∫ (a+b tanh−1(cx))3 ___________________________

3.1.18 \(\int \frac {(a+b \tanh ^{-1}(c x))^3}{d+e x} \, dx\) [18]

Optimal. Leaf size=272 \[ -\frac {\left (a+b \tanh ^{-1}(c x)\right )^3 \log \left (\frac {2}{1+c x}\right )}{e}+\frac {\left (a+b \tanh ^{-1}(c x)\right )^3 \log \left (\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{e}+\frac {3 b \left (a+b \tanh ^{-1}(c x)\right )^2 \text {PolyLog}\left (2,1-\frac {2}{1+c x}\right )}{2 e}-\frac {3 b \left (a+b \tanh ^{-1}(c x)\right )^2 \text {PolyLog}\left (2,1-\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{2 e}+\frac {3 b^2 \left (a+b \tanh ^{-1}(c x)\right ) \text {PolyLog}\left (3,1-\frac {2}{1+c x}\right )}{2 e}-\frac {3 b^2 \left (a+b \tanh ^{-1}(c x)\right ) \text {PolyLog}\left (3,1-\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{2 e}+\frac {3 b^3 \text {PolyLog}\left (4,1-\frac {2}{1+c x}\right )}{4 e}-\frac {3 b^3 \text {PolyLog}\left (4,1-\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{4 e} \]

[Out]

-(a+b*arctanh(c*x))^3*ln(2/(c*x+1))/e+(a+b*arctanh(c*x))^3*ln(2*c*(e*x+d)/(c*d+e)/(c*x+1))/e+3/2*b*(a+b*arctan

h(c*x))^2*polylog(2,1-2/(c*x+1))/e-3/2*b*(a+b*arctanh(c*x))^2*polylog(2,1-2*c*(e*x+d)/(c*d+e)/(c*x+1))/e+3/2*b

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

+1))/e+3/4*b^3*polylog(4,1-2/(c*x+1))/e-3/4*b^3*polylog(4,1-2*c*(e*x+d)/(c*d+e)/(c*x+1))/e

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Rubi [A]
time = 0.04, antiderivative size = 272, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {6061} \begin {gather*} -\frac {3 b^2 \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_3\left (1-\frac {2 c (d+e x)}{(c d+e) (c x+1)}\right )}{2 e}+\frac {3 b^2 \text {Li}_3\left (1-\frac {2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{2 e}-\frac {3 b \left (a+b \tanh ^{-1}(c x)\right )^2 \text {Li}_2\left (1-\frac {2 c (d+e x)}{(c d+e) (c x+1)}\right )}{2 e}+\frac {\left (a+b \tanh ^{-1}(c x)\right )^3 \log \left (\frac {2 c (d+e x)}{(c x+1) (c d+e)}\right )}{e}+\frac {3 b \text {Li}_2\left (1-\frac {2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )^2}{2 e}-\frac {\log \left (\frac {2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )^3}{e}-\frac {3 b^3 \text {Li}_4\left (1-\frac {2 c (d+e x)}{(c d+e) (c x+1)}\right )}{4 e}+\frac {3 b^3 \text {Li}_4\left (1-\frac {2}{c x+1}\right )}{4 e} \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

*x))])/e + (3*b*(a + b*ArcTanh[c*x])^2*PolyLog[2, 1 - 2/(1 + c*x)])/(2*e) - (3*b*(a + b*ArcTanh[c*x])^2*PolyLo

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

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

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

Rule 6061

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^3/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-(a + b*ArcTanh[c*x])^3)*(Lo

g[2/(1 + c*x)]/e), x] + (Simp[(a + b*ArcTanh[c*x])^3*(Log[2*c*((d + e*x)/((c*d + e)*(1 + c*x)))]/e), x] + Simp

[3*b*(a + b*ArcTanh[c*x])^2*(PolyLog[2, 1 - 2/(1 + c*x)]/(2*e)), x] - Simp[3*b*(a + b*ArcTanh[c*x])^2*(PolyLog

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

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

/(2*e)), x] + Simp[3*b^3*(PolyLog[4, 1 - 2/(1 + c*x)]/(4*e)), x] - Simp[3*b^3*(PolyLog[4, 1 - 2*c*((d + e*x)/(

(c*d + e)*(1 + c*x)))]/(4*e)), x]) /; FreeQ[{a, b, c, d, e}, x] && NeQ[c^2*d^2 - e^2, 0]

Rubi steps

\begin {align*} \int \frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{d+e x} \, dx &=-\frac {\left (a+b \tanh ^{-1}(c x)\right )^3 \log \left (\frac {2}{1+c x}\right )}{e}+\frac {\left (a+b \tanh ^{-1}(c x)\right )^3 \log \left (\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{e}+\frac {3 b \left (a+b \tanh ^{-1}(c x)\right )^2 \text {Li}_2\left (1-\frac {2}{1+c x}\right )}{2 e}-\frac {3 b \left (a+b \tanh ^{-1}(c x)\right )^2 \text {Li}_2\left (1-\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{2 e}+\frac {3 b^2 \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_3\left (1-\frac {2}{1+c x}\right )}{2 e}-\frac {3 b^2 \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_3\left (1-\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{2 e}+\frac {3 b^3 \text {Li}_4\left (1-\frac {2}{1+c x}\right )}{4 e}-\frac {3 b^3 \text {Li}_4\left (1-\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{4 e}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 31.92, size = 2179, normalized size = 8.01 \begin {gather*} \text {Result too large to show} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(a^3*Log[d + e*x])/e - ((3*I)*a^2*b*(I*ArcTanh[c*x]*(-Log[1/Sqrt[1 - c^2*x^2]] + Log[I*Sinh[ArcTanh[(c*d)/e] +

ArcTanh[c*x]]]) + ((-I)*(I*ArcTanh[(c*d)/e] + I*ArcTanh[c*x])^2 - (I/4)*(Pi - (2*I)*ArcTanh[c*x])^2 + 2*(I*Ar

cTanh[(c*d)/e] + I*ArcTanh[c*x])*Log[1 - E^((2*I)*(I*ArcTanh[(c*d)/e] + I*ArcTanh[c*x]))] + (Pi - (2*I)*ArcTan

h[c*x])*Log[1 - E^(I*(Pi - (2*I)*ArcTanh[c*x]))] - (Pi - (2*I)*ArcTanh[c*x])*Log[2*Sin[(Pi - (2*I)*ArcTanh[c*x

])/2]] - 2*(I*ArcTanh[(c*d)/e] + I*ArcTanh[c*x])*Log[(2*I)*Sinh[ArcTanh[(c*d)/e] + ArcTanh[c*x]]] - I*PolyLog[

2, E^((2*I)*(I*ArcTanh[(c*d)/e] + I*ArcTanh[c*x]))] - I*PolyLog[2, E^(I*(Pi - (2*I)*ArcTanh[c*x]))])/2))/e + (

a*b^2*(-8*c*d*ArcTanh[c*x]^3 + 4*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*c*d*ArcTanh[c*x]^2*Log[1 + E^(-2*ArcTanh[c*x])] - (6*I)*c*d*Pi*ArcTanh[c*x]*Log[(1 + E^(2*ArcTanh[c*

x]))/(2*E^ArcTanh[c*x])] - 6*c*d*ArcTanh[c*x]^2*Log[1 + ((c*d + e)*E^(2*ArcTanh[c*x]))/(c*d - e)] + 6*c*d*ArcT

anh[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] + A

rcTanh[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*I)*c*d*Pi*ArcTanh[c*x]*Log[1/Sqrt[1 - c^2*x^2]] - 6*c*d*ArcTanh[c*x]^2*Log[(c*d)/Sqrt[1 - c^2*x^2] + (c*e*x

)/Sqrt[1 - c^2*x^2]] - 12*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^(-2*ArcTanh[c*x])] - 6*c*d*ArcTanh[c*x]*PolyLog[2, -(((c*d + e)*E^(2*ArcTanh[c*x

]))/(c*d - e))] + 12*c*d*ArcTanh[c*x]*PolyLog[2, -E^(ArcTanh[(c*d)/e] + ArcTanh[c*x])] + 12*c*d*ArcTanh[c*x]*P

olyLog[2, E^(ArcTanh[(c*d)/e] + ArcTanh[c*x])] + 6*c*d*ArcTanh[c*x]*PolyLog[2, E^(2*(ArcTanh[(c*d)/e] + ArcTan

h[c*x]))] + 3*c*d*PolyLog[3, -E^(-2*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^(ArcTanh[(c*d)/e] + ArcTanh[c*x])] - 12*c*d*PolyLog[3, E^(ArcTanh[(c*d)/e] + ArcTa

nh[c*x])] - 3*c*d*PolyLog[3, E^(2*(ArcTanh[(c*d)/e] + ArcTanh[c*x]))]))/(2*c*d*e) + (b^3*(-5*c*d*ArcTanh[c*x]^

4 + 3*e*ArcTanh[c*x]^4 - (3*Sqrt[1 - (c^2*d^2)/e^2]*e*ArcTanh[c*x]^4)/E^ArcTanh[(c*d)/e] - 4*c*d*ArcTanh[c*x]^

3*Log[1 + E^(-2*ArcTanh[c*x])] - (6*I)*c*d*Pi*ArcTanh[c*x]^2*Log[1 - I*E^ArcTanh[c*x]] - (6*I)*c*d*Pi*ArcTanh[

c*x]^2*Log[1 + I*E^ArcTanh[c*x]] + 4*c*d*ArcTanh[c*x]^3*Log[1 - (Sqrt[c*d + e]*E^ArcTanh[c*x])/Sqrt[-(c*d) + e

]] + 4*c*d*ArcTanh[c*x]^3*Log[1 + (Sqrt[c*d + e]*E^ArcTanh[c*x])/Sqrt[-(c*d) + e]] + (6*I)*c*d*Pi*ArcTanh[c*x]

^2*Log[1 + E^(2*ArcTanh[c*x])] - (6*I)*c*d*Pi*ArcTanh[c*x]^2*Log[(1 + E^(2*ArcTanh[c*x]))/(2*E^ArcTanh[c*x])]

- 12*c*d*ArcTanh[c*x]^3*Log[1 + ((c*d + e)*E^(2*ArcTanh[c*x]))/(c*d - e)] + 12*c*d*ArcTanh[c*x]^3*Log[1 - E^(2

*(ArcTanh[(c*d)/e] + ArcTanh[c*x]))] + 12*c*d*ArcTanh[(c*d)/e]*ArcTanh[c*x]^2*Log[(I/2)*E^(-ArcTanh[(c*d)/e] -

ArcTanh[c*x])*(-1 + E^(2*(ArcTanh[(c*d)/e] + ArcTanh[c*x])))] + 8*c*d*ArcTanh[c*x]^3*Log[(e*(-1 + E^(2*ArcTan

h[c*x])) + c*d*(1 + E^(2*ArcTanh[c*x])))/(2*E^ArcTanh[c*x])] + (6*I)*c*d*Pi*ArcTanh[c*x]^2*Log[1/Sqrt[1 - c^2*

x^2]] - 8*c*d*ArcTanh[c*x]^3*Log[(c*d)/Sqrt[1 - c^2*x^2] + (c*e*x)/Sqrt[1 - c^2*x^2]] - 12*c*d*ArcTanh[(c*d)/e

]*ArcTanh[c*x]^2*Log[I*Sinh[ArcTanh[(c*d)/e] + ArcTanh[c*x]]] + 6*c*d*ArcTanh[c*x]^2*PolyLog[2, -E^(-2*ArcTanh

[c*x])] - (12*I)*c*d*Pi*ArcTanh[c*x]*PolyLog[2, (-I)*E^ArcTanh[c*x]] - (12*I)*c*d*Pi*ArcTanh[c*x]*PolyLog[2, I

*E^ArcTanh[c*x]] + 12*c*d*ArcTanh[c*x]^2*PolyLog[2, -((Sqrt[c*d + e]*E^ArcTanh[c*x])/Sqrt[-(c*d) + e])] + 12*c

*d*ArcTanh[c*x]^2*PolyLog[2, (Sqrt[c*d + e]*E^ArcTanh[c*x])/Sqrt[-(c*d) + e]] + (6*I)*c*d*Pi*ArcTanh[c*x]*Poly

Log[2, -E^(2*ArcTanh[c*x])] - 18*c*d*ArcTanh[c*x]^2*PolyLog[2, -(((c*d + e)*E^(2*ArcTanh[c*x]))/(c*d - e))] +

18*c*d*ArcTanh[c*x]^2*PolyLog[2, E^(2*(ArcTanh[(c*d)/e] + ArcTanh[c*x]))] + 6*c*d*ArcTanh[c*x]*PolyLog[3, -E^(

-2*ArcTanh[c*x])] + (12*I)*c*d*Pi*PolyLog[3, (-I)*E^ArcTanh[c*x]] + (12*I)*c*d*Pi*PolyLog[3, I*E^ArcTanh[c*x]]

- 24*c*d*ArcTanh[c*x]*PolyLog[3, -((Sqrt[c*d + e]*E^ArcTanh[c*x])/Sqrt[-(c*d) + e])] - 24*c*d*ArcTanh[c*x]*Po

lyLog[3, (Sqrt[c*d + e]*E^ArcTanh[c*x])/Sqrt[-(c*d) + e]] - (3*I)*c*d*Pi*PolyLog[3, -E^(2*ArcTanh[c*x])] + 18*

c*d*ArcTanh[c*x]*PolyLog[3, -(((c*d + e)*E^(2*ArcTanh[c*x]))/(c*d - e))] - 18*c*d*ArcTanh[c*x]*PolyLog[3, E^(2

*(ArcTanh[(c*d)/e] + ArcTanh[c*x]))] + 3*c*d*PolyLog[4, -E^(-2*ArcTanh[c*x])] + 24*c*d*PolyLog[4, -((Sqrt[c*d

+ e]*E^ArcTanh[c*x])/Sqrt[-(c*d) + e])] + 24*c*d*PolyLog[4, (Sqrt[c*d + e]*E^ArcTanh[c*x])/Sqrt[-(c*d) + e]] -

9*c*d*PolyLog[4, -(((c*d + e)*E^(2*ArcTanh[c*x]))/(c*d - e))] + 9*c*d*PolyLog[4, E^(2*(ArcTanh[(c*d)/e] + Arc

Tanh[c*x]))]))/(4*c*d*e)

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 11.47, size = 2415, normalized size = 8.88


method	result	size

derivativedivides	\(\text {Expression too large to display}\)	\(2415\)
default	\(\text {Expression too large to display}\)	\(2415\)

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

^2*x^2+1))+3*a*b^2*c/(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*a*b^2*c/(c*d+e)*ar

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

*c/e*ln((c*e*x-e)/(-c*d-e))*ln(c*e*x+c*d)-3/2*a^2*b*c/e*ln((c*e*x+e)/(-c*d+e))*ln(c*e*x+c*d)+b^3*c^2/e*d/(c*d+

e)*arctanh(c*x)^3*ln(1-(c*d+e)*(c*x+1)^2/(-c^2*x^2+1)/(-c*d+e))+3/2*b^3*c^2/e*d/(c*d+e)*arctanh(c*x)^2*polylog

(2,(c*d+e)*(c*x+1)^2/(-c^2*x^2+1)/(-c*d+e))-3/2*b^3*c^2/e*d/(c*d+e)*arctanh(c*x)*polylog(3,(c*d+e)*(c*x+1)^2/(

-c^2*x^2+1)/(-c*d+e))+1/2*I*b^3*c/e*arctanh(c*x)^3*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*a*b^2*c/e*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)))*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/(1+(c*x+1)^2/(-c^2*x^2+1)))*arctanh(c*x)^2-1/2*I*b^3*c/e*arctanh(c*x)^3*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)))^2*csgn(I/(1+(c*x+1)^2/(-c^2*x^2+

1)))-1/2*I*b^3*c/e*arctanh(c*x)^3*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)))^2*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)))+3*a*b^2*c^2/e*

d/(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*a*b^2*c^2/e*d/(c*d+e)*arctanh(c*x)*po

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

*d+e)*(c*x+1)^2/(-c^2*x^2+1)/(-c*d+e))+1/2*I*b^3*c/e*arctanh(c*x)^3*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)))*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/(1+(c*x+1)^2/(-c^2*x^2+1)))-3/2*I*a*b^2*c/e*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)))^2*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)))*arctanh(c*x)^2-3/2*I*a*b^2*c/e*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)))^2*csgn(I/(1+(c*x+1)^2/(-c^2*x^2+1)))*arctanh(c*x)^2+3*a*b^2*c*ln(c*e

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

)*(c*x+1)^2/(-c^2*x^2+1)/(-c*d+e))+3/2*a^2*b*c/e*dilog((c*e*x-e)/(-c*d-e))-3/2*a^2*b*c/e*dilog((c*e*x+e)/(-c*d

+e))+b^3*c*ln(c*e*x+c*d)/e*arctanh(c*x)^3-b^3*c/e*arctanh(c*x)^3*ln(d*c*(1+(c*x+1)^2/(-c^2*x^2+1))+e*((c*x+1)^

2/(-c^2*x^2+1)-1))-3/2*b^3*c/e*arctanh(c*x)^2*polylog(2,-(c*x+1)^2/(-c^2*x^2+1))+3/2*b^3*c/e*arctanh(c*x)*poly

log(3,-(c*x+1)^2/(-c^2*x^2+1))+b^3*c/(c*d+e)*arctanh(c*x)^3*ln(1-(c*d+e)*(c*x+1)^2/(-c^2*x^2+1)/(-c*d+e))+3/2*

b^3*c/(c*d+e)*arctanh(c*x)^2*polylog(2,(c*d+e)*(c*x+1)^2/(-c^2*x^2+1)/(-c*d+e))-3/2*b^3*c/(c*d+e)*arctanh(c*x)

*polylog(3,(c*d+e)*(c*x+1)^2/(-c^2*x^2+1)/(-c*d+e))-3/4*b^3*c/e*polylog(4,-(c*x+1)^2/(-c^2*x^2+1))+3/4*b^3*c/(

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

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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),x, algorithm="maxima")

[Out]

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

1) - log(-c*x + 1))^2/(x*e + d) + 3/2*a^2*b*(log(c*x + 1) - log(-c*x + 1))/(x*e + d), 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),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*e + d), 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}}{d + e x}\, dx \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

[Out]

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

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