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

Optimal. Leaf size=614 \[ -\frac {b^2 e^3 \log \left (\frac {2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{2 c^4}+\frac {3 a b^2 d e^2 x}{c^2}+\frac {b^2 e^3 x^2 \left (a+b \tanh ^{-1}(c x)\right )}{4 c^2}-\frac {3 b^2 e \left (6 c^2 d^2+e^2\right ) \log \left (\frac {2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{2 c^4}-\frac {3 b^2 d \left (c^2 d^2+e^2\right ) \text {Li}_2\left (1-\frac {2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{c^3}+\frac {b e^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{4 c^4}-\frac {3 b d e^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c^3}+\frac {3 b e \left (6 c^2 d^2+e^2\right ) \left (a+b \tanh ^{-1}(c x)\right )^2}{4 c^4}-\frac {\left (c^4 d^4+6 c^2 d^2 e^2+e^4\right ) \left (a+b \tanh ^{-1}(c x)\right )^3}{4 c^4 e}+\frac {d \left (c^2 d^2+e^2\right ) \left (a+b \tanh ^{-1}(c x)\right )^3}{c^3}+\frac {3 b e x \left (6 c^2 d^2+e^2\right ) \left (a+b \tanh ^{-1}(c x)\right )^2}{4 c^3}-\frac {3 b d \left (c^2 d^2+e^2\right ) \log \left (\frac {2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )^2}{c^3}+\frac {3 b d e^2 x^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c}+\frac {(d+e x)^4 \left (a+b \tanh ^{-1}(c x)\right )^3}{4 e}+\frac {b e^3 x^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{4 c}-\frac {b^3 e^3 \text {Li}_2\left (1-\frac {2}{1-c x}\right )}{4 c^4}-\frac {b^3 e^3 \tanh ^{-1}(c x)}{4 c^4}+\frac {b^3 e^3 x}{4 c^3}+\frac {3 b^3 d e^2 x \tanh ^{-1}(c x)}{c^2}-\frac {3 b^3 e \left (6 c^2 d^2+e^2\right ) \text {Li}_2\left (1-\frac {2}{1-c x}\right )}{4 c^4}+\frac {3 b^3 d \left (c^2 d^2+e^2\right ) \text {Li}_3\left (1-\frac {2}{1-c x}\right )}{2 c^3}+\frac {3 b^3 d e^2 \log \left (1-c^2 x^2\right )}{2 c^3} \]

[Out]

3*a*b^2*d*e^2*x/c^2+1/4*b^3*e^3*x/c^3-1/4*b^3*e^3*arctanh(c*x)/c^4+3*b^3*d*e^2*x*arctanh(c*x)/c^2+1/4*b^2*e^3*
x^2*(a+b*arctanh(c*x))/c^2-3/2*b*d*e^2*(a+b*arctanh(c*x))^2/c^3+1/4*b*e^3*(a+b*arctanh(c*x))^2/c^4+3/4*b*e*(6*
c^2*d^2+e^2)*(a+b*arctanh(c*x))^2/c^4+3/4*b*e*(6*c^2*d^2+e^2)*x*(a+b*arctanh(c*x))^2/c^3+3/2*b*d*e^2*x^2*(a+b*
arctanh(c*x))^2/c+1/4*b*e^3*x^3*(a+b*arctanh(c*x))^2/c+d*(c^2*d^2+e^2)*(a+b*arctanh(c*x))^3/c^3-1/4*(c^4*d^4+6
*c^2*d^2*e^2+e^4)*(a+b*arctanh(c*x))^3/c^4/e+1/4*(e*x+d)^4*(a+b*arctanh(c*x))^3/e-1/2*b^2*e^3*(a+b*arctanh(c*x
))*ln(2/(-c*x+1))/c^4-3/2*b^2*e*(6*c^2*d^2+e^2)*(a+b*arctanh(c*x))*ln(2/(-c*x+1))/c^4-3*b*d*(c^2*d^2+e^2)*(a+b
*arctanh(c*x))^2*ln(2/(-c*x+1))/c^3+3/2*b^3*d*e^2*ln(-c^2*x^2+1)/c^3-1/4*b^3*e^3*polylog(2,1-2/(-c*x+1))/c^4-3
/4*b^3*e*(6*c^2*d^2+e^2)*polylog(2,1-2/(-c*x+1))/c^4-3*b^2*d*(c^2*d^2+e^2)*(a+b*arctanh(c*x))*polylog(2,1-2/(-
c*x+1))/c^3+3/2*b^3*d*(c^2*d^2+e^2)*polylog(3,1-2/(-c*x+1))/c^3

________________________________________________________________________________________

Rubi [A]  time = 1.18, antiderivative size = 614, normalized size of antiderivative = 1.00, number of steps used = 29, number of rules used = 15, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.833, Rules used = {5928, 5910, 5984, 5918, 2402, 2315, 5916, 5980, 260, 5948, 321, 206, 6048, 6058, 6610} \[ -\frac {3 b^2 d \left (c^2 d^2+e^2\right ) \text {PolyLog}\left (2,1-\frac {2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{c^3}-\frac {3 b^3 e \left (6 c^2 d^2+e^2\right ) \text {PolyLog}\left (2,1-\frac {2}{1-c x}\right )}{4 c^4}+\frac {3 b^3 d \left (c^2 d^2+e^2\right ) \text {PolyLog}\left (3,1-\frac {2}{1-c x}\right )}{2 c^3}-\frac {b^3 e^3 \text {PolyLog}\left (2,1-\frac {2}{1-c x}\right )}{4 c^4}-\frac {3 b^2 e \left (6 c^2 d^2+e^2\right ) \log \left (\frac {2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{2 c^4}+\frac {3 a b^2 d e^2 x}{c^2}+\frac {b^2 e^3 x^2 \left (a+b \tanh ^{-1}(c x)\right )}{4 c^2}-\frac {b^2 e^3 \log \left (\frac {2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{2 c^4}+\frac {d \left (c^2 d^2+e^2\right ) \left (a+b \tanh ^{-1}(c x)\right )^3}{c^3}-\frac {\left (6 c^2 d^2 e^2+c^4 d^4+e^4\right ) \left (a+b \tanh ^{-1}(c x)\right )^3}{4 c^4 e}+\frac {3 b e \left (6 c^2 d^2+e^2\right ) \left (a+b \tanh ^{-1}(c x)\right )^2}{4 c^4}+\frac {3 b e x \left (6 c^2 d^2+e^2\right ) \left (a+b \tanh ^{-1}(c x)\right )^2}{4 c^3}-\frac {3 b d \left (c^2 d^2+e^2\right ) \log \left (\frac {2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )^2}{c^3}-\frac {3 b d e^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c^3}+\frac {b e^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{4 c^4}+\frac {3 b d e^2 x^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c}+\frac {(d+e x)^4 \left (a+b \tanh ^{-1}(c x)\right )^3}{4 e}+\frac {b e^3 x^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{4 c}+\frac {3 b^3 d e^2 \log \left (1-c^2 x^2\right )}{2 c^3}+\frac {3 b^3 d e^2 x \tanh ^{-1}(c x)}{c^2}+\frac {b^3 e^3 x}{4 c^3}-\frac {b^3 e^3 \tanh ^{-1}(c x)}{4 c^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(3*a*b^2*d*e^2*x)/c^2 + (b^3*e^3*x)/(4*c^3) - (b^3*e^3*ArcTanh[c*x])/(4*c^4) + (3*b^3*d*e^2*x*ArcTanh[c*x])/c^
2 + (b^2*e^3*x^2*(a + b*ArcTanh[c*x]))/(4*c^2) - (3*b*d*e^2*(a + b*ArcTanh[c*x])^2)/(2*c^3) + (b*e^3*(a + b*Ar
cTanh[c*x])^2)/(4*c^4) + (3*b*e*(6*c^2*d^2 + e^2)*(a + b*ArcTanh[c*x])^2)/(4*c^4) + (3*b*e*(6*c^2*d^2 + e^2)*x
*(a + b*ArcTanh[c*x])^2)/(4*c^3) + (3*b*d*e^2*x^2*(a + b*ArcTanh[c*x])^2)/(2*c) + (b*e^3*x^3*(a + b*ArcTanh[c*
x])^2)/(4*c) + (d*(c^2*d^2 + e^2)*(a + b*ArcTanh[c*x])^3)/c^3 - ((c^4*d^4 + 6*c^2*d^2*e^2 + e^4)*(a + b*ArcTan
h[c*x])^3)/(4*c^4*e) + ((d + e*x)^4*(a + b*ArcTanh[c*x])^3)/(4*e) - (b^2*e^3*(a + b*ArcTanh[c*x])*Log[2/(1 - c
*x)])/(2*c^4) - (3*b^2*e*(6*c^2*d^2 + e^2)*(a + b*ArcTanh[c*x])*Log[2/(1 - c*x)])/(2*c^4) - (3*b*d*(c^2*d^2 +
e^2)*(a + b*ArcTanh[c*x])^2*Log[2/(1 - c*x)])/c^3 + (3*b^3*d*e^2*Log[1 - c^2*x^2])/(2*c^3) - (b^3*e^3*PolyLog[
2, 1 - 2/(1 - c*x)])/(4*c^4) - (3*b^3*e*(6*c^2*d^2 + e^2)*PolyLog[2, 1 - 2/(1 - c*x)])/(4*c^4) - (3*b^2*d*(c^2
*d^2 + e^2)*(a + b*ArcTanh[c*x])*PolyLog[2, 1 - 2/(1 - c*x)])/c^3 + (3*b^3*d*(c^2*d^2 + e^2)*PolyLog[3, 1 - 2/
(1 - c*x)])/(2*c^3)

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 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 5928

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

Rule 5948

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

Rule 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]

Rule 6048

Int[(((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*((f_) + (g_.)*(x_))^(m_.))/((d_) + (e_.)*(x_)^2), x_Symbol] :>
Int[ExpandIntegrand[(a + b*ArcTanh[c*x])^p/(d + e*x^2), (f + g*x)^m, x], x] /; FreeQ[{a, b, c, d, e, f, g}, x]
&& IGtQ[p, 0] && EqQ[c^2*d + e, 0] && IGtQ[m, 0]

Rule 6058

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 6610

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

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Mathematica [A]  time = 2.11, size = 830, normalized size = 1.35 \[ \frac {2 a^3 e^3 x^4 c^4+6 a^2 b x \left (4 d^3+6 e x d^2+4 e^2 x^2 d+e^3 x^3\right ) \tanh ^{-1}(c x) c^4+2 a^2 e^2 (4 a c d+b e) x^3 c^3+12 a^2 d e (a c d+b e) x^2 c^3+24 a b^2 d^3 \left (\tanh ^{-1}(c x) \left ((c x-1) \tanh ^{-1}(c x)-2 \log \left (1+e^{-2 \tanh ^{-1}(c x)}\right )\right )+\text {Li}_2\left (-e^{-2 \tanh ^{-1}(c x)}\right )\right ) c^3+8 b^3 d^3 \left (\left ((c x-1) \tanh ^{-1}(c x)-3 \log \left (1+e^{-2 \tanh ^{-1}(c x)}\right )\right ) \tanh ^{-1}(c x)^2+3 \text {Li}_2\left (-e^{-2 \tanh ^{-1}(c x)}\right ) \tanh ^{-1}(c x)+\frac {3}{2} \text {Li}_3\left (-e^{-2 \tanh ^{-1}(c x)}\right )\right ) c^3+36 a b^2 d^2 e \left (\left (c^2 x^2-1\right ) \tanh ^{-1}(c x)^2+2 c x \tanh ^{-1}(c x)+\log \left (1-c^2 x^2\right )\right ) c^2-12 b^3 d^2 e \left (\tanh ^{-1}(c x) \left (\left (1-c^2 x^2\right ) \tanh ^{-1}(c x)^2+(3-3 c x) \tanh ^{-1}(c x)+6 \log \left (1+e^{-2 \tanh ^{-1}(c x)}\right )\right )-3 \text {Li}_2\left (-e^{-2 \tanh ^{-1}(c x)}\right )\right ) c^2+2 a^2 \left (4 a c^3 d^3+3 b e \left (6 c^2 d^2+e^2\right )\right ) x c+24 a b^2 d e^2 \left (\left (c^3 x^3-1\right ) \tanh ^{-1}(c x)^2+\left (c^2 x^2-2 \log \left (1+e^{-2 \tanh ^{-1}(c x)}\right )-1\right ) \tanh ^{-1}(c x)+c x+\text {Li}_2\left (-e^{-2 \tanh ^{-1}(c x)}\right )\right ) c+4 b^3 d e^2 \left (2 c^3 x^3 \tanh ^{-1}(c x)^3-2 \tanh ^{-1}(c x)^3+3 c^2 x^2 \tanh ^{-1}(c x)^2-6 \log \left (1+e^{-2 \tanh ^{-1}(c x)}\right ) \tanh ^{-1}(c x)^2-3 \tanh ^{-1}(c x)^2+6 c x \tanh ^{-1}(c x)+6 \text {Li}_2\left (-e^{-2 \tanh ^{-1}(c x)}\right ) \tanh ^{-1}(c x)+3 \log \left (1-c^2 x^2\right )+3 \text {Li}_3\left (-e^{-2 \tanh ^{-1}(c x)}\right )\right ) c+3 a^2 b \left (4 c^3 d^3+6 c^2 e d^2+4 c e^2 d+e^3\right ) \log (1-c x)+3 a^2 b \left (4 c^3 d^3-6 c^2 e d^2+4 c e^2 d-e^3\right ) \log (c x+1)+2 a b^2 e^3 \left (c^2 x^2+2 c \left (c^2 x^2+3\right ) \tanh ^{-1}(c x) x+3 \left (c^4 x^4-1\right ) \tanh ^{-1}(c x)^2+4 \log \left (1-c^2 x^2\right )-1\right )+2 b^3 e^3 \left (\left (c^4 x^4-1\right ) \tanh ^{-1}(c x)^3+\left (c^3 x^3+3 c x-4\right ) \tanh ^{-1}(c x)^2+\left (c^2 x^2-8 \log \left (1+e^{-2 \tanh ^{-1}(c x)}\right )-1\right ) \tanh ^{-1}(c x)+c x+4 \text {Li}_2\left (-e^{-2 \tanh ^{-1}(c x)}\right )\right )}{8 c^4} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(2*a^2*c*(4*a*c^3*d^3 + 3*b*e*(6*c^2*d^2 + e^2))*x + 12*a^2*c^3*d*e*(a*c*d + b*e)*x^2 + 2*a^2*c^3*e^2*(4*a*c*d
+ b*e)*x^3 + 2*a^3*c^4*e^3*x^4 + 6*a^2*b*c^4*x*(4*d^3 + 6*d^2*e*x + 4*d*e^2*x^2 + e^3*x^3)*ArcTanh[c*x] + 3*a
^2*b*(4*c^3*d^3 + 6*c^2*d^2*e + 4*c*d*e^2 + e^3)*Log[1 - c*x] + 3*a^2*b*(4*c^3*d^3 - 6*c^2*d^2*e + 4*c*d*e^2 -
e^3)*Log[1 + c*x] + 36*a*b^2*c^2*d^2*e*(2*c*x*ArcTanh[c*x] + (-1 + c^2*x^2)*ArcTanh[c*x]^2 + Log[1 - c^2*x^2]
) + 2*a*b^2*e^3*(-1 + c^2*x^2 + 2*c*x*(3 + c^2*x^2)*ArcTanh[c*x] + 3*(-1 + c^4*x^4)*ArcTanh[c*x]^2 + 4*Log[1 -
c^2*x^2]) - 12*b^3*c^2*d^2*e*(ArcTanh[c*x]*((3 - 3*c*x)*ArcTanh[c*x] + (1 - c^2*x^2)*ArcTanh[c*x]^2 + 6*Log[1
+ E^(-2*ArcTanh[c*x])]) - 3*PolyLog[2, -E^(-2*ArcTanh[c*x])]) + 24*a*b^2*c*d*e^2*(c*x + (-1 + c^3*x^3)*ArcTan
h[c*x]^2 + ArcTanh[c*x]*(-1 + c^2*x^2 - 2*Log[1 + E^(-2*ArcTanh[c*x])]) + PolyLog[2, -E^(-2*ArcTanh[c*x])]) +
24*a*b^2*c^3*d^3*(ArcTanh[c*x]*((-1 + c*x)*ArcTanh[c*x] - 2*Log[1 + E^(-2*ArcTanh[c*x])]) + PolyLog[2, -E^(-2*
ArcTanh[c*x])]) + 2*b^3*e^3*(c*x + (-4 + 3*c*x + c^3*x^3)*ArcTanh[c*x]^2 + (-1 + c^4*x^4)*ArcTanh[c*x]^3 + Arc
Tanh[c*x]*(-1 + c^2*x^2 - 8*Log[1 + E^(-2*ArcTanh[c*x])]) + 4*PolyLog[2, -E^(-2*ArcTanh[c*x])]) + 8*b^3*c^3*d^
3*(ArcTanh[c*x]^2*((-1 + c*x)*ArcTanh[c*x] - 3*Log[1 + E^(-2*ArcTanh[c*x])]) + 3*ArcTanh[c*x]*PolyLog[2, -E^(-
2*ArcTanh[c*x])] + (3*PolyLog[3, -E^(-2*ArcTanh[c*x])])/2) + 4*b^3*c*d*e^2*(6*c*x*ArcTanh[c*x] - 3*ArcTanh[c*x
]^2 + 3*c^2*x^2*ArcTanh[c*x]^2 - 2*ArcTanh[c*x]^3 + 2*c^3*x^3*ArcTanh[c*x]^3 - 6*ArcTanh[c*x]^2*Log[1 + E^(-2*
ArcTanh[c*x])] + 3*Log[1 - c^2*x^2] + 6*ArcTanh[c*x]*PolyLog[2, -E^(-2*ArcTanh[c*x])] + 3*PolyLog[3, -E^(-2*Ar
cTanh[c*x])]))/(8*c^4)

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fricas [F]  time = 1.22, size = 0, normalized size = 0.00 \[ {\rm integral}\left (a^{3} e^{3} x^{3} + 3 \, a^{3} d e^{2} x^{2} + 3 \, a^{3} d^{2} e x + a^{3} d^{3} + {\left (b^{3} e^{3} x^{3} + 3 \, b^{3} d e^{2} x^{2} + 3 \, b^{3} d^{2} e x + b^{3} d^{3}\right )} \operatorname {artanh}\left (c x\right )^{3} + 3 \, {\left (a b^{2} e^{3} x^{3} + 3 \, a b^{2} d e^{2} x^{2} + 3 \, a b^{2} d^{2} e x + a b^{2} d^{3}\right )} \operatorname {artanh}\left (c x\right )^{2} + 3 \, {\left (a^{2} b e^{3} x^{3} + 3 \, a^{2} b d e^{2} x^{2} + 3 \, a^{2} b d^{2} e x + a^{2} b d^{3}\right )} \operatorname {artanh}\left (c x\right ), x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 11.29, size = 6104, normalized size = 9.94 \[ \text {output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

result too large to display

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*a^3*e^3*x^4 + a^3*d*e^2*x^3 + 3/2*a^3*d^2*e*x^2 + 9/4*(2*x^2*arctanh(c*x) + c*(2*x/c^2 - log(c*x + 1)/c^3
+ log(c*x - 1)/c^3))*a^2*b*d^2*e + 3/2*(2*x^3*arctanh(c*x) + c*(x^2/c^2 + log(c^2*x^2 - 1)/c^4))*a^2*b*d*e^2 +
1/8*(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^2*b*e^3 + a^
3*d^3*x + 3/2*(2*c*x*arctanh(c*x) + log(-c^2*x^2 + 1))*a^2*b*d^3/c - 1/32*((b^3*c^4*e^3*x^4 + 4*b^3*c^4*d*e^2*
x^3 + 6*b^3*c^4*d^2*e*x^2 + 4*b^3*c^4*d^3*x - (4*c^3*d^3 + 6*c^2*d^2*e + 4*c*d*e^2 + e^3)*b^3)*log(-c*x + 1)^3
- (6*a*b^2*c^4*e^3*x^4 + 2*(12*a*b^2*c^4*d*e^2 + b^3*c^3*e^3)*x^3 + 12*(3*a*b^2*c^4*d^2*e + b^3*c^3*d*e^2)*x^
2 + 6*(4*a*b^2*c^4*d^3 + (6*c^3*d^2*e + c*e^3)*b^3)*x + 3*(b^3*c^4*e^3*x^4 + 4*b^3*c^4*d*e^2*x^3 + 6*b^3*c^4*d
^2*e*x^2 + 4*b^3*c^4*d^3*x + (4*c^3*d^3 - 6*c^2*d^2*e + 4*c*d*e^2 - e^3)*b^3)*log(c*x + 1))*log(-c*x + 1)^2)/c
^4 - integrate(-1/16*(2*(b^3*c^4*e^3*x^4 - b^3*c^3*d^3 + (3*c^4*d*e^2 - c^3*e^3)*b^3*x^3 + 3*(c^4*d^2*e - c^3*
d*e^2)*b^3*x^2 + (c^4*d^3 - 3*c^3*d^2*e)*b^3*x)*log(c*x + 1)^3 + 12*(a*b^2*c^4*e^3*x^4 - a*b^2*c^3*d^3 + (3*c^
4*d*e^2 - c^3*e^3)*a*b^2*x^3 + 3*(c^4*d^2*e - c^3*d*e^2)*a*b^2*x^2 + (c^4*d^3 - 3*c^3*d^2*e)*a*b^2*x)*log(c*x
+ 1)^2 - (6*a*b^2*c^4*e^3*x^4 + 2*(12*a*b^2*c^4*d*e^2 + b^3*c^3*e^3)*x^3 + 12*(3*a*b^2*c^4*d^2*e + b^3*c^3*d*e
^2)*x^2 + 6*(b^3*c^4*e^3*x^4 - b^3*c^3*d^3 + (3*c^4*d*e^2 - c^3*e^3)*b^3*x^3 + 3*(c^4*d^2*e - c^3*d*e^2)*b^3*x
^2 + (c^4*d^3 - 3*c^3*d^2*e)*b^3*x)*log(c*x + 1)^2 + 6*(4*a*b^2*c^4*d^3 + (6*c^3*d^2*e + c*e^3)*b^3)*x - 3*(8*
a*b^2*c^3*d^3 - (8*a*b^2*c^4*e^3 + b^3*c^4*e^3)*x^4 - (4*c^3*d^3 - 6*c^2*d^2*e + 4*c*d*e^2 - e^3)*b^3 - 4*(b^3
*c^4*d*e^2 + 2*(3*c^4*d*e^2 - c^3*e^3)*a*b^2)*x^3 - 6*(b^3*c^4*d^2*e + 4*(c^4*d^2*e - c^3*d*e^2)*a*b^2)*x^2 -
4*(b^3*c^4*d^3 + 2*(c^4*d^3 - 3*c^3*d^2*e)*a*b^2)*x)*log(c*x + 1))*log(-c*x + 1))/(c^4*x - c^3), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((a + b*atanh(c*x))**3*(d + e*x)**3, x)

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