3.79 \(\int \frac{\coth ^{-1}(a+b x)}{c+\frac{d}{x^2}} \, dx\)

Optimal. Leaf size=738 \[ \frac{\sqrt{d} \text{PolyLog}\left (2,\frac{\sqrt{-c} (-a-b x+1)}{(1-a) \sqrt{-c}-b \sqrt{d}}\right )}{4 (-c)^{3/2}}-\frac{\sqrt{d} \text{PolyLog}\left (2,\frac{\sqrt{-c} (-a-b x+1)}{(1-a) \sqrt{-c}+b \sqrt{d}}\right )}{4 (-c)^{3/2}}+\frac{\sqrt{d} \text{PolyLog}\left (2,\frac{\sqrt{-c} (a+b x+1)}{(a+1) \sqrt{-c}-b \sqrt{d}}\right )}{4 (-c)^{3/2}}-\frac{\sqrt{d} \text{PolyLog}\left (2,\frac{\sqrt{-c} (a+b x+1)}{(a+1) \sqrt{-c}+b \sqrt{d}}\right )}{4 (-c)^{3/2}}-\frac{\sqrt{d} \left (\log (a+b x-1)-\log \left (-\frac{-a-b x+1}{a+b x}\right )-\log (a+b x)\right ) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{d}}\right )}{2 c^{3/2}}-\frac{\sqrt{d} \left (\log (a+b x)-\log (a+b x+1)+\log \left (\frac{a+b x+1}{a+b x}\right )\right ) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{d}}\right )}{2 c^{3/2}}+\frac{\sqrt{d} \log (a+b x-1) \log \left (-\frac{b \left (\sqrt{d}-\sqrt{-c} x\right )}{(1-a) \sqrt{-c}-b \sqrt{d}}\right )}{4 (-c)^{3/2}}-\frac{\sqrt{d} \log (a+b x-1) \log \left (\frac{b \left (\sqrt{-c} x+\sqrt{d}\right )}{(1-a) \sqrt{-c}+b \sqrt{d}}\right )}{4 (-c)^{3/2}}-\frac{\sqrt{d} \log (a+b x+1) \log \left (\frac{b \left (\sqrt{d}-\sqrt{-c} x\right )}{(a+1) \sqrt{-c}+b \sqrt{d}}\right )}{4 (-c)^{3/2}}+\frac{\sqrt{d} \log (a+b x+1) \log \left (-\frac{b \left (\sqrt{-c} x+\sqrt{d}\right )}{(a+1) \sqrt{-c}-b \sqrt{d}}\right )}{4 (-c)^{3/2}}+\frac{(-a-b x+1) \log (a+b x-1)}{2 b c}+\frac{x \left (\log (a+b x-1)-\log \left (-\frac{-a-b x+1}{a+b x}\right )-\log (a+b x)\right )}{2 c}+\frac{(a+b x+1) \log (a+b x+1)}{2 b c}+\frac{x \left (\log (a+b x)-\log (a+b x+1)+\log \left (\frac{a+b x+1}{a+b x}\right )\right )}{2 c} \]

[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) - (Sqrt[d]*ArcTan[(Sqrt[c]*x)/Sqrt[d]]*(Log[-1 + a + b*x] - Log[-((1 - a - b*x)/(a + b*x))] - Log
[a + b*x]))/(2*c^(3/2)) + ((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) - (Sqrt[d]*ArcTan[(Sqrt[c]*x)/Sqrt[d]]*(Log[a + b*x] - Log[1 + a + b*x] + Lo
g[(1 + a + b*x)/(a + b*x)]))/(2*c^(3/2)) + (Sqrt[d]*Log[-1 + a + b*x]*Log[-((b*(Sqrt[d] - Sqrt[-c]*x))/((1 - a
)*Sqrt[-c] - b*Sqrt[d]))])/(4*(-c)^(3/2)) - (Sqrt[d]*Log[1 + a + b*x]*Log[(b*(Sqrt[d] - Sqrt[-c]*x))/((1 + a)*
Sqrt[-c] + b*Sqrt[d])])/(4*(-c)^(3/2)) + (Sqrt[d]*Log[1 + a + b*x]*Log[-((b*(Sqrt[d] + Sqrt[-c]*x))/((1 + a)*S
qrt[-c] - b*Sqrt[d]))])/(4*(-c)^(3/2)) - (Sqrt[d]*Log[-1 + a + b*x]*Log[(b*(Sqrt[d] + Sqrt[-c]*x))/((1 - a)*Sq
rt[-c] + b*Sqrt[d])])/(4*(-c)^(3/2)) + (Sqrt[d]*PolyLog[2, (Sqrt[-c]*(1 - a - b*x))/((1 - a)*Sqrt[-c] - b*Sqrt
[d])])/(4*(-c)^(3/2)) - (Sqrt[d]*PolyLog[2, (Sqrt[-c]*(1 - a - b*x))/((1 - a)*Sqrt[-c] + b*Sqrt[d])])/(4*(-c)^
(3/2)) + (Sqrt[d]*PolyLog[2, (Sqrt[-c]*(1 + a + b*x))/((1 + a)*Sqrt[-c] - b*Sqrt[d])])/(4*(-c)^(3/2)) - (Sqrt[
d]*PolyLog[2, (Sqrt[-c]*(1 + a + b*x))/((1 + a)*Sqrt[-c] + b*Sqrt[d])])/(4*(-c)^(3/2))

________________________________________________________________________________________

Rubi [A]  time = 1.53032, antiderivative size = 738, normalized size of antiderivative = 1., number of steps used = 57, number of rules used = 11, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.688, Rules used = {6116, 2513, 2409, 2389, 2295, 2394, 2393, 2391, 193, 321, 205} \[ \frac{\sqrt{d} \text{PolyLog}\left (2,\frac{\sqrt{-c} (-a-b x+1)}{(1-a) \sqrt{-c}-b \sqrt{d}}\right )}{4 (-c)^{3/2}}-\frac{\sqrt{d} \text{PolyLog}\left (2,\frac{\sqrt{-c} (-a-b x+1)}{(1-a) \sqrt{-c}+b \sqrt{d}}\right )}{4 (-c)^{3/2}}+\frac{\sqrt{d} \text{PolyLog}\left (2,\frac{\sqrt{-c} (a+b x+1)}{(a+1) \sqrt{-c}-b \sqrt{d}}\right )}{4 (-c)^{3/2}}-\frac{\sqrt{d} \text{PolyLog}\left (2,\frac{\sqrt{-c} (a+b x+1)}{(a+1) \sqrt{-c}+b \sqrt{d}}\right )}{4 (-c)^{3/2}}-\frac{\sqrt{d} \left (\log (a+b x-1)-\log \left (-\frac{-a-b x+1}{a+b x}\right )-\log (a+b x)\right ) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{d}}\right )}{2 c^{3/2}}-\frac{\sqrt{d} \left (\log (a+b x)-\log (a+b x+1)+\log \left (\frac{a+b x+1}{a+b x}\right )\right ) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{d}}\right )}{2 c^{3/2}}+\frac{\sqrt{d} \log (a+b x-1) \log \left (-\frac{b \left (\sqrt{d}-\sqrt{-c} x\right )}{(1-a) \sqrt{-c}-b \sqrt{d}}\right )}{4 (-c)^{3/2}}-\frac{\sqrt{d} \log (a+b x-1) \log \left (\frac{b \left (\sqrt{-c} x+\sqrt{d}\right )}{(1-a) \sqrt{-c}+b \sqrt{d}}\right )}{4 (-c)^{3/2}}-\frac{\sqrt{d} \log (a+b x+1) \log \left (\frac{b \left (\sqrt{d}-\sqrt{-c} x\right )}{(a+1) \sqrt{-c}+b \sqrt{d}}\right )}{4 (-c)^{3/2}}+\frac{\sqrt{d} \log (a+b x+1) \log \left (-\frac{b \left (\sqrt{-c} x+\sqrt{d}\right )}{(a+1) \sqrt{-c}-b \sqrt{d}}\right )}{4 (-c)^{3/2}}+\frac{(-a-b x+1) \log (a+b x-1)}{2 b c}+\frac{x \left (\log (a+b x-1)-\log \left (-\frac{-a-b x+1}{a+b x}\right )-\log (a+b x)\right )}{2 c}+\frac{(a+b x+1) \log (a+b x+1)}{2 b c}+\frac{x \left (\log (a+b x)-\log (a+b x+1)+\log \left (\frac{a+b x+1}{a+b x}\right )\right )}{2 c} \]

Antiderivative was successfully verified.

[In]

Int[ArcCoth[a + b*x]/(c + d/x^2),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) - (Sqrt[d]*ArcTan[(Sqrt[c]*x)/Sqrt[d]]*(Log[-1 + a + b*x] - Log[-((1 - a - b*x)/(a + b*x))] - Log
[a + b*x]))/(2*c^(3/2)) + ((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) - (Sqrt[d]*ArcTan[(Sqrt[c]*x)/Sqrt[d]]*(Log[a + b*x] - Log[1 + a + b*x] + Lo
g[(1 + a + b*x)/(a + b*x)]))/(2*c^(3/2)) + (Sqrt[d]*Log[-1 + a + b*x]*Log[-((b*(Sqrt[d] - Sqrt[-c]*x))/((1 - a
)*Sqrt[-c] - b*Sqrt[d]))])/(4*(-c)^(3/2)) - (Sqrt[d]*Log[1 + a + b*x]*Log[(b*(Sqrt[d] - Sqrt[-c]*x))/((1 + a)*
Sqrt[-c] + b*Sqrt[d])])/(4*(-c)^(3/2)) + (Sqrt[d]*Log[1 + a + b*x]*Log[-((b*(Sqrt[d] + Sqrt[-c]*x))/((1 + a)*S
qrt[-c] - b*Sqrt[d]))])/(4*(-c)^(3/2)) - (Sqrt[d]*Log[-1 + a + b*x]*Log[(b*(Sqrt[d] + Sqrt[-c]*x))/((1 - a)*Sq
rt[-c] + b*Sqrt[d])])/(4*(-c)^(3/2)) + (Sqrt[d]*PolyLog[2, (Sqrt[-c]*(1 - a - b*x))/((1 - a)*Sqrt[-c] - b*Sqrt
[d])])/(4*(-c)^(3/2)) - (Sqrt[d]*PolyLog[2, (Sqrt[-c]*(1 - a - b*x))/((1 - a)*Sqrt[-c] + b*Sqrt[d])])/(4*(-c)^
(3/2)) + (Sqrt[d]*PolyLog[2, (Sqrt[-c]*(1 + a + b*x))/((1 + a)*Sqrt[-c] - b*Sqrt[d])])/(4*(-c)^(3/2)) - (Sqrt[
d]*PolyLog[2, (Sqrt[-c]*(1 + a + b*x))/((1 + a)*Sqrt[-c] + b*Sqrt[d])])/(4*(-c)^(3/2))

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 321

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^n
)^(p + 1))/(b*(m + n*p + 1)), x] - Dist[(a*c^n*(m - n + 1))/(b*(m + n*p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rubi steps

\begin{align*} \int \frac{\coth ^{-1}(a+b x)}{c+\frac{d}{x^2}} \, dx &=-\left (\frac{1}{2} \int \frac{\log \left (\frac{-1+a+b x}{a+b x}\right )}{c+\frac{d}{x^2}} \, dx\right )+\frac{1}{2} \int \frac{\log \left (\frac{1+a+b x}{a+b x}\right )}{c+\frac{d}{x^2}} \, dx\\ &=-\left (\frac{1}{2} \int \frac{\log (-1+a+b x)}{c+\frac{d}{x^2}} \, dx\right )+\frac{1}{2} \int \frac{\log (1+a+b x)}{c+\frac{d}{x^2}} \, 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^2}} \, 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^2}} \, dx\\ &=-\left (\frac{1}{2} \int \left (\frac{\log (-1+a+b x)}{c}-\frac{d \log (-1+a+b x)}{c \left (d+c x^2\right )}\right ) \, dx\right )+\frac{1}{2} \int \left (\frac{\log (1+a+b x)}{c}-\frac{d \log (1+a+b x)}{c \left (d+c x^2\right )}\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^2}{d+c x^2} \, 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^2}{d+c x^2} \, 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{\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^2} \, dx}{2 c}-\frac{d \int \frac{\log (1+a+b x)}{d+c x^2} \, dx}{2 c}-\frac{\left (d \left (\log (-1+a+b x)-\log \left (\frac{-1+a+b x}{a+b x}\right )-\log (a+b x)\right )\right ) \int \frac{1}{d+c x^2} \, dx}{2 c}+\frac{\left (d \left (-\log (a+b x)+\log (1+a+b x)-\log \left (\frac{1+a+b x}{a+b x}\right )\right )\right ) \int \frac{1}{d+c x^2} \, dx}{2 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{\sqrt{d} \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{d}}\right ) \left (\log (-1+a+b x)-\log \left (-\frac{1-a-b x}{a+b x}\right )-\log (a+b x)\right )}{2 c^{3/2}}+\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{\sqrt{d} \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{d}}\right ) \left (\log (a+b x)-\log (1+a+b x)+\log \left (\frac{1+a+b x}{a+b x}\right )\right )}{2 c^{3/2}}-\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{d \int \left (\frac{\log (-1+a+b x)}{2 \sqrt{d} \left (\sqrt{d}-\sqrt{-c} x\right )}+\frac{\log (-1+a+b x)}{2 \sqrt{d} \left (\sqrt{d}+\sqrt{-c} x\right )}\right ) \, dx}{2 c}-\frac{d \int \left (\frac{\log (1+a+b x)}{2 \sqrt{d} \left (\sqrt{d}-\sqrt{-c} x\right )}+\frac{\log (1+a+b x)}{2 \sqrt{d} \left (\sqrt{d}+\sqrt{-c} x\right )}\right ) \, dx}{2 c}\\ &=\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{\sqrt{d} \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{d}}\right ) \left (\log (-1+a+b x)-\log \left (-\frac{1-a-b x}{a+b x}\right )-\log (a+b x)\right )}{2 c^{3/2}}+\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{\sqrt{d} \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{d}}\right ) \left (\log (a+b x)-\log (1+a+b x)+\log \left (\frac{1+a+b x}{a+b x}\right )\right )}{2 c^{3/2}}+\frac{\sqrt{d} \int \frac{\log (-1+a+b x)}{\sqrt{d}-\sqrt{-c} x} \, dx}{4 c}+\frac{\sqrt{d} \int \frac{\log (-1+a+b x)}{\sqrt{d}+\sqrt{-c} x} \, dx}{4 c}-\frac{\sqrt{d} \int \frac{\log (1+a+b x)}{\sqrt{d}-\sqrt{-c} x} \, dx}{4 c}-\frac{\sqrt{d} \int \frac{\log (1+a+b x)}{\sqrt{d}+\sqrt{-c} x} \, dx}{4 c}\\ &=\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{\sqrt{d} \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{d}}\right ) \left (\log (-1+a+b x)-\log \left (-\frac{1-a-b x}{a+b x}\right )-\log (a+b x)\right )}{2 c^{3/2}}+\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{\sqrt{d} \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{d}}\right ) \left (\log (a+b x)-\log (1+a+b x)+\log \left (\frac{1+a+b x}{a+b x}\right )\right )}{2 c^{3/2}}+\frac{\sqrt{d} \log (-1+a+b x) \log \left (-\frac{b \left (\sqrt{d}-\sqrt{-c} x\right )}{(1-a) \sqrt{-c}-b \sqrt{d}}\right )}{4 (-c)^{3/2}}-\frac{\sqrt{d} \log (1+a+b x) \log \left (\frac{b \left (\sqrt{d}-\sqrt{-c} x\right )}{(1+a) \sqrt{-c}+b \sqrt{d}}\right )}{4 (-c)^{3/2}}+\frac{\sqrt{d} \log (1+a+b x) \log \left (-\frac{b \left (\sqrt{d}+\sqrt{-c} x\right )}{(1+a) \sqrt{-c}-b \sqrt{d}}\right )}{4 (-c)^{3/2}}-\frac{\sqrt{d} \log (-1+a+b x) \log \left (\frac{b \left (\sqrt{d}+\sqrt{-c} x\right )}{(1-a) \sqrt{-c}+b \sqrt{d}}\right )}{4 (-c)^{3/2}}-\frac{\left (b \sqrt{d}\right ) \int \frac{\log \left (\frac{b \left (\sqrt{d}-\sqrt{-c} x\right )}{(-1+a) \sqrt{-c}+b \sqrt{d}}\right )}{-1+a+b x} \, dx}{4 (-c)^{3/2}}+\frac{\left (b \sqrt{d}\right ) \int \frac{\log \left (\frac{b \left (\sqrt{d}-\sqrt{-c} x\right )}{(1+a) \sqrt{-c}+b \sqrt{d}}\right )}{1+a+b x} \, dx}{4 (-c)^{3/2}}+\frac{\left (b \sqrt{d}\right ) \int \frac{\log \left (\frac{b \left (\sqrt{d}+\sqrt{-c} x\right )}{-(-1+a) \sqrt{-c}+b \sqrt{d}}\right )}{-1+a+b x} \, dx}{4 (-c)^{3/2}}-\frac{\left (b \sqrt{d}\right ) \int \frac{\log \left (\frac{b \left (\sqrt{d}+\sqrt{-c} x\right )}{-(1+a) \sqrt{-c}+b \sqrt{d}}\right )}{1+a+b x} \, dx}{4 (-c)^{3/2}}\\ &=\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{\sqrt{d} \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{d}}\right ) \left (\log (-1+a+b x)-\log \left (-\frac{1-a-b x}{a+b x}\right )-\log (a+b x)\right )}{2 c^{3/2}}+\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{\sqrt{d} \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{d}}\right ) \left (\log (a+b x)-\log (1+a+b x)+\log \left (\frac{1+a+b x}{a+b x}\right )\right )}{2 c^{3/2}}+\frac{\sqrt{d} \log (-1+a+b x) \log \left (-\frac{b \left (\sqrt{d}-\sqrt{-c} x\right )}{(1-a) \sqrt{-c}-b \sqrt{d}}\right )}{4 (-c)^{3/2}}-\frac{\sqrt{d} \log (1+a+b x) \log \left (\frac{b \left (\sqrt{d}-\sqrt{-c} x\right )}{(1+a) \sqrt{-c}+b \sqrt{d}}\right )}{4 (-c)^{3/2}}+\frac{\sqrt{d} \log (1+a+b x) \log \left (-\frac{b \left (\sqrt{d}+\sqrt{-c} x\right )}{(1+a) \sqrt{-c}-b \sqrt{d}}\right )}{4 (-c)^{3/2}}-\frac{\sqrt{d} \log (-1+a+b x) \log \left (\frac{b \left (\sqrt{d}+\sqrt{-c} x\right )}{(1-a) \sqrt{-c}+b \sqrt{d}}\right )}{4 (-c)^{3/2}}+\frac{\sqrt{d} \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{\sqrt{-c} x}{-(-1+a) \sqrt{-c}+b \sqrt{d}}\right )}{x} \, dx,x,-1+a+b x\right )}{4 (-c)^{3/2}}-\frac{\sqrt{d} \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{\sqrt{-c} x}{(-1+a) \sqrt{-c}+b \sqrt{d}}\right )}{x} \, dx,x,-1+a+b x\right )}{4 (-c)^{3/2}}-\frac{\sqrt{d} \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{\sqrt{-c} x}{-(1+a) \sqrt{-c}+b \sqrt{d}}\right )}{x} \, dx,x,1+a+b x\right )}{4 (-c)^{3/2}}+\frac{\sqrt{d} \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{\sqrt{-c} x}{(1+a) \sqrt{-c}+b \sqrt{d}}\right )}{x} \, dx,x,1+a+b x\right )}{4 (-c)^{3/2}}\\ &=\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{\sqrt{d} \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{d}}\right ) \left (\log (-1+a+b x)-\log \left (-\frac{1-a-b x}{a+b x}\right )-\log (a+b x)\right )}{2 c^{3/2}}+\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{\sqrt{d} \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{d}}\right ) \left (\log (a+b x)-\log (1+a+b x)+\log \left (\frac{1+a+b x}{a+b x}\right )\right )}{2 c^{3/2}}+\frac{\sqrt{d} \log (-1+a+b x) \log \left (-\frac{b \left (\sqrt{d}-\sqrt{-c} x\right )}{(1-a) \sqrt{-c}-b \sqrt{d}}\right )}{4 (-c)^{3/2}}-\frac{\sqrt{d} \log (1+a+b x) \log \left (\frac{b \left (\sqrt{d}-\sqrt{-c} x\right )}{(1+a) \sqrt{-c}+b \sqrt{d}}\right )}{4 (-c)^{3/2}}+\frac{\sqrt{d} \log (1+a+b x) \log \left (-\frac{b \left (\sqrt{d}+\sqrt{-c} x\right )}{(1+a) \sqrt{-c}-b \sqrt{d}}\right )}{4 (-c)^{3/2}}-\frac{\sqrt{d} \log (-1+a+b x) \log \left (\frac{b \left (\sqrt{d}+\sqrt{-c} x\right )}{(1-a) \sqrt{-c}+b \sqrt{d}}\right )}{4 (-c)^{3/2}}+\frac{\sqrt{d} \text{Li}_2\left (\frac{\sqrt{-c} (1-a-b x)}{(1-a) \sqrt{-c}-b \sqrt{d}}\right )}{4 (-c)^{3/2}}-\frac{\sqrt{d} \text{Li}_2\left (\frac{\sqrt{-c} (1-a-b x)}{(1-a) \sqrt{-c}+b \sqrt{d}}\right )}{4 (-c)^{3/2}}+\frac{\sqrt{d} \text{Li}_2\left (\frac{\sqrt{-c} (1+a+b x)}{(1+a) \sqrt{-c}-b \sqrt{d}}\right )}{4 (-c)^{3/2}}-\frac{\sqrt{d} \text{Li}_2\left (\frac{\sqrt{-c} (1+a+b x)}{(1+a) \sqrt{-c}+b \sqrt{d}}\right )}{4 (-c)^{3/2}}\\ \end{align*}

Mathematica [C]  time = 35.6651, size = 5552, normalized size = 7.52 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[ArcCoth[a + b*x]/(c + d/x^2),x]

[Out]

Result too large to show

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Maple [C]  time = 1.251, size = 19686, normalized size = 26.7 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arccoth(b*x+a)/(c+d/x^2),x)

[Out]

result too large to display

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccoth(b*x+a)/(c+d/x^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{x^{2} \operatorname{arcoth}\left (b x + a\right )}{c x^{2} + d}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccoth(b*x+a)/(c+d/x^2),x, algorithm="fricas")

[Out]

integral(x^2*arccoth(b*x + a)/(c*x^2 + 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**2),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^{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccoth(b*x+a)/(c+d/x^2),x, algorithm="giac")

[Out]

integrate(arccoth(b*x + a)/(c + d/x^2), x)