3.41 \(\int \frac{\coth ^{-1}(a x)}{(c+d x^2)^3} \, dx\)

Optimal. Leaf size=657 \[ \frac{3 i \text{PolyLog}\left (2,\frac{a \left (\sqrt{c}-i \sqrt{d} x\right )}{a \sqrt{c}-i \sqrt{d}}\right )}{32 c^{5/2} \sqrt{d}}-\frac{3 i \text{PolyLog}\left (2,\frac{a \left (\sqrt{c}-i \sqrt{d} x\right )}{a \sqrt{c}+i \sqrt{d}}\right )}{32 c^{5/2} \sqrt{d}}+\frac{3 i \text{PolyLog}\left (2,\frac{a \left (\sqrt{c}+i \sqrt{d} x\right )}{a \sqrt{c}-i \sqrt{d}}\right )}{32 c^{5/2} \sqrt{d}}-\frac{3 i \text{PolyLog}\left (2,\frac{a \left (\sqrt{c}+i \sqrt{d} x\right )}{a \sqrt{c}+i \sqrt{d}}\right )}{32 c^{5/2} \sqrt{d}}+\frac{a \left (5 a^2 c+3 d\right ) \log \left (1-a^2 x^2\right )}{16 c^2 \left (a^2 c+d\right )^2}-\frac{a \left (5 a^2 c+3 d\right ) \log \left (c+d x^2\right )}{16 c^2 \left (a^2 c+d\right )^2}+\frac{a}{8 c \left (a^2 c+d\right ) \left (c+d x^2\right )}+\frac{3 x \coth ^{-1}(a x)}{8 c^2 \left (c+d x^2\right )}+\frac{3 i \log \left (1-\frac{i \sqrt{d} x}{\sqrt{c}}\right ) \log \left (\frac{\sqrt{d} (1-a x)}{\sqrt{d}+i a \sqrt{c}}\right )}{32 c^{5/2} \sqrt{d}}-\frac{3 i \log \left (1-\frac{i \sqrt{d} x}{\sqrt{c}}\right ) \log \left (-\frac{\sqrt{d} (a x+1)}{-\sqrt{d}+i a \sqrt{c}}\right )}{32 c^{5/2} \sqrt{d}}-\frac{3 i \log \left (1+\frac{i \sqrt{d} x}{\sqrt{c}}\right ) \log \left (-\frac{\sqrt{d} (1-a x)}{-\sqrt{d}+i a \sqrt{c}}\right )}{32 c^{5/2} \sqrt{d}}+\frac{3 i \log \left (1+\frac{i \sqrt{d} x}{\sqrt{c}}\right ) \log \left (\frac{\sqrt{d} (a x+1)}{\sqrt{d}+i a \sqrt{c}}\right )}{32 c^{5/2} \sqrt{d}}+\frac{3 \coth ^{-1}(a x) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{8 c^{5/2} \sqrt{d}}+\frac{x \coth ^{-1}(a x)}{4 c \left (c+d x^2\right )^2} \]

[Out]

a/(8*c*(a^2*c + d)*(c + d*x^2)) + (x*ArcCoth[a*x])/(4*c*(c + d*x^2)^2) + (3*x*ArcCoth[a*x])/(8*c^2*(c + d*x^2)
) + (3*ArcCoth[a*x]*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(8*c^(5/2)*Sqrt[d]) + (((3*I)/32)*Log[(Sqrt[d]*(1 - a*x))/(I*
a*Sqrt[c] + Sqrt[d])]*Log[1 - (I*Sqrt[d]*x)/Sqrt[c]])/(c^(5/2)*Sqrt[d]) - (((3*I)/32)*Log[-((Sqrt[d]*(1 + a*x)
)/(I*a*Sqrt[c] - Sqrt[d]))]*Log[1 - (I*Sqrt[d]*x)/Sqrt[c]])/(c^(5/2)*Sqrt[d]) - (((3*I)/32)*Log[-((Sqrt[d]*(1
- a*x))/(I*a*Sqrt[c] - Sqrt[d]))]*Log[1 + (I*Sqrt[d]*x)/Sqrt[c]])/(c^(5/2)*Sqrt[d]) + (((3*I)/32)*Log[(Sqrt[d]
*(1 + a*x))/(I*a*Sqrt[c] + Sqrt[d])]*Log[1 + (I*Sqrt[d]*x)/Sqrt[c]])/(c^(5/2)*Sqrt[d]) + (a*(5*a^2*c + 3*d)*Lo
g[1 - a^2*x^2])/(16*c^2*(a^2*c + d)^2) - (a*(5*a^2*c + 3*d)*Log[c + d*x^2])/(16*c^2*(a^2*c + d)^2) + (((3*I)/3
2)*PolyLog[2, (a*(Sqrt[c] - I*Sqrt[d]*x))/(a*Sqrt[c] - I*Sqrt[d])])/(c^(5/2)*Sqrt[d]) - (((3*I)/32)*PolyLog[2,
(a*(Sqrt[c] - I*Sqrt[d]*x))/(a*Sqrt[c] + I*Sqrt[d])])/(c^(5/2)*Sqrt[d]) + (((3*I)/32)*PolyLog[2, (a*(Sqrt[c]
+ I*Sqrt[d]*x))/(a*Sqrt[c] - I*Sqrt[d])])/(c^(5/2)*Sqrt[d]) - (((3*I)/32)*PolyLog[2, (a*(Sqrt[c] + I*Sqrt[d]*x
))/(a*Sqrt[c] + I*Sqrt[d])])/(c^(5/2)*Sqrt[d])

________________________________________________________________________________________

Rubi [A]  time = 0.950909, antiderivative size = 657, normalized size of antiderivative = 1., number of steps used = 23, number of rules used = 11, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.786, Rules used = {199, 205, 5977, 6725, 571, 77, 4908, 2409, 2394, 2393, 2391} \[ \frac{3 i \text{PolyLog}\left (2,\frac{a \left (\sqrt{c}-i \sqrt{d} x\right )}{a \sqrt{c}-i \sqrt{d}}\right )}{32 c^{5/2} \sqrt{d}}-\frac{3 i \text{PolyLog}\left (2,\frac{a \left (\sqrt{c}-i \sqrt{d} x\right )}{a \sqrt{c}+i \sqrt{d}}\right )}{32 c^{5/2} \sqrt{d}}+\frac{3 i \text{PolyLog}\left (2,\frac{a \left (\sqrt{c}+i \sqrt{d} x\right )}{a \sqrt{c}-i \sqrt{d}}\right )}{32 c^{5/2} \sqrt{d}}-\frac{3 i \text{PolyLog}\left (2,\frac{a \left (\sqrt{c}+i \sqrt{d} x\right )}{a \sqrt{c}+i \sqrt{d}}\right )}{32 c^{5/2} \sqrt{d}}+\frac{a \left (5 a^2 c+3 d\right ) \log \left (1-a^2 x^2\right )}{16 c^2 \left (a^2 c+d\right )^2}-\frac{a \left (5 a^2 c+3 d\right ) \log \left (c+d x^2\right )}{16 c^2 \left (a^2 c+d\right )^2}+\frac{a}{8 c \left (a^2 c+d\right ) \left (c+d x^2\right )}+\frac{3 x \coth ^{-1}(a x)}{8 c^2 \left (c+d x^2\right )}+\frac{3 i \log \left (1-\frac{i \sqrt{d} x}{\sqrt{c}}\right ) \log \left (\frac{\sqrt{d} (1-a x)}{\sqrt{d}+i a \sqrt{c}}\right )}{32 c^{5/2} \sqrt{d}}-\frac{3 i \log \left (1-\frac{i \sqrt{d} x}{\sqrt{c}}\right ) \log \left (-\frac{\sqrt{d} (a x+1)}{-\sqrt{d}+i a \sqrt{c}}\right )}{32 c^{5/2} \sqrt{d}}-\frac{3 i \log \left (1+\frac{i \sqrt{d} x}{\sqrt{c}}\right ) \log \left (-\frac{\sqrt{d} (1-a x)}{-\sqrt{d}+i a \sqrt{c}}\right )}{32 c^{5/2} \sqrt{d}}+\frac{3 i \log \left (1+\frac{i \sqrt{d} x}{\sqrt{c}}\right ) \log \left (\frac{\sqrt{d} (a x+1)}{\sqrt{d}+i a \sqrt{c}}\right )}{32 c^{5/2} \sqrt{d}}+\frac{3 \coth ^{-1}(a x) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{8 c^{5/2} \sqrt{d}}+\frac{x \coth ^{-1}(a x)}{4 c \left (c+d x^2\right )^2} \]

Antiderivative was successfully verified.

[In]

Int[ArcCoth[a*x]/(c + d*x^2)^3,x]

[Out]

a/(8*c*(a^2*c + d)*(c + d*x^2)) + (x*ArcCoth[a*x])/(4*c*(c + d*x^2)^2) + (3*x*ArcCoth[a*x])/(8*c^2*(c + d*x^2)
) + (3*ArcCoth[a*x]*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(8*c^(5/2)*Sqrt[d]) + (((3*I)/32)*Log[(Sqrt[d]*(1 - a*x))/(I*
a*Sqrt[c] + Sqrt[d])]*Log[1 - (I*Sqrt[d]*x)/Sqrt[c]])/(c^(5/2)*Sqrt[d]) - (((3*I)/32)*Log[-((Sqrt[d]*(1 + a*x)
)/(I*a*Sqrt[c] - Sqrt[d]))]*Log[1 - (I*Sqrt[d]*x)/Sqrt[c]])/(c^(5/2)*Sqrt[d]) - (((3*I)/32)*Log[-((Sqrt[d]*(1
- a*x))/(I*a*Sqrt[c] - Sqrt[d]))]*Log[1 + (I*Sqrt[d]*x)/Sqrt[c]])/(c^(5/2)*Sqrt[d]) + (((3*I)/32)*Log[(Sqrt[d]
*(1 + a*x))/(I*a*Sqrt[c] + Sqrt[d])]*Log[1 + (I*Sqrt[d]*x)/Sqrt[c]])/(c^(5/2)*Sqrt[d]) + (a*(5*a^2*c + 3*d)*Lo
g[1 - a^2*x^2])/(16*c^2*(a^2*c + d)^2) - (a*(5*a^2*c + 3*d)*Log[c + d*x^2])/(16*c^2*(a^2*c + d)^2) + (((3*I)/3
2)*PolyLog[2, (a*(Sqrt[c] - I*Sqrt[d]*x))/(a*Sqrt[c] - I*Sqrt[d])])/(c^(5/2)*Sqrt[d]) - (((3*I)/32)*PolyLog[2,
(a*(Sqrt[c] - I*Sqrt[d]*x))/(a*Sqrt[c] + I*Sqrt[d])])/(c^(5/2)*Sqrt[d]) + (((3*I)/32)*PolyLog[2, (a*(Sqrt[c]
+ I*Sqrt[d]*x))/(a*Sqrt[c] - I*Sqrt[d])])/(c^(5/2)*Sqrt[d]) - (((3*I)/32)*PolyLog[2, (a*(Sqrt[c] + I*Sqrt[d]*x
))/(a*Sqrt[c] + I*Sqrt[d])])/(c^(5/2)*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 571

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

Rule 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

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 )^3} \, dx &=\frac{x \coth ^{-1}(a x)}{4 c \left (c+d x^2\right )^2}+\frac{3 x \coth ^{-1}(a x)}{8 c^2 \left (c+d x^2\right )}+\frac{3 \coth ^{-1}(a x) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{8 c^{5/2} \sqrt{d}}-a \int \frac{\frac{x}{4 c \left (c+d x^2\right )^2}+\frac{3 x}{8 c^2 \left (c+d x^2\right )}+\frac{3 \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{8 c^{5/2} \sqrt{d}}}{1-a^2 x^2} \, dx\\ &=\frac{x \coth ^{-1}(a x)}{4 c \left (c+d x^2\right )^2}+\frac{3 x \coth ^{-1}(a x)}{8 c^2 \left (c+d x^2\right )}+\frac{3 \coth ^{-1}(a x) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{8 c^{5/2} \sqrt{d}}-a \int \left (-\frac{x \left (5 c+3 d x^2\right )}{8 c^2 \left (-1+a^2 x^2\right ) \left (c+d x^2\right )^2}-\frac{3 \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{8 c^{5/2} \sqrt{d} \left (-1+a^2 x^2\right )}\right ) \, dx\\ &=\frac{x \coth ^{-1}(a x)}{4 c \left (c+d x^2\right )^2}+\frac{3 x \coth ^{-1}(a x)}{8 c^2 \left (c+d x^2\right )}+\frac{3 \coth ^{-1}(a x) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{8 c^{5/2} \sqrt{d}}+\frac{a \int \frac{x \left (5 c+3 d x^2\right )}{\left (-1+a^2 x^2\right ) \left (c+d x^2\right )^2} \, dx}{8 c^2}+\frac{(3 a) \int \frac{\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{-1+a^2 x^2} \, dx}{8 c^{5/2} \sqrt{d}}\\ &=\frac{x \coth ^{-1}(a x)}{4 c \left (c+d x^2\right )^2}+\frac{3 x \coth ^{-1}(a x)}{8 c^2 \left (c+d x^2\right )}+\frac{3 \coth ^{-1}(a x) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{8 c^{5/2} \sqrt{d}}+\frac{a \operatorname{Subst}\left (\int \frac{5 c+3 d x}{\left (-1+a^2 x\right ) (c+d x)^2} \, dx,x,x^2\right )}{16 c^2}+\frac{(3 i a) \int \frac{\log \left (1-\frac{i \sqrt{d} x}{\sqrt{c}}\right )}{-1+a^2 x^2} \, dx}{16 c^{5/2} \sqrt{d}}-\frac{(3 i a) \int \frac{\log \left (1+\frac{i \sqrt{d} x}{\sqrt{c}}\right )}{-1+a^2 x^2} \, dx}{16 c^{5/2} \sqrt{d}}\\ &=\frac{x \coth ^{-1}(a x)}{4 c \left (c+d x^2\right )^2}+\frac{3 x \coth ^{-1}(a x)}{8 c^2 \left (c+d x^2\right )}+\frac{3 \coth ^{-1}(a x) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{8 c^{5/2} \sqrt{d}}+\frac{a \operatorname{Subst}\left (\int \left (\frac{a^2 \left (5 a^2 c+3 d\right )}{\left (a^2 c+d\right )^2 \left (-1+a^2 x\right )}-\frac{2 c d}{\left (a^2 c+d\right ) (c+d x)^2}-\frac{d \left (5 a^2 c+3 d\right )}{\left (a^2 c+d\right )^2 (c+d x)}\right ) \, dx,x,x^2\right )}{16 c^2}+\frac{(3 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}{16 c^{5/2} \sqrt{d}}-\frac{(3 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}{16 c^{5/2} \sqrt{d}}\\ &=\frac{a}{8 c \left (a^2 c+d\right ) \left (c+d x^2\right )}+\frac{x \coth ^{-1}(a x)}{4 c \left (c+d x^2\right )^2}+\frac{3 x \coth ^{-1}(a x)}{8 c^2 \left (c+d x^2\right )}+\frac{3 \coth ^{-1}(a x) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{8 c^{5/2} \sqrt{d}}+\frac{a \left (5 a^2 c+3 d\right ) \log \left (1-a^2 x^2\right )}{16 c^2 \left (a^2 c+d\right )^2}-\frac{a \left (5 a^2 c+3 d\right ) \log \left (c+d x^2\right )}{16 c^2 \left (a^2 c+d\right )^2}-\frac{(3 i a) \int \frac{\log \left (1-\frac{i \sqrt{d} x}{\sqrt{c}}\right )}{1-a x} \, dx}{32 c^{5/2} \sqrt{d}}-\frac{(3 i a) \int \frac{\log \left (1-\frac{i \sqrt{d} x}{\sqrt{c}}\right )}{1+a x} \, dx}{32 c^{5/2} \sqrt{d}}+\frac{(3 i a) \int \frac{\log \left (1+\frac{i \sqrt{d} x}{\sqrt{c}}\right )}{1-a x} \, dx}{32 c^{5/2} \sqrt{d}}+\frac{(3 i a) \int \frac{\log \left (1+\frac{i \sqrt{d} x}{\sqrt{c}}\right )}{1+a x} \, dx}{32 c^{5/2} \sqrt{d}}\\ &=\frac{a}{8 c \left (a^2 c+d\right ) \left (c+d x^2\right )}+\frac{x \coth ^{-1}(a x)}{4 c \left (c+d x^2\right )^2}+\frac{3 x \coth ^{-1}(a x)}{8 c^2 \left (c+d x^2\right )}+\frac{3 \coth ^{-1}(a x) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{8 c^{5/2} \sqrt{d}}+\frac{3 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 )}{32 c^{5/2} \sqrt{d}}-\frac{3 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 )}{32 c^{5/2} \sqrt{d}}-\frac{3 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 )}{32 c^{5/2} \sqrt{d}}+\frac{3 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 )}{32 c^{5/2} \sqrt{d}}+\frac{a \left (5 a^2 c+3 d\right ) \log \left (1-a^2 x^2\right )}{16 c^2 \left (a^2 c+d\right )^2}-\frac{a \left (5 a^2 c+3 d\right ) \log \left (c+d x^2\right )}{16 c^2 \left (a^2 c+d\right )^2}-\frac{3 \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}{32 c^3}-\frac{3 \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}{32 c^3}+\frac{3 \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}{32 c^3}+\frac{3 \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}{32 c^3}\\ &=\frac{a}{8 c \left (a^2 c+d\right ) \left (c+d x^2\right )}+\frac{x \coth ^{-1}(a x)}{4 c \left (c+d x^2\right )^2}+\frac{3 x \coth ^{-1}(a x)}{8 c^2 \left (c+d x^2\right )}+\frac{3 \coth ^{-1}(a x) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{8 c^{5/2} \sqrt{d}}+\frac{3 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 )}{32 c^{5/2} \sqrt{d}}-\frac{3 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 )}{32 c^{5/2} \sqrt{d}}-\frac{3 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 )}{32 c^{5/2} \sqrt{d}}+\frac{3 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 )}{32 c^{5/2} \sqrt{d}}+\frac{a \left (5 a^2 c+3 d\right ) \log \left (1-a^2 x^2\right )}{16 c^2 \left (a^2 c+d\right )^2}-\frac{a \left (5 a^2 c+3 d\right ) \log \left (c+d x^2\right )}{16 c^2 \left (a^2 c+d\right )^2}+\frac{(3 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 )}{32 c^{5/2} \sqrt{d}}-\frac{(3 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 )}{32 c^{5/2} \sqrt{d}}-\frac{(3 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 )}{32 c^{5/2} \sqrt{d}}+\frac{(3 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 )}{32 c^{5/2} \sqrt{d}}\\ &=\frac{a}{8 c \left (a^2 c+d\right ) \left (c+d x^2\right )}+\frac{x \coth ^{-1}(a x)}{4 c \left (c+d x^2\right )^2}+\frac{3 x \coth ^{-1}(a x)}{8 c^2 \left (c+d x^2\right )}+\frac{3 \coth ^{-1}(a x) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{8 c^{5/2} \sqrt{d}}+\frac{3 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 )}{32 c^{5/2} \sqrt{d}}-\frac{3 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 )}{32 c^{5/2} \sqrt{d}}-\frac{3 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 )}{32 c^{5/2} \sqrt{d}}+\frac{3 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 )}{32 c^{5/2} \sqrt{d}}+\frac{a \left (5 a^2 c+3 d\right ) \log \left (1-a^2 x^2\right )}{16 c^2 \left (a^2 c+d\right )^2}-\frac{a \left (5 a^2 c+3 d\right ) \log \left (c+d x^2\right )}{16 c^2 \left (a^2 c+d\right )^2}+\frac{3 i \text{Li}_2\left (\frac{a \left (\sqrt{c}-i \sqrt{d} x\right )}{a \sqrt{c}-i \sqrt{d}}\right )}{32 c^{5/2} \sqrt{d}}-\frac{3 i \text{Li}_2\left (\frac{a \left (\sqrt{c}-i \sqrt{d} x\right )}{a \sqrt{c}+i \sqrt{d}}\right )}{32 c^{5/2} \sqrt{d}}+\frac{3 i \text{Li}_2\left (\frac{a \left (\sqrt{c}+i \sqrt{d} x\right )}{a \sqrt{c}-i \sqrt{d}}\right )}{32 c^{5/2} \sqrt{d}}-\frac{3 i \text{Li}_2\left (\frac{a \left (\sqrt{c}+i \sqrt{d} x\right )}{a \sqrt{c}+i \sqrt{d}}\right )}{32 c^{5/2} \sqrt{d}}\\ \end{align*}

Mathematica [B]  time = 12.88, size = 1838, normalized size = 2.8 \[ \text{result too large to display} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[ArcCoth[a*x]/(c + d*x^2)^3,x]

[Out]

a^5*((-5*Log[1 + ((a^2*c + d)*Cosh[2*ArcCoth[a*x]])/(-(a^2*c) + d)])/(16*a^2*c*(a^2*c + d)^2) - (3*d*Log[1 + (
(a^2*c + d)*Cosh[2*ArcCoth[a*x]])/(-(a^2*c) + d)])/(16*a^4*c^2*(a^2*c + d)^2) + (3*((-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[1 - ((-(a^2*c) + d - (2*I)*Sqrt[a^2*c*d]
)*(2*d - ((2*I)*Sqrt[a^2*c*d])/(a*x)))/((a^2*c + d)*(2*d + ((2*I)*Sqrt[a^2*c*d])/(a*x)))] + (-ArcCos[-((-(a^2*
c) + d)/(a^2*c + d))] - 2*ArcTan[(a*c)/(Sqrt[a^2*c*d]*x)])*Log[1 - ((-(a^2*c) + d + (2*I)*Sqrt[a^2*c*d])*(2*d
- ((2*I)*Sqrt[a^2*c*d])/(a*x)))/((a^2*c + d)*(2*d + ((2*I)*Sqrt[a^2*c*d])/(a*x)))] + (ArcCos[-((-(a^2*c) + d)/
(a^2*c + d))] + (2*I)*((-I)*ArcTan[(a*c)/(Sqrt[a^2*c*d]*x)] - I*ArcTan[(a*d*x)/Sqrt[a^2*c*d]]))*Log[(Sqrt[2]*S
qrt[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]]])] + (ArcCo
s[-((-(a^2*c) + d)/(a^2*c + d))] - (2*I)*((-I)*ArcTan[(a*c)/(Sqrt[a^2*c*d]*x)] - I*ArcTan[(a*d*x)/Sqrt[a^2*c*d
]]))*Log[(Sqrt[2]*Sqrt[a^2*c*d]*E^ArcCoth[a*x])/(Sqrt[a^2*c + d]*Sqrt[-(a^2*c) + d + (a^2*c + d)*Cosh[2*ArcCot
h[a*x]]])] + I*(PolyLog[2, ((-(a^2*c) + d - (2*I)*Sqrt[a^2*c*d])*(2*d - ((2*I)*Sqrt[a^2*c*d])/(a*x)))/((a^2*c
+ d)*(2*d + ((2*I)*Sqrt[a^2*c*d])/(a*x)))] - PolyLog[2, ((-(a^2*c) + d + (2*I)*Sqrt[a^2*c*d])*(2*d - ((2*I)*Sq
rt[a^2*c*d])/(a*x)))/((a^2*c + d)*(2*d + ((2*I)*Sqrt[a^2*c*d])/(a*x)))])))/(32*a^2*c*Sqrt[a^2*c*d]*(a^2*c + d)
) + (3*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[1 -
((-(a^2*c) + d - (2*I)*Sqrt[a^2*c*d])*(2*d - ((2*I)*Sqrt[a^2*c*d])/(a*x)))/((a^2*c + d)*(2*d + ((2*I)*Sqrt[a^2
*c*d])/(a*x)))] + (-ArcCos[-((-(a^2*c) + d)/(a^2*c + d))] - 2*ArcTan[(a*c)/(Sqrt[a^2*c*d]*x)])*Log[1 - ((-(a^2
*c) + d + (2*I)*Sqrt[a^2*c*d])*(2*d - ((2*I)*Sqrt[a^2*c*d])/(a*x)))/((a^2*c + d)*(2*d + ((2*I)*Sqrt[a^2*c*d])/
(a*x)))] + (ArcCos[-((-(a^2*c) + d)/(a^2*c + d))] + (2*I)*((-I)*ArcTan[(a*c)/(Sqrt[a^2*c*d]*x)] - I*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*I)*((-I)*ArcTan[(a*c)/(Sqrt[a^2*c*d]
*x)] - I*ArcTan[(a*d*x)/Sqrt[a^2*c*d]]))*Log[(Sqrt[2]*Sqrt[a^2*c*d]*E^ArcCoth[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])*(2*d - ((
2*I)*Sqrt[a^2*c*d])/(a*x)))/((a^2*c + d)*(2*d + ((2*I)*Sqrt[a^2*c*d])/(a*x)))] - PolyLog[2, ((-(a^2*c) + d + (
2*I)*Sqrt[a^2*c*d])*(2*d - ((2*I)*Sqrt[a^2*c*d])/(a*x)))/((a^2*c + d)*(2*d + ((2*I)*Sqrt[a^2*c*d])/(a*x)))])))
/(32*a^4*c^2*Sqrt[a^2*c*d]*(a^2*c + d)) - (d*ArcCoth[a*x]*Sinh[2*ArcCoth[a*x]])/(2*a^2*c*(a^2*c + d)*(-(a^2*c)
+ d + a^2*c*Cosh[2*ArcCoth[a*x]] + d*Cosh[2*ArcCoth[a*x]])^2) - (2*a^2*c*d - 5*a^4*c^2*ArcCoth[a*x]*Sinh[2*Ar
cCoth[a*x]] - 8*a^2*c*d*ArcCoth[a*x]*Sinh[2*ArcCoth[a*x]] - 3*d^2*ArcCoth[a*x]*Sinh[2*ArcCoth[a*x]])/(8*a^4*c^
2*(a^2*c + d)^2*(-(a^2*c) + d + a^2*c*Cosh[2*ArcCoth[a*x]] + d*Cosh[2*ArcCoth[a*x]])))

________________________________________________________________________________________

Maple [B]  time = 0.596, size = 4128, normalized size = 6.3 \begin{align*} \text{output 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)^3,x)

[Out]

1/8*a^5/(a^4*c^2+2*a^2*c*d+d^2)/(a^2*d*x^2+a^2*c)^2*d-5/8*a^5/(a^4*c^2+2*a^2*c*d+d^2)/(a^2*c+d)*ln((a*x-1)/(a*
x+1))-5/16*a^5/(a^4*c^2+2*a^2*c*d+d^2)/(a^2*c+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^5*arccoth(a*x)^2/(a^4*c^2+2*a^2*c*d+d^2)^2*(-a^2*c*d)^(1/2)-3
/8*a^5*polylog(2,(a^2*c+d)*(a*x+1)/(a*x-1)/(a^2*c-2*(-a^2*c*d)^(1/2)-d))/(a^4*c^2+2*a^2*c*d+d^2)^2*(-a^2*c*d)^
(1/2)-1/8*a^2*(c*d)^(1/2)/c^2/(a^4*c^2+2*a^2*c*d+d^2)*arctan(a/d*(c*d)^(1/2))-3/16*(c*d)^(1/2)/c^3*d/(a^4*c^2+
2*a^2*c*d+d^2)*arctan(a/d*(c*d)^(1/2))-5/8*a^7/(a^4*c^2+2*a^2*c*d+d^2)/(a^2*d*x^2+a^2*c)^2*c*arccoth(a*x)-9/8*
a^3/c/(a^4*c^2+2*a^2*c*d+d^2)^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)*d-1/8*a^2*(c*d)^(1/2)/c^2/(a^4*c^2+2*a^2*c*d+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*(-a^2*c*d)^(1/2)/c^2/(a^4*c^2+2
*a^2*c*d+d^2)*arccoth(a*x)^2-3/8*a^5/(a^4*c^2+2*a^2*c*d+d^2)/(a^2*d*x^2+a^2*c)^2*d*arccoth(a*x)+3/16*a*(-a^2*c
*d)^(1/2)/c^2/(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))-3/4*a^
5*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^4*c^2+2*a^2*c*d+d^2)^2*(-a^2*c*
d)^(1/2)-3/16*(c*d)^(1/2)/c^3*d/(a^4*c^2+2*a^2*c*d+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))-1/8*a^7/(a^4*c^2+2*a^2*c*d+d^2)/(a^2*d*x^2+a
^2*c)^2*d*x^2-5/8*a^7/(a^4*c^2+2*a^2*c*d+d^2)/(a^2*d*x^2+a^2*c)^2/c*arccoth(a*x)*x^4*d^2-3/4*a/c^2*d^2/(a^4*c^
2+2*a^2*c*d+d^2)^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)-
3/16*a^7/d/(a^4*c^2+2*a^2*c*d+d^2)^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+3/8*a^8/(a^4*c^2+2*a^2*c*d+d^2)/(a^2*d*x^2+a^2*c)^2*arccoth(a*x)*x^3*d-5/4*a^7/(a^4*c^2+2*
a^2*c*d+d^2)/(a^2*d*x^2+a^2*c)^2*arccoth(a*x)*x^2*d+5/4*a^6/(a^4*c^2+2*a^2*c*d+d^2)/(a^2*d*x^2+a^2*c)^2*arccot
h(a*x)*x*d+3/16*(c*d)^(1/2)/c^3*d^2/(a^2*c+d)/(a^4*c^2+2*a^2*c*d+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))-5/16*a^6*(c*d)^(1/2)/d/(a^2*c+
d)/(a^4*c^2+2*a^2*c*d+d^2)*arctan(a/d*(c*d)^(1/2))+5/16*a^4*(c*d)^(1/2)/c/d/(a^4*c^2+2*a^2*c*d+d^2)*arctan(a/d
*(c*d)^(1/2))-3/16*a^4*(c*d)^(1/2)/c/(a^2*c+d)/(a^4*c^2+2*a^2*c*d+d^2)*arctan(a/d*(c*d)^(1/2))+3/16*(c*d)^(1/2
)/c^3*d^2/(a^2*c+d)/(a^4*c^2+2*a^2*c*d+d^2)*arctan(a/d*(c*d)^(1/2))-1/8*a^7/(a^4*c^2+2*a^2*c*d+d^2)/(a^2*d*x^2
+a^2*c)^2/c*d^2*x^4+1/8*a^5/(a^4*c^2+2*a^2*c*d+d^2)/(a^2*d*x^2+a^2*c)^2/c*d^2*x^2+9/8*a^3/c*d/(a^4*c^2+2*a^2*c
*d+d^2)^2*arccoth(a*x)^2*(-a^2*c*d)^(1/2)-9/16*a^3/c*d/(a^4*c^2+2*a^2*c*d+d^2)^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)-3/8*a/c^2*d^2/(a^4*c^2+2*a^2*c*d+d^2)^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)-3/16/a*(-a^2*c*d)^(1/2)/c^3*d/(a^4*c^2+2*a^
2*c*d+d^2)*arccoth(a*x)^2-a^3/(a^4*c^2+2*a^2*c*d+d^2)/c*d/(a^2*c+d)*ln((a*x-1)/(a*x+1))+3/32*a^3*(-a^2*c*d)^(1
/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))-3/32*a^7/d/(
a^4*c^2+2*a^2*c*d+d^2)^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+
3/16/a/c^3*d^3/(a^4*c^2+2*a^2*c*d+d^2)^2*arccoth(a*x)^2*(-a^2*c*d)^(1/2)-5/16*a^6*(c*d)^(1/2)/d/(a^2*c+d)/(a^4
*c^2+2*a^2*c*d+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*(-a^2*c*d)^(1/2)/c^2/(a^4*c^2+2*a^2*c*d+d^2)*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))-3/32/a/c^3*d^3/(a^4*c^2+2*a^2*c*d+d^2)^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)-3/16*a^3*(-a^2*c*d)^(1/2)/c/d/(a^4*c^2+2*a^2*c*d+d^
2)*arccoth(a*x)^2-3/8*a/(a^4*c^2+2*a^2*c*d+d^2)/c^2*d^2/(a^2*c+d)*ln((a*x-1)/(a*x+1))-1/2*a^3/(a^4*c^2+2*a^2*c
*d+d^2)/c*d/(a^2*c+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)-3/16*a/(a^4*c^2+2*a^2*c*d+d^2)/c^2*d^2/(a^2*c+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)+5/8*a^8/(a^4*c^2+2*a^2*c*d+d^2)/(a^2*d*x^2+a^2*c)^2
*c*arccoth(a*x)*x+3/32/a*(-a^2*c*d)^(1/2)/c^3*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))+3/4*a*d^2*arccoth(a*x)^2/c^2/(a^4*c^2+2*a^2*c*d+d^2)^2*(-a^2*c*d)^(1/2)+3/16*a^7/d
/(a^4*c^2+2*a^2*c*d+d^2)^2*arccoth(a*x)^2*(-a^2*c*d)^(1/2)*c-3/16*a^4*(c*d)^(1/2)/c/(a^2*c+d)/(a^4*c^2+2*a^2*c
*d+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))+5/16*a^4*(c*d)^(1/2)/c/d/(a^4*c^2+2*a^2*c*d+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))+3/4*a^6/(a^4*c^2+2*a^2*c*d+d^2)/(
a^2*d*x^2+a^2*c)^2/c*arccoth(a*x)*x^3*d^2+3/16/a*(-a^2*c*d)^(1/2)/c^3*d/(a^4*c^2+2*a^2*c*d+d^2)*arccoth(a*x)*l
n(1-(a^2*c+d)*(a*x+1)/(a*x-1)/(a^2*c+2*(-a^2*c*d)^(1/2)-d))-3/16/a/c^3*d^3/(a^4*c^2+2*a^2*c*d+d^2)^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)+3/8*a^4/(a^4*c^2+2*a^2*c*d+d
^2)/(a^2*d*x^2+a^2*c)^2/c^2*arccoth(a*x)*x^3*d^3+3/16*a^3*(-a^2*c*d)^(1/2)/c/d/(a^4*c^2+2*a^2*c*d+d^2)*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^5/(a^4*c^2+2*a^2*c*d+d^2)/(a^2*d*x^2+
a^2*c)^2/c*arccoth(a*x)*x^2*d^2+5/8*a^4/(a^4*c^2+2*a^2*c*d+d^2)/(a^2*d*x^2+a^2*c)^2/c*arccoth(a*x)*x*d^2+5/16*
a^2*(c*d)^(1/2)/c^2*d/(a^2*c+d)/(a^4*c^2+2*a^2*c*d+d^2)*arctan(a/d*(c*d)^(1/2))-3/8*a^5/(a^4*c^2+2*a^2*c*d+d^2
)/(a^2*d*x^2+a^2*c)^2/c^2*arccoth(a*x)*x^4*d^3+5/16*a^2*(c*d)^(1/2)/c^2*d/(a^2*c+d)/(a^4*c^2+2*a^2*c*d+d^2)*ar
ctan(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))

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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)^3,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^{3} x^{6} + 3 \, c d^{2} x^{4} + 3 \, c^{2} d x^{2} + c^{3}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccoth(a*x)/(d*x^2+c)^3,x, algorithm="fricas")

[Out]

integral(arccoth(a*x)/(d^3*x^6 + 3*c*d^2*x^4 + 3*c^2*d*x^2 + c^3), 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)**3,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 )}^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccoth(a*x)/(d*x^2+c)^3,x, algorithm="giac")

[Out]

integrate(arccoth(a*x)/(d*x^2 + c)^3, x)