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

Optimal. Leaf size=356 \[ -\frac{2 b^2 d^2 \text{PolyLog}\left (2,1-\frac{2}{1-c x}\right )}{5 c^4}+\frac{1}{6} c^2 d^2 x^6 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac{2 b d^2 x^2 \left (a+b \tanh ^{-1}(c x)\right )}{5 c^2}+\frac{5 a b d^2 x}{6 c^3}-\frac{d^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{60 c^4}-\frac{4 b d^2 \log \left (\frac{2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{5 c^4}+\frac{2}{5} c d^2 x^5 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac{1}{15} b c d^2 x^5 \left (a+b \tanh ^{-1}(c x)\right )+\frac{1}{4} d^2 x^4 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac{1}{5} b d^2 x^4 \left (a+b \tanh ^{-1}(c x)\right )+\frac{5 b d^2 x^3 \left (a+b \tanh ^{-1}(c x)\right )}{18 c}+\frac{31 b^2 d^2 x^2}{180 c^2}+\frac{53 b^2 d^2 \log \left (1-c^2 x^2\right )}{90 c^4}+\frac{3 b^2 d^2 x}{5 c^3}+\frac{5 b^2 d^2 x \tanh ^{-1}(c x)}{6 c^3}-\frac{3 b^2 d^2 \tanh ^{-1}(c x)}{5 c^4}+\frac{b^2 d^2 x^3}{15 c}+\frac{1}{60} b^2 d^2 x^4 \]

[Out]

(5*a*b*d^2*x)/(6*c^3) + (3*b^2*d^2*x)/(5*c^3) + (31*b^2*d^2*x^2)/(180*c^2) + (b^2*d^2*x^3)/(15*c) + (b^2*d^2*x
^4)/60 - (3*b^2*d^2*ArcTanh[c*x])/(5*c^4) + (5*b^2*d^2*x*ArcTanh[c*x])/(6*c^3) + (2*b*d^2*x^2*(a + b*ArcTanh[c
*x]))/(5*c^2) + (5*b*d^2*x^3*(a + b*ArcTanh[c*x]))/(18*c) + (b*d^2*x^4*(a + b*ArcTanh[c*x]))/5 + (b*c*d^2*x^5*
(a + b*ArcTanh[c*x]))/15 - (d^2*(a + b*ArcTanh[c*x])^2)/(60*c^4) + (d^2*x^4*(a + b*ArcTanh[c*x])^2)/4 + (2*c*d
^2*x^5*(a + b*ArcTanh[c*x])^2)/5 + (c^2*d^2*x^6*(a + b*ArcTanh[c*x])^2)/6 - (4*b*d^2*(a + b*ArcTanh[c*x])*Log[
2/(1 - c*x)])/(5*c^4) + (53*b^2*d^2*Log[1 - c^2*x^2])/(90*c^4) - (2*b^2*d^2*PolyLog[2, 1 - 2/(1 - c*x)])/(5*c^
4)

________________________________________________________________________________________

Rubi [A]  time = 1.02416, antiderivative size = 356, normalized size of antiderivative = 1., number of steps used = 43, number of rules used = 15, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.682, Rules used = {5940, 5916, 5980, 266, 43, 5910, 260, 5948, 302, 206, 321, 5984, 5918, 2402, 2315} \[ -\frac{2 b^2 d^2 \text{PolyLog}\left (2,1-\frac{2}{1-c x}\right )}{5 c^4}+\frac{1}{6} c^2 d^2 x^6 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac{2 b d^2 x^2 \left (a+b \tanh ^{-1}(c x)\right )}{5 c^2}+\frac{5 a b d^2 x}{6 c^3}-\frac{d^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{60 c^4}-\frac{4 b d^2 \log \left (\frac{2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{5 c^4}+\frac{2}{5} c d^2 x^5 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac{1}{15} b c d^2 x^5 \left (a+b \tanh ^{-1}(c x)\right )+\frac{1}{4} d^2 x^4 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac{1}{5} b d^2 x^4 \left (a+b \tanh ^{-1}(c x)\right )+\frac{5 b d^2 x^3 \left (a+b \tanh ^{-1}(c x)\right )}{18 c}+\frac{31 b^2 d^2 x^2}{180 c^2}+\frac{53 b^2 d^2 \log \left (1-c^2 x^2\right )}{90 c^4}+\frac{3 b^2 d^2 x}{5 c^3}+\frac{5 b^2 d^2 x \tanh ^{-1}(c x)}{6 c^3}-\frac{3 b^2 d^2 \tanh ^{-1}(c x)}{5 c^4}+\frac{b^2 d^2 x^3}{15 c}+\frac{1}{60} b^2 d^2 x^4 \]

Antiderivative was successfully verified.

[In]

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

[Out]

(5*a*b*d^2*x)/(6*c^3) + (3*b^2*d^2*x)/(5*c^3) + (31*b^2*d^2*x^2)/(180*c^2) + (b^2*d^2*x^3)/(15*c) + (b^2*d^2*x
^4)/60 - (3*b^2*d^2*ArcTanh[c*x])/(5*c^4) + (5*b^2*d^2*x*ArcTanh[c*x])/(6*c^3) + (2*b*d^2*x^2*(a + b*ArcTanh[c
*x]))/(5*c^2) + (5*b*d^2*x^3*(a + b*ArcTanh[c*x]))/(18*c) + (b*d^2*x^4*(a + b*ArcTanh[c*x]))/5 + (b*c*d^2*x^5*
(a + b*ArcTanh[c*x]))/15 - (d^2*(a + b*ArcTanh[c*x])^2)/(60*c^4) + (d^2*x^4*(a + b*ArcTanh[c*x])^2)/4 + (2*c*d
^2*x^5*(a + b*ArcTanh[c*x])^2)/5 + (c^2*d^2*x^6*(a + b*ArcTanh[c*x])^2)/6 - (4*b*d^2*(a + b*ArcTanh[c*x])*Log[
2/(1 - c*x)])/(5*c^4) + (53*b^2*d^2*Log[1 - c^2*x^2])/(90*c^4) - (2*b^2*d^2*PolyLog[2, 1 - 2/(1 - c*x)])/(5*c^
4)

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 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 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 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 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 302

Int[(x_)^(m_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Int[PolynomialDivide[x^m, a + b*x^n, x], x] /; FreeQ[{a,
b}, x] && IGtQ[m, 0] && IGtQ[n, 0] && GtQ[m, 2*n - 1]

Rule 206

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

Rule 321

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

Rule 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]

Rubi steps

\begin{align*} \int x^3 (d+c d x)^2 \left (a+b \tanh ^{-1}(c x)\right )^2 \, dx &=\int \left (d^2 x^3 \left (a+b \tanh ^{-1}(c x)\right )^2+2 c d^2 x^4 \left (a+b \tanh ^{-1}(c x)\right )^2+c^2 d^2 x^5 \left (a+b \tanh ^{-1}(c x)\right )^2\right ) \, dx\\ &=d^2 \int x^3 \left (a+b \tanh ^{-1}(c x)\right )^2 \, dx+\left (2 c d^2\right ) \int x^4 \left (a+b \tanh ^{-1}(c x)\right )^2 \, dx+\left (c^2 d^2\right ) \int x^5 \left (a+b \tanh ^{-1}(c x)\right )^2 \, dx\\ &=\frac{1}{4} d^2 x^4 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac{2}{5} c d^2 x^5 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac{1}{6} c^2 d^2 x^6 \left (a+b \tanh ^{-1}(c x)\right )^2-\frac{1}{2} \left (b c d^2\right ) \int \frac{x^4 \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx-\frac{1}{5} \left (4 b c^2 d^2\right ) \int \frac{x^5 \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx-\frac{1}{3} \left (b c^3 d^2\right ) \int \frac{x^6 \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx\\ &=\frac{1}{4} d^2 x^4 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac{2}{5} c d^2 x^5 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac{1}{6} c^2 d^2 x^6 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac{1}{5} \left (4 b d^2\right ) \int x^3 \left (a+b \tanh ^{-1}(c x)\right ) \, dx-\frac{1}{5} \left (4 b d^2\right ) \int \frac{x^3 \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx+\frac{\left (b d^2\right ) \int x^2 \left (a+b \tanh ^{-1}(c x)\right ) \, dx}{2 c}-\frac{\left (b d^2\right ) \int \frac{x^2 \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx}{2 c}+\frac{1}{3} \left (b c d^2\right ) \int x^4 \left (a+b \tanh ^{-1}(c x)\right ) \, dx-\frac{1}{3} \left (b c d^2\right ) \int \frac{x^4 \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx\\ &=\frac{b d^2 x^3 \left (a+b \tanh ^{-1}(c x)\right )}{6 c}+\frac{1}{5} b d^2 x^4 \left (a+b \tanh ^{-1}(c x)\right )+\frac{1}{15} b c d^2 x^5 \left (a+b \tanh ^{-1}(c x)\right )+\frac{1}{4} d^2 x^4 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac{2}{5} c d^2 x^5 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac{1}{6} c^2 d^2 x^6 \left (a+b \tanh ^{-1}(c x)\right )^2-\frac{1}{6} \left (b^2 d^2\right ) \int \frac{x^3}{1-c^2 x^2} \, dx+\frac{\left (b d^2\right ) \int \left (a+b \tanh ^{-1}(c x)\right ) \, dx}{2 c^3}-\frac{\left (b d^2\right ) \int \frac{a+b \tanh ^{-1}(c x)}{1-c^2 x^2} \, dx}{2 c^3}+\frac{\left (4 b d^2\right ) \int x \left (a+b \tanh ^{-1}(c x)\right ) \, dx}{5 c^2}-\frac{\left (4 b d^2\right ) \int \frac{x \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx}{5 c^2}+\frac{\left (b d^2\right ) \int x^2 \left (a+b \tanh ^{-1}(c x)\right ) \, dx}{3 c}-\frac{\left (b d^2\right ) \int \frac{x^2 \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx}{3 c}-\frac{1}{5} \left (b^2 c d^2\right ) \int \frac{x^4}{1-c^2 x^2} \, dx-\frac{1}{15} \left (b^2 c^2 d^2\right ) \int \frac{x^5}{1-c^2 x^2} \, dx\\ &=\frac{a b d^2 x}{2 c^3}+\frac{2 b d^2 x^2 \left (a+b \tanh ^{-1}(c x)\right )}{5 c^2}+\frac{5 b d^2 x^3 \left (a+b \tanh ^{-1}(c x)\right )}{18 c}+\frac{1}{5} b d^2 x^4 \left (a+b \tanh ^{-1}(c x)\right )+\frac{1}{15} b c d^2 x^5 \left (a+b \tanh ^{-1}(c x)\right )+\frac{3 d^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{20 c^4}+\frac{1}{4} d^2 x^4 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac{2}{5} c d^2 x^5 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac{1}{6} c^2 d^2 x^6 \left (a+b \tanh ^{-1}(c x)\right )^2-\frac{1}{12} \left (b^2 d^2\right ) \operatorname{Subst}\left (\int \frac{x}{1-c^2 x} \, dx,x,x^2\right )-\frac{1}{9} \left (b^2 d^2\right ) \int \frac{x^3}{1-c^2 x^2} \, dx+\frac{\left (b d^2\right ) \int \left (a+b \tanh ^{-1}(c x)\right ) \, dx}{3 c^3}-\frac{\left (b d^2\right ) \int \frac{a+b \tanh ^{-1}(c x)}{1-c^2 x^2} \, dx}{3 c^3}-\frac{\left (4 b d^2\right ) \int \frac{a+b \tanh ^{-1}(c x)}{1-c x} \, dx}{5 c^3}+\frac{\left (b^2 d^2\right ) \int \tanh ^{-1}(c x) \, dx}{2 c^3}-\frac{\left (2 b^2 d^2\right ) \int \frac{x^2}{1-c^2 x^2} \, dx}{5 c}-\frac{1}{5} \left (b^2 c d^2\right ) \int \left (-\frac{1}{c^4}-\frac{x^2}{c^2}+\frac{1}{c^4 \left (1-c^2 x^2\right )}\right ) \, dx-\frac{1}{30} \left (b^2 c^2 d^2\right ) \operatorname{Subst}\left (\int \frac{x^2}{1-c^2 x} \, dx,x,x^2\right )\\ &=\frac{5 a b d^2 x}{6 c^3}+\frac{3 b^2 d^2 x}{5 c^3}+\frac{b^2 d^2 x^3}{15 c}+\frac{b^2 d^2 x \tanh ^{-1}(c x)}{2 c^3}+\frac{2 b d^2 x^2 \left (a+b \tanh ^{-1}(c x)\right )}{5 c^2}+\frac{5 b d^2 x^3 \left (a+b \tanh ^{-1}(c x)\right )}{18 c}+\frac{1}{5} b d^2 x^4 \left (a+b \tanh ^{-1}(c x)\right )+\frac{1}{15} b c d^2 x^5 \left (a+b \tanh ^{-1}(c x)\right )-\frac{d^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{60 c^4}+\frac{1}{4} d^2 x^4 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac{2}{5} c d^2 x^5 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac{1}{6} c^2 d^2 x^6 \left (a+b \tanh ^{-1}(c x)\right )^2-\frac{4 b d^2 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac{2}{1-c x}\right )}{5 c^4}-\frac{1}{18} \left (b^2 d^2\right ) \operatorname{Subst}\left (\int \frac{x}{1-c^2 x} \, dx,x,x^2\right )-\frac{1}{12} \left (b^2 d^2\right ) \operatorname{Subst}\left (\int \left (-\frac{1}{c^2}-\frac{1}{c^2 \left (-1+c^2 x\right )}\right ) \, dx,x,x^2\right )-\frac{\left (b^2 d^2\right ) \int \frac{1}{1-c^2 x^2} \, dx}{5 c^3}+\frac{\left (b^2 d^2\right ) \int \tanh ^{-1}(c x) \, dx}{3 c^3}-\frac{\left (2 b^2 d^2\right ) \int \frac{1}{1-c^2 x^2} \, dx}{5 c^3}+\frac{\left (4 b^2 d^2\right ) \int \frac{\log \left (\frac{2}{1-c x}\right )}{1-c^2 x^2} \, dx}{5 c^3}-\frac{\left (b^2 d^2\right ) \int \frac{x}{1-c^2 x^2} \, dx}{2 c^2}-\frac{1}{30} \left (b^2 c^2 d^2\right ) \operatorname{Subst}\left (\int \left (-\frac{1}{c^4}-\frac{x}{c^2}-\frac{1}{c^4 \left (-1+c^2 x\right )}\right ) \, dx,x,x^2\right )\\ &=\frac{5 a b d^2 x}{6 c^3}+\frac{3 b^2 d^2 x}{5 c^3}+\frac{7 b^2 d^2 x^2}{60 c^2}+\frac{b^2 d^2 x^3}{15 c}+\frac{1}{60} b^2 d^2 x^4-\frac{3 b^2 d^2 \tanh ^{-1}(c x)}{5 c^4}+\frac{5 b^2 d^2 x \tanh ^{-1}(c x)}{6 c^3}+\frac{2 b d^2 x^2 \left (a+b \tanh ^{-1}(c x)\right )}{5 c^2}+\frac{5 b d^2 x^3 \left (a+b \tanh ^{-1}(c x)\right )}{18 c}+\frac{1}{5} b d^2 x^4 \left (a+b \tanh ^{-1}(c x)\right )+\frac{1}{15} b c d^2 x^5 \left (a+b \tanh ^{-1}(c x)\right )-\frac{d^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{60 c^4}+\frac{1}{4} d^2 x^4 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac{2}{5} c d^2 x^5 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac{1}{6} c^2 d^2 x^6 \left (a+b \tanh ^{-1}(c x)\right )^2-\frac{4 b d^2 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac{2}{1-c x}\right )}{5 c^4}+\frac{11 b^2 d^2 \log \left (1-c^2 x^2\right )}{30 c^4}-\frac{1}{18} \left (b^2 d^2\right ) \operatorname{Subst}\left (\int \left (-\frac{1}{c^2}-\frac{1}{c^2 \left (-1+c^2 x\right )}\right ) \, dx,x,x^2\right )-\frac{\left (4 b^2 d^2\right ) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1-c x}\right )}{5 c^4}-\frac{\left (b^2 d^2\right ) \int \frac{x}{1-c^2 x^2} \, dx}{3 c^2}\\ &=\frac{5 a b d^2 x}{6 c^3}+\frac{3 b^2 d^2 x}{5 c^3}+\frac{31 b^2 d^2 x^2}{180 c^2}+\frac{b^2 d^2 x^3}{15 c}+\frac{1}{60} b^2 d^2 x^4-\frac{3 b^2 d^2 \tanh ^{-1}(c x)}{5 c^4}+\frac{5 b^2 d^2 x \tanh ^{-1}(c x)}{6 c^3}+\frac{2 b d^2 x^2 \left (a+b \tanh ^{-1}(c x)\right )}{5 c^2}+\frac{5 b d^2 x^3 \left (a+b \tanh ^{-1}(c x)\right )}{18 c}+\frac{1}{5} b d^2 x^4 \left (a+b \tanh ^{-1}(c x)\right )+\frac{1}{15} b c d^2 x^5 \left (a+b \tanh ^{-1}(c x)\right )-\frac{d^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{60 c^4}+\frac{1}{4} d^2 x^4 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac{2}{5} c d^2 x^5 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac{1}{6} c^2 d^2 x^6 \left (a+b \tanh ^{-1}(c x)\right )^2-\frac{4 b d^2 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac{2}{1-c x}\right )}{5 c^4}+\frac{53 b^2 d^2 \log \left (1-c^2 x^2\right )}{90 c^4}-\frac{2 b^2 d^2 \text{Li}_2\left (1-\frac{2}{1-c x}\right )}{5 c^4}\\ \end{align*}

Mathematica [A]  time = 1.04655, size = 329, normalized size = 0.92 \[ \frac{d^2 \left (72 b^2 \text{PolyLog}\left (2,-e^{-2 \tanh ^{-1}(c x)}\right )+30 a^2 c^6 x^6+72 a^2 c^5 x^5+45 a^2 c^4 x^4+12 a b c^5 x^5+36 a b c^4 x^4+50 a b c^3 x^3+72 a b c^2 x^2+72 a b \log \left (c^2 x^2-1\right )+2 b \tanh ^{-1}(c x) \left (3 a c^4 x^4 \left (10 c^2 x^2+24 c x+15\right )+b \left (6 c^5 x^5+18 c^4 x^4+25 c^3 x^3+36 c^2 x^2+75 c x-54\right )-72 b \log \left (e^{-2 \tanh ^{-1}(c x)}+1\right )\right )+150 a b c x+75 a b \log (1-c x)-75 a b \log (c x+1)-108 a b+3 b^2 c^4 x^4+12 b^2 c^3 x^3+31 b^2 c^2 x^2+106 b^2 \log \left (1-c^2 x^2\right )+3 b^2 \left (10 c^6 x^6+24 c^5 x^5+15 c^4 x^4-49\right ) \tanh ^{-1}(c x)^2+108 b^2 c x-34 b^2\right )}{180 c^4} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(d^2*(-108*a*b - 34*b^2 + 150*a*b*c*x + 108*b^2*c*x + 72*a*b*c^2*x^2 + 31*b^2*c^2*x^2 + 50*a*b*c^3*x^3 + 12*b^
2*c^3*x^3 + 45*a^2*c^4*x^4 + 36*a*b*c^4*x^4 + 3*b^2*c^4*x^4 + 72*a^2*c^5*x^5 + 12*a*b*c^5*x^5 + 30*a^2*c^6*x^6
+ 3*b^2*(-49 + 15*c^4*x^4 + 24*c^5*x^5 + 10*c^6*x^6)*ArcTanh[c*x]^2 + 2*b*ArcTanh[c*x]*(3*a*c^4*x^4*(15 + 24*
c*x + 10*c^2*x^2) + b*(-54 + 75*c*x + 36*c^2*x^2 + 25*c^3*x^3 + 18*c^4*x^4 + 6*c^5*x^5) - 72*b*Log[1 + E^(-2*A
rcTanh[c*x])]) + 75*a*b*Log[1 - c*x] - 75*a*b*Log[1 + c*x] + 106*b^2*Log[1 - c^2*x^2] + 72*a*b*Log[-1 + c^2*x^
2] + 72*b^2*PolyLog[2, -E^(-2*ArcTanh[c*x])]))/(180*c^4)

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Maple [A]  time = 0.054, size = 569, normalized size = 1.6 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*d^2*a^2*x^4+3/5*b^2*d^2*x/c^3+31/180*b^2*d^2*x^2/c^2+1/15*b^2*d^2*x^3/c+8/9/c^4*d^2*b^2*ln(c*x-1)+13/45/c^
4*d^2*b^2*ln(c*x+1)+1/6*c^2*d^2*a^2*x^6+2/5*c*d^2*a^2*x^5+1/5*d^2*a*b*x^4-2/5/c^4*d^2*b^2*dilog(1/2+1/2*c*x)+1
/240/c^4*d^2*b^2*ln(c*x+1)^2+49/240/c^4*d^2*b^2*ln(c*x-1)^2+1/5*d^2*b^2*arctanh(c*x)*x^4+1/4*d^2*b^2*arctanh(c
*x)^2*x^4+1/15*c*d^2*b^2*arctanh(c*x)*x^5+5/18/c*d^2*a*b*x^3+1/6*c^2*d^2*b^2*arctanh(c*x)^2*x^6+2/5*c*d^2*b^2*
arctanh(c*x)^2*x^5+1/2*d^2*a*b*arctanh(c*x)*x^4+5/18/c*d^2*b^2*arctanh(c*x)*x^3+2/5/c^2*d^2*b^2*arctanh(c*x)*x
^2-1/60/c^4*d^2*a*b*ln(c*x+1)+49/60/c^4*d^2*b^2*arctanh(c*x)*ln(c*x-1)-1/60/c^4*d^2*b^2*arctanh(c*x)*ln(c*x+1)
-49/120/c^4*d^2*b^2*ln(c*x-1)*ln(1/2+1/2*c*x)+49/60/c^4*d^2*a*b*ln(c*x-1)-1/120/c^4*d^2*b^2*ln(-1/2*c*x+1/2)*l
n(c*x+1)+2/5/c^2*d^2*a*b*x^2+1/15*c*d^2*a*b*x^5+1/60*b^2*d^2*x^4+5/6*a*b*d^2*x/c^3+5/6*b^2*d^2*x*arctanh(c*x)/
c^3+1/120/c^4*d^2*b^2*ln(-1/2*c*x+1/2)*ln(1/2+1/2*c*x)+1/3*c^2*d^2*a*b*arctanh(c*x)*x^6+4/5*c*d^2*a*b*arctanh(
c*x)*x^5

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Maxima [B]  time = 2.24754, size = 1034, normalized size = 2.9 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*a^2*c^2*d^2*x^6 + 2/5*a^2*c*d^2*x^5 + 1/4*b^2*d^2*x^4*arctanh(c*x)^2 + 1/4*a^2*d^2*x^4 + 1/90*(30*x^6*arct
anh(c*x) + c*(2*(3*c^4*x^5 + 5*c^2*x^3 + 15*x)/c^6 - 15*log(c*x + 1)/c^7 + 15*log(c*x - 1)/c^7))*a*b*c^2*d^2 +
1/5*(4*x^5*arctanh(c*x) + c*((c^2*x^4 + 2*x^2)/c^4 + 2*log(c^2*x^2 - 1)/c^6))*a*b*c*d^2 + 1/12*(6*x^4*arctanh
(c*x) + c*(2*(c^2*x^3 + 3*x)/c^4 - 3*log(c*x + 1)/c^5 + 3*log(c*x - 1)/c^5))*a*b*d^2 + 1/48*(4*c*(2*(c^2*x^3 +
3*x)/c^4 - 3*log(c*x + 1)/c^5 + 3*log(c*x - 1)/c^5)*arctanh(c*x) + (4*c^2*x^2 - 2*(3*log(c*x - 1) - 8)*log(c*
x + 1) + 3*log(c*x + 1)^2 + 3*log(c*x - 1)^2 + 16*log(c*x - 1))/c^4)*b^2*d^2 + 2/5*(log(c*x + 1)*log(-1/2*c*x
+ 1/2) + dilog(1/2*c*x + 1/2))*b^2*d^2/c^4 - 2/45*b^2*d^2*log(c*x + 1)/c^4 + 5/9*b^2*d^2*log(c*x - 1)/c^4 + 1/
360*(6*b^2*c^4*d^2*x^4 + 24*b^2*c^3*d^2*x^3 + 32*b^2*c^2*d^2*x^2 + 216*b^2*c*d^2*x + 3*(5*b^2*c^6*d^2*x^6 + 12
*b^2*c^5*d^2*x^5 + 7*b^2*d^2)*log(c*x + 1)^2 + 3*(5*b^2*c^6*d^2*x^6 + 12*b^2*c^5*d^2*x^5 - 17*b^2*d^2)*log(-c*
x + 1)^2 + 4*(3*b^2*c^5*d^2*x^5 + 9*b^2*c^4*d^2*x^4 + 5*b^2*c^3*d^2*x^3 + 18*b^2*c^2*d^2*x^2 + 15*b^2*c*d^2*x)
*log(c*x + 1) - 2*(6*b^2*c^5*d^2*x^5 + 18*b^2*c^4*d^2*x^4 + 10*b^2*c^3*d^2*x^3 + 36*b^2*c^2*d^2*x^2 + 30*b^2*c
*d^2*x + 3*(5*b^2*c^6*d^2*x^6 + 12*b^2*c^5*d^2*x^5 + 7*b^2*d^2)*log(c*x + 1))*log(-c*x + 1))/c^4

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (a^{2} c^{2} d^{2} x^{5} + 2 \, a^{2} c d^{2} x^{4} + a^{2} d^{2} x^{3} +{\left (b^{2} c^{2} d^{2} x^{5} + 2 \, b^{2} c d^{2} x^{4} + b^{2} d^{2} x^{3}\right )} \operatorname{artanh}\left (c x\right )^{2} + 2 \,{\left (a b c^{2} d^{2} x^{5} + 2 \, a b c d^{2} x^{4} + a b d^{2} x^{3}\right )} \operatorname{artanh}\left (c x\right ), x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(a^2*c^2*d^2*x^5 + 2*a^2*c*d^2*x^4 + a^2*d^2*x^3 + (b^2*c^2*d^2*x^5 + 2*b^2*c*d^2*x^4 + b^2*d^2*x^3)*a
rctanh(c*x)^2 + 2*(a*b*c^2*d^2*x^5 + 2*a*b*c*d^2*x^4 + a*b*d^2*x^3)*arctanh(c*x), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} d^{2} \left (\int a^{2} x^{3}\, dx + \int 2 a^{2} c x^{4}\, dx + \int a^{2} c^{2} x^{5}\, dx + \int b^{2} x^{3} \operatorname{atanh}^{2}{\left (c x \right )}\, dx + \int 2 a b x^{3} \operatorname{atanh}{\left (c x \right )}\, dx + \int 2 b^{2} c x^{4} \operatorname{atanh}^{2}{\left (c x \right )}\, dx + \int b^{2} c^{2} x^{5} \operatorname{atanh}^{2}{\left (c x \right )}\, dx + \int 4 a b c x^{4} \operatorname{atanh}{\left (c x \right )}\, dx + \int 2 a b c^{2} x^{5} \operatorname{atanh}{\left (c x \right )}\, dx\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

d**2*(Integral(a**2*x**3, x) + Integral(2*a**2*c*x**4, x) + Integral(a**2*c**2*x**5, x) + Integral(b**2*x**3*a
tanh(c*x)**2, x) + Integral(2*a*b*x**3*atanh(c*x), x) + Integral(2*b**2*c*x**4*atanh(c*x)**2, x) + Integral(b*
*2*c**2*x**5*atanh(c*x)**2, x) + Integral(4*a*b*c*x**4*atanh(c*x), x) + Integral(2*a*b*c**2*x**5*atanh(c*x), x
))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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