\(\int (e+f x) (a+b \coth ^{-1}(c+d x))^3 \, dx\) [115]

Optimal result

Integrand size = 18, antiderivative size = 326 \[ \int (e+f x) \left (a+b \coth ^{-1}(c+d x)\right )^3 \, dx=\frac {3 b f \left (a+b \coth ^{-1}(c+d x)\right )^2}{2 d^2}+\frac {3 b f (c+d x) \left (a+b \coth ^{-1}(c+d x)\right )^2}{2 d^2}+\frac {(d e-c f) \left (a+b \coth ^{-1}(c+d x)\right )^3}{d^2}-\frac {\left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right ) \left (a+b \coth ^{-1}(c+d x)\right )^3}{2 d^2 f}+\frac {(e+f x)^2 \left (a+b \coth ^{-1}(c+d x)\right )^3}{2 f}-\frac {3 b^2 f \left (a+b \coth ^{-1}(c+d x)\right ) \log \left (\frac {2}{1-c-d x}\right )}{d^2}-\frac {3 b (d e-c f) \left (a+b \coth ^{-1}(c+d x)\right )^2 \log \left (\frac {2}{1-c-d x}\right )}{d^2}-\frac {3 b^3 f \operatorname {PolyLog}\left (2,-\frac {1+c+d x}{1-c-d x}\right )}{2 d^2}-\frac {3 b^2 (d e-c f) \left (a+b \coth ^{-1}(c+d x)\right ) \operatorname {PolyLog}\left (2,1-\frac {2}{1-c-d x}\right )}{d^2}+\frac {3 b^3 (d e-c f) \operatorname {PolyLog}\left (3,1-\frac {2}{1-c-d x}\right )}{2 d^2} \]

[Out]

3/2*b*f*(a+b*arccoth(d*x+c))^2/d^2+3/2*b*f*(d*x+c)*(a+b*arccoth(d*x+c))^2/d^2+(-c*f+d*e)*(a+b*arccoth(d*x+c))^
3/d^2-1/2*(d^2*e^2-2*c*d*e*f+(c^2+1)*f^2)*(a+b*arccoth(d*x+c))^3/d^2/f+1/2*(f*x+e)^2*(a+b*arccoth(d*x+c))^3/f-
3*b^2*f*(a+b*arccoth(d*x+c))*ln(2/(-d*x-c+1))/d^2-3*b*(-c*f+d*e)*(a+b*arccoth(d*x+c))^2*ln(2/(-d*x-c+1))/d^2-3
/2*b^3*f*polylog(2,(-d*x-c-1)/(-d*x-c+1))/d^2-3*b^2*(-c*f+d*e)*(a+b*arccoth(d*x+c))*polylog(2,1-2/(-d*x-c+1))/
d^2+3/2*b^3*(-c*f+d*e)*polylog(3,1-2/(-d*x-c+1))/d^2

Rubi [A] (verified)

Time = 0.55 (sec) , antiderivative size = 325, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.611, Rules used = {6247, 6066, 6022, 6132, 6056, 2449, 2352, 6196, 6096, 6206, 6745} \[ \int (e+f x) \left (a+b \coth ^{-1}(c+d x)\right )^3 \, dx=-\frac {3 b^2 (d e-c f) \operatorname {PolyLog}\left (2,1-\frac {2}{-c-d x+1}\right ) \left (a+b \coth ^{-1}(c+d x)\right )}{d^2}-\frac {3 b^2 f \log \left (\frac {2}{-c-d x+1}\right ) \left (a+b \coth ^{-1}(c+d x)\right )}{d^2}+\frac {\left (-\frac {\left (c^2+1\right ) f}{d}+2 c e-\frac {d e^2}{f}\right ) \left (a+b \coth ^{-1}(c+d x)\right )^3}{2 d}+\frac {(d e-c f) \left (a+b \coth ^{-1}(c+d x)\right )^3}{d^2}-\frac {3 b (d e-c f) \log \left (\frac {2}{-c-d x+1}\right ) \left (a+b \coth ^{-1}(c+d x)\right )^2}{d^2}+\frac {3 b f \left (a+b \coth ^{-1}(c+d x)\right )^2}{2 d^2}+\frac {3 b f (c+d x) \left (a+b \coth ^{-1}(c+d x)\right )^2}{2 d^2}+\frac {(e+f x)^2 \left (a+b \coth ^{-1}(c+d x)\right )^3}{2 f}+\frac {3 b^3 (d e-c f) \operatorname {PolyLog}\left (3,1-\frac {2}{-c-d x+1}\right )}{2 d^2}-\frac {3 b^3 f \operatorname {PolyLog}\left (2,-\frac {c+d x+1}{-c-d x+1}\right )}{2 d^2} \]

[In]

Int[(e + f*x)*(a + b*ArcCoth[c + d*x])^3,x]

[Out]

(3*b*f*(a + b*ArcCoth[c + d*x])^2)/(2*d^2) + (3*b*f*(c + d*x)*(a + b*ArcCoth[c + d*x])^2)/(2*d^2) + ((d*e - c*
f)*(a + b*ArcCoth[c + d*x])^3)/d^2 + ((2*c*e - (d*e^2)/f - ((1 + c^2)*f)/d)*(a + b*ArcCoth[c + d*x])^3)/(2*d)
+ ((e + f*x)^2*(a + b*ArcCoth[c + d*x])^3)/(2*f) - (3*b^2*f*(a + b*ArcCoth[c + d*x])*Log[2/(1 - c - d*x)])/d^2
- (3*b*(d*e - c*f)*(a + b*ArcCoth[c + d*x])^2*Log[2/(1 - c - d*x)])/d^2 - (3*b^3*f*PolyLog[2, -((1 + c + d*x)
/(1 - c - d*x))])/(2*d^2) - (3*b^2*(d*e - c*f)*(a + b*ArcCoth[c + d*x])*PolyLog[2, 1 - 2/(1 - c - d*x)])/d^2 +
(3*b^3*(d*e - c*f)*PolyLog[3, 1 - 2/(1 - c - d*x)])/(2*d^2)

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 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 6206

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

Rule 6745

Int[(u_)*PolyLog[n_, v_], x_Symbol] :> With[{w = DerivativeDivides[v, u*v, x]}, Simp[w*PolyLog[n + 1, v], x] /
;  !FalseQ[w]] /; FreeQ[n, x]

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 3.69 (sec) , antiderivative size = 600, normalized size of antiderivative = 1.84 \[ \int (e+f x) \left (a+b \coth ^{-1}(c+d x)\right )^3 \, dx=\frac {2 a^2 (2 a d e+3 b f-2 a c f) (c+d x)+2 a^3 f (c+d x)^2-6 a^2 b (c+d x) (c f-d (2 e+f x)) \coth ^{-1}(c+d x)+3 a^2 b (2 d e+f-2 c f) \log (1-c-d x)+3 a^2 b (2 d e-(1+2 c) f) \log (1+c+d x)+12 a b^2 f \left ((c+d x) \coth ^{-1}(c+d x)+\frac {1}{2} \left (-1+(c+d x)^2\right ) \coth ^{-1}(c+d x)^2-\log \left (\frac {1}{(c+d x) \sqrt {1-\frac {1}{(c+d x)^2}}}\right )\right )+12 a b^2 d e \left (\coth ^{-1}(c+d x) \left ((-1+c+d x) \coth ^{-1}(c+d x)-2 \log \left (1-e^{-2 \coth ^{-1}(c+d x)}\right )\right )+\operatorname {PolyLog}\left (2,e^{-2 \coth ^{-1}(c+d x)}\right )\right )-12 a b^2 c f \left (\coth ^{-1}(c+d x) \left ((-1+c+d x) \coth ^{-1}(c+d x)-2 \log \left (1-e^{-2 \coth ^{-1}(c+d x)}\right )\right )+\operatorname {PolyLog}\left (2,e^{-2 \coth ^{-1}(c+d x)}\right )\right )+2 b^3 f \left (\coth ^{-1}(c+d x) \left (3 (-1+c+d x) \coth ^{-1}(c+d x)+\left (-1+c^2+2 c d x+d^2 x^2\right ) \coth ^{-1}(c+d x)^2-6 \log \left (1-e^{-2 \coth ^{-1}(c+d x)}\right )\right )+3 \operatorname {PolyLog}\left (2,e^{-2 \coth ^{-1}(c+d x)}\right )\right )+4 b^3 d e \left (-\frac {i \pi ^3}{8}+\coth ^{-1}(c+d x)^3+(c+d x) \coth ^{-1}(c+d x)^3-3 \coth ^{-1}(c+d x)^2 \log \left (1-e^{2 \coth ^{-1}(c+d x)}\right )-3 \coth ^{-1}(c+d x) \operatorname {PolyLog}\left (2,e^{2 \coth ^{-1}(c+d x)}\right )+\frac {3}{2} \operatorname {PolyLog}\left (3,e^{2 \coth ^{-1}(c+d x)}\right )\right )-4 b^3 c f \left (-\frac {i \pi ^3}{8}+\coth ^{-1}(c+d x)^3+(c+d x) \coth ^{-1}(c+d x)^3-3 \coth ^{-1}(c+d x)^2 \log \left (1-e^{2 \coth ^{-1}(c+d x)}\right )-3 \coth ^{-1}(c+d x) \operatorname {PolyLog}\left (2,e^{2 \coth ^{-1}(c+d x)}\right )+\frac {3}{2} \operatorname {PolyLog}\left (3,e^{2 \coth ^{-1}(c+d x)}\right )\right )}{4 d^2} \]

[In]

Integrate[(e + f*x)*(a + b*ArcCoth[c + d*x])^3,x]

[Out]

(2*a^2*(2*a*d*e + 3*b*f - 2*a*c*f)*(c + d*x) + 2*a^3*f*(c + d*x)^2 - 6*a^2*b*(c + d*x)*(c*f - d*(2*e + f*x))*A
rcCoth[c + d*x] + 3*a^2*b*(2*d*e + f - 2*c*f)*Log[1 - c - d*x] + 3*a^2*b*(2*d*e - (1 + 2*c)*f)*Log[1 + c + d*x
] + 12*a*b^2*f*((c + d*x)*ArcCoth[c + d*x] + ((-1 + (c + d*x)^2)*ArcCoth[c + d*x]^2)/2 - Log[1/((c + d*x)*Sqrt
[1 - (c + d*x)^(-2)])]) + 12*a*b^2*d*e*(ArcCoth[c + d*x]*((-1 + c + d*x)*ArcCoth[c + d*x] - 2*Log[1 - E^(-2*Ar
cCoth[c + d*x])]) + PolyLog[2, E^(-2*ArcCoth[c + d*x])]) - 12*a*b^2*c*f*(ArcCoth[c + d*x]*((-1 + c + d*x)*ArcC
oth[c + d*x] - 2*Log[1 - E^(-2*ArcCoth[c + d*x])]) + PolyLog[2, E^(-2*ArcCoth[c + d*x])]) + 2*b^3*f*(ArcCoth[c
+ d*x]*(3*(-1 + c + d*x)*ArcCoth[c + d*x] + (-1 + c^2 + 2*c*d*x + d^2*x^2)*ArcCoth[c + d*x]^2 - 6*Log[1 - E^(
-2*ArcCoth[c + d*x])]) + 3*PolyLog[2, E^(-2*ArcCoth[c + d*x])]) + 4*b^3*d*e*((-1/8*I)*Pi^3 + ArcCoth[c + d*x]^
3 + (c + d*x)*ArcCoth[c + d*x]^3 - 3*ArcCoth[c + d*x]^2*Log[1 - E^(2*ArcCoth[c + d*x])] - 3*ArcCoth[c + d*x]*P
olyLog[2, E^(2*ArcCoth[c + d*x])] + (3*PolyLog[3, E^(2*ArcCoth[c + d*x])])/2) - 4*b^3*c*f*((-1/8*I)*Pi^3 + Arc
Coth[c + d*x]^3 + (c + d*x)*ArcCoth[c + d*x]^3 - 3*ArcCoth[c + d*x]^2*Log[1 - E^(2*ArcCoth[c + d*x])] - 3*ArcC
oth[c + d*x]*PolyLog[2, E^(2*ArcCoth[c + d*x])] + (3*PolyLog[3, E^(2*ArcCoth[c + d*x])])/2))/(4*d^2)

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 5.56 (sec) , antiderivative size = 7528, normalized size of antiderivative = 23.09

method	result	size
parts	\(\text {Expression too large to display}\)	\(7528\)
derivativedivides	\(\text {Expression too large to display}\)	\(7638\)
default	\(\text {Expression too large to display}\)	\(7638\)

[In]

int((f*x+e)*(a+b*arccoth(d*x+c))^3,x,method=_RETURNVERBOSE)

[Out]

result too large to display

Fricas [F]

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

[In]

integrate((f*x+e)*(a+b*arccoth(d*x+c))^3,x, algorithm="fricas")

[Out]

integral(a^3*f*x + a^3*e + (b^3*f*x + b^3*e)*arccoth(d*x + c)^3 + 3*(a*b^2*f*x + a*b^2*e)*arccoth(d*x + c)^2 +
3*(a^2*b*f*x + a^2*b*e)*arccoth(d*x + c), x)

Sympy [F]

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

[In]

integrate((f*x+e)*(a+b*acoth(d*x+c))**3,x)

[Out]

Integral((a + b*acoth(c + d*x))**3*(e + f*x), x)

Maxima [F]

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

[In]

integrate((f*x+e)*(a+b*arccoth(d*x+c))^3,x, algorithm="maxima")

[Out]

1/2*a^3*f*x^2 + 3/4*(2*x^2*arccoth(d*x + c) + d*(2*x/d^2 - (c^2 + 2*c + 1)*log(d*x + c + 1)/d^3 + (c^2 - 2*c +
1)*log(d*x + c - 1)/d^3))*a^2*b*f + a^3*e*x + 3/2*(2*(d*x + c)*arccoth(d*x + c) + log(-(d*x + c)^2 + 1))*a^2*
b*e/d + 1/16*((b^3*d^2*f*x^2 + 2*b^3*d^2*e*x - (c^2*f - 2*(d*e - f)*c - 2*d*e + f)*b^3)*log(d*x + c + 1)^3 + 3
*(2*a*b^2*d^2*f*x^2 + 2*(2*a*b^2*d^2*e + b^3*d*f)*x - (b^3*d^2*f*x^2 + 2*b^3*d^2*e*x - (c^2*f - 2*(d*e + f)*c
+ 2*d*e + f)*b^3)*log(d*x + c - 1))*log(d*x + c + 1)^2)/d^2 + integrate(-1/8*((b^3*d^2*f*x^2 + (d^2*e + c*d*f
+ d*f)*b^3*x + (c*d*e + d*e)*b^3)*log(d*x + c - 1)^3 - 6*(a*b^2*d^2*f*x^2 + (d^2*e + c*d*f + d*f)*a*b^2*x + (c
*d*e + d*e)*a*b^2)*log(d*x + c - 1)^2 + 3*(2*a*b^2*d^2*f*x^2 - (b^3*d^2*f*x^2 + (d^2*e + c*d*f + d*f)*b^3*x +
(c*d*e + d*e)*b^3)*log(d*x + c - 1)^2 + 2*(2*a*b^2*d^2*e + b^3*d*f)*x + (4*(c*d*e + d*e)*a*b^2 + (c^2*f - 2*(d
*e + f)*c + 2*d*e + f)*b^3 + (4*a*b^2*d^2*f - b^3*d^2*f)*x^2 - 2*(b^3*d^2*e - 2*(d^2*e + c*d*f + d*f)*a*b^2)*x
)*log(d*x + c - 1))*log(d*x + c + 1))/(d^2*x + c*d + d), x)

Giac [F]

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

[In]

integrate((f*x+e)*(a+b*arccoth(d*x+c))^3,x, algorithm="giac")

[Out]

integrate((f*x + e)*(b*arccoth(d*x + c) + a)^3, x)

Mupad [F(-1)]

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

[In]

int((e + f*x)*(a + b*acoth(c + d*x))^3,x)

[Out]

int((e + f*x)*(a + b*acoth(c + d*x))^3, x)