Integrand size = 12, antiderivative size = 370 \[ \int \frac {\coth ^{-1}(a+b x)^2}{x^3} \, dx=-\frac {b \coth ^{-1}(a+b x)}{\left (1-a^2\right ) x}-\frac {\coth ^{-1}(a+b x)^2}{2 x^2}+\frac {b^2 \log (x)}{\left (1-a^2\right )^2}+\frac {b^2 \coth ^{-1}(a+b x) \log \left (\frac {2}{1-a-b x}\right )}{2 (1-a)^2}-\frac {b^2 \log (1-a-b x)}{2 (1-a)^2 (1+a)}-\frac {b^2 \coth ^{-1}(a+b x) \log \left (\frac {2}{1+a+b x}\right )}{2 (1+a)^2}-\frac {2 a b^2 \coth ^{-1}(a+b x) \log \left (\frac {2}{1+a+b x}\right )}{\left (1-a^2\right )^2}+\frac {2 a b^2 \coth ^{-1}(a+b x) \log \left (\frac {2 b x}{(1-a) (1+a+b x)}\right )}{\left (1-a^2\right )^2}-\frac {b^2 \log (1+a+b x)}{2 (1-a) (1+a)^2}+\frac {b^2 \operatorname {PolyLog}\left (2,-\frac {1+a+b x}{1-a-b x}\right )}{4 (1-a)^2}+\frac {b^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1+a+b x}\right )}{4 (1+a)^2}+\frac {a b^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1+a+b x}\right )}{\left (1-a^2\right )^2}-\frac {a b^2 \operatorname {PolyLog}\left (2,1-\frac {2 b x}{(1-a) (1+a+b x)}\right )}{\left (1-a^2\right )^2} \] Output:
-b*arccoth(b*x+a)/(-a^2+1)/x-1/2*arccoth(b*x+a)^2/x^2+b^2*ln(x)/(-a^2+1)^2 +1/2*b^2*arccoth(b*x+a)*ln(2/(-b*x-a+1))/(1-a)^2-1/2*b^2*ln(-b*x-a+1)/(1-a )^2/(1+a)-1/2*b^2*arccoth(b*x+a)*ln(2/(b*x+a+1))/(1+a)^2-2*a*b^2*arccoth(b *x+a)*ln(2/(b*x+a+1))/(-a^2+1)^2+2*a*b^2*arccoth(b*x+a)*ln(2*b*x/(1-a)/(b* x+a+1))/(-a^2+1)^2-1/2*b^2*ln(b*x+a+1)/(1-a)/(1+a)^2+1/4*b^2*polylog(2,-(b *x+a+1)/(-b*x-a+1))/(1-a)^2+1/4*b^2*polylog(2,1-2/(b*x+a+1))/(1+a)^2+a*b^2 *polylog(2,1-2/(b*x+a+1))/(-a^2+1)^2-a*b^2*polylog(2,1-2*b*x/(1-a)/(b*x+a+ 1))/(-a^2+1)^2
Result contains complex when optimal does not.
Time = 1.54 (sec) , antiderivative size = 291, normalized size of antiderivative = 0.79 \[ \int \frac {\coth ^{-1}(a+b x)^2}{x^3} \, dx=\frac {\left (-1-a^4+b^2 x^2+a^2 \left (2+b^2 \left (-1+2 \sqrt {1-\frac {1}{a^2}} e^{\text {arctanh}\left (\frac {1}{a}\right )}\right ) x^2\right )\right ) \coth ^{-1}(a+b x)^2+2 b x \coth ^{-1}(a+b x) \left (-1+a^2+a b x+i a b \pi x-2 a b x \text {arctanh}\left (\frac {1}{a}\right )+2 a b x \log \left (1-e^{-2 \coth ^{-1}(a+b x)+2 \text {arctanh}\left (\frac {1}{a}\right )}\right )\right )+2 b^2 x^2 \left (-i a \pi \log \left (1+e^{2 \coth ^{-1}(a+b x)}\right )+i a \pi \log \left (\frac {1}{\sqrt {1-\frac {1}{(a+b x)^2}}}\right )+\log \left (-\frac {b x}{(a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}\right )-2 a \text {arctanh}\left (\frac {1}{a}\right ) \left (\log \left (1-e^{-2 \coth ^{-1}(a+b x)+2 \text {arctanh}\left (\frac {1}{a}\right )}\right )-\log \left (i \sinh \left (\coth ^{-1}(a+b x)-\text {arctanh}\left (\frac {1}{a}\right )\right )\right )\right )\right )-2 a b^2 x^2 \operatorname {PolyLog}\left (2,e^{-2 \coth ^{-1}(a+b x)+2 \text {arctanh}\left (\frac {1}{a}\right )}\right )}{2 \left (-1+a^2\right )^2 x^2} \] Input:
Integrate[ArcCoth[a + b*x]^2/x^3,x]
Output:
((-1 - a^4 + b^2*x^2 + a^2*(2 + b^2*(-1 + 2*Sqrt[1 - a^(-2)]*E^ArcTanh[a^( -1)])*x^2))*ArcCoth[a + b*x]^2 + 2*b*x*ArcCoth[a + b*x]*(-1 + a^2 + a*b*x + I*a*b*Pi*x - 2*a*b*x*ArcTanh[a^(-1)] + 2*a*b*x*Log[1 - E^(-2*ArcCoth[a + b*x] + 2*ArcTanh[a^(-1)])]) + 2*b^2*x^2*((-I)*a*Pi*Log[1 + E^(2*ArcCoth[a + b*x])] + I*a*Pi*Log[1/Sqrt[1 - (a + b*x)^(-2)]] + Log[-((b*x)/((a + b*x )*Sqrt[1 - (a + b*x)^(-2)]))] - 2*a*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)]]])) - 2*a*b^2*x^2*PolyLog[2, E^(-2*ArcCoth[a + b*x] + 2*ArcTanh[a^(- 1)])])/(2*(-1 + a^2)^2*x^2)
Time = 1.12 (sec) , antiderivative size = 347, normalized size of antiderivative = 0.94, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6660, 7292, 6672, 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.
| \(\displaystyle \int \frac {\coth ^{-1}(a+b x)^2}{x^3} \, dx\) |
| \(\Big \downarrow \) 6660 |
| \(\displaystyle b \int \frac {\coth ^{-1}(a+b x)}{x^2 \left (1-(a+b x)^2\right )}dx-\frac {\coth ^{-1}(a+b x)^2}{2 x^2}\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle b \int \frac {\coth ^{-1}(a+b x)}{x^2 \left (-a^2-2 b x a-b^2 x^2+1\right )}dx-\frac {\coth ^{-1}(a+b x)^2}{2 x^2}\) |
| \(\Big \downarrow \) 6672 |
| \(\displaystyle \int \frac {\coth ^{-1}(a+b x)}{x^2 \left (1-(a+b x)^2\right )}d(a+b x)-\frac {\coth ^{-1}(a+b x)^2}{2 x^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle b^2 \int \frac {\coth ^{-1}(a+b x)}{b^2 x^2 \left (1-(a+b x)^2\right )}d(a+b x)-\frac {\coth ^{-1}(a+b x)^2}{2 x^2}\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle b^2 \int \left (\frac {2 a \coth ^{-1}(a+b x)}{\left (a^2-1\right )^2 b x}-\frac {\coth ^{-1}(a+b x)}{2 (a-1)^2 (a+b x-1)}+\frac {\coth ^{-1}(a+b x)}{2 (a+1)^2 (a+b x+1)}-\frac {\coth ^{-1}(a+b x)}{\left (a^2-1\right ) b^2 x^2}\right )d(a+b x)-\frac {\coth ^{-1}(a+b x)^2}{2 x^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle b^2 \left (\frac {a \operatorname {PolyLog}\left (2,1-\frac {2}{a+b x+1}\right )}{\left (1-a^2\right )^2}-\frac {a \operatorname {PolyLog}\left (2,1-\frac {2 b x}{(1-a) (a+b x+1)}\right )}{\left (1-a^2\right )^2}+\frac {\log (-b x)}{\left (1-a^2\right )^2}-\frac {\coth ^{-1}(a+b x)}{\left (1-a^2\right ) b x}-\frac {2 a \log \left (\frac {2}{a+b x+1}\right ) \coth ^{-1}(a+b x)}{\left (1-a^2\right )^2}+\frac {2 a \log \left (\frac {2 b x}{(1-a) (a+b x+1)}\right ) \coth ^{-1}(a+b x)}{\left (1-a^2\right )^2}+\frac {\operatorname {PolyLog}\left (2,-\frac {a+b x+1}{-a-b x+1}\right )}{4 (1-a)^2}+\frac {\operatorname {PolyLog}\left (2,1-\frac {2}{a+b x+1}\right )}{4 (a+1)^2}-\frac {\log (-a-b x+1)}{2 (1-a)^2 (a+1)}-\frac {\log (a+b x+1)}{2 (1-a) (a+1)^2}+\frac {\log \left (\frac {2}{-a-b x+1}\right ) \coth ^{-1}(a+b x)}{2 (1-a)^2}-\frac {\log \left (\frac {2}{a+b x+1}\right ) \coth ^{-1}(a+b x)}{2 (a+1)^2}\right )-\frac {\coth ^{-1}(a+b x)^2}{2 x^2}\) |
Input:
Int[ArcCoth[a + b*x]^2/x^3,x]
Output:
-1/2*ArcCoth[a + b*x]^2/x^2 + b^2*(-(ArcCoth[a + b*x]/((1 - a^2)*b*x)) + L og[-(b*x)]/(1 - a^2)^2 + (ArcCoth[a + b*x]*Log[2/(1 - a - b*x)])/(2*(1 - a )^2) - Log[1 - a - b*x]/(2*(1 - a)^2*(1 + a)) - (ArcCoth[a + b*x]*Log[2/(1 + a + b*x)])/(2*(1 + a)^2) - (2*a*ArcCoth[a + b*x]*Log[2/(1 + a + b*x)])/ (1 - a^2)^2 + (2*a*ArcCoth[a + b*x]*Log[(2*b*x)/((1 - a)*(1 + a + b*x))])/ (1 - a^2)^2 - Log[1 + a + b*x]/(2*(1 - a)*(1 + a)^2) + PolyLog[2, -((1 + a + b*x)/(1 - a - b*x))]/(4*(1 - a)^2) + PolyLog[2, 1 - 2/(1 + a + b*x)]/(4 *(1 + a)^2) + (a*PolyLog[2, 1 - 2/(1 + a + b*x)])/(1 - a^2)^2 - (a*PolyLog [2, 1 - (2*b*x)/((1 - a)*(1 + a + b*x))])/(1 - a^2)^2)
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 = 0.62 (sec) , antiderivative size = 386, normalized size of antiderivative = 1.04
| method | result | size |
| parts | \(-\frac {\operatorname {arccoth}\left (b x +a \right )^{2}}{2 x^{2}}-b^{2} \left (\frac {\operatorname {arccoth}\left (b x +a \right ) \ln \left (b x +a -1\right )}{2 \left (a -1\right )^{2}}-\frac {\operatorname {arccoth}\left (b x +a \right )}{\left (a -1\right ) \left (1+a \right ) b x}-\frac {2 \,\operatorname {arccoth}\left (b x +a \right ) a \ln \left (-b x \right )}{\left (a -1\right )^{2} \left (1+a \right )^{2}}-\frac {\operatorname {arccoth}\left (b x +a \right ) \ln \left (b x +a +1\right )}{2 \left (1+a \right )^{2}}+\frac {-\frac {\operatorname {dilog}\left (\frac {b x}{2}+\frac {a}{2}+\frac {1}{2}\right )}{2}-\frac {\ln \left (b x +a -1\right ) \ln \left (\frac {b x}{2}+\frac {a}{2}+\frac {1}{2}\right )}{2}+\frac {\ln \left (b x +a -1\right )^{2}}{4}}{2 \left (a -1\right )^{2}}-\frac {\frac {\left (\ln \left (b x +a +1\right )-\ln \left (\frac {b x}{2}+\frac {a}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {b x}{2}-\frac {a}{2}+\frac {1}{2}\right )}{2}-\frac {\operatorname {dilog}\left (\frac {b x}{2}+\frac {a}{2}+\frac {1}{2}\right )}{2}-\frac {\ln \left (b x +a +1\right )^{2}}{4}}{2 \left (1+a \right )^{2}}+\frac {\frac {\ln \left (b x +a -1\right )}{2 a -2}-\frac {\ln \left (-b x \right )}{\left (a -1\right ) \left (1+a \right )}-\frac {\ln \left (b x +a +1\right )}{2+2 a}}{\left (a -1\right ) \left (1+a \right )}-\frac {2 a \left (\frac {\operatorname {dilog}\left (\frac {-b x -a +1}{1-a}\right )}{2}+\frac {\ln \left (-b x \right ) \ln \left (\frac {-b x -a +1}{1-a}\right )}{2}-\frac {\operatorname {dilog}\left (\frac {-b x -a -1}{-a -1}\right )}{2}-\frac {\ln \left (-b x \right ) \ln \left (\frac {-b x -a -1}{-a -1}\right )}{2}\right )}{\left (a -1\right )^{2} \left (1+a \right )^{2}}\right )\) | \(386\) |
| derivativedivides | \(b^{2} \left (-\frac {\operatorname {arccoth}\left (b x +a \right )^{2}}{2 b^{2} x^{2}}-\frac {\operatorname {arccoth}\left (b x +a \right ) \ln \left (b x +a -1\right )}{2 \left (a -1\right )^{2}}+\frac {\operatorname {arccoth}\left (b x +a \right )}{\left (a -1\right ) \left (1+a \right ) b x}+\frac {2 \,\operatorname {arccoth}\left (b x +a \right ) a \ln \left (-b x \right )}{\left (a -1\right )^{2} \left (1+a \right )^{2}}+\frac {\operatorname {arccoth}\left (b x +a \right ) \ln \left (b x +a +1\right )}{2 \left (1+a \right )^{2}}-\frac {-\frac {\operatorname {dilog}\left (\frac {b x}{2}+\frac {a}{2}+\frac {1}{2}\right )}{2}-\frac {\ln \left (b x +a -1\right ) \ln \left (\frac {b x}{2}+\frac {a}{2}+\frac {1}{2}\right )}{2}+\frac {\ln \left (b x +a -1\right )^{2}}{4}}{2 \left (a -1\right )^{2}}+\frac {\frac {\left (\ln \left (b x +a +1\right )-\ln \left (\frac {b x}{2}+\frac {a}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {b x}{2}-\frac {a}{2}+\frac {1}{2}\right )}{2}-\frac {\operatorname {dilog}\left (\frac {b x}{2}+\frac {a}{2}+\frac {1}{2}\right )}{2}-\frac {\ln \left (b x +a +1\right )^{2}}{4}}{2 \left (1+a \right )^{2}}-\frac {\frac {\ln \left (b x +a -1\right )}{2 a -2}-\frac {\ln \left (-b x \right )}{\left (a -1\right ) \left (1+a \right )}-\frac {\ln \left (b x +a +1\right )}{2+2 a}}{\left (a -1\right ) \left (1+a \right )}+\frac {2 a \left (\frac {\operatorname {dilog}\left (\frac {-b x -a +1}{1-a}\right )}{2}+\frac {\ln \left (-b x \right ) \ln \left (\frac {-b x -a +1}{1-a}\right )}{2}-\frac {\operatorname {dilog}\left (\frac {-b x -a -1}{-a -1}\right )}{2}-\frac {\ln \left (-b x \right ) \ln \left (\frac {-b x -a -1}{-a -1}\right )}{2}\right )}{\left (a -1\right )^{2} \left (1+a \right )^{2}}\right )\) | \(387\) |
| default | \(b^{2} \left (-\frac {\operatorname {arccoth}\left (b x +a \right )^{2}}{2 b^{2} x^{2}}-\frac {\operatorname {arccoth}\left (b x +a \right ) \ln \left (b x +a -1\right )}{2 \left (a -1\right )^{2}}+\frac {\operatorname {arccoth}\left (b x +a \right )}{\left (a -1\right ) \left (1+a \right ) b x}+\frac {2 \,\operatorname {arccoth}\left (b x +a \right ) a \ln \left (-b x \right )}{\left (a -1\right )^{2} \left (1+a \right )^{2}}+\frac {\operatorname {arccoth}\left (b x +a \right ) \ln \left (b x +a +1\right )}{2 \left (1+a \right )^{2}}-\frac {-\frac {\operatorname {dilog}\left (\frac {b x}{2}+\frac {a}{2}+\frac {1}{2}\right )}{2}-\frac {\ln \left (b x +a -1\right ) \ln \left (\frac {b x}{2}+\frac {a}{2}+\frac {1}{2}\right )}{2}+\frac {\ln \left (b x +a -1\right )^{2}}{4}}{2 \left (a -1\right )^{2}}+\frac {\frac {\left (\ln \left (b x +a +1\right )-\ln \left (\frac {b x}{2}+\frac {a}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {b x}{2}-\frac {a}{2}+\frac {1}{2}\right )}{2}-\frac {\operatorname {dilog}\left (\frac {b x}{2}+\frac {a}{2}+\frac {1}{2}\right )}{2}-\frac {\ln \left (b x +a +1\right )^{2}}{4}}{2 \left (1+a \right )^{2}}-\frac {\frac {\ln \left (b x +a -1\right )}{2 a -2}-\frac {\ln \left (-b x \right )}{\left (a -1\right ) \left (1+a \right )}-\frac {\ln \left (b x +a +1\right )}{2+2 a}}{\left (a -1\right ) \left (1+a \right )}+\frac {2 a \left (\frac {\operatorname {dilog}\left (\frac {-b x -a +1}{1-a}\right )}{2}+\frac {\ln \left (-b x \right ) \ln \left (\frac {-b x -a +1}{1-a}\right )}{2}-\frac {\operatorname {dilog}\left (\frac {-b x -a -1}{-a -1}\right )}{2}-\frac {\ln \left (-b x \right ) \ln \left (\frac {-b x -a -1}{-a -1}\right )}{2}\right )}{\left (a -1\right )^{2} \left (1+a \right )^{2}}\right )\) | \(387\) |
Input:
int(arccoth(b*x+a)^2/x^3,x,method=_RETURNVERBOSE)
Output:
-1/2*arccoth(b*x+a)^2/x^2-b^2*(1/2*arccoth(b*x+a)/(a-1)^2*ln(b*x+a-1)-arcc oth(b*x+a)/(a-1)/(1+a)/b/x-2*arccoth(b*x+a)*a/(a-1)^2/(1+a)^2*ln(-b*x)-1/2 *arccoth(b*x+a)/(1+a)^2*ln(b*x+a+1)+1/2/(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)+1/4*ln(b*x+a-1)^2)-1/2/(1+a)^2* (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/(a-1)/(1+a)*(1/(2*a-2)*ln(b*x+a-1) -1/(a-1)/(1+a)*ln(-b*x)-1/(2+2*a)*ln(b*x+a+1))-2*a/(a-1)^2/(1+a)^2*(1/2*di log((-b*x-a+1)/(1-a))+1/2*ln(-b*x)*ln((-b*x-a+1)/(1-a))-1/2*dilog((-b*x-a- 1)/(-a-1))-1/2*ln(-b*x)*ln((-b*x-a-1)/(-a-1))))
\[ \int \frac {\coth ^{-1}(a+b x)^2}{x^3} \, dx=\int { \frac {\operatorname {arcoth}\left (b x + a\right )^{2}}{x^{3}} \,d x } \] Input:
integrate(arccoth(b*x+a)^2/x^3,x, algorithm="fricas")
Output:
integral(arccoth(b*x + a)^2/x^3, x)
\[ \int \frac {\coth ^{-1}(a+b x)^2}{x^3} \, dx=\int \frac {\operatorname {acoth}^{2}{\left (a + b x \right )}}{x^{3}}\, dx \] Input:
integrate(acoth(b*x+a)**2/x**3,x)
Output:
Integral(acoth(a + b*x)**2/x**3, x)
Time = 0.04 (sec) , antiderivative size = 360, normalized size of antiderivative = 0.97 \[ \int \frac {\coth ^{-1}(a+b x)^2}{x^3} \, dx=\frac {1}{8} \, {\left (\frac {8 \, {\left (\log \left (b x + a - 1\right ) \log \left (\frac {1}{2} \, b x + \frac {1}{2} \, a + \frac {1}{2}\right ) + {\rm Li}_2\left (-\frac {1}{2} \, b x - \frac {1}{2} \, a + \frac {1}{2}\right )\right )} a}{a^{4} - 2 \, a^{2} + 1} - \frac {8 \, {\left (\log \left (\frac {b x}{a + 1} + 1\right ) \log \left (x\right ) + {\rm Li}_2\left (-\frac {b x}{a + 1}\right )\right )} a}{a^{4} - 2 \, a^{2} + 1} + \frac {8 \, {\left (\log \left (\frac {b x}{a - 1} + 1\right ) \log \left (x\right ) + {\rm Li}_2\left (-\frac {b x}{a - 1}\right )\right )} a}{a^{4} - 2 \, a^{2} + 1} - \frac {{\left (a^{2} - 2 \, a + 1\right )} \log \left (b x + a + 1\right )^{2} - 2 \, {\left (a^{2} - 2 \, a + 1\right )} \log \left (b x + a + 1\right ) \log \left (b x + a - 1\right ) + {\left (a^{2} + 2 \, a + 1\right )} \log \left (b x + a - 1\right )^{2}}{a^{4} - 2 \, a^{2} + 1} + \frac {4 \, \log \left (b x + a + 1\right )}{a^{3} + a^{2} - a - 1} - \frac {4 \, \log \left (b x + a - 1\right )}{a^{3} - a^{2} - a + 1} + \frac {8 \, \log \left (x\right )}{a^{4} - 2 \, a^{2} + 1}\right )} b^{2} + \frac {1}{2} \, {\left (\frac {4 \, a b \log \left (x\right )}{a^{4} - 2 \, a^{2} + 1} + \frac {b \log \left (b x + a + 1\right )}{a^{2} + 2 \, a + 1} - \frac {b \log \left (b x + a - 1\right )}{a^{2} - 2 \, a + 1} + \frac {2}{{\left (a^{2} - 1\right )} x}\right )} b \operatorname {arcoth}\left (b x + a\right ) - \frac {\operatorname {arcoth}\left (b x + a\right )^{2}}{2 \, x^{2}} \] Input:
integrate(arccoth(b*x+a)^2/x^3,x, algorithm="maxima")
Output:
1/8*(8*(log(b*x + a - 1)*log(1/2*b*x + 1/2*a + 1/2) + dilog(-1/2*b*x - 1/2 *a + 1/2))*a/(a^4 - 2*a^2 + 1) - 8*(log(b*x/(a + 1) + 1)*log(x) + dilog(-b *x/(a + 1)))*a/(a^4 - 2*a^2 + 1) + 8*(log(b*x/(a - 1) + 1)*log(x) + dilog( -b*x/(a - 1)))*a/(a^4 - 2*a^2 + 1) - ((a^2 - 2*a + 1)*log(b*x + a + 1)^2 - 2*(a^2 - 2*a + 1)*log(b*x + a + 1)*log(b*x + a - 1) + (a^2 + 2*a + 1)*log (b*x + a - 1)^2)/(a^4 - 2*a^2 + 1) + 4*log(b*x + a + 1)/(a^3 + a^2 - a - 1 ) - 4*log(b*x + a - 1)/(a^3 - a^2 - a + 1) + 8*log(x)/(a^4 - 2*a^2 + 1))*b ^2 + 1/2*(4*a*b*log(x)/(a^4 - 2*a^2 + 1) + b*log(b*x + a + 1)/(a^2 + 2*a + 1) - b*log(b*x + a - 1)/(a^2 - 2*a + 1) + 2/((a^2 - 1)*x))*b*arccoth(b*x + a) - 1/2*arccoth(b*x + a)^2/x^2
\[ \int \frac {\coth ^{-1}(a+b x)^2}{x^3} \, dx=\int { \frac {\operatorname {arcoth}\left (b x + a\right )^{2}}{x^{3}} \,d x } \] Input:
integrate(arccoth(b*x+a)^2/x^3,x, algorithm="giac")
Output:
integrate(arccoth(b*x + a)^2/x^3, x)
Timed out. \[ \int \frac {\coth ^{-1}(a+b x)^2}{x^3} \, dx=\int \frac {{\mathrm {acoth}\left (a+b\,x\right )}^2}{x^3} \,d x \] Input:
int(acoth(a + b*x)^2/x^3,x)
Output:
int(acoth(a + b*x)^2/x^3, x)
\[ \int \frac {\coth ^{-1}(a+b x)^2}{x^3} \, dx=\frac {-\mathit {acoth} \left (b x +a \right )^{2}-6 \mathit {acoth} \left (b x +a \right ) a^{2} b^{2} x^{2}+\mathit {acoth} \left (b x +a \right )^{2} a^{4} b^{2} x^{2}+\mathit {acoth} \left (b x +a \right )^{2} b^{2} x^{2}+2 a b x +4 \mathit {acoth} \left (b x +a \right ) a^{3}-2 \mathit {acoth} \left (b x +a \right ) a +4 \mathit {acoth} \left (b x +a \right ) a^{3} b^{2} x^{2}-2 \mathit {acoth} \left (b x +a \right )^{2} a^{2} b^{2} x^{2}+2 \mathit {acoth} \left (b x +a \right ) a^{4} b x +2 \mathit {acoth} \left (b x +a \right ) b x +2 \mathit {acoth} \left (b x +a \right ) b^{2} x^{2}-4 \mathit {acoth} \left (b x +a \right ) a^{2} b x +6 \,\mathrm {log}\left (b x +a -1\right ) a^{2} b^{2} x^{2}-6 \,\mathrm {log}\left (x \right ) a^{2} b^{2} x^{2}-3 \mathit {acoth} \left (b x +a \right )^{2} a^{6}-2 \mathit {acoth} \left (b x +a \right ) a^{5}+5 \mathit {acoth} \left (b x +a \right )^{2} a^{4}-\mathit {acoth} \left (b x +a \right )^{2} a^{2}-12 \left (\int \frac {\mathit {acoth} \left (b x +a \right )}{3 a^{2} b^{2} x^{5}+6 a^{3} b \,x^{4}+3 a^{4} x^{3}+b^{2} x^{5}+2 a b \,x^{4}-2 a^{2} x^{3}-x^{3}}d x \right ) a^{9} x^{2}+32 \left (\int \frac {\mathit {acoth} \left (b x +a \right )}{3 a^{2} b^{2} x^{5}+6 a^{3} b \,x^{4}+3 a^{4} x^{3}+b^{2} x^{5}+2 a b \,x^{4}-2 a^{2} x^{3}-x^{3}}d x \right ) a^{7} x^{2}-24 \left (\int \frac {\mathit {acoth} \left (b x +a \right )}{3 a^{2} b^{2} x^{5}+6 a^{3} b \,x^{4}+3 a^{4} x^{3}+b^{2} x^{5}+2 a b \,x^{4}-2 a^{2} x^{3}-x^{3}}d x \right ) a^{5} x^{2}+4 \left (\int \frac {\mathit {acoth} \left (b x +a \right )}{3 a^{2} b^{2} x^{5}+6 a^{3} b \,x^{4}+3 a^{4} x^{3}+b^{2} x^{5}+2 a b \,x^{4}-2 a^{2} x^{3}-x^{3}}d x \right ) a \,x^{2}-2 \,\mathrm {log}\left (b x +a -1\right ) b^{2} x^{2}+2 \,\mathrm {log}\left (x \right ) b^{2} x^{2}-2 a^{3} b x}{2 x^{2} \left (3 a^{6}-5 a^{4}+a^{2}+1\right )} \] Input:
int(acoth(b*x+a)^2/x^3,x)
Output:
( - 3*acoth(a + b*x)**2*a**6 + acoth(a + b*x)**2*a**4*b**2*x**2 + 5*acoth( a + b*x)**2*a**4 - 2*acoth(a + b*x)**2*a**2*b**2*x**2 - acoth(a + b*x)**2* a**2 + acoth(a + b*x)**2*b**2*x**2 - acoth(a + b*x)**2 - 2*acoth(a + b*x)* a**5 + 2*acoth(a + b*x)*a**4*b*x + 4*acoth(a + b*x)*a**3*b**2*x**2 + 4*aco th(a + b*x)*a**3 - 6*acoth(a + b*x)*a**2*b**2*x**2 - 4*acoth(a + b*x)*a**2 *b*x - 2*acoth(a + b*x)*a + 2*acoth(a + b*x)*b**2*x**2 + 2*acoth(a + b*x)* b*x - 12*int(acoth(a + b*x)/(3*a**4*x**3 + 6*a**3*b*x**4 + 3*a**2*b**2*x** 5 - 2*a**2*x**3 + 2*a*b*x**4 + b**2*x**5 - x**3),x)*a**9*x**2 + 32*int(aco th(a + b*x)/(3*a**4*x**3 + 6*a**3*b*x**4 + 3*a**2*b**2*x**5 - 2*a**2*x**3 + 2*a*b*x**4 + b**2*x**5 - x**3),x)*a**7*x**2 - 24*int(acoth(a + b*x)/(3*a **4*x**3 + 6*a**3*b*x**4 + 3*a**2*b**2*x**5 - 2*a**2*x**3 + 2*a*b*x**4 + b **2*x**5 - x**3),x)*a**5*x**2 + 4*int(acoth(a + b*x)/(3*a**4*x**3 + 6*a**3 *b*x**4 + 3*a**2*b**2*x**5 - 2*a**2*x**3 + 2*a*b*x**4 + b**2*x**5 - x**3), x)*a*x**2 + 6*log(a + b*x - 1)*a**2*b**2*x**2 - 2*log(a + b*x - 1)*b**2*x* *2 - 6*log(x)*a**2*b**2*x**2 + 2*log(x)*b**2*x**2 - 2*a**3*b*x + 2*a*b*x)/ (2*x**2*(3*a**6 - 5*a**4 + a**2 + 1))