3.2.14 \(\int (e+f x)^2 (a+b \coth ^{-1}(c+d x))^3 \, dx\) [114]

Optimal. Leaf size=546 \[ \frac {a b^2 f^2 x}{d^2}+\frac {b^3 f^2 (c+d x) \coth ^{-1}(c+d x)}{d^3}-\frac {b f^2 \left (a+b \coth ^{-1}(c+d x)\right )^2}{2 d^3}+\frac {3 b f (d e-c f) \left (a+b \coth ^{-1}(c+d x)\right )^2}{d^3}+\frac {3 b f (d e-c f) (c+d x) \left (a+b \coth ^{-1}(c+d x)\right )^2}{d^3}+\frac {b f^2 (c+d x)^2 \left (a+b \coth ^{-1}(c+d x)\right )^2}{2 d^3}-\frac {(d e-c f) \left (d^2 e^2-2 c d e f+\left (3+c^2\right ) f^2\right ) \left (a+b \coth ^{-1}(c+d x)\right )^3}{3 d^3 f}+\frac {\left (3 d^2 e^2-6 c d e f+\left (1+3 c^2\right ) f^2\right ) \left (a+b \coth ^{-1}(c+d x)\right )^3}{3 d^3}+\frac {(e+f x)^3 \left (a+b \coth ^{-1}(c+d x)\right )^3}{3 f}-\frac {6 b^2 f (d e-c f) \left (a+b \coth ^{-1}(c+d x)\right ) \log \left (\frac {2}{1-c-d x}\right )}{d^3}-\frac {b \left (3 d^2 e^2-6 c d e f+\left (1+3 c^2\right ) f^2\right ) \left (a+b \coth ^{-1}(c+d x)\right )^2 \log \left (\frac {2}{1-c-d x}\right )}{d^3}+\frac {b^3 f^2 \log \left (1-(c+d x)^2\right )}{2 d^3}-\frac {3 b^3 f (d e-c f) \text {PolyLog}\left (2,-\frac {1+c+d x}{1-c-d x}\right )}{d^3}-\frac {b^2 \left (3 d^2 e^2-6 c d e f+\left (1+3 c^2\right ) f^2\right ) \left (a+b \coth ^{-1}(c+d x)\right ) \text {PolyLog}\left (2,1-\frac {2}{1-c-d x}\right )}{d^3}+\frac {b^3 \left (3 d^2 e^2-6 c d e f+\left (1+3 c^2\right ) f^2\right ) \text {PolyLog}\left (3,1-\frac {2}{1-c-d x}\right )}{2 d^3} \]

[Out]

a*b^2*f^2*x/d^2+b^3*f^2*(d*x+c)*arccoth(d*x+c)/d^3-1/2*b*f^2*(a+b*arccoth(d*x+c))^2/d^3+3*b*f*(-c*f+d*e)*(a+b*
arccoth(d*x+c))^2/d^3+3*b*f*(-c*f+d*e)*(d*x+c)*(a+b*arccoth(d*x+c))^2/d^3+1/2*b*f^2*(d*x+c)^2*(a+b*arccoth(d*x
+c))^2/d^3-1/3*(-c*f+d*e)*(d^2*e^2-2*c*d*e*f+(c^2+3)*f^2)*(a+b*arccoth(d*x+c))^3/d^3/f+1/3*(3*d^2*e^2-6*c*d*e*
f+(3*c^2+1)*f^2)*(a+b*arccoth(d*x+c))^3/d^3+1/3*(f*x+e)^3*(a+b*arccoth(d*x+c))^3/f-6*b^2*f*(-c*f+d*e)*(a+b*arc
coth(d*x+c))*ln(2/(-d*x-c+1))/d^3-b*(3*d^2*e^2-6*c*d*e*f+(3*c^2+1)*f^2)*(a+b*arccoth(d*x+c))^2*ln(2/(-d*x-c+1)
)/d^3+1/2*b^3*f^2*ln(1-(d*x+c)^2)/d^3-3*b^3*f*(-c*f+d*e)*polylog(2,(-d*x-c-1)/(-d*x-c+1))/d^3-b^2*(3*d^2*e^2-6
*c*d*e*f+(3*c^2+1)*f^2)*(a+b*arccoth(d*x+c))*polylog(2,1-2/(-d*x-c+1))/d^3+1/2*b^3*(3*d^2*e^2-6*c*d*e*f+(3*c^2
+1)*f^2)*polylog(3,1-2/(-d*x-c+1))/d^3

________________________________________________________________________________________

Rubi [A]
time = 0.71, antiderivative size = 546, normalized size of antiderivative = 1.00, number of steps used = 21, number of rules used = 14, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {6247, 6066, 6022, 6132, 6056, 2449, 2352, 6038, 6128, 266, 6096, 6196, 6206, 6745} \begin {gather*} -\frac {b^2 \left (\left (3 c^2+1\right ) f^2-6 c d e f+3 d^2 e^2\right ) \text {Li}_2\left (1-\frac {2}{-c-d x+1}\right ) \left (a+b \coth ^{-1}(c+d x)\right )}{d^3}-\frac {6 b^2 f (d e-c f) \log \left (\frac {2}{-c-d x+1}\right ) \left (a+b \coth ^{-1}(c+d x)\right )}{d^3}+\frac {a b^2 f^2 x}{d^2}-\frac {(d e-c f) \left (\left (c^2+3\right ) f^2-2 c d e f+d^2 e^2\right ) \left (a+b \coth ^{-1}(c+d x)\right )^3}{3 d^3 f}+\frac {\left (\left (3 c^2+1\right ) f^2-6 c d e f+3 d^2 e^2\right ) \left (a+b \coth ^{-1}(c+d x)\right )^3}{3 d^3}-\frac {b \left (\left (3 c^2+1\right ) f^2-6 c d e f+3 d^2 e^2\right ) \log \left (\frac {2}{-c-d x+1}\right ) \left (a+b \coth ^{-1}(c+d x)\right )^2}{d^3}+\frac {3 b f (d e-c f) \left (a+b \coth ^{-1}(c+d x)\right )^2}{d^3}+\frac {3 b f (c+d x) (d e-c f) \left (a+b \coth ^{-1}(c+d x)\right )^2}{d^3}-\frac {b f^2 \left (a+b \coth ^{-1}(c+d x)\right )^2}{2 d^3}+\frac {b f^2 (c+d x)^2 \left (a+b \coth ^{-1}(c+d x)\right )^2}{2 d^3}+\frac {(e+f x)^3 \left (a+b \coth ^{-1}(c+d x)\right )^3}{3 f}+\frac {b^3 \left (\left (3 c^2+1\right ) f^2-6 c d e f+3 d^2 e^2\right ) \text {Li}_3\left (1-\frac {2}{-c-d x+1}\right )}{2 d^3}-\frac {3 b^3 f (d e-c f) \text {Li}_2\left (-\frac {c+d x+1}{-c-d x+1}\right )}{d^3}+\frac {b^3 f^2 \log \left (1-(c+d x)^2\right )}{2 d^3}+\frac {b^3 f^2 (c+d x) \coth ^{-1}(c+d x)}{d^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(e + f*x)^2*(a + b*ArcCoth[c + d*x])^3,x]

[Out]

(a*b^2*f^2*x)/d^2 + (b^3*f^2*(c + d*x)*ArcCoth[c + d*x])/d^3 - (b*f^2*(a + b*ArcCoth[c + d*x])^2)/(2*d^3) + (3
*b*f*(d*e - c*f)*(a + b*ArcCoth[c + d*x])^2)/d^3 + (3*b*f*(d*e - c*f)*(c + d*x)*(a + b*ArcCoth[c + d*x])^2)/d^
3 + (b*f^2*(c + d*x)^2*(a + b*ArcCoth[c + d*x])^2)/(2*d^3) - ((d*e - c*f)*(d^2*e^2 - 2*c*d*e*f + (3 + c^2)*f^2
)*(a + b*ArcCoth[c + d*x])^3)/(3*d^3*f) + ((3*d^2*e^2 - 6*c*d*e*f + (1 + 3*c^2)*f^2)*(a + b*ArcCoth[c + d*x])^
3)/(3*d^3) + ((e + f*x)^3*(a + b*ArcCoth[c + d*x])^3)/(3*f) - (6*b^2*f*(d*e - c*f)*(a + b*ArcCoth[c + d*x])*Lo
g[2/(1 - c - d*x)])/d^3 - (b*(3*d^2*e^2 - 6*c*d*e*f + (1 + 3*c^2)*f^2)*(a + b*ArcCoth[c + d*x])^2*Log[2/(1 - c
- d*x)])/d^3 + (b^3*f^2*Log[1 - (c + d*x)^2])/(2*d^3) - (3*b^3*f*(d*e - c*f)*PolyLog[2, -((1 + c + d*x)/(1 -
c - d*x))])/d^3 - (b^2*(3*d^2*e^2 - 6*c*d*e*f + (1 + 3*c^2)*f^2)*(a + b*ArcCoth[c + d*x])*PolyLog[2, 1 - 2/(1
- c - d*x)])/d^3 + (b^3*(3*d^2*e^2 - 6*c*d*e*f + (1 + 3*c^2)*f^2)*PolyLog[3, 1 - 2/(1 - c - d*x)])/(2*d^3)

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 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 6128

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 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*} \int (e+f x)^2 \left (a+b \coth ^{-1}(c+d x)\right )^3 \, dx &=\frac {\text {Subst}\left (\int \left (\frac {d e-c f}{d}+\frac {f x}{d}\right )^2 \left (a+b \coth ^{-1}(x)\right )^3 \, dx,x,c+d x\right )}{d}\\ &=\frac {(e+f x)^3 \left (a+b \coth ^{-1}(c+d x)\right )^3}{3 f}-\frac {b \text {Subst}\left (\int \left (-\frac {3 f^2 (d e-c f) \left (a+b \coth ^{-1}(x)\right )^2}{d^3}-\frac {f^3 x \left (a+b \coth ^{-1}(x)\right )^2}{d^3}+\frac {\left ((d e-c f) \left (d^2 e^2-2 c d e f+3 f^2+c^2 f^2\right )+f \left (3 d^2 e^2-6 c d e f+\left (1+3 c^2\right ) f^2\right ) x\right ) \left (a+b \coth ^{-1}(x)\right )^2}{d^3 \left (1-x^2\right )}\right ) \, dx,x,c+d x\right )}{f}\\ &=\frac {(e+f x)^3 \left (a+b \coth ^{-1}(c+d x)\right )^3}{3 f}-\frac {b \text {Subst}\left (\int \frac {\left ((d e-c f) \left (d^2 e^2-2 c d e f+3 f^2+c^2 f^2\right )+f \left (3 d^2 e^2-6 c d e f+\left (1+3 c^2\right ) f^2\right ) x\right ) \left (a+b \coth ^{-1}(x)\right )^2}{1-x^2} \, dx,x,c+d x\right )}{d^3 f}+\frac {\left (b f^2\right ) \text {Subst}\left (\int x \left (a+b \coth ^{-1}(x)\right )^2 \, dx,x,c+d x\right )}{d^3}+\frac {(3 b f (d e-c f)) \text {Subst}\left (\int \left (a+b \coth ^{-1}(x)\right )^2 \, dx,x,c+d x\right )}{d^3}\\ &=\frac {3 b f (d e-c f) (c+d x) \left (a+b \coth ^{-1}(c+d x)\right )^2}{d^3}+\frac {b f^2 (c+d x)^2 \left (a+b \coth ^{-1}(c+d x)\right )^2}{2 d^3}+\frac {(e+f x)^3 \left (a+b \coth ^{-1}(c+d x)\right )^3}{3 f}-\frac {b \text {Subst}\left (\int \left (\frac {(d e-c f) \left (d^2 e^2-2 c d e f+\left (3+c^2\right ) f^2\right ) \left (a+b \coth ^{-1}(x)\right )^2}{1-x^2}+\frac {f \left (3 d^2 e^2-6 c d e f+\left (1+3 c^2\right ) f^2\right ) x \left (a+b \coth ^{-1}(x)\right )^2}{1-x^2}\right ) \, dx,x,c+d x\right )}{d^3 f}-\frac {\left (b^2 f^2\right ) \text {Subst}\left (\int \frac {x^2 \left (a+b \coth ^{-1}(x)\right )}{1-x^2} \, dx,x,c+d x\right )}{d^3}-\frac {\left (6 b^2 f (d e-c 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^3}\\ &=\frac {3 b f (d e-c f) \left (a+b \coth ^{-1}(c+d x)\right )^2}{d^3}+\frac {3 b f (d e-c f) (c+d x) \left (a+b \coth ^{-1}(c+d x)\right )^2}{d^3}+\frac {b f^2 (c+d x)^2 \left (a+b \coth ^{-1}(c+d x)\right )^2}{2 d^3}+\frac {(e+f x)^3 \left (a+b \coth ^{-1}(c+d x)\right )^3}{3 f}+\frac {\left (b^2 f^2\right ) \text {Subst}\left (\int \left (a+b \coth ^{-1}(x)\right ) \, dx,x,c+d x\right )}{d^3}-\frac {\left (b^2 f^2\right ) \text {Subst}\left (\int \frac {a+b \coth ^{-1}(x)}{1-x^2} \, dx,x,c+d x\right )}{d^3}-\frac {\left (6 b^2 f (d e-c f)\right ) \text {Subst}\left (\int \frac {a+b \coth ^{-1}(x)}{1-x} \, dx,x,c+d x\right )}{d^3}-\frac {\left (b (d e-c f) \left (d^2 e^2-2 c d e f+\left (3+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 )}{d^3 f}-\frac {\left (b \left (3 d^2 e^2-6 c d e f+\left (1+3 c^2\right ) f^2\right )\right ) \text {Subst}\left (\int \frac {x \left (a+b \coth ^{-1}(x)\right )^2}{1-x^2} \, dx,x,c+d x\right )}{d^3}\\ &=\frac {a b^2 f^2 x}{d^2}-\frac {b f^2 \left (a+b \coth ^{-1}(c+d x)\right )^2}{2 d^3}+\frac {3 b f (d e-c f) \left (a+b \coth ^{-1}(c+d x)\right )^2}{d^3}+\frac {3 b f (d e-c f) (c+d x) \left (a+b \coth ^{-1}(c+d x)\right )^2}{d^3}+\frac {b f^2 (c+d x)^2 \left (a+b \coth ^{-1}(c+d x)\right )^2}{2 d^3}-\frac {(d e-c f) \left (d^2 e^2-2 c d e f+\left (3+c^2\right ) f^2\right ) \left (a+b \coth ^{-1}(c+d x)\right )^3}{3 d^3 f}+\frac {\left (3 d^2 e^2-6 c d e f+\left (1+3 c^2\right ) f^2\right ) \left (a+b \coth ^{-1}(c+d x)\right )^3}{3 d^3}+\frac {(e+f x)^3 \left (a+b \coth ^{-1}(c+d x)\right )^3}{3 f}-\frac {6 b^2 f (d e-c f) \left (a+b \coth ^{-1}(c+d x)\right ) \log \left (\frac {2}{1-c-d x}\right )}{d^3}+\frac {\left (b^3 f^2\right ) \text {Subst}\left (\int \coth ^{-1}(x) \, dx,x,c+d x\right )}{d^3}+\frac {\left (6 b^3 f (d e-c f)\right ) \text {Subst}\left (\int \frac {\log \left (\frac {2}{1-x}\right )}{1-x^2} \, dx,x,c+d x\right )}{d^3}-\frac {\left (b \left (3 d^2 e^2-6 c d e f+\left (1+3 c^2\right ) f^2\right )\right ) \text {Subst}\left (\int \frac {\left (a+b \coth ^{-1}(x)\right )^2}{1-x} \, dx,x,c+d x\right )}{d^3}\\ &=\frac {a b^2 f^2 x}{d^2}+\frac {b^3 f^2 (c+d x) \coth ^{-1}(c+d x)}{d^3}-\frac {b f^2 \left (a+b \coth ^{-1}(c+d x)\right )^2}{2 d^3}+\frac {3 b f (d e-c f) \left (a+b \coth ^{-1}(c+d x)\right )^2}{d^3}+\frac {3 b f (d e-c f) (c+d x) \left (a+b \coth ^{-1}(c+d x)\right )^2}{d^3}+\frac {b f^2 (c+d x)^2 \left (a+b \coth ^{-1}(c+d x)\right )^2}{2 d^3}-\frac {(d e-c f) \left (d^2 e^2-2 c d e f+\left (3+c^2\right ) f^2\right ) \left (a+b \coth ^{-1}(c+d x)\right )^3}{3 d^3 f}+\frac {\left (3 d^2 e^2-6 c d e f+\left (1+3 c^2\right ) f^2\right ) \left (a+b \coth ^{-1}(c+d x)\right )^3}{3 d^3}+\frac {(e+f x)^3 \left (a+b \coth ^{-1}(c+d x)\right )^3}{3 f}-\frac {6 b^2 f (d e-c f) \left (a+b \coth ^{-1}(c+d x)\right ) \log \left (\frac {2}{1-c-d x}\right )}{d^3}-\frac {b \left (3 d^2 e^2-6 c d e f+\left (1+3 c^2\right ) f^2\right ) \left (a+b \coth ^{-1}(c+d x)\right )^2 \log \left (\frac {2}{1-c-d x}\right )}{d^3}-\frac {\left (b^3 f^2\right ) \text {Subst}\left (\int \frac {x}{1-x^2} \, dx,x,c+d x\right )}{d^3}-\frac {\left (6 b^3 f (d e-c f)\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-c-d x}\right )}{d^3}+\frac {\left (2 b^2 \left (3 d^2 e^2-6 c d e f+\left (1+3 c^2\right ) f^2\right )\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^3}\\ &=\frac {a b^2 f^2 x}{d^2}+\frac {b^3 f^2 (c+d x) \coth ^{-1}(c+d x)}{d^3}-\frac {b f^2 \left (a+b \coth ^{-1}(c+d x)\right )^2}{2 d^3}+\frac {3 b f (d e-c f) \left (a+b \coth ^{-1}(c+d x)\right )^2}{d^3}+\frac {3 b f (d e-c f) (c+d x) \left (a+b \coth ^{-1}(c+d x)\right )^2}{d^3}+\frac {b f^2 (c+d x)^2 \left (a+b \coth ^{-1}(c+d x)\right )^2}{2 d^3}-\frac {(d e-c f) \left (d^2 e^2-2 c d e f+\left (3+c^2\right ) f^2\right ) \left (a+b \coth ^{-1}(c+d x)\right )^3}{3 d^3 f}+\frac {\left (3 d^2 e^2-6 c d e f+\left (1+3 c^2\right ) f^2\right ) \left (a+b \coth ^{-1}(c+d x)\right )^3}{3 d^3}+\frac {(e+f x)^3 \left (a+b \coth ^{-1}(c+d x)\right )^3}{3 f}-\frac {6 b^2 f (d e-c f) \left (a+b \coth ^{-1}(c+d x)\right ) \log \left (\frac {2}{1-c-d x}\right )}{d^3}-\frac {b \left (3 d^2 e^2-6 c d e f+\left (1+3 c^2\right ) f^2\right ) \left (a+b \coth ^{-1}(c+d x)\right )^2 \log \left (\frac {2}{1-c-d x}\right )}{d^3}+\frac {b^3 f^2 \log \left (1-(c+d x)^2\right )}{2 d^3}-\frac {3 b^3 f (d e-c f) \text {Li}_2\left (1-\frac {2}{1-c-d x}\right )}{d^3}-\frac {b^2 \left (3 d^2 e^2-6 c d e f+\left (1+3 c^2\right ) f^2\right ) \left (a+b \coth ^{-1}(c+d x)\right ) \text {Li}_2\left (1-\frac {2}{1-c-d x}\right )}{d^3}+\frac {\left (b^3 \left (3 d^2 e^2-6 c d e f+\left (1+3 c^2\right ) f^2\right )\right ) \text {Subst}\left (\int \frac {\text {Li}_2\left (1-\frac {2}{1-x}\right )}{1-x^2} \, dx,x,c+d x\right )}{d^3}\\ &=\frac {a b^2 f^2 x}{d^2}+\frac {b^3 f^2 (c+d x) \coth ^{-1}(c+d x)}{d^3}-\frac {b f^2 \left (a+b \coth ^{-1}(c+d x)\right )^2}{2 d^3}+\frac {3 b f (d e-c f) \left (a+b \coth ^{-1}(c+d x)\right )^2}{d^3}+\frac {3 b f (d e-c f) (c+d x) \left (a+b \coth ^{-1}(c+d x)\right )^2}{d^3}+\frac {b f^2 (c+d x)^2 \left (a+b \coth ^{-1}(c+d x)\right )^2}{2 d^3}-\frac {(d e-c f) \left (d^2 e^2-2 c d e f+\left (3+c^2\right ) f^2\right ) \left (a+b \coth ^{-1}(c+d x)\right )^3}{3 d^3 f}+\frac {\left (3 d^2 e^2-6 c d e f+\left (1+3 c^2\right ) f^2\right ) \left (a+b \coth ^{-1}(c+d x)\right )^3}{3 d^3}+\frac {(e+f x)^3 \left (a+b \coth ^{-1}(c+d x)\right )^3}{3 f}-\frac {6 b^2 f (d e-c f) \left (a+b \coth ^{-1}(c+d x)\right ) \log \left (\frac {2}{1-c-d x}\right )}{d^3}-\frac {b \left (3 d^2 e^2-6 c d e f+\left (1+3 c^2\right ) f^2\right ) \left (a+b \coth ^{-1}(c+d x)\right )^2 \log \left (\frac {2}{1-c-d x}\right )}{d^3}+\frac {b^3 f^2 \log \left (1-(c+d x)^2\right )}{2 d^3}-\frac {3 b^3 f (d e-c f) \text {Li}_2\left (1-\frac {2}{1-c-d x}\right )}{d^3}-\frac {b^2 \left (3 d^2 e^2-6 c d e f+\left (1+3 c^2\right ) f^2\right ) \left (a+b \coth ^{-1}(c+d x)\right ) \text {Li}_2\left (1-\frac {2}{1-c-d x}\right )}{d^3}+\frac {b^3 \left (3 d^2 e^2-6 c d e f+\left (1+3 c^2\right ) f^2\right ) \text {Li}_3\left (1-\frac {2}{1-c-d x}\right )}{2 d^3}\\ \end {align*}

________________________________________________________________________________________

Mathematica [C] Result contains complex when optimal does not.
time = 8.95, size = 2594, normalized size = 4.75 \begin {gather*} \text {Result too large to show} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[(e + f*x)^2*(a + b*ArcCoth[c + d*x])^3,x]

[Out]

(a^2*(a*d^2*e^2 + 3*b*d*e*f - 2*b*c*f^2)*x)/d^2 + (a^2*f*(2*a*d*e + b*f)*x^2)/(2*d) + (a^3*f^2*x^3)/3 + a^2*b*
x*(3*e^2 + 3*e*f*x + f^2*x^2)*ArcCoth[c + d*x] + ((3*a^2*b*d^2*e^2 - 3*a^2*b*c*d^2*e^2 + 3*a^2*b*d*e*f - 6*a^2
*b*c*d*e*f + 3*a^2*b*c^2*d*e*f + a^2*b*f^2 - 3*a^2*b*c*f^2 + 3*a^2*b*c^2*f^2 - a^2*b*c^3*f^2)*Log[1 - c - d*x]
)/(2*d^3) + ((3*a^2*b*d^2*e^2 + 3*a^2*b*c*d^2*e^2 - 3*a^2*b*d*e*f - 6*a^2*b*c*d*e*f - 3*a^2*b*c^2*d*e*f + a^2*
b*f^2 + 3*a^2*b*c*f^2 + 3*a^2*b*c^2*f^2 + a^2*b*c^3*f^2)*Log[1 + c + d*x])/(2*d^3) + (3*a*b^2*e^2*(1 - (c + d*
x)^2)*(ArcCoth[c + d*x]*(ArcCoth[c + d*x] - (c + d*x)*ArcCoth[c + d*x] + 2*Log[1 - E^(-2*ArcCoth[c + d*x])]) -
PolyLog[2, E^(-2*ArcCoth[c + d*x])]))/(d*(c + d*x)^2*(1 - (c + d*x)^(-2))) - (3*a*b^2*e*f*(1 - (c + d*x)^2)*(
2*c*ArcCoth[c + d*x]^2 + (c + d*x)^2*(1 - (c + d*x)^(-2))*ArcCoth[c + d*x]^2 - 2*(c + d*x)*ArcCoth[c + d*x]*(-
1 + c*ArcCoth[c + d*x]) + 4*c*ArcCoth[c + d*x]*Log[1 - E^(-2*ArcCoth[c + d*x])] - 2*Log[1/((c + d*x)*Sqrt[1 -
(c + d*x)^(-2)])] - 2*c*PolyLog[2, E^(-2*ArcCoth[c + d*x])]))/(d^2*(c + d*x)^2*(1 - (c + d*x)^(-2))) + (b^3*e^
2*(1 - (c + d*x)^2)*((I/8)*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]*PolyLog[2, E^(2*ArcCoth[c + d*x])] - (3*PolyLog[3, E^(2*ArcC
oth[c + d*x])])/2))/(d*(c + d*x)^2*(1 - (c + d*x)^(-2))) - (b^3*e*f*(1 - (c + d*x)^2)*(I*c*Pi^3 - 12*ArcCoth[c
+ d*x]^2 + 12*(c + d*x)*ArcCoth[c + d*x]^2 - 8*c*ArcCoth[c + d*x]^3 - 8*c*(c + d*x)*ArcCoth[c + d*x]^3 + 4*(c
+ d*x)^2*(1 - (c + d*x)^(-2))*ArcCoth[c + d*x]^3 - 24*ArcCoth[c + d*x]*Log[1 - E^(-2*ArcCoth[c + d*x])] + 24*
c*ArcCoth[c + d*x]^2*Log[1 - E^(2*ArcCoth[c + d*x])] + 12*PolyLog[2, E^(-2*ArcCoth[c + d*x])] + 24*c*ArcCoth[c
+ d*x]*PolyLog[2, E^(2*ArcCoth[c + d*x])] - 12*c*PolyLog[3, E^(2*ArcCoth[c + d*x])]))/(4*d^2*(c + d*x)^2*(1 -
(c + d*x)^(-2))) - (a*b^2*f^2*(c + d*x)*Sqrt[1 - (c + d*x)^(-2)]*(1 - (c + d*x)^2)*((4*ArcCoth[c + d*x])/((c
+ d*x)*Sqrt[1 - (c + d*x)^(-2)]) + (3*ArcCoth[c + d*x]^2)/((c + d*x)*Sqrt[1 - (c + d*x)^(-2)]) - (12*c*ArcCoth
[c + d*x]^2)/((c + d*x)*Sqrt[1 - (c + d*x)^(-2)]) + (9*c^2*ArcCoth[c + d*x]^2)/((c + d*x)*Sqrt[1 - (c + d*x)^(
-2)]) + (-1 + 6*c*ArcCoth[c + d*x] + 3*ArcCoth[c + d*x]^2 - 3*c^2*ArcCoth[c + d*x]^2)/Sqrt[1 - (c + d*x)^(-2)]
+ Cosh[3*ArcCoth[c + d*x]] - 6*c*ArcCoth[c + d*x]*Cosh[3*ArcCoth[c + d*x]] + ArcCoth[c + d*x]^2*Cosh[3*ArcCot
h[c + d*x]] + 3*c^2*ArcCoth[c + d*x]^2*Cosh[3*ArcCoth[c + d*x]] + (6*ArcCoth[c + d*x]*Log[1 - E^(-2*ArcCoth[c
+ d*x])])/((c + d*x)*Sqrt[1 - (c + d*x)^(-2)]) + (18*c^2*ArcCoth[c + d*x]*Log[1 - E^(-2*ArcCoth[c + d*x])])/((
c + d*x)*Sqrt[1 - (c + d*x)^(-2)]) - (18*c*Log[1/((c + d*x)*Sqrt[1 - (c + d*x)^(-2)])])/((c + d*x)*Sqrt[1 - (c
+ d*x)^(-2)]) + (4*(1 + 3*c^2)*PolyLog[2, E^(-2*ArcCoth[c + d*x])])/((c + d*x)^3*(1 - (c + d*x)^(-2))^(3/2))
- ArcCoth[c + d*x]^2*Sinh[3*ArcCoth[c + d*x]] - 3*c^2*ArcCoth[c + d*x]^2*Sinh[3*ArcCoth[c + d*x]] - 2*ArcCoth[
c + d*x]*Log[1 - E^(-2*ArcCoth[c + d*x])]*Sinh[3*ArcCoth[c + d*x]] - 6*c^2*ArcCoth[c + d*x]*Log[1 - E^(-2*ArcC
oth[c + d*x])]*Sinh[3*ArcCoth[c + d*x]] + 6*c*Log[1/((c + d*x)*Sqrt[1 - (c + d*x)^(-2)])]*Sinh[3*ArcCoth[c + d
*x]]))/(4*d^3) + (b^3*f^2*(1 - (c + d*x)^2)*(3*c*PolyLog[2, E^(-2*ArcCoth[c + d*x])] + ((c + d*x)^3*(1 - (c +
d*x)^(-2))^(3/2)*(((-3*I)*Pi^3)/((c + d*x)*Sqrt[1 - (c + d*x)^(-2)]) - ((9*I)*c^2*Pi^3)/((c + d*x)*Sqrt[1 - (c
+ d*x)^(-2)]) + (24*ArcCoth[c + d*x])/Sqrt[1 - (c + d*x)^(-2)] - (72*c*ArcCoth[c + d*x]^2)/Sqrt[1 - (c + d*x)
^(-2)] - (48*ArcCoth[c + d*x]^2)/((c + d*x)*Sqrt[1 - (c + d*x)^(-2)]) + (216*c*ArcCoth[c + d*x]^2)/((c + d*x)*
Sqrt[1 - (c + d*x)^(-2)]) - (24*ArcCoth[c + d*x]^3)/Sqrt[1 - (c + d*x)^(-2)] + (24*c^2*ArcCoth[c + d*x]^3)/Sqr
t[1 - (c + d*x)^(-2)] + (24*ArcCoth[c + d*x]^3)/((c + d*x)*Sqrt[1 - (c + d*x)^(-2)]) + (96*c*ArcCoth[c + d*x]^
3)/((c + d*x)*Sqrt[1 - (c + d*x)^(-2)]) + (72*c^2*ArcCoth[c + d*x]^3)/((c + d*x)*Sqrt[1 - (c + d*x)^(-2)]) - 2
4*ArcCoth[c + d*x]*Cosh[3*ArcCoth[c + d*x]] + 72*c*ArcCoth[c + d*x]^2*Cosh[3*ArcCoth[c + d*x]] - 8*ArcCoth[c +
d*x]^3*Cosh[3*ArcCoth[c + d*x]] - 24*c^2*ArcCoth[c + d*x]^3*Cosh[3*ArcCoth[c + d*x]] + (432*c*ArcCoth[c + d*x
]*Log[1 - E^(-2*ArcCoth[c + d*x])])/((c + d*x)*Sqrt[1 - (c + d*x)^(-2)]) - (72*ArcCoth[c + d*x]^2*Log[1 - E^(2
*ArcCoth[c + d*x])])/((c + d*x)*Sqrt[1 - (c + d*x)^(-2)]) - (216*c^2*ArcCoth[c + d*x]^2*Log[1 - E^(2*ArcCoth[c
+ d*x])])/((c + d*x)*Sqrt[1 - (c + d*x)^(-2)]) - (72*Log[1/((c + d*x)*Sqrt[1 - (c + d*x)^(-2)])])/((c + d*x)*
Sqrt[1 - (c + d*x)^(-2)]) + (96*(1 + 3*c^2)*ArcCoth[c + d*x]*PolyLog[2, E^(2*ArcCoth[c + d*x])])/((c + d*x)^3*
(1 - (c + d*x)^(-2))^(3/2)) - (48*(1 + 3*c^2)*PolyLog[3, E^(2*ArcCoth[c + d*x])])/((c + d*x)^3*(1 - (c + d*x)^
(-2))^(3/2)) + I*Pi^3*Sinh[3*ArcCoth[c + d*x]] + (3*I)*c^2*Pi^3*Sinh[3*ArcCoth[c + d*x]] - 72*c*ArcCoth[c + d*
x]^2*Sinh[3*ArcCoth[c + d*x]] - 8*ArcCoth[c + d*x]^3*Sinh[3*ArcCoth[c + d*x]] - 24*c^2*ArcCoth[c + d*x]^3*Sinh
[3*ArcCoth[c + d*x]] - 144*c*ArcCoth[c + d*x]*L...

________________________________________________________________________________________

Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 22.13, size = 10601, normalized size = 19.42

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(10601\)
default	\(\text {Expression too large to display}\)	\(10601\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

result too large to display

________________________________________________________________________________________

Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*a^3*f^2*x^3 + a^3*f*x^2*e + 1/2*(2*x^3*arccoth(d*x + c) + d*((d*x^2 - 4*c*x)/d^3 + (c^3 + 3*c^2 + 3*c + 1)
*log(d*x + c + 1)/d^4 - (c^3 - 3*c^2 + 3*c - 1)*log(d*x + c - 1)/d^4))*a^2*b*f^2 + 3/2*(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*e + a^3*
x*e^2 + 3/2*(2*(d*x + c)*arccoth(d*x + c) + log(-(d*x + c)^2 + 1))*a^2*b*e^2/d + 1/24*((b^3*d^3*f^2*x^3 + 3*b^
3*d^3*f*x^2*e + 3*b^3*d^3*x*e^2 + 3*(c*d^2 + d^2)*b^3*e^2 - 3*(c^2*d*f + 2*c*d*f + d*f)*b^3*e + (c^3*f^2 + 3*c
^2*f^2 + 3*c*f^2 + f^2)*b^3)*log(d*x + c + 1)^3 + 3*(2*a*b^2*d^3*f^2*x^3 + (6*a*b^2*d^3*f*e + b^3*d^2*f^2)*x^2
- 2*(2*b^3*c*d*f^2 - 3*a*b^2*d^3*e^2 - 3*b^3*d^2*f*e)*x - (b^3*d^3*f^2*x^3 + 3*b^3*d^3*f*x^2*e + 3*b^3*d^3*x*
e^2 + 3*(c*d^2 - d^2)*b^3*e^2 - 3*(c^2*d*f - 2*c*d*f + d*f)*b^3*e + (c^3*f^2 - 3*c^2*f^2 + 3*c*f^2 - f^2)*b^3)
*log(d*x + c - 1))*log(d*x + c + 1)^2)/d^3 + integrate(-1/8*((b^3*d^3*f^2*x^3 + (c*d^2 + d^2)*b^3*e^2 + (2*b^3
*d^3*f*e + (c*d^2*f^2 + d^2*f^2)*b^3)*x^2 + (b^3*d^3*e^2 + 2*(c*d^2*f + d^2*f)*b^3*e)*x)*log(d*x + c - 1)^3 -
6*(a*b^2*d^3*f^2*x^3 + (c*d^2 + d^2)*a*b^2*e^2 + (2*a*b^2*d^3*f*e + (c*d^2*f^2 + d^2*f^2)*a*b^2)*x^2 + (a*b^2*
d^3*e^2 + 2*(c*d^2*f + d^2*f)*a*b^2*e)*x)*log(d*x + c - 1)^2 + (4*a*b^2*d^3*f^2*x^3 + 2*(6*a*b^2*d^3*f*e + b^3
*d^2*f^2)*x^2 - 3*(b^3*d^3*f^2*x^3 + (c*d^2 + d^2)*b^3*e^2 + (2*b^3*d^3*f*e + (c*d^2*f^2 + d^2*f^2)*b^3)*x^2 +
(b^3*d^3*e^2 + 2*(c*d^2*f + d^2*f)*b^3*e)*x)*log(d*x + c - 1)^2 - 4*(2*b^3*c*d*f^2 - 3*a*b^2*d^3*e^2 - 3*b^3*
d^2*f*e)*x + 2*(3*(c^2*d*f - 2*c*d*f + d*f)*b^3*e - (c^3*f^2 - 3*c^2*f^2 + 3*c*f^2 - f^2)*b^3 + (6*a*b^2*d^3*f
^2 - b^3*d^3*f^2)*x^3 + 3*(2*(c*d^2*f^2 + d^2*f^2)*a*b^2 + (4*a*b^2*d^3*f - b^3*d^3*f)*e)*x^2 + 3*(4*(c*d^2*f
+ d^2*f)*a*b^2*e + (2*a*b^2*d^3 - b^3*d^3)*e^2)*x + 3*(2*(c*d^2 + d^2)*a*b^2 - (c*d^2 - d^2)*b^3)*e^2)*log(d*x
+ c - 1))*log(d*x + c + 1))/(d^3*x + c*d^2 + d^2), x)

________________________________________________________________________________________

Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (a + b \operatorname {acoth}{\left (c + d x \right )}\right )^{3} \left (e + f x\right )^{2}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x+e)**2*(a+b*acoth(d*x+c))**3,x)

[Out]

Integral((a + b*acoth(c + d*x))**3*(e + f*x)**2, x)

________________________________________________________________________________________

Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int {\left (e+f\,x\right )}^2\,{\left (a+b\,\mathrm {acoth}\left (c+d\,x\right )\right )}^3 \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e + f*x)^2*(a + b*acoth(c + d*x))^3,x)

[Out]

int((e + f*x)^2*(a + b*acoth(c + d*x))^3, x)

________________________________________________________________________________________