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

Optimal. Leaf size=415 \[ -\frac {d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{140 c^4}-\frac {52 b d^3 \log \left (\frac {2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{35 c^4}+\frac {1}{7} c^3 d^3 x^7 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {3 a b d^3 x}{2 c^3}+\frac {1}{2} c^2 d^3 x^6 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {1}{21} b c^2 d^3 x^6 \left (a+b \tanh ^{-1}(c x)\right )+\frac {26 b d^3 x^2 \left (a+b \tanh ^{-1}(c x)\right )}{35 c^2}+\frac {3}{5} c d^3 x^5 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {1}{5} b c d^3 x^5 \left (a+b \tanh ^{-1}(c x)\right )+\frac {1}{4} d^3 x^4 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {13}{35} b d^3 x^4 \left (a+b \tanh ^{-1}(c x)\right )+\frac {b d^3 x^3 \left (a+b \tanh ^{-1}(c x)\right )}{2 c}-\frac {26 b^2 d^3 \text {Li}_2\left (1-\frac {2}{1-c x}\right )}{35 c^4}-\frac {122 b^2 d^3 \tanh ^{-1}(c x)}{105 c^4}+\frac {122 b^2 d^3 x}{105 c^3}+\frac {3 b^2 d^3 x \tanh ^{-1}(c x)}{2 c^3}+\frac {7 b^2 d^3 x^2}{20 c^2}+\frac {11 b^2 d^3 \log \left (1-c^2 x^2\right )}{10 c^4}+\frac {1}{105} b^2 c d^3 x^5+\frac {44 b^2 d^3 x^3}{315 c}+\frac {1}{20} b^2 d^3 x^4 \]

[Out]

3/2*a*b*d^3*x/c^3+122/105*b^2*d^3*x/c^3+7/20*b^2*d^3*x^2/c^2+44/315*b^2*d^3*x^3/c+1/20*b^2*d^3*x^4+1/105*b^2*c
*d^3*x^5-122/105*b^2*d^3*arctanh(c*x)/c^4+3/2*b^2*d^3*x*arctanh(c*x)/c^3+26/35*b*d^3*x^2*(a+b*arctanh(c*x))/c^
2+1/2*b*d^3*x^3*(a+b*arctanh(c*x))/c+13/35*b*d^3*x^4*(a+b*arctanh(c*x))+1/5*b*c*d^3*x^5*(a+b*arctanh(c*x))+1/2
1*b*c^2*d^3*x^6*(a+b*arctanh(c*x))-1/140*d^3*(a+b*arctanh(c*x))^2/c^4+1/4*d^3*x^4*(a+b*arctanh(c*x))^2+3/5*c*d
^3*x^5*(a+b*arctanh(c*x))^2+1/2*c^2*d^3*x^6*(a+b*arctanh(c*x))^2+1/7*c^3*d^3*x^7*(a+b*arctanh(c*x))^2-52/35*b*
d^3*(a+b*arctanh(c*x))*ln(2/(-c*x+1))/c^4+11/10*b^2*d^3*ln(-c^2*x^2+1)/c^4-26/35*b^2*d^3*polylog(2,1-2/(-c*x+1
))/c^4

________________________________________________________________________________________

Rubi [A]  time = 1.46, antiderivative size = 415, normalized size of antiderivative = 1.00, number of steps used = 62, 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} \[ -\frac {26 b^2 d^3 \text {PolyLog}\left (2,1-\frac {2}{1-c x}\right )}{35 c^4}+\frac {1}{7} c^3 d^3 x^7 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {1}{2} c^2 d^3 x^6 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {1}{21} b c^2 d^3 x^6 \left (a+b \tanh ^{-1}(c x)\right )+\frac {26 b d^3 x^2 \left (a+b \tanh ^{-1}(c x)\right )}{35 c^2}+\frac {3 a b d^3 x}{2 c^3}-\frac {d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{140 c^4}-\frac {52 b d^3 \log \left (\frac {2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{35 c^4}+\frac {3}{5} c d^3 x^5 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {1}{5} b c d^3 x^5 \left (a+b \tanh ^{-1}(c x)\right )+\frac {1}{4} d^3 x^4 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {13}{35} b d^3 x^4 \left (a+b \tanh ^{-1}(c x)\right )+\frac {b d^3 x^3 \left (a+b \tanh ^{-1}(c x)\right )}{2 c}+\frac {7 b^2 d^3 x^2}{20 c^2}+\frac {11 b^2 d^3 \log \left (1-c^2 x^2\right )}{10 c^4}+\frac {122 b^2 d^3 x}{105 c^3}+\frac {3 b^2 d^3 x \tanh ^{-1}(c x)}{2 c^3}-\frac {122 b^2 d^3 \tanh ^{-1}(c x)}{105 c^4}+\frac {1}{105} b^2 c d^3 x^5+\frac {44 b^2 d^3 x^3}{315 c}+\frac {1}{20} b^2 d^3 x^4 \]

Antiderivative was successfully verified.

[In]

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

[Out]

(3*a*b*d^3*x)/(2*c^3) + (122*b^2*d^3*x)/(105*c^3) + (7*b^2*d^3*x^2)/(20*c^2) + (44*b^2*d^3*x^3)/(315*c) + (b^2
*d^3*x^4)/20 + (b^2*c*d^3*x^5)/105 - (122*b^2*d^3*ArcTanh[c*x])/(105*c^4) + (3*b^2*d^3*x*ArcTanh[c*x])/(2*c^3)
+ (26*b*d^3*x^2*(a + b*ArcTanh[c*x]))/(35*c^2) + (b*d^3*x^3*(a + b*ArcTanh[c*x]))/(2*c) + (13*b*d^3*x^4*(a +
b*ArcTanh[c*x]))/35 + (b*c*d^3*x^5*(a + b*ArcTanh[c*x]))/5 + (b*c^2*d^3*x^6*(a + b*ArcTanh[c*x]))/21 - (d^3*(a
+ b*ArcTanh[c*x])^2)/(140*c^4) + (d^3*x^4*(a + b*ArcTanh[c*x])^2)/4 + (3*c*d^3*x^5*(a + b*ArcTanh[c*x])^2)/5
+ (c^2*d^3*x^6*(a + b*ArcTanh[c*x])^2)/2 + (c^3*d^3*x^7*(a + b*ArcTanh[c*x])^2)/7 - (52*b*d^3*(a + b*ArcTanh[c
*x])*Log[2/(1 - c*x)])/(35*c^4) + (11*b^2*d^3*Log[1 - c^2*x^2])/(10*c^4) - (26*b^2*d^3*PolyLog[2, 1 - 2/(1 - c
*x)])/(35*c^4)

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^3 (d+c d x)^3 \left (a+b \tanh ^{-1}(c x)\right )^2 \, dx &=\int \left (d^3 x^3 \left (a+b \tanh ^{-1}(c x)\right )^2+3 c d^3 x^4 \left (a+b \tanh ^{-1}(c x)\right )^2+3 c^2 d^3 x^5 \left (a+b \tanh ^{-1}(c x)\right )^2+c^3 d^3 x^6 \left (a+b \tanh ^{-1}(c x)\right )^2\right ) \, dx\\ &=d^3 \int x^3 \left (a+b \tanh ^{-1}(c x)\right )^2 \, dx+\left (3 c d^3\right ) \int x^4 \left (a+b \tanh ^{-1}(c x)\right )^2 \, dx+\left (3 c^2 d^3\right ) \int x^5 \left (a+b \tanh ^{-1}(c x)\right )^2 \, dx+\left (c^3 d^3\right ) \int x^6 \left (a+b \tanh ^{-1}(c x)\right )^2 \, dx\\ &=\frac {1}{4} d^3 x^4 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {3}{5} c d^3 x^5 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {1}{2} c^2 d^3 x^6 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {1}{7} c^3 d^3 x^7 \left (a+b \tanh ^{-1}(c x)\right )^2-\frac {1}{2} \left (b c 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^2 d^3\right ) \int \frac {x^5 \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx-\left (b c^3 d^3\right ) \int \frac {x^6 \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx-\frac {1}{7} \left (2 b c^4 d^3\right ) \int \frac {x^7 \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx\\ &=\frac {1}{4} d^3 x^4 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {3}{5} c d^3 x^5 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {1}{2} c^2 d^3 x^6 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {1}{7} c^3 d^3 x^7 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {1}{5} \left (6 b d^3\right ) \int x^3 \left (a+b \tanh ^{-1}(c x)\right ) \, dx-\frac {1}{5} \left (6 b d^3\right ) \int \frac {x^3 \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx+\frac {\left (b d^3\right ) \int x^2 \left (a+b \tanh ^{-1}(c x)\right ) \, dx}{2 c}-\frac {\left (b d^3\right ) \int \frac {x^2 \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx}{2 c}+\left (b c d^3\right ) \int x^4 \left (a+b \tanh ^{-1}(c x)\right ) \, dx-\left (b c d^3\right ) \int \frac {x^4 \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx+\frac {1}{7} \left (2 b c^2 d^3\right ) \int x^5 \left (a+b \tanh ^{-1}(c x)\right ) \, dx-\frac {1}{7} \left (2 b c^2 d^3\right ) \int \frac {x^5 \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx\\ &=\frac {b d^3 x^3 \left (a+b \tanh ^{-1}(c x)\right )}{6 c}+\frac {3}{10} b d^3 x^4 \left (a+b \tanh ^{-1}(c x)\right )+\frac {1}{5} b c d^3 x^5 \left (a+b \tanh ^{-1}(c x)\right )+\frac {1}{21} b c^2 d^3 x^6 \left (a+b \tanh ^{-1}(c x)\right )+\frac {1}{4} d^3 x^4 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {3}{5} c d^3 x^5 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {1}{2} c^2 d^3 x^6 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {1}{7} c^3 d^3 x^7 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {1}{7} \left (2 b d^3\right ) \int x^3 \left (a+b \tanh ^{-1}(c x)\right ) \, dx-\frac {1}{7} \left (2 b d^3\right ) \int \frac {x^3 \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx-\frac {1}{6} \left (b^2 d^3\right ) \int \frac {x^3}{1-c^2 x^2} \, dx+\frac {\left (b d^3\right ) \int \left (a+b \tanh ^{-1}(c x)\right ) \, dx}{2 c^3}-\frac {\left (b d^3\right ) \int \frac {a+b \tanh ^{-1}(c x)}{1-c^2 x^2} \, dx}{2 c^3}+\frac {\left (6 b d^3\right ) \int x \left (a+b \tanh ^{-1}(c x)\right ) \, dx}{5 c^2}-\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^2}+\frac {\left (b d^3\right ) \int x^2 \left (a+b \tanh ^{-1}(c x)\right ) \, dx}{c}-\frac {\left (b d^3\right ) \int \frac {x^2 \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx}{c}-\frac {1}{10} \left (3 b^2 c d^3\right ) \int \frac {x^4}{1-c^2 x^2} \, dx-\frac {1}{5} \left (b^2 c^2 d^3\right ) \int \frac {x^5}{1-c^2 x^2} \, dx-\frac {1}{21} \left (b^2 c^3 d^3\right ) \int \frac {x^6}{1-c^2 x^2} \, dx\\ &=\frac {a b d^3 x}{2 c^3}+\frac {3 b d^3 x^2 \left (a+b \tanh ^{-1}(c x)\right )}{5 c^2}+\frac {b d^3 x^3 \left (a+b \tanh ^{-1}(c x)\right )}{2 c}+\frac {13}{35} b d^3 x^4 \left (a+b \tanh ^{-1}(c x)\right )+\frac {1}{5} b c d^3 x^5 \left (a+b \tanh ^{-1}(c x)\right )+\frac {1}{21} b c^2 d^3 x^6 \left (a+b \tanh ^{-1}(c x)\right )+\frac {7 d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{20 c^4}+\frac {1}{4} d^3 x^4 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {3}{5} c d^3 x^5 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {1}{2} c^2 d^3 x^6 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {1}{7} c^3 d^3 x^7 \left (a+b \tanh ^{-1}(c x)\right )^2-\frac {1}{12} \left (b^2 d^3\right ) \operatorname {Subst}\left (\int \frac {x}{1-c^2 x} \, dx,x,x^2\right )-\frac {1}{3} \left (b^2 d^3\right ) \int \frac {x^3}{1-c^2 x^2} \, dx+\frac {\left (b d^3\right ) \int \left (a+b \tanh ^{-1}(c x)\right ) \, dx}{c^3}-\frac {\left (b d^3\right ) \int \frac {a+b \tanh ^{-1}(c x)}{1-c^2 x^2} \, dx}{c^3}-\frac {\left (6 b d^3\right ) \int \frac {a+b \tanh ^{-1}(c x)}{1-c x} \, dx}{5 c^3}+\frac {\left (b^2 d^3\right ) \int \tanh ^{-1}(c x) \, dx}{2 c^3}+\frac {\left (2 b d^3\right ) \int x \left (a+b \tanh ^{-1}(c x)\right ) \, dx}{7 c^2}-\frac {\left (2 b d^3\right ) \int \frac {x \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx}{7 c^2}-\frac {\left (3 b^2 d^3\right ) \int \frac {x^2}{1-c^2 x^2} \, dx}{5 c}-\frac {1}{14} \left (b^2 c d^3\right ) \int \frac {x^4}{1-c^2 x^2} \, dx-\frac {1}{10} \left (3 b^2 c 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}{10} \left (b^2 c^2 d^3\right ) \operatorname {Subst}\left (\int \frac {x^2}{1-c^2 x} \, dx,x,x^2\right )-\frac {1}{21} \left (b^2 c^3 d^3\right ) \int \left (-\frac {1}{c^6}-\frac {x^2}{c^4}-\frac {x^4}{c^2}+\frac {1}{c^6 \left (1-c^2 x^2\right )}\right ) \, dx\\ &=\frac {3 a b d^3 x}{2 c^3}+\frac {199 b^2 d^3 x}{210 c^3}+\frac {73 b^2 d^3 x^3}{630 c}+\frac {1}{105} b^2 c d^3 x^5+\frac {b^2 d^3 x \tanh ^{-1}(c x)}{2 c^3}+\frac {26 b d^3 x^2 \left (a+b \tanh ^{-1}(c x)\right )}{35 c^2}+\frac {b d^3 x^3 \left (a+b \tanh ^{-1}(c x)\right )}{2 c}+\frac {13}{35} b d^3 x^4 \left (a+b \tanh ^{-1}(c x)\right )+\frac {1}{5} b c d^3 x^5 \left (a+b \tanh ^{-1}(c x)\right )+\frac {1}{21} b c^2 d^3 x^6 \left (a+b \tanh ^{-1}(c x)\right )-\frac {d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{140 c^4}+\frac {1}{4} d^3 x^4 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {3}{5} c d^3 x^5 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {1}{2} c^2 d^3 x^6 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {1}{7} c^3 d^3 x^7 \left (a+b \tanh ^{-1}(c x)\right )^2-\frac {6 b d^3 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right )}{5 c^4}-\frac {1}{12} \left (b^2 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}{6} \left (b^2 d^3\right ) \operatorname {Subst}\left (\int \frac {x}{1-c^2 x} \, dx,x,x^2\right )-\frac {\left (2 b d^3\right ) \int \frac {a+b \tanh ^{-1}(c x)}{1-c x} \, dx}{7 c^3}-\frac {\left (b^2 d^3\right ) \int \frac {1}{1-c^2 x^2} \, dx}{21 c^3}-\frac {\left (3 b^2 d^3\right ) \int \frac {1}{1-c^2 x^2} \, dx}{10 c^3}-\frac {\left (3 b^2 d^3\right ) \int \frac {1}{1-c^2 x^2} \, dx}{5 c^3}+\frac {\left (b^2 d^3\right ) \int \tanh ^{-1}(c x) \, dx}{c^3}+\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^3}-\frac {\left (b^2 d^3\right ) \int \frac {x}{1-c^2 x^2} \, dx}{2 c^2}-\frac {\left (b^2 d^3\right ) \int \frac {x^2}{1-c^2 x^2} \, dx}{7 c}-\frac {1}{14} \left (b^2 c 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}{10} \left (b^2 c^2 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 {3 a b d^3 x}{2 c^3}+\frac {122 b^2 d^3 x}{105 c^3}+\frac {11 b^2 d^3 x^2}{60 c^2}+\frac {44 b^2 d^3 x^3}{315 c}+\frac {1}{20} b^2 d^3 x^4+\frac {1}{105} b^2 c d^3 x^5-\frac {199 b^2 d^3 \tanh ^{-1}(c x)}{210 c^4}+\frac {3 b^2 d^3 x \tanh ^{-1}(c x)}{2 c^3}+\frac {26 b d^3 x^2 \left (a+b \tanh ^{-1}(c x)\right )}{35 c^2}+\frac {b d^3 x^3 \left (a+b \tanh ^{-1}(c x)\right )}{2 c}+\frac {13}{35} b d^3 x^4 \left (a+b \tanh ^{-1}(c x)\right )+\frac {1}{5} b c d^3 x^5 \left (a+b \tanh ^{-1}(c x)\right )+\frac {1}{21} b c^2 d^3 x^6 \left (a+b \tanh ^{-1}(c x)\right )-\frac {d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{140 c^4}+\frac {1}{4} d^3 x^4 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {3}{5} c d^3 x^5 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {1}{2} c^2 d^3 x^6 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {1}{7} c^3 d^3 x^7 \left (a+b \tanh ^{-1}(c x)\right )^2-\frac {52 b d^3 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right )}{35 c^4}+\frac {13 b^2 d^3 \log \left (1-c^2 x^2\right )}{30 c^4}-\frac {1}{6} \left (b^2 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 {\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^4}-\frac {\left (b^2 d^3\right ) \int \frac {1}{1-c^2 x^2} \, dx}{14 c^3}-\frac {\left (b^2 d^3\right ) \int \frac {1}{1-c^2 x^2} \, dx}{7 c^3}+\frac {\left (2 b^2 d^3\right ) \int \frac {\log \left (\frac {2}{1-c x}\right )}{1-c^2 x^2} \, dx}{7 c^3}-\frac {\left (b^2 d^3\right ) \int \frac {x}{1-c^2 x^2} \, dx}{c^2}\\ &=\frac {3 a b d^3 x}{2 c^3}+\frac {122 b^2 d^3 x}{105 c^3}+\frac {7 b^2 d^3 x^2}{20 c^2}+\frac {44 b^2 d^3 x^3}{315 c}+\frac {1}{20} b^2 d^3 x^4+\frac {1}{105} b^2 c d^3 x^5-\frac {122 b^2 d^3 \tanh ^{-1}(c x)}{105 c^4}+\frac {3 b^2 d^3 x \tanh ^{-1}(c x)}{2 c^3}+\frac {26 b d^3 x^2 \left (a+b \tanh ^{-1}(c x)\right )}{35 c^2}+\frac {b d^3 x^3 \left (a+b \tanh ^{-1}(c x)\right )}{2 c}+\frac {13}{35} b d^3 x^4 \left (a+b \tanh ^{-1}(c x)\right )+\frac {1}{5} b c d^3 x^5 \left (a+b \tanh ^{-1}(c x)\right )+\frac {1}{21} b c^2 d^3 x^6 \left (a+b \tanh ^{-1}(c x)\right )-\frac {d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{140 c^4}+\frac {1}{4} d^3 x^4 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {3}{5} c d^3 x^5 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {1}{2} c^2 d^3 x^6 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {1}{7} c^3 d^3 x^7 \left (a+b \tanh ^{-1}(c x)\right )^2-\frac {52 b d^3 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right )}{35 c^4}+\frac {11 b^2 d^3 \log \left (1-c^2 x^2\right )}{10 c^4}-\frac {3 b^2 d^3 \text {Li}_2\left (1-\frac {2}{1-c x}\right )}{5 c^4}-\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 )}{7 c^4}\\ &=\frac {3 a b d^3 x}{2 c^3}+\frac {122 b^2 d^3 x}{105 c^3}+\frac {7 b^2 d^3 x^2}{20 c^2}+\frac {44 b^2 d^3 x^3}{315 c}+\frac {1}{20} b^2 d^3 x^4+\frac {1}{105} b^2 c d^3 x^5-\frac {122 b^2 d^3 \tanh ^{-1}(c x)}{105 c^4}+\frac {3 b^2 d^3 x \tanh ^{-1}(c x)}{2 c^3}+\frac {26 b d^3 x^2 \left (a+b \tanh ^{-1}(c x)\right )}{35 c^2}+\frac {b d^3 x^3 \left (a+b \tanh ^{-1}(c x)\right )}{2 c}+\frac {13}{35} b d^3 x^4 \left (a+b \tanh ^{-1}(c x)\right )+\frac {1}{5} b c d^3 x^5 \left (a+b \tanh ^{-1}(c x)\right )+\frac {1}{21} b c^2 d^3 x^6 \left (a+b \tanh ^{-1}(c x)\right )-\frac {d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{140 c^4}+\frac {1}{4} d^3 x^4 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {3}{5} c d^3 x^5 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {1}{2} c^2 d^3 x^6 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {1}{7} c^3 d^3 x^7 \left (a+b \tanh ^{-1}(c x)\right )^2-\frac {52 b d^3 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right )}{35 c^4}+\frac {11 b^2 d^3 \log \left (1-c^2 x^2\right )}{10 c^4}-\frac {26 b^2 d^3 \text {Li}_2\left (1-\frac {2}{1-c x}\right )}{35 c^4}\\ \end {align*}

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Mathematica [A]  time = 1.66, size = 385, normalized size = 0.93 \[ \frac {d^3 \left (180 a^2 c^7 x^7+630 a^2 c^6 x^6+756 a^2 c^5 x^5+315 a^2 c^4 x^4+60 a b c^6 x^6+252 a b c^5 x^5+468 a b c^4 x^4+630 a b c^3 x^3+936 a b c^2 x^2+936 a b \log \left (c^2 x^2-1\right )+6 b \tanh ^{-1}(c x) \left (3 a c^4 x^4 \left (20 c^3 x^3+70 c^2 x^2+84 c x+35\right )+b \left (10 c^6 x^6+42 c^5 x^5+78 c^4 x^4+105 c^3 x^3+156 c^2 x^2+315 c x-244\right )-312 b \log \left (e^{-2 \tanh ^{-1}(c x)}+1\right )\right )+1890 a b c x+945 a b \log (1-c x)-945 a b \log (c x+1)-1464 a b+12 b^2 c^5 x^5+63 b^2 c^4 x^4+176 b^2 c^3 x^3+441 b^2 c^2 x^2+1386 b^2 \log \left (1-c^2 x^2\right )+9 b^2 \left (20 c^7 x^7+70 c^6 x^6+84 c^5 x^5+35 c^4 x^4-209\right ) \tanh ^{-1}(c x)^2+936 b^2 \text {Li}_2\left (-e^{-2 \tanh ^{-1}(c x)}\right )+1464 b^2 c x-504 b^2\right )}{1260 c^4} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(d^3*(-1464*a*b - 504*b^2 + 1890*a*b*c*x + 1464*b^2*c*x + 936*a*b*c^2*x^2 + 441*b^2*c^2*x^2 + 630*a*b*c^3*x^3
+ 176*b^2*c^3*x^3 + 315*a^2*c^4*x^4 + 468*a*b*c^4*x^4 + 63*b^2*c^4*x^4 + 756*a^2*c^5*x^5 + 252*a*b*c^5*x^5 + 1
2*b^2*c^5*x^5 + 630*a^2*c^6*x^6 + 60*a*b*c^6*x^6 + 180*a^2*c^7*x^7 + 9*b^2*(-209 + 35*c^4*x^4 + 84*c^5*x^5 + 7
0*c^6*x^6 + 20*c^7*x^7)*ArcTanh[c*x]^2 + 6*b*ArcTanh[c*x]*(3*a*c^4*x^4*(35 + 84*c*x + 70*c^2*x^2 + 20*c^3*x^3)
+ b*(-244 + 315*c*x + 156*c^2*x^2 + 105*c^3*x^3 + 78*c^4*x^4 + 42*c^5*x^5 + 10*c^6*x^6) - 312*b*Log[1 + E^(-2
*ArcTanh[c*x])]) + 945*a*b*Log[1 - c*x] - 945*a*b*Log[1 + c*x] + 1386*b^2*Log[1 - c^2*x^2] + 936*a*b*Log[-1 +
c^2*x^2] + 936*b^2*PolyLog[2, -E^(-2*ArcTanh[c*x])]))/(1260*c^4)

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fricas [F]  time = 0.54, size = 0, normalized size = 0.00 \[ {\rm integral}\left (a^{2} c^{3} d^{3} x^{6} + 3 \, a^{2} c^{2} d^{3} x^{5} + 3 \, a^{2} c d^{3} x^{4} + a^{2} d^{3} x^{3} + {\left (b^{2} c^{3} d^{3} x^{6} + 3 \, b^{2} c^{2} d^{3} x^{5} + 3 \, b^{2} c d^{3} x^{4} + b^{2} d^{3} x^{3}\right )} \operatorname {artanh}\left (c x\right )^{2} + 2 \, {\left (a b c^{3} d^{3} x^{6} + 3 \, a b c^{2} d^{3} x^{5} + 3 \, a b c d^{3} x^{4} + a b d^{3} x^{3}\right )} \operatorname {artanh}\left (c x\right ), x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(a^2*c^3*d^3*x^6 + 3*a^2*c^2*d^3*x^5 + 3*a^2*c*d^3*x^4 + a^2*d^3*x^3 + (b^2*c^3*d^3*x^6 + 3*b^2*c^2*d^
3*x^5 + 3*b^2*c*d^3*x^4 + b^2*d^3*x^3)*arctanh(c*x)^2 + 2*(a*b*c^3*d^3*x^6 + 3*a*b*c^2*d^3*x^5 + 3*a*b*c*d^3*x
^4 + a*b*d^3*x^3)*arctanh(c*x), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(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^3, x)

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maple [A]  time = 0.06, size = 662, normalized size = 1.60 \[ \frac {3 a b \,d^{3} x}{2 c^{3}}+\frac {2 c^{3} d^{3} a b \arctanh \left (c x \right ) x^{7}}{7}+c^{2} d^{3} a b \arctanh \left (c x \right ) x^{6}+\frac {6 c \,d^{3} a b \arctanh \left (c x \right ) x^{5}}{5}+\frac {13 d^{3} a b \,x^{4}}{35}+\frac {353 d^{3} b^{2} \ln \left (c x -1\right )}{210 c^{4}}+\frac {209 d^{3} b^{2} \ln \left (c x -1\right )^{2}}{560 c^{4}}+\frac {d^{3} b^{2} \ln \left (c x +1\right )^{2}}{560 c^{4}}+\frac {109 d^{3} b^{2} \ln \left (c x +1\right )}{210 c^{4}}-\frac {26 d^{3} b^{2} \dilog \left (\frac {1}{2}+\frac {c x}{2}\right )}{35 c^{4}}+\frac {13 d^{3} b^{2} \arctanh \left (c x \right ) x^{4}}{35}+\frac {d^{3} b^{2} \arctanh \left (c x \right )^{2} x^{4}}{4}+\frac {c^{3} d^{3} a^{2} x^{7}}{7}+\frac {c^{2} d^{3} a^{2} x^{6}}{2}+\frac {3 c \,d^{3} a^{2} x^{5}}{5}+\frac {b^{2} d^{3} x^{4}}{20}+\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 )}{280 c^{4}}+\frac {c \,d^{3} a b \,x^{5}}{5}+\frac {d^{3} a b \,x^{3}}{2 c}+\frac {26 d^{3} a b \,x^{2}}{35 c^{2}}+\frac {209 d^{3} a b \ln \left (c x -1\right )}{140 c^{4}}-\frac {d^{3} a b \ln \left (c x +1\right )}{140 c^{4}}+\frac {c^{2} d^{3} a b \,x^{6}}{21}+\frac {c^{3} d^{3} b^{2} \arctanh \left (c x \right )^{2} x^{7}}{7}+\frac {26 d^{3} b^{2} \arctanh \left (c x \right ) x^{2}}{35 c^{2}}+\frac {3 c \,d^{3} b^{2} \arctanh \left (c x \right )^{2} x^{5}}{5}+\frac {c^{2} d^{3} b^{2} \arctanh \left (c x \right ) x^{6}}{21}+\frac {d^{3} b^{2} \arctanh \left (c x \right ) x^{3}}{2 c}-\frac {d^{3} b^{2} \arctanh \left (c x \right ) \ln \left (c x +1\right )}{140 c^{4}}+\frac {d^{3} a b \arctanh \left (c x \right ) x^{4}}{2}+\frac {d^{3} a^{2} x^{4}}{4}+\frac {3 b^{2} d^{3} x \arctanh \left (c x \right )}{2 c^{3}}+\frac {209 d^{3} b^{2} \arctanh \left (c x \right ) \ln \left (c x -1\right )}{140 c^{4}}+\frac {c \,d^{3} b^{2} \arctanh \left (c x \right ) x^{5}}{5}+\frac {c^{2} d^{3} b^{2} \arctanh \left (c x \right )^{2} x^{6}}{2}-\frac {209 d^{3} b^{2} \ln \left (c x -1\right ) \ln \left (\frac {1}{2}+\frac {c x}{2}\right )}{280 c^{4}}-\frac {d^{3} b^{2} \ln \left (-\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (c x +1\right )}{280 c^{4}}+\frac {122 b^{2} d^{3} x}{105 c^{3}}+\frac {7 b^{2} d^{3} x^{2}}{20 c^{2}}+\frac {44 b^{2} d^{3} x^{3}}{315 c}+\frac {b^{2} c \,d^{3} x^{5}}{105} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/2*a*b*d^3*x/c^3+3/2*b^2*d^3*x*arctanh(c*x)/c^3+6/5*c*d^3*a*b*arctanh(c*x)*x^5+2/7*c^3*d^3*a*b*arctanh(c*x)*x
^7+c^2*d^3*a*b*arctanh(c*x)*x^6+13/35*d^3*a*b*x^4-26/35/c^4*d^3*b^2*dilog(1/2+1/2*c*x)+353/210/c^4*d^3*b^2*ln(
c*x-1)+209/560/c^4*d^3*b^2*ln(c*x-1)^2+1/560/c^4*d^3*b^2*ln(c*x+1)^2+109/210/c^4*d^3*b^2*ln(c*x+1)+13/35*d^3*b
^2*arctanh(c*x)*x^4+1/4*d^3*b^2*arctanh(c*x)^2*x^4+1/7*c^3*d^3*a^2*x^7+1/2*c^2*d^3*a^2*x^6+3/5*c*d^3*a^2*x^5+1
/20*b^2*d^3*x^4+1/280/c^4*d^3*b^2*ln(-1/2*c*x+1/2)*ln(1/2+1/2*c*x)+209/140/c^4*d^3*b^2*arctanh(c*x)*ln(c*x-1)+
1/5*c*d^3*a*b*x^5+1/2/c*d^3*a*b*x^3+26/35/c^2*d^3*a*b*x^2+1/5*c*d^3*b^2*arctanh(c*x)*x^5+1/2*c^2*d^3*b^2*arcta
nh(c*x)^2*x^6+1/7*c^3*d^3*b^2*arctanh(c*x)^2*x^7+26/35/c^2*d^3*b^2*arctanh(c*x)*x^2+3/5*c*d^3*b^2*arctanh(c*x)
^2*x^5+1/21*c^2*d^3*b^2*arctanh(c*x)*x^6+209/140/c^4*d^3*a*b*ln(c*x-1)-1/140/c^4*d^3*a*b*ln(c*x+1)+1/21*c^2*d^
3*a*b*x^6+1/2/c*d^3*b^2*arctanh(c*x)*x^3-1/140/c^4*d^3*b^2*arctanh(c*x)*ln(c*x+1)+1/2*d^3*a*b*arctanh(c*x)*x^4
+1/4*d^3*a^2*x^4-209/280/c^4*d^3*b^2*ln(c*x-1)*ln(1/2+1/2*c*x)-1/280/c^4*d^3*b^2*ln(-1/2*c*x+1/2)*ln(c*x+1)+12
2/105*b^2*d^3*x/c^3+7/20*b^2*d^3*x^2/c^2+44/315*b^2*d^3*x^3/c+1/105*b^2*c*d^3*x^5

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maxima [B]  time = 0.70, size = 928, normalized size = 2.24 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*a^2*c^3*d^3*x^7 + 1/2*a^2*c^2*d^3*x^6 + 3/5*a^2*c*d^3*x^5 + 1/4*b^2*d^3*x^4*arctanh(c*x)^2 + 1/42*(12*x^7*
arctanh(c*x) + c*((2*c^4*x^6 + 3*c^2*x^4 + 6*x^2)/c^6 + 6*log(c^2*x^2 - 1)/c^8))*a*b*c^3*d^3 + 1/4*a^2*d^3*x^4
+ 1/30*(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^2*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*d^3
+ 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^3 + 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^3 + 26/35*(log
(c*x + 1)*log(-1/2*c*x + 1/2) + dilog(1/2*c*x + 1/2))*b^2*d^3/c^4 + 13/70*b^2*d^3*log(c*x + 1)/c^4 + 283/210*b
^2*d^3*log(c*x - 1)/c^4 + 1/2520*(24*b^2*c^5*d^3*x^5 + 126*b^2*c^4*d^3*x^4 + 352*b^2*c^3*d^3*x^3 + 672*b^2*c^2
*d^3*x^2 + 2928*b^2*c*d^3*x + 9*(10*b^2*c^7*d^3*x^7 + 35*b^2*c^6*d^3*x^6 + 42*b^2*c^5*d^3*x^5 + 17*b^2*d^3)*lo
g(c*x + 1)^2 + 9*(10*b^2*c^7*d^3*x^7 + 35*b^2*c^6*d^3*x^6 + 42*b^2*c^5*d^3*x^5 - 87*b^2*d^3)*log(-c*x + 1)^2 +
12*(5*b^2*c^6*d^3*x^6 + 21*b^2*c^5*d^3*x^5 + 39*b^2*c^4*d^3*x^4 + 35*b^2*c^3*d^3*x^3 + 78*b^2*c^2*d^3*x^2 + 1
05*b^2*c*d^3*x)*log(c*x + 1) - 6*(10*b^2*c^6*d^3*x^6 + 42*b^2*c^5*d^3*x^5 + 78*b^2*c^4*d^3*x^4 + 70*b^2*c^3*d^
3*x^3 + 156*b^2*c^2*d^3*x^2 + 210*b^2*c*d^3*x + 3*(10*b^2*c^7*d^3*x^7 + 35*b^2*c^6*d^3*x^6 + 42*b^2*c^5*d^3*x^
5 + 17*b^2*d^3)*log(c*x + 1))*log(-c*x + 1))/c^4

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(x^3*(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 \[ d^{3} \left (\int a^{2} x^{3}\, dx + \int 3 a^{2} c x^{4}\, dx + \int 3 a^{2} c^{2} x^{5}\, dx + \int a^{2} c^{3} x^{6}\, dx + \int b^{2} x^{3} \operatorname {atanh}^{2}{\left (c x \right )}\, dx + \int 2 a b x^{3} \operatorname {atanh}{\left (c x \right )}\, dx + \int 3 b^{2} c x^{4} \operatorname {atanh}^{2}{\left (c x \right )}\, dx + \int 3 b^{2} c^{2} x^{5} \operatorname {atanh}^{2}{\left (c x \right )}\, dx + \int b^{2} c^{3} x^{6} \operatorname {atanh}^{2}{\left (c x \right )}\, dx + \int 6 a b c x^{4} \operatorname {atanh}{\left (c x \right )}\, dx + \int 6 a b c^{2} x^{5} \operatorname {atanh}{\left (c x \right )}\, dx + \int 2 a b c^{3} x^{6} \operatorname {atanh}{\left (c x \right )}\, dx\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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