3.2.26 \(\int \frac {(a+b \tanh ^{-1}(c x))^3}{(1+c x)^4} \, dx\) [126]

Optimal. Leaf size=275 \[ -\frac {b^3}{108 c (1+c x)^3}-\frac {19 b^3}{576 c (1+c x)^2}-\frac {85 b^3}{576 c (1+c x)}+\frac {85 b^3 \tanh ^{-1}(c x)}{576 c}-\frac {b^2 \left (a+b \tanh ^{-1}(c x)\right )}{18 c (1+c x)^3}-\frac {5 b^2 \left (a+b \tanh ^{-1}(c x)\right )}{48 c (1+c x)^2}-\frac {11 b^2 \left (a+b \tanh ^{-1}(c x)\right )}{48 c (1+c x)}+\frac {11 b \left (a+b \tanh ^{-1}(c x)\right )^2}{96 c}-\frac {b \left (a+b \tanh ^{-1}(c x)\right )^2}{6 c (1+c x)^3}-\frac {b \left (a+b \tanh ^{-1}(c x)\right )^2}{8 c (1+c x)^2}-\frac {b \left (a+b \tanh ^{-1}(c x)\right )^2}{8 c (1+c x)}+\frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{24 c}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{3 c (1+c x)^3} \]

[Out]

-1/108*b^3/c/(c*x+1)^3-19/576*b^3/c/(c*x+1)^2-85/576*b^3/c/(c*x+1)+85/576*b^3*arctanh(c*x)/c-1/18*b^2*(a+b*arc
tanh(c*x))/c/(c*x+1)^3-5/48*b^2*(a+b*arctanh(c*x))/c/(c*x+1)^2-11/48*b^2*(a+b*arctanh(c*x))/c/(c*x+1)+11/96*b*
(a+b*arctanh(c*x))^2/c-1/6*b*(a+b*arctanh(c*x))^2/c/(c*x+1)^3-1/8*b*(a+b*arctanh(c*x))^2/c/(c*x+1)^2-1/8*b*(a+
b*arctanh(c*x))^2/c/(c*x+1)+1/24*(a+b*arctanh(c*x))^3/c-1/3*(a+b*arctanh(c*x))^3/c/(c*x+1)^3

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Rubi [A]
time = 0.45, antiderivative size = 275, normalized size of antiderivative = 1.00, number of steps used = 42, 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 {11 b^2 \left (a+b \tanh ^{-1}(c x)\right )}{48 c (c x+1)}-\frac {5 b^2 \left (a+b \tanh ^{-1}(c x)\right )}{48 c (c x+1)^2}-\frac {b^2 \left (a+b \tanh ^{-1}(c x)\right )}{18 c (c x+1)^3}+\frac {11 b \left (a+b \tanh ^{-1}(c x)\right )^2}{96 c}-\frac {b \left (a+b \tanh ^{-1}(c x)\right )^2}{8 c (c x+1)}-\frac {b \left (a+b \tanh ^{-1}(c x)\right )^2}{8 c (c x+1)^2}-\frac {b \left (a+b \tanh ^{-1}(c x)\right )^2}{6 c (c x+1)^3}+\frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{24 c}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{3 c (c x+1)^3}-\frac {85 b^3}{576 c (c x+1)}-\frac {19 b^3}{576 c (c x+1)^2}-\frac {b^3}{108 c (c x+1)^3}+\frac {85 b^3 \tanh ^{-1}(c x)}{576 c} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcTanh[c*x])^3/(1 + c*x)^4,x]

[Out]

-1/108*b^3/(c*(1 + c*x)^3) - (19*b^3)/(576*c*(1 + c*x)^2) - (85*b^3)/(576*c*(1 + c*x)) + (85*b^3*ArcTanh[c*x])
/(576*c) - (b^2*(a + b*ArcTanh[c*x]))/(18*c*(1 + c*x)^3) - (5*b^2*(a + b*ArcTanh[c*x]))/(48*c*(1 + c*x)^2) - (
11*b^2*(a + b*ArcTanh[c*x]))/(48*c*(1 + c*x)) + (11*b*(a + b*ArcTanh[c*x])^2)/(96*c) - (b*(a + b*ArcTanh[c*x])
^2)/(6*c*(1 + c*x)^3) - (b*(a + b*ArcTanh[c*x])^2)/(8*c*(1 + c*x)^2) - (b*(a + b*ArcTanh[c*x])^2)/(8*c*(1 + c*
x)) + (a + b*ArcTanh[c*x])^3/(24*c) - (a + b*ArcTanh[c*x])^3/(3*c*(1 + c*x)^3)

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)^4} \, dx &=-\frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{3 c (1+c x)^3}+b \int \left (\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{2 (1+c x)^4}+\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{4 (1+c x)^3}+\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{8 (1+c x)^2}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{8 \left (-1+c^2 x^2\right )}\right ) \, dx\\ &=-\frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{3 c (1+c x)^3}+\frac {1}{8} b \int \frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{(1+c x)^2} \, dx-\frac {1}{8} b \int \frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{-1+c^2 x^2} \, dx+\frac {1}{4} b \int \frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{(1+c x)^3} \, dx+\frac {1}{2} b \int \frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{(1+c x)^4} \, dx\\ &=-\frac {b \left (a+b \tanh ^{-1}(c x)\right )^2}{6 c (1+c x)^3}-\frac {b \left (a+b \tanh ^{-1}(c x)\right )^2}{8 c (1+c x)^2}-\frac {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}{24 c}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{3 c (1+c x)^3}+\frac {1}{4} b^2 \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} b^2 \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 {1}{3} b^2 \int \left (\frac {a+b \tanh ^{-1}(c x)}{2 (1+c x)^4}+\frac {a+b \tanh ^{-1}(c x)}{4 (1+c x)^3}+\frac {a+b \tanh ^{-1}(c x)}{8 (1+c x)^2}-\frac {a+b \tanh ^{-1}(c x)}{8 \left (-1+c^2 x^2\right )}\right ) \, dx\\ &=-\frac {b \left (a+b \tanh ^{-1}(c x)\right )^2}{6 c (1+c x)^3}-\frac {b \left (a+b \tanh ^{-1}(c x)\right )^2}{8 c (1+c x)^2}-\frac {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}{24 c}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{3 c (1+c x)^3}+\frac {1}{24} b^2 \int \frac {a+b \tanh ^{-1}(c x)}{(1+c x)^2} \, dx-\frac {1}{24} b^2 \int \frac {a+b \tanh ^{-1}(c x)}{-1+c^2 x^2} \, dx+\frac {1}{16} b^2 \int \frac {a+b \tanh ^{-1}(c x)}{(1+c x)^2} \, dx-\frac {1}{16} b^2 \int \frac {a+b \tanh ^{-1}(c x)}{-1+c^2 x^2} \, dx+\frac {1}{12} b^2 \int \frac {a+b \tanh ^{-1}(c x)}{(1+c x)^3} \, dx+\frac {1}{8} b^2 \int \frac {a+b \tanh ^{-1}(c x)}{(1+c x)^3} \, dx+\frac {1}{8} b^2 \int \frac {a+b \tanh ^{-1}(c x)}{(1+c x)^2} \, dx-\frac {1}{8} b^2 \int \frac {a+b \tanh ^{-1}(c x)}{-1+c^2 x^2} \, dx+\frac {1}{6} b^2 \int \frac {a+b \tanh ^{-1}(c x)}{(1+c x)^4} \, dx\\ &=-\frac {b^2 \left (a+b \tanh ^{-1}(c x)\right )}{18 c (1+c x)^3}-\frac {5 b^2 \left (a+b \tanh ^{-1}(c x)\right )}{48 c (1+c x)^2}-\frac {11 b^2 \left (a+b \tanh ^{-1}(c x)\right )}{48 c (1+c x)}+\frac {11 b \left (a+b \tanh ^{-1}(c x)\right )^2}{96 c}-\frac {b \left (a+b \tanh ^{-1}(c x)\right )^2}{6 c (1+c x)^3}-\frac {b \left (a+b \tanh ^{-1}(c x)\right )^2}{8 c (1+c x)^2}-\frac {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}{24 c}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{3 c (1+c x)^3}+\frac {1}{24} b^3 \int \frac {1}{(1+c x)^2 \left (1-c^2 x^2\right )} \, dx+\frac {1}{24} b^3 \int \frac {1}{(1+c x) \left (1-c^2 x^2\right )} \, dx+\frac {1}{18} b^3 \int \frac {1}{(1+c x)^3 \left (1-c^2 x^2\right )} \, dx+\frac {1}{16} b^3 \int \frac {1}{(1+c x)^2 \left (1-c^2 x^2\right )} \, dx+\frac {1}{16} b^3 \int \frac {1}{(1+c x) \left (1-c^2 x^2\right )} \, dx+\frac {1}{8} b^3 \int \frac {1}{(1+c x) \left (1-c^2 x^2\right )} \, dx\\ &=-\frac {b^2 \left (a+b \tanh ^{-1}(c x)\right )}{18 c (1+c x)^3}-\frac {5 b^2 \left (a+b \tanh ^{-1}(c x)\right )}{48 c (1+c x)^2}-\frac {11 b^2 \left (a+b \tanh ^{-1}(c x)\right )}{48 c (1+c x)}+\frac {11 b \left (a+b \tanh ^{-1}(c x)\right )^2}{96 c}-\frac {b \left (a+b \tanh ^{-1}(c x)\right )^2}{6 c (1+c x)^3}-\frac {b \left (a+b \tanh ^{-1}(c x)\right )^2}{8 c (1+c x)^2}-\frac {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}{24 c}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{3 c (1+c x)^3}+\frac {1}{24} b^3 \int \frac {1}{(1-c x) (1+c x)^3} \, dx+\frac {1}{24} b^3 \int \frac {1}{(1-c x) (1+c x)^2} \, dx+\frac {1}{18} b^3 \int \frac {1}{(1-c x) (1+c x)^4} \, dx+\frac {1}{16} b^3 \int \frac {1}{(1-c x) (1+c x)^3} \, dx+\frac {1}{16} b^3 \int \frac {1}{(1-c x) (1+c x)^2} \, dx+\frac {1}{8} b^3 \int \frac {1}{(1-c x) (1+c x)^2} \, dx\\ &=-\frac {b^2 \left (a+b \tanh ^{-1}(c x)\right )}{18 c (1+c x)^3}-\frac {5 b^2 \left (a+b \tanh ^{-1}(c x)\right )}{48 c (1+c x)^2}-\frac {11 b^2 \left (a+b \tanh ^{-1}(c x)\right )}{48 c (1+c x)}+\frac {11 b \left (a+b \tanh ^{-1}(c x)\right )^2}{96 c}-\frac {b \left (a+b \tanh ^{-1}(c x)\right )^2}{6 c (1+c x)^3}-\frac {b \left (a+b \tanh ^{-1}(c x)\right )^2}{8 c (1+c x)^2}-\frac {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}{24 c}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{3 c (1+c x)^3}+\frac {1}{24} b^3 \int \left (\frac {1}{2 (1+c x)^2}-\frac {1}{2 \left (-1+c^2 x^2\right )}\right ) \, dx+\frac {1}{24} b^3 \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}{18} b^3 \int \left (\frac {1}{2 (1+c x)^4}+\frac {1}{4 (1+c x)^3}+\frac {1}{8 (1+c x)^2}-\frac {1}{8 \left (-1+c^2 x^2\right )}\right ) \, dx+\frac {1}{16} b^3 \int \left (\frac {1}{2 (1+c x)^2}-\frac {1}{2 \left (-1+c^2 x^2\right )}\right ) \, dx+\frac {1}{16} b^3 \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} b^3 \int \left (\frac {1}{2 (1+c x)^2}-\frac {1}{2 \left (-1+c^2 x^2\right )}\right ) \, dx\\ &=-\frac {b^3}{108 c (1+c x)^3}-\frac {19 b^3}{576 c (1+c x)^2}-\frac {85 b^3}{576 c (1+c x)}-\frac {b^2 \left (a+b \tanh ^{-1}(c x)\right )}{18 c (1+c x)^3}-\frac {5 b^2 \left (a+b \tanh ^{-1}(c x)\right )}{48 c (1+c x)^2}-\frac {11 b^2 \left (a+b \tanh ^{-1}(c x)\right )}{48 c (1+c x)}+\frac {11 b \left (a+b \tanh ^{-1}(c x)\right )^2}{96 c}-\frac {b \left (a+b \tanh ^{-1}(c x)\right )^2}{6 c (1+c x)^3}-\frac {b \left (a+b \tanh ^{-1}(c x)\right )^2}{8 c (1+c x)^2}-\frac {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}{24 c}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{3 c (1+c x)^3}-\frac {1}{144} b^3 \int \frac {1}{-1+c^2 x^2} \, dx-\frac {1}{96} b^3 \int \frac {1}{-1+c^2 x^2} \, dx-\frac {1}{64} b^3 \int \frac {1}{-1+c^2 x^2} \, dx-\frac {1}{48} b^3 \int \frac {1}{-1+c^2 x^2} \, dx-\frac {1}{32} b^3 \int \frac {1}{-1+c^2 x^2} \, dx-\frac {1}{16} b^3 \int \frac {1}{-1+c^2 x^2} \, dx\\ &=-\frac {b^3}{108 c (1+c x)^3}-\frac {19 b^3}{576 c (1+c x)^2}-\frac {85 b^3}{576 c (1+c x)}+\frac {85 b^3 \tanh ^{-1}(c x)}{576 c}-\frac {b^2 \left (a+b \tanh ^{-1}(c x)\right )}{18 c (1+c x)^3}-\frac {5 b^2 \left (a+b \tanh ^{-1}(c x)\right )}{48 c (1+c x)^2}-\frac {11 b^2 \left (a+b \tanh ^{-1}(c x)\right )}{48 c (1+c x)}+\frac {11 b \left (a+b \tanh ^{-1}(c x)\right )^2}{96 c}-\frac {b \left (a+b \tanh ^{-1}(c x)\right )^2}{6 c (1+c x)^3}-\frac {b \left (a+b \tanh ^{-1}(c x)\right )^2}{8 c (1+c x)^2}-\frac {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}{24 c}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{3 c (1+c x)^3}\\ \end {align*}

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Mathematica [A]
time = 0.14, size = 279, normalized size = 1.01 \begin {gather*} -\frac {32 \left (36 a^3+18 a^2 b+6 a b^2+b^3\right )+6 b \left (72 a^2+60 a b+19 b^2\right ) (1+c x)+6 b \left (72 a^2+132 a b+85 b^2\right ) (1+c x)^2+24 b \left (144 a^2+12 a b \left (10+9 c x+3 c^2 x^2\right )+b^2 \left (56+81 c x+33 c^2 x^2\right )\right ) \tanh ^{-1}(c x)-36 b^2 (-1+c x) \left (12 a \left (7+4 c x+c^2 x^2\right )+b \left (29+32 c x+11 c^2 x^2\right )\right ) \tanh ^{-1}(c x)^2-144 b^3 \left (-7+3 c x+3 c^2 x^2+c^3 x^3\right ) \tanh ^{-1}(c x)^3+3 b \left (72 a^2+132 a b+85 b^2\right ) (1+c x)^3 \log (1-c x)-3 b \left (72 a^2+132 a b+85 b^2\right ) (1+c x)^3 \log (1+c x)}{3456 c (1+c x)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*ArcTanh[c*x])^3/(1 + c*x)^4,x]

[Out]

-1/3456*(32*(36*a^3 + 18*a^2*b + 6*a*b^2 + b^3) + 6*b*(72*a^2 + 60*a*b + 19*b^2)*(1 + c*x) + 6*b*(72*a^2 + 132
*a*b + 85*b^2)*(1 + c*x)^2 + 24*b*(144*a^2 + 12*a*b*(10 + 9*c*x + 3*c^2*x^2) + b^2*(56 + 81*c*x + 33*c^2*x^2))
*ArcTanh[c*x] - 36*b^2*(-1 + c*x)*(12*a*(7 + 4*c*x + c^2*x^2) + b*(29 + 32*c*x + 11*c^2*x^2))*ArcTanh[c*x]^2 -
144*b^3*(-7 + 3*c*x + 3*c^2*x^2 + c^3*x^3)*ArcTanh[c*x]^3 + 3*b*(72*a^2 + 132*a*b + 85*b^2)*(1 + c*x)^3*Log[1
- c*x] - 3*b*(72*a^2 + 132*a*b + 85*b^2)*(1 + c*x)^3*Log[1 + c*x])/(c*(1 + c*x)^3)

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 10.51, size = 3557, normalized size = 12.93

method	result	size
risch	\(\frac {b^{3} \left (x^{3} c^{3}+3 c^{2} x^{2}+3 c x -7\right ) \ln \left (c x +1\right )^{3}}{192 \left (c x +1\right )^{3} c}+\frac {b^{2} \left (-6 x^{3} b \ln \left (-c x +1\right ) c^{3}+12 c^{3} x^{3} a +11 b \,c^{3} x^{3}-18 b \,x^{2} \ln \left (-c x +1\right ) c^{2}+36 a \,c^{2} x^{2}+21 b \,c^{2} x^{2}-18 b c x \ln \left (-c x +1\right )+36 c x a -3 b c x +42 b \ln \left (-c x +1\right )-84 a -29 b \right ) \ln \left (c x +1\right )^{2}}{384 \left (c x +1\right )^{3} c}-\frac {b \left (-9 b^{2} c^{3} x^{3} \ln \left (-c x +1\right )^{2}+36 \ln \left (-c x +1\right ) a b \,c^{3} x^{3}+33 b^{2} c^{3} x^{3} \ln \left (-c x +1\right )-27 b^{2} c^{2} x^{2} \ln \left (-c x +1\right )^{2}+108 a b \,c^{2} x^{2} \ln \left (-c x +1\right )+63 b^{2} c^{2} \ln \left (-c x +1\right ) x^{2}+72 a b \,c^{2} x^{2}+66 b^{2} c^{2} x^{2}-27 b^{2} c x \ln \left (-c x +1\right )^{2}+108 a b c x \ln \left (-c x +1\right )-9 b^{2} c x \ln \left (-c x +1\right )+216 a b c x +162 b^{2} c x +63 b^{2} \ln \left (-c x +1\right )^{2}-252 b \ln \left (-c x +1\right ) a -87 b^{2} \ln \left (-c x +1\right )+288 a^{2}+240 a b +112 b^{2}\right ) \ln \left (c x +1\right )}{576 \left (c x +1\right )^{3} c}+\frac {-656 b^{3}-432 a^{2} b \,c^{2} x^{2}+108 \ln \left (-c x +1\right )^{2} a \,b^{2} c^{3} x^{3}+324 \ln \left (-c x +1\right )^{2} a \,b^{2} c^{2} x^{2}+324 \ln \left (-c x +1\right )^{2} a \,b^{2} c x -54 \ln \left (-c x +1\right )^{3} b^{3} c x -18 \ln \left (-c x +1\right )^{3} b^{3} c^{3} x^{3}-54 \ln \left (-c x +1\right )^{3} b^{3} c^{2} x^{2}-756 \ln \left (-c x +1\right )^{2} a \,b^{2}+1728 \ln \left (-c x +1\right ) a^{2} b -510 c^{2} b^{3} x^{2}-216 a^{2} b \ln \left (c x -1\right )-396 a \,b^{2} \ln \left (c x -1\right )-1134 b^{3} c x -1296 a^{2} b c x -792 a \,b^{2} c^{2} x^{2}-1344 a \,b^{2}-1152 a^{3}-1944 b^{2} c x a -648 \ln \left (c x -1\right ) a^{2} b c x -1188 \ln \left (c x -1\right ) a \,b^{2} c x +648 \ln \left (-c x -1\right ) a^{2} b c x +1188 \ln \left (-c x -1\right ) a \,b^{2} c x +216 \ln \left (-c x -1\right ) a^{2} b +396 \ln \left (-c x -1\right ) a \,b^{2}+1440 a \,b^{2} \ln \left (-c x +1\right )+1296 a \,b^{2} c x \ln \left (-c x +1\right )+648 \ln \left (-c x -1\right ) a^{2} b \,c^{2} x^{2}+1188 \ln \left (-c x -1\right ) a \,b^{2} c^{2} x^{2}-648 \ln \left (c x -1\right ) a^{2} b \,c^{2} x^{2}-1188 \ln \left (c x -1\right ) a \,b^{2} c^{2} x^{2}-261 b^{3} \ln \left (-c x +1\right )^{2}+672 b^{3} \ln \left (-c x +1\right )-255 b^{3} \ln \left (c x -1\right )+255 b^{3} \ln \left (-c x -1\right )+189 b^{3} c^{2} x^{2} \ln \left (-c x +1\right )^{2}+972 b^{3} c x \ln \left (-c x +1\right )+765 \ln \left (-c x -1\right ) b^{3} c^{2} x^{2}-765 \ln \left (c x -1\right ) b^{3} c^{2} x^{2}-27 b^{3} c x \ln \left (-c x +1\right )^{2}-765 \ln \left (c x -1\right ) b^{3} c x +765 \ln \left (-c x -1\right ) b^{3} c x -1440 a^{2} b +99 b^{3} c^{3} x^{3} \ln \left (-c x +1\right )^{2}+396 b^{3} c^{2} x^{2} \ln \left (-c x +1\right )+255 \ln \left (-c x -1\right ) b^{3} c^{3} x^{3}-255 \ln \left (c x -1\right ) b^{3} c^{3} x^{3}+126 \ln \left (-c x +1\right )^{3} b^{3}+216 \ln \left (-c x -1\right ) a^{2} b \,c^{3} x^{3}+396 \ln \left (-c x -1\right ) a \,b^{2} c^{3} x^{3}-216 \ln \left (c x -1\right ) a^{2} b \,c^{3} x^{3}-396 \ln \left (c x -1\right ) a \,b^{2} c^{3} x^{3}+432 a \,b^{2} c^{2} x^{2} \ln \left (-c x +1\right )}{3456 \left (c x +1\right )^{3} c}\)	\(1221\)
derivativedivides	\(\text {Expression too large to display}\)	\(3557\)
default	\(\text {Expression too large to display}\)	\(3557\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arctanh(c*x))^3/(c*x+1)^4,x,method=_RETURNVERBOSE)

[Out]

1/c*(1/16*I*b^3/(c*x+1)^3*Pi*arctanh(c*x)^2*csgn(I/(1+(c*x+1)^2/(-c^2*x^2+1)))^3+3/32*I*b^3/(c*x+1)^3*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)))*csgn(I*(c*x
+1)^2/(c^2*x^2-1))*Pi*c^2*x^2+3/32*I*b^3/(c*x+1)^3*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)))*csgn(I*(c*x+1)^2/(c^2*x^2-1))*Pi*c*x+1/32*I*b^3/(c*x+1)^3*arct
anh(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)))*csgn(I*
(c*x+1)^2/(c^2*x^2-1))*Pi*c^3*x^3-3/16*I*b^3/(c*x+1)^3*arctanh(c*x)^2*csgn(I*(c*x+1)^2/(c^2*x^2-1))^2*csgn(I*(
c*x+1)/(-c^2*x^2+1)^(1/2))*Pi*c*x-3/32*I*b^3/(c*x+1)^3*arctanh(c*x)^2*csgn(I*(c*x+1)^2/(c^2*x^2-1))*csgn(I*(c*
x+1)/(-c^2*x^2+1)^(1/2))^2*Pi*c*x-1/32*I*b^3/(c*x+1)^3*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*Pi*c^3*x^3+1/32*I*b^3/(c*x+1)^3*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*csgn(I*(c*x+1)^2/(c^2*x^2-1))*Pi*c^3*x^3-1/32*I*b^3/(c*x+1)^
3*arctanh(c*x)^2*csgn(I*(c*x+1)^2/(c^2*x^2-1))^3*Pi*c^3*x^3+3/16*I*b^3/(c*x+1)^3*arctanh(c*x)^2*csgn(I/(1+(c*x
+1)^2/(-c^2*x^2+1)))^3*Pi*c^2*x^2-1/16*I*b^3/(c*x+1)^3*arctanh(c*x)^2*csgn(I/(1+(c*x+1)^2/(-c^2*x^2+1)))^2*Pi*
c^3*x^3+1/32*I*b^3/(c*x+1)^3*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)))-3/32*I*b^3/(c*x+1)^3*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^2-3/32*I*b^3/(c*x+1)^3*arctanh(c*x)^2*csgn(I*(c*x+1)^2/(c
^2*x^2-1))^3*Pi*c^2*x^2+3/16*I*b^3/(c*x+1)^3*arctanh(c*x)^2*csgn(I/(1+(c*x+1)^2/(-c^2*x^2+1)))^3*Pi*c*x-3/16*I
*b^3/(c*x+1)^3*arctanh(c*x)^2*csgn(I/(1+(c*x+1)^2/(-c^2*x^2+1)))^2*Pi*c^2*x^2-3/32*I*b^3/(c*x+1)^3*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-3/32*I*b^3/(c*x+1)^3*arctanh(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)^3*arctanh(c*x)^2*csgn(I/(1+(c*x+1)^2/(-c^2*x^2+1)))^2*Pi
*c*x+1/16*I*b^3/(c*x+1)^3*arctanh(c*x)^2*csgn(I/(1+(c*x+1)^2/(-c^2*x^2+1)))^3*Pi*c^3*x^3-1/32*I*b^3/(c*x+1)^3*
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^3*x^3-1/32*I*b^3/(c*x+1)^3*Pi*a
rctanh(c*x)^2*csgn(I*(c*x+1)^2/(c^2*x^2-1)/(1+(c*x+1)^2/(-c^2*x^2+1)))^3-1/32*I*b^3/(c*x+1)^3*Pi*arctanh(c*x)^
2*csgn(I*(c*x+1)^2/(c^2*x^2-1))^3-1/16*I*b^3/(c*x+1)^3*Pi*arctanh(c*x)^2*csgn(I/(1+(c*x+1)^2/(-c^2*x^2+1)))^2-
1/16*a^2*b*ln(c*x-1)+1/16*a^2*b*ln(c*x+1)-1/8*b^3*arctanh(c*x)^2*ln((c*x+1)/(-c^2*x^2+1)^(1/2))-1/16*b^3*arcta
nh(c*x)^2*ln(c*x-1)+1/16*b^3*arctanh(c*x)^2*ln(c*x+1)-1/32*a*b^2*ln(c*x-1)^2-1/32*a*b^2*ln(c*x+1)^2-11/96*a*b^
2*ln(c*x-1)+11/96*a*b^2*ln(c*x+1)+1/8*a*b^2*arctanh(c*x)*ln(c*x+1)+1/16*a*b^2*ln(c*x-1)*ln(1/2*c*x+1/2)-1/16*I
*b^3/(c*x+1)^3*arctanh(c*x)^2*csgn(I*(c*x+1)^2/(c^2*x^2-1))^2*csgn(I*(c*x+1)/(-c^2*x^2+1)^(1/2))*Pi*c^3*x^3-1/
32*I*b^3/(c*x+1)^3*arctanh(c*x)^2*csgn(I*(c*x+1)^2/(c^2*x^2-1))*csgn(I*(c*x+1)/(-c^2*x^2+1)^(1/2))^2*Pi*c^3*x^
3-3/32*I*b^3/(c*x+1)^3*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*Pi*c^2*x^2+3/32*I*b^3/(c*x+1)^3*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*csgn(I*(c*x+1)^2/(c^2*x^2-1))*Pi*c^2*x^2-3/16*I*b^3/(c*x+1)^3*arctanh(c*x)^2*csgn(I*(c*x+1)^
2/(c^2*x^2-1))^2*csgn(I*(c*x+1)/(-c^2*x^2+1)^(1/2))*Pi*c^2*x^2-3/32*I*b^3/(c*x+1)^3*arctanh(c*x)^2*csgn(I*(c*x
+1)^2/(c^2*x^2-1))*csgn(I*(c*x+1)/(-c^2*x^2+1)^(1/2))^2*Pi*c^2*x^2-3/32*I*b^3/(c*x+1)^3*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*Pi*c*x+3/32*I*b^3/(c*x+
1)^3*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*csgn(I*(c*x+1)^2/(c^2*x^2-1))*P
i*c*x+1/16*a*b^2*ln(-1/2*c*x+1/2)*ln(c*x+1)-1/16*a*b^2*ln(-1/2*c*x+1/2)*ln(1/2*c*x+1/2)-1/8*a*b^2*arctanh(c*x)
*ln(c*x-1)-1/32*I*b^3/(c*x+1)^3*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+1/32*I*b^3/(c*x+1)^3*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+1/16*I*b^3/(c*x+1)^3*arctanh(c*x)^2*Pi*c^3*x^3+3/16*I*b^3/
(c*x+1)^3*arctanh(c*x)^2*Pi*c^2*x^2+3/16*I*b^3/(c*x+1)^3*arctanh(c*x)^2*Pi*c*x-1/16*I*b^3/(c*x+1)^3*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))^2-1/32*I*b^3/(c*x+1)^3*Pi*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))-5/48*a*b^2/(c*x+1)^2-1/8*a^2*b/(c*x+1)^
2-1/8*b^3/(c*x+1)^2*arctanh(c*x)^2-1/3*a^3/(c*x+1)^3-737/6912*b^3/(c*x+1)^3-a^2*b*arctanh(c*x)/(c*x+1)^3-a*b^2
/(c*x+1)^3*arctanh(c*x)^2-1/3*a*b^2*arctanh(c*x)/(c*x+1)^3+575/6912*b^3/(c*x+1)^3*c^3*x^3+235/2304*b^3/(c*x+1)
^3*c^2*x^2-181/2304*b^3/(c*x+1)^3*c*x-1/8/(c*x+1)*a^2*b-11/48*a*b^2/(c*x+1)-1/18*a*b^2/(c*x+1)^3-7/24*b^3/(c*x
+1)^3*arctanh(c*x)^3-5/96*b^3/(c*x+1)^3*arctanh...

________________________________________________________________________________________

Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 1085 vs. \(2 (249) = 498\).
time = 0.32, size = 1085, normalized size = 3.95 \begin {gather*} -\frac {b^{3} \operatorname {artanh}\left (c x\right )^{3}}{3 \, {\left (c^{4} x^{3} + 3 \, c^{3} x^{2} + 3 \, c^{2} x + c\right )}} - \frac {1}{48} \, {\left (c {\left (\frac {2 \, {\left (3 \, c^{2} x^{2} + 9 \, c x + 10\right )}}{c^{5} x^{3} + 3 \, c^{4} x^{2} + 3 \, c^{3} x + c^{2}} - \frac {3 \, \log \left (c x + 1\right )}{c^{2}} + \frac {3 \, \log \left (c x - 1\right )}{c^{2}}\right )} + \frac {48 \, \operatorname {artanh}\left (c x\right )}{c^{4} x^{3} + 3 \, c^{3} x^{2} + 3 \, c^{2} x + c}\right )} a^{2} b - \frac {1}{288} \, {\left (12 \, c {\left (\frac {2 \, {\left (3 \, c^{2} x^{2} + 9 \, c x + 10\right )}}{c^{5} x^{3} + 3 \, c^{4} x^{2} + 3 \, c^{3} x + c^{2}} - \frac {3 \, \log \left (c x + 1\right )}{c^{2}} + \frac {3 \, \log \left (c x - 1\right )}{c^{2}}\right )} \operatorname {artanh}\left (c x\right ) + \frac {{\left (66 \, c^{2} x^{2} + 9 \, {\left (c^{3} x^{3} + 3 \, c^{2} x^{2} + 3 \, c x + 1\right )} \log \left (c x + 1\right )^{2} + 9 \, {\left (c^{3} x^{3} + 3 \, c^{2} x^{2} + 3 \, c x + 1\right )} \log \left (c x - 1\right )^{2} + 162 \, c x - 3 \, {\left (11 \, c^{3} x^{3} + 33 \, c^{2} x^{2} + 33 \, c x + 6 \, {\left (c^{3} x^{3} + 3 \, c^{2} x^{2} + 3 \, c x + 1\right )} \log \left (c x - 1\right ) + 11\right )} \log \left (c x + 1\right ) + 33 \, {\left (c^{3} x^{3} + 3 \, c^{2} x^{2} + 3 \, c x + 1\right )} \log \left (c x - 1\right ) + 112\right )} c^{2}}{c^{6} x^{3} + 3 \, c^{5} x^{2} + 3 \, c^{4} x + c^{3}}\right )} a b^{2} - \frac {1}{3456} \, {\left (72 \, c {\left (\frac {2 \, {\left (3 \, c^{2} x^{2} + 9 \, c x + 10\right )}}{c^{5} x^{3} + 3 \, c^{4} x^{2} + 3 \, c^{3} x + c^{2}} - \frac {3 \, \log \left (c x + 1\right )}{c^{2}} + \frac {3 \, \log \left (c x - 1\right )}{c^{2}}\right )} \operatorname {artanh}\left (c x\right )^{2} + {\left (\frac {{\left (510 \, c^{2} x^{2} - 18 \, {\left (c^{3} x^{3} + 3 \, c^{2} x^{2} + 3 \, c x + 1\right )} \log \left (c x + 1\right )^{3} + 18 \, {\left (c^{3} x^{3} + 3 \, c^{2} x^{2} + 3 \, c x + 1\right )} \log \left (c x - 1\right )^{3} + 9 \, {\left (11 \, c^{3} x^{3} + 33 \, c^{2} x^{2} + 33 \, c x + 6 \, {\left (c^{3} x^{3} + 3 \, c^{2} x^{2} + 3 \, c x + 1\right )} \log \left (c x - 1\right ) + 11\right )} \log \left (c x + 1\right )^{2} + 99 \, {\left (c^{3} x^{3} + 3 \, c^{2} x^{2} + 3 \, c x + 1\right )} \log \left (c x - 1\right )^{2} + 1134 \, c x - 3 \, {\left (85 \, c^{3} x^{3} + 255 \, c^{2} x^{2} + 18 \, {\left (c^{3} x^{3} + 3 \, c^{2} x^{2} + 3 \, c x + 1\right )} \log \left (c x - 1\right )^{2} + 255 \, c x + 66 \, {\left (c^{3} x^{3} + 3 \, c^{2} x^{2} + 3 \, c x + 1\right )} \log \left (c x - 1\right ) + 85\right )} \log \left (c x + 1\right ) + 255 \, {\left (c^{3} x^{3} + 3 \, c^{2} x^{2} + 3 \, c x + 1\right )} \log \left (c x - 1\right ) + 656\right )} c^{2}}{c^{7} x^{3} + 3 \, c^{6} x^{2} + 3 \, c^{5} x + c^{4}} + \frac {12 \, {\left (66 \, c^{2} x^{2} + 9 \, {\left (c^{3} x^{3} + 3 \, c^{2} x^{2} + 3 \, c x + 1\right )} \log \left (c x + 1\right )^{2} + 9 \, {\left (c^{3} x^{3} + 3 \, c^{2} x^{2} + 3 \, c x + 1\right )} \log \left (c x - 1\right )^{2} + 162 \, c x - 3 \, {\left (11 \, c^{3} x^{3} + 33 \, c^{2} x^{2} + 33 \, c x + 6 \, {\left (c^{3} x^{3} + 3 \, c^{2} x^{2} + 3 \, c x + 1\right )} \log \left (c x - 1\right ) + 11\right )} \log \left (c x + 1\right ) + 33 \, {\left (c^{3} x^{3} + 3 \, c^{2} x^{2} + 3 \, c x + 1\right )} \log \left (c x - 1\right ) + 112\right )} c \operatorname {artanh}\left (c x\right )}{c^{6} x^{3} + 3 \, c^{5} x^{2} + 3 \, c^{4} x + c^{3}}\right )} c\right )} b^{3} - \frac {a b^{2} \operatorname {artanh}\left (c x\right )^{2}}{c^{4} x^{3} + 3 \, c^{3} x^{2} + 3 \, c^{2} x + c} - \frac {a^{3}}{3 \, {\left (c^{4} x^{3} + 3 \, c^{3} x^{2} + 3 \, 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)^4,x, algorithm="maxima")

[Out]

-1/3*b^3*arctanh(c*x)^3/(c^4*x^3 + 3*c^3*x^2 + 3*c^2*x + c) - 1/48*(c*(2*(3*c^2*x^2 + 9*c*x + 10)/(c^5*x^3 + 3
*c^4*x^2 + 3*c^3*x + c^2) - 3*log(c*x + 1)/c^2 + 3*log(c*x - 1)/c^2) + 48*arctanh(c*x)/(c^4*x^3 + 3*c^3*x^2 +
3*c^2*x + c))*a^2*b - 1/288*(12*c*(2*(3*c^2*x^2 + 9*c*x + 10)/(c^5*x^3 + 3*c^4*x^2 + 3*c^3*x + c^2) - 3*log(c*
x + 1)/c^2 + 3*log(c*x - 1)/c^2)*arctanh(c*x) + (66*c^2*x^2 + 9*(c^3*x^3 + 3*c^2*x^2 + 3*c*x + 1)*log(c*x + 1)
^2 + 9*(c^3*x^3 + 3*c^2*x^2 + 3*c*x + 1)*log(c*x - 1)^2 + 162*c*x - 3*(11*c^3*x^3 + 33*c^2*x^2 + 33*c*x + 6*(c
^3*x^3 + 3*c^2*x^2 + 3*c*x + 1)*log(c*x - 1) + 11)*log(c*x + 1) + 33*(c^3*x^3 + 3*c^2*x^2 + 3*c*x + 1)*log(c*x
- 1) + 112)*c^2/(c^6*x^3 + 3*c^5*x^2 + 3*c^4*x + c^3))*a*b^2 - 1/3456*(72*c*(2*(3*c^2*x^2 + 9*c*x + 10)/(c^5*
x^3 + 3*c^4*x^2 + 3*c^3*x + c^2) - 3*log(c*x + 1)/c^2 + 3*log(c*x - 1)/c^2)*arctanh(c*x)^2 + ((510*c^2*x^2 - 1
8*(c^3*x^3 + 3*c^2*x^2 + 3*c*x + 1)*log(c*x + 1)^3 + 18*(c^3*x^3 + 3*c^2*x^2 + 3*c*x + 1)*log(c*x - 1)^3 + 9*(
11*c^3*x^3 + 33*c^2*x^2 + 33*c*x + 6*(c^3*x^3 + 3*c^2*x^2 + 3*c*x + 1)*log(c*x - 1) + 11)*log(c*x + 1)^2 + 99*
(c^3*x^3 + 3*c^2*x^2 + 3*c*x + 1)*log(c*x - 1)^2 + 1134*c*x - 3*(85*c^3*x^3 + 255*c^2*x^2 + 18*(c^3*x^3 + 3*c^
2*x^2 + 3*c*x + 1)*log(c*x - 1)^2 + 255*c*x + 66*(c^3*x^3 + 3*c^2*x^2 + 3*c*x + 1)*log(c*x - 1) + 85)*log(c*x
+ 1) + 255*(c^3*x^3 + 3*c^2*x^2 + 3*c*x + 1)*log(c*x - 1) + 656)*c^2/(c^7*x^3 + 3*c^6*x^2 + 3*c^5*x + c^4) + 1
2*(66*c^2*x^2 + 9*(c^3*x^3 + 3*c^2*x^2 + 3*c*x + 1)*log(c*x + 1)^2 + 9*(c^3*x^3 + 3*c^2*x^2 + 3*c*x + 1)*log(c
*x - 1)^2 + 162*c*x - 3*(11*c^3*x^3 + 33*c^2*x^2 + 33*c*x + 6*(c^3*x^3 + 3*c^2*x^2 + 3*c*x + 1)*log(c*x - 1) +
11)*log(c*x + 1) + 33*(c^3*x^3 + 3*c^2*x^2 + 3*c*x + 1)*log(c*x - 1) + 112)*c*arctanh(c*x)/(c^6*x^3 + 3*c^5*x
^2 + 3*c^4*x + c^3))*c)*b^3 - a*b^2*arctanh(c*x)^2/(c^4*x^3 + 3*c^3*x^2 + 3*c^2*x + c) - 1/3*a^3/(c^4*x^3 + 3*
c^3*x^2 + 3*c^2*x + c)

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Fricas [A]
time = 0.37, size = 345, normalized size = 1.25 \begin {gather*} -\frac {6 \, {\left (72 \, a^{2} b + 132 \, a b^{2} + 85 \, b^{3}\right )} c^{2} x^{2} - 18 \, {\left (b^{3} c^{3} x^{3} + 3 \, b^{3} c^{2} x^{2} + 3 \, b^{3} c x - 7 \, b^{3}\right )} \log \left (-\frac {c x + 1}{c x - 1}\right )^{3} + 1152 \, a^{3} + 1440 \, a^{2} b + 1344 \, a b^{2} + 656 \, b^{3} + 162 \, {\left (8 \, a^{2} b + 12 \, a b^{2} + 7 \, b^{3}\right )} c x - 9 \, {\left ({\left (12 \, a b^{2} + 11 \, b^{3}\right )} c^{3} x^{3} + 3 \, {\left (12 \, a b^{2} + 7 \, b^{3}\right )} c^{2} x^{2} - 84 \, a b^{2} - 29 \, b^{3} + 3 \, {\left (12 \, a b^{2} - b^{3}\right )} c x\right )} \log \left (-\frac {c x + 1}{c x - 1}\right )^{2} - 3 \, {\left ({\left (72 \, a^{2} b + 132 \, a b^{2} + 85 \, b^{3}\right )} c^{3} x^{3} + 3 \, {\left (72 \, a^{2} b + 84 \, a b^{2} + 41 \, b^{3}\right )} c^{2} x^{2} - 504 \, a^{2} b - 348 \, a b^{2} - 139 \, b^{3} + 3 \, {\left (72 \, a^{2} b - 12 \, a b^{2} - 23 \, b^{3}\right )} c x\right )} \log \left (-\frac {c x + 1}{c x - 1}\right )}{3456 \, {\left (c^{4} x^{3} + 3 \, c^{3} x^{2} + 3 \, 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)^4,x, algorithm="fricas")

[Out]

-1/3456*(6*(72*a^2*b + 132*a*b^2 + 85*b^3)*c^2*x^2 - 18*(b^3*c^3*x^3 + 3*b^3*c^2*x^2 + 3*b^3*c*x - 7*b^3)*log(
-(c*x + 1)/(c*x - 1))^3 + 1152*a^3 + 1440*a^2*b + 1344*a*b^2 + 656*b^3 + 162*(8*a^2*b + 12*a*b^2 + 7*b^3)*c*x
- 9*((12*a*b^2 + 11*b^3)*c^3*x^3 + 3*(12*a*b^2 + 7*b^3)*c^2*x^2 - 84*a*b^2 - 29*b^3 + 3*(12*a*b^2 - b^3)*c*x)*
log(-(c*x + 1)/(c*x - 1))^2 - 3*((72*a^2*b + 132*a*b^2 + 85*b^3)*c^3*x^3 + 3*(72*a^2*b + 84*a*b^2 + 41*b^3)*c^
2*x^2 - 504*a^2*b - 348*a*b^2 - 139*b^3 + 3*(72*a^2*b - 12*a*b^2 - 23*b^3)*c*x)*log(-(c*x + 1)/(c*x - 1)))/(c^
4*x^3 + 3*c^3*x^2 + 3*c^2*x + c)

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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 )^{4}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*atanh(c*x))**3/(c*x+1)**4,x)

[Out]

Integral((a + b*atanh(c*x))**3/(c*x + 1)**4, x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 555 vs. \(2 (249) = 498\).
time = 0.43, size = 555, normalized size = 2.02 \begin {gather*} \frac {1}{6912} \, {\left (\frac {36 \, {\left (\frac {3 \, {\left (c x + 1\right )}^{2} b^{3}}{{\left (c x - 1\right )}^{2}} - \frac {3 \, {\left (c x + 1\right )} b^{3}}{c x - 1} + b^{3}\right )} {\left (c x - 1\right )}^{3} \log \left (-\frac {c x + 1}{c x - 1}\right )^{3}}{{\left (c x + 1\right )}^{3} c^{2}} + \frac {18 \, {\left (\frac {36 \, {\left (c x + 1\right )}^{2} a b^{2}}{{\left (c x - 1\right )}^{2}} - \frac {36 \, {\left (c x + 1\right )} a b^{2}}{c x - 1} + 12 \, a b^{2} + \frac {18 \, {\left (c x + 1\right )}^{2} b^{3}}{{\left (c x - 1\right )}^{2}} - \frac {9 \, {\left (c x + 1\right )} b^{3}}{c x - 1} + 2 \, b^{3}\right )} {\left (c x - 1\right )}^{3} \log \left (-\frac {c x + 1}{c x - 1}\right )^{2}}{{\left (c x + 1\right )}^{3} c^{2}} + \frac {6 \, {\left (\frac {216 \, {\left (c x + 1\right )}^{2} a^{2} b}{{\left (c x - 1\right )}^{2}} - \frac {216 \, {\left (c x + 1\right )} a^{2} b}{c x - 1} + 72 \, a^{2} b + \frac {216 \, {\left (c x + 1\right )}^{2} a b^{2}}{{\left (c x - 1\right )}^{2}} - \frac {108 \, {\left (c x + 1\right )} a b^{2}}{c x - 1} + 24 \, a b^{2} + \frac {108 \, {\left (c x + 1\right )}^{2} b^{3}}{{\left (c x - 1\right )}^{2}} - \frac {27 \, {\left (c x + 1\right )} b^{3}}{c x - 1} + 4 \, b^{3}\right )} {\left (c x - 1\right )}^{3} \log \left (-\frac {c x + 1}{c x - 1}\right )}{{\left (c x + 1\right )}^{3} c^{2}} + \frac {{\left (\frac {864 \, {\left (c x + 1\right )}^{2} a^{3}}{{\left (c x - 1\right )}^{2}} - \frac {864 \, {\left (c x + 1\right )} a^{3}}{c x - 1} + 288 \, a^{3} + \frac {1296 \, {\left (c x + 1\right )}^{2} a^{2} b}{{\left (c x - 1\right )}^{2}} - \frac {648 \, {\left (c x + 1\right )} a^{2} b}{c x - 1} + 144 \, a^{2} b + \frac {1296 \, {\left (c x + 1\right )}^{2} a b^{2}}{{\left (c x - 1\right )}^{2}} - \frac {324 \, {\left (c x + 1\right )} a b^{2}}{c x - 1} + 48 \, a b^{2} + \frac {648 \, {\left (c x + 1\right )}^{2} b^{3}}{{\left (c x - 1\right )}^{2}} - \frac {81 \, {\left (c x + 1\right )} b^{3}}{c x - 1} + 8 \, b^{3}\right )} {\left (c x - 1\right )}^{3}}{{\left (c x + 1\right )}^{3} 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)^4,x, algorithm="giac")

[Out]

1/6912*(36*(3*(c*x + 1)^2*b^3/(c*x - 1)^2 - 3*(c*x + 1)*b^3/(c*x - 1) + b^3)*(c*x - 1)^3*log(-(c*x + 1)/(c*x -
1))^3/((c*x + 1)^3*c^2) + 18*(36*(c*x + 1)^2*a*b^2/(c*x - 1)^2 - 36*(c*x + 1)*a*b^2/(c*x - 1) + 12*a*b^2 + 18
*(c*x + 1)^2*b^3/(c*x - 1)^2 - 9*(c*x + 1)*b^3/(c*x - 1) + 2*b^3)*(c*x - 1)^3*log(-(c*x + 1)/(c*x - 1))^2/((c*
x + 1)^3*c^2) + 6*(216*(c*x + 1)^2*a^2*b/(c*x - 1)^2 - 216*(c*x + 1)*a^2*b/(c*x - 1) + 72*a^2*b + 216*(c*x + 1
)^2*a*b^2/(c*x - 1)^2 - 108*(c*x + 1)*a*b^2/(c*x - 1) + 24*a*b^2 + 108*(c*x + 1)^2*b^3/(c*x - 1)^2 - 27*(c*x +
1)*b^3/(c*x - 1) + 4*b^3)*(c*x - 1)^3*log(-(c*x + 1)/(c*x - 1))/((c*x + 1)^3*c^2) + (864*(c*x + 1)^2*a^3/(c*x
- 1)^2 - 864*(c*x + 1)*a^3/(c*x - 1) + 288*a^3 + 1296*(c*x + 1)^2*a^2*b/(c*x - 1)^2 - 648*(c*x + 1)*a^2*b/(c*
x - 1) + 144*a^2*b + 1296*(c*x + 1)^2*a*b^2/(c*x - 1)^2 - 324*(c*x + 1)*a*b^2/(c*x - 1) + 48*a*b^2 + 648*(c*x
+ 1)^2*b^3/(c*x - 1)^2 - 81*(c*x + 1)*b^3/(c*x - 1) + 8*b^3)*(c*x - 1)^3/((c*x + 1)^3*c^2))*c

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Mupad [B]
time = 4.49, size = 1304, normalized size = 4.74 \begin {gather*} \frac {1398\,b^3\,\ln \left (1-c\,x\right )-1398\,b^3\,\ln \left (c\,x+1\right )-1344\,a\,b^2-1440\,a^2\,b-261\,b^3\,{\ln \left (c\,x+1\right )}^2-126\,b^3\,{\ln \left (c\,x+1\right )}^3-261\,b^3\,{\ln \left (1-c\,x\right )}^2+126\,b^3\,{\ln \left (1-c\,x\right )}^3+1962\,b^3\,\mathrm {atanh}\left (c\,x\right )-1152\,a^3-656\,b^3+1584\,a\,b^2\,\mathrm {atanh}\left (c\,x\right )+432\,a^2\,b\,\mathrm {atanh}\left (c\,x\right )+522\,b^3\,\ln \left (c\,x+1\right )\,\ln \left (1-c\,x\right )-1836\,a\,b^2\,\ln \left (c\,x+1\right )-1728\,a^2\,b\,\ln \left (c\,x+1\right )+1836\,a\,b^2\,\ln \left (1-c\,x\right )+1728\,a^2\,b\,\ln \left (1-c\,x\right )-378\,b^3\,\ln \left (c\,x+1\right )\,{\ln \left (1-c\,x\right )}^2+378\,b^3\,{\ln \left (c\,x+1\right )}^2\,\ln \left (1-c\,x\right )-510\,b^3\,c^2\,x^2-756\,a\,b^2\,{\ln \left (c\,x+1\right )}^2-756\,a\,b^2\,{\ln \left (1-c\,x\right )}^2-1134\,b^3\,c\,x-3150\,b^3\,c\,x\,\ln \left (c\,x+1\right )+3150\,b^3\,c\,x\,\ln \left (1-c\,x\right )-792\,a\,b^2\,c^2\,x^2-432\,a^2\,b\,c^2\,x^2+189\,b^3\,c^2\,x^2\,{\ln \left (c\,x+1\right )}^2+54\,b^3\,c^2\,x^2\,{\ln \left (c\,x+1\right )}^3+189\,b^3\,c^2\,x^2\,{\ln \left (1-c\,x\right )}^2-54\,b^3\,c^2\,x^2\,{\ln \left (1-c\,x\right )}^3+99\,b^3\,c^3\,x^3\,{\ln \left (c\,x+1\right )}^2+18\,b^3\,c^3\,x^3\,{\ln \left (c\,x+1\right )}^3+99\,b^3\,c^3\,x^3\,{\ln \left (1-c\,x\right )}^2-18\,b^3\,c^3\,x^3\,{\ln \left (1-c\,x\right )}^3+5886\,b^3\,c^2\,x^2\,\mathrm {atanh}\left (c\,x\right )+1962\,b^3\,c^3\,x^3\,\mathrm {atanh}\left (c\,x\right )-1944\,a\,b^2\,c\,x-1296\,a^2\,b\,c\,x-27\,b^3\,c\,x\,{\ln \left (c\,x+1\right )}^2+54\,b^3\,c\,x\,{\ln \left (c\,x+1\right )}^3-27\,b^3\,c\,x\,{\ln \left (1-c\,x\right )}^2-54\,b^3\,c\,x\,{\ln \left (1-c\,x\right )}^3+1512\,a\,b^2\,\ln \left (c\,x+1\right )\,\ln \left (1-c\,x\right )+5886\,b^3\,c\,x\,\mathrm {atanh}\left (c\,x\right )-2574\,b^3\,c^2\,x^2\,\ln \left (c\,x+1\right )+2574\,b^3\,c^2\,x^2\,\ln \left (1-c\,x\right )-726\,b^3\,c^3\,x^3\,\ln \left (c\,x+1\right )+726\,b^3\,c^3\,x^3\,\ln \left (1-c\,x\right )+54\,b^3\,c\,x\,\ln \left (c\,x+1\right )\,\ln \left (1-c\,x\right )-1620\,a\,b^2\,c^2\,x^2\,\ln \left (c\,x+1\right )+1620\,a\,b^2\,c^2\,x^2\,\ln \left (1-c\,x\right )-396\,a\,b^2\,c^3\,x^3\,\ln \left (c\,x+1\right )+396\,a\,b^2\,c^3\,x^3\,\ln \left (1-c\,x\right )+162\,b^3\,c^2\,x^2\,\ln \left (c\,x+1\right )\,{\ln \left (1-c\,x\right )}^2-162\,b^3\,c^2\,x^2\,{\ln \left (c\,x+1\right )}^2\,\ln \left (1-c\,x\right )+54\,b^3\,c^3\,x^3\,\ln \left (c\,x+1\right )\,{\ln \left (1-c\,x\right )}^2-54\,b^3\,c^3\,x^3\,{\ln \left (c\,x+1\right )}^2\,\ln \left (1-c\,x\right )-2484\,a\,b^2\,c\,x\,\ln \left (c\,x+1\right )+2484\,a\,b^2\,c\,x\,\ln \left (1-c\,x\right )+162\,b^3\,c\,x\,\ln \left (c\,x+1\right )\,{\ln \left (1-c\,x\right )}^2-162\,b^3\,c\,x\,{\ln \left (c\,x+1\right )}^2\,\ln \left (1-c\,x\right )+324\,a\,b^2\,c^2\,x^2\,{\ln \left (c\,x+1\right )}^2+324\,a\,b^2\,c^2\,x^2\,{\ln \left (1-c\,x\right )}^2+108\,a\,b^2\,c^3\,x^3\,{\ln \left (c\,x+1\right )}^2+108\,a\,b^2\,c^3\,x^3\,{\ln \left (1-c\,x\right )}^2+4752\,a\,b^2\,c^2\,x^2\,\mathrm {atanh}\left (c\,x\right )+1296\,a^2\,b\,c^2\,x^2\,\mathrm {atanh}\left (c\,x\right )+1584\,a\,b^2\,c^3\,x^3\,\mathrm {atanh}\left (c\,x\right )+432\,a^2\,b\,c^3\,x^3\,\mathrm {atanh}\left (c\,x\right )+324\,a\,b^2\,c\,x\,{\ln \left (c\,x+1\right )}^2+324\,a\,b^2\,c\,x\,{\ln \left (1-c\,x\right )}^2-378\,b^3\,c^2\,x^2\,\ln \left (c\,x+1\right )\,\ln \left (1-c\,x\right )-198\,b^3\,c^3\,x^3\,\ln \left (c\,x+1\right )\,\ln \left (1-c\,x\right )+4752\,a\,b^2\,c\,x\,\mathrm {atanh}\left (c\,x\right )+1296\,a^2\,b\,c\,x\,\mathrm {atanh}\left (c\,x\right )-648\,a\,b^2\,c^2\,x^2\,\ln \left (c\,x+1\right )\,\ln \left (1-c\,x\right )-216\,a\,b^2\,c^3\,x^3\,\ln \left (c\,x+1\right )\,\ln \left (1-c\,x\right )-648\,a\,b^2\,c\,x\,\ln \left (c\,x+1\right )\,\ln \left (1-c\,x\right )}{3456\,c\,{\left (c\,x+1\right )}^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*atanh(c*x))^3/(c*x + 1)^4,x)

[Out]

(1398*b^3*log(1 - c*x) - 1398*b^3*log(c*x + 1) - 1344*a*b^2 - 1440*a^2*b - 261*b^3*log(c*x + 1)^2 - 126*b^3*lo
g(c*x + 1)^3 - 261*b^3*log(1 - c*x)^2 + 126*b^3*log(1 - c*x)^3 + 1962*b^3*atanh(c*x) - 1152*a^3 - 656*b^3 + 15
84*a*b^2*atanh(c*x) + 432*a^2*b*atanh(c*x) + 522*b^3*log(c*x + 1)*log(1 - c*x) - 1836*a*b^2*log(c*x + 1) - 172
8*a^2*b*log(c*x + 1) + 1836*a*b^2*log(1 - c*x) + 1728*a^2*b*log(1 - c*x) - 378*b^3*log(c*x + 1)*log(1 - c*x)^2
+ 378*b^3*log(c*x + 1)^2*log(1 - c*x) - 510*b^3*c^2*x^2 - 756*a*b^2*log(c*x + 1)^2 - 756*a*b^2*log(1 - c*x)^2
- 1134*b^3*c*x - 3150*b^3*c*x*log(c*x + 1) + 3150*b^3*c*x*log(1 - c*x) - 792*a*b^2*c^2*x^2 - 432*a^2*b*c^2*x^
2 + 189*b^3*c^2*x^2*log(c*x + 1)^2 + 54*b^3*c^2*x^2*log(c*x + 1)^3 + 189*b^3*c^2*x^2*log(1 - c*x)^2 - 54*b^3*c
^2*x^2*log(1 - c*x)^3 + 99*b^3*c^3*x^3*log(c*x + 1)^2 + 18*b^3*c^3*x^3*log(c*x + 1)^3 + 99*b^3*c^3*x^3*log(1 -
c*x)^2 - 18*b^3*c^3*x^3*log(1 - c*x)^3 + 5886*b^3*c^2*x^2*atanh(c*x) + 1962*b^3*c^3*x^3*atanh(c*x) - 1944*a*b
^2*c*x - 1296*a^2*b*c*x - 27*b^3*c*x*log(c*x + 1)^2 + 54*b^3*c*x*log(c*x + 1)^3 - 27*b^3*c*x*log(1 - c*x)^2 -
54*b^3*c*x*log(1 - c*x)^3 + 1512*a*b^2*log(c*x + 1)*log(1 - c*x) + 5886*b^3*c*x*atanh(c*x) - 2574*b^3*c^2*x^2*
log(c*x + 1) + 2574*b^3*c^2*x^2*log(1 - c*x) - 726*b^3*c^3*x^3*log(c*x + 1) + 726*b^3*c^3*x^3*log(1 - c*x) + 5
4*b^3*c*x*log(c*x + 1)*log(1 - c*x) - 1620*a*b^2*c^2*x^2*log(c*x + 1) + 1620*a*b^2*c^2*x^2*log(1 - c*x) - 396*
a*b^2*c^3*x^3*log(c*x + 1) + 396*a*b^2*c^3*x^3*log(1 - c*x) + 162*b^3*c^2*x^2*log(c*x + 1)*log(1 - c*x)^2 - 16
2*b^3*c^2*x^2*log(c*x + 1)^2*log(1 - c*x) + 54*b^3*c^3*x^3*log(c*x + 1)*log(1 - c*x)^2 - 54*b^3*c^3*x^3*log(c*
x + 1)^2*log(1 - c*x) - 2484*a*b^2*c*x*log(c*x + 1) + 2484*a*b^2*c*x*log(1 - c*x) + 162*b^3*c*x*log(c*x + 1)*l
og(1 - c*x)^2 - 162*b^3*c*x*log(c*x + 1)^2*log(1 - c*x) + 324*a*b^2*c^2*x^2*log(c*x + 1)^2 + 324*a*b^2*c^2*x^2
*log(1 - c*x)^2 + 108*a*b^2*c^3*x^3*log(c*x + 1)^2 + 108*a*b^2*c^3*x^3*log(1 - c*x)^2 + 4752*a*b^2*c^2*x^2*ata
nh(c*x) + 1296*a^2*b*c^2*x^2*atanh(c*x) + 1584*a*b^2*c^3*x^3*atanh(c*x) + 432*a^2*b*c^3*x^3*atanh(c*x) + 324*a
*b^2*c*x*log(c*x + 1)^2 + 324*a*b^2*c*x*log(1 - c*x)^2 - 378*b^3*c^2*x^2*log(c*x + 1)*log(1 - c*x) - 198*b^3*c
^3*x^3*log(c*x + 1)*log(1 - c*x) + 4752*a*b^2*c*x*atanh(c*x) + 1296*a^2*b*c*x*atanh(c*x) - 648*a*b^2*c^2*x^2*l
og(c*x + 1)*log(1 - c*x) - 216*a*b^2*c^3*x^3*log(c*x + 1)*log(1 - c*x) - 648*a*b^2*c*x*log(c*x + 1)*log(1 - c*
x))/(3456*c*(c*x + 1)^3)

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