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

Optimal. Leaf size=377 \[ \frac {1}{6} c^3 d^3 x^6 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{60 c^3}-\frac {28 b d^3 \log \left (\frac {2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{15 c^3}+\frac {3}{5} c^2 d^3 x^5 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {1}{15} b c^2 d^3 x^5 \left (a+b \tanh ^{-1}(c x)\right )+\frac {11 a b d^3 x}{6 c^2}+\frac {3}{4} c d^3 x^4 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {3}{10} b c d^3 x^4 \left (a+b \tanh ^{-1}(c x)\right )+\frac {1}{3} d^3 x^3 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {11}{18} b d^3 x^3 \left (a+b \tanh ^{-1}(c x)\right )+\frac {14 b d^3 x^2 \left (a+b \tanh ^{-1}(c x)\right )}{15 c}-\frac {14 b^2 d^3 \text {Li}_2\left (1-\frac {2}{1-c x}\right )}{15 c^3}-\frac {37 b^2 d^3 \tanh ^{-1}(c x)}{30 c^3}+\frac {37 b^2 d^3 x}{30 c^2}+\frac {11 b^2 d^3 x \tanh ^{-1}(c x)}{6 c^2}+\frac {113 b^2 d^3 \log \left (1-c^2 x^2\right )}{90 c^3}+\frac {1}{60} b^2 c d^3 x^4+\frac {61 b^2 d^3 x^2}{180 c}+\frac {1}{10} b^2 d^3 x^3 \]

[Out]

11/6*a*b*d^3*x/c^2+37/30*b^2*d^3*x/c^2+61/180*b^2*d^3*x^2/c+1/10*b^2*d^3*x^3+1/60*b^2*c*d^3*x^4-37/30*b^2*d^3*
arctanh(c*x)/c^3+11/6*b^2*d^3*x*arctanh(c*x)/c^2+14/15*b*d^3*x^2*(a+b*arctanh(c*x))/c+11/18*b*d^3*x^3*(a+b*arc
tanh(c*x))+3/10*b*c*d^3*x^4*(a+b*arctanh(c*x))+1/15*b*c^2*d^3*x^5*(a+b*arctanh(c*x))+1/60*d^3*(a+b*arctanh(c*x
))^2/c^3+1/3*d^3*x^3*(a+b*arctanh(c*x))^2+3/4*c*d^3*x^4*(a+b*arctanh(c*x))^2+3/5*c^2*d^3*x^5*(a+b*arctanh(c*x)
)^2+1/6*c^3*d^3*x^6*(a+b*arctanh(c*x))^2-28/15*b*d^3*(a+b*arctanh(c*x))*ln(2/(-c*x+1))/c^3+113/90*b^2*d^3*ln(-
c^2*x^2+1)/c^3-14/15*b^2*d^3*polylog(2,1-2/(-c*x+1))/c^3

________________________________________________________________________________________

Rubi [A]  time = 1.23, antiderivative size = 377, normalized size of antiderivative = 1.00, number of steps used = 52, number of rules used = 15, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.682, Rules used = {5940, 5916, 5980, 321, 206, 5984, 5918, 2402, 2315, 266, 43, 5910, 260, 5948, 302} \[ -\frac {14 b^2 d^3 \text {PolyLog}\left (2,1-\frac {2}{1-c x}\right )}{15 c^3}+\frac {1}{6} c^3 d^3 x^6 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {3}{5} c^2 d^3 x^5 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {1}{15} b c^2 d^3 x^5 \left (a+b \tanh ^{-1}(c x)\right )+\frac {11 a b d^3 x}{6 c^2}+\frac {d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{60 c^3}-\frac {28 b d^3 \log \left (\frac {2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{15 c^3}+\frac {3}{4} c d^3 x^4 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {3}{10} b c d^3 x^4 \left (a+b \tanh ^{-1}(c x)\right )+\frac {1}{3} d^3 x^3 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {11}{18} b d^3 x^3 \left (a+b \tanh ^{-1}(c x)\right )+\frac {14 b d^3 x^2 \left (a+b \tanh ^{-1}(c x)\right )}{15 c}+\frac {113 b^2 d^3 \log \left (1-c^2 x^2\right )}{90 c^3}+\frac {37 b^2 d^3 x}{30 c^2}+\frac {11 b^2 d^3 x \tanh ^{-1}(c x)}{6 c^2}-\frac {37 b^2 d^3 \tanh ^{-1}(c x)}{30 c^3}+\frac {1}{60} b^2 c d^3 x^4+\frac {61 b^2 d^3 x^2}{180 c}+\frac {1}{10} b^2 d^3 x^3 \]

Antiderivative was successfully verified.

[In]

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

[Out]

(11*a*b*d^3*x)/(6*c^2) + (37*b^2*d^3*x)/(30*c^2) + (61*b^2*d^3*x^2)/(180*c) + (b^2*d^3*x^3)/10 + (b^2*c*d^3*x^
4)/60 - (37*b^2*d^3*ArcTanh[c*x])/(30*c^3) + (11*b^2*d^3*x*ArcTanh[c*x])/(6*c^2) + (14*b*d^3*x^2*(a + b*ArcTan
h[c*x]))/(15*c) + (11*b*d^3*x^3*(a + b*ArcTanh[c*x]))/18 + (3*b*c*d^3*x^4*(a + b*ArcTanh[c*x]))/10 + (b*c^2*d^
3*x^5*(a + b*ArcTanh[c*x]))/15 + (d^3*(a + b*ArcTanh[c*x])^2)/(60*c^3) + (d^3*x^3*(a + b*ArcTanh[c*x])^2)/3 +
(3*c*d^3*x^4*(a + b*ArcTanh[c*x])^2)/4 + (3*c^2*d^3*x^5*(a + b*ArcTanh[c*x])^2)/5 + (c^3*d^3*x^6*(a + b*ArcTan
h[c*x])^2)/6 - (28*b*d^3*(a + b*ArcTanh[c*x])*Log[2/(1 - c*x)])/(15*c^3) + (113*b^2*d^3*Log[1 - c^2*x^2])/(90*
c^3) - (14*b^2*d^3*PolyLog[2, 1 - 2/(1 - c*x)])/(15*c^3)

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 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 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 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 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 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 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 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 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 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 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 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 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 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 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]

Rubi steps

\begin {align*} \int x^2 (d+c d x)^3 \left (a+b \tanh ^{-1}(c x)\right )^2 \, dx &=\int \left (d^3 x^2 \left (a+b \tanh ^{-1}(c x)\right )^2+3 c d^3 x^3 \left (a+b \tanh ^{-1}(c x)\right )^2+3 c^2 d^3 x^4 \left (a+b \tanh ^{-1}(c x)\right )^2+c^3 d^3 x^5 \left (a+b \tanh ^{-1}(c x)\right )^2\right ) \, dx\\ &=d^3 \int x^2 \left (a+b \tanh ^{-1}(c x)\right )^2 \, dx+\left (3 c d^3\right ) \int x^3 \left (a+b \tanh ^{-1}(c x)\right )^2 \, dx+\left (3 c^2 d^3\right ) \int x^4 \left (a+b \tanh ^{-1}(c x)\right )^2 \, dx+\left (c^3 d^3\right ) \int x^5 \left (a+b \tanh ^{-1}(c x)\right )^2 \, dx\\ &=\frac {1}{3} d^3 x^3 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {3}{4} c d^3 x^4 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {3}{5} c^2 d^3 x^5 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {1}{6} c^3 d^3 x^6 \left (a+b \tanh ^{-1}(c x)\right )^2-\frac {1}{3} \left (2 b c d^3\right ) \int \frac {x^3 \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx-\frac {1}{2} \left (3 b c^2 d^3\right ) \int \frac {x^4 \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx-\frac {1}{5} \left (6 b c^3 d^3\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^4 d^3\right ) \int \frac {x^6 \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx\\ &=\frac {1}{3} d^3 x^3 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {3}{4} c d^3 x^4 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {3}{5} c^2 d^3 x^5 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {1}{6} c^3 d^3 x^6 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {1}{2} \left (3 b d^3\right ) \int x^2 \left (a+b \tanh ^{-1}(c x)\right ) \, dx-\frac {1}{2} \left (3 b d^3\right ) \int \frac {x^2 \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx+\frac {\left (2 b d^3\right ) \int x \left (a+b \tanh ^{-1}(c x)\right ) \, dx}{3 c}-\frac {\left (2 b d^3\right ) \int \frac {x \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx}{3 c}+\frac {1}{5} \left (6 b c d^3\right ) \int x^3 \left (a+b \tanh ^{-1}(c x)\right ) \, dx-\frac {1}{5} \left (6 b c d^3\right ) \int \frac {x^3 \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx+\frac {1}{3} \left (b c^2 d^3\right ) \int x^4 \left (a+b \tanh ^{-1}(c x)\right ) \, dx-\frac {1}{3} \left (b c^2 d^3\right ) \int \frac {x^4 \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx\\ &=\frac {b d^3 x^2 \left (a+b \tanh ^{-1}(c x)\right )}{3 c}+\frac {1}{2} b d^3 x^3 \left (a+b \tanh ^{-1}(c x)\right )+\frac {3}{10} b c d^3 x^4 \left (a+b \tanh ^{-1}(c x)\right )+\frac {1}{15} b c^2 d^3 x^5 \left (a+b \tanh ^{-1}(c x)\right )+\frac {d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{3 c^3}+\frac {1}{3} d^3 x^3 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {3}{4} c d^3 x^4 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {3}{5} c^2 d^3 x^5 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {1}{6} c^3 d^3 x^6 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {1}{3} \left (b d^3\right ) \int x^2 \left (a+b \tanh ^{-1}(c x)\right ) \, dx-\frac {1}{3} \left (b d^3\right ) \int \frac {x^2 \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx-\frac {1}{3} \left (b^2 d^3\right ) \int \frac {x^2}{1-c^2 x^2} \, dx-\frac {\left (2 b d^3\right ) \int \frac {a+b \tanh ^{-1}(c x)}{1-c x} \, dx}{3 c^2}+\frac {\left (3 b d^3\right ) \int \left (a+b \tanh ^{-1}(c x)\right ) \, dx}{2 c^2}-\frac {\left (3 b d^3\right ) \int \frac {a+b \tanh ^{-1}(c x)}{1-c^2 x^2} \, dx}{2 c^2}+\frac {\left (6 b d^3\right ) \int x \left (a+b \tanh ^{-1}(c x)\right ) \, dx}{5 c}-\frac {\left (6 b d^3\right ) \int \frac {x \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx}{5 c}-\frac {1}{2} \left (b^2 c d^3\right ) \int \frac {x^3}{1-c^2 x^2} \, dx-\frac {1}{10} \left (3 b^2 c^2 d^3\right ) \int \frac {x^4}{1-c^2 x^2} \, dx-\frac {1}{15} \left (b^2 c^3 d^3\right ) \int \frac {x^5}{1-c^2 x^2} \, dx\\ &=\frac {3 a b d^3 x}{2 c^2}+\frac {b^2 d^3 x}{3 c^2}+\frac {14 b d^3 x^2 \left (a+b \tanh ^{-1}(c x)\right )}{15 c}+\frac {11}{18} b d^3 x^3 \left (a+b \tanh ^{-1}(c x)\right )+\frac {3}{10} b c d^3 x^4 \left (a+b \tanh ^{-1}(c x)\right )+\frac {1}{15} b c^2 d^3 x^5 \left (a+b \tanh ^{-1}(c x)\right )+\frac {11 d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{60 c^3}+\frac {1}{3} d^3 x^3 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {3}{4} c d^3 x^4 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {3}{5} c^2 d^3 x^5 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {1}{6} c^3 d^3 x^6 \left (a+b \tanh ^{-1}(c x)\right )^2-\frac {2 b d^3 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right )}{3 c^3}-\frac {1}{5} \left (3 b^2 d^3\right ) \int \frac {x^2}{1-c^2 x^2} \, dx+\frac {\left (b d^3\right ) \int \left (a+b \tanh ^{-1}(c x)\right ) \, dx}{3 c^2}-\frac {\left (b d^3\right ) \int \frac {a+b \tanh ^{-1}(c x)}{1-c^2 x^2} \, dx}{3 c^2}-\frac {\left (6 b d^3\right ) \int \frac {a+b \tanh ^{-1}(c x)}{1-c x} \, dx}{5 c^2}-\frac {\left (b^2 d^3\right ) \int \frac {1}{1-c^2 x^2} \, dx}{3 c^2}+\frac {\left (2 b^2 d^3\right ) \int \frac {\log \left (\frac {2}{1-c x}\right )}{1-c^2 x^2} \, dx}{3 c^2}+\frac {\left (3 b^2 d^3\right ) \int \tanh ^{-1}(c x) \, dx}{2 c^2}-\frac {1}{9} \left (b^2 c d^3\right ) \int \frac {x^3}{1-c^2 x^2} \, dx-\frac {1}{4} \left (b^2 c d^3\right ) \operatorname {Subst}\left (\int \frac {x}{1-c^2 x} \, dx,x,x^2\right )-\frac {1}{10} \left (3 b^2 c^2 d^3\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^3 d^3\right ) \operatorname {Subst}\left (\int \frac {x^2}{1-c^2 x} \, dx,x,x^2\right )\\ &=\frac {11 a b d^3 x}{6 c^2}+\frac {37 b^2 d^3 x}{30 c^2}+\frac {1}{10} b^2 d^3 x^3-\frac {b^2 d^3 \tanh ^{-1}(c x)}{3 c^3}+\frac {3 b^2 d^3 x \tanh ^{-1}(c x)}{2 c^2}+\frac {14 b d^3 x^2 \left (a+b \tanh ^{-1}(c x)\right )}{15 c}+\frac {11}{18} b d^3 x^3 \left (a+b \tanh ^{-1}(c x)\right )+\frac {3}{10} b c d^3 x^4 \left (a+b \tanh ^{-1}(c x)\right )+\frac {1}{15} b c^2 d^3 x^5 \left (a+b \tanh ^{-1}(c x)\right )+\frac {d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{60 c^3}+\frac {1}{3} d^3 x^3 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {3}{4} c d^3 x^4 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {3}{5} c^2 d^3 x^5 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {1}{6} c^3 d^3 x^6 \left (a+b \tanh ^{-1}(c x)\right )^2-\frac {28 b d^3 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right )}{15 c^3}-\frac {\left (2 b^2 d^3\right ) \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-c x}\right )}{3 c^3}-\frac {\left (3 b^2 d^3\right ) \int \frac {1}{1-c^2 x^2} \, dx}{10 c^2}+\frac {\left (b^2 d^3\right ) \int \tanh ^{-1}(c x) \, dx}{3 c^2}-\frac {\left (3 b^2 d^3\right ) \int \frac {1}{1-c^2 x^2} \, dx}{5 c^2}+\frac {\left (6 b^2 d^3\right ) \int \frac {\log \left (\frac {2}{1-c x}\right )}{1-c^2 x^2} \, dx}{5 c^2}-\frac {\left (3 b^2 d^3\right ) \int \frac {x}{1-c^2 x^2} \, dx}{2 c}-\frac {1}{18} \left (b^2 c d^3\right ) \operatorname {Subst}\left (\int \frac {x}{1-c^2 x} \, dx,x,x^2\right )-\frac {1}{4} \left (b^2 c d^3\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 {1}{30} \left (b^2 c^3 d^3\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 {11 a b d^3 x}{6 c^2}+\frac {37 b^2 d^3 x}{30 c^2}+\frac {17 b^2 d^3 x^2}{60 c}+\frac {1}{10} b^2 d^3 x^3+\frac {1}{60} b^2 c d^3 x^4-\frac {37 b^2 d^3 \tanh ^{-1}(c x)}{30 c^3}+\frac {11 b^2 d^3 x \tanh ^{-1}(c x)}{6 c^2}+\frac {14 b d^3 x^2 \left (a+b \tanh ^{-1}(c x)\right )}{15 c}+\frac {11}{18} b d^3 x^3 \left (a+b \tanh ^{-1}(c x)\right )+\frac {3}{10} b c d^3 x^4 \left (a+b \tanh ^{-1}(c x)\right )+\frac {1}{15} b c^2 d^3 x^5 \left (a+b \tanh ^{-1}(c x)\right )+\frac {d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{60 c^3}+\frac {1}{3} d^3 x^3 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {3}{4} c d^3 x^4 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {3}{5} c^2 d^3 x^5 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {1}{6} c^3 d^3 x^6 \left (a+b \tanh ^{-1}(c x)\right )^2-\frac {28 b d^3 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right )}{15 c^3}+\frac {31 b^2 d^3 \log \left (1-c^2 x^2\right )}{30 c^3}-\frac {b^2 d^3 \text {Li}_2\left (1-\frac {2}{1-c x}\right )}{3 c^3}-\frac {\left (6 b^2 d^3\right ) \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-c x}\right )}{5 c^3}-\frac {\left (b^2 d^3\right ) \int \frac {x}{1-c^2 x^2} \, dx}{3 c}-\frac {1}{18} \left (b^2 c d^3\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 {11 a b d^3 x}{6 c^2}+\frac {37 b^2 d^3 x}{30 c^2}+\frac {61 b^2 d^3 x^2}{180 c}+\frac {1}{10} b^2 d^3 x^3+\frac {1}{60} b^2 c d^3 x^4-\frac {37 b^2 d^3 \tanh ^{-1}(c x)}{30 c^3}+\frac {11 b^2 d^3 x \tanh ^{-1}(c x)}{6 c^2}+\frac {14 b d^3 x^2 \left (a+b \tanh ^{-1}(c x)\right )}{15 c}+\frac {11}{18} b d^3 x^3 \left (a+b \tanh ^{-1}(c x)\right )+\frac {3}{10} b c d^3 x^4 \left (a+b \tanh ^{-1}(c x)\right )+\frac {1}{15} b c^2 d^3 x^5 \left (a+b \tanh ^{-1}(c x)\right )+\frac {d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{60 c^3}+\frac {1}{3} d^3 x^3 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {3}{4} c d^3 x^4 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {3}{5} c^2 d^3 x^5 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {1}{6} c^3 d^3 x^6 \left (a+b \tanh ^{-1}(c x)\right )^2-\frac {28 b d^3 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right )}{15 c^3}+\frac {113 b^2 d^3 \log \left (1-c^2 x^2\right )}{90 c^3}-\frac {14 b^2 d^3 \text {Li}_2\left (1-\frac {2}{1-c x}\right )}{15 c^3}\\ \end {align*}

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Mathematica [A]  time = 1.33, size = 356, normalized size = 0.94 \[ \frac {d^3 \left (30 a^2 c^6 x^6+108 a^2 c^5 x^5+135 a^2 c^4 x^4+60 a^2 c^3 x^3+12 a b c^5 x^5+54 a b c^4 x^4+110 a b c^3 x^3+168 a b c^2 x^2+168 a b \log \left (c^2 x^2-1\right )+2 b \tanh ^{-1}(c x) \left (3 a c^3 x^3 \left (10 c^3 x^3+36 c^2 x^2+45 c x+20\right )+b \left (6 c^5 x^5+27 c^4 x^4+55 c^3 x^3+84 c^2 x^2+165 c x-111\right )-168 b \log \left (e^{-2 \tanh ^{-1}(c x)}+1\right )\right )+330 a b c x+165 a b \log (1-c x)-165 a b \log (c x+1)-162 a b+3 b^2 c^4 x^4+18 b^2 c^3 x^3+61 b^2 c^2 x^2+226 b^2 \log \left (1-c^2 x^2\right )+3 b^2 \left (10 c^6 x^6+36 c^5 x^5+45 c^4 x^4+20 c^3 x^3-111\right ) \tanh ^{-1}(c x)^2+168 b^2 \text {Li}_2\left (-e^{-2 \tanh ^{-1}(c x)}\right )+222 b^2 c x-64 b^2\right )}{180 c^3} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(d^3*(-162*a*b - 64*b^2 + 330*a*b*c*x + 222*b^2*c*x + 168*a*b*c^2*x^2 + 61*b^2*c^2*x^2 + 60*a^2*c^3*x^3 + 110*
a*b*c^3*x^3 + 18*b^2*c^3*x^3 + 135*a^2*c^4*x^4 + 54*a*b*c^4*x^4 + 3*b^2*c^4*x^4 + 108*a^2*c^5*x^5 + 12*a*b*c^5
*x^5 + 30*a^2*c^6*x^6 + 3*b^2*(-111 + 20*c^3*x^3 + 45*c^4*x^4 + 36*c^5*x^5 + 10*c^6*x^6)*ArcTanh[c*x]^2 + 2*b*
ArcTanh[c*x]*(3*a*c^3*x^3*(20 + 45*c*x + 36*c^2*x^2 + 10*c^3*x^3) + b*(-111 + 165*c*x + 84*c^2*x^2 + 55*c^3*x^
3 + 27*c^4*x^4 + 6*c^5*x^5) - 168*b*Log[1 + E^(-2*ArcTanh[c*x])]) + 165*a*b*Log[1 - c*x] - 165*a*b*Log[1 + c*x
] + 226*b^2*Log[1 - c^2*x^2] + 168*a*b*Log[-1 + c^2*x^2] + 168*b^2*PolyLog[2, -E^(-2*ArcTanh[c*x])]))/(180*c^3
)

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fricas [F]  time = 0.69, size = 0, normalized size = 0.00 \[ {\rm integral}\left (a^{2} c^{3} d^{3} x^{5} + 3 \, a^{2} c^{2} d^{3} x^{4} + 3 \, a^{2} c d^{3} x^{3} + a^{2} d^{3} x^{2} + {\left (b^{2} c^{3} d^{3} x^{5} + 3 \, b^{2} c^{2} d^{3} x^{4} + 3 \, b^{2} c d^{3} x^{3} + b^{2} d^{3} x^{2}\right )} \operatorname {artanh}\left (c x\right )^{2} + 2 \, {\left (a b c^{3} d^{3} x^{5} + 3 \, a b c^{2} d^{3} x^{4} + 3 \, a b c d^{3} x^{3} + a b d^{3} x^{2}\right )} \operatorname {artanh}\left (c x\right ), x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (c d x + d\right )}^{3} {\left (b \operatorname {artanh}\left (c x\right ) + a\right )}^{2} x^{2}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.06, size = 618, normalized size = 1.64 \[ \frac {11 a b \,d^{3} x}{6 c^{2}}+\frac {d^{3} b^{2} \arctanh \left (c x \right )^{2} x^{3}}{3}+\frac {11 d^{3} b^{2} \arctanh \left (c x \right ) x^{3}}{18}-\frac {14 d^{3} b^{2} \dilog \left (\frac {1}{2}+\frac {c x}{2}\right )}{15 c^{3}}+\frac {11 d^{3} a b \,x^{3}}{18}+\frac {3 c \,d^{3} a^{2} x^{4}}{4}-\frac {d^{3} b^{2} \ln \left (c x +1\right )^{2}}{240 c^{3}}+\frac {23 d^{3} b^{2} \ln \left (c x +1\right )}{36 c^{3}}+\frac {37 d^{3} b^{2} \ln \left (c x -1\right )^{2}}{80 c^{3}}+\frac {337 d^{3} b^{2} \ln \left (c x -1\right )}{180 c^{3}}+\frac {c^{3} d^{3} a^{2} x^{6}}{6}+\frac {3 c^{2} d^{3} a^{2} x^{5}}{5}+\frac {6 c^{2} d^{3} a b \arctanh \left (c x \right ) x^{5}}{5}+\frac {c^{3} d^{3} a b \arctanh \left (c x \right ) x^{6}}{3}+\frac {3 c \,d^{3} a b \arctanh \left (c x \right ) x^{4}}{2}+\frac {b^{2} d^{3} x^{3}}{10}+\frac {d^{3} a b \ln \left (c x +1\right )}{60 c^{3}}+\frac {37 d^{3} a b \ln \left (c x -1\right )}{20 c^{3}}+\frac {c^{2} d^{3} a b \,x^{5}}{15}+\frac {3 c \,d^{3} a b \,x^{4}}{10}+\frac {14 d^{3} a b \,x^{2}}{15 c}+\frac {11 b^{2} d^{3} x \arctanh \left (c x \right )}{6 c^{2}}+\frac {d^{3} b^{2} \arctanh \left (c x \right ) \ln \left (c x +1\right )}{60 c^{3}}-\frac {37 d^{3} b^{2} \ln \left (c x -1\right ) \ln \left (\frac {1}{2}+\frac {c x}{2}\right )}{40 c^{3}}+\frac {d^{3} b^{2} \ln \left (-\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (c x +1\right )}{120 c^{3}}-\frac {d^{3} b^{2} \ln \left (-\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (\frac {1}{2}+\frac {c x}{2}\right )}{120 c^{3}}+\frac {d^{3} a^{2} x^{3}}{3}+\frac {c^{2} d^{3} b^{2} \arctanh \left (c x \right ) x^{5}}{15}+\frac {2 d^{3} a b \arctanh \left (c x \right ) x^{3}}{3}+\frac {14 d^{3} b^{2} \arctanh \left (c x \right ) x^{2}}{15 c}+\frac {3 c^{2} d^{3} b^{2} \arctanh \left (c x \right )^{2} x^{5}}{5}+\frac {3 c \,d^{3} b^{2} \arctanh \left (c x \right )^{2} x^{4}}{4}+\frac {3 c \,d^{3} b^{2} \arctanh \left (c x \right ) x^{4}}{10}+\frac {37 d^{3} b^{2} \arctanh \left (c x \right ) \ln \left (c x -1\right )}{20 c^{3}}+\frac {c^{3} d^{3} b^{2} \arctanh \left (c x \right )^{2} x^{6}}{6}+\frac {37 b^{2} d^{3} x}{30 c^{2}}+\frac {61 b^{2} d^{3} x^{2}}{180 c}+\frac {b^{2} c \,d^{3} x^{4}}{60} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

11/6*a*b*d^3*x/c^2+11/6*b^2*d^3*x*arctanh(c*x)/c^2+11/18*d^3*a*b*x^3+3/4*c*d^3*a^2*x^4-1/240/c^3*d^3*b^2*ln(c*
x+1)^2+1/3*d^3*b^2*arctanh(c*x)^2*x^3+11/18*d^3*b^2*arctanh(c*x)*x^3+23/36/c^3*d^3*b^2*ln(c*x+1)-14/15/c^3*d^3
*b^2*dilog(1/2+1/2*c*x)+37/80/c^3*d^3*b^2*ln(c*x-1)^2+337/180/c^3*d^3*b^2*ln(c*x-1)+1/6*c^3*d^3*a^2*x^6+3/5*c^
2*d^3*a^2*x^5+6/5*c^2*d^3*a*b*arctanh(c*x)*x^5+1/3*c^3*d^3*a*b*arctanh(c*x)*x^6+3/2*c*d^3*a*b*arctanh(c*x)*x^4
+1/10*b^2*d^3*x^3+1/60/c^3*d^3*b^2*arctanh(c*x)*ln(c*x+1)+1/60/c^3*d^3*a*b*ln(c*x+1)+37/20/c^3*d^3*a*b*ln(c*x-
1)+1/15*c^2*d^3*b^2*arctanh(c*x)*x^5+2/3*d^3*a*b*arctanh(c*x)*x^3+1/15*c^2*d^3*a*b*x^5+3/10*c*d^3*a*b*x^4+14/1
5/c*d^3*a*b*x^2+14/15/c*d^3*b^2*arctanh(c*x)*x^2+3/5*c^2*d^3*b^2*arctanh(c*x)^2*x^5+3/4*c*d^3*b^2*arctanh(c*x)
^2*x^4+3/10*c*d^3*b^2*arctanh(c*x)*x^4-37/40/c^3*d^3*b^2*ln(c*x-1)*ln(1/2+1/2*c*x)+1/120/c^3*d^3*b^2*ln(-1/2*c
*x+1/2)*ln(c*x+1)-1/120/c^3*d^3*b^2*ln(-1/2*c*x+1/2)*ln(1/2+1/2*c*x)+37/20/c^3*d^3*b^2*arctanh(c*x)*ln(c*x-1)+
1/6*c^3*d^3*b^2*arctanh(c*x)^2*x^6+1/3*d^3*a^2*x^3+37/30*b^2*d^3*x/c^2+61/180*b^2*d^3*x^2/c+1/60*b^2*c*d^3*x^4

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maxima [B]  time = 0.70, size = 775, normalized size = 2.06 \[ \frac {1}{6} \, a^{2} c^{3} d^{3} x^{6} + \frac {3}{5} \, a^{2} c^{2} d^{3} x^{5} + \frac {3}{4} \, a^{2} c d^{3} x^{4} + \frac {1}{90} \, {\left (30 \, x^{6} \operatorname {artanh}\left (c x\right ) + c {\left (\frac {2 \, {\left (3 \, c^{4} x^{5} + 5 \, c^{2} x^{3} + 15 \, x\right )}}{c^{6}} - \frac {15 \, \log \left (c x + 1\right )}{c^{7}} + \frac {15 \, \log \left (c x - 1\right )}{c^{7}}\right )}\right )} a b c^{3} d^{3} + \frac {3}{10} \, {\left (4 \, x^{5} \operatorname {artanh}\left (c x\right ) + c {\left (\frac {c^{2} x^{4} + 2 \, x^{2}}{c^{4}} + \frac {2 \, \log \left (c^{2} x^{2} - 1\right )}{c^{6}}\right )}\right )} a b c^{2} d^{3} + \frac {1}{3} \, a^{2} d^{3} x^{3} + \frac {1}{4} \, {\left (6 \, x^{4} \operatorname {artanh}\left (c x\right ) + c {\left (\frac {2 \, {\left (c^{2} x^{3} + 3 \, x\right )}}{c^{4}} - \frac {3 \, \log \left (c x + 1\right )}{c^{5}} + \frac {3 \, \log \left (c x - 1\right )}{c^{5}}\right )}\right )} a b c d^{3} + \frac {1}{3} \, {\left (2 \, x^{3} \operatorname {artanh}\left (c x\right ) + c {\left (\frac {x^{2}}{c^{2}} + \frac {\log \left (c^{2} x^{2} - 1\right )}{c^{4}}\right )}\right )} a b d^{3} + \frac {14 \, {\left (\log \left (c x + 1\right ) \log \left (-\frac {1}{2} \, c x + \frac {1}{2}\right ) + {\rm Li}_2\left (\frac {1}{2} \, c x + \frac {1}{2}\right )\right )} b^{2} d^{3}}{15 \, c^{3}} + \frac {23 \, b^{2} d^{3} \log \left (c x + 1\right )}{36 \, c^{3}} + \frac {337 \, b^{2} d^{3} \log \left (c x - 1\right )}{180 \, c^{3}} + \frac {12 \, b^{2} c^{4} d^{3} x^{4} + 72 \, b^{2} c^{3} d^{3} x^{3} + 244 \, b^{2} c^{2} d^{3} x^{2} + 888 \, b^{2} c d^{3} x + 3 \, {\left (10 \, b^{2} c^{6} d^{3} x^{6} + 36 \, b^{2} c^{5} d^{3} x^{5} + 45 \, b^{2} c^{4} d^{3} x^{4} + 20 \, b^{2} c^{3} d^{3} x^{3} + b^{2} d^{3}\right )} \log \left (c x + 1\right )^{2} + 3 \, {\left (10 \, b^{2} c^{6} d^{3} x^{6} + 36 \, b^{2} c^{5} d^{3} x^{5} + 45 \, b^{2} c^{4} d^{3} x^{4} + 20 \, b^{2} c^{3} d^{3} x^{3} - 111 \, b^{2} d^{3}\right )} \log \left (-c x + 1\right )^{2} + 4 \, {\left (6 \, b^{2} c^{5} d^{3} x^{5} + 27 \, b^{2} c^{4} d^{3} x^{4} + 55 \, b^{2} c^{3} d^{3} x^{3} + 84 \, b^{2} c^{2} d^{3} x^{2} + 165 \, b^{2} c d^{3} x\right )} \log \left (c x + 1\right ) - 2 \, {\left (12 \, b^{2} c^{5} d^{3} x^{5} + 54 \, b^{2} c^{4} d^{3} x^{4} + 110 \, b^{2} c^{3} d^{3} x^{3} + 168 \, b^{2} c^{2} d^{3} x^{2} + 330 \, b^{2} c d^{3} x + 3 \, {\left (10 \, b^{2} c^{6} d^{3} x^{6} + 36 \, b^{2} c^{5} d^{3} x^{5} + 45 \, b^{2} c^{4} d^{3} x^{4} + 20 \, b^{2} c^{3} d^{3} x^{3} + b^{2} d^{3}\right )} \log \left (c x + 1\right )\right )} \log \left (-c x + 1\right )}{720 \, c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*a^2*c^3*d^3*x^6 + 3/5*a^2*c^2*d^3*x^5 + 3/4*a^2*c*d^3*x^4 + 1/90*(30*x^6*arctanh(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^3*d^3 + 3/10*(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^2*d^3 + 1/3*a^2*d^3*x^3 + 1/4*(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*c*d^3 + 1/3*(2*x^3*arctanh(c*x) + c*(x
^2/c^2 + log(c^2*x^2 - 1)/c^4))*a*b*d^3 + 14/15*(log(c*x + 1)*log(-1/2*c*x + 1/2) + dilog(1/2*c*x + 1/2))*b^2*
d^3/c^3 + 23/36*b^2*d^3*log(c*x + 1)/c^3 + 337/180*b^2*d^3*log(c*x - 1)/c^3 + 1/720*(12*b^2*c^4*d^3*x^4 + 72*b
^2*c^3*d^3*x^3 + 244*b^2*c^2*d^3*x^2 + 888*b^2*c*d^3*x + 3*(10*b^2*c^6*d^3*x^6 + 36*b^2*c^5*d^3*x^5 + 45*b^2*c
^4*d^3*x^4 + 20*b^2*c^3*d^3*x^3 + b^2*d^3)*log(c*x + 1)^2 + 3*(10*b^2*c^6*d^3*x^6 + 36*b^2*c^5*d^3*x^5 + 45*b^
2*c^4*d^3*x^4 + 20*b^2*c^3*d^3*x^3 - 111*b^2*d^3)*log(-c*x + 1)^2 + 4*(6*b^2*c^5*d^3*x^5 + 27*b^2*c^4*d^3*x^4
+ 55*b^2*c^3*d^3*x^3 + 84*b^2*c^2*d^3*x^2 + 165*b^2*c*d^3*x)*log(c*x + 1) - 2*(12*b^2*c^5*d^3*x^5 + 54*b^2*c^4
*d^3*x^4 + 110*b^2*c^3*d^3*x^3 + 168*b^2*c^2*d^3*x^2 + 330*b^2*c*d^3*x + 3*(10*b^2*c^6*d^3*x^6 + 36*b^2*c^5*d^
3*x^5 + 45*b^2*c^4*d^3*x^4 + 20*b^2*c^3*d^3*x^3 + b^2*d^3)*log(c*x + 1))*log(-c*x + 1))/c^3

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int x^2\,{\left (a+b\,\mathrm {atanh}\left (c\,x\right )\right )}^2\,{\left (d+c\,d\,x\right )}^3 \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ d^{3} \left (\int a^{2} x^{2}\, dx + \int 3 a^{2} c x^{3}\, dx + \int 3 a^{2} c^{2} x^{4}\, dx + \int a^{2} c^{3} x^{5}\, dx + \int b^{2} x^{2} \operatorname {atanh}^{2}{\left (c x \right )}\, dx + \int 2 a b x^{2} \operatorname {atanh}{\left (c x \right )}\, dx + \int 3 b^{2} c x^{3} \operatorname {atanh}^{2}{\left (c x \right )}\, dx + \int 3 b^{2} c^{2} x^{4} \operatorname {atanh}^{2}{\left (c x \right )}\, dx + \int b^{2} c^{3} x^{5} \operatorname {atanh}^{2}{\left (c x \right )}\, dx + \int 6 a b c x^{3} \operatorname {atanh}{\left (c x \right )}\, dx + \int 6 a b c^{2} x^{4} \operatorname {atanh}{\left (c x \right )}\, dx + \int 2 a b c^{3} x^{5} \operatorname {atanh}{\left (c x \right )}\, dx\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

d**3*(Integral(a**2*x**2, x) + Integral(3*a**2*c*x**3, x) + Integral(3*a**2*c**2*x**4, x) + Integral(a**2*c**3
*x**5, x) + Integral(b**2*x**2*atanh(c*x)**2, x) + Integral(2*a*b*x**2*atanh(c*x), x) + Integral(3*b**2*c*x**3
*atanh(c*x)**2, x) + Integral(3*b**2*c**2*x**4*atanh(c*x)**2, x) + Integral(b**2*c**3*x**5*atanh(c*x)**2, x) +
Integral(6*a*b*c*x**3*atanh(c*x), x) + Integral(6*a*b*c**2*x**4*atanh(c*x), x) + Integral(2*a*b*c**3*x**5*ata
nh(c*x), x))

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