3.1.19 \(\int \frac {(a+b \tanh ^{-1}(c x))^3}{(d+e x)^2} \, dx\) [19]

Optimal. Leaf size=517 \[ -\frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{e (d+e x)}+\frac {3 b c \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2}{1-c x}\right )}{2 e (c d+e)}-\frac {3 b c \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2}{1+c x}\right )}{2 (c d-e) e}+\frac {3 b c \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2}{1+c x}\right )}{c^2 d^2-e^2}-\frac {3 b c \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{c^2 d^2-e^2}+\frac {3 b^2 c \left (a+b \tanh ^{-1}(c x)\right ) \text {PolyLog}\left (2,1-\frac {2}{1-c x}\right )}{2 e (c d+e)}+\frac {3 b^2 c \left (a+b \tanh ^{-1}(c x)\right ) \text {PolyLog}\left (2,1-\frac {2}{1+c x}\right )}{2 (c d-e) e}-\frac {3 b^2 c \left (a+b \tanh ^{-1}(c x)\right ) \text {PolyLog}\left (2,1-\frac {2}{1+c x}\right )}{c^2 d^2-e^2}+\frac {3 b^2 c \left (a+b \tanh ^{-1}(c x)\right ) \text {PolyLog}\left (2,1-\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{c^2 d^2-e^2}-\frac {3 b^3 c \text {PolyLog}\left (3,1-\frac {2}{1-c x}\right )}{4 e (c d+e)}+\frac {3 b^3 c \text {PolyLog}\left (3,1-\frac {2}{1+c x}\right )}{4 (c d-e) e}-\frac {3 b^3 c \text {PolyLog}\left (3,1-\frac {2}{1+c x}\right )}{2 \left (c^2 d^2-e^2\right )}+\frac {3 b^3 c \text {PolyLog}\left (3,1-\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{2 \left (c^2 d^2-e^2\right )} \]

[Out]

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

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Rubi [A]
time = 0.38, antiderivative size = 517, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.389, Rules used = {6065, 6055, 6095, 6205, 6745, 6203, 6059} \begin {gather*} -\frac {3 b^2 c \text {Li}_2\left (1-\frac {2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{c^2 d^2-e^2}+\frac {3 b^2 c \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2 c (d+e x)}{(c d+e) (c x+1)}\right )}{c^2 d^2-e^2}+\frac {3 b^2 c \text {Li}_2\left (1-\frac {2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{2 e (c d+e)}+\frac {3 b^2 c \text {Li}_2\left (1-\frac {2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{2 e (c d-e)}+\frac {3 b c \log \left (\frac {2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )^2}{c^2 d^2-e^2}-\frac {3 b c \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2 c (d+e x)}{(c x+1) (c d+e)}\right )}{c^2 d^2-e^2}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{e (d+e x)}+\frac {3 b c \log \left (\frac {2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )^2}{2 e (c d+e)}-\frac {3 b c \log \left (\frac {2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )^2}{2 e (c d-e)}-\frac {3 b^3 c \text {Li}_3\left (1-\frac {2}{c x+1}\right )}{2 \left (c^2 d^2-e^2\right )}+\frac {3 b^3 c \text {Li}_3\left (1-\frac {2 c (d+e x)}{(c d+e) (c x+1)}\right )}{2 \left (c^2 d^2-e^2\right )}-\frac {3 b^3 c \text {Li}_3\left (1-\frac {2}{1-c x}\right )}{4 e (c d+e)}+\frac {3 b^3 c \text {Li}_3\left (1-\frac {2}{c x+1}\right )}{4 e (c d-e)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-((a + b*ArcTanh[c*x])^3/(e*(d + e*x))) + (3*b*c*(a + b*ArcTanh[c*x])^2*Log[2/(1 - c*x)])/(2*e*(c*d + e)) - (3
*b*c*(a + b*ArcTanh[c*x])^2*Log[2/(1 + c*x)])/(2*(c*d - e)*e) + (3*b*c*(a + b*ArcTanh[c*x])^2*Log[2/(1 + c*x)]
)/(c^2*d^2 - e^2) - (3*b*c*(a + b*ArcTanh[c*x])^2*Log[(2*c*(d + e*x))/((c*d + e)*(1 + c*x))])/(c^2*d^2 - e^2)
+ (3*b^2*c*(a + b*ArcTanh[c*x])*PolyLog[2, 1 - 2/(1 - c*x)])/(2*e*(c*d + e)) + (3*b^2*c*(a + b*ArcTanh[c*x])*P
olyLog[2, 1 - 2/(1 + c*x)])/(2*(c*d - e)*e) - (3*b^2*c*(a + b*ArcTanh[c*x])*PolyLog[2, 1 - 2/(1 + c*x)])/(c^2*
d^2 - e^2) + (3*b^2*c*(a + b*ArcTanh[c*x])*PolyLog[2, 1 - (2*c*(d + e*x))/((c*d + e)*(1 + c*x))])/(c^2*d^2 - e
^2) - (3*b^3*c*PolyLog[3, 1 - 2/(1 - c*x)])/(4*e*(c*d + e)) + (3*b^3*c*PolyLog[3, 1 - 2/(1 + c*x)])/(4*(c*d -
e)*e) - (3*b^3*c*PolyLog[3, 1 - 2/(1 + c*x)])/(2*(c^2*d^2 - e^2)) + (3*b^3*c*PolyLog[3, 1 - (2*c*(d + e*x))/((
c*d + e)*(1 + c*x))])/(2*(c^2*d^2 - e^2))

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 6059

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^2/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-(a + b*ArcTanh[c*x])^2)*(Lo
g[2/(1 + c*x)]/e), x] + (Simp[(a + b*ArcTanh[c*x])^2*(Log[2*c*((d + e*x)/((c*d + e)*(1 + c*x)))]/e), x] + Simp
[b*(a + b*ArcTanh[c*x])*(PolyLog[2, 1 - 2/(1 + c*x)]/e), x] - Simp[b*(a + b*ArcTanh[c*x])*(PolyLog[2, 1 - 2*c*
((d + e*x)/((c*d + e)*(1 + c*x)))]/e), x] + Simp[b^2*(PolyLog[3, 1 - 2/(1 + c*x)]/(2*e)), x] - Simp[b^2*(PolyL
og[3, 1 - 2*c*((d + e*x)/((c*d + e)*(1 + c*x)))]/(2*e)), x]) /; FreeQ[{a, b, c, d, e}, x] && NeQ[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 6203

Int[(Log[u_]*((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[(a + b*ArcTan
h[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 6205

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

Rule 6745

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

Rubi steps

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

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Mathematica [C] Result contains complex when optimal does not.
time = 11.58, size = 990, normalized size = 1.91 \begin {gather*} -\frac {a^3}{e (d+e x)}-\frac {3 a^2 b \tanh ^{-1}(c x)}{e (d+e x)}-\frac {3 a^2 b c \log (1-c x)}{2 e (c d+e)}+\frac {3 a^2 b c \log (1+c x)}{2 c d e-2 e^2}-\frac {3 a^2 b c \log (d+e x)}{c^2 d^2-e^2}+\frac {3 a b^2 \left (-\frac {e^{-\tanh ^{-1}\left (\frac {c d}{e}\right )} \tanh ^{-1}(c x)^2}{\sqrt {1-\frac {c^2 d^2}{e^2}} e}+\frac {x \tanh ^{-1}(c x)^2}{d+e x}+\frac {c d \left (i \pi \log \left (1+e^{2 \tanh ^{-1}(c x)}\right )-2 \tanh ^{-1}(c x) \log \left (1-e^{-2 \left (\tanh ^{-1}\left (\frac {c d}{e}\right )+\tanh ^{-1}(c x)\right )}\right )-i \pi \left (\tanh ^{-1}(c x)-\frac {1}{2} \log \left (1-c^2 x^2\right )\right )-2 \tanh ^{-1}\left (\frac {c d}{e}\right ) \left (\tanh ^{-1}(c x)+\log \left (1-e^{-2 \left (\tanh ^{-1}\left (\frac {c d}{e}\right )+\tanh ^{-1}(c x)\right )}\right )-\log \left (i \sinh \left (\tanh ^{-1}\left (\frac {c d}{e}\right )+\tanh ^{-1}(c x)\right )\right )\right )+\text {PolyLog}\left (2,e^{-2 \left (\tanh ^{-1}\left (\frac {c d}{e}\right )+\tanh ^{-1}(c x)\right )}\right )\right )}{c^2 d^2-e^2}\right )}{d}+\frac {b^3 \left (\frac {x \tanh ^{-1}(c x)^3}{d+e x}+\frac {3 \left (6 c d \tanh ^{-1}(c x)^3-2 e \tanh ^{-1}(c x)^3+4 \sqrt {1-\frac {c^2 d^2}{e^2}} e e^{-\tanh ^{-1}\left (\frac {c d}{e}\right )} \tanh ^{-1}(c x)^3+6 i c d \pi \tanh ^{-1}(c x) \log \left (\frac {1}{2} \left (e^{-\tanh ^{-1}(c x)}+e^{\tanh ^{-1}(c x)}\right )\right )+6 c d \tanh ^{-1}(c x)^2 \log \left (1+\frac {(c d+e) e^{2 \tanh ^{-1}(c x)}}{c d-e}\right )-6 c d \tanh ^{-1}(c x)^2 \log \left (1-e^{\tanh ^{-1}\left (\frac {c d}{e}\right )+\tanh ^{-1}(c x)}\right )-6 c d \tanh ^{-1}(c x)^2 \log \left (1+e^{\tanh ^{-1}\left (\frac {c d}{e}\right )+\tanh ^{-1}(c x)}\right )-6 c d \tanh ^{-1}(c x)^2 \log \left (1-e^{2 \left (\tanh ^{-1}\left (\frac {c d}{e}\right )+\tanh ^{-1}(c x)\right )}\right )-12 c d \tanh ^{-1}\left (\frac {c d}{e}\right ) \tanh ^{-1}(c x) \log \left (\frac {1}{2} i e^{-\tanh ^{-1}\left (\frac {c d}{e}\right )-\tanh ^{-1}(c x)} \left (-1+e^{2 \left (\tanh ^{-1}\left (\frac {c d}{e}\right )+\tanh ^{-1}(c x)\right )}\right )\right )-6 c d \tanh ^{-1}(c x)^2 \log \left (\frac {1}{2} e^{-\tanh ^{-1}(c x)} \left (e \left (-1+e^{2 \tanh ^{-1}(c x)}\right )+c d \left (1+e^{2 \tanh ^{-1}(c x)}\right )\right )\right )+6 c d \tanh ^{-1}(c x)^2 \log \left (\frac {c (d+e x)}{\sqrt {1-c^2 x^2}}\right )+3 i c d \pi \tanh ^{-1}(c x) \log \left (1-c^2 x^2\right )+12 c d \tanh ^{-1}\left (\frac {c d}{e}\right ) \tanh ^{-1}(c x) \log \left (i \sinh \left (\tanh ^{-1}\left (\frac {c d}{e}\right )+\tanh ^{-1}(c x)\right )\right )+6 c d \tanh ^{-1}(c x) \text {PolyLog}\left (2,-\frac {(c d+e) e^{2 \tanh ^{-1}(c x)}}{c d-e}\right )-12 c d \tanh ^{-1}(c x) \text {PolyLog}\left (2,-e^{\tanh ^{-1}\left (\frac {c d}{e}\right )+\tanh ^{-1}(c x)}\right )-12 c d \tanh ^{-1}(c x) \text {PolyLog}\left (2,e^{\tanh ^{-1}\left (\frac {c d}{e}\right )+\tanh ^{-1}(c x)}\right )-6 c d \tanh ^{-1}(c x) \text {PolyLog}\left (2,e^{2 \left (\tanh ^{-1}\left (\frac {c d}{e}\right )+\tanh ^{-1}(c x)\right )}\right )-3 c d \text {PolyLog}\left (3,-\frac {(c d+e) e^{2 \tanh ^{-1}(c x)}}{c d-e}\right )+12 c d \text {PolyLog}\left (3,-e^{\tanh ^{-1}\left (\frac {c d}{e}\right )+\tanh ^{-1}(c x)}\right )+12 c d \text {PolyLog}\left (3,e^{\tanh ^{-1}\left (\frac {c d}{e}\right )+\tanh ^{-1}(c x)}\right )+3 c d \text {PolyLog}\left (3,e^{2 \left (\tanh ^{-1}\left (\frac {c d}{e}\right )+\tanh ^{-1}(c x)\right )}\right )\right )}{6 c^2 d^2-6 e^2}\right )}{d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-(a^3/(e*(d + e*x))) - (3*a^2*b*ArcTanh[c*x])/(e*(d + e*x)) - (3*a^2*b*c*Log[1 - c*x])/(2*e*(c*d + e)) + (3*a^
2*b*c*Log[1 + c*x])/(2*c*d*e - 2*e^2) - (3*a^2*b*c*Log[d + e*x])/(c^2*d^2 - e^2) + (3*a*b^2*(-(ArcTanh[c*x]^2/
(Sqrt[1 - (c^2*d^2)/e^2]*e*E^ArcTanh[(c*d)/e])) + (x*ArcTanh[c*x]^2)/(d + e*x) + (c*d*(I*Pi*Log[1 + E^(2*ArcTa
nh[c*x])] - 2*ArcTanh[c*x]*Log[1 - E^(-2*(ArcTanh[(c*d)/e] + ArcTanh[c*x]))] - I*Pi*(ArcTanh[c*x] - Log[1 - c^
2*x^2]/2) - 2*ArcTanh[(c*d)/e]*(ArcTanh[c*x] + Log[1 - E^(-2*(ArcTanh[(c*d)/e] + ArcTanh[c*x]))] - Log[I*Sinh[
ArcTanh[(c*d)/e] + ArcTanh[c*x]]]) + PolyLog[2, E^(-2*(ArcTanh[(c*d)/e] + ArcTanh[c*x]))]))/(c^2*d^2 - e^2)))/
d + (b^3*((x*ArcTanh[c*x]^3)/(d + e*x) + (3*(6*c*d*ArcTanh[c*x]^3 - 2*e*ArcTanh[c*x]^3 + (4*Sqrt[1 - (c^2*d^2)
/e^2]*e*ArcTanh[c*x]^3)/E^ArcTanh[(c*d)/e] + (6*I)*c*d*Pi*ArcTanh[c*x]*Log[(E^(-ArcTanh[c*x]) + E^ArcTanh[c*x]
)/2] + 6*c*d*ArcTanh[c*x]^2*Log[1 + ((c*d + e)*E^(2*ArcTanh[c*x]))/(c*d - e)] - 6*c*d*ArcTanh[c*x]^2*Log[1 - E
^(ArcTanh[(c*d)/e] + ArcTanh[c*x])] - 6*c*d*ArcTanh[c*x]^2*Log[1 + E^(ArcTanh[(c*d)/e] + ArcTanh[c*x])] - 6*c*
d*ArcTanh[c*x]^2*Log[1 - E^(2*(ArcTanh[(c*d)/e] + ArcTanh[c*x]))] - 12*c*d*ArcTanh[(c*d)/e]*ArcTanh[c*x]*Log[(
I/2)*E^(-ArcTanh[(c*d)/e] - ArcTanh[c*x])*(-1 + E^(2*(ArcTanh[(c*d)/e] + ArcTanh[c*x])))] - 6*c*d*ArcTanh[c*x]
^2*Log[(e*(-1 + E^(2*ArcTanh[c*x])) + c*d*(1 + E^(2*ArcTanh[c*x])))/(2*E^ArcTanh[c*x])] + 6*c*d*ArcTanh[c*x]^2
*Log[(c*(d + e*x))/Sqrt[1 - c^2*x^2]] + (3*I)*c*d*Pi*ArcTanh[c*x]*Log[1 - c^2*x^2] + 12*c*d*ArcTanh[(c*d)/e]*A
rcTanh[c*x]*Log[I*Sinh[ArcTanh[(c*d)/e] + ArcTanh[c*x]]] + 6*c*d*ArcTanh[c*x]*PolyLog[2, -(((c*d + e)*E^(2*Arc
Tanh[c*x]))/(c*d - e))] - 12*c*d*ArcTanh[c*x]*PolyLog[2, -E^(ArcTanh[(c*d)/e] + ArcTanh[c*x])] - 12*c*d*ArcTan
h[c*x]*PolyLog[2, E^(ArcTanh[(c*d)/e] + ArcTanh[c*x])] - 6*c*d*ArcTanh[c*x]*PolyLog[2, E^(2*(ArcTanh[(c*d)/e]
+ ArcTanh[c*x]))] - 3*c*d*PolyLog[3, -(((c*d + e)*E^(2*ArcTanh[c*x]))/(c*d - e))] + 12*c*d*PolyLog[3, -E^(ArcT
anh[(c*d)/e] + ArcTanh[c*x])] + 12*c*d*PolyLog[3, E^(ArcTanh[(c*d)/e] + ArcTanh[c*x])] + 3*c*d*PolyLog[3, E^(2
*(ArcTanh[(c*d)/e] + ArcTanh[c*x]))]))/(6*c^2*d^2 - 6*e^2)))/d

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 8.04, size = 3591, normalized size = 6.95

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(3591\)
default	\(\text {Expression too large to display}\)	\(3591\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/c*(-3*a^2*b*c^2/(c*d+e)/(c*d-e)*ln(c*e*x+c*d)-3/4*I*b^3*c^2/(c*d+e)/(c*d-e)*arctanh(c*x)^2*Pi*csgn(I*(c*x+1)
^2/(c^2*x^2-1))^3-3/2*I*b^3*c^2/(c*d+e)/(c*d-e)*arctanh(c*x)^2*Pi*csgn(I*(d*c*(1+(c*x+1)^2/(-c^2*x^2+1))+e*((c
*x+1)^2/(-c^2*x^2+1)-1))/(1+(c*x+1)^2/(-c^2*x^2+1)))^3-3/2*I*b^3*c^2/(c*d+e)/(c*d-e)*arctanh(c*x)^2*Pi*csgn(I/
(1+(c*x+1)^2/(-c^2*x^2+1)))^3+3/2*I*b^3*c^2/(c*d+e)/(c*d-e)*arctanh(c*x)^2*Pi*csgn(I/(1+(c*x+1)^2/(-c^2*x^2+1)
))^2-3/4*I*b^3*c^2/(c*d+e)/(c*d-e)*arctanh(c*x)^2*Pi*csgn(I*(c*x+1)^2/(c^2*x^2-1)/(1+(c*x+1)^2/(-c^2*x^2+1)))^
3-3*a^2*b*c^2/(c*e*x+c*d)/e*arctanh(c*x)-3*a^2*b*c^2/e/(2*c*d+2*e)*ln(c*x-1)+3*a^2*b*c^2/e/(2*c*d-2*e)*ln(c*x+
1)+3/2*b^3*c^3/(c*d+e)^2/(c*d-e)*d*polylog(3,(c*d+e)*(c*x+1)^2/(-c^2*x^2+1)/(-c*d+e))-3*b^3*c^2*arctanh(c*x)^2
/(c*d+e)/(c*d-e)*ln(c*e*x+c*d)+3*b^3*c^2*arctanh(c*x)^2/(c*d+e)/(c*d-e)*ln(d*c*(1+(c*x+1)^2/(-c^2*x^2+1))+e*((
c*x+1)^2/(-c^2*x^2+1)-1))-3*b^3*c^2/(c*d+e)/(c*d-e)*arctanh(c*x)^2*ln(2)+3/2*b^3*c^2*e/(c*d+e)^2/(c*d-e)*polyl
og(3,(c*d+e)*(c*x+1)^2/(-c^2*x^2+1)/(-c*d+e))+3*b^3*c^2/e*arctanh(c*x)^2/(2*c*d-2*e)*ln(c*x+1)-3*b^3*c^2/e*arc
tanh(c*x)^2/(2*c*d+2*e)*ln(c*x-1)-3*b^3*c^2/e*arctanh(c*x)^2/(c*d-e)*ln((c*x+1)/(-c^2*x^2+1)^(1/2))-3*a*b^2*c^
2/(c*e*x+c*d)/e*arctanh(c*x)^2-3*a*b^2*c^2/(c*d+e)/(c*d-e)*dilog((c*e*x-e)/(-c*d-e))+3*a*b^2*c^2/(c*d+e)/(c*d-
e)*dilog((c*e*x+e)/(-c*d+e))-3/2*a*b^2*c^2/e/(c*d-e)*dilog(1/2*c*x+1/2)-3/4*a*b^2*c^2/e/(c*d-e)*ln(c*x+1)^2-3/
4*a*b^2*c^2/e/(c*d+e)*ln(c*x-1)^2+3/2*a*b^2*c^2/e/(c*d+e)*dilog(1/2*c*x+1/2)+3/4*I*b^3*c^2/(c*d+e)/(c*d-e)*arc
tanh(c*x)^2*Pi*csgn(I/(1+(c*x+1)^2/(-c^2*x^2+1)))*csgn(I*(c*x+1)^2/(c^2*x^2-1))*csgn(I*(c*x+1)^2/(c^2*x^2-1)/(
1+(c*x+1)^2/(-c^2*x^2+1)))-3/4*I*b^3*c^3/e/(c*d+e)/(c*d-e)*arctanh(c*x)^2*Pi*d*csgn(I*(c*x+1)^2/(c^2*x^2-1)/(1
+(c*x+1)^2/(-c^2*x^2+1)))^3-3/2*I*b^3*c^2/(c*d+e)/(c*d-e)*arctanh(c*x)^2*Pi*csgn(I/(1+(c*x+1)^2/(-c^2*x^2+1)))
*csgn(I*(d*c*(1+(c*x+1)^2/(-c^2*x^2+1))+e*((c*x+1)^2/(-c^2*x^2+1)-1)))*csgn(I*(d*c*(1+(c*x+1)^2/(-c^2*x^2+1))+
e*((c*x+1)^2/(-c^2*x^2+1)-1))/(1+(c*x+1)^2/(-c^2*x^2+1)))+3/2*I*b^3*c^3/e/(c*d+e)/(c*d-e)*arctanh(c*x)^2*Pi*d*
csgn(I/(1+(c*x+1)^2/(-c^2*x^2+1)))^3-3/4*I*b^3*c^3/e/(c*d+e)/(c*d-e)*arctanh(c*x)^2*Pi*d*csgn(I*(c*x+1)^2/(c^2
*x^2-1))^3-3/2*I*b^3*c^3/e/(c*d+e)/(c*d-e)*arctanh(c*x)^2*Pi*d*csgn(I/(1+(c*x+1)^2/(-c^2*x^2+1)))^2+3/2*I*b^3*
c^2/(c*d+e)/(c*d-e)*arctanh(c*x)^2*Pi*csgn(I/(1+(c*x+1)^2/(-c^2*x^2+1)))*csgn(I*(d*c*(1+(c*x+1)^2/(-c^2*x^2+1)
)+e*((c*x+1)^2/(-c^2*x^2+1)-1))/(1+(c*x+1)^2/(-c^2*x^2+1)))^2+3/4*I*b^3*c^2/(c*d+e)/(c*d-e)*arctanh(c*x)^2*Pi*
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)))^2-3/2*I*b^3*c^2/(c*d+e)
/(c*d-e)*arctanh(c*x)^2*Pi*csgn(I*(c*x+1)/(-c^2*x^2+1)^(1/2))*csgn(I*(c*x+1)^2/(c^2*x^2-1))^2-3/4*I*b^3*c^2/(c
*d+e)/(c*d-e)*arctanh(c*x)^2*Pi*csgn(I*(c*x+1)/(-c^2*x^2+1)^(1/2))^2*csgn(I*(c*x+1)^2/(c^2*x^2-1))-3/4*I*b^3*c
^2/(c*d+e)/(c*d-e)*arctanh(c*x)^2*Pi*csgn(I/(1+(c*x+1)^2/(-c^2*x^2+1)))*csgn(I*(c*x+1)^2/(c^2*x^2-1)/(1+(c*x+1
)^2/(-c^2*x^2+1)))^2+3/2*I*b^3*c^2/(c*d+e)/(c*d-e)*arctanh(c*x)^2*Pi*csgn(I*(d*c*(1+(c*x+1)^2/(-c^2*x^2+1))+e*
((c*x+1)^2/(-c^2*x^2+1)-1)))*csgn(I*(d*c*(1+(c*x+1)^2/(-c^2*x^2+1))+e*((c*x+1)^2/(-c^2*x^2+1)-1))/(1+(c*x+1)^2
/(-c^2*x^2+1)))^2+3/2*I*b^3*c^3/e/(c*d+e)/(c*d-e)*arctanh(c*x)^2*Pi*d-6*a*b^2*c^2/e*arctanh(c*x)/(2*c*d+2*e)*l
n(c*x-1)+6*a*b^2*c^2/e*arctanh(c*x)/(2*c*d-2*e)*ln(c*x+1)-6*a*b^2*c^2*arctanh(c*x)/(c*d+e)/(c*d-e)*ln(c*e*x+c*
d)-3*a*b^2*c^2/(c*d+e)/(c*d-e)*ln(c*e*x+c*d)*ln((c*e*x-e)/(-c*d-e))+3*a*b^2*c^2/(c*d+e)/(c*d-e)*ln(c*e*x+c*d)*
ln((c*e*x+e)/(-c*d+e))+3/2*a*b^2*c^2/e/(c*d-e)*ln(-1/2*c*x+1/2)*ln(c*x+1)-3/2*a*b^2*c^2/e/(c*d-e)*ln(-1/2*c*x+
1/2)*ln(1/2*c*x+1/2)+3/2*a*b^2*c^2/e/(c*d+e)*ln(c*x-1)*ln(1/2*c*x+1/2)-3*b^3*c^2*e/(c*d+e)^2/(c*d-e)*arctanh(c
*x)*polylog(2,(c*d+e)*(c*x+1)^2/(-c^2*x^2+1)/(-c*d+e))-3*b^3*c^2*e/(c*d+e)^2/(c*d-e)*arctanh(c*x)^2*ln(1-(c*d+
e)*(c*x+1)^2/(-c^2*x^2+1)/(-c*d+e))-3*b^3*c^3/(c*d+e)^2/(c*d-e)*d*arctanh(c*x)^2*ln(1-(c*d+e)*(c*x+1)^2/(-c^2*
x^2+1)/(-c*d+e))-3*b^3*c^3/(c*d+e)^2/(c*d-e)*d*arctanh(c*x)*polylog(2,(c*d+e)*(c*x+1)^2/(-c^2*x^2+1)/(-c*d+e))
-3/2*I*b^3*c^2/(c*d+e)/(c*d-e)*arctanh(c*x)^2*Pi-3/2*I*b^3*c^3/e/(c*d+e)/(c*d-e)*arctanh(c*x)^2*Pi*d*csgn(I*(c
*x+1)/(-c^2*x^2+1)^(1/2))*csgn(I*(c*x+1)^2/(c^2*x^2-1))^2-3/4*I*b^3*c^3/e/(c*d+e)/(c*d-e)*arctanh(c*x)^2*Pi*d*
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)))^2-3/4*I*b^3*c^3/e/
(c*d+e)/(c*d-e)*arctanh(c*x)^2*Pi*d*csgn(I*(c*x+1)/(-c^2*x^2+1)^(1/2))^2*csgn(I*(c*x+1)^2/(c^2*x^2-1))+3/4*I*b
^3*c^3/e/(c*d+e)/(c*d-e)*arctanh(c*x)^2*Pi*d*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-b^3*c^2/(c*e*x+c*d)/e*arctanh(c*x)^3+b^3*c^2/e*arctanh(c*x)^3/(c*d-e)-a^3*c^2/(c*e*x+c
*d)/e+3/4*I*b^3*c^3/e/(c*d+e)/(c*d-e)*arctanh(c*x)^2*Pi*d*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))))

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

[Out]

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

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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 (d + e x\right )^{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((a + b*atanh(c*x))**3/(d + e*x)**2, 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((a+b*arctanh(c*x))^3/(e*x+d)^2,x, algorithm="giac")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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