3.2.25 \(\int \frac {(a+b \tanh ^{-1}(c x))^3}{(1+c x)^3} \, dx\) [125]

Optimal. Leaf size=208 \[ -\frac {3 b^3}{64 c (1+c x)^2}-\frac {21 b^3}{64 c (1+c x)}+\frac {21 b^3 \tanh ^{-1}(c x)}{64 c}-\frac {3 b^2 \left (a+b \tanh ^{-1}(c x)\right )}{16 c (1+c x)^2}-\frac {9 b^2 \left (a+b \tanh ^{-1}(c x)\right )}{16 c (1+c x)}+\frac {9 b \left (a+b \tanh ^{-1}(c x)\right )^2}{32 c}-\frac {3 b \left (a+b \tanh ^{-1}(c x)\right )^2}{8 c (1+c x)^2}-\frac {3 b \left (a+b \tanh ^{-1}(c x)\right )^2}{8 c (1+c x)}+\frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{8 c}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{2 c (1+c x)^2} \]

[Out]

-3/64*b^3/c/(c*x+1)^2-21/64*b^3/c/(c*x+1)+21/64*b^3*arctanh(c*x)/c-3/16*b^2*(a+b*arctanh(c*x))/c/(c*x+1)^2-9/1
6*b^2*(a+b*arctanh(c*x))/c/(c*x+1)+9/32*b*(a+b*arctanh(c*x))^2/c-3/8*b*(a+b*arctanh(c*x))^2/c/(c*x+1)^2-3/8*b*
(a+b*arctanh(c*x))^2/c/(c*x+1)+1/8*(a+b*arctanh(c*x))^3/c-1/2*(a+b*arctanh(c*x))^3/c/(c*x+1)^2

________________________________________________________________________________________

Rubi [A]
time = 0.28, antiderivative size = 208, normalized size of antiderivative = 1.00, number of steps used = 24, number of rules used = 6, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6065, 6063, 641, 46, 213, 6095} \begin {gather*} -\frac {9 b^2 \left (a+b \tanh ^{-1}(c x)\right )}{16 c (c x+1)}-\frac {3 b^2 \left (a+b \tanh ^{-1}(c x)\right )}{16 c (c x+1)^2}+\frac {9 b \left (a+b \tanh ^{-1}(c x)\right )^2}{32 c}-\frac {3 b \left (a+b \tanh ^{-1}(c x)\right )^2}{8 c (c x+1)}-\frac {3 b \left (a+b \tanh ^{-1}(c x)\right )^2}{8 c (c x+1)^2}+\frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{8 c}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{2 c (c x+1)^2}-\frac {21 b^3}{64 c (c x+1)}-\frac {3 b^3}{64 c (c x+1)^2}+\frac {21 b^3 \tanh ^{-1}(c x)}{64 c} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*b^3)/(64*c*(1 + c*x)^2) - (21*b^3)/(64*c*(1 + c*x)) + (21*b^3*ArcTanh[c*x])/(64*c) - (3*b^2*(a + b*ArcTanh
[c*x]))/(16*c*(1 + c*x)^2) - (9*b^2*(a + b*ArcTanh[c*x]))/(16*c*(1 + c*x)) + (9*b*(a + b*ArcTanh[c*x])^2)/(32*
c) - (3*b*(a + b*ArcTanh[c*x])^2)/(8*c*(1 + c*x)^2) - (3*b*(a + b*ArcTanh[c*x])^2)/(8*c*(1 + c*x)) + (a + b*Ar
cTanh[c*x])^3/(8*c) - (a + b*ArcTanh[c*x])^3/(2*c*(1 + c*x)^2)

Rule 46

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}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && Lt
Q[m + n + 2, 0])

Rule 213

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

Rule 641

Int[((d_) + (e_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[(d + e*x)^(m + p)*(a/d + (c/e)*x)^
p, x] /; FreeQ[{a, c, d, e, m, p}, x] && EqQ[c*d^2 + a*e^2, 0] && (IntegerQ[p] || (GtQ[a, 0] && GtQ[d, 0] && I
ntegerQ[m + p]))

Rule 6063

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

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]

Rubi steps

\begin {align*} \int \frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{(1+c x)^3} \, dx &=-\frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{2 c (1+c x)^2}+\frac {1}{2} (3 b) \int \left (\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{2 (1+c x)^3}+\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{4 (1+c x)^2}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{4 \left (-1+c^2 x^2\right )}\right ) \, dx\\ &=-\frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{2 c (1+c x)^2}+\frac {1}{8} (3 b) \int \frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{(1+c x)^2} \, dx-\frac {1}{8} (3 b) \int \frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{-1+c^2 x^2} \, dx+\frac {1}{4} (3 b) \int \frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{(1+c x)^3} \, dx\\ &=-\frac {3 b \left (a+b \tanh ^{-1}(c x)\right )^2}{8 c (1+c x)^2}-\frac {3 b \left (a+b \tanh ^{-1}(c x)\right )^2}{8 c (1+c x)}+\frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{8 c}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{2 c (1+c x)^2}+\frac {1}{4} \left (3 b^2\right ) \int \left (\frac {a+b \tanh ^{-1}(c x)}{2 (1+c x)^2}-\frac {a+b \tanh ^{-1}(c x)}{2 \left (-1+c^2 x^2\right )}\right ) \, dx+\frac {1}{4} \left (3 b^2\right ) \int \left (\frac {a+b \tanh ^{-1}(c x)}{2 (1+c x)^3}+\frac {a+b \tanh ^{-1}(c x)}{4 (1+c x)^2}-\frac {a+b \tanh ^{-1}(c x)}{4 \left (-1+c^2 x^2\right )}\right ) \, dx\\ &=-\frac {3 b \left (a+b \tanh ^{-1}(c x)\right )^2}{8 c (1+c x)^2}-\frac {3 b \left (a+b \tanh ^{-1}(c x)\right )^2}{8 c (1+c x)}+\frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{8 c}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{2 c (1+c x)^2}+\frac {1}{16} \left (3 b^2\right ) \int \frac {a+b \tanh ^{-1}(c x)}{(1+c x)^2} \, dx-\frac {1}{16} \left (3 b^2\right ) \int \frac {a+b \tanh ^{-1}(c x)}{-1+c^2 x^2} \, dx+\frac {1}{8} \left (3 b^2\right ) \int \frac {a+b \tanh ^{-1}(c x)}{(1+c x)^3} \, dx+\frac {1}{8} \left (3 b^2\right ) \int \frac {a+b \tanh ^{-1}(c x)}{(1+c x)^2} \, dx-\frac {1}{8} \left (3 b^2\right ) \int \frac {a+b \tanh ^{-1}(c x)}{-1+c^2 x^2} \, dx\\ &=-\frac {3 b^2 \left (a+b \tanh ^{-1}(c x)\right )}{16 c (1+c x)^2}-\frac {9 b^2 \left (a+b \tanh ^{-1}(c x)\right )}{16 c (1+c x)}+\frac {9 b \left (a+b \tanh ^{-1}(c x)\right )^2}{32 c}-\frac {3 b \left (a+b \tanh ^{-1}(c x)\right )^2}{8 c (1+c x)^2}-\frac {3 b \left (a+b \tanh ^{-1}(c x)\right )^2}{8 c (1+c x)}+\frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{8 c}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{2 c (1+c x)^2}+\frac {1}{16} \left (3 b^3\right ) \int \frac {1}{(1+c x)^2 \left (1-c^2 x^2\right )} \, dx+\frac {1}{16} \left (3 b^3\right ) \int \frac {1}{(1+c x) \left (1-c^2 x^2\right )} \, dx+\frac {1}{8} \left (3 b^3\right ) \int \frac {1}{(1+c x) \left (1-c^2 x^2\right )} \, dx\\ &=-\frac {3 b^2 \left (a+b \tanh ^{-1}(c x)\right )}{16 c (1+c x)^2}-\frac {9 b^2 \left (a+b \tanh ^{-1}(c x)\right )}{16 c (1+c x)}+\frac {9 b \left (a+b \tanh ^{-1}(c x)\right )^2}{32 c}-\frac {3 b \left (a+b \tanh ^{-1}(c x)\right )^2}{8 c (1+c x)^2}-\frac {3 b \left (a+b \tanh ^{-1}(c x)\right )^2}{8 c (1+c x)}+\frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{8 c}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{2 c (1+c x)^2}+\frac {1}{16} \left (3 b^3\right ) \int \frac {1}{(1-c x) (1+c x)^3} \, dx+\frac {1}{16} \left (3 b^3\right ) \int \frac {1}{(1-c x) (1+c x)^2} \, dx+\frac {1}{8} \left (3 b^3\right ) \int \frac {1}{(1-c x) (1+c x)^2} \, dx\\ &=-\frac {3 b^2 \left (a+b \tanh ^{-1}(c x)\right )}{16 c (1+c x)^2}-\frac {9 b^2 \left (a+b \tanh ^{-1}(c x)\right )}{16 c (1+c x)}+\frac {9 b \left (a+b \tanh ^{-1}(c x)\right )^2}{32 c}-\frac {3 b \left (a+b \tanh ^{-1}(c x)\right )^2}{8 c (1+c x)^2}-\frac {3 b \left (a+b \tanh ^{-1}(c x)\right )^2}{8 c (1+c x)}+\frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{8 c}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{2 c (1+c x)^2}+\frac {1}{16} \left (3 b^3\right ) \int \left (\frac {1}{2 (1+c x)^2}-\frac {1}{2 \left (-1+c^2 x^2\right )}\right ) \, dx+\frac {1}{16} \left (3 b^3\right ) \int \left (\frac {1}{2 (1+c x)^3}+\frac {1}{4 (1+c x)^2}-\frac {1}{4 \left (-1+c^2 x^2\right )}\right ) \, dx+\frac {1}{8} \left (3 b^3\right ) \int \left (\frac {1}{2 (1+c x)^2}-\frac {1}{2 \left (-1+c^2 x^2\right )}\right ) \, dx\\ &=-\frac {3 b^3}{64 c (1+c x)^2}-\frac {21 b^3}{64 c (1+c x)}-\frac {3 b^2 \left (a+b \tanh ^{-1}(c x)\right )}{16 c (1+c x)^2}-\frac {9 b^2 \left (a+b \tanh ^{-1}(c x)\right )}{16 c (1+c x)}+\frac {9 b \left (a+b \tanh ^{-1}(c x)\right )^2}{32 c}-\frac {3 b \left (a+b \tanh ^{-1}(c x)\right )^2}{8 c (1+c x)^2}-\frac {3 b \left (a+b \tanh ^{-1}(c x)\right )^2}{8 c (1+c x)}+\frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{8 c}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{2 c (1+c x)^2}-\frac {1}{64} \left (3 b^3\right ) \int \frac {1}{-1+c^2 x^2} \, dx-\frac {1}{32} \left (3 b^3\right ) \int \frac {1}{-1+c^2 x^2} \, dx-\frac {1}{16} \left (3 b^3\right ) \int \frac {1}{-1+c^2 x^2} \, dx\\ &=-\frac {3 b^3}{64 c (1+c x)^2}-\frac {21 b^3}{64 c (1+c x)}+\frac {21 b^3 \tanh ^{-1}(c x)}{64 c}-\frac {3 b^2 \left (a+b \tanh ^{-1}(c x)\right )}{16 c (1+c x)^2}-\frac {9 b^2 \left (a+b \tanh ^{-1}(c x)\right )}{16 c (1+c x)}+\frac {9 b \left (a+b \tanh ^{-1}(c x)\right )^2}{32 c}-\frac {3 b \left (a+b \tanh ^{-1}(c x)\right )^2}{8 c (1+c x)^2}-\frac {3 b \left (a+b \tanh ^{-1}(c x)\right )^2}{8 c (1+c x)}+\frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{8 c}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{2 c (1+c x)^2}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.12, size = 215, normalized size = 1.03 \begin {gather*} \frac {-2 \left (32 a^3+24 a^2 b+12 a b^2+3 b^3\right )-6 b \left (8 a^2+12 a b+7 b^2\right ) (1+c x)-24 b \left (8 a^2+4 a b (2+c x)+b^2 (4+3 c x)\right ) \tanh ^{-1}(c x)+12 b^2 (-1+c x) (4 a (3+c x)+b (5+3 c x)) \tanh ^{-1}(c x)^2+16 b^3 \left (-3+2 c x+c^2 x^2\right ) \tanh ^{-1}(c x)^3-3 b \left (8 a^2+12 a b+7 b^2\right ) (1+c x)^2 \log (1-c x)+3 b \left (8 a^2+12 a b+7 b^2\right ) (1+c x)^2 \log (1+c x)}{128 c (1+c x)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(32*a^3 + 24*a^2*b + 12*a*b^2 + 3*b^3) - 6*b*(8*a^2 + 12*a*b + 7*b^2)*(1 + c*x) - 24*b*(8*a^2 + 4*a*b*(2 +
c*x) + b^2*(4 + 3*c*x))*ArcTanh[c*x] + 12*b^2*(-1 + c*x)*(4*a*(3 + c*x) + b*(5 + 3*c*x))*ArcTanh[c*x]^2 + 16*
b^3*(-3 + 2*c*x + c^2*x^2)*ArcTanh[c*x]^3 - 3*b*(8*a^2 + 12*a*b + 7*b^2)*(1 + c*x)^2*Log[1 - c*x] + 3*b*(8*a^2
+ 12*a*b + 7*b^2)*(1 + c*x)^2*Log[1 + c*x])/(128*c*(1 + c*x)^2)

________________________________________________________________________________________

Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 7.26, size = 2684, normalized size = 12.90

method	result	size
risch	\(\frac {b^{3} \left (c^{2} x^{2}+2 c x -3\right ) \ln \left (c x +1\right )^{3}}{64 \left (c x +1\right )^{2} c}+\frac {3 b^{2} \left (-2 b \,x^{2} \ln \left (-c x +1\right ) c^{2}+4 a \,c^{2} x^{2}+3 b \,c^{2} x^{2}-4 b c x \ln \left (-c x +1\right )+8 c x a +2 b c x +6 b \ln \left (-c x +1\right )-12 a -5 b \right ) \ln \left (c x +1\right )^{2}}{128 \left (c x +1\right )^{2} c}-\frac {3 b \left (-b^{2} c^{2} x^{2} \ln \left (-c x +1\right )^{2}+4 a b \,c^{2} x^{2} \ln \left (-c x +1\right )+3 b^{2} c^{2} \ln \left (-c x +1\right ) x^{2}-2 b^{2} c x \ln \left (-c x +1\right )^{2}+8 a b c x \ln \left (-c x +1\right )+2 b^{2} c x \ln \left (-c x +1\right )+8 a b c x +6 b^{2} c x +3 b^{2} \ln \left (-c x +1\right )^{2}-12 b \ln \left (-c x +1\right ) a -5 b^{2} \ln \left (-c x +1\right )+16 a^{2}+16 a b +8 b^{2}\right ) \ln \left (c x +1\right )}{64 \left (c x +1\right )^{2} c}+\frac {-48 b^{3}+12 \ln \left (-c x +1\right )^{2} a \,b^{2} c^{2} x^{2}+24 \ln \left (-c x +1\right )^{2} a \,b^{2} c x -4 \ln \left (-c x +1\right )^{3} b^{3} c x -2 \ln \left (-c x +1\right )^{3} b^{3} c^{2} x^{2}-36 \ln \left (-c x +1\right )^{2} a \,b^{2}+96 \ln \left (-c x +1\right ) a^{2} b -24 a^{2} b \ln \left (c x -1\right )-36 a \,b^{2} \ln \left (c x -1\right )-42 b^{3} c x -48 a^{2} b c x -96 a \,b^{2}-64 a^{3}-72 b^{2} c x a -48 \ln \left (c x -1\right ) a^{2} b c x -72 \ln \left (c x -1\right ) a \,b^{2} c x +48 \ln \left (-c x -1\right ) a^{2} b c x +72 \ln \left (-c x -1\right ) a \,b^{2} c x +24 \ln \left (-c x -1\right ) a^{2} b +36 \ln \left (-c x -1\right ) a \,b^{2}+96 a \,b^{2} \ln \left (-c x +1\right )+48 a \,b^{2} c x \ln \left (-c x +1\right )+24 \ln \left (-c x -1\right ) a^{2} b \,c^{2} x^{2}+36 \ln \left (-c x -1\right ) a \,b^{2} c^{2} x^{2}-24 \ln \left (c x -1\right ) a^{2} b \,c^{2} x^{2}-36 \ln \left (c x -1\right ) a \,b^{2} c^{2} x^{2}-15 b^{3} \ln \left (-c x +1\right )^{2}+48 b^{3} \ln \left (-c x +1\right )-21 b^{3} \ln \left (c x -1\right )+21 b^{3} \ln \left (-c x -1\right )+9 b^{3} c^{2} x^{2} \ln \left (-c x +1\right )^{2}+36 b^{3} c x \ln \left (-c x +1\right )+21 \ln \left (-c x -1\right ) b^{3} c^{2} x^{2}-21 \ln \left (c x -1\right ) b^{3} c^{2} x^{2}+6 b^{3} c x \ln \left (-c x +1\right )^{2}-42 \ln \left (c x -1\right ) b^{3} c x +42 \ln \left (-c x -1\right ) b^{3} c x -96 a^{2} b +6 \ln \left (-c x +1\right )^{3} b^{3}}{128 \left (c x +1\right )^{2} c}\)	\(861\)
derivativedivides	\(\text {Expression too large to display}\)	\(2684\)
default	\(\text {Expression too large to display}\)	\(2684\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/c*(3/16*I*b^3/(c*x+1)^2*Pi*arctanh(c*x)^2*csgn(I/(1+(c*x+1)^2/(-c^2*x^2+1)))^3+3/16*I*b^3/(c*x+1)^2*csgn(I/(
1+(c*x+1)^2/(-c^2*x^2+1)))*arctanh(c*x)^2*csgn(I*(c*x+1)^2/(c^2*x^2-1))*csgn(I*(c*x+1)^2/(c^2*x^2-1)/(1+(c*x+1
)^2/(-c^2*x^2+1)))*Pi*c*x+3/32*I*b^3/(c*x+1)^2*csgn(I/(1+(c*x+1)^2/(-c^2*x^2+1)))*arctanh(c*x)^2*csgn(I*(c*x+1
)^2/(c^2*x^2-1))*csgn(I*(c*x+1)^2/(c^2*x^2-1)/(1+(c*x+1)^2/(-c^2*x^2+1)))*Pi*c^2*x^2-3/32*I*b^3/(c*x+1)^2*csgn
(I/(1+(c*x+1)^2/(-c^2*x^2+1)))*arctanh(c*x)^2*csgn(I*(c*x+1)^2/(c^2*x^2-1)/(1+(c*x+1)^2/(-c^2*x^2+1)))^2*Pi*c^
2*x^2-3/32*I*b^3/(c*x+1)^2*arctanh(c*x)^2*csgn(I*(c*x+1)/(-c^2*x^2+1)^(1/2))^2*csgn(I*(c*x+1)^2/(c^2*x^2-1))*P
i*c^2*x^2-3/16*I*b^3/(c*x+1)^2*arctanh(c*x)^2*csgn(I*(c*x+1)/(-c^2*x^2+1)^(1/2))*csgn(I*(c*x+1)^2/(c^2*x^2-1))
^2*Pi*c^2*x^2+3/32*I*b^3/(c*x+1)^2*arctanh(c*x)^2*csgn(I*(c*x+1)^2/(c^2*x^2-1))*csgn(I*(c*x+1)^2/(c^2*x^2-1)/(
1+(c*x+1)^2/(-c^2*x^2+1)))^2*Pi*c^2*x^2-3/16*I*b^3/(c*x+1)^2*csgn(I/(1+(c*x+1)^2/(-c^2*x^2+1)))*arctanh(c*x)^2
*csgn(I*(c*x+1)^2/(c^2*x^2-1)/(1+(c*x+1)^2/(-c^2*x^2+1)))^2*Pi*c*x-3/16*I*b^3/(c*x+1)^2*arctanh(c*x)^2*csgn(I*
(c*x+1)/(-c^2*x^2+1)^(1/2))^2*csgn(I*(c*x+1)^2/(c^2*x^2-1))*Pi*c*x-3/8*I*b^3/(c*x+1)^2*arctanh(c*x)^2*csgn(I*(
c*x+1)/(-c^2*x^2+1)^(1/2))*csgn(I*(c*x+1)^2/(c^2*x^2-1))^2*Pi*c*x+3/16*I*b^3/(c*x+1)^2*arctanh(c*x)^2*csgn(I*(
c*x+1)^2/(c^2*x^2-1))*csgn(I*(c*x+1)^2/(c^2*x^2-1)/(1+(c*x+1)^2/(-c^2*x^2+1)))^2*Pi*c*x-3/16*a^2*b*ln(c*x-1)+3
/16*a^2*b*ln(c*x+1)-3/8*b^3*arctanh(c*x)^2*ln((c*x+1)/(-c^2*x^2+1)^(1/2))-3/16*b^3*arctanh(c*x)^2*ln(c*x-1)+3/
16*b^3*arctanh(c*x)^2*ln(c*x+1)-3/32*a*b^2*ln(c*x-1)^2-3/32*a*b^2*ln(c*x+1)^2-9/32*a*b^2*ln(c*x-1)+9/32*a*b^2*
ln(c*x+1)+3/8*a*b^2*arctanh(c*x)*ln(c*x+1)+3/16*a*b^2*ln(c*x-1)*ln(1/2*c*x+1/2)+3/16*a*b^2*ln(-1/2*c*x+1/2)*ln
(c*x+1)-3/16*a*b^2*ln(-1/2*c*x+1/2)*ln(1/2*c*x+1/2)-3/8*a*b^2*arctanh(c*x)*ln(c*x-1)-3/32*I*b^3/(c*x+1)^2*Pi*a
rctanh(c*x)^2*csgn(I*(c*x+1)^2/(c^2*x^2-1))^3-3/32*I*b^3/(c*x+1)^2*Pi*arctanh(c*x)^2*csgn(I*(c*x+1)^2/(c^2*x^2
-1)/(1+(c*x+1)^2/(-c^2*x^2+1)))^3-3/16*I*b^3/(c*x+1)^2*Pi*arctanh(c*x)^2*csgn(I/(1+(c*x+1)^2/(-c^2*x^2+1)))^2-
3/16*I*b^3/(c*x+1)^2*csgn(I/(1+(c*x+1)^2/(-c^2*x^2+1)))^2*arctanh(c*x)^2*Pi*c^2*x^2-3/16*I*b^3/(c*x+1)^2*arcta
nh(c*x)^2*csgn(I*(c*x+1)^2/(c^2*x^2-1))^3*Pi*c*x-3/16*I*b^3/(c*x+1)^2*arctanh(c*x)^2*csgn(I*(c*x+1)^2/(c^2*x^2
-1)/(1+(c*x+1)^2/(-c^2*x^2+1)))^3*Pi*c*x-3/8*I*b^3/(c*x+1)^2*csgn(I/(1+(c*x+1)^2/(-c^2*x^2+1)))^2*arctanh(c*x)
^2*Pi*c*x+3/32*I*b^3/(c*x+1)^2*Pi*arctanh(c*x)^2*csgn(I/(1+(c*x+1)^2/(-c^2*x^2+1)))*csgn(I*(c*x+1)^2/(c^2*x^2-
1))*csgn(I*(c*x+1)^2/(c^2*x^2-1)/(1+(c*x+1)^2/(-c^2*x^2+1)))+3/16*I*b^3/(c*x+1)^2*csgn(I/(1+(c*x+1)^2/(-c^2*x^
2+1)))^3*arctanh(c*x)^2*Pi*c^2*x^2-3/32*I*b^3/(c*x+1)^2*arctanh(c*x)^2*csgn(I*(c*x+1)^2/(c^2*x^2-1))^3*Pi*c^2*
x^2-3/32*I*b^3/(c*x+1)^2*arctanh(c*x)^2*csgn(I*(c*x+1)^2/(c^2*x^2-1)/(1+(c*x+1)^2/(-c^2*x^2+1)))^3*Pi*c^2*x^2+
3/8*I*b^3/(c*x+1)^2*csgn(I/(1+(c*x+1)^2/(-c^2*x^2+1)))^3*arctanh(c*x)^2*Pi*c*x+3/16*I*b^3/(c*x+1)^2*arctanh(c*
x)^2*Pi*c^2*x^2+3/8*I*b^3/(c*x+1)^2*arctanh(c*x)^2*Pi*c*x-3/32*I*b^3/(c*x+1)^2*Pi*arctanh(c*x)^2*csgn(I/(1+(c*
x+1)^2/(-c^2*x^2+1)))*csgn(I*(c*x+1)^2/(c^2*x^2-1)/(1+(c*x+1)^2/(-c^2*x^2+1)))^2-3/32*I*b^3/(c*x+1)^2*Pi*arcta
nh(c*x)^2*csgn(I*(c*x+1)/(-c^2*x^2+1)^(1/2))^2*csgn(I*(c*x+1)^2/(c^2*x^2-1))-3/16*I*b^3/(c*x+1)^2*Pi*arctanh(c
*x)^2*csgn(I*(c*x+1)/(-c^2*x^2+1)^(1/2))*csgn(I*(c*x+1)^2/(c^2*x^2-1))^2+3/32*I*b^3/(c*x+1)^2*Pi*arctanh(c*x)^
2*csgn(I*(c*x+1)^2/(c^2*x^2-1))*csgn(I*(c*x+1)^2/(c^2*x^2-1)/(1+(c*x+1)^2/(-c^2*x^2+1)))^2-3/16*a*b^2/(c*x+1)^
2-3/8*a^2*b/(c*x+1)^2-27/64*b^3/(c*x+1)^2*arctanh(c*x)-3/8*b^3/(c*x+1)^2*arctanh(c*x)^3-3/32*b^3/(c*x+1)^2*arc
tanh(c*x)^2+3/16*I*b^3/(c*x+1)^2*Pi*arctanh(c*x)^2+1/8*b^3/(c*x+1)^2*arctanh(c*x)^3*c^2*x^2+1/4*b^3/(c*x+1)^2*
arctanh(c*x)^3*c*x+9/32*b^3/(c*x+1)^2*arctanh(c*x)^2*c^2*x^2+9/16*b^3/(c*x+1)^2*arctanh(c*x)^2*c*x+21/64*b^3/(
c*x+1)^2*arctanh(c*x)*c^2*x^2+3/32*b^3/(c*x+1)^2*arctanh(c*x)*c*x-3/8/(c*x+1)*a^2*b-51/256*b^3/(c*x+1)^2-1/2*a
^3/(c*x+1)^2-9/16*a*b^2/(c*x+1)-3/2*a^2*b*arctanh(c*x)/(c*x+1)^2-3/2*a*b^2/(c*x+1)^2*arctanh(c*x)^2-3/4*a*b^2*
arctanh(c*x)/(c*x+1)^2+45/256*b^3/(c*x+1)^2*c^2*x^2+3/128*b^3/(c*x+1)^2*c*x-3/4*a*b^2*arctanh(c*x)/(c*x+1)-3/8
*b^3/(c*x+1)*arctanh(c*x)^2)

________________________________________________________________________________________

Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 796 vs. \(2 (188) = 376\).
time = 0.47, size = 796, normalized size = 3.83 \begin {gather*} -\frac {b^{3} \operatorname {artanh}\left (c x\right )^{3}}{2 \, {\left (c^{3} x^{2} + 2 \, c^{2} x + c\right )}} - \frac {3}{16} \, {\left (c {\left (\frac {2 \, {\left (c x + 2\right )}}{c^{4} x^{2} + 2 \, c^{3} x + c^{2}} - \frac {\log \left (c x + 1\right )}{c^{2}} + \frac {\log \left (c x - 1\right )}{c^{2}}\right )} + \frac {8 \, \operatorname {artanh}\left (c x\right )}{c^{3} x^{2} + 2 \, c^{2} x + c}\right )} a^{2} b - \frac {3}{32} \, {\left (4 \, c {\left (\frac {2 \, {\left (c x + 2\right )}}{c^{4} x^{2} + 2 \, c^{3} x + c^{2}} - \frac {\log \left (c x + 1\right )}{c^{2}} + \frac {\log \left (c x - 1\right )}{c^{2}}\right )} \operatorname {artanh}\left (c x\right ) + \frac {{\left ({\left (c^{2} x^{2} + 2 \, c x + 1\right )} \log \left (c x + 1\right )^{2} + {\left (c^{2} x^{2} + 2 \, c x + 1\right )} \log \left (c x - 1\right )^{2} + 6 \, c x - {\left (3 \, c^{2} x^{2} + 6 \, c x + 2 \, {\left (c^{2} x^{2} + 2 \, c x + 1\right )} \log \left (c x - 1\right ) + 3\right )} \log \left (c x + 1\right ) + 3 \, {\left (c^{2} x^{2} + 2 \, c x + 1\right )} \log \left (c x - 1\right ) + 8\right )} c^{2}}{c^{5} x^{2} + 2 \, c^{4} x + c^{3}}\right )} a b^{2} - \frac {1}{128} \, {\left (24 \, c {\left (\frac {2 \, {\left (c x + 2\right )}}{c^{4} x^{2} + 2 \, c^{3} x + c^{2}} - \frac {\log \left (c x + 1\right )}{c^{2}} + \frac {\log \left (c x - 1\right )}{c^{2}}\right )} \operatorname {artanh}\left (c x\right )^{2} - {\left (\frac {{\left (2 \, {\left (c^{2} x^{2} + 2 \, c x + 1\right )} \log \left (c x + 1\right )^{3} - 2 \, {\left (c^{2} x^{2} + 2 \, c x + 1\right )} \log \left (c x - 1\right )^{3} - 3 \, {\left (3 \, c^{2} x^{2} + 6 \, c x + 2 \, {\left (c^{2} x^{2} + 2 \, c x + 1\right )} \log \left (c x - 1\right ) + 3\right )} \log \left (c x + 1\right )^{2} - 9 \, {\left (c^{2} x^{2} + 2 \, c x + 1\right )} \log \left (c x - 1\right )^{2} - 42 \, c x + 3 \, {\left (7 \, c^{2} x^{2} + 2 \, {\left (c^{2} x^{2} + 2 \, c x + 1\right )} \log \left (c x - 1\right )^{2} + 14 \, c x + 6 \, {\left (c^{2} x^{2} + 2 \, c x + 1\right )} \log \left (c x - 1\right ) + 7\right )} \log \left (c x + 1\right ) - 21 \, {\left (c^{2} x^{2} + 2 \, c x + 1\right )} \log \left (c x - 1\right ) - 48\right )} c^{2}}{c^{6} x^{2} + 2 \, c^{5} x + c^{4}} - \frac {12 \, {\left ({\left (c^{2} x^{2} + 2 \, c x + 1\right )} \log \left (c x + 1\right )^{2} + {\left (c^{2} x^{2} + 2 \, c x + 1\right )} \log \left (c x - 1\right )^{2} + 6 \, c x - {\left (3 \, c^{2} x^{2} + 6 \, c x + 2 \, {\left (c^{2} x^{2} + 2 \, c x + 1\right )} \log \left (c x - 1\right ) + 3\right )} \log \left (c x + 1\right ) + 3 \, {\left (c^{2} x^{2} + 2 \, c x + 1\right )} \log \left (c x - 1\right ) + 8\right )} c \operatorname {artanh}\left (c x\right )}{c^{5} x^{2} + 2 \, c^{4} x + c^{3}}\right )} c\right )} b^{3} - \frac {3 \, a b^{2} \operatorname {artanh}\left (c x\right )^{2}}{2 \, {\left (c^{3} x^{2} + 2 \, c^{2} x + c\right )}} - \frac {a^{3}}{2 \, {\left (c^{3} x^{2} + 2 \, c^{2} x + c\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*b^3*arctanh(c*x)^3/(c^3*x^2 + 2*c^2*x + c) - 3/16*(c*(2*(c*x + 2)/(c^4*x^2 + 2*c^3*x + c^2) - log(c*x + 1
)/c^2 + log(c*x - 1)/c^2) + 8*arctanh(c*x)/(c^3*x^2 + 2*c^2*x + c))*a^2*b - 3/32*(4*c*(2*(c*x + 2)/(c^4*x^2 +
2*c^3*x + c^2) - log(c*x + 1)/c^2 + log(c*x - 1)/c^2)*arctanh(c*x) + ((c^2*x^2 + 2*c*x + 1)*log(c*x + 1)^2 + (
c^2*x^2 + 2*c*x + 1)*log(c*x - 1)^2 + 6*c*x - (3*c^2*x^2 + 6*c*x + 2*(c^2*x^2 + 2*c*x + 1)*log(c*x - 1) + 3)*l
og(c*x + 1) + 3*(c^2*x^2 + 2*c*x + 1)*log(c*x - 1) + 8)*c^2/(c^5*x^2 + 2*c^4*x + c^3))*a*b^2 - 1/128*(24*c*(2*
(c*x + 2)/(c^4*x^2 + 2*c^3*x + c^2) - log(c*x + 1)/c^2 + log(c*x - 1)/c^2)*arctanh(c*x)^2 - ((2*(c^2*x^2 + 2*c
*x + 1)*log(c*x + 1)^3 - 2*(c^2*x^2 + 2*c*x + 1)*log(c*x - 1)^3 - 3*(3*c^2*x^2 + 6*c*x + 2*(c^2*x^2 + 2*c*x +
1)*log(c*x - 1) + 3)*log(c*x + 1)^2 - 9*(c^2*x^2 + 2*c*x + 1)*log(c*x - 1)^2 - 42*c*x + 3*(7*c^2*x^2 + 2*(c^2*
x^2 + 2*c*x + 1)*log(c*x - 1)^2 + 14*c*x + 6*(c^2*x^2 + 2*c*x + 1)*log(c*x - 1) + 7)*log(c*x + 1) - 21*(c^2*x^
2 + 2*c*x + 1)*log(c*x - 1) - 48)*c^2/(c^6*x^2 + 2*c^5*x + c^4) - 12*((c^2*x^2 + 2*c*x + 1)*log(c*x + 1)^2 + (
c^2*x^2 + 2*c*x + 1)*log(c*x - 1)^2 + 6*c*x - (3*c^2*x^2 + 6*c*x + 2*(c^2*x^2 + 2*c*x + 1)*log(c*x - 1) + 3)*l
og(c*x + 1) + 3*(c^2*x^2 + 2*c*x + 1)*log(c*x - 1) + 8)*c*arctanh(c*x)/(c^5*x^2 + 2*c^4*x + c^3))*c)*b^3 - 3/2
*a*b^2*arctanh(c*x)^2/(c^3*x^2 + 2*c^2*x + c) - 1/2*a^3/(c^3*x^2 + 2*c^2*x + c)

________________________________________________________________________________________

Fricas [A]
time = 0.35, size = 250, normalized size = 1.20 \begin {gather*} \frac {2 \, {\left (b^{3} c^{2} x^{2} + 2 \, b^{3} c x - 3 \, b^{3}\right )} \log \left (-\frac {c x + 1}{c x - 1}\right )^{3} - 64 \, a^{3} - 96 \, a^{2} b - 96 \, a b^{2} - 48 \, b^{3} - 6 \, {\left (8 \, a^{2} b + 12 \, a b^{2} + 7 \, b^{3}\right )} c x + 3 \, {\left ({\left (4 \, a b^{2} + 3 \, b^{3}\right )} c^{2} x^{2} - 12 \, a b^{2} - 5 \, b^{3} + 2 \, {\left (4 \, a b^{2} + b^{3}\right )} c x\right )} \log \left (-\frac {c x + 1}{c x - 1}\right )^{2} + 3 \, {\left ({\left (8 \, a^{2} b + 12 \, a b^{2} + 7 \, b^{3}\right )} c^{2} x^{2} - 24 \, a^{2} b - 20 \, a b^{2} - 9 \, b^{3} + 2 \, {\left (8 \, a^{2} b + 4 \, a b^{2} + b^{3}\right )} c x\right )} \log \left (-\frac {c x + 1}{c x - 1}\right )}{128 \, {\left (c^{3} x^{2} + 2 \, c^{2} x + c\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/128*(2*(b^3*c^2*x^2 + 2*b^3*c*x - 3*b^3)*log(-(c*x + 1)/(c*x - 1))^3 - 64*a^3 - 96*a^2*b - 96*a*b^2 - 48*b^3
- 6*(8*a^2*b + 12*a*b^2 + 7*b^3)*c*x + 3*((4*a*b^2 + 3*b^3)*c^2*x^2 - 12*a*b^2 - 5*b^3 + 2*(4*a*b^2 + b^3)*c*
x)*log(-(c*x + 1)/(c*x - 1))^2 + 3*((8*a^2*b + 12*a*b^2 + 7*b^3)*c^2*x^2 - 24*a^2*b - 20*a*b^2 - 9*b^3 + 2*(8*
a^2*b + 4*a*b^2 + b^3)*c*x)*log(-(c*x + 1)/(c*x - 1)))/(c^3*x^2 + 2*c^2*x + c)

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\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((a+b*atanh(c*x))**3/(c*x+1)**3,x)

[Out]

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

________________________________________________________________________________________

Giac [A]
time = 0.41, size = 362, normalized size = 1.74 \begin {gather*} \frac {1}{256} \, {\left (\frac {4 \, {\left (\frac {2 \, {\left (c x + 1\right )} b^{3}}{c x - 1} - b^{3}\right )} {\left (c x - 1\right )}^{2} \log \left (-\frac {c x + 1}{c x - 1}\right )^{3}}{{\left (c x + 1\right )}^{2} c^{2}} + \frac {6 \, {\left (\frac {8 \, {\left (c x + 1\right )} a b^{2}}{c x - 1} - 4 \, a b^{2} + \frac {4 \, {\left (c x + 1\right )} b^{3}}{c x - 1} - b^{3}\right )} {\left (c x - 1\right )}^{2} \log \left (-\frac {c x + 1}{c x - 1}\right )^{2}}{{\left (c x + 1\right )}^{2} c^{2}} + \frac {6 \, {\left (\frac {16 \, {\left (c x + 1\right )} a^{2} b}{c x - 1} - 8 \, a^{2} b + \frac {16 \, {\left (c x + 1\right )} a b^{2}}{c x - 1} - 4 \, a b^{2} + \frac {8 \, {\left (c x + 1\right )} b^{3}}{c x - 1} - b^{3}\right )} {\left (c x - 1\right )}^{2} \log \left (-\frac {c x + 1}{c x - 1}\right )}{{\left (c x + 1\right )}^{2} c^{2}} + \frac {{\left (\frac {64 \, {\left (c x + 1\right )} a^{3}}{c x - 1} - 32 \, a^{3} + \frac {96 \, {\left (c x + 1\right )} a^{2} b}{c x - 1} - 24 \, a^{2} b + \frac {96 \, {\left (c x + 1\right )} a b^{2}}{c x - 1} - 12 \, a b^{2} + \frac {48 \, {\left (c x + 1\right )} b^{3}}{c x - 1} - 3 \, b^{3}\right )} {\left (c x - 1\right )}^{2}}{{\left (c x + 1\right )}^{2} c^{2}}\right )} c \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/256*(4*(2*(c*x + 1)*b^3/(c*x - 1) - b^3)*(c*x - 1)^2*log(-(c*x + 1)/(c*x - 1))^3/((c*x + 1)^2*c^2) + 6*(8*(c
*x + 1)*a*b^2/(c*x - 1) - 4*a*b^2 + 4*(c*x + 1)*b^3/(c*x - 1) - b^3)*(c*x - 1)^2*log(-(c*x + 1)/(c*x - 1))^2/(
(c*x + 1)^2*c^2) + 6*(16*(c*x + 1)*a^2*b/(c*x - 1) - 8*a^2*b + 16*(c*x + 1)*a*b^2/(c*x - 1) - 4*a*b^2 + 8*(c*x
+ 1)*b^3/(c*x - 1) - b^3)*(c*x - 1)^2*log(-(c*x + 1)/(c*x - 1))/((c*x + 1)^2*c^2) + (64*(c*x + 1)*a^3/(c*x -
1) - 32*a^3 + 96*(c*x + 1)*a^2*b/(c*x - 1) - 24*a^2*b + 96*(c*x + 1)*a*b^2/(c*x - 1) - 12*a*b^2 + 48*(c*x + 1)
*b^3/(c*x - 1) - 3*b^3)*(c*x - 1)^2/((c*x + 1)^2*c^2))*c

________________________________________________________________________________________

Mupad [B]
time = 3.46, size = 930, normalized size = 4.47 \begin {gather*} \frac {102\,b^3\,\ln \left (1-c\,x\right )-102\,b^3\,\ln \left (c\,x+1\right )-96\,a\,b^2-96\,a^2\,b-15\,b^3\,{\ln \left (c\,x+1\right )}^2-6\,b^3\,{\ln \left (c\,x+1\right )}^3-15\,b^3\,{\ln \left (1-c\,x\right )}^2+6\,b^3\,{\ln \left (1-c\,x\right )}^3+150\,b^3\,\mathrm {atanh}\left (c\,x\right )-64\,a^3-48\,b^3+144\,a\,b^2\,\mathrm {atanh}\left (c\,x\right )+48\,a^2\,b\,\mathrm {atanh}\left (c\,x\right )+30\,b^3\,\ln \left (c\,x+1\right )\,\ln \left (1-c\,x\right )-132\,a\,b^2\,\ln \left (c\,x+1\right )-96\,a^2\,b\,\ln \left (c\,x+1\right )+132\,a\,b^2\,\ln \left (1-c\,x\right )+96\,a^2\,b\,\ln \left (1-c\,x\right )-18\,b^3\,\ln \left (c\,x+1\right )\,{\ln \left (1-c\,x\right )}^2+18\,b^3\,{\ln \left (c\,x+1\right )}^2\,\ln \left (1-c\,x\right )-36\,a\,b^2\,{\ln \left (c\,x+1\right )}^2-36\,a\,b^2\,{\ln \left (1-c\,x\right )}^2-42\,b^3\,c\,x-144\,b^3\,c\,x\,\ln \left (c\,x+1\right )+144\,b^3\,c\,x\,\ln \left (1-c\,x\right )+9\,b^3\,c^2\,x^2\,{\ln \left (c\,x+1\right )}^2+2\,b^3\,c^2\,x^2\,{\ln \left (c\,x+1\right )}^3+9\,b^3\,c^2\,x^2\,{\ln \left (1-c\,x\right )}^2-2\,b^3\,c^2\,x^2\,{\ln \left (1-c\,x\right )}^3+150\,b^3\,c^2\,x^2\,\mathrm {atanh}\left (c\,x\right )-72\,a\,b^2\,c\,x-48\,a^2\,b\,c\,x+6\,b^3\,c\,x\,{\ln \left (c\,x+1\right )}^2+4\,b^3\,c\,x\,{\ln \left (c\,x+1\right )}^3+6\,b^3\,c\,x\,{\ln \left (1-c\,x\right )}^2-4\,b^3\,c\,x\,{\ln \left (1-c\,x\right )}^3+72\,a\,b^2\,\ln \left (c\,x+1\right )\,\ln \left (1-c\,x\right )+300\,b^3\,c\,x\,\mathrm {atanh}\left (c\,x\right )-54\,b^3\,c^2\,x^2\,\ln \left (c\,x+1\right )+54\,b^3\,c^2\,x^2\,\ln \left (1-c\,x\right )-12\,b^3\,c\,x\,\ln \left (c\,x+1\right )\,\ln \left (1-c\,x\right )-36\,a\,b^2\,c^2\,x^2\,\ln \left (c\,x+1\right )+36\,a\,b^2\,c^2\,x^2\,\ln \left (1-c\,x\right )+6\,b^3\,c^2\,x^2\,\ln \left (c\,x+1\right )\,{\ln \left (1-c\,x\right )}^2-6\,b^3\,c^2\,x^2\,{\ln \left (c\,x+1\right )}^2\,\ln \left (1-c\,x\right )-120\,a\,b^2\,c\,x\,\ln \left (c\,x+1\right )+120\,a\,b^2\,c\,x\,\ln \left (1-c\,x\right )+12\,b^3\,c\,x\,\ln \left (c\,x+1\right )\,{\ln \left (1-c\,x\right )}^2-12\,b^3\,c\,x\,{\ln \left (c\,x+1\right )}^2\,\ln \left (1-c\,x\right )+12\,a\,b^2\,c^2\,x^2\,{\ln \left (c\,x+1\right )}^2+12\,a\,b^2\,c^2\,x^2\,{\ln \left (1-c\,x\right )}^2+144\,a\,b^2\,c^2\,x^2\,\mathrm {atanh}\left (c\,x\right )+48\,a^2\,b\,c^2\,x^2\,\mathrm {atanh}\left (c\,x\right )+24\,a\,b^2\,c\,x\,{\ln \left (c\,x+1\right )}^2+24\,a\,b^2\,c\,x\,{\ln \left (1-c\,x\right )}^2-18\,b^3\,c^2\,x^2\,\ln \left (c\,x+1\right )\,\ln \left (1-c\,x\right )+288\,a\,b^2\,c\,x\,\mathrm {atanh}\left (c\,x\right )+96\,a^2\,b\,c\,x\,\mathrm {atanh}\left (c\,x\right )-24\,a\,b^2\,c^2\,x^2\,\ln \left (c\,x+1\right )\,\ln \left (1-c\,x\right )-48\,a\,b^2\,c\,x\,\ln \left (c\,x+1\right )\,\ln \left (1-c\,x\right )}{128\,c\,{\left (c\,x+1\right )}^2} \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]

(102*b^3*log(1 - c*x) - 102*b^3*log(c*x + 1) - 96*a*b^2 - 96*a^2*b - 15*b^3*log(c*x + 1)^2 - 6*b^3*log(c*x + 1
)^3 - 15*b^3*log(1 - c*x)^2 + 6*b^3*log(1 - c*x)^3 + 150*b^3*atanh(c*x) - 64*a^3 - 48*b^3 + 144*a*b^2*atanh(c*
x) + 48*a^2*b*atanh(c*x) + 30*b^3*log(c*x + 1)*log(1 - c*x) - 132*a*b^2*log(c*x + 1) - 96*a^2*b*log(c*x + 1) +
132*a*b^2*log(1 - c*x) + 96*a^2*b*log(1 - c*x) - 18*b^3*log(c*x + 1)*log(1 - c*x)^2 + 18*b^3*log(c*x + 1)^2*l
og(1 - c*x) - 36*a*b^2*log(c*x + 1)^2 - 36*a*b^2*log(1 - c*x)^2 - 42*b^3*c*x - 144*b^3*c*x*log(c*x + 1) + 144*
b^3*c*x*log(1 - c*x) + 9*b^3*c^2*x^2*log(c*x + 1)^2 + 2*b^3*c^2*x^2*log(c*x + 1)^3 + 9*b^3*c^2*x^2*log(1 - c*x
)^2 - 2*b^3*c^2*x^2*log(1 - c*x)^3 + 150*b^3*c^2*x^2*atanh(c*x) - 72*a*b^2*c*x - 48*a^2*b*c*x + 6*b^3*c*x*log(
c*x + 1)^2 + 4*b^3*c*x*log(c*x + 1)^3 + 6*b^3*c*x*log(1 - c*x)^2 - 4*b^3*c*x*log(1 - c*x)^3 + 72*a*b^2*log(c*x
+ 1)*log(1 - c*x) + 300*b^3*c*x*atanh(c*x) - 54*b^3*c^2*x^2*log(c*x + 1) + 54*b^3*c^2*x^2*log(1 - c*x) - 12*b
^3*c*x*log(c*x + 1)*log(1 - c*x) - 36*a*b^2*c^2*x^2*log(c*x + 1) + 36*a*b^2*c^2*x^2*log(1 - c*x) + 6*b^3*c^2*x
^2*log(c*x + 1)*log(1 - c*x)^2 - 6*b^3*c^2*x^2*log(c*x + 1)^2*log(1 - c*x) - 120*a*b^2*c*x*log(c*x + 1) + 120*
a*b^2*c*x*log(1 - c*x) + 12*b^3*c*x*log(c*x + 1)*log(1 - c*x)^2 - 12*b^3*c*x*log(c*x + 1)^2*log(1 - c*x) + 12*
a*b^2*c^2*x^2*log(c*x + 1)^2 + 12*a*b^2*c^2*x^2*log(1 - c*x)^2 + 144*a*b^2*c^2*x^2*atanh(c*x) + 48*a^2*b*c^2*x
^2*atanh(c*x) + 24*a*b^2*c*x*log(c*x + 1)^2 + 24*a*b^2*c*x*log(1 - c*x)^2 - 18*b^3*c^2*x^2*log(c*x + 1)*log(1
- c*x) + 288*a*b^2*c*x*atanh(c*x) + 96*a^2*b*c*x*atanh(c*x) - 24*a*b^2*c^2*x^2*log(c*x + 1)*log(1 - c*x) - 48*
a*b^2*c*x*log(c*x + 1)*log(1 - c*x))/(128*c*(c*x + 1)^2)

________________________________________________________________________________________