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

Optimal. Leaf size=673 \[ \frac{\text{PolyLog}\left (2,-\frac{(-a-b x+1) \left (a^2 d+b^2 c\right )}{(a+b x) \left (-(1-a) a d+b^2 c-b \sqrt{-c} \sqrt{d}\right )}\right )}{4 \sqrt{-c} \sqrt{d}}-\frac{\text{PolyLog}\left (2,-\frac{(-a-b x+1) \left (a^2 d+b^2 c\right )}{(a+b x) \left (-(1-a) a d+b^2 c+b \sqrt{-c} \sqrt{d}\right )}\right )}{4 \sqrt{-c} \sqrt{d}}+\frac{\text{PolyLog}\left (2,\frac{(a+b x+1) \left (a^2 d+b^2 c\right )}{(a+b x) \left (a (a+1) d+b^2 c-b \sqrt{-c} \sqrt{d}\right )}\right )}{4 \sqrt{-c} \sqrt{d}}-\frac{\text{PolyLog}\left (2,\frac{(a+b x+1) \left (a^2 d+b^2 c\right )}{(a+b x) \left (a (a+1) d+b^2 c+b \sqrt{-c} \sqrt{d}\right )}\right )}{4 \sqrt{-c} \sqrt{d}}+\frac{\log \left (-\frac{-a-b x+1}{a+b x}\right ) \log \left (\frac{(-a-b x+1) \left (a^2 d+b^2 c\right )}{(a+b x) \left (-(1-a) a d+b^2 c-b \sqrt{-c} \sqrt{d}\right )}+1\right )}{4 \sqrt{-c} \sqrt{d}}-\frac{\log \left (-\frac{-a-b x+1}{a+b x}\right ) \log \left (\frac{(-a-b x+1) \left (a^2 d+b^2 c\right )}{(a+b x) \left (-(1-a) a d+b^2 c+b \sqrt{-c} \sqrt{d}\right )}+1\right )}{4 \sqrt{-c} \sqrt{d}}+\frac{\log \left (\frac{a+b x+1}{a+b x}\right ) \log \left (1-\frac{(a+b x+1) \left (a^2 d+b^2 c\right )}{(a+b x) \left (a (a+1) d+b^2 c-b \sqrt{-c} \sqrt{d}\right )}\right )}{4 \sqrt{-c} \sqrt{d}}-\frac{\log \left (\frac{a+b x+1}{a+b x}\right ) \log \left (1-\frac{(a+b x+1) \left (a^2 d+b^2 c\right )}{(a+b x) \left (a (a+1) d+b^2 c+b \sqrt{-c} \sqrt{d}\right )}\right )}{4 \sqrt{-c} \sqrt{d}} \]

[Out]

(Log[-((1 - a - b*x)/(a + b*x))]*Log[1 + ((b^2*c + a^2*d)*(1 - a - b*x))/((b^2*c - b*Sqrt[-c]*Sqrt[d] - (1 - a
)*a*d)*(a + b*x))])/(4*Sqrt[-c]*Sqrt[d]) - (Log[-((1 - a - b*x)/(a + b*x))]*Log[1 + ((b^2*c + a^2*d)*(1 - a -
b*x))/((b^2*c + b*Sqrt[-c]*Sqrt[d] - (1 - a)*a*d)*(a + b*x))])/(4*Sqrt[-c]*Sqrt[d]) + (Log[(1 + a + b*x)/(a +
b*x)]*Log[1 - ((b^2*c + a^2*d)*(1 + a + b*x))/((b^2*c - b*Sqrt[-c]*Sqrt[d] + a*(1 + a)*d)*(a + b*x))])/(4*Sqrt
[-c]*Sqrt[d]) - (Log[(1 + a + b*x)/(a + b*x)]*Log[1 - ((b^2*c + a^2*d)*(1 + a + b*x))/((b^2*c + b*Sqrt[-c]*Sqr
t[d] + a*(1 + a)*d)*(a + b*x))])/(4*Sqrt[-c]*Sqrt[d]) + PolyLog[2, -(((b^2*c + a^2*d)*(1 - a - b*x))/((b^2*c -
b*Sqrt[-c]*Sqrt[d] - (1 - a)*a*d)*(a + b*x)))]/(4*Sqrt[-c]*Sqrt[d]) - PolyLog[2, -(((b^2*c + a^2*d)*(1 - a -
b*x))/((b^2*c + b*Sqrt[-c]*Sqrt[d] - (1 - a)*a*d)*(a + b*x)))]/(4*Sqrt[-c]*Sqrt[d]) + PolyLog[2, ((b^2*c + a^2
*d)*(1 + a + b*x))/((b^2*c - b*Sqrt[-c]*Sqrt[d] + a*(1 + a)*d)*(a + b*x))]/(4*Sqrt[-c]*Sqrt[d]) - PolyLog[2, (
(b^2*c + a^2*d)*(1 + a + b*x))/((b^2*c + b*Sqrt[-c]*Sqrt[d] + a*(1 + a)*d)*(a + b*x))]/(4*Sqrt[-c]*Sqrt[d])

________________________________________________________________________________________

Rubi [A]  time = 1.09947, antiderivative size = 597, normalized size of antiderivative = 0.89, number of steps used = 37, number of rules used = 7, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.438, Rules used = {6116, 2513, 2409, 2394, 2393, 2391, 205} \[ -\frac{\text{PolyLog}\left (2,-\frac{\sqrt{d} (-a-b x+1)}{b \sqrt{-c}-(1-a) \sqrt{d}}\right )}{4 \sqrt{-c} \sqrt{d}}+\frac{\text{PolyLog}\left (2,\frac{\sqrt{d} (-a-b x+1)}{(1-a) \sqrt{d}+b \sqrt{-c}}\right )}{4 \sqrt{-c} \sqrt{d}}-\frac{\text{PolyLog}\left (2,-\frac{\sqrt{d} (a+b x+1)}{b \sqrt{-c}-(a+1) \sqrt{d}}\right )}{4 \sqrt{-c} \sqrt{d}}+\frac{\text{PolyLog}\left (2,\frac{\sqrt{d} (a+b x+1)}{(a+1) \sqrt{d}+b \sqrt{-c}}\right )}{4 \sqrt{-c} \sqrt{d}}-\frac{\log (a+b x-1) \log \left (\frac{b \left (\sqrt{-c}-\sqrt{d} x\right )}{b \sqrt{-c}-(1-a) \sqrt{d}}\right )}{4 \sqrt{-c} \sqrt{d}}+\frac{\log (a+b x+1) \log \left (\frac{b \left (\sqrt{-c}-\sqrt{d} x\right )}{(a+1) \sqrt{d}+b \sqrt{-c}}\right )}{4 \sqrt{-c} \sqrt{d}}+\frac{\log (a+b x-1) \log \left (\frac{b \left (\sqrt{-c}+\sqrt{d} x\right )}{(1-a) \sqrt{d}+b \sqrt{-c}}\right )}{4 \sqrt{-c} \sqrt{d}}-\frac{\log (a+b x+1) \log \left (\frac{b \left (\sqrt{-c}+\sqrt{d} x\right )}{b \sqrt{-c}-(a+1) \sqrt{d}}\right )}{4 \sqrt{-c} \sqrt{d}}+\frac{\left (\log (a+b x-1)-\log \left (-\frac{-a-b x+1}{a+b x}\right )-\log (a+b x)\right ) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{2 \sqrt{c} \sqrt{d}}+\frac{\left (\log (a+b x)-\log (a+b x+1)+\log \left (\frac{a+b x+1}{a+b x}\right )\right ) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{2 \sqrt{c} \sqrt{d}} \]

Warning: Unable to verify antiderivative.

[In]

Int[ArcCoth[a + b*x]/(c + d*x^2),x]

[Out]

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

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

Mathematica [A]  time = 0.58637, size = 529, normalized size = 0.79 \[ \frac{\text{PolyLog}\left (2,\frac{b \left (\sqrt{-c}-\sqrt{d} x\right )}{(a-1) \sqrt{d}+b \sqrt{-c}}\right )-\text{PolyLog}\left (2,\frac{b \left (\sqrt{-c}-\sqrt{d} x\right )}{(a+1) \sqrt{d}+b \sqrt{-c}}\right )-\text{PolyLog}\left (2,\frac{b \left (\sqrt{-c}+\sqrt{d} x\right )}{b \sqrt{-c}-(a-1) \sqrt{d}}\right )+\text{PolyLog}\left (2,\frac{b \left (\sqrt{-c}+\sqrt{d} x\right )}{b \sqrt{-c}-(a+1) \sqrt{d}}\right )+\log \left (\sqrt{-c}-\sqrt{d} x\right ) \log \left (\frac{\sqrt{d} (a+b x-1)}{(a-1) \sqrt{d}+b \sqrt{-c}}\right )-\log \left (\frac{a+b x-1}{a+b x}\right ) \log \left (\sqrt{-c}-\sqrt{d} x\right )-\log \left (\sqrt{-c}-\sqrt{d} x\right ) \log \left (\frac{\sqrt{d} (a+b x+1)}{(a+1) \sqrt{d}+b \sqrt{-c}}\right )+\log \left (\frac{a+b x+1}{a+b x}\right ) \log \left (\sqrt{-c}-\sqrt{d} x\right )-\log \left (\sqrt{-c}+\sqrt{d} x\right ) \log \left (-\frac{\sqrt{d} (a+b x-1)}{b \sqrt{-c}-(a-1) \sqrt{d}}\right )+\log \left (\frac{a+b x-1}{a+b x}\right ) \log \left (\sqrt{-c}+\sqrt{d} x\right )+\log \left (\sqrt{-c}+\sqrt{d} x\right ) \log \left (-\frac{\sqrt{d} (a+b x+1)}{b \sqrt{-c}-(a+1) \sqrt{d}}\right )-\log \left (\frac{a+b x+1}{a+b x}\right ) \log \left (\sqrt{-c}+\sqrt{d} x\right )}{4 \sqrt{-c} \sqrt{d}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(Log[(Sqrt[d]*(-1 + a + b*x))/(b*Sqrt[-c] + (-1 + a)*Sqrt[d])]*Log[Sqrt[-c] - Sqrt[d]*x] - Log[(-1 + a + b*x)/
(a + b*x)]*Log[Sqrt[-c] - Sqrt[d]*x] - Log[(Sqrt[d]*(1 + a + b*x))/(b*Sqrt[-c] + (1 + a)*Sqrt[d])]*Log[Sqrt[-c
] - Sqrt[d]*x] + Log[(1 + a + b*x)/(a + b*x)]*Log[Sqrt[-c] - Sqrt[d]*x] - Log[-((Sqrt[d]*(-1 + a + b*x))/(b*Sq
rt[-c] - (-1 + a)*Sqrt[d]))]*Log[Sqrt[-c] + Sqrt[d]*x] + Log[(-1 + a + b*x)/(a + b*x)]*Log[Sqrt[-c] + Sqrt[d]*
x] + Log[-((Sqrt[d]*(1 + a + b*x))/(b*Sqrt[-c] - (1 + a)*Sqrt[d]))]*Log[Sqrt[-c] + Sqrt[d]*x] - Log[(1 + a + b
*x)/(a + b*x)]*Log[Sqrt[-c] + Sqrt[d]*x] + PolyLog[2, (b*(Sqrt[-c] - Sqrt[d]*x))/(b*Sqrt[-c] + (-1 + a)*Sqrt[d
])] - PolyLog[2, (b*(Sqrt[-c] - Sqrt[d]*x))/(b*Sqrt[-c] + (1 + a)*Sqrt[d])] - PolyLog[2, (b*(Sqrt[-c] + Sqrt[d
]*x))/(b*Sqrt[-c] - (-1 + a)*Sqrt[d])] + PolyLog[2, (b*(Sqrt[-c] + Sqrt[d]*x))/(b*Sqrt[-c] - (1 + a)*Sqrt[d])]
)/(4*Sqrt[-c]*Sqrt[d])

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Maple [B]  time = 0.49, size = 1230, normalized size = 1.8 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2/b*(-b^2*c*d)^(1/2)/c/d*arccoth(b*x+a)*ln(1-(a^2*d+b^2*c-2*a*d+d)*(b*x+a+1)/(b*x+a-1)/(a^2*d+b^2*c+2*(-b^2*
c*d)^(1/2)-d))-1/2/b*(-b^2*c*d)^(1/2)/c/d*arccoth(b*x+a)^2+1/4/b*(-b^2*c*d)^(1/2)/c/d*polylog(2,(a^2*d+b^2*c-2
*a*d+d)*(b*x+a+1)/(b*x+a-1)/(a^2*d+b^2*c+2*(-b^2*c*d)^(1/2)-d))-1/2/b*(-b^2*c*d)^(1/2)/c*ln(1-(a^2*d+b^2*c-2*a
*d+d)*(b*x+a+1)/(b*x+a-1)/(a^2*d+b^2*c-2*(-b^2*c*d)^(1/2)-d))*arccoth(b*x+a)*a^2/(a^2*d+b^2*c-2*(-b^2*c*d)^(1/
2)-d)-1/2*b*(-b^2*c*d)^(1/2)/d*ln(1-(a^2*d+b^2*c-2*a*d+d)*(b*x+a+1)/(b*x+a-1)/(a^2*d+b^2*c-2*(-b^2*c*d)^(1/2)-
d))*arccoth(b*x+a)/(a^2*d+b^2*c-2*(-b^2*c*d)^(1/2)-d)+1/2/b*(-b^2*c*d)^(1/2)/c*arccoth(b*x+a)^2*a^2/(a^2*d+b^2
*c-2*(-b^2*c*d)^(1/2)-d)+1/2*b*(-b^2*c*d)^(1/2)/d*arccoth(b*x+a)^2/(a^2*d+b^2*c-2*(-b^2*c*d)^(1/2)-d)-1/4/b*(-
b^2*c*d)^(1/2)/c*polylog(2,(a^2*d+b^2*c-2*a*d+d)*(b*x+a+1)/(b*x+a-1)/(a^2*d+b^2*c-2*(-b^2*c*d)^(1/2)-d))*a^2/(
a^2*d+b^2*c-2*(-b^2*c*d)^(1/2)-d)-1/4*b*(-b^2*c*d)^(1/2)/d*polylog(2,(a^2*d+b^2*c-2*a*d+d)*(b*x+a+1)/(b*x+a-1)
/(a^2*d+b^2*c-2*(-b^2*c*d)^(1/2)-d))/(a^2*d+b^2*c-2*(-b^2*c*d)^(1/2)-d)-b/(a^2*d+b^2*c-2*(-b^2*c*d)^(1/2)-d)*l
n(1-(a^2*d+b^2*c-2*a*d+d)*(b*x+a+1)/(b*x+a-1)/(a^2*d+b^2*c-2*(-b^2*c*d)^(1/2)-d))*arccoth(b*x+a)+b/(a^2*d+b^2*
c-2*(-b^2*c*d)^(1/2)-d)*arccoth(b*x+a)^2+1/2/b*(-b^2*c*d)^(1/2)/c*ln(1-(a^2*d+b^2*c-2*a*d+d)*(b*x+a+1)/(b*x+a-
1)/(a^2*d+b^2*c-2*(-b^2*c*d)^(1/2)-d))*arccoth(b*x+a)/(a^2*d+b^2*c-2*(-b^2*c*d)^(1/2)-d)-1/2/b*(-b^2*c*d)^(1/2
)/c*arccoth(b*x+a)^2/(a^2*d+b^2*c-2*(-b^2*c*d)^(1/2)-d)-1/2*b/(a^2*d+b^2*c-2*(-b^2*c*d)^(1/2)-d)*polylog(2,(a^
2*d+b^2*c-2*a*d+d)*(b*x+a+1)/(b*x+a-1)/(a^2*d+b^2*c-2*(-b^2*c*d)^(1/2)-d))+1/4/b*(-b^2*c*d)^(1/2)/c*polylog(2,
(a^2*d+b^2*c-2*a*d+d)*(b*x+a+1)/(b*x+a-1)/(a^2*d+b^2*c-2*(-b^2*c*d)^(1/2)-d))/(a^2*d+b^2*c-2*(-b^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(b*x+a)/(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 (b x + a\right )}{d x^{2} + c}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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