3.30 \(\int \frac{(a+b \tanh ^{-1}(c x))^3}{x} \, dx\)

Optimal. Leaf size=184 \[ \frac{3}{2} b^2 \text{PolyLog}\left (3,1-\frac{2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )-\frac{3}{2} b^2 \text{PolyLog}\left (3,\frac{2}{1-c x}-1\right ) \left (a+b \tanh ^{-1}(c x)\right )-\frac{3}{2} b \text{PolyLog}\left (2,1-\frac{2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )^2+\frac{3}{2} b \text{PolyLog}\left (2,\frac{2}{1-c x}-1\right ) \left (a+b \tanh ^{-1}(c x)\right )^2-\frac{3}{4} b^3 \text{PolyLog}\left (4,1-\frac{2}{1-c x}\right )+\frac{3}{4} b^3 \text{PolyLog}\left (4,\frac{2}{1-c x}-1\right )+2 \tanh ^{-1}\left (1-\frac{2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )^3 \]

[Out]

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

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Rubi [A]  time = 0.448718, antiderivative size = 184, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429, Rules used = {5914, 6052, 5948, 6058, 6062, 6610} \[ \frac{3}{2} b^2 \text{PolyLog}\left (3,1-\frac{2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )-\frac{3}{2} b^2 \text{PolyLog}\left (3,\frac{2}{1-c x}-1\right ) \left (a+b \tanh ^{-1}(c x)\right )-\frac{3}{2} b \text{PolyLog}\left (2,1-\frac{2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )^2+\frac{3}{2} b \text{PolyLog}\left (2,\frac{2}{1-c x}-1\right ) \left (a+b \tanh ^{-1}(c x)\right )^2-\frac{3}{4} b^3 \text{PolyLog}\left (4,1-\frac{2}{1-c x}\right )+\frac{3}{4} b^3 \text{PolyLog}\left (4,\frac{2}{1-c x}-1\right )+2 \tanh ^{-1}\left (1-\frac{2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )^3 \]

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcTanh[c*x])^3/x,x]

[Out]

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

Rule 5914

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

Rule 6052

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

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 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 6062

Int[(((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*PolyLog[k_, u_])/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[((a +
b*ArcTanh[c*x])^p*PolyLog[k + 1, u])/(2*c*d), x] - Dist[(b*p)/2, Int[((a + b*ArcTanh[c*x])^(p - 1)*PolyLog[k
+ 1, u])/(d + e*x^2), x], x] /; FreeQ[{a, b, c, d, e, k}, x] && IGtQ[p, 0] && EqQ[c^2*d + e, 0] && EqQ[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}{x} \, dx &=2 \left (a+b \tanh ^{-1}(c x)\right )^3 \tanh ^{-1}\left (1-\frac{2}{1-c x}\right )-(6 b c) \int \frac{\left (a+b \tanh ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac{2}{1-c x}\right )}{1-c^2 x^2} \, dx\\ &=2 \left (a+b \tanh ^{-1}(c x)\right )^3 \tanh ^{-1}\left (1-\frac{2}{1-c x}\right )+(3 b c) \int \frac{\left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac{2}{1-c x}\right )}{1-c^2 x^2} \, dx-(3 b c) \int \frac{\left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (2-\frac{2}{1-c x}\right )}{1-c^2 x^2} \, dx\\ &=2 \left (a+b \tanh ^{-1}(c x)\right )^3 \tanh ^{-1}\left (1-\frac{2}{1-c x}\right )-\frac{3}{2} b \left (a+b \tanh ^{-1}(c x)\right )^2 \text{Li}_2\left (1-\frac{2}{1-c x}\right )+\frac{3}{2} b \left (a+b \tanh ^{-1}(c x)\right )^2 \text{Li}_2\left (-1+\frac{2}{1-c x}\right )+\left (3 b^2 c\right ) \int \frac{\left (a+b \tanh ^{-1}(c x)\right ) \text{Li}_2\left (1-\frac{2}{1-c x}\right )}{1-c^2 x^2} \, dx-\left (3 b^2 c\right ) \int \frac{\left (a+b \tanh ^{-1}(c x)\right ) \text{Li}_2\left (-1+\frac{2}{1-c x}\right )}{1-c^2 x^2} \, dx\\ &=2 \left (a+b \tanh ^{-1}(c x)\right )^3 \tanh ^{-1}\left (1-\frac{2}{1-c x}\right )-\frac{3}{2} b \left (a+b \tanh ^{-1}(c x)\right )^2 \text{Li}_2\left (1-\frac{2}{1-c x}\right )+\frac{3}{2} b \left (a+b \tanh ^{-1}(c x)\right )^2 \text{Li}_2\left (-1+\frac{2}{1-c x}\right )+\frac{3}{2} b^2 \left (a+b \tanh ^{-1}(c x)\right ) \text{Li}_3\left (1-\frac{2}{1-c x}\right )-\frac{3}{2} b^2 \left (a+b \tanh ^{-1}(c x)\right ) \text{Li}_3\left (-1+\frac{2}{1-c x}\right )-\frac{1}{2} \left (3 b^3 c\right ) \int \frac{\text{Li}_3\left (1-\frac{2}{1-c x}\right )}{1-c^2 x^2} \, dx+\frac{1}{2} \left (3 b^3 c\right ) \int \frac{\text{Li}_3\left (-1+\frac{2}{1-c x}\right )}{1-c^2 x^2} \, dx\\ &=2 \left (a+b \tanh ^{-1}(c x)\right )^3 \tanh ^{-1}\left (1-\frac{2}{1-c x}\right )-\frac{3}{2} b \left (a+b \tanh ^{-1}(c x)\right )^2 \text{Li}_2\left (1-\frac{2}{1-c x}\right )+\frac{3}{2} b \left (a+b \tanh ^{-1}(c x)\right )^2 \text{Li}_2\left (-1+\frac{2}{1-c x}\right )+\frac{3}{2} b^2 \left (a+b \tanh ^{-1}(c x)\right ) \text{Li}_3\left (1-\frac{2}{1-c x}\right )-\frac{3}{2} b^2 \left (a+b \tanh ^{-1}(c x)\right ) \text{Li}_3\left (-1+\frac{2}{1-c x}\right )-\frac{3}{4} b^3 \text{Li}_4\left (1-\frac{2}{1-c x}\right )+\frac{3}{4} b^3 \text{Li}_4\left (-1+\frac{2}{1-c x}\right )\\ \end{align*}

Mathematica [A]  time = 0.127977, size = 178, normalized size = 0.97 \[ \frac{3}{4} b \left (2 \text{PolyLog}\left (2,\frac{c x+1}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )^2-2 \text{PolyLog}\left (2,\frac{c x+1}{c x-1}\right ) \left (a+b \tanh ^{-1}(c x)\right )^2+b \left (-2 \text{PolyLog}\left (3,\frac{c x+1}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )+2 \text{PolyLog}\left (3,\frac{c x+1}{c x-1}\right ) \left (a+b \tanh ^{-1}(c x)\right )+b \left (\text{PolyLog}\left (4,\frac{c x+1}{1-c x}\right )-\text{PolyLog}\left (4,\frac{c x+1}{c x-1}\right )\right )\right )\right )+2 \tanh ^{-1}\left (\frac{c x+1}{c x-1}\right ) \left (a+b \tanh ^{-1}(c x)\right )^3 \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*ArcTanh[c*x])^3/x,x]

[Out]

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

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Maple [C]  time = 0.113, size = 1470, normalized size = 8. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/2*a*b^2*polylog(3,-(c*x+1)^2/(-c^2*x^2+1))-6*a*b^2*polylog(3,(c*x+1)/(-c^2*x^2+1)^(1/2))-6*a*b^2*polylog(3,-
(c*x+1)/(-c^2*x^2+1)^(1/2))+b^3*ln(c*x)*arctanh(c*x)^3-b^3*arctanh(c*x)^3*ln((c*x+1)^2/(-c^2*x^2+1)-1)-3/2*b^3
*arctanh(c*x)^2*polylog(2,-(c*x+1)^2/(-c^2*x^2+1))-3/2*a^2*b*dilog(c*x)+3/2*b^3*arctanh(c*x)*polylog(3,-(c*x+1
)^2/(-c^2*x^2+1))+b^3*arctanh(c*x)^3*ln(1-(c*x+1)/(-c^2*x^2+1)^(1/2))+3*b^3*arctanh(c*x)^2*polylog(2,(c*x+1)/(
-c^2*x^2+1)^(1/2))-6*b^3*arctanh(c*x)*polylog(3,(c*x+1)/(-c^2*x^2+1)^(1/2))+b^3*arctanh(c*x)^3*ln(1+(c*x+1)/(-
c^2*x^2+1)^(1/2))+3*b^3*arctanh(c*x)^2*polylog(2,-(c*x+1)/(-c^2*x^2+1)^(1/2))-6*b^3*arctanh(c*x)*polylog(3,-(c
*x+1)/(-c^2*x^2+1)^(1/2))-3/2*a^2*b*dilog(c*x+1)+3/2*I*a*b^2*Pi*csgn(I*((c*x+1)^2/(-c^2*x^2+1)-1))*csgn(I/((c*
x+1)^2/(-c^2*x^2+1)+1))*csgn(I*((c*x+1)^2/(-c^2*x^2+1)-1)/((c*x+1)^2/(-c^2*x^2+1)+1))*arctanh(c*x)^2-1/2*I*b^3
*Pi*csgn(I*((c*x+1)^2/(-c^2*x^2+1)-1))*csgn(I*((c*x+1)^2/(-c^2*x^2+1)-1)/((c*x+1)^2/(-c^2*x^2+1)+1))^2*arctanh
(c*x)^3+3/2*I*a*b^2*Pi*csgn(I*((c*x+1)^2/(-c^2*x^2+1)-1)/((c*x+1)^2/(-c^2*x^2+1)+1))^3*arctanh(c*x)^2-1/2*I*b^
3*Pi*csgn(I/((c*x+1)^2/(-c^2*x^2+1)+1))*csgn(I*((c*x+1)^2/(-c^2*x^2+1)-1)/((c*x+1)^2/(-c^2*x^2+1)+1))^2*arctan
h(c*x)^3-3/2*I*a*b^2*Pi*csgn(I*((c*x+1)^2/(-c^2*x^2+1)-1))*csgn(I*((c*x+1)^2/(-c^2*x^2+1)-1)/((c*x+1)^2/(-c^2*
x^2+1)+1))^2*arctanh(c*x)^2-3/2*I*a*b^2*Pi*csgn(I/((c*x+1)^2/(-c^2*x^2+1)+1))*csgn(I*((c*x+1)^2/(-c^2*x^2+1)-1
)/((c*x+1)^2/(-c^2*x^2+1)+1))^2*arctanh(c*x)^2+1/2*I*b^3*Pi*csgn(I*((c*x+1)^2/(-c^2*x^2+1)-1))*csgn(I/((c*x+1)
^2/(-c^2*x^2+1)+1))*csgn(I*((c*x+1)^2/(-c^2*x^2+1)-1)/((c*x+1)^2/(-c^2*x^2+1)+1))*arctanh(c*x)^3-3*a*b^2*arcta
nh(c*x)*polylog(2,-(c*x+1)^2/(-c^2*x^2+1))-3*a*b^2*arctanh(c*x)^2*ln((c*x+1)^2/(-c^2*x^2+1)-1)+3*a*b^2*arctanh
(c*x)^2*ln(1-(c*x+1)/(-c^2*x^2+1)^(1/2))+6*a*b^2*arctanh(c*x)*polylog(2,(c*x+1)/(-c^2*x^2+1)^(1/2))+3*a*b^2*ar
ctanh(c*x)^2*ln(1+(c*x+1)/(-c^2*x^2+1)^(1/2))+6*a*b^2*arctanh(c*x)*polylog(2,-(c*x+1)/(-c^2*x^2+1)^(1/2))+3*a^
2*b*ln(c*x)*arctanh(c*x)+3*a*b^2*ln(c*x)*arctanh(c*x)^2+a^3*ln(c*x)-3/4*b^3*polylog(4,-(c*x+1)^2/(-c^2*x^2+1))
+6*b^3*polylog(4,(c*x+1)/(-c^2*x^2+1)^(1/2))+6*b^3*polylog(4,-(c*x+1)/(-c^2*x^2+1)^(1/2))+1/2*I*b^3*Pi*csgn(I*
((c*x+1)^2/(-c^2*x^2+1)-1)/((c*x+1)^2/(-c^2*x^2+1)+1))^3*arctanh(c*x)^3-3/2*a^2*b*ln(c*x)*ln(c*x+1)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} a^{3} \log \left (x\right ) + \int \frac{b^{3}{\left (\log \left (c x + 1\right ) - \log \left (-c x + 1\right )\right )}^{3}}{8 \, x} + \frac{3 \, a b^{2}{\left (\log \left (c x + 1\right ) - \log \left (-c x + 1\right )\right )}^{2}}{4 \, x} + \frac{3 \, a^{2} b{\left (\log \left (c x + 1\right ) - \log \left (-c x + 1\right )\right )}}{2 \, x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\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}}{x}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arctanh(c*x))^3/x,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, x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (a + b \operatorname{atanh}{\left (c x \right )}\right )^{3}}{x}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((a + b*atanh(c*x))**3/x, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \operatorname{artanh}\left (c x\right ) + a\right )}^{3}}{x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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