^{2} + 1 \right )}}{3} - \frac{2 a e x^{3}}{9} - \frac{2 a e x}{3 c^{2}} + \frac{2 a e \operatorname{acoth}{\left (c x \right )}}{3 c^{3}} + \frac{b d x^{3} \operatorname{acoth}{\left (c x \right )}}{3} + \frac{b e x^{3} \log{\left (- c^{2} x^{2} + 1 \right )} \operatorname{acoth}{\left (c x \right )}}{3} - \frac{2 b e x^{3} \operatorname{acoth}{\left (c x \right )}}{9} + \frac{b d x^{2}}{6 c} + \frac{b e x^{2} \log{\left (- c^{2} x^{2} + 1 \right )}}{6 c} - \frac{5 b e x^{2}}{18 c} - \frac{2 b e x \operatorname{acoth}{\left (c x \right )}}{3 c^{2}} + \frac{b d \log{\left (- c^{2} x^{2} + 1 \right )}}{6 c^{3}} + \frac{b e \log{\left (- c^{2} x^{2} + 1 \right )}^{2}}{12 c^{3}} - \frac{11 b e \log{\left (- c^{2} x^{2} + 1 \right )}}{18 c^{3}} + \frac{b e \operatorname{acoth}^{2}{\left (c x \right )}}{3 c^{3}} & \text{for}\: c \neq 0 \\\frac{d x^{3} \left (a + \frac{i \pi b}{2}\right )}{3} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((a*d*x**3/3 + a*e*x**3*log(-c**2*x**2 + 1)/3 - 2*a*e*x**3/9 - 2*a*e*x/(3*c**2) + 2*a*e*acoth(c*x)/(3
*c**3) + b*d*x**3*acoth(c*x)/3 + b*e*x**3*log(-c**2*x**2 + 1)*acoth(c*x)/3 - 2*b*e*x**3*acoth(c*x)/9 + b*d*x**
2/(6*c) + b*e*x**2*log(-c**2*x**2 + 1)/(6*c) - 5*b*e*x**2/(18*c) - 2*b*e*x*acoth(c*x)/(3*c**2) + b*d*log(-c**2
*x**2 + 1)/(6*c**3) + b*e*log(-c**2*x**2 + 1)**2/(12*c**3) - 11*b*e*log(-c**2*x**2 + 1)/(18*c**3) + b*e*acoth(
c*x)**2/(3*c**3), Ne(c, 0)), (d*x**3*(a + I*pi*b/2)/3, 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^{2}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(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^2, x)