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

Optimal. Leaf size=590 \[ \frac{i \text{PolyLog}\left (2,\frac{a \left (\sqrt{c}-i \sqrt{d} x\right )}{a \sqrt{c}-i \sqrt{d}}\right )}{8 c^{3/2} \sqrt{d}}-\frac{i \text{PolyLog}\left (2,\frac{a \left (\sqrt{c}-i \sqrt{d} x\right )}{a \sqrt{c}+i \sqrt{d}}\right )}{8 c^{3/2} \sqrt{d}}+\frac{i \text{PolyLog}\left (2,\frac{a \left (\sqrt{c}+i \sqrt{d} x\right )}{a \sqrt{c}-i \sqrt{d}}\right )}{8 c^{3/2} \sqrt{d}}-\frac{i \text{PolyLog}\left (2,\frac{a \left (\sqrt{c}+i \sqrt{d} x\right )}{a \sqrt{c}+i \sqrt{d}}\right )}{8 c^{3/2} \sqrt{d}}+\frac{a \log \left (1-a^2 x^2\right )}{4 c \left (a^2 c+d\right )}-\frac{a \log \left (c+d x^2\right )}{4 c \left (a^2 c+d\right )}+\frac{i \log \left (1-\frac{i \sqrt{d} x}{\sqrt{c}}\right ) \log \left (\frac{\sqrt{d} (1-a x)}{\sqrt{d}+i a \sqrt{c}}\right )}{8 c^{3/2} \sqrt{d}}-\frac{i \log \left (1-\frac{i \sqrt{d} x}{\sqrt{c}}\right ) \log \left (-\frac{\sqrt{d} (a x+1)}{-\sqrt{d}+i a \sqrt{c}}\right )}{8 c^{3/2} \sqrt{d}}-\frac{i \log \left (1+\frac{i \sqrt{d} x}{\sqrt{c}}\right ) \log \left (-\frac{\sqrt{d} (1-a x)}{-\sqrt{d}+i a \sqrt{c}}\right )}{8 c^{3/2} \sqrt{d}}+\frac{i \log \left (1+\frac{i \sqrt{d} x}{\sqrt{c}}\right ) \log \left (\frac{\sqrt{d} (a x+1)}{\sqrt{d}+i a \sqrt{c}}\right )}{8 c^{3/2} \sqrt{d}}+\frac{\coth ^{-1}(a x) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{2 c^{3/2} \sqrt{d}}+\frac{x \coth ^{-1}(a x)}{2 c \left (c+d x^2\right )} \]

[Out]

(x*ArcCoth[a*x])/(2*c*(c + d*x^2)) + (ArcCoth[a*x]*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(2*c^(3/2)*Sqrt[d]) + ((I/8)*L
og[(Sqrt[d]*(1 - a*x))/(I*a*Sqrt[c] + Sqrt[d])]*Log[1 - (I*Sqrt[d]*x)/Sqrt[c]])/(c^(3/2)*Sqrt[d]) - ((I/8)*Log
[-((Sqrt[d]*(1 + a*x))/(I*a*Sqrt[c] - Sqrt[d]))]*Log[1 - (I*Sqrt[d]*x)/Sqrt[c]])/(c^(3/2)*Sqrt[d]) - ((I/8)*Lo
g[-((Sqrt[d]*(1 - a*x))/(I*a*Sqrt[c] - Sqrt[d]))]*Log[1 + (I*Sqrt[d]*x)/Sqrt[c]])/(c^(3/2)*Sqrt[d]) + ((I/8)*L
og[(Sqrt[d]*(1 + a*x))/(I*a*Sqrt[c] + Sqrt[d])]*Log[1 + (I*Sqrt[d]*x)/Sqrt[c]])/(c^(3/2)*Sqrt[d]) + (a*Log[1 -
a^2*x^2])/(4*c*(a^2*c + d)) - (a*Log[c + d*x^2])/(4*c*(a^2*c + d)) + ((I/8)*PolyLog[2, (a*(Sqrt[c] - I*Sqrt[d
]*x))/(a*Sqrt[c] - I*Sqrt[d])])/(c^(3/2)*Sqrt[d]) - ((I/8)*PolyLog[2, (a*(Sqrt[c] - I*Sqrt[d]*x))/(a*Sqrt[c] +
I*Sqrt[d])])/(c^(3/2)*Sqrt[d]) + ((I/8)*PolyLog[2, (a*(Sqrt[c] + I*Sqrt[d]*x))/(a*Sqrt[c] - I*Sqrt[d])])/(c^(
3/2)*Sqrt[d]) - ((I/8)*PolyLog[2, (a*(Sqrt[c] + I*Sqrt[d]*x))/(a*Sqrt[c] + I*Sqrt[d])])/(c^(3/2)*Sqrt[d])

________________________________________________________________________________________

Rubi [A]  time = 0.869703, antiderivative size = 590, normalized size of antiderivative = 1., number of steps used = 25, number of rules used = 13, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.929, Rules used = {199, 205, 5977, 6725, 517, 444, 36, 31, 4908, 2409, 2394, 2393, 2391} \[ \frac{i \text{PolyLog}\left (2,\frac{a \left (\sqrt{c}-i \sqrt{d} x\right )}{a \sqrt{c}-i \sqrt{d}}\right )}{8 c^{3/2} \sqrt{d}}-\frac{i \text{PolyLog}\left (2,\frac{a \left (\sqrt{c}-i \sqrt{d} x\right )}{a \sqrt{c}+i \sqrt{d}}\right )}{8 c^{3/2} \sqrt{d}}+\frac{i \text{PolyLog}\left (2,\frac{a \left (\sqrt{c}+i \sqrt{d} x\right )}{a \sqrt{c}-i \sqrt{d}}\right )}{8 c^{3/2} \sqrt{d}}-\frac{i \text{PolyLog}\left (2,\frac{a \left (\sqrt{c}+i \sqrt{d} x\right )}{a \sqrt{c}+i \sqrt{d}}\right )}{8 c^{3/2} \sqrt{d}}+\frac{a \log \left (1-a^2 x^2\right )}{4 c \left (a^2 c+d\right )}-\frac{a \log \left (c+d x^2\right )}{4 c \left (a^2 c+d\right )}+\frac{i \log \left (1-\frac{i \sqrt{d} x}{\sqrt{c}}\right ) \log \left (\frac{\sqrt{d} (1-a x)}{\sqrt{d}+i a \sqrt{c}}\right )}{8 c^{3/2} \sqrt{d}}-\frac{i \log \left (1-\frac{i \sqrt{d} x}{\sqrt{c}}\right ) \log \left (-\frac{\sqrt{d} (a x+1)}{-\sqrt{d}+i a \sqrt{c}}\right )}{8 c^{3/2} \sqrt{d}}-\frac{i \log \left (1+\frac{i \sqrt{d} x}{\sqrt{c}}\right ) \log \left (-\frac{\sqrt{d} (1-a x)}{-\sqrt{d}+i a \sqrt{c}}\right )}{8 c^{3/2} \sqrt{d}}+\frac{i \log \left (1+\frac{i \sqrt{d} x}{\sqrt{c}}\right ) \log \left (\frac{\sqrt{d} (a x+1)}{\sqrt{d}+i a \sqrt{c}}\right )}{8 c^{3/2} \sqrt{d}}+\frac{\coth ^{-1}(a x) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{2 c^{3/2} \sqrt{d}}+\frac{x \coth ^{-1}(a x)}{2 c \left (c+d x^2\right )} \]

Antiderivative was successfully verified.

[In]

Int[ArcCoth[a*x]/(c + d*x^2)^2,x]

[Out]

(x*ArcCoth[a*x])/(2*c*(c + d*x^2)) + (ArcCoth[a*x]*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(2*c^(3/2)*Sqrt[d]) + ((I/8)*L
og[(Sqrt[d]*(1 - a*x))/(I*a*Sqrt[c] + Sqrt[d])]*Log[1 - (I*Sqrt[d]*x)/Sqrt[c]])/(c^(3/2)*Sqrt[d]) - ((I/8)*Log
[-((Sqrt[d]*(1 + a*x))/(I*a*Sqrt[c] - Sqrt[d]))]*Log[1 - (I*Sqrt[d]*x)/Sqrt[c]])/(c^(3/2)*Sqrt[d]) - ((I/8)*Lo
g[-((Sqrt[d]*(1 - a*x))/(I*a*Sqrt[c] - Sqrt[d]))]*Log[1 + (I*Sqrt[d]*x)/Sqrt[c]])/(c^(3/2)*Sqrt[d]) + ((I/8)*L
og[(Sqrt[d]*(1 + a*x))/(I*a*Sqrt[c] + Sqrt[d])]*Log[1 + (I*Sqrt[d]*x)/Sqrt[c]])/(c^(3/2)*Sqrt[d]) + (a*Log[1 -
a^2*x^2])/(4*c*(a^2*c + d)) - (a*Log[c + d*x^2])/(4*c*(a^2*c + d)) + ((I/8)*PolyLog[2, (a*(Sqrt[c] - I*Sqrt[d
]*x))/(a*Sqrt[c] - I*Sqrt[d])])/(c^(3/2)*Sqrt[d]) - ((I/8)*PolyLog[2, (a*(Sqrt[c] - I*Sqrt[d]*x))/(a*Sqrt[c] +
I*Sqrt[d])])/(c^(3/2)*Sqrt[d]) + ((I/8)*PolyLog[2, (a*(Sqrt[c] + I*Sqrt[d]*x))/(a*Sqrt[c] - I*Sqrt[d])])/(c^(
3/2)*Sqrt[d]) - ((I/8)*PolyLog[2, (a*(Sqrt[c] + I*Sqrt[d]*x))/(a*Sqrt[c] + I*Sqrt[d])])/(c^(3/2)*Sqrt[d])

Rule 199

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Simp[(x*(a + b*x^n)^(p + 1))/(a*n*(p + 1)), x] + Dist[(n*(p +
1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[p, -1] && (In
tegerQ[2*p] || (n == 2 && IntegerQ[4*p]) || (n == 2 && IntegerQ[3*p]) || Denominator[p + 1/n] < Denominator[p]
)

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]

Rule 5977

Int[((a_.) + ArcCoth[(c_.)*(x_)]*(b_.))*((d_.) + (e_.)*(x_)^2)^(q_.), x_Symbol] :> With[{u = IntHide[(d + e*x^
2)^q, x]}, Dist[a + b*ArcCoth[c*x], u, x] - Dist[b*c, Int[u/(1 - c^2*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x
] && (IntegerQ[q] || ILtQ[q + 1/2, 0])

Rule 6725

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a + b*x^n), x]}, Int[v, x]
/; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ[n, 0]

Rule 517

Int[(u_.)*((c_) + (d_.)*(x_)^(n_.))^(q_.)*((a1_) + (b1_.)*(x_)^(non2_.))^(p_.)*((a2_) + (b2_.)*(x_)^(non2_.))^
(p_.), x_Symbol] :> Int[u*(a1*a2 + b1*b2*x^n)^p*(c + d*x^n)^q, x] /; FreeQ[{a1, b1, a2, b2, c, d, n, p, q}, x]
&& EqQ[non2, n/2] && EqQ[a2*b1 + a1*b2, 0] && (IntegerQ[p] || (GtQ[a1, 0] && GtQ[a2, 0]))

Rule 444

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && EqQ[m
- n + 1, 0]

Rule 36

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Dist[b/(b*c - a*d), Int[1/(a + b*x), x], x] -
Dist[d/(b*c - a*d), Int[1/(c + d*x), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 4908

Int[ArcTan[(c_.)*(x_)]/((d_.) + (e_.)*(x_)^2), x_Symbol] :> Dist[I/2, Int[Log[1 - I*c*x]/(d + e*x^2), x], x] -
Dist[I/2, Int[Log[1 + I*c*x]/(d + e*x^2), x], x] /; FreeQ[{c, d, e}, x]

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 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]

Rubi steps

\begin{align*} \int \frac{\coth ^{-1}(a x)}{\left (c+d x^2\right )^2} \, dx &=\frac{x \coth ^{-1}(a x)}{2 c \left (c+d x^2\right )}+\frac{\coth ^{-1}(a x) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{2 c^{3/2} \sqrt{d}}-a \int \frac{\frac{x}{2 c \left (c+d x^2\right )}+\frac{\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{2 c^{3/2} \sqrt{d}}}{1-a^2 x^2} \, dx\\ &=\frac{x \coth ^{-1}(a x)}{2 c \left (c+d x^2\right )}+\frac{\coth ^{-1}(a x) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{2 c^{3/2} \sqrt{d}}-a \int \left (-\frac{x}{2 c (-1+a x) (1+a x) \left (c+d x^2\right )}-\frac{\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{2 c^{3/2} \sqrt{d} \left (-1+a^2 x^2\right )}\right ) \, dx\\ &=\frac{x \coth ^{-1}(a x)}{2 c \left (c+d x^2\right )}+\frac{\coth ^{-1}(a x) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{2 c^{3/2} \sqrt{d}}+\frac{a \int \frac{x}{(-1+a x) (1+a x) \left (c+d x^2\right )} \, dx}{2 c}+\frac{a \int \frac{\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{-1+a^2 x^2} \, dx}{2 c^{3/2} \sqrt{d}}\\ &=\frac{x \coth ^{-1}(a x)}{2 c \left (c+d x^2\right )}+\frac{\coth ^{-1}(a x) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{2 c^{3/2} \sqrt{d}}+\frac{a \int \frac{x}{\left (-1+a^2 x^2\right ) \left (c+d x^2\right )} \, dx}{2 c}+\frac{(i a) \int \frac{\log \left (1-\frac{i \sqrt{d} x}{\sqrt{c}}\right )}{-1+a^2 x^2} \, dx}{4 c^{3/2} \sqrt{d}}-\frac{(i a) \int \frac{\log \left (1+\frac{i \sqrt{d} x}{\sqrt{c}}\right )}{-1+a^2 x^2} \, dx}{4 c^{3/2} \sqrt{d}}\\ &=\frac{x \coth ^{-1}(a x)}{2 c \left (c+d x^2\right )}+\frac{\coth ^{-1}(a x) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{2 c^{3/2} \sqrt{d}}+\frac{a \operatorname{Subst}\left (\int \frac{1}{\left (-1+a^2 x\right ) (c+d x)} \, dx,x,x^2\right )}{4 c}+\frac{(i a) \int \left (-\frac{\log \left (1-\frac{i \sqrt{d} x}{\sqrt{c}}\right )}{2 (1-a x)}-\frac{\log \left (1-\frac{i \sqrt{d} x}{\sqrt{c}}\right )}{2 (1+a x)}\right ) \, dx}{4 c^{3/2} \sqrt{d}}-\frac{(i a) \int \left (-\frac{\log \left (1+\frac{i \sqrt{d} x}{\sqrt{c}}\right )}{2 (1-a x)}-\frac{\log \left (1+\frac{i \sqrt{d} x}{\sqrt{c}}\right )}{2 (1+a x)}\right ) \, dx}{4 c^{3/2} \sqrt{d}}\\ &=\frac{x \coth ^{-1}(a x)}{2 c \left (c+d x^2\right )}+\frac{\coth ^{-1}(a x) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{2 c^{3/2} \sqrt{d}}-\frac{(i a) \int \frac{\log \left (1-\frac{i \sqrt{d} x}{\sqrt{c}}\right )}{1-a x} \, dx}{8 c^{3/2} \sqrt{d}}-\frac{(i a) \int \frac{\log \left (1-\frac{i \sqrt{d} x}{\sqrt{c}}\right )}{1+a x} \, dx}{8 c^{3/2} \sqrt{d}}+\frac{(i a) \int \frac{\log \left (1+\frac{i \sqrt{d} x}{\sqrt{c}}\right )}{1-a x} \, dx}{8 c^{3/2} \sqrt{d}}+\frac{(i a) \int \frac{\log \left (1+\frac{i \sqrt{d} x}{\sqrt{c}}\right )}{1+a x} \, dx}{8 c^{3/2} \sqrt{d}}+\frac{a^3 \operatorname{Subst}\left (\int \frac{1}{-1+a^2 x} \, dx,x,x^2\right )}{4 c \left (a^2 c+d\right )}-\frac{(a d) \operatorname{Subst}\left (\int \frac{1}{c+d x} \, dx,x,x^2\right )}{4 c \left (a^2 c+d\right )}\\ &=\frac{x \coth ^{-1}(a x)}{2 c \left (c+d x^2\right )}+\frac{\coth ^{-1}(a x) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{2 c^{3/2} \sqrt{d}}+\frac{i \log \left (\frac{\sqrt{d} (1-a x)}{i a \sqrt{c}+\sqrt{d}}\right ) \log \left (1-\frac{i \sqrt{d} x}{\sqrt{c}}\right )}{8 c^{3/2} \sqrt{d}}-\frac{i \log \left (-\frac{\sqrt{d} (1+a x)}{i a \sqrt{c}-\sqrt{d}}\right ) \log \left (1-\frac{i \sqrt{d} x}{\sqrt{c}}\right )}{8 c^{3/2} \sqrt{d}}-\frac{i \log \left (-\frac{\sqrt{d} (1-a x)}{i a \sqrt{c}-\sqrt{d}}\right ) \log \left (1+\frac{i \sqrt{d} x}{\sqrt{c}}\right )}{8 c^{3/2} \sqrt{d}}+\frac{i \log \left (\frac{\sqrt{d} (1+a x)}{i a \sqrt{c}+\sqrt{d}}\right ) \log \left (1+\frac{i \sqrt{d} x}{\sqrt{c}}\right )}{8 c^{3/2} \sqrt{d}}+\frac{a \log \left (1-a^2 x^2\right )}{4 c \left (a^2 c+d\right )}-\frac{a \log \left (c+d x^2\right )}{4 c \left (a^2 c+d\right )}-\frac{\int \frac{\log \left (-\frac{i \sqrt{d} (1-a x)}{\sqrt{c} \left (a-\frac{i \sqrt{d}}{\sqrt{c}}\right )}\right )}{1-\frac{i \sqrt{d} x}{\sqrt{c}}} \, dx}{8 c^2}-\frac{\int \frac{\log \left (\frac{i \sqrt{d} (1-a x)}{\sqrt{c} \left (a+\frac{i \sqrt{d}}{\sqrt{c}}\right )}\right )}{1+\frac{i \sqrt{d} x}{\sqrt{c}}} \, dx}{8 c^2}+\frac{\int \frac{\log \left (-\frac{i \sqrt{d} (1+a x)}{\sqrt{c} \left (-a-\frac{i \sqrt{d}}{\sqrt{c}}\right )}\right )}{1-\frac{i \sqrt{d} x}{\sqrt{c}}} \, dx}{8 c^2}+\frac{\int \frac{\log \left (\frac{i \sqrt{d} (1+a x)}{\sqrt{c} \left (-a+\frac{i \sqrt{d}}{\sqrt{c}}\right )}\right )}{1+\frac{i \sqrt{d} x}{\sqrt{c}}} \, dx}{8 c^2}\\ &=\frac{x \coth ^{-1}(a x)}{2 c \left (c+d x^2\right )}+\frac{\coth ^{-1}(a x) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{2 c^{3/2} \sqrt{d}}+\frac{i \log \left (\frac{\sqrt{d} (1-a x)}{i a \sqrt{c}+\sqrt{d}}\right ) \log \left (1-\frac{i \sqrt{d} x}{\sqrt{c}}\right )}{8 c^{3/2} \sqrt{d}}-\frac{i \log \left (-\frac{\sqrt{d} (1+a x)}{i a \sqrt{c}-\sqrt{d}}\right ) \log \left (1-\frac{i \sqrt{d} x}{\sqrt{c}}\right )}{8 c^{3/2} \sqrt{d}}-\frac{i \log \left (-\frac{\sqrt{d} (1-a x)}{i a \sqrt{c}-\sqrt{d}}\right ) \log \left (1+\frac{i \sqrt{d} x}{\sqrt{c}}\right )}{8 c^{3/2} \sqrt{d}}+\frac{i \log \left (\frac{\sqrt{d} (1+a x)}{i a \sqrt{c}+\sqrt{d}}\right ) \log \left (1+\frac{i \sqrt{d} x}{\sqrt{c}}\right )}{8 c^{3/2} \sqrt{d}}+\frac{a \log \left (1-a^2 x^2\right )}{4 c \left (a^2 c+d\right )}-\frac{a \log \left (c+d x^2\right )}{4 c \left (a^2 c+d\right )}+\frac{i \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{a x}{-a-\frac{i \sqrt{d}}{\sqrt{c}}}\right )}{x} \, dx,x,1-\frac{i \sqrt{d} x}{\sqrt{c}}\right )}{8 c^{3/2} \sqrt{d}}-\frac{i \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{a x}{a-\frac{i \sqrt{d}}{\sqrt{c}}}\right )}{x} \, dx,x,1-\frac{i \sqrt{d} x}{\sqrt{c}}\right )}{8 c^{3/2} \sqrt{d}}-\frac{i \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{a x}{-a+\frac{i \sqrt{d}}{\sqrt{c}}}\right )}{x} \, dx,x,1+\frac{i \sqrt{d} x}{\sqrt{c}}\right )}{8 c^{3/2} \sqrt{d}}+\frac{i \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{a x}{a+\frac{i \sqrt{d}}{\sqrt{c}}}\right )}{x} \, dx,x,1+\frac{i \sqrt{d} x}{\sqrt{c}}\right )}{8 c^{3/2} \sqrt{d}}\\ &=\frac{x \coth ^{-1}(a x)}{2 c \left (c+d x^2\right )}+\frac{\coth ^{-1}(a x) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{2 c^{3/2} \sqrt{d}}+\frac{i \log \left (\frac{\sqrt{d} (1-a x)}{i a \sqrt{c}+\sqrt{d}}\right ) \log \left (1-\frac{i \sqrt{d} x}{\sqrt{c}}\right )}{8 c^{3/2} \sqrt{d}}-\frac{i \log \left (-\frac{\sqrt{d} (1+a x)}{i a \sqrt{c}-\sqrt{d}}\right ) \log \left (1-\frac{i \sqrt{d} x}{\sqrt{c}}\right )}{8 c^{3/2} \sqrt{d}}-\frac{i \log \left (-\frac{\sqrt{d} (1-a x)}{i a \sqrt{c}-\sqrt{d}}\right ) \log \left (1+\frac{i \sqrt{d} x}{\sqrt{c}}\right )}{8 c^{3/2} \sqrt{d}}+\frac{i \log \left (\frac{\sqrt{d} (1+a x)}{i a \sqrt{c}+\sqrt{d}}\right ) \log \left (1+\frac{i \sqrt{d} x}{\sqrt{c}}\right )}{8 c^{3/2} \sqrt{d}}+\frac{a \log \left (1-a^2 x^2\right )}{4 c \left (a^2 c+d\right )}-\frac{a \log \left (c+d x^2\right )}{4 c \left (a^2 c+d\right )}+\frac{i \text{Li}_2\left (\frac{a \left (\sqrt{c}-i \sqrt{d} x\right )}{a \sqrt{c}-i \sqrt{d}}\right )}{8 c^{3/2} \sqrt{d}}-\frac{i \text{Li}_2\left (\frac{a \left (\sqrt{c}-i \sqrt{d} x\right )}{a \sqrt{c}+i \sqrt{d}}\right )}{8 c^{3/2} \sqrt{d}}+\frac{i \text{Li}_2\left (\frac{a \left (\sqrt{c}+i \sqrt{d} x\right )}{a \sqrt{c}-i \sqrt{d}}\right )}{8 c^{3/2} \sqrt{d}}-\frac{i \text{Li}_2\left (\frac{a \left (\sqrt{c}+i \sqrt{d} x\right )}{a \sqrt{c}+i \sqrt{d}}\right )}{8 c^{3/2} \sqrt{d}}\\ \end{align*}

Mathematica [A]  time = 7.81475, size = 753, normalized size = 1.28 \[ -\frac{a \left (\frac{i \left (\text{PolyLog}\left (2,\frac{\left (-2 i \sqrt{a^2 c d}+a^2 c-d\right ) \left (\sqrt{a^2 c d}+i a d x\right )}{\left (a^2 c+d\right ) \left (\sqrt{a^2 c d}-i a d x\right )}\right )-\text{PolyLog}\left (2,\frac{\left (2 i \sqrt{a^2 c d}+a^2 c-d\right ) \left (\sqrt{a^2 c d}+i a d x\right )}{\left (a^2 c+d\right ) \left (\sqrt{a^2 c d}-i a d x\right )}\right )\right )+2 i \cos ^{-1}\left (\frac{a^2 c-d}{a^2 c+d}\right ) \tan ^{-1}\left (\frac{a c}{x \sqrt{a^2 c d}}\right )-4 \coth ^{-1}(a x) \tan ^{-1}\left (\frac{a d x}{\sqrt{a^2 c d}}\right )+\log \left (\frac{2 d (a x-1) \left (a^2 c-i \sqrt{a^2 c d}\right )}{\left (a^2 c+d\right ) \left (a d x+i \sqrt{a^2 c d}\right )}\right ) \left (2 \tan ^{-1}\left (\frac{a c}{x \sqrt{a^2 c d}}\right )+\cos ^{-1}\left (\frac{a^2 c-d}{a^2 c+d}\right )\right )+\log \left (\frac{2 d (a x+1) \left (a^2 c+i \sqrt{a^2 c d}\right )}{\left (a^2 c+d\right ) \left (a d x+i \sqrt{a^2 c d}\right )}\right ) \left (\cos ^{-1}\left (\frac{a^2 c-d}{a^2 c+d}\right )-2 \tan ^{-1}\left (\frac{a c}{x \sqrt{a^2 c d}}\right )\right )-\left (2 \left (\tan ^{-1}\left (\frac{a c}{x \sqrt{a^2 c d}}\right )+\tan ^{-1}\left (\frac{a d x}{\sqrt{a^2 c d}}\right )\right )+\cos ^{-1}\left (\frac{a^2 c-d}{a^2 c+d}\right )\right ) \log \left (\frac{\sqrt{2} \sqrt{a^2 c d} e^{-\coth ^{-1}(a x)}}{\sqrt{a^2 c+d} \sqrt{\left (a^2 c+d\right ) \cosh \left (2 \coth ^{-1}(a x)\right )+a^2 (-c)+d}}\right )-\left (\cos ^{-1}\left (\frac{a^2 c-d}{a^2 c+d}\right )-2 \left (\tan ^{-1}\left (\frac{a c}{x \sqrt{a^2 c d}}\right )+\tan ^{-1}\left (\frac{a d x}{\sqrt{a^2 c d}}\right )\right )\right ) \log \left (\frac{\sqrt{2} \sqrt{a^2 c d} e^{\coth ^{-1}(a x)}}{\sqrt{a^2 c+d} \sqrt{\left (a^2 c+d\right ) \cosh \left (2 \coth ^{-1}(a x)\right )+a^2 (-c)+d}}\right )}{\sqrt{a^2 c d}}+\frac{2 \log \left (\frac{\left (a^2 c+d\right ) \cosh \left (2 \coth ^{-1}(a x)\right )}{d-a^2 c}+1\right )}{a^2 c+d}-\frac{4 \coth ^{-1}(a x) \sinh \left (2 \coth ^{-1}(a x)\right )}{\left (a^2 c+d\right ) \cosh \left (2 \coth ^{-1}(a x)\right )+a^2 (-c)+d}\right )}{8 c} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-(a*((2*Log[1 + ((a^2*c + d)*Cosh[2*ArcCoth[a*x]])/(-(a^2*c) + d)])/(a^2*c + d) + ((2*I)*ArcCos[(a^2*c - d)/(a
^2*c + d)]*ArcTan[(a*c)/(Sqrt[a^2*c*d]*x)] - 4*ArcCoth[a*x]*ArcTan[(a*d*x)/Sqrt[a^2*c*d]] + (ArcCos[(a^2*c - d
)/(a^2*c + d)] + 2*ArcTan[(a*c)/(Sqrt[a^2*c*d]*x)])*Log[(2*d*(a^2*c - I*Sqrt[a^2*c*d])*(-1 + a*x))/((a^2*c + d
)*(I*Sqrt[a^2*c*d] + a*d*x))] + (ArcCos[(a^2*c - d)/(a^2*c + d)] - 2*ArcTan[(a*c)/(Sqrt[a^2*c*d]*x)])*Log[(2*d
*(a^2*c + I*Sqrt[a^2*c*d])*(1 + a*x))/((a^2*c + d)*(I*Sqrt[a^2*c*d] + a*d*x))] - (ArcCos[(a^2*c - d)/(a^2*c +
d)] + 2*(ArcTan[(a*c)/(Sqrt[a^2*c*d]*x)] + ArcTan[(a*d*x)/Sqrt[a^2*c*d]]))*Log[(Sqrt[2]*Sqrt[a^2*c*d])/(Sqrt[a
^2*c + d]*E^ArcCoth[a*x]*Sqrt[-(a^2*c) + d + (a^2*c + d)*Cosh[2*ArcCoth[a*x]]])] - (ArcCos[(a^2*c - d)/(a^2*c
+ d)] - 2*(ArcTan[(a*c)/(Sqrt[a^2*c*d]*x)] + ArcTan[(a*d*x)/Sqrt[a^2*c*d]]))*Log[(Sqrt[2]*Sqrt[a^2*c*d]*E^ArcC
oth[a*x])/(Sqrt[a^2*c + d]*Sqrt[-(a^2*c) + d + (a^2*c + d)*Cosh[2*ArcCoth[a*x]]])] + I*(PolyLog[2, ((a^2*c - d
- (2*I)*Sqrt[a^2*c*d])*(Sqrt[a^2*c*d] + I*a*d*x))/((a^2*c + d)*(Sqrt[a^2*c*d] - I*a*d*x))] - PolyLog[2, ((a^2
*c - d + (2*I)*Sqrt[a^2*c*d])*(Sqrt[a^2*c*d] + I*a*d*x))/((a^2*c + d)*(Sqrt[a^2*c*d] - I*a*d*x))]))/Sqrt[a^2*c
*d] - (4*ArcCoth[a*x]*Sinh[2*ArcCoth[a*x]])/(-(a^2*c) + d + (a^2*c + d)*Cosh[2*ArcCoth[a*x]])))/(8*c)

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Maple [B]  time = 0.41, size = 2218, normalized size = 3.8 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*a^3/(a^2*c+d)^2*ln(a^2*c/(a*x-1)^2*(a*x+1)^2-2*a^2*c*(a*x+1)/(a*x-1)+d/(a*x-1)^2*(a*x+1)^2+a^2*c+2*(a*x+1
)/(a*x-1)*d+d)-1/2*a^3/(a^2*c+d)^2*ln((a*x-1)/(a*x+1))-1/4*a^4*(c*d)^(1/2)/d/(a^2*c+d)^2*arctan(a/d*(c*d)^(1/2
))+1/4*(c*d)^(1/2)/c^2*d/(a^2*c+d)^2*arctan(a/d*(c*d)^(1/2))-3/4*a*d*ln(1-(a^2*c+d)*(a*x+1)/(a*x-1)/(a^2*c-2*(
-a^2*c*d)^(1/2)-d))*arccoth(a*x)/(a^2*c+d)/c/(a^4*c^2+2*a^2*c*d+d^2)*(-a^2*c*d)^(1/2)-1/4*a^5/d/(a^2*c+d)/(a^4
*c^2+2*a^2*c*d+d^2)*ln(1-(a^2*c+d)*(a*x+1)/(a*x-1)/(a^2*c-2*(-a^2*c*d)^(1/2)-d))*arccoth(a*x)*(-a^2*c*d)^(1/2)
*c-1/4/a/c^2*d^2/(a^2*c+d)/(a^4*c^2+2*a^2*c*d+d^2)*ln(1-(a^2*c+d)*(a*x+1)/(a*x-1)/(a^2*c-2*(-a^2*c*d)^(1/2)-d)
)*arccoth(a*x)*(-a^2*c*d)^(1/2)+1/4*a^2*(c*d)^(1/2)/c/d/(a^2*c+d)*arctan(a/d*(c*d)^(1/2))-1/4*a*(-a^2*c*d)^(1/
2)/c/d/(a^2*c+d)*arccoth(a*x)^2+1/8*a*(-a^2*c*d)^(1/2)/c/d/(a^2*c+d)*polylog(2,(a^2*c+d)*(a*x+1)/(a*x-1)/(a^2*
c+2*(-a^2*c*d)^(1/2)-d))+1/4/a*(-a^2*c*d)^(1/2)/c^2/(a^2*c+d)*arccoth(a*x)*ln(1-(a^2*c+d)*(a*x+1)/(a*x-1)/(a^2
*c+2*(-a^2*c*d)^(1/2)-d))-3/4*a^3/(a^2*c+d)/(a^4*c^2+2*a^2*c*d+d^2)*ln(1-(a^2*c+d)*(a*x+1)/(a*x-1)/(a^2*c-2*(-
a^2*c*d)^(1/2)-d))*arccoth(a*x)*(-a^2*c*d)^(1/2)+1/4*a^2*(c*d)^(1/2)/c/d/(a^2*c+d)*arctan(1/(a^2*c+d)*d^2/(c*d
)^(1/2)*x+1/(a^2*c+d)*d/(c*d)^(1/2)*a*c+a^2/(a^2*c+d)*(c*d)^(1/2)*x-a/(a^2*c+d)*(c*d)^(1/2))-1/2*a/c/(a^2*c+d)
^2*d*ln((a*x-1)/(a*x+1))+1/2*a^4*arccoth(a*x)/(a^2*c+d)/(a^2*d*x^2+a^2*c)*x+1/8/a*(-a^2*c*d)^(1/2)/c^2/(a^2*c+
d)*polylog(2,(a^2*c+d)*(a*x+1)/(a*x-1)/(a^2*c+2*(-a^2*c*d)^(1/2)-d))-1/4/a*(-a^2*c*d)^(1/2)/c^2/(a^2*c+d)*arcc
oth(a*x)^2-1/4*a/c/(a^2*c+d)^2*d*ln(a^2*c/(a*x-1)^2*(a*x+1)^2-2*a^2*c*(a*x+1)/(a*x-1)+d/(a*x-1)^2*(a*x+1)^2+a^
2*c+2*(a*x+1)/(a*x-1)*d+d)+3/4*a^3/(a^2*c+d)/(a^4*c^2+2*a^2*c*d+d^2)*arccoth(a*x)^2*(-a^2*c*d)^(1/2)-3/8*a^3/(
a^2*c+d)/(a^4*c^2+2*a^2*c*d+d^2)*polylog(2,(a^2*c+d)*(a*x+1)/(a*x-1)/(a^2*c-2*(-a^2*c*d)^(1/2)-d))*(-a^2*c*d)^
(1/2)+1/4*(c*d)^(1/2)/c^2*d/(a^2*c+d)^2*arctan(1/(a^2*c+d)*d^2/(c*d)^(1/2)*x+1/(a^2*c+d)*d/(c*d)^(1/2)*a*c+a^2
/(a^2*c+d)*(c*d)^(1/2)*x-a/(a^2*c+d)*(c*d)^(1/2))-1/4*(c*d)^(1/2)/c^2/(a^2*c+d)*arctan(1/(a^2*c+d)*d^2/(c*d)^(
1/2)*x+1/(a^2*c+d)*d/(c*d)^(1/2)*a*c+a^2/(a^2*c+d)*(c*d)^(1/2)*x-a/(a^2*c+d)*(c*d)^(1/2))-1/2*a^3*arccoth(a*x)
/(a^2*c+d)/(a^2*d*x^2+a^2*c)-1/4*(c*d)^(1/2)/c^2/(a^2*c+d)*arctan(a/d*(c*d)^(1/2))-1/4*a^4*(c*d)^(1/2)/d/(a^2*
c+d)^2*arctan(1/(a^2*c+d)*d^2/(c*d)^(1/2)*x+1/(a^2*c+d)*d/(c*d)^(1/2)*a*c+a^2/(a^2*c+d)*(c*d)^(1/2)*x-a/(a^2*c
+d)*(c*d)^(1/2))-3/8*a/c*d/(a^2*c+d)/(a^4*c^2+2*a^2*c*d+d^2)*polylog(2,(a^2*c+d)*(a*x+1)/(a*x-1)/(a^2*c-2*(-a^
2*c*d)^(1/2)-d))*(-a^2*c*d)^(1/2)-1/8/a/c^2*d^2/(a^2*c+d)/(a^4*c^2+2*a^2*c*d+d^2)*polylog(2,(a^2*c+d)*(a*x+1)/
(a*x-1)/(a^2*c-2*(-a^2*c*d)^(1/2)-d))*(-a^2*c*d)^(1/2)+1/4/a/c^2*d^2/(a^2*c+d)/(a^4*c^2+2*a^2*c*d+d^2)*arccoth
(a*x)^2*(-a^2*c*d)^(1/2)+3/4*a/c*d/(a^2*c+d)/(a^4*c^2+2*a^2*c*d+d^2)*arccoth(a*x)^2*(-a^2*c*d)^(1/2)-1/2*a^3*a
rccoth(a*x)/c/(a^2*c+d)/(a^2*d*x^2+a^2*c)*d*x^2+1/2*a^2*arccoth(a*x)/c/(a^2*c+d)/(a^2*d*x^2+a^2*c)*x*d+1/4*a*(
-a^2*c*d)^(1/2)/c/d/(a^2*c+d)*arccoth(a*x)*ln(1-(a^2*c+d)*(a*x+1)/(a*x-1)/(a^2*c+2*(-a^2*c*d)^(1/2)-d))-1/8*a^
5/d/(a^2*c+d)/(a^4*c^2+2*a^2*c*d+d^2)*polylog(2,(a^2*c+d)*(a*x+1)/(a*x-1)/(a^2*c-2*(-a^2*c*d)^(1/2)-d))*(-a^2*
c*d)^(1/2)*c+1/4*a^5/d/(a^2*c+d)/(a^4*c^2+2*a^2*c*d+d^2)*arccoth(a*x)^2*(-a^2*c*d)^(1/2)*c

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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(a*x)/(d*x^2+c)^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{\operatorname{arcoth}\left (a x\right )}{d^{2} x^{4} + 2 \, c d x^{2} + c^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(arccoth(a*x)/(d^2*x^4 + 2*c*d*x^2 + c^2), 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(a*x)/(d*x**2+c)**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 (a x\right )}{{\left (d x^{2} + c\right )}^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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