Integrand size = 12, antiderivative size = 251 \[ \int \frac {\coth ^{-1}(a+b x)^2}{x^2} \, dx=-\frac {\coth ^{-1}(a+b x)^2}{x}+\frac {b \coth ^{-1}(a+b x) \log \left (\frac {2}{1-a-b x}\right )}{1-a}+\frac {b \coth ^{-1}(a+b x) \log \left (\frac {2}{1+a+b x}\right )}{1+a}-\frac {2 b \coth ^{-1}(a+b x) \log \left (\frac {2}{1+a+b x}\right )}{1-a^2}+\frac {2 b \coth ^{-1}(a+b x) \log \left (\frac {2 b x}{(1-a) (1+a+b x)}\right )}{1-a^2}+\frac {b \operatorname {PolyLog}\left (2,-\frac {1+a+b x}{1-a-b x}\right )}{2 (1-a)}-\frac {b \operatorname {PolyLog}\left (2,1-\frac {2}{1+a+b x}\right )}{2 (1+a)}+\frac {b \operatorname {PolyLog}\left (2,1-\frac {2}{1+a+b x}\right )}{1-a^2}-\frac {b \operatorname {PolyLog}\left (2,1-\frac {2 b x}{(1-a) (1+a+b x)}\right )}{1-a^2} \]
-arccoth(b*x+a)^2/x+b*arccoth(b*x+a)*ln(2/(-b*x-a+1))/(1-a)+b*arccoth(b*x+ a)*ln(2/(b*x+a+1))/(1+a)-2*b*arccoth(b*x+a)*ln(2/(b*x+a+1))/(-a^2+1)+2*b*a rccoth(b*x+a)*ln(2*b*x/(1-a)/(b*x+a+1))/(-a^2+1)+1/2*b*polylog(2,(-b*x-a-1 )/(-b*x-a+1))/(1-a)-1/2*b*polylog(2,1-2/(b*x+a+1))/(1+a)+b*polylog(2,1-2/( b*x+a+1))/(-a^2+1)-b*polylog(2,1-2*b*x/(1-a)/(b*x+a+1))/(-a^2+1)
Result contains complex when optimal does not.
Time = 0.82 (sec) , antiderivative size = 206, normalized size of antiderivative = 0.82 \[ \int \frac {\coth ^{-1}(a+b x)^2}{x^2} \, dx=\frac {-\left (\left (-1+a^2+\sqrt {1-\frac {1}{a^2}} a b e^{\text {arctanh}\left (\frac {1}{a}\right )} x\right ) \coth ^{-1}(a+b x)^2\right )+b x \coth ^{-1}(a+b x) \left (-i \pi +2 \text {arctanh}\left (\frac {1}{a}\right )-2 \log \left (1-e^{-2 \coth ^{-1}(a+b x)+2 \text {arctanh}\left (\frac {1}{a}\right )}\right )\right )+b x \left (i \pi \left (\log \left (1+e^{2 \coth ^{-1}(a+b x)}\right )-\log \left (\frac {1}{\sqrt {1-\frac {1}{(a+b x)^2}}}\right )\right )+2 \text {arctanh}\left (\frac {1}{a}\right ) \left (\log \left (1-e^{-2 \coth ^{-1}(a+b x)+2 \text {arctanh}\left (\frac {1}{a}\right )}\right )-\log \left (i \sinh \left (\coth ^{-1}(a+b x)-\text {arctanh}\left (\frac {1}{a}\right )\right )\right )\right )\right )+b x \operatorname {PolyLog}\left (2,e^{-2 \coth ^{-1}(a+b x)+2 \text {arctanh}\left (\frac {1}{a}\right )}\right )}{\left (-1+a^2\right ) x} \]
(-((-1 + a^2 + Sqrt[1 - a^(-2)]*a*b*E^ArcTanh[a^(-1)]*x)*ArcCoth[a + b*x]^ 2) + b*x*ArcCoth[a + b*x]*((-I)*Pi + 2*ArcTanh[a^(-1)] - 2*Log[1 - E^(-2*A rcCoth[a + b*x] + 2*ArcTanh[a^(-1)])]) + b*x*(I*Pi*(Log[1 + E^(2*ArcCoth[a + b*x])] - Log[1/Sqrt[1 - (a + b*x)^(-2)]]) + 2*ArcTanh[a^(-1)]*(Log[1 - E^(-2*ArcCoth[a + b*x] + 2*ArcTanh[a^(-1)])] - Log[I*Sinh[ArcCoth[a + b*x] - ArcTanh[a^(-1)]]])) + b*x*PolyLog[2, E^(-2*ArcCoth[a + b*x] + 2*ArcTanh [a^(-1)])])/((-1 + a^2)*x)
Time = 1.02 (sec) , antiderivative size = 257, normalized size of antiderivative = 1.02, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.583, Rules used = {6660, 7292, 6672, 25, 27, 7276, 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.
| \(\displaystyle \int \frac {\coth ^{-1}(a+b x)^2}{x^2} \, dx\) |
| \(\Big \downarrow \) 6660 |
| \(\displaystyle 2 b \int \frac {\coth ^{-1}(a+b x)}{x \left (1-(a+b x)^2\right )}dx-\frac {\coth ^{-1}(a+b x)^2}{x}\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle 2 b \int \frac {\coth ^{-1}(a+b x)}{x \left (-a^2-2 b x a-b^2 x^2+1\right )}dx-\frac {\coth ^{-1}(a+b x)^2}{x}\) |
| \(\Big \downarrow \) 6672 |
| \(\displaystyle 2 \int \frac {\coth ^{-1}(a+b x)}{x \left (1-(a+b x)^2\right )}d(a+b x)-\frac {\coth ^{-1}(a+b x)^2}{x}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -2 \int -\frac {\coth ^{-1}(a+b x)}{x \left (1-(a+b x)^2\right )}d(a+b x)-\frac {\coth ^{-1}(a+b x)^2}{x}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -2 b \int -\frac {\coth ^{-1}(a+b x)}{b x \left (1-(a+b x)^2\right )}d(a+b x)-\frac {\coth ^{-1}(a+b x)^2}{x}\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle -2 b \int \left (\frac {\coth ^{-1}(a+b x)}{\left (a^2-1\right ) b x}-\frac {\coth ^{-1}(a+b x)}{2 (a-1) (a+b x-1)}+\frac {\coth ^{-1}(a+b x)}{2 (a+1) (a+b x+1)}\right )d(a+b x)-\frac {\coth ^{-1}(a+b x)^2}{x}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -2 b \left (-\frac {\operatorname {PolyLog}\left (2,1-\frac {2}{a+b x+1}\right )}{2 \left (1-a^2\right )}+\frac {\operatorname {PolyLog}\left (2,1-\frac {2 b x}{(1-a) (a+b x+1)}\right )}{2 \left (1-a^2\right )}+\frac {\log \left (\frac {2}{a+b x+1}\right ) \coth ^{-1}(a+b x)}{1-a^2}-\frac {\log \left (\frac {2 b x}{(1-a) (a+b x+1)}\right ) \coth ^{-1}(a+b x)}{1-a^2}-\frac {\operatorname {PolyLog}\left (2,-\frac {a+b x+1}{-a-b x+1}\right )}{4 (1-a)}+\frac {\operatorname {PolyLog}\left (2,1-\frac {2}{a+b x+1}\right )}{4 (a+1)}-\frac {\log \left (\frac {2}{-a-b x+1}\right ) \coth ^{-1}(a+b x)}{2 (1-a)}-\frac {\log \left (\frac {2}{a+b x+1}\right ) \coth ^{-1}(a+b x)}{2 (a+1)}\right )-\frac {\coth ^{-1}(a+b x)^2}{x}\) |
-(ArcCoth[a + b*x]^2/x) - 2*b*(-1/2*(ArcCoth[a + b*x]*Log[2/(1 - a - b*x)] )/(1 - a) - (ArcCoth[a + b*x]*Log[2/(1 + a + b*x)])/(2*(1 + a)) + (ArcCoth [a + b*x]*Log[2/(1 + a + b*x)])/(1 - a^2) - (ArcCoth[a + b*x]*Log[(2*b*x)/ ((1 - a)*(1 + a + b*x))])/(1 - a^2) - PolyLog[2, -((1 + a + b*x)/(1 - a - b*x))]/(4*(1 - a)) + PolyLog[2, 1 - 2/(1 + a + b*x)]/(4*(1 + a)) - PolyLog [2, 1 - 2/(1 + a + b*x)]/(2*(1 - a^2)) + PolyLog[2, 1 - (2*b*x)/((1 - a)*( 1 + a + b*x))]/(2*(1 - a^2)))
3.1.74.3.1 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[((a_.) + ArcCoth[(c_) + (d_.)*(x_)]*(b_.))^(p_.)*((e_.) + (f_.)*(x_))^( m_), x_Symbol] :> Simp[(e + f*x)^(m + 1)*((a + b*ArcCoth[c + d*x])^p/(f*(m + 1))), x] - Simp[b*d*(p/(f*(m + 1))) Int[(e + f*x)^(m + 1)*((a + b*ArcCo th[c + d*x])^(p - 1)/(1 - (c + d*x)^2)), x], x] /; FreeQ[{a, b, c, d, e, f} , x] && IGtQ[p, 0] && ILtQ[m, -1]
Int[((a_.) + ArcCoth[(c_) + (d_.)*(x_)]*(b_.))^(p_.)*((e_.) + (f_.)*(x_))^( m_.)*((A_.) + (B_.)*(x_) + (C_.)*(x_)^2)^(q_.), x_Symbol] :> Simp[1/d Sub st[Int[((d*e - c*f)/d + f*(x/d))^m*(-C/d^2 + (C/d^2)*x^2)^q*(a + b*ArcCoth[ x])^p, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, m, p, q}, x ] && EqQ[B*(1 - c^2) + 2*A*c*d, 0] && EqQ[2*c*C - B*d, 0]
Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionE xpand[u/(a + b*x^n), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ [n, 0]
Time = 0.14 (sec) , antiderivative size = 299, normalized size of antiderivative = 1.19
| method | result | size |
| parts | \(-\frac {\operatorname {arccoth}\left (b x +a \right )^{2}}{x}-2 b \left (\frac {\operatorname {arccoth}\left (b x +a \right ) \ln \left (b x +a +1\right )}{2 a +2}-\frac {\operatorname {arccoth}\left (b x +a \right ) \ln \left (b x +a -1\right )}{-2+2 a}+\frac {\operatorname {arccoth}\left (b x +a \right ) \ln \left (-b x \right )}{\left (-1+a \right ) \left (1+a \right )}-\frac {\frac {\ln \left (b x +a -1\right )^{2}}{4}-\frac {\operatorname {dilog}\left (\frac {b x}{2}+\frac {a}{2}+\frac {1}{2}\right )}{2}-\frac {\ln \left (b x +a -1\right ) \ln \left (\frac {b x}{2}+\frac {a}{2}+\frac {1}{2}\right )}{2}}{2 \left (-1+a \right )}+\frac {\frac {\left (\ln \left (b x +a +1\right )-\ln \left (\frac {b x}{2}+\frac {a}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {b x}{2}-\frac {a}{2}+\frac {1}{2}\right )}{2}-\frac {\operatorname {dilog}\left (\frac {b x}{2}+\frac {a}{2}+\frac {1}{2}\right )}{2}-\frac {\ln \left (b x +a +1\right )^{2}}{4}}{2 a +2}+\frac {\frac {\operatorname {dilog}\left (\frac {-b x -a +1}{1-a}\right )}{2}+\frac {\ln \left (-b x \right ) \ln \left (\frac {-b x -a +1}{1-a}\right )}{2}-\frac {\operatorname {dilog}\left (\frac {-b x -a -1}{-1-a}\right )}{2}-\frac {\ln \left (-b x \right ) \ln \left (\frac {-b x -a -1}{-1-a}\right )}{2}}{\left (-1+a \right ) \left (1+a \right )}\right )\) | \(299\) |
| derivativedivides | \(b \left (-\frac {\operatorname {arccoth}\left (b x +a \right )^{2}}{b x}-\frac {2 \,\operatorname {arccoth}\left (b x +a \right ) \ln \left (b x +a +1\right )}{2 a +2}+\frac {2 \,\operatorname {arccoth}\left (b x +a \right ) \ln \left (b x +a -1\right )}{-2+2 a}-\frac {2 \,\operatorname {arccoth}\left (b x +a \right ) \ln \left (-b x \right )}{\left (-1+a \right ) \left (1+a \right )}+\frac {\frac {\ln \left (b x +a -1\right )^{2}}{4}-\frac {\operatorname {dilog}\left (\frac {b x}{2}+\frac {a}{2}+\frac {1}{2}\right )}{2}-\frac {\ln \left (b x +a -1\right ) \ln \left (\frac {b x}{2}+\frac {a}{2}+\frac {1}{2}\right )}{2}}{-1+a}-\frac {\frac {\left (\ln \left (b x +a +1\right )-\ln \left (\frac {b x}{2}+\frac {a}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {b x}{2}-\frac {a}{2}+\frac {1}{2}\right )}{2}-\frac {\operatorname {dilog}\left (\frac {b x}{2}+\frac {a}{2}+\frac {1}{2}\right )}{2}-\frac {\ln \left (b x +a +1\right )^{2}}{4}}{1+a}-\frac {2 \left (\frac {\operatorname {dilog}\left (\frac {-b x -a +1}{1-a}\right )}{2}+\frac {\ln \left (-b x \right ) \ln \left (\frac {-b x -a +1}{1-a}\right )}{2}-\frac {\operatorname {dilog}\left (\frac {-b x -a -1}{-1-a}\right )}{2}-\frac {\ln \left (-b x \right ) \ln \left (\frac {-b x -a -1}{-1-a}\right )}{2}\right )}{\left (-1+a \right ) \left (1+a \right )}\right )\) | \(302\) |
| default | \(b \left (-\frac {\operatorname {arccoth}\left (b x +a \right )^{2}}{b x}-\frac {2 \,\operatorname {arccoth}\left (b x +a \right ) \ln \left (b x +a +1\right )}{2 a +2}+\frac {2 \,\operatorname {arccoth}\left (b x +a \right ) \ln \left (b x +a -1\right )}{-2+2 a}-\frac {2 \,\operatorname {arccoth}\left (b x +a \right ) \ln \left (-b x \right )}{\left (-1+a \right ) \left (1+a \right )}+\frac {\frac {\ln \left (b x +a -1\right )^{2}}{4}-\frac {\operatorname {dilog}\left (\frac {b x}{2}+\frac {a}{2}+\frac {1}{2}\right )}{2}-\frac {\ln \left (b x +a -1\right ) \ln \left (\frac {b x}{2}+\frac {a}{2}+\frac {1}{2}\right )}{2}}{-1+a}-\frac {\frac {\left (\ln \left (b x +a +1\right )-\ln \left (\frac {b x}{2}+\frac {a}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {b x}{2}-\frac {a}{2}+\frac {1}{2}\right )}{2}-\frac {\operatorname {dilog}\left (\frac {b x}{2}+\frac {a}{2}+\frac {1}{2}\right )}{2}-\frac {\ln \left (b x +a +1\right )^{2}}{4}}{1+a}-\frac {2 \left (\frac {\operatorname {dilog}\left (\frac {-b x -a +1}{1-a}\right )}{2}+\frac {\ln \left (-b x \right ) \ln \left (\frac {-b x -a +1}{1-a}\right )}{2}-\frac {\operatorname {dilog}\left (\frac {-b x -a -1}{-1-a}\right )}{2}-\frac {\ln \left (-b x \right ) \ln \left (\frac {-b x -a -1}{-1-a}\right )}{2}\right )}{\left (-1+a \right ) \left (1+a \right )}\right )\) | \(302\) |
-arccoth(b*x+a)^2/x-2*b*(arccoth(b*x+a)/(2*a+2)*ln(b*x+a+1)-arccoth(b*x+a) /(-2+2*a)*ln(b*x+a-1)+arccoth(b*x+a)/(-1+a)/(1+a)*ln(-b*x)-1/2/(-1+a)*(1/4 *ln(b*x+a-1)^2-1/2*dilog(1/2*b*x+1/2*a+1/2)-1/2*ln(b*x+a-1)*ln(1/2*b*x+1/2 *a+1/2))+1/2/(1+a)*(1/2*(ln(b*x+a+1)-ln(1/2*b*x+1/2*a+1/2))*ln(-1/2*b*x-1/ 2*a+1/2)-1/2*dilog(1/2*b*x+1/2*a+1/2)-1/4*ln(b*x+a+1)^2)+1/(-1+a)/(1+a)*(1 /2*dilog(1/(1-a)*(-b*x-a+1))+1/2*ln(-b*x)*ln(1/(1-a)*(-b*x-a+1))-1/2*dilog ((-b*x-a-1)/(-1-a))-1/2*ln(-b*x)*ln((-b*x-a-1)/(-1-a))))
\[ \int \frac {\coth ^{-1}(a+b x)^2}{x^2} \, dx=\int { \frac {\operatorname {arcoth}\left (b x + a\right )^{2}}{x^{2}} \,d x } \]
\[ \int \frac {\coth ^{-1}(a+b x)^2}{x^2} \, dx=\int \frac {\operatorname {acoth}^{2}{\left (a + b x \right )}}{x^{2}}\, dx \]
Time = 0.22 (sec) , antiderivative size = 244, normalized size of antiderivative = 0.97 \[ \int \frac {\coth ^{-1}(a+b x)^2}{x^2} \, dx=\frac {1}{4} \, b^{2} {\left (\frac {{\left (a - 1\right )} \log \left (b x + a + 1\right )^{2} - 2 \, {\left (a - 1\right )} \log \left (b x + a + 1\right ) \log \left (b x + a - 1\right ) + {\left (a + 1\right )} \log \left (b x + a - 1\right )^{2}}{a^{2} b - b} - \frac {4 \, {\left (\log \left (b x + a - 1\right ) \log \left (\frac {1}{2} \, b x + \frac {1}{2} \, a + \frac {1}{2}\right ) + {\rm Li}_2\left (-\frac {1}{2} \, b x - \frac {1}{2} \, a + \frac {1}{2}\right )\right )}}{a^{2} b - b} + \frac {4 \, {\left (\log \left (\frac {b x}{a + 1} + 1\right ) \log \left (x\right ) + {\rm Li}_2\left (-\frac {b x}{a + 1}\right )\right )}}{a^{2} b - b} - \frac {4 \, {\left (\log \left (\frac {b x}{a - 1} + 1\right ) \log \left (x\right ) + {\rm Li}_2\left (-\frac {b x}{a - 1}\right )\right )}}{a^{2} b - b}\right )} - b {\left (\frac {\log \left (b x + a + 1\right )}{a + 1} - \frac {\log \left (b x + a - 1\right )}{a - 1} + \frac {2 \, \log \left (x\right )}{a^{2} - 1}\right )} \operatorname {arcoth}\left (b x + a\right ) - \frac {\operatorname {arcoth}\left (b x + a\right )^{2}}{x} \]
1/4*b^2*(((a - 1)*log(b*x + a + 1)^2 - 2*(a - 1)*log(b*x + a + 1)*log(b*x + a - 1) + (a + 1)*log(b*x + a - 1)^2)/(a^2*b - b) - 4*(log(b*x + a - 1)*l og(1/2*b*x + 1/2*a + 1/2) + dilog(-1/2*b*x - 1/2*a + 1/2))/(a^2*b - b) + 4 *(log(b*x/(a + 1) + 1)*log(x) + dilog(-b*x/(a + 1)))/(a^2*b - b) - 4*(log( b*x/(a - 1) + 1)*log(x) + dilog(-b*x/(a - 1)))/(a^2*b - b)) - b*(log(b*x + a + 1)/(a + 1) - log(b*x + a - 1)/(a - 1) + 2*log(x)/(a^2 - 1))*arccoth(b *x + a) - arccoth(b*x + a)^2/x
\[ \int \frac {\coth ^{-1}(a+b x)^2}{x^2} \, dx=\int { \frac {\operatorname {arcoth}\left (b x + a\right )^{2}}{x^{2}} \,d x } \]
Timed out. \[ \int \frac {\coth ^{-1}(a+b x)^2}{x^2} \, dx=\int \frac {{\mathrm {acoth}\left (a+b\,x\right )}^2}{x^2} \,d x \]