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

Optimal. Leaf size=517 \[ -\frac {3 b^2 c \text {Li}_2\left (1-\frac {2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{c^2 d^2-e^2}+\frac {3 b^2 c \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2 c (d+e x)}{(c d+e) (c x+1)}\right )}{c^2 d^2-e^2}+\frac {3 b^2 c \text {Li}_2\left (1-\frac {2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{2 e (c d+e)}+\frac {3 b^2 c \text {Li}_2\left (1-\frac {2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{2 e (c d-e)}+\frac {3 b c \log \left (\frac {2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )^2}{c^2 d^2-e^2}-\frac {3 b c \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2 c (d+e x)}{(c x+1) (c d+e)}\right )}{c^2 d^2-e^2}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{e (d+e x)}+\frac {3 b c \log \left (\frac {2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )^2}{2 e (c d+e)}-\frac {3 b c \log \left (\frac {2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )^2}{2 e (c d-e)}-\frac {3 b^3 c \text {Li}_3\left (1-\frac {2}{c x+1}\right )}{2 \left (c^2 d^2-e^2\right )}+\frac {3 b^3 c \text {Li}_3\left (1-\frac {2 c (d+e x)}{(c d+e) (c x+1)}\right )}{2 \left (c^2 d^2-e^2\right )}-\frac {3 b^3 c \text {Li}_3\left (1-\frac {2}{1-c x}\right )}{4 e (c d+e)}+\frac {3 b^3 c \text {Li}_3\left (1-\frac {2}{c x+1}\right )}{4 e (c d-e)} \]

[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 = 0.52, 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 = {5928, 5918, 5948, 6058, 6610, 6056, 5922} \[ -\frac {3 b^2 c \text {PolyLog}\left (2,1-\frac {2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{c^2 d^2-e^2}+\frac {3 b^2 c \left (a+b \tanh ^{-1}(c x)\right ) \text {PolyLog}\left (2,1-\frac {2 c (d+e x)}{(c x+1) (c d+e)}\right )}{c^2 d^2-e^2}+\frac {3 b^2 c \text {PolyLog}\left (2,1-\frac {2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{2 e (c d+e)}+\frac {3 b^2 c \text {PolyLog}\left (2,1-\frac {2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{2 e (c d-e)}-\frac {3 b^3 c \text {PolyLog}\left (3,1-\frac {2}{c x+1}\right )}{2 \left (c^2 d^2-e^2\right )}+\frac {3 b^3 c \text {PolyLog}\left (3,1-\frac {2 c (d+e x)}{(c x+1) (c d+e)}\right )}{2 \left (c^2 d^2-e^2\right )}-\frac {3 b^3 c \text {PolyLog}\left (3,1-\frac {2}{1-c x}\right )}{4 e (c d+e)}+\frac {3 b^3 c \text {PolyLog}\left (3,1-\frac {2}{c x+1}\right )}{4 e (c d-e)}+\frac {3 b c \log \left (\frac {2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )^2}{c^2 d^2-e^2}-\frac {3 b c \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2 c (d+e x)}{(c x+1) (c d+e)}\right )}{c^2 d^2-e^2}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{e (d+e x)}+\frac {3 b c \log \left (\frac {2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )^2}{2 e (c d+e)}-\frac {3 b c \log \left (\frac {2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )^2}{2 e (c d-e)} \]

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 5918

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

Rule 5922

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^2/((d_) + (e_.)*(x_)), x_Symbol] :> -Simp[((a + b*ArcTanh[c*x])^2*Log[
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*PolyLog
[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 5928

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

Rule 5948

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

Rule 6056

Int[(Log[u_]*((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[((a + b*ArcTa
nh[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 6058

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

Rule 6610

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

Rubi steps

\begin {align*} \int \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]  time = 13.85, size = 813, normalized size = 1.57 \[ -\frac {a^3}{e (d+e x)}-\frac {3 b \tanh ^{-1}(c x) a^2}{e (d+e x)}-\frac {3 b c \log (1-c x) a^2}{2 e (c d+e)}+\frac {3 b c \log (c x+1) a^2}{2 c d e-2 e^2}-\frac {3 b c \log (d+e x) a^2}{c^2 d^2-e^2}+\frac {3 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 {Li}_2\left (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 ) a}{d}+\frac {b^3 \left (\frac {x \tanh ^{-1}(c x)^3}{d+e x}+\frac {3 \left (2 \sqrt {1-\frac {c^2 d^2}{e^2}} e e^{-\tanh ^{-1}\left (\frac {c d}{e}\right )} \tanh ^{-1}(c x)^3+3 c d \tanh ^{-1}(c x)^3-e \tanh ^{-1}(c x)^3-3 c d \log \left (1-e^{\tanh ^{-1}\left (\frac {c d}{e}\right )+\tanh ^{-1}(c x)}\right ) \tanh ^{-1}(c x)^2-3 c d \log \left (1+e^{\tanh ^{-1}\left (\frac {c d}{e}\right )+\tanh ^{-1}(c x)}\right ) \tanh ^{-1}(c x)^2-3 c d \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 ) \tanh ^{-1}(c x)^2+3 c d \log \left (\frac {c (d+e x)}{\sqrt {1-c^2 x^2}}\right ) \tanh ^{-1}(c x)^2+3 i c d \pi \log \left (\frac {1}{2} \left (e^{-\tanh ^{-1}(c x)}+e^{\tanh ^{-1}(c x)}\right )\right ) \tanh ^{-1}(c x)-6 c d \tanh ^{-1}\left (\frac {c d}{e}\right ) \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 ) \tanh ^{-1}(c x)+\frac {3}{2} i c d \pi \log \left (1-c^2 x^2\right ) \tanh ^{-1}(c x)+6 c d \tanh ^{-1}\left (\frac {c d}{e}\right ) \log \left (i \sinh \left (\tanh ^{-1}\left (\frac {c d}{e}\right )+\tanh ^{-1}(c x)\right )\right ) \tanh ^{-1}(c x)-6 c d \text {Li}_2\left (-e^{\tanh ^{-1}\left (\frac {c d}{e}\right )+\tanh ^{-1}(c x)}\right ) \tanh ^{-1}(c x)-6 c d \text {Li}_2\left (e^{\tanh ^{-1}\left (\frac {c d}{e}\right )+\tanh ^{-1}(c x)}\right ) \tanh ^{-1}(c x)+6 c d \text {Li}_3\left (-e^{\tanh ^{-1}\left (\frac {c d}{e}\right )+\tanh ^{-1}(c x)}\right )+6 c d \text {Li}_3\left (e^{\tanh ^{-1}\left (\frac {c d}{e}\right )+\tanh ^{-1}(c x)}\right )\right )}{3 c^2 d^2-3 e^2}\right )}{d} \]

Warning: Unable to verify antiderivative.

[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*(3*c*d*ArcTanh[c*x]^3 - e*ArcTanh[c*x]^3 + (2*Sqrt[1 - (c^2*d^2)/e
^2]*e*ArcTanh[c*x]^3)/E^ArcTanh[(c*d)/e] + (3*I)*c*d*Pi*ArcTanh[c*x]*Log[(E^(-ArcTanh[c*x]) + E^ArcTanh[c*x])/
2] - 3*c*d*ArcTanh[c*x]^2*Log[1 - E^(ArcTanh[(c*d)/e] + ArcTanh[c*x])] - 3*c*d*ArcTanh[c*x]^2*Log[1 + E^(ArcTa
nh[(c*d)/e] + ArcTanh[c*x])] - 6*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])))] - 3*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])] + 3*c*d*ArcTanh[c*x]^2*Log[(c*(d + e*x))/Sqrt[1 - c^2*x^2]]
+ ((3*I)/2)*c*d*Pi*ArcTanh[c*x]*Log[1 - c^2*x^2] + 6*c*d*ArcTanh[(c*d)/e]*ArcTanh[c*x]*Log[I*Sinh[ArcTanh[(c*d
)/e] + ArcTanh[c*x]]] - 6*c*d*ArcTanh[c*x]*PolyLog[2, -E^(ArcTanh[(c*d)/e] + ArcTanh[c*x])] - 6*c*d*ArcTanh[c*
x]*PolyLog[2, E^(ArcTanh[(c*d)/e] + ArcTanh[c*x])] + 6*c*d*PolyLog[3, -E^(ArcTanh[(c*d)/e] + ArcTanh[c*x])] +
6*c*d*PolyLog[3, E^(ArcTanh[(c*d)/e] + ArcTanh[c*x])]))/(3*c^2*d^2 - 3*e^2)))/d

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fricas [F]  time = 0.60, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b^{3} \operatorname {artanh}\left (c x\right )^{3} + 3 \, a b^{2} \operatorname {artanh}\left (c x\right )^{2} + 3 \, a^{2} b \operatorname {artanh}\left (c x\right ) + a^{3}}{e^{2} x^{2} + 2 \, d e x + d^{2}}, x\right ) \]

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)/(e^2*x^2 + 2*d*e*x + d^2),
x)

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

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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maple [C]  time = 0.89, size = 3497, normalized size = 6.76 \[ \text {output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {3}{2} \, {\left (c {\left (\frac {\log \left (c x + 1\right )}{c d e - e^{2}} - \frac {\log \left (c x - 1\right )}{c d e + e^{2}} - \frac {2 \, \log \left (e x + d\right )}{c^{2} d^{2} - e^{2}}\right )} - \frac {2 \, \operatorname {artanh}\left (c x\right )}{e^{2} x + d e}\right )} a^{2} b - \frac {a^{3}}{e^{2} x + d e} - \frac {{\left ({\left (c^{2} d e - c e^{2}\right )} b^{3} x - {\left (c d e - e^{2}\right )} b^{3}\right )} \log \left (-c x + 1\right )^{3} + 3 \, {\left (2 \, {\left (c^{2} d^{2} - e^{2}\right )} a b^{2} - {\left ({\left (c^{2} d e + c e^{2}\right )} b^{3} x + {\left (c d e + e^{2}\right )} b^{3}\right )} \log \left (c x + 1\right )\right )} \log \left (-c x + 1\right )^{2}}{8 \, {\left (c^{2} d^{3} e - d e^{3} + {\left (c^{2} d^{2} e^{2} - e^{4}\right )} x\right )}} - \int \frac {{\left ({\left (c^{2} d e - c e^{2}\right )} b^{3} x - {\left (c d e - e^{2}\right )} b^{3}\right )} \log \left (c x + 1\right )^{3} + 6 \, {\left ({\left (c^{2} d e - c e^{2}\right )} a b^{2} x - {\left (c d e - e^{2}\right )} a b^{2}\right )} \log \left (c x + 1\right )^{2} + 3 \, {\left (4 \, {\left (c^{2} d e - c e^{2}\right )} a b^{2} x + 4 \, {\left (c^{2} d^{2} - c d e\right )} a b^{2} - {\left ({\left (c^{2} d e - c e^{2}\right )} b^{3} x - {\left (c d e - e^{2}\right )} b^{3}\right )} \log \left (c x + 1\right )^{2} - 2 \, {\left (b^{3} c^{2} e^{2} x^{2} + b^{3} c d e - 2 \, {\left (c d e - e^{2}\right )} a b^{2} + {\left (2 \, {\left (c^{2} d e - c e^{2}\right )} a b^{2} + {\left (c^{2} d e + c e^{2}\right )} b^{3}\right )} x\right )} \log \left (c x + 1\right )\right )} \log \left (-c x + 1\right )}{8 \, {\left (c d^{3} e - d^{2} e^{2} - {\left (c^{2} d e^{3} - c e^{4}\right )} x^{3} - {\left (2 \, c^{2} d^{2} e^{2} - 3 \, c d e^{3} + e^{4}\right )} x^{2} - {\left (c^{2} d^{3} e - 3 \, c d^{2} e^{2} + 2 \, d e^{3}\right )} x\right )}}\,{d x} \]

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

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

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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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a + b \operatorname {atanh}{\left (c x \right )}\right )^{3}}{\left (d + e x\right )^{2}}\, dx \]

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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