Integrand size = 14, antiderivative size = 120 \[ \int \frac {\coth ^{-1}(a+b x)}{c+d x} \, dx=-\frac {\coth ^{-1}(a+b x) \log \left (\frac {2}{1+a+b x}\right )}{d}+\frac {\coth ^{-1}(a+b x) \log \left (\frac {2 b (c+d x)}{(b c+d-a d) (1+a+b x)}\right )}{d}+\frac {\operatorname {PolyLog}\left (2,1-\frac {2}{1+a+b x}\right )}{2 d}-\frac {\operatorname {PolyLog}\left (2,1-\frac {2 b (c+d x)}{(b c+d-a d) (1+a+b x)}\right )}{2 d} \]
-arccoth(b*x+a)*ln(2/(b*x+a+1))/d+arccoth(b*x+a)*ln(2*b*(d*x+c)/(-a*d+b*c+ d)/(b*x+a+1))/d+1/2*polylog(2,1-2/(b*x+a+1))/d-1/2*polylog(2,1-2*b*(d*x+c) /(-a*d+b*c+d)/(b*x+a+1))/d
Time = 0.05 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.54 \[ \int \frac {\coth ^{-1}(a+b x)}{c+d x} \, dx=\frac {\log \left (\frac {d (1-a-b x)}{b c+d-a d}\right ) \log (c+d x)}{2 d}-\frac {\log \left (\frac {-1+a+b x}{a+b x}\right ) \log (c+d x)}{2 d}-\frac {\log \left (-\frac {d (1+a+b x)}{b c-d-a d}\right ) \log (c+d x)}{2 d}+\frac {\log \left (\frac {1+a+b x}{a+b x}\right ) \log (c+d x)}{2 d}-\frac {\operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c-d-a d}\right )}{2 d}+\frac {\operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c+d-a d}\right )}{2 d} \]
(Log[(d*(1 - a - b*x))/(b*c + d - a*d)]*Log[c + d*x])/(2*d) - (Log[(-1 + a + b*x)/(a + b*x)]*Log[c + d*x])/(2*d) - (Log[-((d*(1 + a + b*x))/(b*c - d - a*d))]*Log[c + d*x])/(2*d) + (Log[(1 + a + b*x)/(a + b*x)]*Log[c + d*x] )/(2*d) - PolyLog[2, (b*(c + d*x))/(b*c - d - a*d)]/(2*d) + PolyLog[2, (b* (c + d*x))/(b*c + d - a*d)]/(2*d)
Time = 0.53 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.15, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {6662, 27, 6473, 2849, 2752, 2897}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\coth ^{-1}(a+b x)}{c+d x} \, dx\) |
| \(\Big \downarrow \) 6662 |
| \(\displaystyle \frac {\int \frac {b \coth ^{-1}(a+b x)}{b \left (c-\frac {a d}{b}\right )+d (a+b x)}d(a+b x)}{b}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \int \frac {\coth ^{-1}(a+b x)}{d (a+b x)-a d+b c}d(a+b x)\) |
| \(\Big \downarrow \) 6473 |
| \(\displaystyle -\frac {\int \frac {\log \left (\frac {2 (b c-a d+d (a+b x))}{(b c-a d+d) (a+b x+1)}\right )}{1-(a+b x)^2}d(a+b x)}{d}+\frac {\int \frac {\log \left (\frac {2}{a+b x+1}\right )}{1-(a+b x)^2}d(a+b x)}{d}+\frac {\coth ^{-1}(a+b x) \log \left (\frac {2 (d (a+b x)-a d+b c)}{(a+b x+1) (-a d+b c+d)}\right )}{d}-\frac {\log \left (\frac {2}{a+b x+1}\right ) \coth ^{-1}(a+b x)}{d}\) |
| \(\Big \downarrow \) 2849 |
| \(\displaystyle -\frac {\int \frac {\log \left (\frac {2 (b c-a d+d (a+b x))}{(b c-a d+d) (a+b x+1)}\right )}{1-(a+b x)^2}d(a+b x)}{d}+\frac {\int \frac {\log \left (\frac {2}{a+b x+1}\right )}{1-\frac {2}{a+b x+1}}d\frac {1}{a+b x+1}}{d}+\frac {\coth ^{-1}(a+b x) \log \left (\frac {2 (d (a+b x)-a d+b c)}{(a+b x+1) (-a d+b c+d)}\right )}{d}-\frac {\log \left (\frac {2}{a+b x+1}\right ) \coth ^{-1}(a+b x)}{d}\) |
| \(\Big \downarrow \) 2752 |
| \(\displaystyle -\frac {\int \frac {\log \left (\frac {2 (b c-a d+d (a+b x))}{(b c-a d+d) (a+b x+1)}\right )}{1-(a+b x)^2}d(a+b x)}{d}+\frac {\coth ^{-1}(a+b x) \log \left (\frac {2 (d (a+b x)-a d+b c)}{(a+b x+1) (-a d+b c+d)}\right )}{d}+\frac {\operatorname {PolyLog}\left (2,1-\frac {2}{a+b x+1}\right )}{2 d}-\frac {\log \left (\frac {2}{a+b x+1}\right ) \coth ^{-1}(a+b x)}{d}\) |
| \(\Big \downarrow \) 2897 |
| \(\displaystyle -\frac {\operatorname {PolyLog}\left (2,1-\frac {2 (b c-a d+d (a+b x))}{(b c-a d+d) (a+b x+1)}\right )}{2 d}+\frac {\coth ^{-1}(a+b x) \log \left (\frac {2 (d (a+b x)-a d+b c)}{(a+b x+1) (-a d+b c+d)}\right )}{d}+\frac {\operatorname {PolyLog}\left (2,1-\frac {2}{a+b x+1}\right )}{2 d}-\frac {\log \left (\frac {2}{a+b x+1}\right ) \coth ^{-1}(a+b x)}{d}\) |
-((ArcCoth[a + b*x]*Log[2/(1 + a + b*x)])/d) + (ArcCoth[a + b*x]*Log[(2*(b *c - a*d + d*(a + b*x)))/((b*c + d - a*d)*(1 + a + b*x))])/d + PolyLog[2, 1 - 2/(1 + a + b*x)]/(2*d) - PolyLog[2, 1 - (2*(b*c - a*d + d*(a + b*x)))/ ((b*c + d - a*d)*(1 + a + b*x))]/(2*d)
3.1.77.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-e^(-1))*PolyLo g[2, 1 - c*x], x] /; FreeQ[{c, d, e}, x] && EqQ[e + c*d, 0]
Int[Log[(c_.)/((d_) + (e_.)*(x_))]/((f_) + (g_.)*(x_)^2), x_Symbol] :> Simp [-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]
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[RationalFunctionExponents[u, x][[2]], Expon[Pq, x]]
Int[((a_.) + ArcCoth[(c_.)*(x_)]*(b_.))/((d_) + (e_.)*(x_)), x_Symbol] :> S imp[(-(a + b*ArcCoth[c*x]))*(Log[2/(1 + c*x)]/e), x] + (Simp[(a + b*ArcCoth [c*x])*(Log[2*c*((d + e*x)/((c*d + e)*(1 + c*x)))]/e), x] + Simp[b*(c/e) Int[Log[2/(1 + c*x)]/(1 - c^2*x^2), x], x] - Simp[b*(c/e) Int[Log[2*c*((d + e*x)/((c*d + e)*(1 + c*x)))]/(1 - c^2*x^2), x], x]) /; FreeQ[{a, b, c, d , e}, x] && NeQ[c^2*d^2 - e^2, 0]
Int[((a_.) + ArcCoth[(c_) + (d_.)*(x_)]*(b_.))^(p_.)*((e_.) + (f_.)*(x_))^( m_.), x_Symbol] :> Simp[1/d Subst[Int[((d*e - c*f)/d + f*(x/d))^m*(a + b* ArcCoth[x])^p, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && IG tQ[p, 0]
Time = 1.30 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.37
| method | result | size |
| risch | \(-\frac {\operatorname {dilog}\left (\frac {\left (b x +a -1\right ) d -a d +b c +d}{-a d +b c +d}\right )}{2 d}-\frac {\ln \left (b x +a -1\right ) \ln \left (\frac {\left (b x +a -1\right ) d -a d +b c +d}{-a d +b c +d}\right )}{2 d}+\frac {\operatorname {dilog}\left (\frac {\left (b x +a +1\right ) d -a d +b c -d}{-a d +b c -d}\right )}{2 d}+\frac {\ln \left (b x +a +1\right ) \ln \left (\frac {\left (b x +a +1\right ) d -a d +b c -d}{-a d +b c -d}\right )}{2 d}\) | \(164\) |
| parts | \(\frac {\ln \left (d x +c \right ) \operatorname {arccoth}\left (b x +a \right )}{d}+\frac {b \left (\frac {d \left (\frac {\operatorname {dilog}\left (\frac {a d -b c +b \left (d x +c \right )-d}{a d -b c -d}\right )}{b}+\frac {\ln \left (d x +c \right ) \ln \left (\frac {a d -b c +b \left (d x +c \right )-d}{a d -b c -d}\right )}{b}\right )}{2}-\frac {d \left (\frac {\operatorname {dilog}\left (\frac {a d -b c +b \left (d x +c \right )+d}{a d -b c +d}\right )}{b}+\frac {\ln \left (d x +c \right ) \ln \left (\frac {a d -b c +b \left (d x +c \right )+d}{a d -b c +d}\right )}{b}\right )}{2}\right )}{d^{2}}\) | \(184\) |
| derivativedivides | \(\frac {\frac {b \ln \left (a d -b c -d \left (b x +a \right )\right ) \operatorname {arccoth}\left (b x +a \right )}{d}-\frac {b \left (\frac {d \left (\operatorname {dilog}\left (\frac {-d \left (b x +a \right )-d}{-a d +b c -d}\right )+\ln \left (a d -b c -d \left (b x +a \right )\right ) \ln \left (\frac {-d \left (b x +a \right )-d}{-a d +b c -d}\right )\right )}{2}-\frac {d \left (\operatorname {dilog}\left (\frac {-d \left (b x +a \right )+d}{-a d +b c +d}\right )+\ln \left (a d -b c -d \left (b x +a \right )\right ) \ln \left (\frac {-d \left (b x +a \right )+d}{-a d +b c +d}\right )\right )}{2}\right )}{d^{2}}}{b}\) | \(185\) |
| default | \(\frac {\frac {b \ln \left (a d -b c -d \left (b x +a \right )\right ) \operatorname {arccoth}\left (b x +a \right )}{d}-\frac {b \left (\frac {d \left (\operatorname {dilog}\left (\frac {-d \left (b x +a \right )-d}{-a d +b c -d}\right )+\ln \left (a d -b c -d \left (b x +a \right )\right ) \ln \left (\frac {-d \left (b x +a \right )-d}{-a d +b c -d}\right )\right )}{2}-\frac {d \left (\operatorname {dilog}\left (\frac {-d \left (b x +a \right )+d}{-a d +b c +d}\right )+\ln \left (a d -b c -d \left (b x +a \right )\right ) \ln \left (\frac {-d \left (b x +a \right )+d}{-a d +b c +d}\right )\right )}{2}\right )}{d^{2}}}{b}\) | \(185\) |
-1/2*dilog(((b*x+a-1)*d-a*d+b*c+d)/(-a*d+b*c+d))/d-1/2*ln(b*x+a-1)*ln(((b* x+a-1)*d-a*d+b*c+d)/(-a*d+b*c+d))/d+1/2*dilog(((b*x+a+1)*d-a*d+b*c-d)/(-a* d+b*c-d))/d+1/2*ln(b*x+a+1)*ln(((b*x+a+1)*d-a*d+b*c-d)/(-a*d+b*c-d))/d
\[ \int \frac {\coth ^{-1}(a+b x)}{c+d x} \, dx=\int { \frac {\operatorname {arcoth}\left (b x + a\right )}{d x + c} \,d x } \]
\[ \int \frac {\coth ^{-1}(a+b x)}{c+d x} \, dx=\int \frac {\operatorname {acoth}{\left (a + b x \right )}}{c + d x}\, dx \]
Time = 0.23 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.60 \[ \int \frac {\coth ^{-1}(a+b x)}{c+d x} \, dx=-\frac {1}{2} \, b {\left (\frac {\log \left (b x + a - 1\right ) \log \left (\frac {b d x + a d - d}{b c - a d + d} + 1\right ) + {\rm Li}_2\left (-\frac {b d x + a d - d}{b c - a d + d}\right )}{b d} - \frac {\log \left (b x + a + 1\right ) \log \left (\frac {b d x + a d + d}{b c - a d - d} + 1\right ) + {\rm Li}_2\left (-\frac {b d x + a d + d}{b c - a d - d}\right )}{b d}\right )} - \frac {b {\left (\frac {\log \left (b x + a + 1\right )}{b} - \frac {\log \left (b x + a - 1\right )}{b}\right )} \log \left (d x + c\right )}{2 \, d} + \frac {\operatorname {arcoth}\left (b x + a\right ) \log \left (d x + c\right )}{d} \]
-1/2*b*((log(b*x + a - 1)*log((b*d*x + a*d - d)/(b*c - a*d + d) + 1) + dil og(-(b*d*x + a*d - d)/(b*c - a*d + d)))/(b*d) - (log(b*x + a + 1)*log((b*d *x + a*d + d)/(b*c - a*d - d) + 1) + dilog(-(b*d*x + a*d + d)/(b*c - a*d - d)))/(b*d)) - 1/2*b*(log(b*x + a + 1)/b - log(b*x + a - 1)/b)*log(d*x + c )/d + arccoth(b*x + a)*log(d*x + c)/d
\[ \int \frac {\coth ^{-1}(a+b x)}{c+d x} \, dx=\int { \frac {\operatorname {arcoth}\left (b x + a\right )}{d x + c} \,d x } \]
Timed out. \[ \int \frac {\coth ^{-1}(a+b x)}{c+d x} \, dx=\int \frac {\mathrm {acoth}\left (a+b\,x\right )}{c+d\,x} \,d x \]