3.69 \(\int x^3 \coth ^{-1}(a+b x)^2 \, dx\)

Optimal. Leaf size=263 \[ \frac{a \left (a^2+1\right ) \text{PolyLog}\left (2,-\frac{a+b x+1}{-a-b x+1}\right )}{b^4}+\frac{\left (6 a^2+1\right ) \log \left (1-(a+b x)^2\right )}{4 b^4}-\frac{a \left (a^2+1\right ) \coth ^{-1}(a+b x)^2}{b^4}-\frac{\left (a^4+6 a^2+1\right ) \coth ^{-1}(a+b x)^2}{4 b^4}+\frac{\left (6 a^2+1\right ) (a+b x) \coth ^{-1}(a+b x)}{2 b^4}+\frac{2 a \left (a^2+1\right ) \log \left (\frac{2}{-a-b x+1}\right ) \coth ^{-1}(a+b x)}{b^4}-\frac{a x}{b^3}+\frac{(a+b x)^2}{12 b^4}+\frac{\log \left (1-(a+b x)^2\right )}{12 b^4}+\frac{a \tanh ^{-1}(a+b x)}{b^4}+\frac{(a+b x)^3 \coth ^{-1}(a+b x)}{6 b^4}-\frac{a (a+b x)^2 \coth ^{-1}(a+b x)}{b^4}+\frac{1}{4} x^4 \coth ^{-1}(a+b x)^2 \]

[Out]

-((a*x)/b^3) + (a + b*x)^2/(12*b^4) + ((1 + 6*a^2)*(a + b*x)*ArcCoth[a + b*x])/(2*b^4) - (a*(a + b*x)^2*ArcCot
h[a + b*x])/b^4 + ((a + b*x)^3*ArcCoth[a + b*x])/(6*b^4) - (a*(1 + a^2)*ArcCoth[a + b*x]^2)/b^4 - ((1 + 6*a^2
+ a^4)*ArcCoth[a + b*x]^2)/(4*b^4) + (x^4*ArcCoth[a + b*x]^2)/4 + (a*ArcTanh[a + b*x])/b^4 + (2*a*(1 + a^2)*Ar
cCoth[a + b*x]*Log[2/(1 - a - b*x)])/b^4 + Log[1 - (a + b*x)^2]/(12*b^4) + ((1 + 6*a^2)*Log[1 - (a + b*x)^2])/
(4*b^4) + (a*(1 + a^2)*PolyLog[2, -((1 + a + b*x)/(1 - a - b*x))])/b^4

________________________________________________________________________________________

Rubi [A]  time = 0.348779, antiderivative size = 263, normalized size of antiderivative = 1., number of steps used = 19, number of rules used = 15, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 1.25, Rules used = {6112, 5929, 5911, 260, 5917, 321, 206, 266, 43, 6049, 5949, 5985, 5919, 2402, 2315} \[ \frac{a \left (a^2+1\right ) \text{PolyLog}\left (2,-\frac{a+b x+1}{-a-b x+1}\right )}{b^4}+\frac{\left (6 a^2+1\right ) \log \left (1-(a+b x)^2\right )}{4 b^4}-\frac{a \left (a^2+1\right ) \coth ^{-1}(a+b x)^2}{b^4}-\frac{\left (a^4+6 a^2+1\right ) \coth ^{-1}(a+b x)^2}{4 b^4}+\frac{\left (6 a^2+1\right ) (a+b x) \coth ^{-1}(a+b x)}{2 b^4}+\frac{2 a \left (a^2+1\right ) \log \left (\frac{2}{-a-b x+1}\right ) \coth ^{-1}(a+b x)}{b^4}-\frac{a x}{b^3}+\frac{(a+b x)^2}{12 b^4}+\frac{\log \left (1-(a+b x)^2\right )}{12 b^4}+\frac{a \tanh ^{-1}(a+b x)}{b^4}+\frac{(a+b x)^3 \coth ^{-1}(a+b x)}{6 b^4}-\frac{a (a+b x)^2 \coth ^{-1}(a+b x)}{b^4}+\frac{1}{4} x^4 \coth ^{-1}(a+b x)^2 \]

Antiderivative was successfully verified.

[In]

Int[x^3*ArcCoth[a + b*x]^2,x]

[Out]

-((a*x)/b^3) + (a + b*x)^2/(12*b^4) + ((1 + 6*a^2)*(a + b*x)*ArcCoth[a + b*x])/(2*b^4) - (a*(a + b*x)^2*ArcCot
h[a + b*x])/b^4 + ((a + b*x)^3*ArcCoth[a + b*x])/(6*b^4) - (a*(1 + a^2)*ArcCoth[a + b*x]^2)/b^4 - ((1 + 6*a^2
+ a^4)*ArcCoth[a + b*x]^2)/(4*b^4) + (x^4*ArcCoth[a + b*x]^2)/4 + (a*ArcTanh[a + b*x])/b^4 + (2*a*(1 + a^2)*Ar
cCoth[a + b*x]*Log[2/(1 - a - b*x)])/b^4 + Log[1 - (a + b*x)^2]/(12*b^4) + ((1 + 6*a^2)*Log[1 - (a + b*x)^2])/
(4*b^4) + (a*(1 + a^2)*PolyLog[2, -((1 + a + b*x)/(1 - a - b*x))])/b^4

Rule 6112

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

Rule 5929

Int[((a_.) + ArcCoth[(c_.)*(x_)]*(b_.))^(p_)*((d_) + (e_.)*(x_))^(q_.), x_Symbol] :> Simp[((d + e*x)^(q + 1)*(
a + b*ArcCoth[c*x])^p)/(e*(q + 1)), x] - Dist[(b*c*p)/(e*(q + 1)), Int[ExpandIntegrand[(a + b*ArcCoth[c*x])^(p
- 1), (d + e*x)^(q + 1)/(1 - c^2*x^2), x], x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 1] && IntegerQ[q] &
& NeQ[q, -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 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 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 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 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 6049

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

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 5985

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

Rule 5919

Int[((a_.) + ArcCoth[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol] :> -Simp[((a + b*ArcCoth[c*x])^p*
Log[2/(1 + (e*x)/d)])/e, x] + Dist[(b*c*p)/e, Int[((a + b*ArcCoth[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 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 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]

Rubi steps

\begin{align*} \int x^3 \coth ^{-1}(a+b x)^2 \, dx &=\frac{\operatorname{Subst}\left (\int \left (-\frac{a}{b}+\frac{x}{b}\right )^3 \coth ^{-1}(x)^2 \, dx,x,a+b x\right )}{b}\\ &=\frac{1}{4} x^4 \coth ^{-1}(a+b x)^2-\frac{1}{2} \operatorname{Subst}\left (\int \left (-\frac{\left (1+6 a^2\right ) \coth ^{-1}(x)}{b^4}+\frac{4 a x \coth ^{-1}(x)}{b^4}-\frac{x^2 \coth ^{-1}(x)}{b^4}+\frac{\left (1+6 a^2+a^4-4 a \left (1+a^2\right ) x\right ) \coth ^{-1}(x)}{b^4 \left (1-x^2\right )}\right ) \, dx,x,a+b x\right )\\ &=\frac{1}{4} x^4 \coth ^{-1}(a+b x)^2+\frac{\operatorname{Subst}\left (\int x^2 \coth ^{-1}(x) \, dx,x,a+b x\right )}{2 b^4}-\frac{\operatorname{Subst}\left (\int \frac{\left (1+6 a^2+a^4-4 a \left (1+a^2\right ) x\right ) \coth ^{-1}(x)}{1-x^2} \, dx,x,a+b x\right )}{2 b^4}-\frac{(2 a) \operatorname{Subst}\left (\int x \coth ^{-1}(x) \, dx,x,a+b x\right )}{b^4}+\frac{\left (1+6 a^2\right ) \operatorname{Subst}\left (\int \coth ^{-1}(x) \, dx,x,a+b x\right )}{2 b^4}\\ &=\frac{\left (1+6 a^2\right ) (a+b x) \coth ^{-1}(a+b x)}{2 b^4}-\frac{a (a+b x)^2 \coth ^{-1}(a+b x)}{b^4}+\frac{(a+b x)^3 \coth ^{-1}(a+b x)}{6 b^4}+\frac{1}{4} x^4 \coth ^{-1}(a+b x)^2-\frac{\operatorname{Subst}\left (\int \frac{x^3}{1-x^2} \, dx,x,a+b x\right )}{6 b^4}-\frac{\operatorname{Subst}\left (\int \left (\frac{\left (1+a^2 \left (6+a^2\right )\right ) \coth ^{-1}(x)}{1-x^2}-\frac{4 a \left (1+a^2\right ) x \coth ^{-1}(x)}{1-x^2}\right ) \, dx,x,a+b x\right )}{2 b^4}+\frac{a \operatorname{Subst}\left (\int \frac{x^2}{1-x^2} \, dx,x,a+b x\right )}{b^4}-\frac{\left (1+6 a^2\right ) \operatorname{Subst}\left (\int \frac{x}{1-x^2} \, dx,x,a+b x\right )}{2 b^4}\\ &=-\frac{a x}{b^3}+\frac{\left (1+6 a^2\right ) (a+b x) \coth ^{-1}(a+b x)}{2 b^4}-\frac{a (a+b x)^2 \coth ^{-1}(a+b x)}{b^4}+\frac{(a+b x)^3 \coth ^{-1}(a+b x)}{6 b^4}+\frac{1}{4} x^4 \coth ^{-1}(a+b x)^2+\frac{\left (1+6 a^2\right ) \log \left (1-(a+b x)^2\right )}{4 b^4}-\frac{\operatorname{Subst}\left (\int \frac{x}{1-x} \, dx,x,(a+b x)^2\right )}{12 b^4}+\frac{a \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,a+b x\right )}{b^4}+\frac{\left (2 a \left (1+a^2\right )\right ) \operatorname{Subst}\left (\int \frac{x \coth ^{-1}(x)}{1-x^2} \, dx,x,a+b x\right )}{b^4}-\frac{\left (1+6 a^2+a^4\right ) \operatorname{Subst}\left (\int \frac{\coth ^{-1}(x)}{1-x^2} \, dx,x,a+b x\right )}{2 b^4}\\ &=-\frac{a x}{b^3}+\frac{\left (1+6 a^2\right ) (a+b x) \coth ^{-1}(a+b x)}{2 b^4}-\frac{a (a+b x)^2 \coth ^{-1}(a+b x)}{b^4}+\frac{(a+b x)^3 \coth ^{-1}(a+b x)}{6 b^4}-\frac{a \left (1+a^2\right ) \coth ^{-1}(a+b x)^2}{b^4}-\frac{\left (1+6 a^2+a^4\right ) \coth ^{-1}(a+b x)^2}{4 b^4}+\frac{1}{4} x^4 \coth ^{-1}(a+b x)^2+\frac{a \tanh ^{-1}(a+b x)}{b^4}+\frac{\left (1+6 a^2\right ) \log \left (1-(a+b x)^2\right )}{4 b^4}-\frac{\operatorname{Subst}\left (\int \left (-1+\frac{1}{1-x}\right ) \, dx,x,(a+b x)^2\right )}{12 b^4}+\frac{\left (2 a \left (1+a^2\right )\right ) \operatorname{Subst}\left (\int \frac{\coth ^{-1}(x)}{1-x} \, dx,x,a+b x\right )}{b^4}\\ &=-\frac{a x}{b^3}+\frac{(a+b x)^2}{12 b^4}+\frac{\left (1+6 a^2\right ) (a+b x) \coth ^{-1}(a+b x)}{2 b^4}-\frac{a (a+b x)^2 \coth ^{-1}(a+b x)}{b^4}+\frac{(a+b x)^3 \coth ^{-1}(a+b x)}{6 b^4}-\frac{a \left (1+a^2\right ) \coth ^{-1}(a+b x)^2}{b^4}-\frac{\left (1+6 a^2+a^4\right ) \coth ^{-1}(a+b x)^2}{4 b^4}+\frac{1}{4} x^4 \coth ^{-1}(a+b x)^2+\frac{a \tanh ^{-1}(a+b x)}{b^4}+\frac{2 a \left (1+a^2\right ) \coth ^{-1}(a+b x) \log \left (\frac{2}{1-a-b x}\right )}{b^4}+\frac{\log \left (1-(a+b x)^2\right )}{12 b^4}+\frac{\left (1+6 a^2\right ) \log \left (1-(a+b x)^2\right )}{4 b^4}-\frac{\left (2 a \left (1+a^2\right )\right ) \operatorname{Subst}\left (\int \frac{\log \left (\frac{2}{1-x}\right )}{1-x^2} \, dx,x,a+b x\right )}{b^4}\\ &=-\frac{a x}{b^3}+\frac{(a+b x)^2}{12 b^4}+\frac{\left (1+6 a^2\right ) (a+b x) \coth ^{-1}(a+b x)}{2 b^4}-\frac{a (a+b x)^2 \coth ^{-1}(a+b x)}{b^4}+\frac{(a+b x)^3 \coth ^{-1}(a+b x)}{6 b^4}-\frac{a \left (1+a^2\right ) \coth ^{-1}(a+b x)^2}{b^4}-\frac{\left (1+6 a^2+a^4\right ) \coth ^{-1}(a+b x)^2}{4 b^4}+\frac{1}{4} x^4 \coth ^{-1}(a+b x)^2+\frac{a \tanh ^{-1}(a+b x)}{b^4}+\frac{2 a \left (1+a^2\right ) \coth ^{-1}(a+b x) \log \left (\frac{2}{1-a-b x}\right )}{b^4}+\frac{\log \left (1-(a+b x)^2\right )}{12 b^4}+\frac{\left (1+6 a^2\right ) \log \left (1-(a+b x)^2\right )}{4 b^4}+\frac{\left (2 a \left (1+a^2\right )\right ) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1-a-b x}\right )}{b^4}\\ &=-\frac{a x}{b^3}+\frac{(a+b x)^2}{12 b^4}+\frac{\left (1+6 a^2\right ) (a+b x) \coth ^{-1}(a+b x)}{2 b^4}-\frac{a (a+b x)^2 \coth ^{-1}(a+b x)}{b^4}+\frac{(a+b x)^3 \coth ^{-1}(a+b x)}{6 b^4}-\frac{a \left (1+a^2\right ) \coth ^{-1}(a+b x)^2}{b^4}-\frac{\left (1+6 a^2+a^4\right ) \coth ^{-1}(a+b x)^2}{4 b^4}+\frac{1}{4} x^4 \coth ^{-1}(a+b x)^2+\frac{a \tanh ^{-1}(a+b x)}{b^4}+\frac{2 a \left (1+a^2\right ) \coth ^{-1}(a+b x) \log \left (\frac{2}{1-a-b x}\right )}{b^4}+\frac{\log \left (1-(a+b x)^2\right )}{12 b^4}+\frac{\left (1+6 a^2\right ) \log \left (1-(a+b x)^2\right )}{4 b^4}+\frac{a \left (1+a^2\right ) \text{Li}_2\left (1-\frac{2}{1-a-b x}\right )}{b^4}\\ \end{align*}

Mathematica [A]  time = 1.56441, size = 203, normalized size = 0.77 \[ -\frac{12 \left (a^3+a\right ) \text{PolyLog}\left (2,e^{-2 \coth ^{-1}(a+b x)}\right )+3 \left (a^4-4 a^3+6 a^2-4 a-b^4 x^4+1\right ) \coth ^{-1}(a+b x)^2-2 \coth ^{-1}(a+b x) \left (9 a^2 b x+12 \left (a^3+a\right ) \log \left (1-e^{-2 \coth ^{-1}(a+b x)}\right )+13 a^3-3 a b^2 x^2+9 a+b^3 x^3+3 b x\right )+36 a^2 \log \left (\frac{1}{(a+b x) \sqrt{1-\frac{1}{(a+b x)^2}}}\right )+11 a^2+10 a b x+8 \log \left (\frac{1}{(a+b x) \sqrt{1-\frac{1}{(a+b x)^2}}}\right )-b^2 x^2+1}{12 b^4} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[x^3*ArcCoth[a + b*x]^2,x]

[Out]

-(1 + 11*a^2 + 10*a*b*x - b^2*x^2 + 3*(1 - 4*a + 6*a^2 - 4*a^3 + a^4 - b^4*x^4)*ArcCoth[a + b*x]^2 - 2*ArcCoth
[a + b*x]*(9*a + 13*a^3 + 3*b*x + 9*a^2*b*x - 3*a*b^2*x^2 + b^3*x^3 + 12*(a + a^3)*Log[1 - E^(-2*ArcCoth[a + b
*x])]) + 8*Log[1/((a + b*x)*Sqrt[1 - (a + b*x)^(-2)])] + 36*a^2*Log[1/((a + b*x)*Sqrt[1 - (a + b*x)^(-2)])] +
12*(a + a^3)*PolyLog[2, E^(-2*ArcCoth[a + b*x])])/(12*b^4)

________________________________________________________________________________________

Maple [B]  time = 0.056, size = 967, normalized size = 3.7 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*arccoth(b*x+a)^2,x)

[Out]

1/4/b^4*arccoth(b*x+a)*ln(b*x+a-1)*a^4+1/8/b^4*ln(-1/2*b*x-1/2*a+1/2)*ln(1/2+1/2*b*x+1/2*a)*a^4+1/2/b^4*ln(-1/
2*b*x-1/2*a+1/2)*ln(1/2+1/2*b*x+1/2*a)*a^3-1/b^4*arccoth(b*x+a)*ln(b*x+a+1)*a-1/b^4*arccoth(b*x+a)*ln(b*x+a+1)
*a^3-1/4/b^4*arccoth(b*x+a)*ln(b*x+a+1)*a^4-11/12/b^4*a^2-1/2/b^2*arccoth(b*x+a)*x^2*a+1/2/b^4*ln(1/2+1/2*b*x+
1/2*a)*ln(b*x+a-1)*a^3+3/2/b^4*arccoth(b*x+a)*ln(b*x+a-1)*a^2-1/8/b^4*ln(1/2+1/2*b*x+1/2*a)*ln(b*x+a-1)*a^4+3/
2/b^3*arccoth(b*x+a)*x*a^2-1/b^4*arccoth(b*x+a)*ln(b*x+a-1)*a-1/2/b^4*ln(-1/2*b*x-1/2*a+1/2)*ln(b*x+a+1)*a-1/b
^4*arccoth(b*x+a)*ln(b*x+a-1)*a^3-3/2/b^4*arccoth(b*x+a)*ln(b*x+a+1)*a^2-3/4/b^4*ln(1/2+1/2*b*x+1/2*a)*ln(b*x+
a-1)*a^2-1/2/b^4*ln(-1/2*b*x-1/2*a+1/2)*ln(b*x+a+1)*a^3+3/2/b^4*ln(b*x+a-1)*a^2-1/2/b^4*ln(b*x+a-1)*a+3/2/b^4*
ln(b*x+a+1)*a^2+1/2/b^4*ln(b*x+a+1)*a-5/6*a*x/b^3+1/2/b^4*ln(-1/2*b*x-1/2*a+1/2)*ln(1/2+1/2*b*x+1/2*a)*a-1/8/b
^4*ln(-1/2*b*x-1/2*a+1/2)*ln(b*x+a+1)*a^4+3/4/b^4*ln(-1/2*b*x-1/2*a+1/2)*ln(1/2+1/2*b*x+1/2*a)*a^2+1/2/b^4*ln(
1/2+1/2*b*x+1/2*a)*ln(b*x+a-1)*a-3/4/b^4*ln(-1/2*b*x-1/2*a+1/2)*ln(b*x+a+1)*a^2+1/4*x^4*arccoth(b*x+a)^2+1/b^4
*dilog(1/2+1/2*b*x+1/2*a)*a+1/2/b^4*arccoth(b*x+a)*a+1/4/b^4*arccoth(b*x+a)*ln(b*x+a-1)+1/2/b^3*arccoth(b*x+a)
*x+3/8/b^4*ln(b*x+a-1)^2*a^2-1/4/b^4*ln(b*x+a-1)^2*a+1/16/b^4*ln(b*x+a+1)^2*a^4+3/8/b^4*ln(b*x+a+1)^2*a^2+1/4/
b^4*ln(b*x+a+1)^2*a+1/16/b^4*ln(b*x+a-1)^2*a^4-1/4/b^4*ln(b*x+a-1)^2*a^3+1/b^4*dilog(1/2+1/2*b*x+1/2*a)*a^3-1/
8/b^4*ln(-1/2*b*x-1/2*a+1/2)*ln(b*x+a+1)-1/4/b^4*arccoth(b*x+a)*ln(b*x+a+1)-1/8/b^4*ln(b*x+a-1)*ln(1/2+1/2*b*x
+1/2*a)+13/6/b^4*arccoth(b*x+a)*a^3+1/6/b*arccoth(b*x+a)*x^3+1/4/b^4*ln(b*x+a+1)^2*a^3+1/8/b^4*ln(-1/2*b*x-1/2
*a+1/2)*ln(1/2+1/2*b*x+1/2*a)+1/16/b^4*ln(b*x+a+1)^2+1/16/b^4*ln(b*x+a-1)^2+1/3/b^4*ln(b*x+a+1)+1/3/b^4*ln(b*x
+a-1)+1/12*x^2/b^2

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Maxima [A]  time = 0.988167, size = 432, normalized size = 1.64 \begin{align*} \frac{1}{4} \, x^{4} \operatorname{arcoth}\left (b x + a\right )^{2} + \frac{1}{48} \, b^{2}{\left (\frac{48 \,{\left (a^{3} + a\right )}{\left (\log \left (b x + a - 1\right ) \log \left (\frac{1}{2} \, b x + \frac{1}{2} \, a + \frac{1}{2}\right ) +{\rm Li}_2\left (-\frac{1}{2} \, b x - \frac{1}{2} \, a + \frac{1}{2}\right )\right )}}{b^{6}} + \frac{4 \,{\left (13 \, a^{3} + 18 \, a^{2} + 9 \, a + 4\right )} \log \left (b x + a + 1\right )}{b^{6}} + \frac{4 \, b^{2} x^{2} - 40 \, a b x + 3 \,{\left (a^{4} + 4 \, a^{3} + 6 \, a^{2} + 4 \, a + 1\right )} \log \left (b x + a + 1\right )^{2} - 6 \,{\left (a^{4} + 4 \, a^{3} + 6 \, a^{2} + 4 \, a + 1\right )} \log \left (b x + a + 1\right ) \log \left (b x + a - 1\right ) + 3 \,{\left (a^{4} - 4 \, a^{3} + 6 \, a^{2} - 4 \, a + 1\right )} \log \left (b x + a - 1\right )^{2} - 4 \,{\left (13 \, a^{3} - 18 \, a^{2} + 9 \, a - 4\right )} \log \left (b x + a - 1\right )}{b^{6}}\right )} + \frac{1}{12} \, b{\left (\frac{2 \,{\left (b^{2} x^{3} - 3 \, a b x^{2} + 3 \,{\left (3 \, a^{2} + 1\right )} x\right )}}{b^{4}} - \frac{3 \,{\left (a^{4} + 4 \, a^{3} + 6 \, a^{2} + 4 \, a + 1\right )} \log \left (b x + a + 1\right )}{b^{5}} + \frac{3 \,{\left (a^{4} - 4 \, a^{3} + 6 \, a^{2} - 4 \, a + 1\right )} \log \left (b x + a - 1\right )}{b^{5}}\right )} \operatorname{arcoth}\left (b x + a\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*arccoth(b*x+a)^2,x, algorithm="maxima")

[Out]

1/4*x^4*arccoth(b*x + a)^2 + 1/48*b^2*(48*(a^3 + a)*(log(b*x + a - 1)*log(1/2*b*x + 1/2*a + 1/2) + dilog(-1/2*
b*x - 1/2*a + 1/2))/b^6 + 4*(13*a^3 + 18*a^2 + 9*a + 4)*log(b*x + a + 1)/b^6 + (4*b^2*x^2 - 40*a*b*x + 3*(a^4
+ 4*a^3 + 6*a^2 + 4*a + 1)*log(b*x + a + 1)^2 - 6*(a^4 + 4*a^3 + 6*a^2 + 4*a + 1)*log(b*x + a + 1)*log(b*x + a
- 1) + 3*(a^4 - 4*a^3 + 6*a^2 - 4*a + 1)*log(b*x + a - 1)^2 - 4*(13*a^3 - 18*a^2 + 9*a - 4)*log(b*x + a - 1))
/b^6) + 1/12*b*(2*(b^2*x^3 - 3*a*b*x^2 + 3*(3*a^2 + 1)*x)/b^4 - 3*(a^4 + 4*a^3 + 6*a^2 + 4*a + 1)*log(b*x + a
+ 1)/b^5 + 3*(a^4 - 4*a^3 + 6*a^2 - 4*a + 1)*log(b*x + a - 1)/b^5)*arccoth(b*x + a)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (x^{3} \operatorname{arcoth}\left (b x + a\right )^{2}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*arccoth(b*x+a)^2,x, algorithm="fricas")

[Out]

integral(x^3*arccoth(b*x + a)^2, x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{3} \operatorname{acoth}^{2}{\left (a + b x \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*acoth(b*x+a)**2,x)

[Out]

Integral(x**3*acoth(a + b*x)**2, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{3} \operatorname{arcoth}\left (b x + a\right )^{2}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*arccoth(b*x+a)^2,x, algorithm="giac")

[Out]

integrate(x^3*arccoth(b*x + a)^2, x)