3.16 \(\int (d+e x)^2 (a+b \tanh ^{-1}(c x))^3 \, dx\)

Optimal. Leaf size=387 \[ -\frac{b^2 \left (3 c^2 d^2+e^2\right ) \text{PolyLog}\left (2,1-\frac{2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{c^3}+\frac{b^3 \left (3 c^2 d^2+e^2\right ) \text{PolyLog}\left (3,1-\frac{2}{1-c x}\right )}{2 c^3}-\frac{3 b^3 d e \text{PolyLog}\left (2,1-\frac{2}{1-c x}\right )}{c^2}-\frac{6 b^2 d e \log \left (\frac{2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{c^2}+\frac{a b^2 e^2 x}{c^2}+\frac{\left (3 c^2 d^2+e^2\right ) \left (a+b \tanh ^{-1}(c x)\right )^3}{3 c^3}-\frac{d \left (\frac{3 e^2}{c^2}+d^2\right ) \left (a+b \tanh ^{-1}(c x)\right )^3}{3 e}-\frac{b \left (3 c^2 d^2+e^2\right ) \log \left (\frac{2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )^2}{c^3}+\frac{3 b d e \left (a+b \tanh ^{-1}(c x)\right )^2}{c^2}-\frac{b e^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c^3}+\frac{3 b d e x \left (a+b \tanh ^{-1}(c x)\right )^2}{c}+\frac{(d+e x)^3 \left (a+b \tanh ^{-1}(c x)\right )^3}{3 e}+\frac{b e^2 x^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c}+\frac{b^3 e^2 \log \left (1-c^2 x^2\right )}{2 c^3}+\frac{b^3 e^2 x \tanh ^{-1}(c x)}{c^2} \]

[Out]

(a*b^2*e^2*x)/c^2 + (b^3*e^2*x*ArcTanh[c*x])/c^2 + (3*b*d*e*(a + b*ArcTanh[c*x])^2)/c^2 - (b*e^2*(a + b*ArcTan
h[c*x])^2)/(2*c^3) + (3*b*d*e*x*(a + b*ArcTanh[c*x])^2)/c + (b*e^2*x^2*(a + b*ArcTanh[c*x])^2)/(2*c) + ((3*c^2
*d^2 + e^2)*(a + b*ArcTanh[c*x])^3)/(3*c^3) - (d*(d^2 + (3*e^2)/c^2)*(a + b*ArcTanh[c*x])^3)/(3*e) + ((d + e*x
)^3*(a + b*ArcTanh[c*x])^3)/(3*e) - (6*b^2*d*e*(a + b*ArcTanh[c*x])*Log[2/(1 - c*x)])/c^2 - (b*(3*c^2*d^2 + e^
2)*(a + b*ArcTanh[c*x])^2*Log[2/(1 - c*x)])/c^3 + (b^3*e^2*Log[1 - c^2*x^2])/(2*c^3) - (3*b^3*d*e*PolyLog[2, 1
- 2/(1 - c*x)])/c^2 - (b^2*(3*c^2*d^2 + e^2)*(a + b*ArcTanh[c*x])*PolyLog[2, 1 - 2/(1 - c*x)])/c^3 + (b^3*(3*
c^2*d^2 + e^2)*PolyLog[3, 1 - 2/(1 - c*x)])/(2*c^3)

________________________________________________________________________________________

Rubi [A]  time = 0.803702, antiderivative size = 387, normalized size of antiderivative = 1., number of steps used = 20, number of rules used = 13, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.722, Rules used = {5928, 5910, 5984, 5918, 2402, 2315, 5916, 5980, 260, 5948, 6048, 6058, 6610} \[ -\frac{b^2 \left (3 c^2 d^2+e^2\right ) \text{PolyLog}\left (2,1-\frac{2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{c^3}+\frac{b^3 \left (3 c^2 d^2+e^2\right ) \text{PolyLog}\left (3,1-\frac{2}{1-c x}\right )}{2 c^3}-\frac{3 b^3 d e \text{PolyLog}\left (2,1-\frac{2}{1-c x}\right )}{c^2}-\frac{6 b^2 d e \log \left (\frac{2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{c^2}+\frac{a b^2 e^2 x}{c^2}+\frac{\left (3 c^2 d^2+e^2\right ) \left (a+b \tanh ^{-1}(c x)\right )^3}{3 c^3}-\frac{d \left (\frac{3 e^2}{c^2}+d^2\right ) \left (a+b \tanh ^{-1}(c x)\right )^3}{3 e}-\frac{b \left (3 c^2 d^2+e^2\right ) \log \left (\frac{2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )^2}{c^3}+\frac{3 b d e \left (a+b \tanh ^{-1}(c x)\right )^2}{c^2}-\frac{b e^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c^3}+\frac{3 b d e x \left (a+b \tanh ^{-1}(c x)\right )^2}{c}+\frac{(d+e x)^3 \left (a+b \tanh ^{-1}(c x)\right )^3}{3 e}+\frac{b e^2 x^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c}+\frac{b^3 e^2 \log \left (1-c^2 x^2\right )}{2 c^3}+\frac{b^3 e^2 x \tanh ^{-1}(c x)}{c^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a*b^2*e^2*x)/c^2 + (b^3*e^2*x*ArcTanh[c*x])/c^2 + (3*b*d*e*(a + b*ArcTanh[c*x])^2)/c^2 - (b*e^2*(a + b*ArcTan
h[c*x])^2)/(2*c^3) + (3*b*d*e*x*(a + b*ArcTanh[c*x])^2)/c + (b*e^2*x^2*(a + b*ArcTanh[c*x])^2)/(2*c) + ((3*c^2
*d^2 + e^2)*(a + b*ArcTanh[c*x])^3)/(3*c^3) - (d*(d^2 + (3*e^2)/c^2)*(a + b*ArcTanh[c*x])^3)/(3*e) + ((d + e*x
)^3*(a + b*ArcTanh[c*x])^3)/(3*e) - (6*b^2*d*e*(a + b*ArcTanh[c*x])*Log[2/(1 - c*x)])/c^2 - (b*(3*c^2*d^2 + e^
2)*(a + b*ArcTanh[c*x])^2*Log[2/(1 - c*x)])/c^3 + (b^3*e^2*Log[1 - c^2*x^2])/(2*c^3) - (3*b^3*d*e*PolyLog[2, 1
- 2/(1 - c*x)])/c^2 - (b^2*(3*c^2*d^2 + e^2)*(a + b*ArcTanh[c*x])*PolyLog[2, 1 - 2/(1 - c*x)])/c^3 + (b^3*(3*
c^2*d^2 + e^2)*PolyLog[3, 1 - 2/(1 - c*x)])/(2*c^3)

Rule 5928

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_)*((d_) + (e_.)*(x_))^(q_.), x_Symbol] :> Simp[((d + e*x)^(q + 1)*(
a + b*ArcTanh[c*x])^p)/(e*(q + 1)), x] - Dist[(b*c*p)/(e*(q + 1)), Int[ExpandIntegrand[(a + b*ArcTanh[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 5910

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

Rule 5984

Int[(((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*(x_))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[(a + b*ArcTanh[c
*x])^(p + 1)/(b*e*(p + 1)), x] + Dist[1/(c*d), Int[(a + b*ArcTanh[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 5918

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

Rule 5916

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*ArcT
anh[c*x])^p)/(d*(m + 1)), x] - Dist[(b*c*p)/(d*(m + 1)), Int[((d*x)^(m + 1)*(a + b*ArcTanh[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 5980

Int[(((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Dist[f^2
/e, Int[(f*x)^(m - 2)*(a + b*ArcTanh[c*x])^p, x], x] - Dist[(d*f^2)/e, Int[((f*x)^(m - 2)*(a + b*ArcTanh[c*x])
^p)/(d + e*x^2), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[p, 0] && GtQ[m, 1]

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 5948

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[(a + b*ArcTanh[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 6048

Int[(((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*((f_) + (g_.)*(x_))^(m_.))/((d_) + (e_.)*(x_)^2), x_Symbol] :>
Int[ExpandIntegrand[(a + b*ArcTanh[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 6058

Int[(Log[u_]*((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.))/((d_) + (e_.)*(x_)^2), x_Symbol] :> -Simp[((a + b*ArcT
anh[c*x])^p*PolyLog[2, 1 - u])/(2*c*d), x] + Dist[(b*p)/2, Int[((a + b*ArcTanh[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 6610

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

Mathematica [A]  time = 1.29567, size = 591, normalized size = 1.53 \[ \frac{18 a b^2 c^2 d^2 \left (\text{PolyLog}\left (2,-e^{-2 \tanh ^{-1}(c x)}\right )+\tanh ^{-1}(c x) \left ((c x-1) \tanh ^{-1}(c x)-2 \log \left (e^{-2 \tanh ^{-1}(c x)}+1\right )\right )\right )+6 a b^2 e^2 \left (\text{PolyLog}\left (2,-e^{-2 \tanh ^{-1}(c x)}\right )+\left (c^3 x^3-1\right ) \tanh ^{-1}(c x)^2+\tanh ^{-1}(c x) \left (c^2 x^2-2 \log \left (e^{-2 \tanh ^{-1}(c x)}+1\right )-1\right )+c x\right )+6 b^3 c^2 d^2 \left (3 \tanh ^{-1}(c x) \text{PolyLog}\left (2,-e^{-2 \tanh ^{-1}(c x)}\right )+\frac{3}{2} \text{PolyLog}\left (3,-e^{-2 \tanh ^{-1}(c x)}\right )+\tanh ^{-1}(c x)^2 \left ((c x-1) \tanh ^{-1}(c x)-3 \log \left (e^{-2 \tanh ^{-1}(c x)}+1\right )\right )\right )-6 b^3 c d e \left (\tanh ^{-1}(c x) \left (\left (1-c^2 x^2\right ) \tanh ^{-1}(c x)^2+(3-3 c x) \tanh ^{-1}(c x)+6 \log \left (e^{-2 \tanh ^{-1}(c x)}+1\right )\right )-3 \text{PolyLog}\left (2,-e^{-2 \tanh ^{-1}(c x)}\right )\right )+b^3 e^2 \left (6 \tanh ^{-1}(c x) \text{PolyLog}\left (2,-e^{-2 \tanh ^{-1}(c x)}\right )+3 \text{PolyLog}\left (3,-e^{-2 \tanh ^{-1}(c x)}\right )+3 \log \left (1-c^2 x^2\right )+2 c^3 x^3 \tanh ^{-1}(c x)^3+3 c^2 x^2 \tanh ^{-1}(c x)^2-2 \tanh ^{-1}(c x)^3-3 \tanh ^{-1}(c x)^2+6 c x \tanh ^{-1}(c x)-6 \tanh ^{-1}(c x)^2 \log \left (e^{-2 \tanh ^{-1}(c x)}+1\right )\right )+6 a^2 b c^3 x \tanh ^{-1}(c x) \left (3 d^2+3 d e x+e^2 x^2\right )+3 a^2 b \left (3 c^2 d^2+3 c d e+e^2\right ) \log (1-c x)+3 a^2 b \left (3 c^2 d^2-3 c d e+e^2\right ) \log (c x+1)+3 a^2 c^2 e x^2 (2 a c d+b e)+6 a^2 c^2 d x (a c d+3 b e)+2 a^3 c^3 e^2 x^3+18 a b^2 c d e \left (\log \left (1-c^2 x^2\right )+\left (c^2 x^2-1\right ) \tanh ^{-1}(c x)^2+2 c x \tanh ^{-1}(c x)\right )}{6 c^3} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(6*a^2*c^2*d*(a*c*d + 3*b*e)*x + 3*a^2*c^2*e*(2*a*c*d + b*e)*x^2 + 2*a^3*c^3*e^2*x^3 + 6*a^2*b*c^3*x*(3*d^2 +
3*d*e*x + e^2*x^2)*ArcTanh[c*x] + 3*a^2*b*(3*c^2*d^2 + 3*c*d*e + e^2)*Log[1 - c*x] + 3*a^2*b*(3*c^2*d^2 - 3*c*
d*e + e^2)*Log[1 + c*x] + 18*a*b^2*c*d*e*(2*c*x*ArcTanh[c*x] + (-1 + c^2*x^2)*ArcTanh[c*x]^2 + Log[1 - c^2*x^2
]) - 6*b^3*c*d*e*(ArcTanh[c*x]*((3 - 3*c*x)*ArcTanh[c*x] + (1 - c^2*x^2)*ArcTanh[c*x]^2 + 6*Log[1 + E^(-2*ArcT
anh[c*x])]) - 3*PolyLog[2, -E^(-2*ArcTanh[c*x])]) + 6*a*b^2*e^2*(c*x + (-1 + c^3*x^3)*ArcTanh[c*x]^2 + ArcTanh
[c*x]*(-1 + c^2*x^2 - 2*Log[1 + E^(-2*ArcTanh[c*x])]) + PolyLog[2, -E^(-2*ArcTanh[c*x])]) + 18*a*b^2*c^2*d^2*(
ArcTanh[c*x]*((-1 + c*x)*ArcTanh[c*x] - 2*Log[1 + E^(-2*ArcTanh[c*x])]) + PolyLog[2, -E^(-2*ArcTanh[c*x])]) +
6*b^3*c^2*d^2*(ArcTanh[c*x]^2*((-1 + c*x)*ArcTanh[c*x] - 3*Log[1 + E^(-2*ArcTanh[c*x])]) + 3*ArcTanh[c*x]*Poly
Log[2, -E^(-2*ArcTanh[c*x])] + (3*PolyLog[3, -E^(-2*ArcTanh[c*x])])/2) + b^3*e^2*(6*c*x*ArcTanh[c*x] - 3*ArcTa
nh[c*x]^2 + 3*c^2*x^2*ArcTanh[c*x]^2 - 2*ArcTanh[c*x]^3 + 2*c^3*x^3*ArcTanh[c*x]^3 - 6*ArcTanh[c*x]^2*Log[1 +
E^(-2*ArcTanh[c*x])] + 3*Log[1 - c^2*x^2] + 6*ArcTanh[c*x]*PolyLog[2, -E^(-2*ArcTanh[c*x])] + 3*PolyLog[3, -E^
(-2*ArcTanh[c*x])]))/(6*c^3)

________________________________________________________________________________________

Maple [C]  time = 2.211, size = 4600, normalized size = 11.9 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

d*x^2*a^3*e-3/4*I/c^2*b^3*e*Pi*d*csgn(I/((c*x+1)^2/(-c^2*x^2+1)+1))*csgn(I*(c*x+1)^2/(c^2*x^2-1))*csgn(I*(c*x+
1)^2/(c^2*x^2-1)/((c*x+1)^2/(-c^2*x^2+1)+1))*arctanh(c*x)^2+a*b^2*e^2*x/c^2+b^3*e^2*x*arctanh(c*x)/c^2-6/c^2*b
^3*e*d*arctanh(c*x)*ln(1+I*(c*x+1)/(-c^2*x^2+1)^(1/2))-6/c^2*b^3*e*d*arctanh(c*x)*ln(1-I*(c*x+1)/(-c^2*x^2+1)^
(1/2))+3/c^2*a*b^2*e*ln(c*x-1)*d+3/c^2*a*b^2*e*ln(c*x+1)*d+3/4/c^2*a*b^2*e*ln(c*x+1)^2*d+3/4/c^2*a*b^2*e*ln(c*
x-1)^2*d+1/c*a*b^2*e^2*arctanh(c*x)*x^2+3/c*b^3*e*arctanh(c*x)^2*x*d+3/c*a*b^2*arctanh(c*x)*ln(c*x-1)*d^2+3/c*
a*b^2*arctanh(c*x)*ln(c*x+1)*d^2-3/2/c*a*b^2*ln(c*x-1)*ln(1/2+1/2*c*x)*d^2+a*b^2/e*arctanh(c*x)*ln(c*x-1)*d^3-
a*b^2/e*arctanh(c*x)*ln(c*x+1)*d^3-1/2*a*b^2/e*ln(c*x+1)*ln(-1/2*c*x+1/2)*d^3+1/2*a*b^2/e*ln(-1/2*c*x+1/2)*ln(
1/2+1/2*c*x)*d^3-1/2*a*b^2/e*ln(c*x-1)*ln(1/2+1/2*c*x)*d^3+3*a*b^2*e*arctanh(c*x)^2*x^2*d+3*a^2*b*e*arctanh(c*
x)*x^2*d-3/2*I/c*b^3*Pi*d^2*arctanh(c*x)^2-1/2*I/c^3*b^3*e^2*Pi*arctanh(c*x)^2-1/2*I*b^3/e*Pi*d^3*arctanh(c*x)
^2-1/4*I*b^3/e*Pi*d^3*csgn(I/((c*x+1)^2/(-c^2*x^2+1)+1))*csgn(I*(c*x+1)^2/(c^2*x^2-1))*csgn(I*(c*x+1)^2/(c^2*x
^2-1)/((c*x+1)^2/(-c^2*x^2+1)+1))*arctanh(c*x)^2-3/4*I/c^2*b^3*e*Pi*d*csgn(I*(c*x+1)^2/(c^2*x^2-1))*csgn(I*(c*
x+1)^2/(c^2*x^2-1)/((c*x+1)^2/(-c^2*x^2+1)+1))^2*arctanh(c*x)^2+3/4*I/c^2*b^3*e*Pi*d*csgn(I*(c*x+1)/(-c^2*x^2+
1)^(1/2))^2*csgn(I*(c*x+1)^2/(c^2*x^2-1))*arctanh(c*x)^2+3/2*I/c^2*b^3*e*Pi*d*csgn(I*(c*x+1)/(-c^2*x^2+1)^(1/2
))*csgn(I*(c*x+1)^2/(c^2*x^2-1))^2*arctanh(c*x)^2+3/4*I/c^2*b^3*e*Pi*d*csgn(I/((c*x+1)^2/(-c^2*x^2+1)+1))*csgn
(I*(c*x+1)^2/(c^2*x^2-1)/((c*x+1)^2/(-c^2*x^2+1)+1))^2*arctanh(c*x)^2+a^3*x*d^2+1/3*a^3*e^2*x^3+3*a^2*b/c*x*d*
e+3/2/c*a*b^2*ln(c*x+1)*ln(-1/2*c*x+1/2)*d^2-3/2/c*a*b^2*ln(-1/2*c*x+1/2)*ln(1/2+1/2*c*x)*d^2+3/2/c^2*a^2*b*e*
ln(c*x-1)*d-3/2/c^2*a^2*b*e*ln(c*x+1)*d+1/2/c^3*a*b^2*e^2*ln(c*x+1)*ln(-1/2*c*x+1/2)-3/2/c^2*b^3*e*arctanh(c*x
)^2*ln(c*x+1)*d-1/2/c^3*a*b^2*e^2*ln(-1/2*c*x+1/2)*ln(1/2+1/2*c*x)+1/c^3*a*b^2*e^2*arctanh(c*x)*ln(c*x-1)+1/c^
3*a*b^2*e^2*arctanh(c*x)*ln(c*x+1)-1/2/c^3*a*b^2*e^2*ln(c*x-1)*ln(1/2+1/2*c*x)+3/2/c^2*b^3*e*arctanh(c*x)^2*ln
(c*x-1)*d+3/c^2*b^3*e*ln((c*x+1)/(-c^2*x^2+1)^(1/2))*arctanh(c*x)^2*d+1/4*I/c^3*b^3*e^2*Pi*csgn(I/((c*x+1)^2/(
-c^2*x^2+1)+1))*csgn(I*(c*x+1)^2/(c^2*x^2-1))*csgn(I*(c*x+1)^2/(c^2*x^2-1)/((c*x+1)^2/(-c^2*x^2+1)+1))*arctanh
(c*x)^2+3/4*I/c*b^3*Pi*d^2*csgn(I/((c*x+1)^2/(-c^2*x^2+1)+1))*csgn(I*(c*x+1)^2/(c^2*x^2-1))*csgn(I*(c*x+1)^2/(
c^2*x^2-1)/((c*x+1)^2/(-c^2*x^2+1)+1))*arctanh(c*x)^2-1/4*I/c^3*b^3*e^2*Pi*csgn(I*(c*x+1)^2/(c^2*x^2-1)/((c*x+
1)^2/(-c^2*x^2+1)+1))^3*arctanh(c*x)^2-1/4*I/c^3*b^3*e^2*Pi*csgn(I*(c*x+1)^2/(c^2*x^2-1))^3*arctanh(c*x)^2-3/2
*I/c^2*b^3*e*Pi*d*arctanh(c*x)^2+1/2*I*b^3/e*Pi*d^3*csgn(I/((c*x+1)^2/(-c^2*x^2+1)+1))^2*arctanh(c*x)^2-1/2*I*
b^3/e*Pi*d^3*csgn(I/((c*x+1)^2/(-c^2*x^2+1)+1))^3*arctanh(c*x)^2+1/4*I*b^3/e*Pi*d^3*csgn(I*(c*x+1)^2/(c^2*x^2-
1)/((c*x+1)^2/(-c^2*x^2+1)+1))^3*arctanh(c*x)^2+1/4*I*b^3/e*Pi*d^3*csgn(I*(c*x+1)^2/(c^2*x^2-1))^3*arctanh(c*x
)^2+6/c*a*b^2*e*arctanh(c*x)*x*d+3/c^2*a*b^2*e*arctanh(c*x)*ln(c*x-1)*d-3/c^2*a*b^2*e*arctanh(c*x)*ln(c*x+1)*d
-3/2/c^2*a*b^2*e*ln(c*x-1)*ln(1/2+1/2*c*x)*d-3/2/c^2*a*b^2*e*ln(c*x+1)*ln(-1/2*c*x+1/2)*d+3/2/c^2*a*b^2*e*ln(-
1/2*c*x+1/2)*ln(1/2+1/2*c*x)*d-3/2*I/c*b^3*Pi*d^2*csgn(I/((c*x+1)^2/(-c^2*x^2+1)+1))^3*arctanh(c*x)^2-3/4*I/c*
b^3*Pi*d^2*csgn(I*(c*x+1)^2/(c^2*x^2-1))^3*arctanh(c*x)^2-3/4*I/c*b^3*Pi*d^2*csgn(I*(c*x+1)^2/(c^2*x^2-1)/((c*
x+1)^2/(-c^2*x^2+1)+1))^3*arctanh(c*x)^2+3/2*I/c*b^3*Pi*d^2*csgn(I/((c*x+1)^2/(-c^2*x^2+1)+1))^2*arctanh(c*x)^
2-1/2*I/c^3*b^3*e^2*Pi*csgn(I/((c*x+1)^2/(-c^2*x^2+1)+1))^3*arctanh(c*x)^2+1/2*I/c^3*b^3*e^2*Pi*csgn(I/((c*x+1
)^2/(-c^2*x^2+1)+1))^2*arctanh(c*x)^2+1/2/c*a^2*b*x^2*e^2+a^2*b/e*arctanh(c*x)*d^3+1/2*a^2*b/e*ln(c*x-1)*d^3-1
/2*a^2*b/e*ln(c*x+1)*d^3+a*b^2/e*arctanh(c*x)^2*d^3+b^3/e*ln((c*x+1)/(-c^2*x^2+1)^(1/2))*arctanh(c*x)^2*d^3+1/
2*b^3/e*arctanh(c*x)^2*ln(c*x-1)*d^3-1/2*b^3/e*arctanh(c*x)^2*ln(c*x+1)*d^3+3*a^2*b*arctanh(c*x)*x*d^2+3*a*b^2
*arctanh(c*x)^2*x*d^2+a^2*b*e^2*arctanh(c*x)*x^3+a*b^2*e^2*arctanh(c*x)^2*x^3+b^3*e*arctanh(c*x)^3*x^2*d+1/4*a
*b^2/e*ln(c*x+1)^2*d^3+1/4*a*b^2/e*ln(c*x-1)^2*d^3-1/c^3*b^3*e^2*polylog(2,-(c*x+1)^2/(-c^2*x^2+1))*arctanh(c*
x)-1/c^3*b^3*e^2*ln((c*x+1)/(-c^2*x^2+1)^(1/2))*arctanh(c*x)^2+1/2/c^3*b^3*e^2*arctanh(c*x)^2*ln(c*x-1)+1/2/c^
3*b^3*e^2*arctanh(c*x)^2*ln(c*x+1)-1/4/c^3*a*b^2*e^2*ln(c*x+1)^2-1/c^2*b^3*e*arctanh(c*x)^3*d+3/c^2*b^3*e*arct
anh(c*x)^2*d-6/c^2*b^3*e*d*dilog(1+I*(c*x+1)/(-c^2*x^2+1)^(1/2))-6/c^2*b^3*e*d*dilog(1-I*(c*x+1)/(-c^2*x^2+1)^
(1/2))+1/2/c^3*a^2*b*e^2*ln(c*x-1)+1/2/c^3*a^2*b*e^2*ln(c*x+1)-1/c^3*b^3*e^2*ln(2)*arctanh(c*x)^2-1/2/c^3*a*b^
2*e^2*ln(c*x+1)+1/2/c^3*a*b^2*e^2*ln(c*x-1)-1/c^3*a*b^2*e^2*dilog(1/2+1/2*c*x)+1/4/c^3*a*b^2*e^2*ln(c*x-1)^2+3
/2/c*a^2*b*ln(c*x-1)*d^2+3/2/c*a^2*b*ln(c*x+1)*d^2-3/4/c*a*b^2*ln(c*x+1)^2*d^2-3/c*a*b^2*dilog(1/2+1/2*c*x)*d^
2+3/4/c*a*b^2*ln(c*x-1)^2*d^2+1/2/c*b^3*e^2*arctanh(c*x)^2*x^2+3/2/c*b^3*arctanh(c*x)^2*ln(c*x-1)*d^2+3/2/c*b^
3*arctanh(c*x)^2*ln(c*x+1)*d^2-3/c*b^3*polylog(2,-(c*x+1)^2/(-c^2*x^2+1))*arctanh(c*x)*d^2-3/c*b^3*ln((c*x+1)/
(-c^2*x^2+1)^(1/2))*arctanh(c*x)^2*d^2-3/c*b^3*ln(2)*d^2*arctanh(c*x)^2+3/2/c*b^3*polylog(3,-(c*x+1)^2/(-c^2*x
^2+1))*d^2+1/c*b^3*arctanh(c*x)^3*d^2+1/c^3*b^3*e^2*arctanh(c*x)+1/3/c^3*b^3*e^2*arctanh(c*x)^3+1/2/c^3*b^3*e^
2*polylog(3,-(c*x+1)^2/(-c^2*x^2+1))-1/2/c^3*b^3*e^2*arctanh(c*x)^2-1/c^3*b^3*e^2*ln((c*x+1)^2/(-c^2*x^2+1)+1)
+b^3*arctanh(c*x)^3*x*d^2+1/3*b^3*e^2*arctanh(c*x)^3*x^3+1/3*a^3/e*d^3+1/2*I*b^3/e*Pi*d^3*csgn(I*(c*x+1)/(-c^2
*x^2+1)^(1/2))*csgn(I*(c*x+1)^2/(c^2*x^2-1))^2*arctanh(c*x)^2-1/4*I*b^3/e*Pi*d^3*csgn(I*(c*x+1)^2/(c^2*x^2-1))
*csgn(I*(c*x+1)^2/(c^2*x^2-1)/((c*x+1)^2/(-c^2*x^2+1)+1))^2*arctanh(c*x)^2+1/4*I*b^3/e*Pi*d^3*csgn(I/((c*x+1)^
2/(-c^2*x^2+1)+1))*csgn(I*(c*x+1)^2/(c^2*x^2-1)/((c*x+1)^2/(-c^2*x^2+1)+1))^2*arctanh(c*x)^2+1/4*I*b^3/e*Pi*d^
3*csgn(I*(c*x+1)/(-c^2*x^2+1)^(1/2))^2*csgn(I*(c*x+1)^2/(c^2*x^2-1))*arctanh(c*x)^2-3/4*I/c*b^3*Pi*d^2*csgn(I*
(c*x+1)/(-c^2*x^2+1)^(1/2))^2*csgn(I*(c*x+1)^2/(c^2*x^2-1))*arctanh(c*x)^2-3/4*I/c*b^3*Pi*d^2*csgn(I/((c*x+1)^
2/(-c^2*x^2+1)+1))*csgn(I*(c*x+1)^2/(c^2*x^2-1)/((c*x+1)^2/(-c^2*x^2+1)+1))^2*arctanh(c*x)^2+3/4*I/c*b^3*Pi*d^
2*csgn(I*(c*x+1)^2/(c^2*x^2-1))*csgn(I*(c*x+1)^2/(c^2*x^2-1)/((c*x+1)^2/(-c^2*x^2+1)+1))^2*arctanh(c*x)^2-3/2*
I/c*b^3*Pi*d^2*csgn(I*(c*x+1)/(-c^2*x^2+1)^(1/2))*csgn(I*(c*x+1)^2/(c^2*x^2-1))^2*arctanh(c*x)^2-1/2*I/c^3*b^3
*e^2*Pi*csgn(I*(c*x+1)/(-c^2*x^2+1)^(1/2))*csgn(I*(c*x+1)^2/(c^2*x^2-1))^2*arctanh(c*x)^2+1/4*I/c^3*b^3*e^2*Pi
*csgn(I*(c*x+1)^2/(c^2*x^2-1))*csgn(I*(c*x+1)^2/(c^2*x^2-1)/((c*x+1)^2/(-c^2*x^2+1)+1))^2*arctanh(c*x)^2-1/4*I
/c^3*b^3*e^2*Pi*csgn(I/((c*x+1)^2/(-c^2*x^2+1)+1))*csgn(I*(c*x+1)^2/(c^2*x^2-1)/((c*x+1)^2/(-c^2*x^2+1)+1))^2*
arctanh(c*x)^2-1/4*I/c^3*b^3*e^2*Pi*csgn(I*(c*x+1)/(-c^2*x^2+1)^(1/2))^2*csgn(I*(c*x+1)^2/(c^2*x^2-1))*arctanh
(c*x)^2+3/4*I/c^2*b^3*e*Pi*d*csgn(I*(c*x+1)^2/(c^2*x^2-1)/((c*x+1)^2/(-c^2*x^2+1)+1))^3*arctanh(c*x)^2+3/4*I/c
^2*b^3*e*Pi*d*csgn(I*(c*x+1)^2/(c^2*x^2-1))^3*arctanh(c*x)^2-3/2*I/c^2*b^3*e*Pi*d*csgn(I/((c*x+1)^2/(-c^2*x^2+
1)+1))^3*arctanh(c*x)^2+3/2*I/c^2*b^3*e*Pi*d*csgn(I/((c*x+1)^2/(-c^2*x^2+1)+1))^2*arctanh(c*x)^2

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*a^3*e^2*x^3 + a^3*d*e*x^2 + 3/2*(2*x^2*arctanh(c*x) + c*(2*x/c^2 - log(c*x + 1)/c^3 + log(c*x - 1)/c^3))*a
^2*b*d*e + 1/2*(2*x^3*arctanh(c*x) + c*(x^2/c^2 + log(c^2*x^2 - 1)/c^4))*a^2*b*e^2 + a^3*d^2*x + 3/2*(2*c*x*ar
ctanh(c*x) + log(-c^2*x^2 + 1))*a^2*b*d^2/c - 1/24*((b^3*c^3*e^2*x^3 + 3*b^3*c^3*d*e*x^2 + 3*b^3*c^3*d^2*x - (
3*c^2*d^2 + 3*c*d*e + e^2)*b^3)*log(-c*x + 1)^3 - 3*(2*a*b^2*c^3*e^2*x^3 + (6*a*b^2*c^3*d*e + b^3*c^2*e^2)*x^2
+ 6*(a*b^2*c^3*d^2 + b^3*c^2*d*e)*x + (b^3*c^3*e^2*x^3 + 3*b^3*c^3*d*e*x^2 + 3*b^3*c^3*d^2*x + (3*c^2*d^2 - 3
*c*d*e + e^2)*b^3)*log(c*x + 1))*log(-c*x + 1)^2)/c^3 - integrate(-1/8*((b^3*c^3*e^2*x^3 - b^3*c^2*d^2 + (2*c^
3*d*e - c^2*e^2)*b^3*x^2 + (c^3*d^2 - 2*c^2*d*e)*b^3*x)*log(c*x + 1)^3 + 6*(a*b^2*c^3*e^2*x^3 - a*b^2*c^2*d^2
+ (2*c^3*d*e - c^2*e^2)*a*b^2*x^2 + (c^3*d^2 - 2*c^2*d*e)*a*b^2*x)*log(c*x + 1)^2 - (4*a*b^2*c^3*e^2*x^3 + 2*(
6*a*b^2*c^3*d*e + b^3*c^2*e^2)*x^2 + 3*(b^3*c^3*e^2*x^3 - b^3*c^2*d^2 + (2*c^3*d*e - c^2*e^2)*b^3*x^2 + (c^3*d
^2 - 2*c^2*d*e)*b^3*x)*log(c*x + 1)^2 + 12*(a*b^2*c^3*d^2 + b^3*c^2*d*e)*x - 2*(6*a*b^2*c^2*d^2 - (3*c^2*d^2 -
3*c*d*e + e^2)*b^3 - (6*a*b^2*c^3*e^2 + b^3*c^3*e^2)*x^3 - 3*(b^3*c^3*d*e + 2*(2*c^3*d*e - c^2*e^2)*a*b^2)*x^
2 - 3*(b^3*c^3*d^2 + 2*(c^3*d^2 - 2*c^2*d*e)*a*b^2)*x)*log(c*x + 1))*log(-c*x + 1))/(c^3*x - c^2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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