3.1.76 \(\int x^3 (d+c d x)^2 (a+b \tanh ^{-1}(c x))^2 \, dx\) [76]

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

[Out]

5/6*a*b*d^2*x/c^3+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+1/60*b^2*d^2*x^4-3/5*b^2*d^2*arc
tanh(c*x)/c^4+5/6*b^2*d^2*x*arctanh(c*x)/c^3+2/5*b*d^2*x^2*(a+b*arctanh(c*x))/c^2+5/18*b*d^2*x^3*(a+b*arctanh(
c*x))/c+1/5*b*d^2*x^4*(a+b*arctanh(c*x))+1/15*b*c*d^2*x^5*(a+b*arctanh(c*x))-1/60*d^2*(a+b*arctanh(c*x))^2/c^4
+1/4*d^2*x^4*(a+b*arctanh(c*x))^2+2/5*c*d^2*x^5*(a+b*arctanh(c*x))^2+1/6*c^2*d^2*x^6*(a+b*arctanh(c*x))^2-4/5*
b*d^2*(a+b*arctanh(c*x))*ln(2/(-c*x+1))/c^4+53/90*b^2*d^2*ln(-c^2*x^2+1)/c^4-2/5*b^2*d^2*polylog(2,1-2/(-c*x+1
))/c^4

________________________________________________________________________________________

Rubi [A]
time = 0.72, antiderivative size = 356, normalized size of antiderivative = 1.00, 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 = {6087, 6037, 6127, 272, 45, 6021, 266, 6095, 308, 212, 327, 6131, 6055, 2449, 2352} \begin {gather*} -\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 {5 a b d^2 x}{6 c^3}+\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 {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 {2 b^2 d^2 \text {Li}_2\left (1-\frac {2}{1-c x}\right )}{5 c^4}-\frac {3 b^2 d^2 \tanh ^{-1}(c x)}{5 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 {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 {b^2 d^2 x^3}{15 c}+\frac {1}{60} b^2 d^2 x^4 \end {gather*}

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 45

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 212

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

Rule 266

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 272

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 308

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 327

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 2352

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-e^(-1))*PolyLog[2, 1 - c*x], x] /; FreeQ[{c, d, e
}, x] && EqQ[e + c*d, 0]

Rule 2449

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 6021

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

Rule 6037

Int[((a_.) + ArcTanh[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)*((a + b*ArcTanh[c*
x^n])^p/(m + 1)), x] - Dist[b*c*n*(p/(m + 1)), Int[x^(m + n)*((a + b*ArcTanh[c*x^n])^(p - 1)/(1 - c^2*x^(2*n))
), x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0] && (EqQ[p, 1] || (EqQ[n, 1] && IntegerQ[m])) && NeQ[m, -1
]

Rule 6055

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 6087

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 6095

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 6127

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 6131

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)^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 ) \text {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 ) \text {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 ) \text {Subst}\left (\int \frac {x}{1-c^2 x} \, dx,x,x^2\right )-\frac {1}{12} \left (b^2 d^2\right ) \text {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 ) \text {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 ) \text {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 ) \text {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*}

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Mathematica [A]
time = 0.70, size = 329, normalized size = 0.92 \begin {gather*} \frac {d^2 \left (-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 \left (-49+15 c^4 x^4+24 c^5 x^5+10 c^6 x^6\right ) \tanh ^{-1}(c x)^2+2 b \tanh ^{-1}(c x) \left (3 a c^4 x^4 \left (15+24 c x+10 c^2 x^2\right )+b \left (-54+75 c x+36 c^2 x^2+25 c^3 x^3+18 c^4 x^4+6 c^5 x^5\right )-72 b \log \left (1+e^{-2 \tanh ^{-1}(c x)}\right )\right )+75 a b \log (1-c x)-75 a b \log (1+c x)+106 b^2 \log \left (1-c^2 x^2\right )+72 a b \log \left (-1+c^2 x^2\right )+72 b^2 \text {PolyLog}\left (2,-e^{-2 \tanh ^{-1}(c x)}\right )\right )}{180 c^4} \end {gather*}

Antiderivative was successfully verified.

[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 = 0.44, size = 549, normalized size = 1.54 Too large to display

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,method=_RETURNVERBOSE)

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 766 vs. \(2 (317) = 634\).
time = 0.49, size = 766, normalized size = 2.15 \begin {gather*} \frac {1}{6} \, a^{2} c^{2} d^{2} x^{6} + \frac {2}{5} \, a^{2} c d^{2} x^{5} + \frac {1}{4} \, b^{2} d^{2} x^{4} \operatorname {artanh}\left (c x\right )^{2} + \frac {1}{4} \, a^{2} d^{2} 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^{2} d^{2} + \frac {1}{5} \, {\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 d^{2} + \frac {1}{12} \, {\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 d^{2} + \frac {1}{48} \, {\left (4 \, 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 )} \operatorname {artanh}\left (c x\right ) + \frac {4 \, c^{2} x^{2} - 2 \, {\left (3 \, \log \left (c x - 1\right ) - 8\right )} \log \left (c x + 1\right ) + 3 \, \log \left (c x + 1\right )^{2} + 3 \, \log \left (c x - 1\right )^{2} + 16 \, \log \left (c x - 1\right )}{c^{4}}\right )} b^{2} d^{2} + \frac {2 \, {\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^{2}}{5 \, c^{4}} - \frac {2 \, b^{2} d^{2} \log \left (c x + 1\right )}{45 \, c^{4}} + \frac {5 \, b^{2} d^{2} \log \left (c x - 1\right )}{9 \, c^{4}} + \frac {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 \, {\left (5 \, b^{2} c^{6} d^{2} x^{6} + 12 \, b^{2} c^{5} d^{2} x^{5} + 7 \, b^{2} d^{2}\right )} \log \left (c x + 1\right )^{2} + 3 \, {\left (5 \, b^{2} c^{6} d^{2} x^{6} + 12 \, b^{2} c^{5} d^{2} x^{5} - 17 \, b^{2} d^{2}\right )} \log \left (-c x + 1\right )^{2} + 4 \, {\left (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\right )} \log \left (c x + 1\right ) - 2 \, {\left (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 \, {\left (5 \, b^{2} c^{6} d^{2} x^{6} + 12 \, b^{2} c^{5} d^{2} x^{5} + 7 \, b^{2} d^{2}\right )} \log \left (c x + 1\right )\right )} \log \left (-c x + 1\right )}{360 \, c^{4}} \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} 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 {gather*}

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 [B] Leaf count of result is larger than twice the leaf count of optimal. 1135 vs. \(2 (317) = 634\).
time = 2.25, size = 1135, normalized size = 3.19 \begin {gather*} \frac {1}{63} \, {\left (\frac {84 \, {\left (\frac {{\left (c x + 1\right )}^{5} b^{2} d^{2}}{{\left (c x - 1\right )}^{5}} + \frac {{\left (c x + 1\right )}^{4} b^{2} d^{2}}{{\left (c x - 1\right )}^{4}} + \frac {{\left (c x + 1\right )}^{3} b^{2} d^{2}}{{\left (c x - 1\right )}^{3}}\right )} \log \left (-\frac {c x + 1}{c x - 1}\right )^{2}}{\frac {{\left (c x + 1\right )}^{8} c^{7}}{{\left (c x - 1\right )}^{8}} - \frac {8 \, {\left (c x + 1\right )}^{7} c^{7}}{{\left (c x - 1\right )}^{7}} + \frac {28 \, {\left (c x + 1\right )}^{6} c^{7}}{{\left (c x - 1\right )}^{6}} - \frac {56 \, {\left (c x + 1\right )}^{5} c^{7}}{{\left (c x - 1\right )}^{5}} + \frac {70 \, {\left (c x + 1\right )}^{4} c^{7}}{{\left (c x - 1\right )}^{4}} - \frac {56 \, {\left (c x + 1\right )}^{3} c^{7}}{{\left (c x - 1\right )}^{3}} + \frac {28 \, {\left (c x + 1\right )}^{2} c^{7}}{{\left (c x - 1\right )}^{2}} - \frac {8 \, {\left (c x + 1\right )} c^{7}}{c x - 1} + c^{7}} + \frac {2 \, {\left (\frac {168 \, {\left (c x + 1\right )}^{5} a b d^{2}}{{\left (c x - 1\right )}^{5}} + \frac {168 \, {\left (c x + 1\right )}^{4} a b d^{2}}{{\left (c x - 1\right )}^{4}} + \frac {168 \, {\left (c x + 1\right )}^{3} a b d^{2}}{{\left (c x - 1\right )}^{3}} + \frac {28 \, {\left (c x + 1\right )}^{5} b^{2} d^{2}}{{\left (c x - 1\right )}^{5}} - \frac {35 \, {\left (c x + 1\right )}^{4} b^{2} d^{2}}{{\left (c x - 1\right )}^{4}} + \frac {28 \, {\left (c x + 1\right )}^{3} b^{2} d^{2}}{{\left (c x - 1\right )}^{3}} - \frac {28 \, {\left (c x + 1\right )}^{2} b^{2} d^{2}}{{\left (c x - 1\right )}^{2}} + \frac {8 \, {\left (c x + 1\right )} b^{2} d^{2}}{c x - 1} - b^{2} d^{2}\right )} \log \left (-\frac {c x + 1}{c x - 1}\right )}{\frac {{\left (c x + 1\right )}^{8} c^{7}}{{\left (c x - 1\right )}^{8}} - \frac {8 \, {\left (c x + 1\right )}^{7} c^{7}}{{\left (c x - 1\right )}^{7}} + \frac {28 \, {\left (c x + 1\right )}^{6} c^{7}}{{\left (c x - 1\right )}^{6}} - \frac {56 \, {\left (c x + 1\right )}^{5} c^{7}}{{\left (c x - 1\right )}^{5}} + \frac {70 \, {\left (c x + 1\right )}^{4} c^{7}}{{\left (c x - 1\right )}^{4}} - \frac {56 \, {\left (c x + 1\right )}^{3} c^{7}}{{\left (c x - 1\right )}^{3}} + \frac {28 \, {\left (c x + 1\right )}^{2} c^{7}}{{\left (c x - 1\right )}^{2}} - \frac {8 \, {\left (c x + 1\right )} c^{7}}{c x - 1} + c^{7}} + \frac {\frac {336 \, {\left (c x + 1\right )}^{5} a^{2} d^{2}}{{\left (c x - 1\right )}^{5}} + \frac {336 \, {\left (c x + 1\right )}^{4} a^{2} d^{2}}{{\left (c x - 1\right )}^{4}} + \frac {336 \, {\left (c x + 1\right )}^{3} a^{2} d^{2}}{{\left (c x - 1\right )}^{3}} + \frac {112 \, {\left (c x + 1\right )}^{5} a b d^{2}}{{\left (c x - 1\right )}^{5}} - \frac {140 \, {\left (c x + 1\right )}^{4} a b d^{2}}{{\left (c x - 1\right )}^{4}} + \frac {112 \, {\left (c x + 1\right )}^{3} a b d^{2}}{{\left (c x - 1\right )}^{3}} - \frac {112 \, {\left (c x + 1\right )}^{2} a b d^{2}}{{\left (c x - 1\right )}^{2}} + \frac {32 \, {\left (c x + 1\right )} a b d^{2}}{c x - 1} - 4 \, a b d^{2} - \frac {2 \, {\left (c x + 1\right )}^{7} b^{2} d^{2}}{{\left (c x - 1\right )}^{7}} + \frac {15 \, {\left (c x + 1\right )}^{6} b^{2} d^{2}}{{\left (c x - 1\right )}^{6}} - \frac {30 \, {\left (c x + 1\right )}^{5} b^{2} d^{2}}{{\left (c x - 1\right )}^{5}} + \frac {34 \, {\left (c x + 1\right )}^{4} b^{2} d^{2}}{{\left (c x - 1\right )}^{4}} - \frac {30 \, {\left (c x + 1\right )}^{3} b^{2} d^{2}}{{\left (c x - 1\right )}^{3}} + \frac {15 \, {\left (c x + 1\right )}^{2} b^{2} d^{2}}{{\left (c x - 1\right )}^{2}} - \frac {2 \, {\left (c x + 1\right )} b^{2} d^{2}}{c x - 1}}{\frac {{\left (c x + 1\right )}^{8} c^{7}}{{\left (c x - 1\right )}^{8}} - \frac {8 \, {\left (c x + 1\right )}^{7} c^{7}}{{\left (c x - 1\right )}^{7}} + \frac {28 \, {\left (c x + 1\right )}^{6} c^{7}}{{\left (c x - 1\right )}^{6}} - \frac {56 \, {\left (c x + 1\right )}^{5} c^{7}}{{\left (c x - 1\right )}^{5}} + \frac {70 \, {\left (c x + 1\right )}^{4} c^{7}}{{\left (c x - 1\right )}^{4}} - \frac {56 \, {\left (c x + 1\right )}^{3} c^{7}}{{\left (c x - 1\right )}^{3}} + \frac {28 \, {\left (c x + 1\right )}^{2} c^{7}}{{\left (c x - 1\right )}^{2}} - \frac {8 \, {\left (c x + 1\right )} c^{7}}{c x - 1} + c^{7}} - \frac {2 \, b^{2} d^{2} \log \left (-\frac {c x + 1}{c x - 1} + 1\right )}{c^{7}} + \frac {2 \, b^{2} d^{2} \log \left (-\frac {c x + 1}{c x - 1}\right )}{c^{7}}\right )} c^{2} \end {gather*}

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]

1/63*(84*((c*x + 1)^5*b^2*d^2/(c*x - 1)^5 + (c*x + 1)^4*b^2*d^2/(c*x - 1)^4 + (c*x + 1)^3*b^2*d^2/(c*x - 1)^3)
*log(-(c*x + 1)/(c*x - 1))^2/((c*x + 1)^8*c^7/(c*x - 1)^8 - 8*(c*x + 1)^7*c^7/(c*x - 1)^7 + 28*(c*x + 1)^6*c^7
/(c*x - 1)^6 - 56*(c*x + 1)^5*c^7/(c*x - 1)^5 + 70*(c*x + 1)^4*c^7/(c*x - 1)^4 - 56*(c*x + 1)^3*c^7/(c*x - 1)^
3 + 28*(c*x + 1)^2*c^7/(c*x - 1)^2 - 8*(c*x + 1)*c^7/(c*x - 1) + c^7) + 2*(168*(c*x + 1)^5*a*b*d^2/(c*x - 1)^5
+ 168*(c*x + 1)^4*a*b*d^2/(c*x - 1)^4 + 168*(c*x + 1)^3*a*b*d^2/(c*x - 1)^3 + 28*(c*x + 1)^5*b^2*d^2/(c*x - 1
)^5 - 35*(c*x + 1)^4*b^2*d^2/(c*x - 1)^4 + 28*(c*x + 1)^3*b^2*d^2/(c*x - 1)^3 - 28*(c*x + 1)^2*b^2*d^2/(c*x -
1)^2 + 8*(c*x + 1)*b^2*d^2/(c*x - 1) - b^2*d^2)*log(-(c*x + 1)/(c*x - 1))/((c*x + 1)^8*c^7/(c*x - 1)^8 - 8*(c*
x + 1)^7*c^7/(c*x - 1)^7 + 28*(c*x + 1)^6*c^7/(c*x - 1)^6 - 56*(c*x + 1)^5*c^7/(c*x - 1)^5 + 70*(c*x + 1)^4*c^
7/(c*x - 1)^4 - 56*(c*x + 1)^3*c^7/(c*x - 1)^3 + 28*(c*x + 1)^2*c^7/(c*x - 1)^2 - 8*(c*x + 1)*c^7/(c*x - 1) +
c^7) + (336*(c*x + 1)^5*a^2*d^2/(c*x - 1)^5 + 336*(c*x + 1)^4*a^2*d^2/(c*x - 1)^4 + 336*(c*x + 1)^3*a^2*d^2/(c
*x - 1)^3 + 112*(c*x + 1)^5*a*b*d^2/(c*x - 1)^5 - 140*(c*x + 1)^4*a*b*d^2/(c*x - 1)^4 + 112*(c*x + 1)^3*a*b*d^
2/(c*x - 1)^3 - 112*(c*x + 1)^2*a*b*d^2/(c*x - 1)^2 + 32*(c*x + 1)*a*b*d^2/(c*x - 1) - 4*a*b*d^2 - 2*(c*x + 1)
^7*b^2*d^2/(c*x - 1)^7 + 15*(c*x + 1)^6*b^2*d^2/(c*x - 1)^6 - 30*(c*x + 1)^5*b^2*d^2/(c*x - 1)^5 + 34*(c*x + 1
)^4*b^2*d^2/(c*x - 1)^4 - 30*(c*x + 1)^3*b^2*d^2/(c*x - 1)^3 + 15*(c*x + 1)^2*b^2*d^2/(c*x - 1)^2 - 2*(c*x + 1
)*b^2*d^2/(c*x - 1))/((c*x + 1)^8*c^7/(c*x - 1)^8 - 8*(c*x + 1)^7*c^7/(c*x - 1)^7 + 28*(c*x + 1)^6*c^7/(c*x -
1)^6 - 56*(c*x + 1)^5*c^7/(c*x - 1)^5 + 70*(c*x + 1)^4*c^7/(c*x - 1)^4 - 56*(c*x + 1)^3*c^7/(c*x - 1)^3 + 28*(
c*x + 1)^2*c^7/(c*x - 1)^2 - 8*(c*x + 1)*c^7/(c*x - 1) + c^7) - 2*b^2*d^2*log(-(c*x + 1)/(c*x - 1) + 1)/c^7 +
2*b^2*d^2*log(-(c*x + 1)/(c*x - 1))/c^7)*c^2

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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