\(\int \frac {\coth ^{-1}(a+b x)}{c+d x^2} \, dx\) [76]

Optimal result

Integrand size = 16, antiderivative size = 673 \[ \int \frac {\coth ^{-1}(a+b x)}{c+d x^2} \, dx=\frac {\log \left (-\frac {1-a-b x}{a+b x}\right ) \log \left (1+\frac {\left (b^2 c+a^2 d\right ) (1-a-b x)}{\left (b^2 c-b \sqrt {-c} \sqrt {d}-(1-a) a d\right ) (a+b x)}\right )}{4 \sqrt {-c} \sqrt {d}}-\frac {\log \left (-\frac {1-a-b x}{a+b x}\right ) \log \left (1+\frac {\left (b^2 c+a^2 d\right ) (1-a-b x)}{\left (b^2 c+b \sqrt {-c} \sqrt {d}-(1-a) a d\right ) (a+b x)}\right )}{4 \sqrt {-c} \sqrt {d}}+\frac {\log \left (\frac {1+a+b x}{a+b x}\right ) \log \left (1-\frac {\left (b^2 c+a^2 d\right ) (1+a+b x)}{\left (b^2 c-b \sqrt {-c} \sqrt {d}+a (1+a) d\right ) (a+b x)}\right )}{4 \sqrt {-c} \sqrt {d}}-\frac {\log \left (\frac {1+a+b x}{a+b x}\right ) \log \left (1-\frac {\left (b^2 c+a^2 d\right ) (1+a+b x)}{\left (b^2 c+b \sqrt {-c} \sqrt {d}+a (1+a) d\right ) (a+b x)}\right )}{4 \sqrt {-c} \sqrt {d}}+\frac {\operatorname {PolyLog}\left (2,-\frac {\left (b^2 c+a^2 d\right ) (1-a-b x)}{\left (b^2 c-b \sqrt {-c} \sqrt {d}-(1-a) a d\right ) (a+b x)}\right )}{4 \sqrt {-c} \sqrt {d}}-\frac {\operatorname {PolyLog}\left (2,-\frac {\left (b^2 c+a^2 d\right ) (1-a-b x)}{\left (b^2 c+b \sqrt {-c} \sqrt {d}-(1-a) a d\right ) (a+b x)}\right )}{4 \sqrt {-c} \sqrt {d}}+\frac {\operatorname {PolyLog}\left (2,\frac {\left (b^2 c+a^2 d\right ) (1+a+b x)}{\left (b^2 c-b \sqrt {-c} \sqrt {d}+a (1+a) d\right ) (a+b x)}\right )}{4 \sqrt {-c} \sqrt {d}}-\frac {\operatorname {PolyLog}\left (2,\frac {\left (b^2 c+a^2 d\right ) (1+a+b x)}{\left (b^2 c+b \sqrt {-c} \sqrt {d}+a (1+a) d\right ) (a+b x)}\right )}{4 \sqrt {-c} \sqrt {d}} \]

[Out]

1/4*ln((b*x+a-1)/(b*x+a))*ln(1+(a^2*d+b^2*c)*(-b*x-a+1)/(b*x+a)/(b^2*c-(1-a)*a*d-b*(-c)^(1/2)*d^(1/2)))/(-c)^(
1/2)/d^(1/2)+1/4*ln((b*x+a+1)/(b*x+a))*ln(1-(a^2*d+b^2*c)*(b*x+a+1)/(b*x+a)/(b^2*c+a*(1+a)*d-b*(-c)^(1/2)*d^(1
/2)))/(-c)^(1/2)/d^(1/2)-1/4*ln((b*x+a-1)/(b*x+a))*ln(1+(a^2*d+b^2*c)*(-b*x-a+1)/(b*x+a)/(b^2*c-(1-a)*a*d+b*(-
c)^(1/2)*d^(1/2)))/(-c)^(1/2)/d^(1/2)-1/4*ln((b*x+a+1)/(b*x+a))*ln(1-(a^2*d+b^2*c)*(b*x+a+1)/(b*x+a)/(b^2*c+a*
(1+a)*d+b*(-c)^(1/2)*d^(1/2)))/(-c)^(1/2)/d^(1/2)+1/4*polylog(2,-(a^2*d+b^2*c)*(-b*x-a+1)/(b*x+a)/(b^2*c-(1-a)
*a*d-b*(-c)^(1/2)*d^(1/2)))/(-c)^(1/2)/d^(1/2)+1/4*polylog(2,(a^2*d+b^2*c)*(b*x+a+1)/(b*x+a)/(b^2*c+a*(1+a)*d-
b*(-c)^(1/2)*d^(1/2)))/(-c)^(1/2)/d^(1/2)-1/4*polylog(2,-(a^2*d+b^2*c)*(-b*x-a+1)/(b*x+a)/(b^2*c-(1-a)*a*d+b*(
-c)^(1/2)*d^(1/2)))/(-c)^(1/2)/d^(1/2)-1/4*polylog(2,(a^2*d+b^2*c)*(b*x+a+1)/(b*x+a)/(b^2*c+a*(1+a)*d+b*(-c)^(
1/2)*d^(1/2)))/(-c)^(1/2)/d^(1/2)

Rubi [A] (verified)

Time = 0.76 (sec) , antiderivative size = 673, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {6251, 2576, 2404, 2354, 2438} \[ \int \frac {\coth ^{-1}(a+b x)}{c+d x^2} \, dx=\frac {\operatorname {PolyLog}\left (2,-\frac {\left (d a^2+b^2 c\right ) (-a-b x+1)}{\left (c b^2-\sqrt {-c} \sqrt {d} b-(1-a) a d\right ) (a+b x)}\right )}{4 \sqrt {-c} \sqrt {d}}-\frac {\operatorname {PolyLog}\left (2,-\frac {\left (d a^2+b^2 c\right ) (-a-b x+1)}{\left (c b^2+\sqrt {-c} \sqrt {d} b-(1-a) a d\right ) (a+b x)}\right )}{4 \sqrt {-c} \sqrt {d}}+\frac {\operatorname {PolyLog}\left (2,\frac {\left (d a^2+b^2 c\right ) (a+b x+1)}{\left (c b^2-\sqrt {-c} \sqrt {d} b+a (a+1) d\right ) (a+b x)}\right )}{4 \sqrt {-c} \sqrt {d}}-\frac {\operatorname {PolyLog}\left (2,\frac {\left (d a^2+b^2 c\right ) (a+b x+1)}{\left (c b^2+\sqrt {-c} \sqrt {d} b+a (a+1) d\right ) (a+b x)}\right )}{4 \sqrt {-c} \sqrt {d}}+\frac {\log \left (-\frac {-a-b x+1}{a+b x}\right ) \log \left (\frac {(-a-b x+1) \left (a^2 d+b^2 c\right )}{(a+b x) \left (-(1-a) a d+b^2 c-b \sqrt {-c} \sqrt {d}\right )}+1\right )}{4 \sqrt {-c} \sqrt {d}}-\frac {\log \left (-\frac {-a-b x+1}{a+b x}\right ) \log \left (\frac {(-a-b x+1) \left (a^2 d+b^2 c\right )}{(a+b x) \left (-(1-a) a d+b^2 c+b \sqrt {-c} \sqrt {d}\right )}+1\right )}{4 \sqrt {-c} \sqrt {d}}+\frac {\log \left (\frac {a+b x+1}{a+b x}\right ) \log \left (1-\frac {(a+b x+1) \left (a^2 d+b^2 c\right )}{(a+b x) \left (a (a+1) d+b^2 c-b \sqrt {-c} \sqrt {d}\right )}\right )}{4 \sqrt {-c} \sqrt {d}}-\frac {\log \left (\frac {a+b x+1}{a+b x}\right ) \log \left (1-\frac {(a+b x+1) \left (a^2 d+b^2 c\right )}{(a+b x) \left (a (a+1) d+b^2 c+b \sqrt {-c} \sqrt {d}\right )}\right )}{4 \sqrt {-c} \sqrt {d}} \]

[In]

Int[ArcCoth[a + b*x]/(c + d*x^2),x]

[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])

Rule 2354

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

Rule 2404

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(RFx_), x_Symbol] :> With[{u = ExpandIntegrand[(a + b*Log[c*x^
n])^p, RFx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, n}, x] && RationalFunctionQ[RFx, x] && IGtQ[p, 0]

Rule 2438

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 2576

Int[((A_.) + Log[(e_.)*(((a_.) + (b_.)*(x_))/((c_.) + (d_.)*(x_)))^(n_.)]*(B_.))^(p_.)*(P2x_)^(m_.), x_Symbol]
:> With[{f = Coeff[P2x, x, 0], g = Coeff[P2x, x, 1], h = Coeff[P2x, x, 2]}, Dist[b*c - a*d, Subst[Int[(b^2*f
- a*b*g + a^2*h - (2*b*d*f - b*c*g - a*d*g + 2*a*c*h)*x + (d^2*f - c*d*g + c^2*h)*x^2)^m*((A + B*Log[e*x^n])^p
/(b - d*x)^(2*(m + 1))), x], x, (a + b*x)/(c + d*x)], x]] /; FreeQ[{a, b, c, d, e, A, B, n}, x] && PolyQ[P2x,
x, 2] && NeQ[b*c - a*d, 0] && IntegerQ[m] && IGtQ[p, 0]

Rule 6251

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]

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

Mathematica [A] (verified)

Time = 0.41 (sec) , antiderivative size = 529, normalized size of antiderivative = 0.79 \[ \int \frac {\coth ^{-1}(a+b x)}{c+d x^2} \, dx=\frac {\log \left (\frac {\sqrt {d} (-1+a+b x)}{b \sqrt {-c}+(-1+a) \sqrt {d}}\right ) \log \left (\sqrt {-c}-\sqrt {d} x\right )-\log \left (\frac {-1+a+b x}{a+b x}\right ) \log \left (\sqrt {-c}-\sqrt {d} x\right )-\log \left (\frac {\sqrt {d} (1+a+b x)}{b \sqrt {-c}+(1+a) \sqrt {d}}\right ) \log \left (\sqrt {-c}-\sqrt {d} x\right )+\log \left (\frac {1+a+b x}{a+b x}\right ) \log \left (\sqrt {-c}-\sqrt {d} x\right )-\log \left (-\frac {\sqrt {d} (-1+a+b x)}{b \sqrt {-c}-(-1+a) \sqrt {d}}\right ) \log \left (\sqrt {-c}+\sqrt {d} x\right )+\log \left (\frac {-1+a+b x}{a+b x}\right ) \log \left (\sqrt {-c}+\sqrt {d} x\right )+\log \left (-\frac {\sqrt {d} (1+a+b x)}{b \sqrt {-c}-(1+a) \sqrt {d}}\right ) \log \left (\sqrt {-c}+\sqrt {d} x\right )-\log \left (\frac {1+a+b x}{a+b x}\right ) \log \left (\sqrt {-c}+\sqrt {d} x\right )+\operatorname {PolyLog}\left (2,\frac {b \left (\sqrt {-c}-\sqrt {d} x\right )}{b \sqrt {-c}+(-1+a) \sqrt {d}}\right )-\operatorname {PolyLog}\left (2,\frac {b \left (\sqrt {-c}-\sqrt {d} x\right )}{b \sqrt {-c}+(1+a) \sqrt {d}}\right )-\operatorname {PolyLog}\left (2,\frac {b \left (\sqrt {-c}+\sqrt {d} x\right )}{b \sqrt {-c}-(-1+a) \sqrt {d}}\right )+\operatorname {PolyLog}\left (2,\frac {b \left (\sqrt {-c}+\sqrt {d} x\right )}{b \sqrt {-c}-(1+a) \sqrt {d}}\right )}{4 \sqrt {-c} \sqrt {d}} \]

[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])

Maple [A] (verified)

Time = 0.89 (sec) , antiderivative size = 426, normalized size of antiderivative = 0.63

method	result	size
risch	\(-\frac {\ln \left (b x +a -1\right ) \ln \left (\frac {b \sqrt {-c d}-\left (b x +a -1\right ) d +a d -d}{b \sqrt {-c d}+a d -d}\right )}{4 \sqrt {-c d}}+\frac {\ln \left (b x +a -1\right ) \ln \left (\frac {b \sqrt {-c d}+\left (b x +a -1\right ) d -a d +d}{b \sqrt {-c d}-a d +d}\right )}{4 \sqrt {-c d}}-\frac {\operatorname {dilog}\left (\frac {b \sqrt {-c d}-\left (b x +a -1\right ) d +a d -d}{b \sqrt {-c d}+a d -d}\right )}{4 \sqrt {-c d}}+\frac {\operatorname {dilog}\left (\frac {b \sqrt {-c d}+\left (b x +a -1\right ) d -a d +d}{b \sqrt {-c d}-a d +d}\right )}{4 \sqrt {-c d}}+\frac {\ln \left (b x +a +1\right ) \ln \left (\frac {b \sqrt {-c d}-\left (b x +a +1\right ) d +a d +d}{b \sqrt {-c d}+a d +d}\right )}{4 \sqrt {-c d}}-\frac {\ln \left (b x +a +1\right ) \ln \left (\frac {b \sqrt {-c d}+\left (b x +a +1\right ) d -a d -d}{b \sqrt {-c d}-a d -d}\right )}{4 \sqrt {-c d}}+\frac {\operatorname {dilog}\left (\frac {b \sqrt {-c d}-\left (b x +a +1\right ) d +a d +d}{b \sqrt {-c d}+a d +d}\right )}{4 \sqrt {-c d}}-\frac {\operatorname {dilog}\left (\frac {b \sqrt {-c d}+\left (b x +a +1\right ) d -a d -d}{b \sqrt {-c d}-a d -d}\right )}{4 \sqrt {-c d}}\)	\(426\)
derivativedivides	\(\text {Expression too large to display}\)	\(1219\)
default	\(\text {Expression too large to display}\)	\(1219\)

[In]

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

[Out]

-1/4*ln(b*x+a-1)/(-c*d)^(1/2)*ln((b*(-c*d)^(1/2)-(b*x+a-1)*d+a*d-d)/(b*(-c*d)^(1/2)+a*d-d))+1/4*ln(b*x+a-1)/(-
c*d)^(1/2)*ln((b*(-c*d)^(1/2)+(b*x+a-1)*d-a*d+d)/(b*(-c*d)^(1/2)-a*d+d))-1/4/(-c*d)^(1/2)*dilog((b*(-c*d)^(1/2
)-(b*x+a-1)*d+a*d-d)/(b*(-c*d)^(1/2)+a*d-d))+1/4/(-c*d)^(1/2)*dilog((b*(-c*d)^(1/2)+(b*x+a-1)*d-a*d+d)/(b*(-c*
d)^(1/2)-a*d+d))+1/4*ln(b*x+a+1)/(-c*d)^(1/2)*ln((b*(-c*d)^(1/2)-(b*x+a+1)*d+a*d+d)/(b*(-c*d)^(1/2)+a*d+d))-1/
4*ln(b*x+a+1)/(-c*d)^(1/2)*ln((b*(-c*d)^(1/2)+(b*x+a+1)*d-a*d-d)/(b*(-c*d)^(1/2)-a*d-d))+1/4/(-c*d)^(1/2)*dilo
g((b*(-c*d)^(1/2)-(b*x+a+1)*d+a*d+d)/(b*(-c*d)^(1/2)+a*d+d))-1/4/(-c*d)^(1/2)*dilog((b*(-c*d)^(1/2)+(b*x+a+1)*
d-a*d-d)/(b*(-c*d)^(1/2)-a*d-d))

Fricas [F]

\[ \int \frac {\coth ^{-1}(a+b x)}{c+d x^2} \, dx=\int { \frac {\operatorname {arcoth}\left (b x + a\right )}{d x^{2} + c} \,d x } \]

[In]

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

[Out]

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

Sympy [F(-1)]

Timed out. \[ \int \frac {\coth ^{-1}(a+b x)}{c+d x^2} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.42 (sec) , antiderivative size = 591, normalized size of antiderivative = 0.88 \[ \int \frac {\coth ^{-1}(a+b x)}{c+d x^2} \, dx=\frac {\operatorname {arcoth}\left (b x + a\right ) \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{\sqrt {c d}} + \frac {{\left (\arctan \left (\frac {{\left (b^{2} x + {\left (a + 1\right )} b\right )} \sqrt {c} \sqrt {d}}{b^{2} c + {\left (a^{2} + 2 \, a + 1\right )} d}, \frac {{\left (a + 1\right )} b d x + {\left (a^{2} + 2 \, a + 1\right )} d}{b^{2} c + {\left (a^{2} + 2 \, a + 1\right )} d}\right ) - \arctan \left (\frac {{\left (b^{2} x + {\left (a - 1\right )} b\right )} \sqrt {c} \sqrt {d}}{b^{2} c + {\left (a^{2} - 2 \, a + 1\right )} d}, \frac {{\left (a - 1\right )} b d x + {\left (a^{2} - 2 \, a + 1\right )} d}{b^{2} c + {\left (a^{2} - 2 \, a + 1\right )} d}\right )\right )} \log \left (d x^{2} + c\right ) - \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \log \left (\frac {b^{2} d x^{2} + 2 \, {\left (a + 1\right )} b d x + {\left (a^{2} + 2 \, a + 1\right )} d}{b^{2} c + {\left (a^{2} + 2 \, a + 1\right )} d}\right ) + \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \log \left (\frac {b^{2} d x^{2} + 2 \, {\left (a - 1\right )} b d x + {\left (a^{2} - 2 \, a + 1\right )} d}{b^{2} c + {\left (a^{2} - 2 \, a + 1\right )} d}\right ) - i \, {\rm Li}_2\left (\frac {{\left (a - 1\right )} b d x + b^{2} c + {\left (i \, b^{2} x + {\left (-i \, a + i\right )} b\right )} \sqrt {c} \sqrt {d}}{b^{2} c + 2 \, {\left (-i \, a + i\right )} b \sqrt {c} \sqrt {d} - {\left (a^{2} - 2 \, a + 1\right )} d}\right ) + i \, {\rm Li}_2\left (\frac {{\left (a - 1\right )} b d x + b^{2} c - {\left (i \, b^{2} x + {\left (-i \, a + i\right )} b\right )} \sqrt {c} \sqrt {d}}{b^{2} c - 2 \, {\left (-i \, a + i\right )} b \sqrt {c} \sqrt {d} - {\left (a^{2} - 2 \, a + 1\right )} d}\right ) + i \, {\rm Li}_2\left (\frac {{\left (a + 1\right )} b d x + b^{2} c + {\left (i \, b^{2} x + {\left (-i \, a - i\right )} b\right )} \sqrt {c} \sqrt {d}}{b^{2} c + 2 \, {\left (-i \, a - i\right )} b \sqrt {c} \sqrt {d} - {\left (a^{2} + 2 \, a + 1\right )} d}\right ) - i \, {\rm Li}_2\left (\frac {{\left (a + 1\right )} b d x + b^{2} c - {\left (i \, b^{2} x + {\left (-i \, a - i\right )} b\right )} \sqrt {c} \sqrt {d}}{b^{2} c - 2 \, {\left (-i \, a - i\right )} b \sqrt {c} \sqrt {d} - {\left (a^{2} + 2 \, a + 1\right )} d}\right )}{4 \, \sqrt {c d}} \]

[In]

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

[Out]

arccoth(b*x + a)*arctan(d*x/sqrt(c*d))/sqrt(c*d) + 1/4*((arctan2((b^2*x + (a + 1)*b)*sqrt(c)*sqrt(d)/(b^2*c +
(a^2 + 2*a + 1)*d), ((a + 1)*b*d*x + (a^2 + 2*a + 1)*d)/(b^2*c + (a^2 + 2*a + 1)*d)) - arctan2((b^2*x + (a - 1
)*b)*sqrt(c)*sqrt(d)/(b^2*c + (a^2 - 2*a + 1)*d), ((a - 1)*b*d*x + (a^2 - 2*a + 1)*d)/(b^2*c + (a^2 - 2*a + 1)
*d)))*log(d*x^2 + c) - arctan(sqrt(d)*x/sqrt(c))*log((b^2*d*x^2 + 2*(a + 1)*b*d*x + (a^2 + 2*a + 1)*d)/(b^2*c
+ (a^2 + 2*a + 1)*d)) + arctan(sqrt(d)*x/sqrt(c))*log((b^2*d*x^2 + 2*(a - 1)*b*d*x + (a^2 - 2*a + 1)*d)/(b^2*c
+ (a^2 - 2*a + 1)*d)) - I*dilog(((a - 1)*b*d*x + b^2*c + (I*b^2*x + (-I*a + I)*b)*sqrt(c)*sqrt(d))/(b^2*c + 2
*(-I*a + I)*b*sqrt(c)*sqrt(d) - (a^2 - 2*a + 1)*d)) + I*dilog(((a - 1)*b*d*x + b^2*c - (I*b^2*x + (-I*a + I)*b
)*sqrt(c)*sqrt(d))/(b^2*c - 2*(-I*a + I)*b*sqrt(c)*sqrt(d) - (a^2 - 2*a + 1)*d)) + I*dilog(((a + 1)*b*d*x + b^
2*c + (I*b^2*x + (-I*a - I)*b)*sqrt(c)*sqrt(d))/(b^2*c + 2*(-I*a - I)*b*sqrt(c)*sqrt(d) - (a^2 + 2*a + 1)*d))
- I*dilog(((a + 1)*b*d*x + b^2*c - (I*b^2*x + (-I*a - I)*b)*sqrt(c)*sqrt(d))/(b^2*c - 2*(-I*a - I)*b*sqrt(c)*s
qrt(d) - (a^2 + 2*a + 1)*d)))/sqrt(c*d)

Giac [F]

\[ \int \frac {\coth ^{-1}(a+b x)}{c+d x^2} \, dx=\int { \frac {\operatorname {arcoth}\left (b x + a\right )}{d x^{2} + c} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\coth ^{-1}(a+b x)}{c+d x^2} \, dx=\int \frac {\mathrm {acoth}\left (a+b\,x\right )}{d\,x^2+c} \,d x \]

[In]

int(acoth(a + b*x)/(c + d*x^2),x)

[Out]

int(acoth(a + b*x)/(c + d*x^2), x)