3.1.88 \(\int \frac {(d+c d x)^3 (a+b \tanh ^{-1}(c x))^2}{x} \, dx\) [88]

Optimal. Leaf size=355 \[ 3 a b c d^3 x+\frac {1}{3} b^2 c d^3 x-\frac {1}{3} b^2 d^3 \tanh ^{-1}(c x)+3 b^2 c d^3 x \tanh ^{-1}(c x)+\frac {1}{3} b c^2 d^3 x^2 \left (a+b \tanh ^{-1}(c x)\right )+\frac {11}{6} d^3 \left (a+b \tanh ^{-1}(c x)\right )^2+3 c d^3 x \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {3}{2} c^2 d^3 x^2 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {1}{3} c^3 d^3 x^3 \left (a+b \tanh ^{-1}(c x)\right )^2+2 d^3 \left (a+b \tanh ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac {2}{1-c x}\right )-\frac {20}{3} b d^3 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right )+\frac {3}{2} b^2 d^3 \log \left (1-c^2 x^2\right )-\frac {10}{3} b^2 d^3 \text {PolyLog}\left (2,1-\frac {2}{1-c x}\right )-b d^3 \left (a+b \tanh ^{-1}(c x)\right ) \text {PolyLog}\left (2,1-\frac {2}{1-c x}\right )+b d^3 \left (a+b \tanh ^{-1}(c x)\right ) \text {PolyLog}\left (2,-1+\frac {2}{1-c x}\right )+\frac {1}{2} b^2 d^3 \text {PolyLog}\left (3,1-\frac {2}{1-c x}\right )-\frac {1}{2} b^2 d^3 \text {PolyLog}\left (3,-1+\frac {2}{1-c x}\right ) \]

[Out]

3*a*b*c*d^3*x+1/3*b^2*c*d^3*x-1/3*b^2*d^3*arctanh(c*x)+3*b^2*c*d^3*x*arctanh(c*x)+1/3*b*c^2*d^3*x^2*(a+b*arcta
nh(c*x))+11/6*d^3*(a+b*arctanh(c*x))^2+3*c*d^3*x*(a+b*arctanh(c*x))^2+3/2*c^2*d^3*x^2*(a+b*arctanh(c*x))^2+1/3
*c^3*d^3*x^3*(a+b*arctanh(c*x))^2-2*d^3*(a+b*arctanh(c*x))^2*arctanh(-1+2/(-c*x+1))-20/3*b*d^3*(a+b*arctanh(c*
x))*ln(2/(-c*x+1))+3/2*b^2*d^3*ln(-c^2*x^2+1)-10/3*b^2*d^3*polylog(2,1-2/(-c*x+1))-b*d^3*(a+b*arctanh(c*x))*po
lylog(2,1-2/(-c*x+1))+b*d^3*(a+b*arctanh(c*x))*polylog(2,-1+2/(-c*x+1))+1/2*b^2*d^3*polylog(3,1-2/(-c*x+1))-1/
2*b^2*d^3*polylog(3,-1+2/(-c*x+1))

________________________________________________________________________________________

Rubi [A]
time = 0.59, antiderivative size = 355, normalized size of antiderivative = 1.00, number of steps used = 28, number of rules used = 16, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.727, Rules used = {6087, 6021, 6131, 6055, 2449, 2352, 6033, 6199, 6095, 6205, 6745, 6037, 6127, 266, 327, 212} \begin {gather*} \frac {1}{3} c^3 d^3 x^3 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {3}{2} c^2 d^3 x^2 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {1}{3} b c^2 d^3 x^2 \left (a+b \tanh ^{-1}(c x)\right )-b d^3 \text {Li}_2\left (1-\frac {2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )+b d^3 \text {Li}_2\left (\frac {2}{1-c x}-1\right ) \left (a+b \tanh ^{-1}(c x)\right )+3 a b c d^3 x+3 c d^3 x \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {11}{6} d^3 \left (a+b \tanh ^{-1}(c x)\right )^2+2 d^3 \tanh ^{-1}\left (1-\frac {2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )^2-\frac {20}{3} b d^3 \log \left (\frac {2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )+\frac {3}{2} b^2 d^3 \log \left (1-c^2 x^2\right )-\frac {10}{3} b^2 d^3 \text {Li}_2\left (1-\frac {2}{1-c x}\right )+\frac {1}{2} b^2 d^3 \text {Li}_3\left (1-\frac {2}{1-c x}\right )-\frac {1}{2} b^2 d^3 \text {Li}_3\left (\frac {2}{1-c x}-1\right )+\frac {1}{3} b^2 c d^3 x-\frac {1}{3} b^2 d^3 \tanh ^{-1}(c x)+3 b^2 c d^3 x \tanh ^{-1}(c x) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

3*a*b*c*d^3*x + (b^2*c*d^3*x)/3 - (b^2*d^3*ArcTanh[c*x])/3 + 3*b^2*c*d^3*x*ArcTanh[c*x] + (b*c^2*d^3*x^2*(a +
b*ArcTanh[c*x]))/3 + (11*d^3*(a + b*ArcTanh[c*x])^2)/6 + 3*c*d^3*x*(a + b*ArcTanh[c*x])^2 + (3*c^2*d^3*x^2*(a
+ b*ArcTanh[c*x])^2)/2 + (c^3*d^3*x^3*(a + b*ArcTanh[c*x])^2)/3 + 2*d^3*(a + b*ArcTanh[c*x])^2*ArcTanh[1 - 2/(
1 - c*x)] - (20*b*d^3*(a + b*ArcTanh[c*x])*Log[2/(1 - c*x)])/3 + (3*b^2*d^3*Log[1 - c^2*x^2])/2 - (10*b^2*d^3*
PolyLog[2, 1 - 2/(1 - c*x)])/3 - b*d^3*(a + b*ArcTanh[c*x])*PolyLog[2, 1 - 2/(1 - c*x)] + b*d^3*(a + b*ArcTanh
[c*x])*PolyLog[2, -1 + 2/(1 - c*x)] + (b^2*d^3*PolyLog[3, 1 - 2/(1 - c*x)])/2 - (b^2*d^3*PolyLog[3, -1 + 2/(1
- c*x)])/2

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 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 6033

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

Rule 6199

Int[(ArcTanh[u_]*((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Dist[1/2, Int[
Log[1 + u]*((a + b*ArcTanh[c*x])^p/(d + e*x^2)), x], x] - Dist[1/2, Int[Log[1 - u]*((a + b*ArcTanh[c*x])^p/(d
+ e*x^2)), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[c^2*d + e, 0] && EqQ[u^2 - (1 - 2/(1 - c*x
))^2, 0]

Rule 6205

Int[(Log[u_]*((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[(-(a + b*ArcT
anh[c*x])^p)*(PolyLog[2, 1 - u]/(2*c*d)), x] + Dist[b*(p/2), Int[(a + b*ArcTanh[c*x])^(p - 1)*(PolyLog[2, 1 -
u]/(d + e*x^2)), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[c^2*d + e, 0] && EqQ[(1 - u)^2 - (1
- 2/(1 - c*x))^2, 0]

Rule 6745

Int[(u_)*PolyLog[n_, v_], x_Symbol] :> With[{w = DerivativeDivides[v, u*v, x]}, Simp[w*PolyLog[n + 1, v], x] /
;  !FalseQ[w]] /; FreeQ[n, x]

Rubi steps

\begin {align*} \int \frac {(d+c d x)^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{x} \, dx &=\int \left (3 c d^3 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{x}+3 c^2 d^3 x \left (a+b \tanh ^{-1}(c x)\right )^2+c^3 d^3 x^2 \left (a+b \tanh ^{-1}(c x)\right )^2\right ) \, dx\\ &=d^3 \int \frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{x} \, dx+\left (3 c d^3\right ) \int \left (a+b \tanh ^{-1}(c x)\right )^2 \, dx+\left (3 c^2 d^3\right ) \int x \left (a+b \tanh ^{-1}(c x)\right )^2 \, dx+\left (c^3 d^3\right ) \int x^2 \left (a+b \tanh ^{-1}(c x)\right )^2 \, dx\\ &=3 c d^3 x \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {3}{2} c^2 d^3 x^2 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {1}{3} c^3 d^3 x^3 \left (a+b \tanh ^{-1}(c x)\right )^2+2 d^3 \left (a+b \tanh ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac {2}{1-c x}\right )-\left (4 b c d^3\right ) \int \frac {\left (a+b \tanh ^{-1}(c x)\right ) \tanh ^{-1}\left (1-\frac {2}{1-c x}\right )}{1-c^2 x^2} \, dx-\left (6 b c^2 d^3\right ) \int \frac {x \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx-\left (3 b c^3 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 (2 b c^4 d^3\right ) \int \frac {x^3 \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx\\ &=3 d^3 \left (a+b \tanh ^{-1}(c x)\right )^2+3 c d^3 x \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {3}{2} c^2 d^3 x^2 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {1}{3} c^3 d^3 x^3 \left (a+b \tanh ^{-1}(c x)\right )^2+2 d^3 \left (a+b \tanh ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac {2}{1-c x}\right )+\left (2 b c d^3\right ) \int \frac {\left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right )}{1-c^2 x^2} \, dx-\left (2 b c d^3\right ) \int \frac {\left (a+b \tanh ^{-1}(c x)\right ) \log \left (2-\frac {2}{1-c x}\right )}{1-c^2 x^2} \, dx+\left (3 b c d^3\right ) \int \left (a+b \tanh ^{-1}(c x)\right ) \, dx-\left (3 b c d^3\right ) \int \frac {a+b \tanh ^{-1}(c x)}{1-c^2 x^2} \, dx-\left (6 b c d^3\right ) \int \frac {a+b \tanh ^{-1}(c x)}{1-c x} \, dx+\frac {1}{3} \left (2 b c^2 d^3\right ) \int x \left (a+b \tanh ^{-1}(c x)\right ) \, dx-\frac {1}{3} \left (2 b c^2 d^3\right ) \int \frac {x \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx\\ &=3 a b c d^3 x+\frac {1}{3} b c^2 d^3 x^2 \left (a+b \tanh ^{-1}(c x)\right )+\frac {11}{6} d^3 \left (a+b \tanh ^{-1}(c x)\right )^2+3 c d^3 x \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {3}{2} c^2 d^3 x^2 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {1}{3} c^3 d^3 x^3 \left (a+b \tanh ^{-1}(c x)\right )^2+2 d^3 \left (a+b \tanh ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac {2}{1-c x}\right )-6 b d^3 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right )-b d^3 \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1-c x}\right )+b d^3 \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (-1+\frac {2}{1-c x}\right )-\frac {1}{3} \left (2 b c d^3\right ) \int \frac {a+b \tanh ^{-1}(c x)}{1-c x} \, dx+\left (b^2 c d^3\right ) \int \frac {\text {Li}_2\left (1-\frac {2}{1-c x}\right )}{1-c^2 x^2} \, dx-\left (b^2 c d^3\right ) \int \frac {\text {Li}_2\left (-1+\frac {2}{1-c x}\right )}{1-c^2 x^2} \, dx+\left (3 b^2 c d^3\right ) \int \tanh ^{-1}(c x) \, dx+\left (6 b^2 c d^3\right ) \int \frac {\log \left (\frac {2}{1-c x}\right )}{1-c^2 x^2} \, dx-\frac {1}{3} \left (b^2 c^3 d^3\right ) \int \frac {x^2}{1-c^2 x^2} \, dx\\ &=3 a b c d^3 x+\frac {1}{3} b^2 c d^3 x+3 b^2 c d^3 x \tanh ^{-1}(c x)+\frac {1}{3} b c^2 d^3 x^2 \left (a+b \tanh ^{-1}(c x)\right )+\frac {11}{6} d^3 \left (a+b \tanh ^{-1}(c x)\right )^2+3 c d^3 x \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {3}{2} c^2 d^3 x^2 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {1}{3} c^3 d^3 x^3 \left (a+b \tanh ^{-1}(c x)\right )^2+2 d^3 \left (a+b \tanh ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac {2}{1-c x}\right )-\frac {20}{3} b d^3 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right )-b d^3 \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1-c x}\right )+b d^3 \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (-1+\frac {2}{1-c x}\right )+\frac {1}{2} b^2 d^3 \text {Li}_3\left (1-\frac {2}{1-c x}\right )-\frac {1}{2} b^2 d^3 \text {Li}_3\left (-1+\frac {2}{1-c x}\right )-\left (6 b^2 d^3\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-c x}\right )-\frac {1}{3} \left (b^2 c d^3\right ) \int \frac {1}{1-c^2 x^2} \, dx+\frac {1}{3} \left (2 b^2 c d^3\right ) \int \frac {\log \left (\frac {2}{1-c x}\right )}{1-c^2 x^2} \, dx-\left (3 b^2 c^2 d^3\right ) \int \frac {x}{1-c^2 x^2} \, dx\\ &=3 a b c d^3 x+\frac {1}{3} b^2 c d^3 x-\frac {1}{3} b^2 d^3 \tanh ^{-1}(c x)+3 b^2 c d^3 x \tanh ^{-1}(c x)+\frac {1}{3} b c^2 d^3 x^2 \left (a+b \tanh ^{-1}(c x)\right )+\frac {11}{6} d^3 \left (a+b \tanh ^{-1}(c x)\right )^2+3 c d^3 x \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {3}{2} c^2 d^3 x^2 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {1}{3} c^3 d^3 x^3 \left (a+b \tanh ^{-1}(c x)\right )^2+2 d^3 \left (a+b \tanh ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac {2}{1-c x}\right )-\frac {20}{3} b d^3 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right )+\frac {3}{2} b^2 d^3 \log \left (1-c^2 x^2\right )-3 b^2 d^3 \text {Li}_2\left (1-\frac {2}{1-c x}\right )-b d^3 \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1-c x}\right )+b d^3 \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (-1+\frac {2}{1-c x}\right )+\frac {1}{2} b^2 d^3 \text {Li}_3\left (1-\frac {2}{1-c x}\right )-\frac {1}{2} b^2 d^3 \text {Li}_3\left (-1+\frac {2}{1-c x}\right )-\frac {1}{3} \left (2 b^2 d^3\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-c x}\right )\\ &=3 a b c d^3 x+\frac {1}{3} b^2 c d^3 x-\frac {1}{3} b^2 d^3 \tanh ^{-1}(c x)+3 b^2 c d^3 x \tanh ^{-1}(c x)+\frac {1}{3} b c^2 d^3 x^2 \left (a+b \tanh ^{-1}(c x)\right )+\frac {11}{6} d^3 \left (a+b \tanh ^{-1}(c x)\right )^2+3 c d^3 x \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {3}{2} c^2 d^3 x^2 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {1}{3} c^3 d^3 x^3 \left (a+b \tanh ^{-1}(c x)\right )^2+2 d^3 \left (a+b \tanh ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac {2}{1-c x}\right )-\frac {20}{3} b d^3 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right )+\frac {3}{2} b^2 d^3 \log \left (1-c^2 x^2\right )-\frac {10}{3} b^2 d^3 \text {Li}_2\left (1-\frac {2}{1-c x}\right )-b d^3 \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1-c x}\right )+b d^3 \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (-1+\frac {2}{1-c x}\right )+\frac {1}{2} b^2 d^3 \text {Li}_3\left (1-\frac {2}{1-c x}\right )-\frac {1}{2} b^2 d^3 \text {Li}_3\left (-1+\frac {2}{1-c x}\right )\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 0.44, size = 448, normalized size = 1.26 \begin {gather*} \frac {1}{24} d^3 \left (i b^2 \pi ^3+72 a^2 c x+72 a b c x+8 b^2 c x+36 a^2 c^2 x^2+8 a b c^2 x^2+8 a^2 c^3 x^3-8 b^2 \tanh ^{-1}(c x)+144 a b c x \tanh ^{-1}(c x)+72 b^2 c x \tanh ^{-1}(c x)+72 a b c^2 x^2 \tanh ^{-1}(c x)+8 b^2 c^2 x^2 \tanh ^{-1}(c x)+16 a b c^3 x^3 \tanh ^{-1}(c x)-116 b^2 \tanh ^{-1}(c x)^2+72 b^2 c x \tanh ^{-1}(c x)^2+36 b^2 c^2 x^2 \tanh ^{-1}(c x)^2+8 b^2 c^3 x^3 \tanh ^{-1}(c x)^2-16 b^2 \tanh ^{-1}(c x)^3-160 b^2 \tanh ^{-1}(c x) \log \left (1+e^{-2 \tanh ^{-1}(c x)}\right )-24 b^2 \tanh ^{-1}(c x)^2 \log \left (1+e^{-2 \tanh ^{-1}(c x)}\right )+24 b^2 \tanh ^{-1}(c x)^2 \log \left (1-e^{2 \tanh ^{-1}(c x)}\right )+24 a^2 \log (c x)+36 a b \log (1-c x)-36 a b \log (1+c x)+72 a b \log \left (1-c^2 x^2\right )+36 b^2 \log \left (1-c^2 x^2\right )+8 a b \log \left (-1+c^2 x^2\right )+8 b^2 \left (10+3 \tanh ^{-1}(c x)\right ) \text {PolyLog}\left (2,-e^{-2 \tanh ^{-1}(c x)}\right )+24 b^2 \tanh ^{-1}(c x) \text {PolyLog}\left (2,e^{2 \tanh ^{-1}(c x)}\right )-24 a b \text {PolyLog}(2,-c x)+24 a b \text {PolyLog}(2,c x)+12 b^2 \text {PolyLog}\left (3,-e^{-2 \tanh ^{-1}(c x)}\right )-12 b^2 \text {PolyLog}\left (3,e^{2 \tanh ^{-1}(c x)}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(d^3*(I*b^2*Pi^3 + 72*a^2*c*x + 72*a*b*c*x + 8*b^2*c*x + 36*a^2*c^2*x^2 + 8*a*b*c^2*x^2 + 8*a^2*c^3*x^3 - 8*b^
2*ArcTanh[c*x] + 144*a*b*c*x*ArcTanh[c*x] + 72*b^2*c*x*ArcTanh[c*x] + 72*a*b*c^2*x^2*ArcTanh[c*x] + 8*b^2*c^2*
x^2*ArcTanh[c*x] + 16*a*b*c^3*x^3*ArcTanh[c*x] - 116*b^2*ArcTanh[c*x]^2 + 72*b^2*c*x*ArcTanh[c*x]^2 + 36*b^2*c
^2*x^2*ArcTanh[c*x]^2 + 8*b^2*c^3*x^3*ArcTanh[c*x]^2 - 16*b^2*ArcTanh[c*x]^3 - 160*b^2*ArcTanh[c*x]*Log[1 + E^
(-2*ArcTanh[c*x])] - 24*b^2*ArcTanh[c*x]^2*Log[1 + E^(-2*ArcTanh[c*x])] + 24*b^2*ArcTanh[c*x]^2*Log[1 - E^(2*A
rcTanh[c*x])] + 24*a^2*Log[c*x] + 36*a*b*Log[1 - c*x] - 36*a*b*Log[1 + c*x] + 72*a*b*Log[1 - c^2*x^2] + 36*b^2
*Log[1 - c^2*x^2] + 8*a*b*Log[-1 + c^2*x^2] + 8*b^2*(10 + 3*ArcTanh[c*x])*PolyLog[2, -E^(-2*ArcTanh[c*x])] + 2
4*b^2*ArcTanh[c*x]*PolyLog[2, E^(2*ArcTanh[c*x])] - 24*a*b*PolyLog[2, -(c*x)] + 24*a*b*PolyLog[2, c*x] + 12*b^
2*PolyLog[3, -E^(-2*ArcTanh[c*x])] - 12*b^2*PolyLog[3, E^(2*ArcTanh[c*x])]))/24

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 6.67, size = 1186, normalized size = 3.34

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(1186\)
default	\(\text {Expression too large to display}\)	\(1186\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*I*d^3*b^2*arctanh(c*x)^2*Pi*csgn(I*((c*x+1)^2/(-c^2*x^2+1)-1)/(1+(c*x+1)^2/(-c^2*x^2+1)))^3+6*d^3*a*b*arct
anh(c*x)*c*x+1/2*I*d^3*b^2*arctanh(c*x)^2*Pi*csgn(I*((c*x+1)^2/(-c^2*x^2+1)-1))*csgn(I/(1+(c*x+1)^2/(-c^2*x^2+
1)))*csgn(I*((c*x+1)^2/(-c^2*x^2+1)-1)/(1+(c*x+1)^2/(-c^2*x^2+1)))-1/3*d^3*b^2+3*d^3*a*b*arctanh(c*x)*c^2*x^2+
2/3*d^3*a*b*arctanh(c*x)*c^3*x^3+3*d^3*b^2*arctanh(c*x)^2*c*x+3*a*b*c*d^3*x+3*b^2*c*d^3*x*arctanh(c*x)+1/3*d^3
*b^2*arctanh(c*x)^2*c^3*x^3+3/2*d^3*b^2*arctanh(c*x)^2*c^2*x^2+1/3*d^3*b^2*arctanh(c*x)*c^2*x^2+11/6*d^3*b^2*a
rctanh(c*x)^2+29/6*d^3*a*b*ln(c*x-1)+11/6*d^3*a*b*ln(c*x+1)+1/3*d^3*a*b*c^2*x^2+1/3*b^2*c*d^3*x+8/3*b^2*d^3*ar
ctanh(c*x)+2*d^3*a*b*arctanh(c*x)*ln(c*x)-d^3*a*b*ln(c*x)*ln(c*x+1)+3*d^3*a^2*c*x+1/3*d^3*a^2*c^3*x^3+3/2*d^3*
a^2*c^2*x^2-20/3*d^3*b^2*dilog(1-I*(c*x+1)/(-c^2*x^2+1)^(1/2))-3*d^3*b^2*ln(1+(c*x+1)^2/(-c^2*x^2+1))-20/3*d^3
*b^2*dilog(1+I*(c*x+1)/(-c^2*x^2+1)^(1/2))+1/2*d^3*b^2*polylog(3,-(c*x+1)^2/(-c^2*x^2+1))-2*d^3*b^2*polylog(3,
(c*x+1)/(-c^2*x^2+1)^(1/2))-2*d^3*b^2*polylog(3,-(c*x+1)/(-c^2*x^2+1)^(1/2))+d^3*a^2*ln(c*x)-1/2*I*d^3*b^2*arc
tanh(c*x)^2*Pi*csgn(I*((c*x+1)^2/(-c^2*x^2+1)-1))*csgn(I*((c*x+1)^2/(-c^2*x^2+1)-1)/(1+(c*x+1)^2/(-c^2*x^2+1))
)^2-1/2*I*d^3*b^2*arctanh(c*x)^2*Pi*csgn(I/(1+(c*x+1)^2/(-c^2*x^2+1)))*csgn(I*((c*x+1)^2/(-c^2*x^2+1)-1)/(1+(c
*x+1)^2/(-c^2*x^2+1)))^2-d^3*a*b*dilog(c*x)-d^3*a*b*dilog(c*x+1)+d^3*b^2*arctanh(c*x)^2*ln(c*x)-20/3*d^3*b^2*a
rctanh(c*x)*ln(1+I*(c*x+1)/(-c^2*x^2+1)^(1/2))-20/3*d^3*b^2*arctanh(c*x)*ln(1-I*(c*x+1)/(-c^2*x^2+1)^(1/2))-d^
3*b^2*arctanh(c*x)^2*ln((c*x+1)^2/(-c^2*x^2+1)-1)+d^3*b^2*arctanh(c*x)^2*ln(1+(c*x+1)/(-c^2*x^2+1)^(1/2))+2*d^
3*b^2*arctanh(c*x)*polylog(2,-(c*x+1)/(-c^2*x^2+1)^(1/2))+d^3*b^2*arctanh(c*x)^2*ln(1-(c*x+1)/(-c^2*x^2+1)^(1/
2))+2*d^3*b^2*arctanh(c*x)*polylog(2,(c*x+1)/(-c^2*x^2+1)^(1/2))-d^3*b^2*arctanh(c*x)*polylog(2,-(c*x+1)^2/(-c
^2*x^2+1))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*a^2*c^3*d^3*x^3 + 3/2*a^2*c^2*d^3*x^2 + 3*a^2*c*d^3*x + 3*(2*c*x*arctanh(c*x) + log(-c^2*x^2 + 1))*a*b*d^3
+ a^2*d^3*log(x) + 1/24*(2*b^2*c^3*d^3*x^3 + 9*b^2*c^2*d^3*x^2 + 18*b^2*c*d^3*x)*log(-c*x + 1)^2 - integrate(
-1/12*(3*(b^2*c^4*d^3*x^4 + 2*b^2*c^3*d^3*x^3 - 2*b^2*c*d^3*x - b^2*d^3)*log(c*x + 1)^2 + 12*(a*b*c^4*d^3*x^4
+ 2*a*b*c^3*d^3*x^3 - 3*a*b*c^2*d^3*x^2 + a*b*c*d^3*x - a*b*d^3)*log(c*x + 1) - (12*a*b*c*d^3*x - 12*a*b*d^3 +
2*(6*a*b*c^4*d^3 + b^2*c^4*d^3)*x^4 + 3*(8*a*b*c^3*d^3 + 3*b^2*c^3*d^3)*x^3 - 18*(2*a*b*c^2*d^3 - b^2*c^2*d^3
)*x^2 + 6*(b^2*c^4*d^3*x^4 + 2*b^2*c^3*d^3*x^3 - 2*b^2*c*d^3*x - b^2*d^3)*log(c*x + 1))*log(-c*x + 1))/(c*x^2
- x), x)

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

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} d^{3} \left (\int 3 a^{2} c\, dx + \int \frac {a^{2}}{x}\, dx + \int 3 a^{2} c^{2} x\, dx + \int a^{2} c^{3} x^{2}\, dx + \int 3 b^{2} c \operatorname {atanh}^{2}{\left (c x \right )}\, dx + \int \frac {b^{2} \operatorname {atanh}^{2}{\left (c x \right )}}{x}\, dx + \int 6 a b c \operatorname {atanh}{\left (c x \right )}\, dx + \int \frac {2 a b \operatorname {atanh}{\left (c x \right )}}{x}\, dx + \int 3 b^{2} c^{2} x \operatorname {atanh}^{2}{\left (c x \right )}\, dx + \int b^{2} c^{3} x^{2} \operatorname {atanh}^{2}{\left (c x \right )}\, dx + \int 6 a b c^{2} x \operatorname {atanh}{\left (c x \right )}\, dx + \int 2 a b c^{3} x^{2} \operatorname {atanh}{\left (c x \right )}\, dx\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

d**3*(Integral(3*a**2*c, x) + Integral(a**2/x, x) + Integral(3*a**2*c**2*x, x) + Integral(a**2*c**3*x**2, x) +
Integral(3*b**2*c*atanh(c*x)**2, x) + Integral(b**2*atanh(c*x)**2/x, x) + Integral(6*a*b*c*atanh(c*x), x) + I
ntegral(2*a*b*atanh(c*x)/x, x) + Integral(3*b**2*c**2*x*atanh(c*x)**2, x) + Integral(b**2*c**3*x**2*atanh(c*x)
**2, x) + Integral(6*a*b*c**2*x*atanh(c*x), x) + Integral(2*a*b*c**3*x**2*atanh(c*x), x))

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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