Integrand size = 20, antiderivative size = 401 \[ \int \frac {\left (a+b \coth ^{-1}(c+d x)\right )^2}{(e+f x)^2} \, dx=-\frac {\left (a+b \coth ^{-1}(c+d x)\right )^2}{f (e+f x)}+\frac {b d \left (a+b \coth ^{-1}(c+d x)\right ) \log \left (\frac {2}{1-c-d x}\right )}{f (d e+f-c f)}-\frac {b d \left (a+b \coth ^{-1}(c+d x)\right ) \log \left (\frac {2}{1+c+d x}\right )}{f (d e-(1+c) f)}+\frac {2 b d \left (a+b \coth ^{-1}(c+d x)\right ) \log \left (\frac {2}{1+c+d x}\right )}{(d e+f-c f) (d e-(1+c) f)}-\frac {2 b d \left (a+b \coth ^{-1}(c+d x)\right ) \log \left (\frac {2 d (e+f x)}{(d e+f-c f) (1+c+d x)}\right )}{(d e+f-c f) (d e-(1+c) f)}+\frac {b^2 d \operatorname {PolyLog}\left (2,-\frac {1+c+d x}{1-c-d x}\right )}{2 f (d e+f-c f)}+\frac {b^2 d \operatorname {PolyLog}\left (2,1-\frac {2}{1+c+d x}\right )}{2 f (d e-(1+c) f)}-\frac {b^2 d \operatorname {PolyLog}\left (2,1-\frac {2}{1+c+d x}\right )}{(d e+f-c f) (d e-(1+c) f)}+\frac {b^2 d \operatorname {PolyLog}\left (2,1-\frac {2 d (e+f x)}{(d e+f-c f) (1+c+d x)}\right )}{(d e+f-c f) (d e-(1+c) f)} \] Output:
-(a+b*arccoth(d*x+c))^2/f/(f*x+e)+b*d*(a+b*arccoth(d*x+c))*ln(2/(-d*x-c+1) )/f/(-c*f+d*e+f)-b*d*(a+b*arccoth(d*x+c))*ln(2/(d*x+c+1))/f/(d*e-(1+c)*f)+ 2*b*d*(a+b*arccoth(d*x+c))*ln(2/(d*x+c+1))/(-c*f+d*e+f)/(d*e-(1+c)*f)-2*b* d*(a+b*arccoth(d*x+c))*ln(2*d*(f*x+e)/(-c*f+d*e+f)/(d*x+c+1))/(-c*f+d*e+f) /(d*e-(1+c)*f)+1/2*b^2*d*polylog(2,-(d*x+c+1)/(-d*x-c+1))/f/(-c*f+d*e+f)+1 /2*b^2*d*polylog(2,1-2/(d*x+c+1))/f/(d*e-(1+c)*f)-b^2*d*polylog(2,1-2/(d*x +c+1))/(-c*f+d*e+f)/(d*e-(1+c)*f)+b^2*d*polylog(2,1-2*d*(f*x+e)/(-c*f+d*e+ f)/(d*x+c+1))/(-c*f+d*e+f)/(d*e-(1+c)*f)
Result contains complex when optimal does not.
Time = 5.57 (sec) , antiderivative size = 470, normalized size of antiderivative = 1.17 \[ \int \frac {\left (a+b \coth ^{-1}(c+d x)\right )^2}{(e+f x)^2} \, dx=\frac {-\frac {a^2}{f}+\frac {2 a b \left (\left (f-c^2 f+d^2 e x+c d (e-f x)\right ) \coth ^{-1}(c+d x)-d (e+f x) \log \left (-\frac {d (e+f x)}{(c+d x) \sqrt {1-\frac {1}{(c+d x)^2}}}\right )\right )}{(d e+f-c f) (d e-(1+c) f)}+\frac {b^2 d (e+f x) \left (1-(c+d x)^2\right ) \left (\frac {e^{\text {arctanh}\left (\frac {f}{-d e+c f}\right )} \coth ^{-1}(c+d x)^2}{(-d e+c f) \sqrt {1-\frac {f^2}{(d e-c f)^2}}}+\frac {\coth ^{-1}(c+d x)^2}{d e+d f x}+\frac {f \left (-i \pi \log \left (1+e^{2 \coth ^{-1}(c+d x)}\right )-2 \text {arctanh}\left (\frac {f}{-d e+c f}\right ) \log \left (1-e^{-2 \left (\coth ^{-1}(c+d x)+\text {arctanh}\left (\frac {f}{d e-c f}\right )\right )}\right )+\coth ^{-1}(c+d x) \left (i \pi +2 \text {arctanh}\left (\frac {f}{d e-c f}\right )+2 \log \left (1-e^{-2 \left (\coth ^{-1}(c+d x)+\text {arctanh}\left (\frac {f}{d e-c f}\right )\right )}\right )\right )+i \pi \log \left (\frac {1}{\sqrt {1-\frac {1}{(c+d x)^2}}}\right )+2 \text {arctanh}\left (\frac {f}{-d e+c f}\right ) \log \left (i \sinh \left (\coth ^{-1}(c+d x)+\text {arctanh}\left (\frac {f}{d e-c f}\right )\right )\right )-\operatorname {PolyLog}\left (2,e^{-2 \left (\coth ^{-1}(c+d x)+\text {arctanh}\left (\frac {f}{d e-c f}\right )\right )}\right )\right )}{d^2 e^2-2 c d e f+\left (-1+c^2\right ) f^2}\right )}{(c+d x)^2 \left (f-\frac {f}{(c+d x)^2}\right )}}{e+f x} \] Input:
Integrate[(a + b*ArcCoth[c + d*x])^2/(e + f*x)^2,x]
Output:
(-(a^2/f) + (2*a*b*((f - c^2*f + d^2*e*x + c*d*(e - f*x))*ArcCoth[c + d*x] - d*(e + f*x)*Log[-((d*(e + f*x))/((c + d*x)*Sqrt[1 - (c + d*x)^(-2)]))]) )/((d*e + f - c*f)*(d*e - (1 + c)*f)) + (b^2*d*(e + f*x)*(1 - (c + d*x)^2) *((E^ArcTanh[f/(-(d*e) + c*f)]*ArcCoth[c + d*x]^2)/((-(d*e) + c*f)*Sqrt[1 - f^2/(d*e - c*f)^2]) + ArcCoth[c + d*x]^2/(d*e + d*f*x) + (f*((-I)*Pi*Log [1 + E^(2*ArcCoth[c + d*x])] - 2*ArcTanh[f/(-(d*e) + c*f)]*Log[1 - E^(-2*( ArcCoth[c + d*x] + ArcTanh[f/(d*e - c*f)]))] + ArcCoth[c + d*x]*(I*Pi + 2* ArcTanh[f/(d*e - c*f)] + 2*Log[1 - E^(-2*(ArcCoth[c + d*x] + ArcTanh[f/(d* e - c*f)]))]) + I*Pi*Log[1/Sqrt[1 - (c + d*x)^(-2)]] + 2*ArcTanh[f/(-(d*e) + c*f)]*Log[I*Sinh[ArcCoth[c + d*x] + ArcTanh[f/(d*e - c*f)]]] - PolyLog[ 2, E^(-2*(ArcCoth[c + d*x] + ArcTanh[f/(d*e - c*f)]))]))/(d^2*e^2 - 2*c*d* e*f + (-1 + c^2)*f^2)))/((c + d*x)^2*(f - f/(c + d*x)^2)))/(e + f*x)
Time = 1.90 (sec) , antiderivative size = 491, normalized size of antiderivative = 1.22, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, 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.
| \(\displaystyle \int \frac {\left (a+b \coth ^{-1}(c+d x)\right )^2}{(e+f x)^2} \, dx\) |
| \(\Big \downarrow \) 6660 |
| \(\displaystyle \frac {2 b d \int \frac {a+b \coth ^{-1}(c+d x)}{(e+f x) \left (1-(c+d x)^2\right )}dx}{f}-\frac {\left (a+b \coth ^{-1}(c+d x)\right )^2}{f (e+f x)}\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \frac {2 b d \int \frac {a+b \coth ^{-1}(c+d x)}{(e+f x) \left (-c^2-2 d x c-d^2 x^2+1\right )}dx}{f}-\frac {\left (a+b \coth ^{-1}(c+d x)\right )^2}{f (e+f x)}\) |
| \(\Big \downarrow \) 6672 |
| \(\displaystyle \frac {2 b \int \frac {d \left (a+b \coth ^{-1}(c+d x)\right )}{\left (d \left (e-\frac {c f}{d}\right )+f (c+d x)\right ) \left (1-(c+d x)^2\right )}d(c+d x)}{f}-\frac {\left (a+b \coth ^{-1}(c+d x)\right )^2}{f (e+f x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 b d \int \frac {a+b \coth ^{-1}(c+d x)}{(d e-c f+f (c+d x)) \left (1-(c+d x)^2\right )}d(c+d x)}{f}-\frac {\left (a+b \coth ^{-1}(c+d x)\right )^2}{f (e+f x)}\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \frac {2 b d \int \left (-\frac {a}{(c+d x-1) (c+d x+1) (d e-c f+f (c+d x))}-\frac {b \coth ^{-1}(c+d x)}{(c+d x-1) (c+d x+1) (d e-c f+f (c+d x))}\right )d(c+d x)}{f}-\frac {\left (a+b \coth ^{-1}(c+d x)\right )^2}{f (e+f x)}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2 b d \left (-\frac {a \log (-c-d x+1)}{2 (-c f+d e+f)}+\frac {a \log (c+d x+1)}{2 (d e-(c+1) f)}-\frac {a f \log (f (c+d x)-c f+d e)}{(-c f+d e+f) (d e-(c+1) f)}+\frac {b \operatorname {PolyLog}\left (2,-\frac {c+d x+1}{-c-d x+1}\right )}{4 (-c f+d e+f)}+\frac {b \operatorname {PolyLog}\left (2,1-\frac {2}{c+d x+1}\right )}{4 (-c f+d e-f)}-\frac {b f \operatorname {PolyLog}\left (2,1-\frac {2}{c+d x+1}\right )}{2 (-c f+d e+f) (d e-(c+1) f)}+\frac {b f \operatorname {PolyLog}\left (2,1-\frac {2 (d e-c f+f (c+d x))}{(d e-c f+f) (c+d x+1)}\right )}{2 (-c f+d e+f) (d e-(c+1) f)}+\frac {b \log \left (\frac {2}{-c-d x+1}\right ) \coth ^{-1}(c+d x)}{2 (-c f+d e+f)}-\frac {b \log \left (\frac {2}{c+d x+1}\right ) \coth ^{-1}(c+d x)}{2 (-c f+d e-f)}+\frac {b f \log \left (\frac {2}{c+d x+1}\right ) \coth ^{-1}(c+d x)}{(-c f+d e+f) (d e-(c+1) f)}-\frac {b f \coth ^{-1}(c+d x) \log \left (\frac {2 (f (c+d x)-c f+d e)}{(c+d x+1) (-c f+d e+f)}\right )}{(-c f+d e+f) (d e-(c+1) f)}\right )}{f}-\frac {\left (a+b \coth ^{-1}(c+d x)\right )^2}{f (e+f x)}\) |
Input:
Int[(a + b*ArcCoth[c + d*x])^2/(e + f*x)^2,x]
Output:
-((a + b*ArcCoth[c + d*x])^2/(f*(e + f*x))) + (2*b*d*((b*ArcCoth[c + d*x]* Log[2/(1 - c - d*x)])/(2*(d*e + f - c*f)) - (a*Log[1 - c - d*x])/(2*(d*e + f - c*f)) - (b*ArcCoth[c + d*x]*Log[2/(1 + c + d*x)])/(2*(d*e - f - c*f)) + (b*f*ArcCoth[c + d*x]*Log[2/(1 + c + d*x)])/((d*e + f - c*f)*(d*e - (1 + c)*f)) + (a*Log[1 + c + d*x])/(2*(d*e - (1 + c)*f)) - (a*f*Log[d*e - c*f + f*(c + d*x)])/((d*e + f - c*f)*(d*e - (1 + c)*f)) - (b*f*ArcCoth[c + d* x]*Log[(2*(d*e - c*f + f*(c + d*x)))/((d*e + f - c*f)*(1 + c + d*x))])/((d *e + f - c*f)*(d*e - (1 + c)*f)) + (b*PolyLog[2, -((1 + c + d*x)/(1 - c - d*x))])/(4*(d*e + f - c*f)) + (b*PolyLog[2, 1 - 2/(1 + c + d*x)])/(4*(d*e - f - c*f)) - (b*f*PolyLog[2, 1 - 2/(1 + c + d*x)])/(2*(d*e + f - c*f)*(d* e - (1 + c)*f)) + (b*f*PolyLog[2, 1 - (2*(d*e - c*f + f*(c + d*x)))/((d*e + f - c*f)*(1 + c + d*x))])/(2*(d*e + f - c*f)*(d*e - (1 + c)*f))))/f
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 = 2.02 (sec) , antiderivative size = 590, normalized size of antiderivative = 1.47
| method | result | size |
| parts | \(-\frac {a^{2}}{\left (f x +e \right ) f}+\frac {b^{2} \left (-\frac {d^{2} \operatorname {arccoth}\left (d x +c \right )^{2}}{\left (f \left (d x +c \right )-c f +d e \right ) f}-\frac {2 d^{2} \left (\frac {\operatorname {arccoth}\left (d x +c \right ) f \ln \left (f \left (d x +c \right )-c f +d e \right )}{\left (c f -d e -f \right ) \left (c f -d e +f \right )}-\frac {\operatorname {arccoth}\left (d x +c \right ) \ln \left (d x +c -1\right )}{2 c f -2 d e -2 f}+\frac {\operatorname {arccoth}\left (d x +c \right ) \ln \left (d x +c +1\right )}{2 c f -2 d e +2 f}+\frac {\frac {f \left (\operatorname {dilog}\left (\frac {f \left (d x +c \right )-f}{c f -d e -f}\right )+\ln \left (f \left (d x +c \right )-c f +d e \right ) \ln \left (\frac {f \left (d x +c \right )-f}{c f -d e -f}\right )\right )}{2}-\frac {f \left (\operatorname {dilog}\left (\frac {f \left (d x +c \right )+f}{c f -d e +f}\right )+\ln \left (f \left (d x +c \right )-c f +d e \right ) \ln \left (\frac {f \left (d x +c \right )+f}{c f -d e +f}\right )\right )}{2}}{\left (c f -d e -f \right ) \left (c f -d e +f \right )}-\frac {\frac {\ln \left (d x +c -1\right )^{2}}{4}-\frac {\operatorname {dilog}\left (\frac {d x}{2}+\frac {c}{2}+\frac {1}{2}\right )}{2}-\frac {\ln \left (d x +c -1\right ) \ln \left (\frac {d x}{2}+\frac {c}{2}+\frac {1}{2}\right )}{2}}{2 \left (c f -d e -f \right )}+\frac {-\frac {\ln \left (d x +c +1\right )^{2}}{4}+\frac {\left (\ln \left (d x +c +1\right )-\ln \left (\frac {d x}{2}+\frac {c}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {d x}{2}-\frac {c}{2}+\frac {1}{2}\right )}{2}-\frac {\operatorname {dilog}\left (\frac {d x}{2}+\frac {c}{2}+\frac {1}{2}\right )}{2}}{2 c f -2 d e +2 f}\right )}{f}\right )}{d}-\frac {2 a b d \,\operatorname {arccoth}\left (d x +c \right )}{\left (d f x +d e \right ) f}-\frac {2 a b d \ln \left (f \left (d x +c \right )-c f +d e \right )}{\left (c f -d e -f \right ) \left (c f -d e +f \right )}+\frac {2 a b d \ln \left (d x +c -1\right )}{f \left (2 c f -2 d e -2 f \right )}-\frac {2 a b d \ln \left (d x +c +1\right )}{f \left (2 c f -2 d e +2 f \right )}\) | \(590\) |
| derivativedivides | \(\frac {\frac {a^{2} d^{2}}{\left (c f -d e -f \left (d x +c \right )\right ) f}+b^{2} d^{2} \left (\frac {\operatorname {arccoth}\left (d x +c \right )^{2}}{\left (c f -d e -f \left (d x +c \right )\right ) f}+\frac {-\frac {2 \,\operatorname {arccoth}\left (d x +c \right ) f \ln \left (c f -d e -f \left (d x +c \right )\right )}{\left (c f -d e -f \right ) \left (c f -d e +f \right )}-\frac {2 \,\operatorname {arccoth}\left (d x +c \right ) \ln \left (d x +c +1\right )}{2 c f -2 d e +2 f}+\frac {2 \,\operatorname {arccoth}\left (d x +c \right ) \ln \left (d x +c -1\right )}{2 c f -2 d e -2 f}+\frac {2 \left (\frac {\ln \left (d x +c -1\right )^{2}}{4}-\frac {\operatorname {dilog}\left (\frac {d x}{2}+\frac {c}{2}+\frac {1}{2}\right )}{2}-\frac {\ln \left (d x +c -1\right ) \ln \left (\frac {d x}{2}+\frac {c}{2}+\frac {1}{2}\right )}{2}\right )}{2 c f -2 d e -2 f}-\frac {-\frac {\ln \left (d x +c +1\right )^{2}}{4}+\frac {\left (\ln \left (d x +c +1\right )-\ln \left (\frac {d x}{2}+\frac {c}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {d x}{2}-\frac {c}{2}+\frac {1}{2}\right )}{2}-\frac {\operatorname {dilog}\left (\frac {d x}{2}+\frac {c}{2}+\frac {1}{2}\right )}{2}}{c f -d e +f}+\frac {2 \left (-\frac {f \left (\operatorname {dilog}\left (\frac {-f \left (d x +c \right )+f}{-c f +d e +f}\right )+\ln \left (c f -d e -f \left (d x +c \right )\right ) \ln \left (\frac {-f \left (d x +c \right )+f}{-c f +d e +f}\right )\right )}{2}+\frac {f \left (\operatorname {dilog}\left (\frac {-f \left (d x +c \right )-f}{-c f +d e -f}\right )+\ln \left (c f -d e -f \left (d x +c \right )\right ) \ln \left (\frac {-f \left (d x +c \right )-f}{-c f +d e -f}\right )\right )}{2}\right )}{\left (c f -d e -f \right ) \left (c f -d e +f \right )}}{f}\right )+2 a b \,d^{2} \left (\frac {\operatorname {arccoth}\left (d x +c \right )}{\left (c f -d e -f \left (d x +c \right )\right ) f}+\frac {-\frac {\ln \left (d x +c +1\right )}{2 c f -2 d e +2 f}-\frac {f \ln \left (c f -d e -f \left (d x +c \right )\right )}{\left (c f -d e -f \right ) \left (c f -d e +f \right )}+\frac {\ln \left (d x +c -1\right )}{2 c f -2 d e -2 f}}{f}\right )}{d}\) | \(612\) |
| default | \(\frac {\frac {a^{2} d^{2}}{\left (c f -d e -f \left (d x +c \right )\right ) f}+b^{2} d^{2} \left (\frac {\operatorname {arccoth}\left (d x +c \right )^{2}}{\left (c f -d e -f \left (d x +c \right )\right ) f}+\frac {-\frac {2 \,\operatorname {arccoth}\left (d x +c \right ) f \ln \left (c f -d e -f \left (d x +c \right )\right )}{\left (c f -d e -f \right ) \left (c f -d e +f \right )}-\frac {2 \,\operatorname {arccoth}\left (d x +c \right ) \ln \left (d x +c +1\right )}{2 c f -2 d e +2 f}+\frac {2 \,\operatorname {arccoth}\left (d x +c \right ) \ln \left (d x +c -1\right )}{2 c f -2 d e -2 f}+\frac {2 \left (\frac {\ln \left (d x +c -1\right )^{2}}{4}-\frac {\operatorname {dilog}\left (\frac {d x}{2}+\frac {c}{2}+\frac {1}{2}\right )}{2}-\frac {\ln \left (d x +c -1\right ) \ln \left (\frac {d x}{2}+\frac {c}{2}+\frac {1}{2}\right )}{2}\right )}{2 c f -2 d e -2 f}-\frac {-\frac {\ln \left (d x +c +1\right )^{2}}{4}+\frac {\left (\ln \left (d x +c +1\right )-\ln \left (\frac {d x}{2}+\frac {c}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {d x}{2}-\frac {c}{2}+\frac {1}{2}\right )}{2}-\frac {\operatorname {dilog}\left (\frac {d x}{2}+\frac {c}{2}+\frac {1}{2}\right )}{2}}{c f -d e +f}+\frac {2 \left (-\frac {f \left (\operatorname {dilog}\left (\frac {-f \left (d x +c \right )+f}{-c f +d e +f}\right )+\ln \left (c f -d e -f \left (d x +c \right )\right ) \ln \left (\frac {-f \left (d x +c \right )+f}{-c f +d e +f}\right )\right )}{2}+\frac {f \left (\operatorname {dilog}\left (\frac {-f \left (d x +c \right )-f}{-c f +d e -f}\right )+\ln \left (c f -d e -f \left (d x +c \right )\right ) \ln \left (\frac {-f \left (d x +c \right )-f}{-c f +d e -f}\right )\right )}{2}\right )}{\left (c f -d e -f \right ) \left (c f -d e +f \right )}}{f}\right )+2 a b \,d^{2} \left (\frac {\operatorname {arccoth}\left (d x +c \right )}{\left (c f -d e -f \left (d x +c \right )\right ) f}+\frac {-\frac {\ln \left (d x +c +1\right )}{2 c f -2 d e +2 f}-\frac {f \ln \left (c f -d e -f \left (d x +c \right )\right )}{\left (c f -d e -f \right ) \left (c f -d e +f \right )}+\frac {\ln \left (d x +c -1\right )}{2 c f -2 d e -2 f}}{f}\right )}{d}\) | \(612\) |
Input:
int((a+b*arccoth(d*x+c))^2/(f*x+e)^2,x,method=_RETURNVERBOSE)
Output:
-a^2/(f*x+e)/f+b^2/d*(-d^2/(f*(d*x+c)-c*f+d*e)/f*arccoth(d*x+c)^2-2*d^2/f* (arccoth(d*x+c)*f/(c*f-d*e-f)/(c*f-d*e+f)*ln(f*(d*x+c)-c*f+d*e)-arccoth(d* x+c)/(2*c*f-2*d*e-2*f)*ln(d*x+c-1)+arccoth(d*x+c)/(2*c*f-2*d*e+2*f)*ln(d*x +c+1)+1/(c*f-d*e-f)/(c*f-d*e+f)*(1/2*f*(dilog((f*(d*x+c)-f)/(c*f-d*e-f))+l n(f*(d*x+c)-c*f+d*e)*ln((f*(d*x+c)-f)/(c*f-d*e-f)))-1/2*f*(dilog((f*(d*x+c )+f)/(c*f-d*e+f))+ln(f*(d*x+c)-c*f+d*e)*ln((f*(d*x+c)+f)/(c*f-d*e+f))))-1/ 2/(c*f-d*e-f)*(1/4*ln(d*x+c-1)^2-1/2*dilog(1/2*d*x+1/2*c+1/2)-1/2*ln(d*x+c -1)*ln(1/2*d*x+1/2*c+1/2))+1/2/(c*f-d*e+f)*(-1/4*ln(d*x+c+1)^2+1/2*(ln(d*x +c+1)-ln(1/2*d*x+1/2*c+1/2))*ln(-1/2*d*x-1/2*c+1/2)-1/2*dilog(1/2*d*x+1/2* c+1/2))))-2*a*b*d/(d*f*x+d*e)/f*arccoth(d*x+c)-2*a*b*d/(c*f-d*e-f)/(c*f-d* e+f)*ln(f*(d*x+c)-c*f+d*e)+2*a*b*d/f/(2*c*f-2*d*e-2*f)*ln(d*x+c-1)-2*a*b*d /f/(2*c*f-2*d*e+2*f)*ln(d*x+c+1)
\[ \int \frac {\left (a+b \coth ^{-1}(c+d x)\right )^2}{(e+f x)^2} \, dx=\int { \frac {{\left (b \operatorname {arcoth}\left (d x + c\right ) + a\right )}^{2}}{{\left (f x + e\right )}^{2}} \,d x } \] Input:
integrate((a+b*arccoth(d*x+c))^2/(f*x+e)^2,x, algorithm="fricas")
Output:
integral((b^2*arccoth(d*x + c)^2 + 2*a*b*arccoth(d*x + c) + a^2)/(f^2*x^2 + 2*e*f*x + e^2), x)
\[ \int \frac {\left (a+b \coth ^{-1}(c+d x)\right )^2}{(e+f x)^2} \, dx=\int \frac {\left (a + b \operatorname {acoth}{\left (c + d x \right )}\right )^{2}}{\left (e + f x\right )^{2}}\, dx \] Input:
integrate((a+b*acoth(d*x+c))**2/(f*x+e)**2,x)
Output:
Integral((a + b*acoth(c + d*x))**2/(e + f*x)**2, x)
\[ \int \frac {\left (a+b \coth ^{-1}(c+d x)\right )^2}{(e+f x)^2} \, dx=\int { \frac {{\left (b \operatorname {arcoth}\left (d x + c\right ) + a\right )}^{2}}{{\left (f x + e\right )}^{2}} \,d x } \] Input:
integrate((a+b*arccoth(d*x+c))^2/(f*x+e)^2,x, algorithm="maxima")
Output:
(d*(log(d*x + c + 1)/(d*e*f - (c + 1)*f^2) - log(d*x + c - 1)/(d*e*f - (c - 1)*f^2) - 2*log(f*x + e)/(d^2*e^2 - 2*c*d*e*f + (c^2 - 1)*f^2)) - 2*arcc oth(d*x + c)/(f^2*x + e*f))*a*b - 1/4*b^2*(log(d*x + c + 1)^2/(f^2*x + e*f ) + integrate(-((d*f*x + c*f + f)*log(d*x + c - 1)^2 + 2*(d*f*x + d*e - (d *f*x + c*f + f)*log(d*x + c - 1))*log(d*x + c + 1))/(d*f^3*x^3 + c*e^2*f + e^2*f + (2*d*e*f^2 + c*f^3 + f^3)*x^2 + (d*e^2*f + 2*c*e*f^2 + 2*e*f^2)*x ), x)) - a^2/(f^2*x + e*f)
\[ \int \frac {\left (a+b \coth ^{-1}(c+d x)\right )^2}{(e+f x)^2} \, dx=\int { \frac {{\left (b \operatorname {arcoth}\left (d x + c\right ) + a\right )}^{2}}{{\left (f x + e\right )}^{2}} \,d x } \] Input:
integrate((a+b*arccoth(d*x+c))^2/(f*x+e)^2,x, algorithm="giac")
Output:
integrate((b*arccoth(d*x + c) + a)^2/(f*x + e)^2, x)
Timed out. \[ \int \frac {\left (a+b \coth ^{-1}(c+d x)\right )^2}{(e+f x)^2} \, dx=\int \frac {{\left (a+b\,\mathrm {acoth}\left (c+d\,x\right )\right )}^2}{{\left (e+f\,x\right )}^2} \,d x \] Input:
int((a + b*acoth(c + d*x))^2/(e + f*x)^2,x)
Output:
int((a + b*acoth(c + d*x))^2/(e + f*x)^2, x)
\[ \int \frac {\left (a+b \coth ^{-1}(c+d x)\right )^2}{(e+f x)^2} \, dx=\text {too large to display} \] Input:
int((a+b*acoth(d*x+c))^2/(f*x+e)^2,x)
Output:
( - acoth(c + d*x)**2*b**2*c**4*e*f**3 + 2*acoth(c + d*x)**2*b**2*c**3*d*e **2*f**2 - acoth(c + d*x)**2*b**2*c**2*d**2*e**3*f - acoth(c + d*x)**2*b** 2*c**2*d**2*e**2*f**2*x + 2*acoth(c + d*x)**2*b**2*c**2*e*f**3 + 2*acoth(c + d*x)**2*b**2*c*d**3*e**3*f*x - 2*acoth(c + d*x)**2*b**2*c*d*e**2*f**2 - acoth(c + d*x)**2*b**2*d**4*e**4*x + acoth(c + d*x)**2*b**2*d**2*e**3*f + acoth(c + d*x)**2*b**2*d**2*e**2*f**2*x - acoth(c + d*x)**2*b**2*e*f**3 + 2*acoth(c + d*x)*a*b*c**4*f**4*x - 4*acoth(c + d*x)*a*b*c**3*d*e*f**3*x - 4*acoth(c + d*x)*a*b*c**2*f**4*x + 4*acoth(c + d*x)*a*b*c*d**3*e**3*f*x + 4*acoth(c + d*x)*a*b*c*d*e*f**3*x - 2*acoth(c + d*x)*a*b*d**4*e**4*x + 2* acoth(c + d*x)*a*b*f**4*x + 2*acoth(c + d*x)*b**2*c**2*d*e*f**3*x - 4*acot h(c + d*x)*b**2*c*d**2*e**2*f**2*x + 2*acoth(c + d*x)*b**2*d**3*e**3*f*x - 2*acoth(c + d*x)*b**2*d*e*f**3*x + 2*int((acoth(c + d*x)*x)/(c**4*e**2*f* *2 + 2*c**4*e*f**3*x + c**4*f**4*x**2 + 2*c**3*d*e**2*f**2*x + 4*c**3*d*e* f**3*x**2 + 2*c**3*d*f**4*x**3 - c**2*d**2*e**4 - 2*c**2*d**2*e**3*f*x + 2 *c**2*d**2*e*f**3*x**3 + c**2*d**2*f**4*x**4 - 2*c**2*e**2*f**2 - 4*c**2*e *f**3*x - 2*c**2*f**4*x**2 - 2*c*d**3*e**4*x - 4*c*d**3*e**3*f*x**2 - 2*c* d**3*e**2*f**2*x**3 - 2*c*d*e**2*f**2*x - 4*c*d*e*f**3*x**2 - 2*c*d*f**4*x **3 - d**4*e**4*x**2 - 2*d**4*e**3*f*x**3 - d**4*e**2*f**2*x**4 + d**2*e** 4 + 2*d**2*e**3*f*x - 2*d**2*e*f**3*x**3 - d**2*f**4*x**4 + e**2*f**2 + 2* e*f**3*x + f**4*x**2),x)*b**2*c**6*d*e**2*f**6 + 2*int((acoth(c + d*x)*...