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

Optimal. Leaf size=265 \[ \frac{b \text{PolyLog}\left (2,1-\frac{2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{c^3 d^3}+\frac{b^2 \text{PolyLog}\left (3,1-\frac{2}{c x+1}\right )}{2 c^3 d^3}+\frac{7 b \left (a+b \tanh ^{-1}(c x)\right )}{4 c^3 d^3 (c x+1)}-\frac{b \left (a+b \tanh ^{-1}(c x)\right )}{4 c^3 d^3 (c x+1)^2}+\frac{2 \left (a+b \tanh ^{-1}(c x)\right )^2}{c^3 d^3 (c x+1)}-\frac{\left (a+b \tanh ^{-1}(c x)\right )^2}{2 c^3 d^3 (c x+1)^2}-\frac{7 \left (a+b \tanh ^{-1}(c x)\right )^2}{8 c^3 d^3}-\frac{\log \left (\frac{2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )^2}{c^3 d^3}+\frac{13 b^2}{16 c^3 d^3 (c x+1)}-\frac{b^2}{16 c^3 d^3 (c x+1)^2}-\frac{13 b^2 \tanh ^{-1}(c x)}{16 c^3 d^3} \]

[Out]

-b^2/(16*c^3*d^3*(1 + c*x)^2) + (13*b^2)/(16*c^3*d^3*(1 + c*x)) - (13*b^2*ArcTanh[c*x])/(16*c^3*d^3) - (b*(a +
b*ArcTanh[c*x]))/(4*c^3*d^3*(1 + c*x)^2) + (7*b*(a + b*ArcTanh[c*x]))/(4*c^3*d^3*(1 + c*x)) - (7*(a + b*ArcTa
nh[c*x])^2)/(8*c^3*d^3) - (a + b*ArcTanh[c*x])^2/(2*c^3*d^3*(1 + c*x)^2) + (2*(a + b*ArcTanh[c*x])^2)/(c^3*d^3
*(1 + c*x)) - ((a + b*ArcTanh[c*x])^2*Log[2/(1 + c*x)])/(c^3*d^3) + (b*(a + b*ArcTanh[c*x])*PolyLog[2, 1 - 2/(
1 + c*x)])/(c^3*d^3) + (b^2*PolyLog[3, 1 - 2/(1 + c*x)])/(2*c^3*d^3)

________________________________________________________________________________________

Rubi [A]  time = 0.543568, antiderivative size = 265, normalized size of antiderivative = 1., number of steps used = 26, number of rules used = 10, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.454, Rules used = {5940, 5928, 5926, 627, 44, 207, 5948, 5918, 6056, 6610} \[ \frac{b \text{PolyLog}\left (2,1-\frac{2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{c^3 d^3}+\frac{b^2 \text{PolyLog}\left (3,1-\frac{2}{c x+1}\right )}{2 c^3 d^3}+\frac{7 b \left (a+b \tanh ^{-1}(c x)\right )}{4 c^3 d^3 (c x+1)}-\frac{b \left (a+b \tanh ^{-1}(c x)\right )}{4 c^3 d^3 (c x+1)^2}+\frac{2 \left (a+b \tanh ^{-1}(c x)\right )^2}{c^3 d^3 (c x+1)}-\frac{\left (a+b \tanh ^{-1}(c x)\right )^2}{2 c^3 d^3 (c x+1)^2}-\frac{7 \left (a+b \tanh ^{-1}(c x)\right )^2}{8 c^3 d^3}-\frac{\log \left (\frac{2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )^2}{c^3 d^3}+\frac{13 b^2}{16 c^3 d^3 (c x+1)}-\frac{b^2}{16 c^3 d^3 (c x+1)^2}-\frac{13 b^2 \tanh ^{-1}(c x)}{16 c^3 d^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-b^2/(16*c^3*d^3*(1 + c*x)^2) + (13*b^2)/(16*c^3*d^3*(1 + c*x)) - (13*b^2*ArcTanh[c*x])/(16*c^3*d^3) - (b*(a +
b*ArcTanh[c*x]))/(4*c^3*d^3*(1 + c*x)^2) + (7*b*(a + b*ArcTanh[c*x]))/(4*c^3*d^3*(1 + c*x)) - (7*(a + b*ArcTa
nh[c*x])^2)/(8*c^3*d^3) - (a + b*ArcTanh[c*x])^2/(2*c^3*d^3*(1 + c*x)^2) + (2*(a + b*ArcTanh[c*x])^2)/(c^3*d^3
*(1 + c*x)) - ((a + b*ArcTanh[c*x])^2*Log[2/(1 + c*x)])/(c^3*d^3) + (b*(a + b*ArcTanh[c*x])*PolyLog[2, 1 - 2/(
1 + c*x)])/(c^3*d^3) + (b^2*PolyLog[3, 1 - 2/(1 + c*x)])/(2*c^3*d^3)

Rule 5940

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_))^(q_.), x_Symbol] :> Int[E
xpandIntegrand[(a + b*ArcTanh[c*x])^p, (f*x)^m*(d + e*x)^q, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && IGtQ[
p, 0] && IntegerQ[q] && (GtQ[q, 0] || NeQ[a, 0] || IntegerQ[m])

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 5926

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

Rule 627

Int[((d_) + (e_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[(d + e*x)^(m + p)*(a/d + (c*x)/e)^
p, x] /; FreeQ[{a, c, d, e, m, p}, x] && EqQ[c*d^2 + a*e^2, 0] && (IntegerQ[p] || (GtQ[a, 0] && GtQ[d, 0] && I
ntegerQ[m + p]))

Rule 44

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}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

Rule 207

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

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 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 6056

Int[(Log[u_]*((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[((a + b*ArcTa
nh[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 \frac{x^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{(d+c d x)^3} \, dx &=\int \left (\frac{\left (a+b \tanh ^{-1}(c x)\right )^2}{c^2 d^3 (1+c x)^3}-\frac{2 \left (a+b \tanh ^{-1}(c x)\right )^2}{c^2 d^3 (1+c x)^2}+\frac{\left (a+b \tanh ^{-1}(c x)\right )^2}{c^2 d^3 (1+c x)}\right ) \, dx\\ &=\frac{\int \frac{\left (a+b \tanh ^{-1}(c x)\right )^2}{(1+c x)^3} \, dx}{c^2 d^3}+\frac{\int \frac{\left (a+b \tanh ^{-1}(c x)\right )^2}{1+c x} \, dx}{c^2 d^3}-\frac{2 \int \frac{\left (a+b \tanh ^{-1}(c x)\right )^2}{(1+c x)^2} \, dx}{c^2 d^3}\\ &=-\frac{\left (a+b \tanh ^{-1}(c x)\right )^2}{2 c^3 d^3 (1+c x)^2}+\frac{2 \left (a+b \tanh ^{-1}(c x)\right )^2}{c^3 d^3 (1+c x)}-\frac{\left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac{2}{1+c x}\right )}{c^3 d^3}+\frac{b \int \left (\frac{a+b \tanh ^{-1}(c x)}{2 (1+c x)^3}+\frac{a+b \tanh ^{-1}(c x)}{4 (1+c x)^2}-\frac{a+b \tanh ^{-1}(c x)}{4 \left (-1+c^2 x^2\right )}\right ) \, dx}{c^2 d^3}+\frac{(2 b) \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 d^3}-\frac{(4 b) \int \left (\frac{a+b \tanh ^{-1}(c x)}{2 (1+c x)^2}-\frac{a+b \tanh ^{-1}(c x)}{2 \left (-1+c^2 x^2\right )}\right ) \, dx}{c^2 d^3}\\ &=-\frac{\left (a+b \tanh ^{-1}(c x)\right )^2}{2 c^3 d^3 (1+c x)^2}+\frac{2 \left (a+b \tanh ^{-1}(c x)\right )^2}{c^3 d^3 (1+c x)}-\frac{\left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac{2}{1+c x}\right )}{c^3 d^3}+\frac{b \left (a+b \tanh ^{-1}(c x)\right ) \text{Li}_2\left (1-\frac{2}{1+c x}\right )}{c^3 d^3}+\frac{b \int \frac{a+b \tanh ^{-1}(c x)}{(1+c x)^2} \, dx}{4 c^2 d^3}-\frac{b \int \frac{a+b \tanh ^{-1}(c x)}{-1+c^2 x^2} \, dx}{4 c^2 d^3}+\frac{b \int \frac{a+b \tanh ^{-1}(c x)}{(1+c x)^3} \, dx}{2 c^2 d^3}-\frac{(2 b) \int \frac{a+b \tanh ^{-1}(c x)}{(1+c x)^2} \, dx}{c^2 d^3}+\frac{(2 b) \int \frac{a+b \tanh ^{-1}(c x)}{-1+c^2 x^2} \, dx}{c^2 d^3}-\frac{b^2 \int \frac{\text{Li}_2\left (1-\frac{2}{1+c x}\right )}{1-c^2 x^2} \, dx}{c^2 d^3}\\ &=-\frac{b \left (a+b \tanh ^{-1}(c x)\right )}{4 c^3 d^3 (1+c x)^2}+\frac{7 b \left (a+b \tanh ^{-1}(c x)\right )}{4 c^3 d^3 (1+c x)}-\frac{7 \left (a+b \tanh ^{-1}(c x)\right )^2}{8 c^3 d^3}-\frac{\left (a+b \tanh ^{-1}(c x)\right )^2}{2 c^3 d^3 (1+c x)^2}+\frac{2 \left (a+b \tanh ^{-1}(c x)\right )^2}{c^3 d^3 (1+c x)}-\frac{\left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac{2}{1+c x}\right )}{c^3 d^3}+\frac{b \left (a+b \tanh ^{-1}(c x)\right ) \text{Li}_2\left (1-\frac{2}{1+c x}\right )}{c^3 d^3}+\frac{b^2 \text{Li}_3\left (1-\frac{2}{1+c x}\right )}{2 c^3 d^3}+\frac{b^2 \int \frac{1}{(1+c x)^2 \left (1-c^2 x^2\right )} \, dx}{4 c^2 d^3}+\frac{b^2 \int \frac{1}{(1+c x) \left (1-c^2 x^2\right )} \, dx}{4 c^2 d^3}-\frac{\left (2 b^2\right ) \int \frac{1}{(1+c x) \left (1-c^2 x^2\right )} \, dx}{c^2 d^3}\\ &=-\frac{b \left (a+b \tanh ^{-1}(c x)\right )}{4 c^3 d^3 (1+c x)^2}+\frac{7 b \left (a+b \tanh ^{-1}(c x)\right )}{4 c^3 d^3 (1+c x)}-\frac{7 \left (a+b \tanh ^{-1}(c x)\right )^2}{8 c^3 d^3}-\frac{\left (a+b \tanh ^{-1}(c x)\right )^2}{2 c^3 d^3 (1+c x)^2}+\frac{2 \left (a+b \tanh ^{-1}(c x)\right )^2}{c^3 d^3 (1+c x)}-\frac{\left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac{2}{1+c x}\right )}{c^3 d^3}+\frac{b \left (a+b \tanh ^{-1}(c x)\right ) \text{Li}_2\left (1-\frac{2}{1+c x}\right )}{c^3 d^3}+\frac{b^2 \text{Li}_3\left (1-\frac{2}{1+c x}\right )}{2 c^3 d^3}+\frac{b^2 \int \frac{1}{(1-c x) (1+c x)^3} \, dx}{4 c^2 d^3}+\frac{b^2 \int \frac{1}{(1-c x) (1+c x)^2} \, dx}{4 c^2 d^3}-\frac{\left (2 b^2\right ) \int \frac{1}{(1-c x) (1+c x)^2} \, dx}{c^2 d^3}\\ &=-\frac{b \left (a+b \tanh ^{-1}(c x)\right )}{4 c^3 d^3 (1+c x)^2}+\frac{7 b \left (a+b \tanh ^{-1}(c x)\right )}{4 c^3 d^3 (1+c x)}-\frac{7 \left (a+b \tanh ^{-1}(c x)\right )^2}{8 c^3 d^3}-\frac{\left (a+b \tanh ^{-1}(c x)\right )^2}{2 c^3 d^3 (1+c x)^2}+\frac{2 \left (a+b \tanh ^{-1}(c x)\right )^2}{c^3 d^3 (1+c x)}-\frac{\left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac{2}{1+c x}\right )}{c^3 d^3}+\frac{b \left (a+b \tanh ^{-1}(c x)\right ) \text{Li}_2\left (1-\frac{2}{1+c x}\right )}{c^3 d^3}+\frac{b^2 \text{Li}_3\left (1-\frac{2}{1+c x}\right )}{2 c^3 d^3}+\frac{b^2 \int \left (\frac{1}{2 (1+c x)^2}-\frac{1}{2 \left (-1+c^2 x^2\right )}\right ) \, dx}{4 c^2 d^3}+\frac{b^2 \int \left (\frac{1}{2 (1+c x)^3}+\frac{1}{4 (1+c x)^2}-\frac{1}{4 \left (-1+c^2 x^2\right )}\right ) \, dx}{4 c^2 d^3}-\frac{\left (2 b^2\right ) \int \left (\frac{1}{2 (1+c x)^2}-\frac{1}{2 \left (-1+c^2 x^2\right )}\right ) \, dx}{c^2 d^3}\\ &=-\frac{b^2}{16 c^3 d^3 (1+c x)^2}+\frac{13 b^2}{16 c^3 d^3 (1+c x)}-\frac{b \left (a+b \tanh ^{-1}(c x)\right )}{4 c^3 d^3 (1+c x)^2}+\frac{7 b \left (a+b \tanh ^{-1}(c x)\right )}{4 c^3 d^3 (1+c x)}-\frac{7 \left (a+b \tanh ^{-1}(c x)\right )^2}{8 c^3 d^3}-\frac{\left (a+b \tanh ^{-1}(c x)\right )^2}{2 c^3 d^3 (1+c x)^2}+\frac{2 \left (a+b \tanh ^{-1}(c x)\right )^2}{c^3 d^3 (1+c x)}-\frac{\left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac{2}{1+c x}\right )}{c^3 d^3}+\frac{b \left (a+b \tanh ^{-1}(c x)\right ) \text{Li}_2\left (1-\frac{2}{1+c x}\right )}{c^3 d^3}+\frac{b^2 \text{Li}_3\left (1-\frac{2}{1+c x}\right )}{2 c^3 d^3}-\frac{b^2 \int \frac{1}{-1+c^2 x^2} \, dx}{16 c^2 d^3}-\frac{b^2 \int \frac{1}{-1+c^2 x^2} \, dx}{8 c^2 d^3}+\frac{b^2 \int \frac{1}{-1+c^2 x^2} \, dx}{c^2 d^3}\\ &=-\frac{b^2}{16 c^3 d^3 (1+c x)^2}+\frac{13 b^2}{16 c^3 d^3 (1+c x)}-\frac{13 b^2 \tanh ^{-1}(c x)}{16 c^3 d^3}-\frac{b \left (a+b \tanh ^{-1}(c x)\right )}{4 c^3 d^3 (1+c x)^2}+\frac{7 b \left (a+b \tanh ^{-1}(c x)\right )}{4 c^3 d^3 (1+c x)}-\frac{7 \left (a+b \tanh ^{-1}(c x)\right )^2}{8 c^3 d^3}-\frac{\left (a+b \tanh ^{-1}(c x)\right )^2}{2 c^3 d^3 (1+c x)^2}+\frac{2 \left (a+b \tanh ^{-1}(c x)\right )^2}{c^3 d^3 (1+c x)}-\frac{\left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac{2}{1+c x}\right )}{c^3 d^3}+\frac{b \left (a+b \tanh ^{-1}(c x)\right ) \text{Li}_2\left (1-\frac{2}{1+c x}\right )}{c^3 d^3}+\frac{b^2 \text{Li}_3\left (1-\frac{2}{1+c x}\right )}{2 c^3 d^3}\\ \end{align*}

Mathematica [A]  time = 1.39504, size = 310, normalized size = 1.17 \[ \frac{a b \left (16 \text{PolyLog}\left (2,-e^{-2 \tanh ^{-1}(c x)}\right )-12 \sinh \left (2 \tanh ^{-1}(c x)\right )+\sinh \left (4 \tanh ^{-1}(c x)\right )+12 \cosh \left (2 \tanh ^{-1}(c x)\right )-\cosh \left (4 \tanh ^{-1}(c x)\right )+4 \tanh ^{-1}(c x) \left (-8 \log \left (e^{-2 \tanh ^{-1}(c x)}+1\right )-6 \sinh \left (2 \tanh ^{-1}(c x)\right )+\sinh \left (4 \tanh ^{-1}(c x)\right )+6 \cosh \left (2 \tanh ^{-1}(c x)\right )-\cosh \left (4 \tanh ^{-1}(c x)\right )\right )\right )+16 b^2 \left (\tanh ^{-1}(c x) \text{PolyLog}\left (2,-e^{-2 \tanh ^{-1}(c x)}\right )+\frac{1}{2} \text{PolyLog}\left (3,-e^{-2 \tanh ^{-1}(c x)}\right )+\frac{1}{64} \left (\sinh \left (2 \tanh ^{-1}(c x)\right )-\cosh \left (2 \tanh ^{-1}(c x)\right )\right ) \left (-\sinh \left (2 \tanh ^{-1}(c x)\right )+\cosh \left (2 \tanh ^{-1}(c x)\right )+4 \tanh ^{-1}(c x) \left (-\sinh \left (2 \tanh ^{-1}(c x)\right )+\cosh \left (2 \tanh ^{-1}(c x)\right )-12\right )+8 \tanh ^{-1}(c x)^2 \left (\left (8 \log \left (e^{-2 \tanh ^{-1}(c x)}+1\right )-1\right ) \sinh \left (2 \tanh ^{-1}(c x)\right )+\left (8 \log \left (e^{-2 \tanh ^{-1}(c x)}+1\right )+1\right ) \cosh \left (2 \tanh ^{-1}(c x)\right )-6\right )-24\right )\right )+\frac{32 a^2}{c x+1}-\frac{8 a^2}{(c x+1)^2}+16 a^2 \log (c x+1)}{16 c^3 d^3} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((-8*a^2)/(1 + c*x)^2 + (32*a^2)/(1 + c*x) + 16*a^2*Log[1 + c*x] + 16*b^2*(ArcTanh[c*x]*PolyLog[2, -E^(-2*ArcT
anh[c*x])] + PolyLog[3, -E^(-2*ArcTanh[c*x])]/2 + ((-Cosh[2*ArcTanh[c*x]] + Sinh[2*ArcTanh[c*x]])*(-24 + Cosh[
2*ArcTanh[c*x]] + 4*ArcTanh[c*x]*(-12 + Cosh[2*ArcTanh[c*x]] - Sinh[2*ArcTanh[c*x]]) - Sinh[2*ArcTanh[c*x]] +
8*ArcTanh[c*x]^2*(-6 + Cosh[2*ArcTanh[c*x]]*(1 + 8*Log[1 + E^(-2*ArcTanh[c*x])]) + (-1 + 8*Log[1 + E^(-2*ArcTa
nh[c*x])])*Sinh[2*ArcTanh[c*x]])))/64) + a*b*(12*Cosh[2*ArcTanh[c*x]] - Cosh[4*ArcTanh[c*x]] + 16*PolyLog[2, -
E^(-2*ArcTanh[c*x])] - 12*Sinh[2*ArcTanh[c*x]] + Sinh[4*ArcTanh[c*x]] + 4*ArcTanh[c*x]*(6*Cosh[2*ArcTanh[c*x]]
- Cosh[4*ArcTanh[c*x]] - 8*Log[1 + E^(-2*ArcTanh[c*x])] - 6*Sinh[2*ArcTanh[c*x]] + Sinh[4*ArcTanh[c*x]])))/(1
6*c^3*d^3)

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Maple [C]  time = 0.345, size = 1241, normalized size = 4.7 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*I/c^3*b^2/d^3*Pi*arctanh(c*x)^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))-1/64*b^2/c^3/d^3/(c*x+1)^2+3/8*b^2/c^3/d^3/(c*x+1)-1/2/c^3*a^2
/d^3/(c*x+1)^2+2/c^3*a^2/d^3/(c*x+1)+1/2/c^3*b^2/d^3*polylog(3,-(c*x+1)^2/(-c^2*x^2+1))+1/c^3*a^2/d^3*ln(c*x+1
)-7/8/c^3*b^2/d^3*arctanh(c*x)^2+2/3/c^3*b^2/d^3*arctanh(c*x)^3-1/2*I/c^3*b^2/d^3*Pi*arctanh(c*x)^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-I/c^3*b^2/d^3*Pi*arctanh(c
*x)^2*csgn(I*(c*x+1)^2/(c^2*x^2-1))^2*csgn(I*(c*x+1)/(-c^2*x^2+1)^(1/2))+1/2*I/c^3*b^2/d^3*Pi*arctanh(c*x)^2*c
sgn(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))^2-1/2*I/c^3*b^2/d^3*Pi*a
rctanh(c*x)^2*csgn(I*(c*x+1)^2/(c^2*x^2-1))*csgn(I*(c*x+1)/(-c^2*x^2+1)^(1/2))^2-3/8/c^2*b^2/d^3/(c*x+1)*x-1/2
/c^3*a*b/d^3*ln(c*x+1)^2+7/8/c^3*a*b/d^3*ln(c*x-1)-7/8/c^3*a*b/d^3*ln(c*x+1)-1/c^3*b^2/d^3*arctanh(c*x)^2*ln(2
)-2/c^3*b^2/d^3*arctanh(c*x)^2*ln((c*x+1)/(-c^2*x^2+1)^(1/2))+1/c^3*b^2/d^3*arctanh(c*x)^2*ln(c*x+1)-1/2/c^3*b
^2/d^3*arctanh(c*x)^2/(c*x+1)^2+2/c^3*b^2/d^3*arctanh(c*x)^2/(c*x+1)-1/c^3*b^2/d^3*arctanh(c*x)*polylog(2,-(c*
x+1)^2/(-c^2*x^2+1))+3/4/c^3*b^2/d^3*arctanh(c*x)/(c*x+1)-1/16/c^3*b^2/d^3*arctanh(c*x)/(c*x+1)^2-1/c^3*a*b/d^
3*dilog(1/2+1/2*c*x)-1/4/c^3*a*b/d^3/(c*x+1)^2+7/4/c^3*a*b/d^3/(c*x+1)-1/64/c*b^2/d^3/(c*x+1)^2*x^2+1/32/c^2*b
^2/d^3/(c*x+1)^2*x-1/2*I/c^3*b^2/d^3*Pi*arctanh(c*x)^2*csgn(I*(c*x+1)^2/(c^2*x^2-1))^3-1/2*I/c^3*b^2/d^3*Pi*ar
ctanh(c*x)^2*csgn(I*(c*x+1)^2/(c^2*x^2-1)/((c*x+1)^2/(-c^2*x^2+1)+1))^3-1/c^3*a*b/d^3*ln(-1/2*c*x+1/2)*ln(1/2+
1/2*c*x)-3/4/c^2*b^2/d^3*arctanh(c*x)/(c*x+1)*x-1/16/c*b^2/d^3*arctanh(c*x)/(c*x+1)^2*x^2+1/8/c^2*b^2/d^3*arct
anh(c*x)/(c*x+1)^2*x-1/c^3*a*b/d^3*arctanh(c*x)/(c*x+1)^2+4/c^3*a*b/d^3*arctanh(c*x)/(c*x+1)+2/c^3*a*b/d^3*arc
tanh(c*x)*ln(c*x+1)+1/c^3*a*b/d^3*ln(-1/2*c*x+1/2)*ln(c*x+1)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{2} \, a^{2}{\left (\frac{4 \, c x + 3}{c^{5} d^{3} x^{2} + 2 \, c^{4} d^{3} x + c^{3} d^{3}} + \frac{2 \, \log \left (c x + 1\right )}{c^{3} d^{3}}\right )} + \frac{{\left (4 \, b^{2} c x + 3 \, b^{2} + 2 \,{\left (b^{2} c^{2} x^{2} + 2 \, b^{2} c x + b^{2}\right )} \log \left (c x + 1\right )\right )} \log \left (-c x + 1\right )^{2}}{8 \,{\left (c^{5} d^{3} x^{2} + 2 \, c^{4} d^{3} x + c^{3} d^{3}\right )}} - \int -\frac{{\left (b^{2} c^{3} x^{3} - b^{2} c^{2} x^{2}\right )} \log \left (c x + 1\right )^{2} + 4 \,{\left (a b c^{3} x^{3} - a b c^{2} x^{2}\right )} \log \left (c x + 1\right ) -{\left (4 \, a b c^{3} x^{3} + 7 \, b^{2} c x - 4 \,{\left (a b c^{2} - b^{2} c^{2}\right )} x^{2} + 3 \, b^{2} + 2 \,{\left (2 \, b^{2} c^{3} x^{3} + 2 \, b^{2} c^{2} x^{2} + 3 \, b^{2} c x + b^{2}\right )} \log \left (c x + 1\right )\right )} \log \left (-c x + 1\right )}{4 \,{\left (c^{6} d^{3} x^{4} + 2 \, c^{5} d^{3} x^{3} - 2 \, c^{3} d^{3} x - c^{2} d^{3}\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*a^2*((4*c*x + 3)/(c^5*d^3*x^2 + 2*c^4*d^3*x + c^3*d^3) + 2*log(c*x + 1)/(c^3*d^3)) + 1/8*(4*b^2*c*x + 3*b^
2 + 2*(b^2*c^2*x^2 + 2*b^2*c*x + b^2)*log(c*x + 1))*log(-c*x + 1)^2/(c^5*d^3*x^2 + 2*c^4*d^3*x + c^3*d^3) - in
tegrate(-1/4*((b^2*c^3*x^3 - b^2*c^2*x^2)*log(c*x + 1)^2 + 4*(a*b*c^3*x^3 - a*b*c^2*x^2)*log(c*x + 1) - (4*a*b
*c^3*x^3 + 7*b^2*c*x - 4*(a*b*c^2 - b^2*c^2)*x^2 + 3*b^2 + 2*(2*b^2*c^3*x^3 + 2*b^2*c^2*x^2 + 3*b^2*c*x + b^2)
*log(c*x + 1))*log(-c*x + 1))/(c^6*d^3*x^4 + 2*c^5*d^3*x^3 - 2*c^3*d^3*x - c^2*d^3), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{a^{2} x^{2}}{c^{3} x^{3} + 3 c^{2} x^{2} + 3 c x + 1}\, dx + \int \frac{b^{2} x^{2} \operatorname{atanh}^{2}{\left (c x \right )}}{c^{3} x^{3} + 3 c^{2} x^{2} + 3 c x + 1}\, dx + \int \frac{2 a b x^{2} \operatorname{atanh}{\left (c x \right )}}{c^{3} x^{3} + 3 c^{2} x^{2} + 3 c x + 1}\, dx}{d^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(Integral(a**2*x**2/(c**3*x**3 + 3*c**2*x**2 + 3*c*x + 1), x) + Integral(b**2*x**2*atanh(c*x)**2/(c**3*x**3 +
3*c**2*x**2 + 3*c*x + 1), x) + Integral(2*a*b*x**2*atanh(c*x)/(c**3*x**3 + 3*c**2*x**2 + 3*c*x + 1), x))/d**3

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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