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

Optimal. Leaf size=208 \[ -\frac{3}{2} b^2 \text{PolyLog}\left (3,1-\frac{2}{1-\frac{c}{x}}\right ) \left (a+b \coth ^{-1}\left (\frac{x}{c}\right )\right )+\frac{3}{2} b^2 \text{PolyLog}\left (3,\frac{2}{1-\frac{c}{x}}-1\right ) \left (a+b \coth ^{-1}\left (\frac{x}{c}\right )\right )+\frac{3}{2} b \text{PolyLog}\left (2,1-\frac{2}{1-\frac{c}{x}}\right ) \left (a+b \coth ^{-1}\left (\frac{x}{c}\right )\right )^2-\frac{3}{2} b \text{PolyLog}\left (2,\frac{2}{1-\frac{c}{x}}-1\right ) \left (a+b \coth ^{-1}\left (\frac{x}{c}\right )\right )^2+\frac{3}{4} b^3 \text{PolyLog}\left (4,1-\frac{2}{1-\frac{c}{x}}\right )-\frac{3}{4} b^3 \text{PolyLog}\left (4,\frac{2}{1-\frac{c}{x}}-1\right )-2 \tanh ^{-1}\left (1-\frac{2}{1-\frac{c}{x}}\right ) \left (a+b \coth ^{-1}\left (\frac{x}{c}\right )\right )^3 \]

[Out]

-2*(a + b*ArcCoth[x/c])^3*ArcTanh[1 - 2/(1 - c/x)] + (3*b*(a + b*ArcCoth[x/c])^2*PolyLog[2, 1 - 2/(1 - c/x)])/
2 - (3*b*(a + b*ArcCoth[x/c])^2*PolyLog[2, -1 + 2/(1 - c/x)])/2 - (3*b^2*(a + b*ArcCoth[x/c])*PolyLog[3, 1 - 2
/(1 - c/x)])/2 + (3*b^2*(a + b*ArcCoth[x/c])*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 = 0.508605, antiderivative size = 208, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.438, Rules used = {6095, 5914, 6052, 5948, 6058, 6062, 6610} \[ -\frac{3}{2} b^2 \text{PolyLog}\left (3,1-\frac{2}{1-\frac{c}{x}}\right ) \left (a+b \coth ^{-1}\left (\frac{x}{c}\right )\right )+\frac{3}{2} b^2 \text{PolyLog}\left (3,\frac{2}{1-\frac{c}{x}}-1\right ) \left (a+b \coth ^{-1}\left (\frac{x}{c}\right )\right )+\frac{3}{2} b \text{PolyLog}\left (2,1-\frac{2}{1-\frac{c}{x}}\right ) \left (a+b \coth ^{-1}\left (\frac{x}{c}\right )\right )^2-\frac{3}{2} b \text{PolyLog}\left (2,\frac{2}{1-\frac{c}{x}}-1\right ) \left (a+b \coth ^{-1}\left (\frac{x}{c}\right )\right )^2+\frac{3}{4} b^3 \text{PolyLog}\left (4,1-\frac{2}{1-\frac{c}{x}}\right )-\frac{3}{4} b^3 \text{PolyLog}\left (4,\frac{2}{1-\frac{c}{x}}-1\right )-2 \tanh ^{-1}\left (1-\frac{2}{1-\frac{c}{x}}\right ) \left (a+b \coth ^{-1}\left (\frac{x}{c}\right )\right )^3 \]

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcTanh[c/x])^3/x,x]

[Out]

-2*(a + b*ArcCoth[x/c])^3*ArcTanh[1 - 2/(1 - c/x)] + (3*b*(a + b*ArcCoth[x/c])^2*PolyLog[2, 1 - 2/(1 - c/x)])/
2 - (3*b*(a + b*ArcCoth[x/c])^2*PolyLog[2, -1 + 2/(1 - c/x)])/2 - (3*b^2*(a + b*ArcCoth[x/c])*PolyLog[3, 1 - 2
/(1 - c/x)])/2 + (3*b^2*(a + b*ArcCoth[x/c])*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 6095

Int[((a_.) + ArcTanh[(c_.)*(x_)^(n_)]*(b_.))^(p_.)/(x_), x_Symbol] :> Dist[1/n, Subst[Int[(a + b*ArcTanh[c*x])
^p/x, x], x, x^n], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[p, 0]

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}\left (\frac{c}{x}\right )\right )^3}{x} \, dx &=-\operatorname{Subst}\left (\int \frac{\left (a+b \tanh ^{-1}(c x)\right )^3}{x} \, dx,x,\frac{1}{x}\right )\\ &=-2 \left (a+b \coth ^{-1}\left (\frac{x}{c}\right )\right )^3 \tanh ^{-1}\left (1-\frac{2}{1-\frac{c}{x}}\right )+(6 b c) \operatorname{Subst}\left (\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,x,\frac{1}{x}\right )\\ &=-2 \left (a+b \coth ^{-1}\left (\frac{x}{c}\right )\right )^3 \tanh ^{-1}\left (1-\frac{2}{1-\frac{c}{x}}\right )-(3 b c) \operatorname{Subst}\left (\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,x,\frac{1}{x}\right )+(3 b c) \operatorname{Subst}\left (\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,x,\frac{1}{x}\right )\\ &=-2 \left (a+b \coth ^{-1}\left (\frac{x}{c}\right )\right )^3 \tanh ^{-1}\left (1-\frac{2}{1-\frac{c}{x}}\right )+\frac{3}{2} b \left (a+b \coth ^{-1}\left (\frac{x}{c}\right )\right )^2 \text{Li}_2\left (1-\frac{2}{1-\frac{c}{x}}\right )-\frac{3}{2} b \left (a+b \coth ^{-1}\left (\frac{x}{c}\right )\right )^2 \text{Li}_2\left (-1+\frac{2}{1-\frac{c}{x}}\right )-\left (3 b^2 c\right ) \operatorname{Subst}\left (\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,x,\frac{1}{x}\right )+\left (3 b^2 c\right ) \operatorname{Subst}\left (\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,x,\frac{1}{x}\right )\\ &=-2 \left (a+b \coth ^{-1}\left (\frac{x}{c}\right )\right )^3 \tanh ^{-1}\left (1-\frac{2}{1-\frac{c}{x}}\right )+\frac{3}{2} b \left (a+b \coth ^{-1}\left (\frac{x}{c}\right )\right )^2 \text{Li}_2\left (1-\frac{2}{1-\frac{c}{x}}\right )-\frac{3}{2} b \left (a+b \coth ^{-1}\left (\frac{x}{c}\right )\right )^2 \text{Li}_2\left (-1+\frac{2}{1-\frac{c}{x}}\right )-\frac{3}{2} b^2 \left (a+b \coth ^{-1}\left (\frac{x}{c}\right )\right ) \text{Li}_3\left (1-\frac{2}{1-\frac{c}{x}}\right )+\frac{3}{2} b^2 \left (a+b \coth ^{-1}\left (\frac{x}{c}\right )\right ) \text{Li}_3\left (-1+\frac{2}{1-\frac{c}{x}}\right )+\frac{1}{2} \left (3 b^3 c\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_3\left (1-\frac{2}{1-c x}\right )}{1-c^2 x^2} \, dx,x,\frac{1}{x}\right )-\frac{1}{2} \left (3 b^3 c\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_3\left (-1+\frac{2}{1-c x}\right )}{1-c^2 x^2} \, dx,x,\frac{1}{x}\right )\\ &=-2 \left (a+b \coth ^{-1}\left (\frac{x}{c}\right )\right )^3 \tanh ^{-1}\left (1-\frac{2}{1-\frac{c}{x}}\right )+\frac{3}{2} b \left (a+b \coth ^{-1}\left (\frac{x}{c}\right )\right )^2 \text{Li}_2\left (1-\frac{2}{1-\frac{c}{x}}\right )-\frac{3}{2} b \left (a+b \coth ^{-1}\left (\frac{x}{c}\right )\right )^2 \text{Li}_2\left (-1+\frac{2}{1-\frac{c}{x}}\right )-\frac{3}{2} b^2 \left (a+b \coth ^{-1}\left (\frac{x}{c}\right )\right ) \text{Li}_3\left (1-\frac{2}{1-\frac{c}{x}}\right )+\frac{3}{2} b^2 \left (a+b \coth ^{-1}\left (\frac{x}{c}\right )\right ) \text{Li}_3\left (-1+\frac{2}{1-\frac{c}{x}}\right )+\frac{3}{4} b^3 \text{Li}_4\left (1-\frac{2}{1-\frac{c}{x}}\right )-\frac{3}{4} b^3 \text{Li}_4\left (-1+\frac{2}{1-\frac{c}{x}}\right )\\ \end{align*}

Mathematica [A]  time = 0.191137, size = 171, normalized size = 0.82 \[ \frac{3}{4} b \left (2 \text{PolyLog}\left (2,\frac{c+x}{c-x}\right ) \left (a+b \tanh ^{-1}\left (\frac{c}{x}\right )\right )^2-2 \text{PolyLog}\left (2,\frac{c+x}{x-c}\right ) \left (a+b \tanh ^{-1}\left (\frac{c}{x}\right )\right )^2+b \left (-2 \text{PolyLog}\left (3,\frac{c+x}{c-x}\right ) \left (a+b \tanh ^{-1}\left (\frac{c}{x}\right )\right )+2 \text{PolyLog}\left (3,\frac{c+x}{x-c}\right ) \left (a+b \tanh ^{-1}\left (\frac{c}{x}\right )\right )+b \left (\text{PolyLog}\left (4,\frac{c+x}{c-x}\right )-\text{PolyLog}\left (4,\frac{c+x}{x-c}\right )\right )\right )\right )-2 \tanh ^{-1}\left (\frac{c+x}{c-x}\right ) \left (a+b \tanh ^{-1}\left (\frac{c}{x}\right )\right )^3 \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(a + b*ArcTanh[c/x])^3/x,x]

[Out]

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

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Maple [C]  time = 0.07, size = 1631, normalized size = 7.8 \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^2*b*dilog(1+c/x)-3/2*a*b^2*polylog(3,-(1+c/x)^2/(1-c^2/x^2))+6*a*b^2*polylog(3,(1+c/x)/(1-c^2/x^2)^(1/2)
)+6*a*b^2*polylog(3,-(1+c/x)/(1-c^2/x^2)^(1/2))-b^3*arctanh(c/x)^3*ln(1-(1+c/x)/(1-c^2/x^2)^(1/2))-3*b^3*arcta
nh(c/x)^2*polylog(2,(1+c/x)/(1-c^2/x^2)^(1/2))+6*b^3*arctanh(c/x)*polylog(3,(1+c/x)/(1-c^2/x^2)^(1/2))-b^3*arc
tanh(c/x)^3*ln(1+(1+c/x)/(1-c^2/x^2)^(1/2))-3*b^3*arctanh(c/x)^2*polylog(2,-(1+c/x)/(1-c^2/x^2)^(1/2))+6*b^3*a
rctanh(c/x)*polylog(3,-(1+c/x)/(1-c^2/x^2)^(1/2))-b^3*ln(c/x)*arctanh(c/x)^3+b^3*arctanh(c/x)^3*ln((1+c/x)^2/(
1-c^2/x^2)-1)+3/2*b^3*arctanh(c/x)^2*polylog(2,-(1+c/x)^2/(1-c^2/x^2))-3/2*b^3*arctanh(c/x)*polylog(3,-(1+c/x)
^2/(1-c^2/x^2))+3/2*a^2*b*dilog(c/x)-3/2*I*a*b^2*Pi*csgn(I*((1+c/x)^2/(1-c^2/x^2)-1))*csgn(I/((1+c/x)^2/(1-c^2
/x^2)+1))*csgn(I*((1+c/x)^2/(1-c^2/x^2)-1)/((1+c/x)^2/(1-c^2/x^2)+1))*arctanh(c/x)^2-3*a*b^2*arctanh(c/x)^2*ln
(1+(1+c/x)/(1-c^2/x^2)^(1/2))-6*a*b^2*arctanh(c/x)*polylog(2,-(1+c/x)/(1-c^2/x^2)^(1/2))-3*a*b^2*ln(c/x)*arcta
nh(c/x)^2+3*a*b^2*arctanh(c/x)*polylog(2,-(1+c/x)^2/(1-c^2/x^2))-3*a^2*b*ln(c/x)*arctanh(c/x)+3/2*a^2*b*ln(c/x
)*ln(1+c/x)+3*a*b^2*arctanh(c/x)^2*ln((1+c/x)^2/(1-c^2/x^2)-1)-3*a*b^2*arctanh(c/x)^2*ln(1-(1+c/x)/(1-c^2/x^2)
^(1/2))-6*a*b^2*arctanh(c/x)*polylog(2,(1+c/x)/(1-c^2/x^2)^(1/2))+1/2*I*b^3*Pi*csgn(I*((1+c/x)^2/(1-c^2/x^2)-1
))*csgn(I*((1+c/x)^2/(1-c^2/x^2)-1)/((1+c/x)^2/(1-c^2/x^2)+1))^2*arctanh(c/x)^3+1/2*I*b^3*Pi*csgn(I/((1+c/x)^2
/(1-c^2/x^2)+1))*csgn(I*((1+c/x)^2/(1-c^2/x^2)-1)/((1+c/x)^2/(1-c^2/x^2)+1))^2*arctanh(c/x)^3-3/2*I*a*b^2*Pi*c
sgn(I*((1+c/x)^2/(1-c^2/x^2)-1)/((1+c/x)^2/(1-c^2/x^2)+1))^3*arctanh(c/x)^2+3/2*I*a*b^2*Pi*csgn(I*((1+c/x)^2/(
1-c^2/x^2)-1))*csgn(I*((1+c/x)^2/(1-c^2/x^2)-1)/((1+c/x)^2/(1-c^2/x^2)+1))^2*arctanh(c/x)^2+3/2*I*a*b^2*Pi*csg
n(I/((1+c/x)^2/(1-c^2/x^2)+1))*csgn(I*((1+c/x)^2/(1-c^2/x^2)-1)/((1+c/x)^2/(1-c^2/x^2)+1))^2*arctanh(c/x)^2-1/
2*I*b^3*Pi*csgn(I*((1+c/x)^2/(1-c^2/x^2)-1))*csgn(I/((1+c/x)^2/(1-c^2/x^2)+1))*csgn(I*((1+c/x)^2/(1-c^2/x^2)-1
)/((1+c/x)^2/(1-c^2/x^2)+1))*arctanh(c/x)^3-a^3*ln(c/x)+3/4*b^3*polylog(4,-(1+c/x)^2/(1-c^2/x^2))-6*b^3*polylo
g(4,(1+c/x)/(1-c^2/x^2)^(1/2))-6*b^3*polylog(4,-(1+c/x)/(1-c^2/x^2)^(1/2))-1/2*I*b^3*Pi*csgn(I*((1+c/x)^2/(1-c
^2/x^2)-1)/((1+c/x)^2/(1-c^2/x^2)+1))^3*arctanh(c/x)^3

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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 (\frac{c}{x} + 1\right ) - \log \left (-\frac{c}{x} + 1\right )\right )}^{3}}{8 \, x} + \frac{3 \, a b^{2}{\left (\log \left (\frac{c}{x} + 1\right ) - \log \left (-\frac{c}{x} + 1\right )\right )}^{2}}{4 \, x} + \frac{3 \, a^{2} b{\left (\log \left (\frac{c}{x} + 1\right ) - \log \left (-\frac{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 (\frac{c}{x}\right )^{3} + 3 \, a b^{2} \operatorname{artanh}\left (\frac{c}{x}\right )^{2} + 3 \, a^{2} b \operatorname{artanh}\left (\frac{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 (\frac{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 (\frac{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)