3.78 \(\int \frac{\coth ^{-1}(a+b x)}{c+\frac{d}{x}} \, dx\)

Optimal. Leaf size=292 \[ \frac{d \text{PolyLog}\left (2,-\frac{b (c x+d)}{a c-b d+c}\right )}{2 c^2}-\frac{d \text{PolyLog}\left (2,\frac{b (c x+d)}{-a c+b d+c}\right )}{2 c^2}+\frac{d \log \left (-\frac{-a-b x+1}{a+b x}\right ) \log (c x+d)}{2 c^2}-\frac{d \log (c x+d) \log \left (\frac{c (-a-b x+1)}{-a c+b d+c}\right )}{2 c^2}+\frac{d \log (c x+d) \log \left (\frac{c (a+b x+1)}{a c-b d+c}\right )}{2 c^2}-\frac{d \log \left (\frac{a+b x+1}{a+b x}\right ) \log (c x+d)}{2 c^2}+\frac{(-a-b x+1) \log \left (-\frac{-a-b x+1}{a+b x}\right )}{2 b c}+\frac{\log (a+b x)}{2 b c}+\frac{\log (a+b x+1)}{2 b c}+\frac{(a+b x) \log \left (\frac{a+b x+1}{a+b x}\right )}{2 b c} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.499929, antiderivative size = 360, normalized size of antiderivative = 1.23, number of steps used = 37, number of rules used = 10, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.625, Rules used = {6116, 2513, 2409, 2389, 2295, 2394, 2393, 2391, 193, 43} \[ \frac{d \text{PolyLog}\left (2,\frac{c (-a-b x+1)}{-a c+b d+c}\right )}{2 c^2}-\frac{d \text{PolyLog}\left (2,\frac{c (a+b x+1)}{a c-b d+c}\right )}{2 c^2}+\frac{d \log (a+b x-1) \log \left (\frac{b (c x+d)}{-a c+b d+c}\right )}{2 c^2}-\frac{d \left (\log (a+b x-1)-\log \left (-\frac{-a-b x+1}{a+b x}\right )-\log (a+b x)\right ) \log (c x+d)}{2 c^2}-\frac{d \left (\log (a+b x)-\log (a+b x+1)+\log \left (\frac{a+b x+1}{a+b x}\right )\right ) \log (c x+d)}{2 c^2}-\frac{d \log (a+b x+1) \log \left (-\frac{b (c x+d)}{a c-b d+c}\right )}{2 c^2}+\frac{(-a-b x+1) \log (a+b x-1)}{2 b c}+\frac{x \left (\log (a+b x-1)-\log \left (-\frac{-a-b x+1}{a+b x}\right )-\log (a+b x)\right )}{2 c}+\frac{(a+b x+1) \log (a+b x+1)}{2 b c}+\frac{x \left (\log (a+b x)-\log (a+b x+1)+\log \left (\frac{a+b x+1}{a+b x}\right )\right )}{2 c} \]

Warning: Unable to verify antiderivative.

[In]

Int[ArcCoth[a + b*x]/(c + d/x),x]

[Out]

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

Rule 6116

Int[ArcCoth[(c_) + (d_.)*(x_)]/((e_) + (f_.)*(x_)^(n_.)), x_Symbol] :> Dist[1/2, Int[Log[(1 + c + d*x)/(c + d*
x)]/(e + f*x^n), x], x] - Dist[1/2, Int[Log[(-1 + c + d*x)/(c + d*x)]/(e + f*x^n), x], x] /; FreeQ[{c, d, e, f
}, x] && RationalQ[n]

Rule 2513

Int[Log[(e_.)*((f_.)*((a_.) + (b_.)*(x_))^(p_.)*((c_.) + (d_.)*(x_))^(q_.))^(r_.)]*(RFx_.), x_Symbol] :> Dist[
p*r, Int[RFx*Log[a + b*x], x], x] + (Dist[q*r, Int[RFx*Log[c + d*x], x], x] - Dist[p*r*Log[a + b*x] + q*r*Log[
c + d*x] - Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r], Int[RFx, x], x]) /; FreeQ[{a, b, c, d, e, f, p, q, r}, x] &&
RationalFunctionQ[RFx, x] && NeQ[b*c - a*d, 0] &&  !MatchQ[RFx, (u_.)*(a + b*x)^(m_.)*(c + d*x)^(n_.) /; Integ
ersQ[m, n]]

Rule 2409

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_) + (g_.)*(x_)^(r_))^(q_.), x_Symbol] :> In
t[ExpandIntegrand[(a + b*Log[c*(d + e*x)^n])^p, (f + g*x^r)^q, x], x] /; FreeQ[{a, b, c, d, e, f, g, n, r}, x]
&& IGtQ[p, 0] && IntegerQ[q] && (GtQ[q, 0] || (IntegerQ[r] && NeQ[r, 1]))

Rule 2389

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.), x_Symbol] :> Dist[1/e, Subst[Int[(a + b*Log[c*
x^n])^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, n, p}, x]

Rule 2295

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{c, n}, x]

Rule 2394

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))/((f_.) + (g_.)*(x_)), x_Symbol] :> Simp[(Log[(e*(f +
g*x))/(e*f - d*g)]*(a + b*Log[c*(d + e*x)^n]))/g, x] - Dist[(b*e*n)/g, Int[Log[(e*(f + g*x))/(e*f - d*g)]/(d +
e*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && NeQ[e*f - d*g, 0]

Rule 2393

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))]*(b_.))/((f_.) + (g_.)*(x_)), x_Symbol] :> Dist[1/g, Subst[Int[(a +
b*Log[1 + (c*e*x)/g])/x, x], x, f + g*x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && EqQ[g
+ c*(e*f - d*g), 0]

Rule 2391

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> -Simp[PolyLog[2, -(c*e*x^n)]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 193

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[x^(n*p)*(b + a/x^n)^p, x] /; FreeQ[{a, b}, x] && LtQ[n, 0]
&& IntegerQ[p]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{\coth ^{-1}(a+b x)}{c+\frac{d}{x}} \, dx &=-\left (\frac{1}{2} \int \frac{\log \left (\frac{-1+a+b x}{a+b x}\right )}{c+\frac{d}{x}} \, dx\right )+\frac{1}{2} \int \frac{\log \left (\frac{1+a+b x}{a+b x}\right )}{c+\frac{d}{x}} \, dx\\ &=-\left (\frac{1}{2} \int \frac{\log (-1+a+b x)}{c+\frac{d}{x}} \, dx\right )+\frac{1}{2} \int \frac{\log (1+a+b x)}{c+\frac{d}{x}} \, dx-\frac{1}{2} \left (-\log (-1+a+b x)+\log \left (\frac{-1+a+b x}{a+b x}\right )+\log (a+b x)\right ) \int \frac{1}{c+\frac{d}{x}} \, dx+\frac{1}{2} \left (\log (a+b x)-\log (1+a+b x)+\log \left (\frac{1+a+b x}{a+b x}\right )\right ) \int \frac{1}{c+\frac{d}{x}} \, dx\\ &=-\left (\frac{1}{2} \int \left (\frac{\log (-1+a+b x)}{c}-\frac{d \log (-1+a+b x)}{c (d+c x)}\right ) \, dx\right )+\frac{1}{2} \int \left (\frac{\log (1+a+b x)}{c}-\frac{d \log (1+a+b x)}{c (d+c x)}\right ) \, dx-\frac{1}{2} \left (-\log (-1+a+b x)+\log \left (\frac{-1+a+b x}{a+b x}\right )+\log (a+b x)\right ) \int \frac{x}{d+c x} \, dx+\frac{1}{2} \left (\log (a+b x)-\log (1+a+b x)+\log \left (\frac{1+a+b x}{a+b x}\right )\right ) \int \frac{x}{d+c x} \, dx\\ &=-\frac{\int \log (-1+a+b x) \, dx}{2 c}+\frac{\int \log (1+a+b x) \, dx}{2 c}+\frac{d \int \frac{\log (-1+a+b x)}{d+c x} \, dx}{2 c}-\frac{d \int \frac{\log (1+a+b x)}{d+c x} \, dx}{2 c}-\frac{1}{2} \left (-\log (-1+a+b x)+\log \left (\frac{-1+a+b x}{a+b x}\right )+\log (a+b x)\right ) \int \left (\frac{1}{c}-\frac{d}{c (d+c x)}\right ) \, dx+\frac{1}{2} \left (\log (a+b x)-\log (1+a+b x)+\log \left (\frac{1+a+b x}{a+b x}\right )\right ) \int \left (\frac{1}{c}-\frac{d}{c (d+c x)}\right ) \, dx\\ &=\frac{x \left (\log (-1+a+b x)-\log \left (-\frac{1-a-b x}{a+b x}\right )-\log (a+b x)\right )}{2 c}+\frac{x \left (\log (a+b x)-\log (1+a+b x)+\log \left (\frac{1+a+b x}{a+b x}\right )\right )}{2 c}-\frac{d \left (\log (-1+a+b x)-\log \left (-\frac{1-a-b x}{a+b x}\right )-\log (a+b x)\right ) \log (d+c x)}{2 c^2}-\frac{d \left (\log (a+b x)-\log (1+a+b x)+\log \left (\frac{1+a+b x}{a+b x}\right )\right ) \log (d+c x)}{2 c^2}-\frac{d \log (1+a+b x) \log \left (-\frac{b (d+c x)}{c+a c-b d}\right )}{2 c^2}+\frac{d \log (-1+a+b x) \log \left (\frac{b (d+c x)}{c-a c+b d}\right )}{2 c^2}-\frac{\operatorname{Subst}(\int \log (x) \, dx,x,-1+a+b x)}{2 b c}+\frac{\operatorname{Subst}(\int \log (x) \, dx,x,1+a+b x)}{2 b c}-\frac{(b d) \int \frac{\log \left (\frac{b (d+c x)}{-(-1+a) c+b d}\right )}{-1+a+b x} \, dx}{2 c^2}+\frac{(b d) \int \frac{\log \left (\frac{b (d+c x)}{-(1+a) c+b d}\right )}{1+a+b x} \, dx}{2 c^2}\\ &=\frac{(1-a-b x) \log (-1+a+b x)}{2 b c}+\frac{x \left (\log (-1+a+b x)-\log \left (-\frac{1-a-b x}{a+b x}\right )-\log (a+b x)\right )}{2 c}+\frac{(1+a+b x) \log (1+a+b x)}{2 b c}+\frac{x \left (\log (a+b x)-\log (1+a+b x)+\log \left (\frac{1+a+b x}{a+b x}\right )\right )}{2 c}-\frac{d \left (\log (-1+a+b x)-\log \left (-\frac{1-a-b x}{a+b x}\right )-\log (a+b x)\right ) \log (d+c x)}{2 c^2}-\frac{d \left (\log (a+b x)-\log (1+a+b x)+\log \left (\frac{1+a+b x}{a+b x}\right )\right ) \log (d+c x)}{2 c^2}-\frac{d \log (1+a+b x) \log \left (-\frac{b (d+c x)}{c+a c-b d}\right )}{2 c^2}+\frac{d \log (-1+a+b x) \log \left (\frac{b (d+c x)}{c-a c+b d}\right )}{2 c^2}-\frac{d \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{c x}{-(-1+a) c+b d}\right )}{x} \, dx,x,-1+a+b x\right )}{2 c^2}+\frac{d \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{c x}{-(1+a) c+b d}\right )}{x} \, dx,x,1+a+b x\right )}{2 c^2}\\ &=\frac{(1-a-b x) \log (-1+a+b x)}{2 b c}+\frac{x \left (\log (-1+a+b x)-\log \left (-\frac{1-a-b x}{a+b x}\right )-\log (a+b x)\right )}{2 c}+\frac{(1+a+b x) \log (1+a+b x)}{2 b c}+\frac{x \left (\log (a+b x)-\log (1+a+b x)+\log \left (\frac{1+a+b x}{a+b x}\right )\right )}{2 c}-\frac{d \left (\log (-1+a+b x)-\log \left (-\frac{1-a-b x}{a+b x}\right )-\log (a+b x)\right ) \log (d+c x)}{2 c^2}-\frac{d \left (\log (a+b x)-\log (1+a+b x)+\log \left (\frac{1+a+b x}{a+b x}\right )\right ) \log (d+c x)}{2 c^2}-\frac{d \log (1+a+b x) \log \left (-\frac{b (d+c x)}{c+a c-b d}\right )}{2 c^2}+\frac{d \log (-1+a+b x) \log \left (\frac{b (d+c x)}{c-a c+b d}\right )}{2 c^2}+\frac{d \text{Li}_2\left (\frac{c (1-a-b x)}{c-a c+b d}\right )}{2 c^2}-\frac{d \text{Li}_2\left (\frac{c (1+a+b x)}{c+a c-b d}\right )}{2 c^2}\\ \end{align*}

Mathematica [C]  time = 4.25848, size = 502, normalized size = 1.72 \[ \frac{b c d \text{PolyLog}\left (2,\exp \left (2 \tanh ^{-1}\left (\frac{c}{a c-b d}\right )-2 \coth ^{-1}(a+b x)\right )\right )-b c d \text{PolyLog}\left (2,e^{-2 \coth ^{-1}(a+b x)}\right )+b^2 d^2 \sqrt{1-\frac{c^2}{(a c-b d)^2}} \coth ^{-1}(a+b x)^2 e^{\tanh ^{-1}\left (\frac{c}{a c-b d}\right )}-b^2 d^2 \coth ^{-1}(a+b x)^2-a b c d \sqrt{1-\frac{c^2}{(a c-b d)^2}} \coth ^{-1}(a+b x)^2 e^{\tanh ^{-1}\left (\frac{c}{a c-b d}\right )}-2 c^2 \log \left (\frac{1}{(a+b x) \sqrt{1-\frac{1}{(a+b x)^2}}}\right )+2 a c^2 \coth ^{-1}(a+b x)+2 b c^2 x \coth ^{-1}(a+b x)-2 b c d \coth ^{-1}(a+b x) \log \left (1-\exp \left (2 \tanh ^{-1}\left (\frac{c}{a c-b d}\right )-2 \coth ^{-1}(a+b x)\right )\right )+2 b c d \tanh ^{-1}\left (\frac{c}{a c-b d}\right ) \log \left (1-\exp \left (2 \tanh ^{-1}\left (\frac{c}{a c-b d}\right )-2 \coth ^{-1}(a+b x)\right )\right )-i \pi b c d \log \left (\frac{1}{\sqrt{1-\frac{1}{(a+b x)^2}}}\right )+a b c d \coth ^{-1}(a+b x)^2+b c d \coth ^{-1}(a+b x)^2-i \pi b c d \coth ^{-1}(a+b x)+2 b c d \coth ^{-1}(a+b x) \log \left (1-e^{-2 \coth ^{-1}(a+b x)}\right )+i \pi b c d \log \left (e^{2 \coth ^{-1}(a+b x)}+1\right )+2 b c d \coth ^{-1}(a+b x) \tanh ^{-1}\left (\frac{c}{a c-b d}\right )-2 b c d \tanh ^{-1}\left (\frac{c}{a c-b d}\right ) \log \left (i \sinh \left (\coth ^{-1}(a+b x)-\tanh ^{-1}\left (\frac{c}{a c-b d}\right )\right )\right )}{2 b c^3} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[ArcCoth[a + b*x]/(c + d/x),x]

[Out]

(2*a*c^2*ArcCoth[a + b*x] - I*b*c*d*Pi*ArcCoth[a + b*x] + 2*b*c^2*x*ArcCoth[a + b*x] + b*c*d*ArcCoth[a + b*x]^
2 + a*b*c*d*ArcCoth[a + b*x]^2 - b^2*d^2*ArcCoth[a + b*x]^2 - a*b*c*d*Sqrt[1 - c^2/(a*c - b*d)^2]*E^ArcTanh[c/
(a*c - b*d)]*ArcCoth[a + b*x]^2 + b^2*d^2*Sqrt[1 - c^2/(a*c - b*d)^2]*E^ArcTanh[c/(a*c - b*d)]*ArcCoth[a + b*x
]^2 + 2*b*c*d*ArcCoth[a + b*x]*ArcTanh[c/(a*c - b*d)] + 2*b*c*d*ArcCoth[a + b*x]*Log[1 - E^(-2*ArcCoth[a + b*x
])] + I*b*c*d*Pi*Log[1 + E^(2*ArcCoth[a + b*x])] - 2*b*c*d*ArcCoth[a + b*x]*Log[1 - E^(-2*ArcCoth[a + b*x] + 2
*ArcTanh[c/(a*c - b*d)])] + 2*b*c*d*ArcTanh[c/(a*c - b*d)]*Log[1 - E^(-2*ArcCoth[a + b*x] + 2*ArcTanh[c/(a*c -
b*d)])] - I*b*c*d*Pi*Log[1/Sqrt[1 - (a + b*x)^(-2)]] - 2*c^2*Log[1/((a + b*x)*Sqrt[1 - (a + b*x)^(-2)])] - 2*
b*c*d*ArcTanh[c/(a*c - b*d)]*Log[I*Sinh[ArcCoth[a + b*x] - ArcTanh[c/(a*c - b*d)]]] - b*c*d*PolyLog[2, E^(-2*A
rcCoth[a + b*x])] + b*c*d*PolyLog[2, E^(-2*ArcCoth[a + b*x] + 2*ArcTanh[c/(a*c - b*d)])])/(2*b*c^3)

________________________________________________________________________________________

Maple [A]  time = 0.142, size = 297, normalized size = 1. \begin{align*}{\frac{x{\rm arccoth} \left (bx+a\right )}{c}}+{\frac{{\rm arccoth} \left (bx+a\right )a}{bc}}-{\frac{{\rm arccoth} \left (bx+a\right )d\ln \left ( c \left ( bx+a \right ) -ac+bd \right ) }{{c}^{2}}}+{\frac{\ln \left ({a}^{2}{c}^{2}-2\,abcd+{b}^{2}{d}^{2}+2\, \left ( c \left ( bx+a \right ) -ac+bd \right ) ac-2\, \left ( c \left ( bx+a \right ) -ac+bd \right ) bd+ \left ( c \left ( bx+a \right ) -ac+bd \right ) ^{2}-{c}^{2} \right ) }{2\,bc}}-{\frac{d\ln \left ( c \left ( bx+a \right ) -ac+bd \right ) }{2\,{c}^{2}}\ln \left ({\frac{c \left ( bx+a \right ) -c}{ac-bd-c}} \right ) }-{\frac{d}{2\,{c}^{2}}{\it dilog} \left ({\frac{c \left ( bx+a \right ) -c}{ac-bd-c}} \right ) }+{\frac{d\ln \left ( c \left ( bx+a \right ) -ac+bd \right ) }{2\,{c}^{2}}\ln \left ({\frac{c \left ( bx+a \right ) +c}{ac-bd+c}} \right ) }+{\frac{d}{2\,{c}^{2}}{\it dilog} \left ({\frac{c \left ( bx+a \right ) +c}{ac-bd+c}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arccoth(b*x+a)/(c+d/x),x)

[Out]

arccoth(b*x+a)/c*x+1/b*arccoth(b*x+a)/c*a-arccoth(b*x+a)*d/c^2*ln(c*(b*x+a)-a*c+b*d)+1/2/b/c*ln(a^2*c^2-2*a*b*
c*d+b^2*d^2+2*(c*(b*x+a)-a*c+b*d)*a*c-2*(c*(b*x+a)-a*c+b*d)*b*d+(c*(b*x+a)-a*c+b*d)^2-c^2)-1/2/c^2*d*ln(c*(b*x
+a)-a*c+b*d)*ln((c*(b*x+a)-c)/(a*c-b*d-c))-1/2/c^2*d*dilog((c*(b*x+a)-c)/(a*c-b*d-c))+1/2/c^2*d*ln(c*(b*x+a)-a
*c+b*d)*ln((c*(b*x+a)+c)/(a*c-b*d+c))+1/2/c^2*d*dilog((c*(b*x+a)+c)/(a*c-b*d+c))

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Maxima [A]  time = 0.996411, size = 259, normalized size = 0.89 \begin{align*} \frac{1}{2} \, b{\left (\frac{{\left (\log \left (c x + d\right ) \log \left (\frac{b c x + b d}{a c - b d + c} + 1\right ) +{\rm Li}_2\left (-\frac{b c x + b d}{a c - b d + c}\right )\right )} d}{b c^{2}} - \frac{{\left (\log \left (c x + d\right ) \log \left (\frac{b c x + b d}{a c - b d - c} + 1\right ) +{\rm Li}_2\left (-\frac{b c x + b d}{a c - b d - c}\right )\right )} d}{b c^{2}} + \frac{{\left (a + 1\right )} \log \left (b x + a + 1\right )}{b^{2} c} - \frac{{\left (a - 1\right )} \log \left (b x + a - 1\right )}{b^{2} c}\right )} +{\left (\frac{x}{c} - \frac{d \log \left (c x + d\right )}{c^{2}}\right )} \operatorname{arcoth}\left (b x + a\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccoth(b*x+a)/(c+d/x),x, algorithm="maxima")

[Out]

1/2*b*((log(c*x + d)*log((b*c*x + b*d)/(a*c - b*d + c) + 1) + dilog(-(b*c*x + b*d)/(a*c - b*d + c)))*d/(b*c^2)
- (log(c*x + d)*log((b*c*x + b*d)/(a*c - b*d - c) + 1) + dilog(-(b*c*x + b*d)/(a*c - b*d - c)))*d/(b*c^2) + (
a + 1)*log(b*x + a + 1)/(b^2*c) - (a - 1)*log(b*x + a - 1)/(b^2*c)) + (x/c - d*log(c*x + d)/c^2)*arccoth(b*x +
a)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{x \operatorname{arcoth}\left (b x + a\right )}{c x + d}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccoth(b*x+a)/(c+d/x),x, algorithm="fricas")

[Out]

integral(x*arccoth(b*x + a)/(c*x + d), x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(acoth(b*x+a)/(c+d/x),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccoth(b*x+a)/(c+d/x),x, algorithm="giac")

[Out]

integrate(arccoth(b*x + a)/(c + d/x), x)