3.2.20 \(\int (1+c x)^3 (a+b \tanh ^{-1}(c x))^3 \, dx\) [120]

Optimal. Leaf size=306 \[ 3 a b^2 x+\frac {b^3 x}{4}-\frac {b^3 \tanh ^{-1}(c x)}{4 c}+3 b^3 x \tanh ^{-1}(c x)+\frac {1}{4} b^2 c x^2 \left (a+b \tanh ^{-1}(c x)\right )+\frac {4 b \left (a+b \tanh ^{-1}(c x)\right )^2}{c}+\frac {21}{4} b x \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {3}{2} b c x^2 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {1}{4} b c^2 x^3 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {(1+c x)^4 \left (a+b \tanh ^{-1}(c x)\right )^3}{4 c}-\frac {11 b^2 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right )}{c}-\frac {6 b \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2}{1-c x}\right )}{c}+\frac {3 b^3 \log \left (1-c^2 x^2\right )}{2 c}-\frac {11 b^3 \text {PolyLog}\left (2,1-\frac {2}{1-c x}\right )}{2 c}-\frac {6 b^2 \left (a+b \tanh ^{-1}(c x)\right ) \text {PolyLog}\left (2,1-\frac {2}{1-c x}\right )}{c}+\frac {3 b^3 \text {PolyLog}\left (3,1-\frac {2}{1-c x}\right )}{c} \]

[Out]

3*a*b^2*x+1/4*b^3*x-1/4*b^3*arctanh(c*x)/c+3*b^3*x*arctanh(c*x)+1/4*b^2*c*x^2*(a+b*arctanh(c*x))+4*b*(a+b*arct
anh(c*x))^2/c+21/4*b*x*(a+b*arctanh(c*x))^2+3/2*b*c*x^2*(a+b*arctanh(c*x))^2+1/4*b*c^2*x^3*(a+b*arctanh(c*x))^
2+1/4*(c*x+1)^4*(a+b*arctanh(c*x))^3/c-11*b^2*(a+b*arctanh(c*x))*ln(2/(-c*x+1))/c-6*b*(a+b*arctanh(c*x))^2*ln(
2/(-c*x+1))/c+3/2*b^3*ln(-c^2*x^2+1)/c-11/2*b^3*polylog(2,1-2/(-c*x+1))/c-6*b^2*(a+b*arctanh(c*x))*polylog(2,1
-2/(-c*x+1))/c+3*b^3*polylog(3,1-2/(-c*x+1))/c

________________________________________________________________________________________

Rubi [A]
time = 0.48, antiderivative size = 306, normalized size of antiderivative = 1.00, number of steps used = 26, number of rules used = 15, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.833, Rules used = {6065, 6021, 6131, 6055, 2449, 2352, 6037, 6127, 266, 6095, 327, 212, 1600, 6205, 6745} \begin {gather*} -\frac {6 b^2 \text {Li}_2\left (1-\frac {2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{c}+\frac {1}{4} b^2 c x^2 \left (a+b \tanh ^{-1}(c x)\right )-\frac {11 b^2 \log \left (\frac {2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{c}+3 a b^2 x+\frac {1}{4} b c^2 x^3 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {3}{2} b c x^2 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {(c x+1)^4 \left (a+b \tanh ^{-1}(c x)\right )^3}{4 c}+\frac {21}{4} b x \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {4 b \left (a+b \tanh ^{-1}(c x)\right )^2}{c}-\frac {6 b \log \left (\frac {2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )^2}{c}+\frac {3 b^3 \log \left (1-c^2 x^2\right )}{2 c}-\frac {11 b^3 \text {Li}_2\left (1-\frac {2}{1-c x}\right )}{2 c}+\frac {3 b^3 \text {Li}_3\left (1-\frac {2}{1-c x}\right )}{c}+3 b^3 x \tanh ^{-1}(c x)-\frac {b^3 \tanh ^{-1}(c x)}{4 c}+\frac {b^3 x}{4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 + c*x)^3*(a + b*ArcTanh[c*x])^3,x]

[Out]

3*a*b^2*x + (b^3*x)/4 - (b^3*ArcTanh[c*x])/(4*c) + 3*b^3*x*ArcTanh[c*x] + (b^2*c*x^2*(a + b*ArcTanh[c*x]))/4 +
(4*b*(a + b*ArcTanh[c*x])^2)/c + (21*b*x*(a + b*ArcTanh[c*x])^2)/4 + (3*b*c*x^2*(a + b*ArcTanh[c*x])^2)/2 + (
b*c^2*x^3*(a + b*ArcTanh[c*x])^2)/4 + ((1 + c*x)^4*(a + b*ArcTanh[c*x])^3)/(4*c) - (11*b^2*(a + b*ArcTanh[c*x]
)*Log[2/(1 - c*x)])/c - (6*b*(a + b*ArcTanh[c*x])^2*Log[2/(1 - c*x)])/c + (3*b^3*Log[1 - c^2*x^2])/(2*c) - (11
*b^3*PolyLog[2, 1 - 2/(1 - c*x)])/(2*c) - (6*b^2*(a + b*ArcTanh[c*x])*PolyLog[2, 1 - 2/(1 - c*x)])/c + (3*b^3*
PolyLog[3, 1 - 2/(1 - c*x)])/c

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 1600

Int[(u_.)*(Px_)^(p_.)*(Qx_)^(q_.), x_Symbol] :> Int[u*PolynomialQuotient[Px, Qx, x]^p*Qx^(p + q), x] /; FreeQ[
q, x] && PolyQ[Px, x] && PolyQ[Qx, x] && EqQ[PolynomialRemainder[Px, Qx, x], 0] && IntegerQ[p] && LtQ[p*q, 0]

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 6065

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 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 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 (1+c x)^3 \left (a+b \tanh ^{-1}(c x)\right )^3 \, dx &=\frac {(1+c x)^4 \left (a+b \tanh ^{-1}(c x)\right )^3}{4 c}-\frac {1}{4} (3 b) \int \left (-7 \left (a+b \tanh ^{-1}(c x)\right )^2-4 c x \left (a+b \tanh ^{-1}(c x)\right )^2-c^2 x^2 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {8 (1+c x) \left (a+b \tanh ^{-1}(c x)\right )^2}{1-c^2 x^2}\right ) \, dx\\ &=\frac {(1+c x)^4 \left (a+b \tanh ^{-1}(c x)\right )^3}{4 c}+\frac {1}{4} (21 b) \int \left (a+b \tanh ^{-1}(c x)\right )^2 \, dx-(6 b) \int \frac {(1+c x) \left (a+b \tanh ^{-1}(c x)\right )^2}{1-c^2 x^2} \, dx+(3 b c) \int x \left (a+b \tanh ^{-1}(c x)\right )^2 \, dx+\frac {1}{4} \left (3 b c^2\right ) \int x^2 \left (a+b \tanh ^{-1}(c x)\right )^2 \, dx\\ &=\frac {21}{4} b x \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {3}{2} b c x^2 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {1}{4} b c^2 x^3 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {(1+c x)^4 \left (a+b \tanh ^{-1}(c x)\right )^3}{4 c}-(6 b) \int \frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{1-c x} \, dx-\frac {1}{2} \left (21 b^2 c\right ) \int \frac {x \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx-\left (3 b^2 c^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 c^3\right ) \int \frac {x^3 \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx\\ &=\frac {21 b \left (a+b \tanh ^{-1}(c x)\right )^2}{4 c}+\frac {21}{4} b x \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {3}{2} b c x^2 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {1}{4} b c^2 x^3 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {(1+c x)^4 \left (a+b \tanh ^{-1}(c x)\right )^3}{4 c}-\frac {6 b \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2}{1-c x}\right )}{c}+\left (3 b^2\right ) \int \left (a+b \tanh ^{-1}(c x)\right ) \, dx-\left (3 b^2\right ) \int \frac {a+b \tanh ^{-1}(c x)}{1-c^2 x^2} \, dx-\frac {1}{2} \left (21 b^2\right ) \int \frac {a+b \tanh ^{-1}(c x)}{1-c x} \, dx+\left (12 b^2\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+\frac {1}{2} \left (b^2 c\right ) \int x \left (a+b \tanh ^{-1}(c x)\right ) \, dx-\frac {1}{2} \left (b^2 c\right ) \int \frac {x \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx\\ &=3 a b^2 x+\frac {1}{4} b^2 c x^2 \left (a+b \tanh ^{-1}(c x)\right )+\frac {4 b \left (a+b \tanh ^{-1}(c x)\right )^2}{c}+\frac {21}{4} b x \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {3}{2} b c x^2 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {1}{4} b c^2 x^3 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {(1+c x)^4 \left (a+b \tanh ^{-1}(c x)\right )^3}{4 c}-\frac {21 b^2 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right )}{2 c}-\frac {6 b \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2}{1-c x}\right )}{c}-\frac {6 b^2 \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1-c x}\right )}{c}-\frac {1}{2} b^2 \int \frac {a+b \tanh ^{-1}(c x)}{1-c x} \, dx+\left (3 b^3\right ) \int \tanh ^{-1}(c x) \, dx+\left (6 b^3\right ) \int \frac {\text {Li}_2\left (1-\frac {2}{1-c x}\right )}{1-c^2 x^2} \, dx+\frac {1}{2} \left (21 b^3\right ) \int \frac {\log \left (\frac {2}{1-c x}\right )}{1-c^2 x^2} \, dx-\frac {1}{4} \left (b^3 c^2\right ) \int \frac {x^2}{1-c^2 x^2} \, dx\\ &=3 a b^2 x+\frac {b^3 x}{4}+3 b^3 x \tanh ^{-1}(c x)+\frac {1}{4} b^2 c x^2 \left (a+b \tanh ^{-1}(c x)\right )+\frac {4 b \left (a+b \tanh ^{-1}(c x)\right )^2}{c}+\frac {21}{4} b x \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {3}{2} b c x^2 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {1}{4} b c^2 x^3 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {(1+c x)^4 \left (a+b \tanh ^{-1}(c x)\right )^3}{4 c}-\frac {11 b^2 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right )}{c}-\frac {6 b \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2}{1-c x}\right )}{c}-\frac {6 b^2 \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1-c x}\right )}{c}+\frac {3 b^3 \text {Li}_3\left (1-\frac {2}{1-c x}\right )}{c}-\frac {1}{4} b^3 \int \frac {1}{1-c^2 x^2} \, dx+\frac {1}{2} b^3 \int \frac {\log \left (\frac {2}{1-c x}\right )}{1-c^2 x^2} \, dx-\frac {\left (21 b^3\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-c x}\right )}{2 c}-\left (3 b^3 c\right ) \int \frac {x}{1-c^2 x^2} \, dx\\ &=3 a b^2 x+\frac {b^3 x}{4}-\frac {b^3 \tanh ^{-1}(c x)}{4 c}+3 b^3 x \tanh ^{-1}(c x)+\frac {1}{4} b^2 c x^2 \left (a+b \tanh ^{-1}(c x)\right )+\frac {4 b \left (a+b \tanh ^{-1}(c x)\right )^2}{c}+\frac {21}{4} b x \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {3}{2} b c x^2 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {1}{4} b c^2 x^3 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {(1+c x)^4 \left (a+b \tanh ^{-1}(c x)\right )^3}{4 c}-\frac {11 b^2 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right )}{c}-\frac {6 b \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2}{1-c x}\right )}{c}+\frac {3 b^3 \log \left (1-c^2 x^2\right )}{2 c}-\frac {21 b^3 \text {Li}_2\left (1-\frac {2}{1-c x}\right )}{4 c}-\frac {6 b^2 \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1-c x}\right )}{c}+\frac {3 b^3 \text {Li}_3\left (1-\frac {2}{1-c x}\right )}{c}-\frac {b^3 \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-c x}\right )}{2 c}\\ &=3 a b^2 x+\frac {b^3 x}{4}-\frac {b^3 \tanh ^{-1}(c x)}{4 c}+3 b^3 x \tanh ^{-1}(c x)+\frac {1}{4} b^2 c x^2 \left (a+b \tanh ^{-1}(c x)\right )+\frac {4 b \left (a+b \tanh ^{-1}(c x)\right )^2}{c}+\frac {21}{4} b x \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {3}{2} b c x^2 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {1}{4} b c^2 x^3 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {(1+c x)^4 \left (a+b \tanh ^{-1}(c x)\right )^3}{4 c}-\frac {11 b^2 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right )}{c}-\frac {6 b \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2}{1-c x}\right )}{c}+\frac {3 b^3 \log \left (1-c^2 x^2\right )}{2 c}-\frac {11 b^3 \text {Li}_2\left (1-\frac {2}{1-c x}\right )}{2 c}-\frac {6 b^2 \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1-c x}\right )}{c}+\frac {3 b^3 \text {Li}_3\left (1-\frac {2}{1-c x}\right )}{c}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(644\) vs. \(2(306)=612\).
time = 0.89, size = 644, normalized size = 2.10 \begin {gather*} \frac {-2 a b^2+8 a^3 c x+42 a^2 b c x+24 a b^2 c x+2 b^3 c x+12 a^3 c^2 x^2+12 a^2 b c^2 x^2+2 a b^2 c^2 x^2+8 a^3 c^3 x^3+2 a^2 b c^3 x^3+2 a^3 c^4 x^4-24 a b^2 \tanh ^{-1}(c x)-2 b^3 \tanh ^{-1}(c x)+24 a^2 b c x \tanh ^{-1}(c x)+84 a b^2 c x \tanh ^{-1}(c x)+24 b^3 c x \tanh ^{-1}(c x)+36 a^2 b c^2 x^2 \tanh ^{-1}(c x)+24 a b^2 c^2 x^2 \tanh ^{-1}(c x)+2 b^3 c^2 x^2 \tanh ^{-1}(c x)+24 a^2 b c^3 x^3 \tanh ^{-1}(c x)+4 a b^2 c^3 x^3 \tanh ^{-1}(c x)+6 a^2 b c^4 x^4 \tanh ^{-1}(c x)-90 a b^2 \tanh ^{-1}(c x)^2-56 b^3 \tanh ^{-1}(c x)^2+24 a b^2 c x \tanh ^{-1}(c x)^2+42 b^3 c x \tanh ^{-1}(c x)^2+36 a b^2 c^2 x^2 \tanh ^{-1}(c x)^2+12 b^3 c^2 x^2 \tanh ^{-1}(c x)^2+24 a b^2 c^3 x^3 \tanh ^{-1}(c x)^2+2 b^3 c^3 x^3 \tanh ^{-1}(c x)^2+6 a b^2 c^4 x^4 \tanh ^{-1}(c x)^2-30 b^3 \tanh ^{-1}(c x)^3+8 b^3 c x \tanh ^{-1}(c x)^3+12 b^3 c^2 x^2 \tanh ^{-1}(c x)^3+8 b^3 c^3 x^3 \tanh ^{-1}(c x)^3+2 b^3 c^4 x^4 \tanh ^{-1}(c x)^3-96 a b^2 \tanh ^{-1}(c x) \log \left (1+e^{-2 \tanh ^{-1}(c x)}\right )-88 b^3 \tanh ^{-1}(c x) \log \left (1+e^{-2 \tanh ^{-1}(c x)}\right )-48 b^3 \tanh ^{-1}(c x)^2 \log \left (1+e^{-2 \tanh ^{-1}(c x)}\right )+45 a^2 b \log (1-c x)+3 a^2 b \log (1+c x)+44 a b^2 \log \left (1-c^2 x^2\right )+12 b^3 \log \left (1-c^2 x^2\right )+4 b^2 \left (12 a+11 b+12 b \tanh ^{-1}(c x)\right ) \text {PolyLog}\left (2,-e^{-2 \tanh ^{-1}(c x)}\right )+24 b^3 \text {PolyLog}\left (3,-e^{-2 \tanh ^{-1}(c x)}\right )}{8 c} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 + c*x)^3*(a + b*ArcTanh[c*x])^3,x]

[Out]

(-2*a*b^2 + 8*a^3*c*x + 42*a^2*b*c*x + 24*a*b^2*c*x + 2*b^3*c*x + 12*a^3*c^2*x^2 + 12*a^2*b*c^2*x^2 + 2*a*b^2*
c^2*x^2 + 8*a^3*c^3*x^3 + 2*a^2*b*c^3*x^3 + 2*a^3*c^4*x^4 - 24*a*b^2*ArcTanh[c*x] - 2*b^3*ArcTanh[c*x] + 24*a^
2*b*c*x*ArcTanh[c*x] + 84*a*b^2*c*x*ArcTanh[c*x] + 24*b^3*c*x*ArcTanh[c*x] + 36*a^2*b*c^2*x^2*ArcTanh[c*x] + 2
4*a*b^2*c^2*x^2*ArcTanh[c*x] + 2*b^3*c^2*x^2*ArcTanh[c*x] + 24*a^2*b*c^3*x^3*ArcTanh[c*x] + 4*a*b^2*c^3*x^3*Ar
cTanh[c*x] + 6*a^2*b*c^4*x^4*ArcTanh[c*x] - 90*a*b^2*ArcTanh[c*x]^2 - 56*b^3*ArcTanh[c*x]^2 + 24*a*b^2*c*x*Arc
Tanh[c*x]^2 + 42*b^3*c*x*ArcTanh[c*x]^2 + 36*a*b^2*c^2*x^2*ArcTanh[c*x]^2 + 12*b^3*c^2*x^2*ArcTanh[c*x]^2 + 24
*a*b^2*c^3*x^3*ArcTanh[c*x]^2 + 2*b^3*c^3*x^3*ArcTanh[c*x]^2 + 6*a*b^2*c^4*x^4*ArcTanh[c*x]^2 - 30*b^3*ArcTanh
[c*x]^3 + 8*b^3*c*x*ArcTanh[c*x]^3 + 12*b^3*c^2*x^2*ArcTanh[c*x]^3 + 8*b^3*c^3*x^3*ArcTanh[c*x]^3 + 2*b^3*c^4*
x^4*ArcTanh[c*x]^3 - 96*a*b^2*ArcTanh[c*x]*Log[1 + E^(-2*ArcTanh[c*x])] - 88*b^3*ArcTanh[c*x]*Log[1 + E^(-2*Ar
cTanh[c*x])] - 48*b^3*ArcTanh[c*x]^2*Log[1 + E^(-2*ArcTanh[c*x])] + 45*a^2*b*Log[1 - c*x] + 3*a^2*b*Log[1 + c*
x] + 44*a*b^2*Log[1 - c^2*x^2] + 12*b^3*Log[1 - c^2*x^2] + 4*b^2*(12*a + 11*b + 12*b*ArcTanh[c*x])*PolyLog[2,
-E^(-2*ArcTanh[c*x])] + 24*b^3*PolyLog[3, -E^(-2*ArcTanh[c*x])])/(8*c)

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 5.49, size = 883, normalized size = 2.89

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(883\)
default	\(\text {Expression too large to display}\)	\(883\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x+1)^3*(a+b*arctanh(c*x))^3,x,method=_RETURNVERBOSE)

[Out]

1/c*(-1/4*b^3+3/2*a^2*b*c^2*x^2+3/2*b^3*arctanh(c*x)^2*c^2*x^2+3*b^3*arctanh(c*x)*c*x-6*a*b^2*dilog(1/2*c*x+1/
2)-6*b^3*arctanh(c*x)*polylog(2,-(c*x+1)^2/(-c^2*x^2+1))-6*b^3*ln(2)*arctanh(c*x)^2+3*a*b^2*c^2*x^2*arctanh(c*
x)-3*b^3*ln(1+(c*x+1)^2/(-c^2*x^2+1))+3*b^3*polylog(3,-(c*x+1)^2/(-c^2*x^2+1))+3/4*a^2*b*arctanh(c*x)-11*b^3*a
rctanh(c*x)*ln(1+I*(c*x+1)/(-c^2*x^2+1)^(1/2))-11*b^3*arctanh(c*x)*ln(1-I*(c*x+1)/(-c^2*x^2+1)^(1/2))+3/4*a*b^
2*arctanh(c*x)^2+1/4*b^3*arctanh(c*x)^2*c^3*x^3+21/4*b^3*arctanh(c*x)^2*c*x+1/4*b^3*arctanh(c*x)^3*c^4*x^4+b^3
*arctanh(c*x)^3*c^3*x^3+3/2*b^3*arctanh(c*x)^3*c^2*x^2+b^3*arctanh(c*x)^3*c*x+1/4*b^3*arctanh(c*x)*c^2*x^2+6*a
^2*b*ln(c*x-1)+6*b^3*arctanh(c*x)^2*ln(c*x-1)+3*a*b^2*ln(c*x-1)^2+7*a*b^2*ln(c*x-1)+4*a*b^2*ln(c*x+1)+1/4*b^3*
c*x-6*a*b^2*ln(c*x-1)*ln(1/2*c*x+1/2)+1/4*a^2*b*c^3*x^3+21/4*a^2*b*c*x+1/4*a*b^2*c^2*x^2+12*a*b^2*arctanh(c*x)
*ln(c*x-1)-13/4*a*b^2+11/4*b^3*arctanh(c*x)+4*b^3*arctanh(c*x)^2+1/4*b^3*arctanh(c*x)^3-6*I*b^3*Pi*csgn(I/(1+(
c*x+1)^2/(-c^2*x^2+1)))^3*arctanh(c*x)^2+6*I*b^3*Pi*csgn(I/(1+(c*x+1)^2/(-c^2*x^2+1)))^2*arctanh(c*x)^2+3/4*a^
2*b*arctanh(c*x)*c^4*x^4+3*a^2*b*arctanh(c*x)*c^3*x^3+9/2*a^2*b*arctanh(c*x)*c^2*x^2+3*a^2*b*arctanh(c*x)*c*x+
3/4*a*b^2*arctanh(c*x)^2*c^4*x^4+3*a*b^2*arctanh(c*x)^2*c^3*x^3+9/2*a*b^2*arctanh(c*x)^2*c^2*x^2+3*a*b^2*arcta
nh(c*x)^2*c*x+1/2*a*b^2*arctanh(c*x)*c^3*x^3+21/2*a*b^2*arctanh(c*x)*c*x-6*I*b^3*Pi*arctanh(c*x)^2+3*b^2*c*x*a
-11*b^3*dilog(1+I*(c*x+1)/(-c^2*x^2+1)^(1/2))-11*b^3*dilog(1-I*(c*x+1)/(-c^2*x^2+1)^(1/2))+1/4*(c*x+1)^4*a^3)

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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*x+1)^3*(a+b*arctanh(c*x))^3,x, algorithm="maxima")

[Out]

1/4*a^3*c^3*x^4 + a^3*c^2*x^3 + 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*lo
g(c*x - 1)/c^5))*a^2*b*c^3 + 3/2*(2*x^3*arctanh(c*x) + c*(x^2/c^2 + log(c^2*x^2 - 1)/c^4))*a^2*b*c^2 + 3/2*a^3
*c*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*c + a^3*x + 3/2*(2
*c*x*arctanh(c*x) + log(-c^2*x^2 + 1))*a^2*b/c - 1/32*((b^3*c^4*x^4 + 4*b^3*c^3*x^3 + 6*b^3*c^2*x^2 + 4*b^3*c*
x - 15*b^3)*log(-c*x + 1)^3 - (6*a*b^2*c^4*x^4 + 2*(12*a*b^2*c^3 + b^3*c^3)*x^3 + 12*(3*a*b^2*c^2 + b^3*c^2)*x
^2 + 6*(4*a*b^2*c + 7*b^3*c)*x + 3*(b^3*c^4*x^4 + 4*b^3*c^3*x^3 + 6*b^3*c^2*x^2 + 4*b^3*c*x + b^3)*log(c*x + 1
))*log(-c*x + 1)^2)/c - integrate(-1/16*(2*(b^3*c^4*x^4 + 2*b^3*c^3*x^3 - 2*b^3*c*x - b^3)*log(c*x + 1)^3 + 12
*(a*b^2*c^4*x^4 + 2*a*b^2*c^3*x^3 - 2*a*b^2*c*x - a*b^2)*log(c*x + 1)^2 - (6*a*b^2*c^4*x^4 + 2*(12*a*b^2*c^3 +
b^3*c^3)*x^3 + 12*(3*a*b^2*c^2 + b^3*c^2)*x^2 + 6*(b^3*c^4*x^4 + 2*b^3*c^3*x^3 - 2*b^3*c*x - b^3)*log(c*x + 1
)^2 + 6*(4*a*b^2*c + 7*b^3*c)*x + 3*(6*b^3*c^2*x^2 + (8*a*b^2*c^4 + b^3*c^4)*x^4 + 4*(4*a*b^2*c^3 + b^3*c^3)*x
^3 - 8*a*b^2 + b^3 - 4*(4*a*b^2*c - b^3*c)*x)*log(c*x + 1))*log(-c*x + 1))/(c*x - 1), 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*x+1)^3*(a+b*arctanh(c*x))^3,x, algorithm="fricas")

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (a + b \operatorname {atanh}{\left (c x \right )}\right )^{3} \left (c x + 1\right )^{3}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x+1)**3*(a+b*atanh(c*x))**3,x)

[Out]

Integral((a + b*atanh(c*x))**3*(c*x + 1)**3, 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*x+1)^3*(a+b*arctanh(c*x))^3,x, algorithm="giac")

[Out]

integrate((c*x + 1)^3*(b*arctanh(c*x) + a)^3, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*atanh(c*x))^3*(c*x + 1)^3,x)

[Out]

int((a + b*atanh(c*x))^3*(c*x + 1)^3, x)

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