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

Optimal. Leaf size=390 \[ -\frac{i \text{PolyLog}\left (2,1+\frac{2 \sqrt{c} \sqrt{d} (1-a x)}{\left (-\sqrt{d}+i a \sqrt{c}\right ) \left (\sqrt{c}-i \sqrt{d} x\right )}\right )}{4 \sqrt{c} \sqrt{d}}+\frac{i \text{PolyLog}\left (2,1-\frac{2 \sqrt{c} \sqrt{d} (a x+1)}{\left (\sqrt{d}+i a \sqrt{c}\right ) \left (\sqrt{c}-i \sqrt{d} x\right )}\right )}{4 \sqrt{c} \sqrt{d}}-\frac{\log \left (1-\frac{1}{a x}\right ) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{2 \sqrt{c} \sqrt{d}}+\frac{\log \left (\frac{1}{a x}+1\right ) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{2 \sqrt{c} \sqrt{d}}+\frac{\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right ) \log \left (-\frac{2 \sqrt{c} \sqrt{d} (1-a x)}{\left (-\sqrt{d}+i a \sqrt{c}\right ) \left (\sqrt{c}-i \sqrt{d} x\right )}\right )}{2 \sqrt{c} \sqrt{d}}-\frac{\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right ) \log \left (\frac{2 \sqrt{c} \sqrt{d} (a x+1)}{\left (\sqrt{d}+i a \sqrt{c}\right ) \left (\sqrt{c}-i \sqrt{d} x\right )}\right )}{2 \sqrt{c} \sqrt{d}} \]

[Out]

-(ArcTan[(Sqrt[d]*x)/Sqrt[c]]*Log[1 - 1/(a*x)])/(2*Sqrt[c]*Sqrt[d]) + (ArcTan[(Sqrt[d]*x)/Sqrt[c]]*Log[1 + 1/(
a*x)])/(2*Sqrt[c]*Sqrt[d]) + (ArcTan[(Sqrt[d]*x)/Sqrt[c]]*Log[(-2*Sqrt[c]*Sqrt[d]*(1 - a*x))/((I*a*Sqrt[c] - S
qrt[d])*(Sqrt[c] - I*Sqrt[d]*x))])/(2*Sqrt[c]*Sqrt[d]) - (ArcTan[(Sqrt[d]*x)/Sqrt[c]]*Log[(2*Sqrt[c]*Sqrt[d]*(
1 + a*x))/((I*a*Sqrt[c] + Sqrt[d])*(Sqrt[c] - I*Sqrt[d]*x))])/(2*Sqrt[c]*Sqrt[d]) - ((I/4)*PolyLog[2, 1 + (2*S
qrt[c]*Sqrt[d]*(1 - a*x))/((I*a*Sqrt[c] - Sqrt[d])*(Sqrt[c] - I*Sqrt[d]*x))])/(Sqrt[c]*Sqrt[d]) + ((I/4)*PolyL
og[2, 1 - (2*Sqrt[c]*Sqrt[d]*(1 + a*x))/((I*a*Sqrt[c] + Sqrt[d])*(Sqrt[c] - I*Sqrt[d]*x))])/(Sqrt[c]*Sqrt[d])

________________________________________________________________________________________

Rubi [A]  time = 0.946685, antiderivative size = 390, normalized size of antiderivative = 1., number of steps used = 27, number of rules used = 13, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.929, Rules used = {5973, 205, 2470, 12, 260, 6688, 4876, 4848, 2391, 4856, 2402, 2315, 2447} \[ -\frac{i \text{PolyLog}\left (2,1+\frac{2 \sqrt{c} \sqrt{d} (1-a x)}{\left (-\sqrt{d}+i a \sqrt{c}\right ) \left (\sqrt{c}-i \sqrt{d} x\right )}\right )}{4 \sqrt{c} \sqrt{d}}+\frac{i \text{PolyLog}\left (2,1-\frac{2 \sqrt{c} \sqrt{d} (a x+1)}{\left (\sqrt{d}+i a \sqrt{c}\right ) \left (\sqrt{c}-i \sqrt{d} x\right )}\right )}{4 \sqrt{c} \sqrt{d}}-\frac{\log \left (1-\frac{1}{a x}\right ) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{2 \sqrt{c} \sqrt{d}}+\frac{\log \left (\frac{1}{a x}+1\right ) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{2 \sqrt{c} \sqrt{d}}+\frac{\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right ) \log \left (-\frac{2 \sqrt{c} \sqrt{d} (1-a x)}{\left (-\sqrt{d}+i a \sqrt{c}\right ) \left (\sqrt{c}-i \sqrt{d} x\right )}\right )}{2 \sqrt{c} \sqrt{d}}-\frac{\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right ) \log \left (\frac{2 \sqrt{c} \sqrt{d} (a x+1)}{\left (\sqrt{d}+i a \sqrt{c}\right ) \left (\sqrt{c}-i \sqrt{d} x\right )}\right )}{2 \sqrt{c} \sqrt{d}} \]

Antiderivative was successfully verified.

[In]

Int[ArcCoth[a*x]/(c + d*x^2),x]

[Out]

-(ArcTan[(Sqrt[d]*x)/Sqrt[c]]*Log[1 - 1/(a*x)])/(2*Sqrt[c]*Sqrt[d]) + (ArcTan[(Sqrt[d]*x)/Sqrt[c]]*Log[1 + 1/(
a*x)])/(2*Sqrt[c]*Sqrt[d]) + (ArcTan[(Sqrt[d]*x)/Sqrt[c]]*Log[(-2*Sqrt[c]*Sqrt[d]*(1 - a*x))/((I*a*Sqrt[c] - S
qrt[d])*(Sqrt[c] - I*Sqrt[d]*x))])/(2*Sqrt[c]*Sqrt[d]) - (ArcTan[(Sqrt[d]*x)/Sqrt[c]]*Log[(2*Sqrt[c]*Sqrt[d]*(
1 + a*x))/((I*a*Sqrt[c] + Sqrt[d])*(Sqrt[c] - I*Sqrt[d]*x))])/(2*Sqrt[c]*Sqrt[d]) - ((I/4)*PolyLog[2, 1 + (2*S
qrt[c]*Sqrt[d]*(1 - a*x))/((I*a*Sqrt[c] - Sqrt[d])*(Sqrt[c] - I*Sqrt[d]*x))])/(Sqrt[c]*Sqrt[d]) + ((I/4)*PolyL
og[2, 1 - (2*Sqrt[c]*Sqrt[d]*(1 + a*x))/((I*a*Sqrt[c] + Sqrt[d])*(Sqrt[c] - I*Sqrt[d]*x))])/(Sqrt[c]*Sqrt[d])

Rule 5973

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

Rule 2470

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))/((f_) + (g_.)*(x_)^2), x_Symbol] :> With[{u = In
tHide[1/(f + g*x^2), x]}, Simp[u*(a + b*Log[c*(d + e*x^n)^p]), x] - Dist[b*e*n*p, Int[(u*x^(n - 1))/(d + e*x^n
), x], x]] /; FreeQ[{a, b, c, d, e, f, g, n, p}, x] && IntegerQ[n]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 260

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

Rule 6688

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

Rule 4876

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_))^(q_.), x_Symbol] :> Int[Ex
pandIntegrand[(a + b*ArcTan[c*x])^p, (f*x)^m*(d + e*x)^q, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && IGtQ[p,
0] && IntegerQ[q] && (GtQ[q, 0] || NeQ[a, 0] || IntegerQ[m])

Rule 4848

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))/(x_), x_Symbol] :> Simp[a*Log[x], x] + (Dist[(I*b)/2, Int[Log[1 - I*c*x
]/x, x], x] - Dist[(I*b)/2, Int[Log[1 + I*c*x]/x, x], x]) /; FreeQ[{a, b, c}, x]

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 4856

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))/((d_) + (e_.)*(x_)), x_Symbol] :> -Simp[((a + b*ArcTan[c*x])*Log[2/(1 -
I*c*x)])/e, x] + (Dist[(b*c)/e, Int[Log[2/(1 - I*c*x)]/(1 + c^2*x^2), x], x] - Dist[(b*c)/e, Int[Log[(2*c*(d
+ e*x))/((c*d + I*e)*(1 - I*c*x))]/(1 + c^2*x^2), x], x] + Simp[((a + b*ArcTan[c*x])*Log[(2*c*(d + e*x))/((c*d
+ I*e)*(1 - I*c*x))])/e, x]) /; FreeQ[{a, b, c, d, e}, x] && NeQ[c^2*d^2 + e^2, 0]

Rule 2402

Int[Log[(c_.)/((d_) + (e_.)*(x_))]/((f_) + (g_.)*(x_)^2), x_Symbol] :> -Dist[e/g, Subst[Int[Log[2*d*x]/(1 - 2*
d*x), x], x, 1/(d + e*x)], x] /; FreeQ[{c, d, e, f, g}, x] && EqQ[c, 2*d] && EqQ[e^2*f + d^2*g, 0]

Rule 2315

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> -Simp[PolyLog[2, 1 - c*x]/e, x] /; FreeQ[{c, d, e}, x] &
& EqQ[e + c*d, 0]

Rule 2447

Int[Log[u_]*(Pq_)^(m_.), x_Symbol] :> With[{C = FullSimplify[(Pq^m*(1 - u))/D[u, x]]}, Simp[C*PolyLog[2, 1 - u
], x] /; FreeQ[C, x]] /; IntegerQ[m] && PolyQ[Pq, x] && RationalFunctionQ[u, x] && LeQ[RationalFunctionExponen
ts[u, x][[2]], Expon[Pq, x]]

Rubi steps

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

Mathematica [A]  time = 1.34434, size = 671, normalized size = 1.72 \[ \frac{a \left (i \left (\text{PolyLog}\left (2,\frac{\left (2 i \sqrt{a^2 c d}+a^2 c-d\right ) \left (\sqrt{a^2 c d}+i a d x\right )}{\left (a^2 c+d\right ) \left (\sqrt{a^2 c d}-i a d x\right )}\right )-\text{PolyLog}\left (2,\frac{\left (-2 i \sqrt{a^2 c d}+a^2 c-d\right ) \left (\sqrt{a^2 c d}+i a d x\right )}{\left (a^2 c+d\right ) \left (\sqrt{a^2 c d}-i a d x\right )}\right )\right )-2 i \cos ^{-1}\left (\frac{a^2 c-d}{a^2 c+d}\right ) \tan ^{-1}\left (\frac{a c}{x \sqrt{a^2 c d}}\right )+4 \coth ^{-1}(a x) \tan ^{-1}\left (\frac{a d x}{\sqrt{a^2 c d}}\right )-\log \left (\frac{2 d (a x-1) \left (a^2 c-i \sqrt{a^2 c d}\right )}{\left (a^2 c+d\right ) \left (a d x+i \sqrt{a^2 c d}\right )}\right ) \left (2 \tan ^{-1}\left (\frac{a c}{x \sqrt{a^2 c d}}\right )+\cos ^{-1}\left (\frac{a^2 c-d}{a^2 c+d}\right )\right )-\log \left (\frac{2 d (a x+1) \left (a^2 c+i \sqrt{a^2 c d}\right )}{\left (a^2 c+d\right ) \left (a d x+i \sqrt{a^2 c d}\right )}\right ) \left (\cos ^{-1}\left (\frac{a^2 c-d}{a^2 c+d}\right )-2 \tan ^{-1}\left (\frac{a c}{x \sqrt{a^2 c d}}\right )\right )+\left (2 \left (\tan ^{-1}\left (\frac{a c}{x \sqrt{a^2 c d}}\right )+\tan ^{-1}\left (\frac{a d x}{\sqrt{a^2 c d}}\right )\right )+\cos ^{-1}\left (\frac{a^2 c-d}{a^2 c+d}\right )\right ) \log \left (\frac{\sqrt{2} \sqrt{a^2 c d} e^{-\coth ^{-1}(a x)}}{\sqrt{a^2 c+d} \sqrt{\left (a^2 c+d\right ) \cosh \left (2 \coth ^{-1}(a x)\right )+a^2 (-c)+d}}\right )+\left (\cos ^{-1}\left (\frac{a^2 c-d}{a^2 c+d}\right )-2 \left (\tan ^{-1}\left (\frac{a c}{x \sqrt{a^2 c d}}\right )+\tan ^{-1}\left (\frac{a d x}{\sqrt{a^2 c d}}\right )\right )\right ) \log \left (\frac{\sqrt{2} \sqrt{a^2 c d} e^{\coth ^{-1}(a x)}}{\sqrt{a^2 c+d} \sqrt{\left (a^2 c+d\right ) \cosh \left (2 \coth ^{-1}(a x)\right )+a^2 (-c)+d}}\right )\right )}{4 \sqrt{a^2 c d}} \]

Antiderivative was successfully verified.

[In]

Integrate[ArcCoth[a*x]/(c + d*x^2),x]

[Out]

(a*((-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)/Sqr
t[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]]))*L
og[(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])*(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))])))/(4*Sqrt[a^2*c*d])

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Maple [B]  time = 0.218, size = 785, normalized size = 2. \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),x)

[Out]

-1/2*a^3/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-a/(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))*arcco
th(a*x)*(-a^2*c*d)^(1/2)+1/2*a^3/d/(a^4*c^2+2*a^2*c*d+d^2)*arccoth(a*x)^2*(-a^2*c*d)^(1/2)*c+a/(a^4*c^2+2*a^2*
c*d+d^2)*arccoth(a*x)^2*(-a^2*c*d)^(1/2)-1/4*a^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))*(-a^2*c*d)^(1/2)*c-1/2*a/(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)-1/2/a/c/(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)*d+1/2/a/c/(a^4*c^2+2*a^2*c*d+d^2)*arccoth(a*
x)^2*(-a^2*c*d)^(1/2)*d-1/4/a/c/(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)*d+1/2/a*(-a^2*c*d)^(1/2)/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))-1/2/a*(-a^2*c*d)^(1/2)/c/d*arccoth(a*x)^2+1/4/a*(-a^2*c*d)^(1/2)/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))

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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),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 x^{2} + c}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(arccoth(a*x)/(d*x^2 + c), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{acoth}{\left (a x \right )}}{c + d x^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(acoth(a*x)/(c + d*x**2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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