\(\int x^3 \coth ^{-1}(a+b x)^2 \, dx\) [69]

Optimal result

Integrand size = 12, antiderivative size = 263 \[ \int x^3 \coth ^{-1}(a+b x)^2 \, dx=-\frac {a x}{b^3}+\frac {(a+b x)^2}{12 b^4}+\frac {\left (1+6 a^2\right ) (a+b x) \coth ^{-1}(a+b x)}{2 b^4}-\frac {a (a+b x)^2 \coth ^{-1}(a+b x)}{b^4}+\frac {(a+b x)^3 \coth ^{-1}(a+b x)}{6 b^4}-\frac {a \left (1+a^2\right ) \coth ^{-1}(a+b x)^2}{b^4}-\frac {\left (1+6 a^2+a^4\right ) \coth ^{-1}(a+b x)^2}{4 b^4}+\frac {1}{4} x^4 \coth ^{-1}(a+b x)^2+\frac {a \text {arctanh}(a+b x)}{b^4}+\frac {2 a \left (1+a^2\right ) \coth ^{-1}(a+b x) \log \left (\frac {2}{1-a-b x}\right )}{b^4}+\frac {\log \left (1-(a+b x)^2\right )}{12 b^4}+\frac {\left (1+6 a^2\right ) \log \left (1-(a+b x)^2\right )}{4 b^4}+\frac {a \left (1+a^2\right ) \operatorname {PolyLog}\left (2,-\frac {1+a+b x}{1-a-b x}\right )}{b^4} \]

[Out]

-a*x/b^3+1/12*(b*x+a)^2/b^4+1/2*(6*a^2+1)*(b*x+a)*arccoth(b*x+a)/b^4-a*(b*x+a)^2*arccoth(b*x+a)/b^4+1/6*(b*x+a
)^3*arccoth(b*x+a)/b^4-a*(a^2+1)*arccoth(b*x+a)^2/b^4-1/4*(a^4+6*a^2+1)*arccoth(b*x+a)^2/b^4+1/4*x^4*arccoth(b
*x+a)^2+a*arctanh(b*x+a)/b^4+2*a*(a^2+1)*arccoth(b*x+a)*ln(2/(-b*x-a+1))/b^4+1/12*ln(1-(b*x+a)^2)/b^4+1/4*(6*a
^2+1)*ln(1-(b*x+a)^2)/b^4+a*(a^2+1)*polylog(2,(-b*x-a-1)/(-b*x-a+1))/b^4

Rubi [A] (verified)

Time = 0.29 (sec) , antiderivative size = 263, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.250, Rules used = {6247, 6066, 6022, 266, 6038, 327, 212, 272, 45, 6196, 6096, 6132, 6056, 2449, 2352} \[ \int x^3 \coth ^{-1}(a+b x)^2 \, dx=\frac {a \left (a^2+1\right ) \operatorname {PolyLog}\left (2,-\frac {a+b x+1}{-a-b x+1}\right )}{b^4}+\frac {\left (6 a^2+1\right ) \log \left (1-(a+b x)^2\right )}{4 b^4}-\frac {a \left (a^2+1\right ) \coth ^{-1}(a+b x)^2}{b^4}+\frac {\left (6 a^2+1\right ) (a+b x) \coth ^{-1}(a+b x)}{2 b^4}+\frac {2 a \left (a^2+1\right ) \log \left (\frac {2}{-a-b x+1}\right ) \coth ^{-1}(a+b x)}{b^4}-\frac {\left (a^4+6 a^2+1\right ) \coth ^{-1}(a+b x)^2}{4 b^4}+\frac {a \text {arctanh}(a+b x)}{b^4}+\frac {(a+b x)^2}{12 b^4}+\frac {\log \left (1-(a+b x)^2\right )}{12 b^4}+\frac {(a+b x)^3 \coth ^{-1}(a+b x)}{6 b^4}-\frac {a (a+b x)^2 \coth ^{-1}(a+b x)}{b^4}-\frac {a x}{b^3}+\frac {1}{4} x^4 \coth ^{-1}(a+b x)^2 \]

[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 45

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 212

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

Rule 266

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 272

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 327

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 2352

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

Rule 2449

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 6022

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

Rule 6038

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

Rule 6056

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 6066

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 6096

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 6132

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 6196

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 6247

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]

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

Mathematica [A] (verified)

Time = 1.49 (sec) , antiderivative size = 210, normalized size of antiderivative = 0.80 \[ \int x^3 \coth ^{-1}(a+b x)^2 \, dx=-\frac {1+11 a^2+10 a b x-b^2 x^2+3 \left (1-4 a+6 a^2-4 a^3+a^4-b^4 x^4\right ) \coth ^{-1}(a+b x)^2-2 \coth ^{-1}(a+b x) \left (9 a+13 a^3+3 b x+9 a^2 b x-3 a b^2 x^2+b^3 x^3+12 \left (a+a^3\right ) \log \left (1-e^{-2 \coth ^{-1}(a+b x)}\right )\right )+8 \log \left (\frac {1}{a+b x}\right )+36 a^2 \log \left (\frac {1}{a+b x}\right )+8 \log \left (\frac {1}{\sqrt {1-\frac {1}{(a+b x)^2}}}\right )+36 a^2 \log \left (\frac {1}{\sqrt {1-\frac {1}{(a+b x)^2}}}\right )+12 \left (a+a^3\right ) \operatorname {PolyLog}\left (2,e^{-2 \coth ^{-1}(a+b x)}\right )}{12 b^4} \]

[In]

Integrate[x^3*ArcCoth[a + b*x]^2,x]

[Out]

-1/12*(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*Ar
cCoth[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[(a + b*x)^(-1)] + 36*a^2*Log[(a + b*x)^(-1)] + 8*Log[1/Sqrt[1 - (a + b*x)^(-2)]] + 36*a^2*
Log[1/Sqrt[1 - (a + b*x)^(-2)]] + 12*(a + a^3)*PolyLog[2, E^(-2*ArcCoth[a + b*x])])/b^4

Maple [A] (verified)

Time = 0.24 (sec) , antiderivative size = 448, normalized size of antiderivative = 1.70

method	result	size
parts	\(\frac {x^{4} \operatorname {arccoth}\left (b x +a \right )^{2}}{4}+\frac {6 \,\operatorname {arccoth}\left (b x +a \right ) a^{2} \left (b x +a \right )-2 \,\operatorname {arccoth}\left (b x +a \right ) a \left (b x +a \right )^{2}+\frac {\operatorname {arccoth}\left (b x +a \right ) \left (b x +a \right )^{3}}{3}+\left (b x +a \right ) \operatorname {arccoth}\left (b x +a \right )+\frac {\operatorname {arccoth}\left (b x +a \right ) \ln \left (b x +a -1\right ) a^{4}}{2}-2 \,\operatorname {arccoth}\left (b x +a \right ) \ln \left (b x +a -1\right ) a^{3}+3 \,\operatorname {arccoth}\left (b x +a \right ) \ln \left (b x +a -1\right ) a^{2}-2 \,\operatorname {arccoth}\left (b x +a \right ) \ln \left (b x +a -1\right ) a +\frac {\operatorname {arccoth}\left (b x +a \right ) \ln \left (b x +a -1\right )}{2}-\frac {\operatorname {arccoth}\left (b x +a \right ) \ln \left (b x +a +1\right ) a^{4}}{2}-2 \,\operatorname {arccoth}\left (b x +a \right ) \ln \left (b x +a +1\right ) a^{3}-3 \,\operatorname {arccoth}\left (b x +a \right ) \ln \left (b x +a +1\right ) a^{2}-2 \,\operatorname {arccoth}\left (b x +a \right ) \ln \left (b x +a +1\right ) a -\frac {\operatorname {arccoth}\left (b x +a \right ) \ln \left (b x +a +1\right )}{2}-2 \left (b x +a \right ) a +\frac {\left (b x +a \right )^{2}}{6}+\frac {\left (18 a^{2}-6 a +4\right ) \ln \left (b x +a -1\right )}{6}-\frac {\left (-18 a^{2}-6 a -4\right ) \ln \left (b x +a +1\right )}{6}+\frac {\left (3 a^{4}-12 a^{3}+18 a^{2}-12 a +3\right ) \left (\frac {\ln \left (b x +a -1\right )^{2}}{4}-\frac {\operatorname {dilog}\left (\frac {b x}{2}+\frac {a}{2}+\frac {1}{2}\right )}{2}-\frac {\ln \left (b x +a -1\right ) \ln \left (\frac {b x}{2}+\frac {a}{2}+\frac {1}{2}\right )}{2}\right )}{6}+\frac {\left (-3 a^{4}-12 a^{3}-18 a^{2}-12 a -3\right ) \left (\frac {\left (\ln \left (b x +a +1\right )-\ln \left (\frac {b x}{2}+\frac {a}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {b x}{2}-\frac {a}{2}+\frac {1}{2}\right )}{2}-\frac {\operatorname {dilog}\left (\frac {b x}{2}+\frac {a}{2}+\frac {1}{2}\right )}{2}-\frac {\ln \left (b x +a +1\right )^{2}}{4}\right )}{6}}{2 b^{4}}\)	\(448\)
derivativedivides	\(\frac {3 \,\operatorname {arccoth}\left (b x +a \right ) a^{2} \left (b x +a \right )+\frac {3 \operatorname {arccoth}\left (b x +a \right )^{2} a^{2} \left (b x +a \right )^{2}}{2}-\operatorname {arccoth}\left (b x +a \right ) a \left (b x +a \right )^{2}-\operatorname {arccoth}\left (b x +a \right )^{2} a \left (b x +a \right )^{3}-\operatorname {arccoth}\left (b x +a \right ) \ln \left (b x +a +1\right ) a^{3}-\operatorname {arccoth}\left (b x +a \right )^{2} a^{3} \left (b x +a \right )+\frac {\left (b x +a \right )^{2}}{12}+\frac {\operatorname {arccoth}\left (b x +a \right ) \left (b x +a \right )^{3}}{6}-\left (b x +a \right ) a -\operatorname {arccoth}\left (b x +a \right ) \ln \left (b x +a +1\right ) a +\frac {\operatorname {arccoth}\left (b x +a \right ) \ln \left (b x +a -1\right ) a^{4}}{4}-\operatorname {arccoth}\left (b x +a \right ) \ln \left (b x +a -1\right ) a^{3}+\frac {3 \,\operatorname {arccoth}\left (b x +a \right ) \ln \left (b x +a -1\right ) a^{2}}{2}-\operatorname {arccoth}\left (b x +a \right ) \ln \left (b x +a -1\right ) a -\frac {\operatorname {arccoth}\left (b x +a \right ) \ln \left (b x +a +1\right ) a^{4}}{4}-\frac {3 \,\operatorname {arccoth}\left (b x +a \right ) \ln \left (b x +a +1\right ) a^{2}}{2}-\frac {\operatorname {arccoth}\left (b x +a \right ) \ln \left (b x +a +1\right )}{4}+\frac {\left (-3 a^{4}-12 a^{3}-18 a^{2}-12 a -3\right ) \left (\frac {\left (\ln \left (b x +a +1\right )-\ln \left (\frac {b x}{2}+\frac {a}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {b x}{2}-\frac {a}{2}+\frac {1}{2}\right )}{2}-\frac {\operatorname {dilog}\left (\frac {b x}{2}+\frac {a}{2}+\frac {1}{2}\right )}{2}-\frac {\ln \left (b x +a +1\right )^{2}}{4}\right )}{12}+\frac {\left (3 a^{4}-12 a^{3}+18 a^{2}-12 a +3\right ) \left (\frac {\ln \left (b x +a -1\right )^{2}}{4}-\frac {\operatorname {dilog}\left (\frac {b x}{2}+\frac {a}{2}+\frac {1}{2}\right )}{2}-\frac {\ln \left (b x +a -1\right ) \ln \left (\frac {b x}{2}+\frac {a}{2}+\frac {1}{2}\right )}{2}\right )}{12}+\frac {\left (18 a^{2}-6 a +4\right ) \ln \left (b x +a -1\right )}{12}-\frac {\left (-18 a^{2}-6 a -4\right ) \ln \left (b x +a +1\right )}{12}+\frac {\operatorname {arccoth}\left (b x +a \right )^{2} a^{4}}{4}+\frac {\operatorname {arccoth}\left (b x +a \right )^{2} \left (b x +a \right )^{4}}{4}+\frac {\operatorname {arccoth}\left (b x +a \right ) \ln \left (b x +a -1\right )}{4}+\frac {\left (b x +a \right ) \operatorname {arccoth}\left (b x +a \right )}{2}}{b^{4}}\)	\(520\)
default	\(\frac {3 \,\operatorname {arccoth}\left (b x +a \right ) a^{2} \left (b x +a \right )+\frac {3 \operatorname {arccoth}\left (b x +a \right )^{2} a^{2} \left (b x +a \right )^{2}}{2}-\operatorname {arccoth}\left (b x +a \right ) a \left (b x +a \right )^{2}-\operatorname {arccoth}\left (b x +a \right )^{2} a \left (b x +a \right )^{3}-\operatorname {arccoth}\left (b x +a \right ) \ln \left (b x +a +1\right ) a^{3}-\operatorname {arccoth}\left (b x +a \right )^{2} a^{3} \left (b x +a \right )+\frac {\left (b x +a \right )^{2}}{12}+\frac {\operatorname {arccoth}\left (b x +a \right ) \left (b x +a \right )^{3}}{6}-\left (b x +a \right ) a -\operatorname {arccoth}\left (b x +a \right ) \ln \left (b x +a +1\right ) a +\frac {\operatorname {arccoth}\left (b x +a \right ) \ln \left (b x +a -1\right ) a^{4}}{4}-\operatorname {arccoth}\left (b x +a \right ) \ln \left (b x +a -1\right ) a^{3}+\frac {3 \,\operatorname {arccoth}\left (b x +a \right ) \ln \left (b x +a -1\right ) a^{2}}{2}-\operatorname {arccoth}\left (b x +a \right ) \ln \left (b x +a -1\right ) a -\frac {\operatorname {arccoth}\left (b x +a \right ) \ln \left (b x +a +1\right ) a^{4}}{4}-\frac {3 \,\operatorname {arccoth}\left (b x +a \right ) \ln \left (b x +a +1\right ) a^{2}}{2}-\frac {\operatorname {arccoth}\left (b x +a \right ) \ln \left (b x +a +1\right )}{4}+\frac {\left (-3 a^{4}-12 a^{3}-18 a^{2}-12 a -3\right ) \left (\frac {\left (\ln \left (b x +a +1\right )-\ln \left (\frac {b x}{2}+\frac {a}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {b x}{2}-\frac {a}{2}+\frac {1}{2}\right )}{2}-\frac {\operatorname {dilog}\left (\frac {b x}{2}+\frac {a}{2}+\frac {1}{2}\right )}{2}-\frac {\ln \left (b x +a +1\right )^{2}}{4}\right )}{12}+\frac {\left (3 a^{4}-12 a^{3}+18 a^{2}-12 a +3\right ) \left (\frac {\ln \left (b x +a -1\right )^{2}}{4}-\frac {\operatorname {dilog}\left (\frac {b x}{2}+\frac {a}{2}+\frac {1}{2}\right )}{2}-\frac {\ln \left (b x +a -1\right ) \ln \left (\frac {b x}{2}+\frac {a}{2}+\frac {1}{2}\right )}{2}\right )}{12}+\frac {\left (18 a^{2}-6 a +4\right ) \ln \left (b x +a -1\right )}{12}-\frac {\left (-18 a^{2}-6 a -4\right ) \ln \left (b x +a +1\right )}{12}+\frac {\operatorname {arccoth}\left (b x +a \right )^{2} a^{4}}{4}+\frac {\operatorname {arccoth}\left (b x +a \right )^{2} \left (b x +a \right )^{4}}{4}+\frac {\operatorname {arccoth}\left (b x +a \right ) \ln \left (b x +a -1\right )}{4}+\frac {\left (b x +a \right ) \operatorname {arccoth}\left (b x +a \right )}{2}}{b^{4}}\)	\(520\)
risch	\(-\frac {1}{12 b^{4}}-\frac {5 a x}{6 b^{3}}-\frac {\ln \left (b x +a -1\right ) x}{4 b^{3}}+\frac {\ln \left (b x +a -1\right )^{2} a^{3}}{4 b^{4}}-\frac {3 \ln \left (b x +a -1\right )^{2} a^{2}}{8 b^{4}}+\frac {\ln \left (b x +a -1\right )^{2} a}{4 b^{4}}-\frac {\ln \left (b x +a -1\right )^{2} a^{4}}{16 b^{4}}-\frac {\ln \left (b x +a -1\right ) x^{3}}{12 b}-\frac {\left (-b^{4} x^{4}+a^{4}+4 a^{3}+6 a^{2}+4 a +1\right ) \ln \left (b x +a +1\right )^{2}}{16 b^{4}}+\frac {a}{b^{4}}-\frac {11 a^{2}}{12 b^{4}}+\frac {x^{2}}{12 b^{2}}+\frac {\ln \left (b x +a +1\right )}{3 b^{4}}+\frac {\ln \left (b x +a -1\right )^{2} x^{4}}{16}-\frac {\ln \left (b x +a -1\right )^{2}}{16 b^{4}}+\frac {\operatorname {dilog}\left (\frac {b x}{2}+\frac {a}{2}+\frac {1}{2}\right ) a^{3}}{b^{4}}+\frac {\operatorname {dilog}\left (\frac {b x}{2}+\frac {a}{2}+\frac {1}{2}\right ) a}{b^{4}}+\frac {3 \ln \left (b x +a +1\right ) a^{2}}{2 b^{4}}+\frac {3 \ln \left (b x +a +1\right ) a}{4 b^{4}}+\frac {13 \ln \left (b x +a +1\right ) a^{3}}{12 b^{4}}-\frac {13 \ln \left (b x +a -1\right ) a^{3}}{12 b^{4}}+\frac {3 \ln \left (b x +a -1\right ) a^{2}}{2 b^{4}}-\frac {3 \ln \left (b x +a -1\right ) a}{4 b^{4}}+\left (-\frac {x^{4} \ln \left (b x +a -1\right )}{8}-\frac {-2 b^{3} x^{3}-3 a^{4} \ln \left (b x +a -1\right )+6 a \,b^{2} x^{2}+12 \ln \left (b x +a -1\right ) a^{3}-18 a^{2} b x -18 \ln \left (b x +a -1\right ) a^{2}+12 \ln \left (b x +a -1\right ) a -6 b x -3 \ln \left (b x +a -1\right )}{24 b^{4}}\right ) \ln \left (b x +a +1\right )+\frac {\ln \left (b x +a -1\right )}{3 b^{4}}-\frac {3 \ln \left (b x +a -1\right ) x \,a^{2}}{4 b^{3}}+\frac {\ln \left (b x +a -1\right ) x^{2} a}{4 b^{2}}+\frac {\ln \left (b x +a -1\right ) \ln \left (\frac {b x}{2}+\frac {a}{2}+\frac {1}{2}\right ) a}{b^{4}}+\frac {\ln \left (b x +a -1\right ) \ln \left (\frac {b x}{2}+\frac {a}{2}+\frac {1}{2}\right ) a^{3}}{b^{4}}\)	\(527\)

[In]

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

[Out]

1/4*x^4*arccoth(b*x+a)^2+1/2/b^4*(6*arccoth(b*x+a)*a^2*(b*x+a)-2*arccoth(b*x+a)*a*(b*x+a)^2+1/3*arccoth(b*x+a)
*(b*x+a)^3+(b*x+a)*arccoth(b*x+a)+1/2*arccoth(b*x+a)*ln(b*x+a-1)*a^4-2*arccoth(b*x+a)*ln(b*x+a-1)*a^3+3*arccot
h(b*x+a)*ln(b*x+a-1)*a^2-2*arccoth(b*x+a)*ln(b*x+a-1)*a+1/2*arccoth(b*x+a)*ln(b*x+a-1)-1/2*arccoth(b*x+a)*ln(b
*x+a+1)*a^4-2*arccoth(b*x+a)*ln(b*x+a+1)*a^3-3*arccoth(b*x+a)*ln(b*x+a+1)*a^2-2*arccoth(b*x+a)*ln(b*x+a+1)*a-1
/2*arccoth(b*x+a)*ln(b*x+a+1)-2*(b*x+a)*a+1/6*(b*x+a)^2+1/6*(18*a^2-6*a+4)*ln(b*x+a-1)-1/6*(-18*a^2-6*a-4)*ln(
b*x+a+1)+1/6*(3*a^4-12*a^3+18*a^2-12*a+3)*(1/4*ln(b*x+a-1)^2-1/2*dilog(1/2*b*x+1/2*a+1/2)-1/2*ln(b*x+a-1)*ln(1
/2*b*x+1/2*a+1/2))+1/6*(-3*a^4-12*a^3-18*a^2-12*a-3)*(1/2*(ln(b*x+a+1)-ln(1/2*b*x+1/2*a+1/2))*ln(-1/2*b*x-1/2*
a+1/2)-1/2*dilog(1/2*b*x+1/2*a+1/2)-1/4*ln(b*x+a+1)^2))

Fricas [F]

\[ \int x^3 \coth ^{-1}(a+b x)^2 \, dx=\int { x^{3} \operatorname {arcoth}\left (b x + a\right )^{2} \,d x } \]

[In]

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

[Out]

integral(x^3*arccoth(b*x + a)^2, x)

Sympy [F]

\[ \int x^3 \coth ^{-1}(a+b x)^2 \, dx=\int x^{3} \operatorname {acoth}^{2}{\left (a + b x \right )}\, dx \]

[In]

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

[Out]

Integral(x**3*acoth(a + b*x)**2, x)

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 320, normalized size of antiderivative = 1.22 \[ \int x^3 \coth ^{-1}(a+b x)^2 \, dx=\frac {1}{4} \, x^{4} \operatorname {arcoth}\left (b x + a\right )^{2} + \frac {1}{48} \, b^{2} {\left (\frac {48 \, {\left (a^{3} + a\right )} {\left (\log \left (b x + a - 1\right ) \log \left (\frac {1}{2} \, b x + \frac {1}{2} \, a + \frac {1}{2}\right ) + {\rm Li}_2\left (-\frac {1}{2} \, b x - \frac {1}{2} \, a + \frac {1}{2}\right )\right )}}{b^{6}} + \frac {4 \, {\left (13 \, a^{3} + 18 \, a^{2} + 9 \, a + 4\right )} \log \left (b x + a + 1\right )}{b^{6}} + \frac {4 \, b^{2} x^{2} - 40 \, a b x + 3 \, {\left (a^{4} + 4 \, a^{3} + 6 \, a^{2} + 4 \, a + 1\right )} \log \left (b x + a + 1\right )^{2} - 6 \, {\left (a^{4} + 4 \, a^{3} + 6 \, a^{2} + 4 \, a + 1\right )} \log \left (b x + a + 1\right ) \log \left (b x + a - 1\right ) + 3 \, {\left (a^{4} - 4 \, a^{3} + 6 \, a^{2} - 4 \, a + 1\right )} \log \left (b x + a - 1\right )^{2} - 4 \, {\left (13 \, a^{3} - 18 \, a^{2} + 9 \, a - 4\right )} \log \left (b x + a - 1\right )}{b^{6}}\right )} + \frac {1}{12} \, b {\left (\frac {2 \, {\left (b^{2} x^{3} - 3 \, a b x^{2} + 3 \, {\left (3 \, a^{2} + 1\right )} x\right )}}{b^{4}} - \frac {3 \, {\left (a^{4} + 4 \, a^{3} + 6 \, a^{2} + 4 \, a + 1\right )} \log \left (b x + a + 1\right )}{b^{5}} + \frac {3 \, {\left (a^{4} - 4 \, a^{3} + 6 \, a^{2} - 4 \, a + 1\right )} \log \left (b x + a - 1\right )}{b^{5}}\right )} \operatorname {arcoth}\left (b x + a\right ) \]

[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)

Giac [F]

\[ \int x^3 \coth ^{-1}(a+b x)^2 \, dx=\int { x^{3} \operatorname {arcoth}\left (b x + a\right )^{2} \,d x } \]

[In]

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

[Out]

integrate(x^3*arccoth(b*x + a)^2, x)

Mupad [F(-1)]

Timed out. \[ \int x^3 \coth ^{-1}(a+b x)^2 \, dx=\int x^3\,{\mathrm {acoth}\left (a+b\,x\right )}^2 \,d x \]

[In]

int(x^3*acoth(a + b*x)^2,x)

[Out]

int(x^3*acoth(a + b*x)^2, x)