Integrand size = 16, antiderivative size = 292 \[ \int \frac {\coth ^{-1}(a+b x)}{c+\frac {d}{x}} \, dx=\frac {(1-a-b x) \log \left (-\frac {1-a-b x}{a+b x}\right )}{2 b c}+\frac {\log (a+b x)}{2 b c}+\frac {\log (1+a+b x)}{2 b c}+\frac {(a+b x) \log \left (\frac {1+a+b x}{a+b x}\right )}{2 b c}-\frac {d \log \left (\frac {c (1-a-b x)}{c-a c+b d}\right ) \log (d+c x)}{2 c^2}+\frac {d \log \left (-\frac {1-a-b x}{a+b x}\right ) \log (d+c x)}{2 c^2}+\frac {d \log \left (\frac {c (1+a+b x)}{c+a c-b d}\right ) \log (d+c x)}{2 c^2}-\frac {d \log \left (\frac {1+a+b x}{a+b x}\right ) \log (d+c x)}{2 c^2}+\frac {d \operatorname {PolyLog}\left (2,-\frac {b (d+c x)}{c+a c-b d}\right )}{2 c^2}-\frac {d \operatorname {PolyLog}\left (2,\frac {b (d+c x)}{c-a c+b d}\right )}{2 c^2} \] Output:
1/2*(-b*x-a+1)*ln(-(-b*x-a+1)/(b*x+a))/b/c+1/2*ln(b*x+a)/b/c+1/2*ln(b*x+a+ 1)/b/c+1/2*(b*x+a)*ln((b*x+a+1)/(b*x+a))/b/c-1/2*d*ln(c*(-b*x-a+1)/(-a*c+b *d+c))*ln(c*x+d)/c^2+1/2*d*ln(-(-b*x-a+1)/(b*x+a))*ln(c*x+d)/c^2+1/2*d*ln( c*(b*x+a+1)/(a*c-b*d+c))*ln(c*x+d)/c^2-1/2*d*ln((b*x+a+1)/(b*x+a))*ln(c*x+ d)/c^2+1/2*d*polylog(2,-b*(c*x+d)/(a*c-b*d+c))/c^2-1/2*d*polylog(2,b*(c*x+ d)/(-a*c+b*d+c))/c^2
Result contains complex when optimal does not.
Time = 3.51 (sec) , antiderivative size = 502, normalized size of antiderivative = 1.72 \[ \int \frac {\coth ^{-1}(a+b x)}{c+\frac {d}{x}} \, dx=\frac {2 a c^2 \coth ^{-1}(a+b x)-i b c d \pi \coth ^{-1}(a+b x)+2 b c^2 x \coth ^{-1}(a+b x)+b c d \coth ^{-1}(a+b x)^2+a b c d \coth ^{-1}(a+b x)^2-b^2 d^2 \coth ^{-1}(a+b x)^2-a b c d \sqrt {1-\frac {c^2}{(a c-b d)^2}} e^{\text {arctanh}\left (\frac {c}{a c-b d}\right )} \coth ^{-1}(a+b x)^2+b^2 d^2 \sqrt {1-\frac {c^2}{(a c-b d)^2}} e^{\text {arctanh}\left (\frac {c}{a c-b d}\right )} \coth ^{-1}(a+b x)^2+2 b c d \coth ^{-1}(a+b x) \text {arctanh}\left (\frac {c}{a c-b d}\right )+2 b c d \coth ^{-1}(a+b x) \log \left (1-e^{-2 \coth ^{-1}(a+b x)}\right )+i b c d \pi \log \left (1+e^{2 \coth ^{-1}(a+b x)}\right )-2 b c d \coth ^{-1}(a+b x) \log \left (1-e^{-2 \coth ^{-1}(a+b x)+2 \text {arctanh}\left (\frac {c}{a c-b d}\right )}\right )+2 b c d \text {arctanh}\left (\frac {c}{a c-b d}\right ) \log \left (1-e^{-2 \coth ^{-1}(a+b x)+2 \text {arctanh}\left (\frac {c}{a c-b d}\right )}\right )-i b c d \pi \log \left (\frac {1}{\sqrt {1-\frac {1}{(a+b x)^2}}}\right )-2 c^2 \log \left (\frac {1}{(a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}\right )-2 b c d \text {arctanh}\left (\frac {c}{a c-b d}\right ) \log \left (i \sinh \left (\coth ^{-1}(a+b x)-\text {arctanh}\left (\frac {c}{a c-b d}\right )\right )\right )-b c d \operatorname {PolyLog}\left (2,e^{-2 \coth ^{-1}(a+b x)}\right )+b c d \operatorname {PolyLog}\left (2,e^{-2 \coth ^{-1}(a+b x)+2 \text {arctanh}\left (\frac {c}{a c-b d}\right )}\right )}{2 b c^3} \] Input:
Integrate[ArcCoth[a + b*x]/(c + d/x),x]
Output:
(2*a*c^2*ArcCoth[a + b*x] - I*b*c*d*Pi*ArcCoth[a + b*x] + 2*b*c^2*x*ArcCot h[a + b*x] + b*c*d*ArcCoth[a + b*x]^2 + a*b*c*d*ArcCoth[a + b*x]^2 - b^2*d ^2*ArcCoth[a + b*x]^2 - a*b*c*d*Sqrt[1 - c^2/(a*c - b*d)^2]*E^ArcTanh[c/(a *c - b*d)]*ArcCoth[a + b*x]^2 + b^2*d^2*Sqrt[1 - c^2/(a*c - b*d)^2]*E^ArcT anh[c/(a*c - b*d)]*ArcCoth[a + b*x]^2 + 2*b*c*d*ArcCoth[a + b*x]*ArcTanh[c /(a*c - b*d)] + 2*b*c*d*ArcCoth[a + b*x]*Log[1 - E^(-2*ArcCoth[a + b*x])] + I*b*c*d*Pi*Log[1 + E^(2*ArcCoth[a + b*x])] - 2*b*c*d*ArcCoth[a + b*x]*Lo g[1 - E^(-2*ArcCoth[a + b*x] + 2*ArcTanh[c/(a*c - b*d)])] + 2*b*c*d*ArcTan h[c/(a*c - b*d)]*Log[1 - E^(-2*ArcCoth[a + b*x] + 2*ArcTanh[c/(a*c - b*d)] )] - I*b*c*d*Pi*Log[1/Sqrt[1 - (a + b*x)^(-2)]] - 2*c^2*Log[1/((a + b*x)*S qrt[1 - (a + b*x)^(-2)])] - 2*b*c*d*ArcTanh[c/(a*c - b*d)]*Log[I*Sinh[ArcC oth[a + b*x] - ArcTanh[c/(a*c - b*d)]]] - b*c*d*PolyLog[2, E^(-2*ArcCoth[a + b*x])] + b*c*d*PolyLog[2, E^(-2*ArcCoth[a + b*x] + 2*ArcTanh[c/(a*c - b *d)])])/(2*b*c^3)
Time = 1.04 (sec) , antiderivative size = 436, normalized size of antiderivative = 1.49, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.438, Rules used = {6666, 2993, 772, 49, 2009, 2856, 2009}
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.
| \(\displaystyle \int \frac {\coth ^{-1}(a+b x)}{c+\frac {d}{x}} \, dx\) |
| \(\Big \downarrow \) 6666 |
| \(\displaystyle \frac {1}{2} \int \frac {\log \left (\frac {a+b x+1}{a+b x}\right )}{c+\frac {d}{x}}dx-\frac {1}{2} \int \frac {\log \left (-\frac {-a-b x+1}{a+b x}\right )}{c+\frac {d}{x}}dx\) |
| \(\Big \downarrow \) 2993 |
| \(\displaystyle \frac {1}{2} \left (\left (\log (-a-b x+1)-\log \left (-\frac {-a-b x+1}{a+b x}\right )-\log (a+b x)\right ) \int \frac {1}{c+\frac {d}{x}}dx-\int \frac {\log (-a-b x+1)}{c+\frac {d}{x}}dx+\int \frac {\log (a+b x)}{c+\frac {d}{x}}dx\right )+\frac {1}{2} \left (\left (\log (a+b x)-\log (a+b x+1)+\log \left (\frac {a+b x+1}{a+b x}\right )\right ) \int \frac {1}{c+\frac {d}{x}}dx-\int \frac {\log (a+b x)}{c+\frac {d}{x}}dx+\int \frac {\log (a+b x+1)}{c+\frac {d}{x}}dx\right )\) |
| \(\Big \downarrow \) 772 |
| \(\displaystyle \frac {1}{2} \left (\left (\log (-a-b x+1)-\log \left (-\frac {-a-b x+1}{a+b x}\right )-\log (a+b x)\right ) \int \frac {x}{d+c x}dx-\int \frac {\log (-a-b x+1)}{c+\frac {d}{x}}dx+\int \frac {\log (a+b x)}{c+\frac {d}{x}}dx\right )+\frac {1}{2} \left (\left (\log (a+b x)-\log (a+b x+1)+\log \left (\frac {a+b x+1}{a+b x}\right )\right ) \int \frac {x}{d+c x}dx-\int \frac {\log (a+b x)}{c+\frac {d}{x}}dx+\int \frac {\log (a+b x+1)}{c+\frac {d}{x}}dx\right )\) |
| \(\Big \downarrow \) 49 |
| \(\displaystyle \frac {1}{2} \left (\left (\log (-a-b x+1)-\log \left (-\frac {-a-b x+1}{a+b x}\right )-\log (a+b x)\right ) \int \left (\frac {1}{c}-\frac {d}{c (d+c x)}\right )dx-\int \frac {\log (-a-b x+1)}{c+\frac {d}{x}}dx+\int \frac {\log (a+b x)}{c+\frac {d}{x}}dx\right )+\frac {1}{2} \left (\left (\log (a+b x)-\log (a+b x+1)+\log \left (\frac {a+b x+1}{a+b x}\right )\right ) \int \left (\frac {1}{c}-\frac {d}{c (d+c x)}\right )dx-\int \frac {\log (a+b x)}{c+\frac {d}{x}}dx+\int \frac {\log (a+b x+1)}{c+\frac {d}{x}}dx\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{2} \left (-\int \frac {\log (-a-b x+1)}{c+\frac {d}{x}}dx+\int \frac {\log (a+b x)}{c+\frac {d}{x}}dx+\left (\log (-a-b x+1)-\log \left (-\frac {-a-b x+1}{a+b x}\right )-\log (a+b x)\right ) \left (\frac {x}{c}-\frac {d \log (c x+d)}{c^2}\right )\right )+\frac {1}{2} \left (-\int \frac {\log (a+b x)}{c+\frac {d}{x}}dx+\int \frac {\log (a+b x+1)}{c+\frac {d}{x}}dx+\left (\log (a+b x)-\log (a+b x+1)+\log \left (\frac {a+b x+1}{a+b x}\right )\right ) \left (\frac {x}{c}-\frac {d \log (c x+d)}{c^2}\right )\right )\) |
| \(\Big \downarrow \) 2856 |
| \(\displaystyle \frac {1}{2} \left (-\int \left (\frac {\log (-a-b x+1)}{c}-\frac {d \log (-a-b x+1)}{c (d+c x)}\right )dx+\int \left (\frac {\log (a+b x)}{c}-\frac {d \log (a+b x)}{c (d+c x)}\right )dx+\left (\log (-a-b x+1)-\log \left (-\frac {-a-b x+1}{a+b x}\right )-\log (a+b x)\right ) \left (\frac {x}{c}-\frac {d \log (c x+d)}{c^2}\right )\right )+\frac {1}{2} \left (-\int \left (\frac {\log (a+b x)}{c}-\frac {d \log (a+b x)}{c (d+c x)}\right )dx+\int \left (\frac {\log (a+b x+1)}{c}-\frac {d \log (a+b x+1)}{c (d+c x)}\right )dx+\left (\log (a+b x)-\log (a+b x+1)+\log \left (\frac {a+b x+1}{a+b x}\right )\right ) \left (\frac {x}{c}-\frac {d \log (c x+d)}{c^2}\right )\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{2} \left (\frac {d \operatorname {PolyLog}\left (2,\frac {c (-a-b x+1)}{-a c+c+b d}\right )}{c^2}-\frac {d \operatorname {PolyLog}\left (2,\frac {c (a+b x)}{a c-b d}\right )}{c^2}+\frac {d \log (-a-b x+1) \log \left (\frac {b (c x+d)}{-a c+b d+c}\right )}{c^2}+\left (\log (-a-b x+1)-\log \left (-\frac {-a-b x+1}{a+b x}\right )-\log (a+b x)\right ) \left (\frac {x}{c}-\frac {d \log (c x+d)}{c^2}\right )-\frac {d \log (a+b x) \log \left (-\frac {b (c x+d)}{a c-b d}\right )}{c^2}+\frac {(-a-b x+1) \log (-a-b x+1)}{b c}+\frac {(a+b x) \log (a+b x)}{b c}\right )+\frac {1}{2} \left (\frac {d \operatorname {PolyLog}\left (2,\frac {c (a+b x)}{a c-b d}\right )}{c^2}-\frac {d \operatorname {PolyLog}\left (2,\frac {c (a+b x+1)}{a c+c-b d}\right )}{c^2}+\frac {d \log (a+b x) \log \left (-\frac {b (c x+d)}{a c-b d}\right )}{c^2}+\left (\log (a+b x)-\log (a+b x+1)+\log \left (\frac {a+b x+1}{a+b x}\right )\right ) \left (\frac {x}{c}-\frac {d \log (c x+d)}{c^2}\right )-\frac {d \log (a+b x+1) \log \left (-\frac {b (c x+d)}{a c-b d+c}\right )}{c^2}-\frac {(a+b x) \log (a+b x)}{b c}+\frac {(a+b x+1) \log (a+b x+1)}{b c}\right )\) |
Input:
Int[ArcCoth[a + b*x]/(c + d/x),x]
Output:
(((1 - a - b*x)*Log[1 - a - b*x])/(b*c) + ((a + b*x)*Log[a + b*x])/(b*c) + (Log[1 - a - b*x] - Log[-((1 - a - b*x)/(a + b*x))] - Log[a + b*x])*(x/c - (d*Log[d + c*x])/c^2) - (d*Log[a + b*x]*Log[-((b*(d + c*x))/(a*c - b*d)) ])/c^2 + (d*Log[1 - a - b*x]*Log[(b*(d + c*x))/(c - a*c + b*d)])/c^2 + (d* PolyLog[2, (c*(1 - a - b*x))/(c - a*c + b*d)])/c^2 - (d*PolyLog[2, (c*(a + b*x))/(a*c - b*d)])/c^2)/2 + (-(((a + b*x)*Log[a + b*x])/(b*c)) + ((1 + a + b*x)*Log[1 + a + b*x])/(b*c) + (Log[a + b*x] - Log[1 + a + b*x] + Log[( 1 + a + b*x)/(a + b*x)])*(x/c - (d*Log[d + c*x])/c^2) + (d*Log[a + b*x]*Lo g[-((b*(d + c*x))/(a*c - b*d))])/c^2 - (d*Log[1 + a + b*x]*Log[-((b*(d + c *x))/(c + a*c - b*d))])/c^2 + (d*PolyLog[2, (c*(a + b*x))/(a*c - b*d)])/c^ 2 - (d*PolyLog[2, (c*(1 + a + b*x))/(c + a*c - b*d)])/c^2)/2
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int [ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 0] && IGtQ[m + n + 2, 0]
Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[x^(n*p)*(b + a/x^n)^p, x] /; FreeQ[{a, b}, x] && ILtQ[n, 0] && IntegerQ[p]
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_) + (g_. )*(x_)^(r_))^(q_.), x_Symbol] :> Int[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] && I GtQ[p, 0] && IntegerQ[q] && (GtQ[q, 0] || (IntegerQ[r] && NeQ[r, 1]))
Int[Log[(e_.)*((f_.)*((a_.) + (b_.)*(x_))^(p_.)*((c_.) + (d_.)*(x_))^(q_.)) ^(r_.)]*(RFx_.), x_Symbol] :> Simp[p*r Int[RFx*Log[a + b*x], x], x] + (Si mp[q*r Int[RFx*Log[c + d*x], x], x] - Simp[(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_.) /; IntegersQ[ m, n]]
Int[ArcCoth[(c_) + (d_.)*(x_)]/((e_) + (f_.)*(x_)^(n_.)), x_Symbol] :> Simp [1/2 Int[Log[(1 + c + d*x)/(c + d*x)]/(e + f*x^n), x], x] - Simp[1/2 In t[Log[(-1 + c + d*x)/(c + d*x)]/(e + f*x^n), x], x] /; FreeQ[{c, d, e, f}, x] && RationalQ[n]
Time = 1.89 (sec) , antiderivative size = 259, normalized size of antiderivative = 0.89
| method | result | size |
| parts | \(\frac {\operatorname {arccoth}\left (b x +a \right ) x}{c}-\frac {\operatorname {arccoth}\left (b x +a \right ) d \ln \left (c x +d \right )}{c^{2}}+\frac {b \left (-\frac {\left (a -1\right ) \ln \left (a c -b d +b \left (c x +d \right )-c \right )}{2 b^{2}}-\frac {\left (-a -1\right ) \ln \left (a c -b d +b \left (c x +d \right )+c \right )}{2 b^{2}}-d \left (\frac {\frac {\operatorname {dilog}\left (\frac {a c -b d +b \left (c x +d \right )-c}{a c -b d -c}\right )}{b}+\frac {\ln \left (c x +d \right ) \ln \left (\frac {a c -b d +b \left (c x +d \right )-c}{a c -b d -c}\right )}{b}}{2 c}-\frac {\frac {\operatorname {dilog}\left (\frac {a c -b d +b \left (c x +d \right )+c}{a c -b d +c}\right )}{b}+\frac {\ln \left (c x +d \right ) \ln \left (\frac {a c -b d +b \left (c x +d \right )+c}{a c -b d +c}\right )}{b}}{2 c}\right )\right )}{c}\) | \(259\) |
| risch | \(\frac {\ln \left (b x +a +1\right ) x}{2 c}+\frac {\ln \left (b x +a +1\right ) a}{2 b c}+\frac {\ln \left (b x +a +1\right )}{2 b c}-\frac {1}{b c}-\frac {d \operatorname {dilog}\left (\frac {c \left (b x +a +1\right )-a c +b d -c}{-a c +b d -c}\right )}{2 c^{2}}-\frac {d \ln \left (b x +a +1\right ) \ln \left (\frac {c \left (b x +a +1\right )-a c +b d -c}{-a c +b d -c}\right )}{2 c^{2}}-\frac {\ln \left (b x +a -1\right ) x}{2 c}-\frac {\ln \left (b x +a -1\right ) a}{2 b c}+\frac {\ln \left (b x +a -1\right )}{2 b c}+\frac {d \operatorname {dilog}\left (\frac {c \left (b x +a -1\right )-a c +b d +c}{-a c +b d +c}\right )}{2 c^{2}}+\frac {d \ln \left (b x +a -1\right ) \ln \left (\frac {c \left (b x +a -1\right )-a c +b d +c}{-a c +b d +c}\right )}{2 c^{2}}\) | \(264\) |
| derivativedivides | \(\frac {\frac {\operatorname {arccoth}\left (b x +a \right ) \left (b x +a \right )}{c}-\frac {\operatorname {arccoth}\left (b x +a \right ) d b \ln \left (a c -b d -c \left (b x +a \right )\right )}{c^{2}}-\frac {-\frac {\ln \left (a^{2} c^{2}-2 a b c d +b^{2} d^{2}-2 a c \left (a c -b d -c \left (b x +a \right )\right )+2 b d \left (a c -b d -c \left (b x +a \right )\right )-c^{2}+\left (a c -b d -c \left (b x +a \right )\right )^{2}\right )}{2}-b d \left (-\frac {\operatorname {dilog}\left (\frac {-c \left (b x +a \right )+c}{-a c +b d +c}\right )+\ln \left (a c -b d -c \left (b x +a \right )\right ) \ln \left (\frac {-c \left (b x +a \right )+c}{-a c +b d +c}\right )}{2 c}+\frac {\operatorname {dilog}\left (\frac {-c \left (b x +a \right )-c}{-a c +b d -c}\right )+\ln \left (a c -b d -c \left (b x +a \right )\right ) \ln \left (\frac {-c \left (b x +a \right )-c}{-a c +b d -c}\right )}{2 c}\right )}{c}}{b}\) | \(297\) |
| default | \(\frac {\frac {\operatorname {arccoth}\left (b x +a \right ) \left (b x +a \right )}{c}-\frac {\operatorname {arccoth}\left (b x +a \right ) d b \ln \left (a c -b d -c \left (b x +a \right )\right )}{c^{2}}-\frac {-\frac {\ln \left (a^{2} c^{2}-2 a b c d +b^{2} d^{2}-2 a c \left (a c -b d -c \left (b x +a \right )\right )+2 b d \left (a c -b d -c \left (b x +a \right )\right )-c^{2}+\left (a c -b d -c \left (b x +a \right )\right )^{2}\right )}{2}-b d \left (-\frac {\operatorname {dilog}\left (\frac {-c \left (b x +a \right )+c}{-a c +b d +c}\right )+\ln \left (a c -b d -c \left (b x +a \right )\right ) \ln \left (\frac {-c \left (b x +a \right )+c}{-a c +b d +c}\right )}{2 c}+\frac {\operatorname {dilog}\left (\frac {-c \left (b x +a \right )-c}{-a c +b d -c}\right )+\ln \left (a c -b d -c \left (b x +a \right )\right ) \ln \left (\frac {-c \left (b x +a \right )-c}{-a c +b d -c}\right )}{2 c}\right )}{c}}{b}\) | \(297\) |
Input:
int(arccoth(b*x+a)/(c+d/x),x,method=_RETURNVERBOSE)
Output:
arccoth(b*x+a)*x/c-arccoth(b*x+a)/c^2*d*ln(c*x+d)+b/c*(-1/2*(a-1)/b^2*ln(a *c-b*d+b*(c*x+d)-c)-1/2*(-a-1)/b^2*ln(a*c-b*d+b*(c*x+d)+c)-d*(1/2/c*(dilog ((a*c-b*d+b*(c*x+d)-c)/(a*c-b*d-c))/b+ln(c*x+d)*ln((a*c-b*d+b*(c*x+d)-c)/( a*c-b*d-c))/b)-1/2/c*(dilog((a*c-b*d+b*(c*x+d)+c)/(a*c-b*d+c))/b+ln(c*x+d) *ln((a*c-b*d+b*(c*x+d)+c)/(a*c-b*d+c))/b)))
\[ \int \frac {\coth ^{-1}(a+b x)}{c+\frac {d}{x}} \, dx=\int { \frac {\operatorname {arcoth}\left (b x + a\right )}{c + \frac {d}{x}} \,d x } \] Input:
integrate(arccoth(b*x+a)/(c+d/x),x, algorithm="fricas")
Output:
integral(x*arccoth(b*x + a)/(c*x + d), x)
\[ \int \frac {\coth ^{-1}(a+b x)}{c+\frac {d}{x}} \, dx=\int \frac {x \operatorname {acoth}{\left (a + b x \right )}}{c x + d}\, dx \] Input:
integrate(acoth(b*x+a)/(c+d/x),x)
Output:
Integral(x*acoth(a + b*x)/(c*x + d), x)
Time = 0.04 (sec) , antiderivative size = 192, normalized size of antiderivative = 0.66 \[ \int \frac {\coth ^{-1}(a+b x)}{c+\frac {d}{x}} \, dx=\frac {1}{2} \, b {\left (\frac {{\left (\log \left (c x + d\right ) \log \left (\frac {b c x + b d}{a c - b d + c} + 1\right ) + {\rm Li}_2\left (-\frac {b c x + b d}{a c - b d + c}\right )\right )} d}{b c^{2}} - \frac {{\left (\log \left (c x + d\right ) \log \left (\frac {b c x + b d}{a c - b d - c} + 1\right ) + {\rm Li}_2\left (-\frac {b c x + b d}{a c - b d - c}\right )\right )} d}{b c^{2}} + \frac {{\left (a + 1\right )} \log \left (b x + a + 1\right )}{b^{2} c} - \frac {{\left (a - 1\right )} \log \left (b x + a - 1\right )}{b^{2} c}\right )} + {\left (\frac {x}{c} - \frac {d \log \left (c x + d\right )}{c^{2}}\right )} \operatorname {arcoth}\left (b x + a\right ) \] Input:
integrate(arccoth(b*x+a)/(c+d/x),x, algorithm="maxima")
Output:
1/2*b*((log(c*x + d)*log((b*c*x + b*d)/(a*c - b*d + c) + 1) + dilog(-(b*c* x + b*d)/(a*c - b*d + c)))*d/(b*c^2) - (log(c*x + d)*log((b*c*x + b*d)/(a* c - b*d - c) + 1) + dilog(-(b*c*x + b*d)/(a*c - b*d - c)))*d/(b*c^2) + (a + 1)*log(b*x + a + 1)/(b^2*c) - (a - 1)*log(b*x + a - 1)/(b^2*c)) + (x/c - d*log(c*x + d)/c^2)*arccoth(b*x + a)
\[ \int \frac {\coth ^{-1}(a+b x)}{c+\frac {d}{x}} \, dx=\int { \frac {\operatorname {arcoth}\left (b x + a\right )}{c + \frac {d}{x}} \,d x } \] Input:
integrate(arccoth(b*x+a)/(c+d/x),x, algorithm="giac")
Output:
integrate(arccoth(b*x + a)/(c + d/x), x)
Timed out. \[ \int \frac {\coth ^{-1}(a+b x)}{c+\frac {d}{x}} \, dx=\int \frac {\mathrm {acoth}\left (a+b\,x\right )}{c+\frac {d}{x}} \,d x \] Input:
int(acoth(a + b*x)/(c + d/x),x)
Output:
int(acoth(a + b*x)/(c + d/x), x)
\[ \int \frac {\coth ^{-1}(a+b x)}{c+\frac {d}{x}} \, dx=\int \frac {\mathit {acoth} \left (b x +a \right ) x}{c x +d}d x \] Input:
int(acoth(b*x+a)/(c+d/x),x)
Output:
int((acoth(a + b*x)*x)/(c*x + d),x)