\(\int \frac {(a+b \coth ^{-1}(c+d x))^3}{(e+f x)^2} \, dx\) [37]

Optimal result

Integrand size = 20, antiderivative size = 634 \[ \int \frac {\left (a+b \coth ^{-1}(c+d x)\right )^3}{(e+f x)^2} \, dx=-\frac {\left (a+b \coth ^{-1}(c+d x)\right )^3}{f (e+f x)}+\frac {3 b d \left (a+b \coth ^{-1}(c+d x)\right )^2 \log \left (\frac {2}{1-c-d x}\right )}{2 f (d e+f-c f)}-\frac {3 b d \left (a+b \coth ^{-1}(c+d x)\right )^2 \log \left (\frac {2}{1+c+d x}\right )}{2 f (d e-(1+c) f)}+\frac {3 b d \left (a+b \coth ^{-1}(c+d x)\right )^2 \log \left (\frac {2}{1+c+d x}\right )}{(d e+f-c f) (d e-(1+c) f)}-\frac {3 b d \left (a+b \coth ^{-1}(c+d x)\right )^2 \log \left (\frac {2 d (e+f x)}{(d e+f-c f) (1+c+d x)}\right )}{(d e+f-c f) (d e-(1+c) f)}+\frac {3 b^2 d \left (a+b \coth ^{-1}(c+d x)\right ) \operatorname {PolyLog}\left (2,1-\frac {2}{1-c-d x}\right )}{2 f (d e+f-c f)}+\frac {3 b^2 d \left (a+b \coth ^{-1}(c+d x)\right ) \operatorname {PolyLog}\left (2,1-\frac {2}{1+c+d x}\right )}{2 f (d e-(1+c) f)}-\frac {3 b^2 d \left (a+b \coth ^{-1}(c+d x)\right ) \operatorname {PolyLog}\left (2,1-\frac {2}{1+c+d x}\right )}{(d e+f-c f) (d e-(1+c) f)}+\frac {3 b^2 d \left (a+b \coth ^{-1}(c+d x)\right ) \operatorname {PolyLog}\left (2,1-\frac {2 d (e+f x)}{(d e+f-c f) (1+c+d x)}\right )}{(d e+f-c f) (d e-(1+c) f)}-\frac {3 b^3 d \operatorname {PolyLog}\left (3,1-\frac {2}{1-c-d x}\right )}{4 f (d e+f-c f)}+\frac {3 b^3 d \operatorname {PolyLog}\left (3,1-\frac {2}{1+c+d x}\right )}{4 f (d e-(1+c) f)}-\frac {3 b^3 d \operatorname {PolyLog}\left (3,1-\frac {2}{1+c+d x}\right )}{2 (d e+f-c f) (d e-(1+c) f)}+\frac {3 b^3 d \operatorname {PolyLog}\left (3,1-\frac {2 d (e+f x)}{(d e+f-c f) (1+c+d x)}\right )}{2 (d e+f-c f) (d e-(1+c) f)} \] Output:

-(a+b*arccoth(d*x+c))^3/f/(f*x+e)+3/2*b*d*(a+b*arccoth(d*x+c))^2*ln(2/(-d* 
x-c+1))/f/(-c*f+d*e+f)-3/2*b*d*(a+b*arccoth(d*x+c))^2*ln(2/(d*x+c+1))/f/(d 
*e-(1+c)*f)+3*b*d*(a+b*arccoth(d*x+c))^2*ln(2/(d*x+c+1))/(-c*f+d*e+f)/(d*e 
-(1+c)*f)-3*b*d*(a+b*arccoth(d*x+c))^2*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)+3/2*b^2*d*(a+b*arccoth(d*x+c))*polylog(2,1- 
2/(-d*x-c+1))/f/(-c*f+d*e+f)+3/2*b^2*d*(a+b*arccoth(d*x+c))*polylog(2,1-2/ 
(d*x+c+1))/f/(d*e-(1+c)*f)-3*b^2*d*(a+b*arccoth(d*x+c))*polylog(2,1-2/(d*x 
+c+1))/(-c*f+d*e+f)/(d*e-(1+c)*f)+3*b^2*d*(a+b*arccoth(d*x+c))*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)-3/4*b^3*d* 
polylog(3,1-2/(-d*x-c+1))/f/(-c*f+d*e+f)+3/4*b^3*d*polylog(3,1-2/(d*x+c+1) 
)/f/(d*e-(1+c)*f)-3/2*b^3*d*polylog(3,1-2/(d*x+c+1))/(-c*f+d*e+f)/(d*e-(1+ 
c)*f)+3/2*b^3*d*polylog(3,1-2*d*(f*x+e)/(-c*f+d*e+f)/(d*x+c+1))/(-c*f+d*e+ 
f)/(d*e-(1+c)*f)
 

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 13.13 (sec) , antiderivative size = 1945, normalized size of antiderivative = 3.07 \[ \int \frac {\left (a+b \coth ^{-1}(c+d x)\right )^3}{(e+f x)^2} \, dx =\text {Too large to display} \] Input:

Integrate[(a + b*ArcCoth[c + d*x])^3/(e + f*x)^2,x]
 

Output:

-(a^3/(f*(e + f*x))) - (3*a^2*b*ArcCoth[c + d*x])/(f*(e + f*x)) + (3*a^2*b 
*d*Log[1 - c - d*x])/(2*f*(-(d*e) - f + c*f)) - (3*a^2*b*d*Log[1 + c + d*x 
])/(2*f*(-(d*e) + f + c*f)) - (3*a^2*b*d*Log[e + f*x])/(d^2*e^2 - 2*c*d*e* 
f - f^2 + c^2*f^2) + (3*a*b^2*(1 - (c + d*x)^2)*(f/Sqrt[1 - (c + d*x)^(-2) 
] + (d*e - c*f)/((c + d*x)*Sqrt[1 - (c + d*x)^(-2)]))^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/((c + d*x)*Sqrt[1 - (c + d*x)^(-2)]*(f/Sqrt[1 - (c 
 + d*x)^(-2)] + (d*e - c*f)/((c + d*x)*Sqrt[1 - (c + d*x)^(-2)]))) + (f*(I 
*Pi*ArcCoth[c + d*x] + 2*ArcCoth[c + d*x]*ArcTanh[f/(d*e - c*f)] - I*Pi*Lo 
g[1 + E^(2*ArcCoth[c + d*x])] + 2*ArcCoth[c + d*x]*Log[1 - E^(-2*(ArcCoth[ 
c + d*x] + ArcTanh[f/(d*e - c*f)]))] - 2*ArcTanh[f/(-(d*e) + c*f)]*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] + ArcTan 
h[f/(d*e - c*f)]))]))/(d^2*e^2 - 2*c*d*e*f + (-1 + c^2)*f^2)))/(d*f*(e + f 
*x)^2) - (b^3*(1 - (c + d*x)^2)*(f/Sqrt[1 - (c + d*x)^(-2)] + (d*e - c*f)/ 
((c + d*x)*Sqrt[1 - (c + d*x)^(-2)]))^2*((d*ArcCoth[c + d*x]^3)/(f*(c + d* 
x)*Sqrt[1 - (c + d*x)^(-2)]*(-(f/Sqrt[1 - (c + d*x)^(-2)]) - (d*e)/((c + d 
*x)*Sqrt[1 - (c + d*x)^(-2)]) + (c*f)/((c + d*x)*Sqrt[1 - (c + d*x)^(-2)]) 
)) - (d*(2*d*e*ArcCoth[c + d*x]^3 - 6*f*ArcCoth[c + d*x]^3 - 2*c*f*ArcC...
 

Rubi [A] (verified)

Time = 2.88 (sec) , antiderivative size = 1085, normalized size of antiderivative = 1.71, 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.

\(\displaystyle \int \frac {\left (a+b \coth ^{-1}(c+d x)\right )^3}{(e+f x)^2} \, dx\)

\(\Big \downarrow \) 6660

\(\displaystyle \frac {3 b d \int \frac {\left (a+b \coth ^{-1}(c+d x)\right )^2}{(e+f x) \left (1-(c+d x)^2\right )}dx}{f}-\frac {\left (a+b \coth ^{-1}(c+d x)\right )^3}{f (e+f x)}\)

\(\Big \downarrow \) 7292

\(\displaystyle \frac {3 b d \int \frac {\left (a+b \coth ^{-1}(c+d x)\right )^2}{(e+f x) \left (-c^2-2 d x c-d^2 x^2+1\right )}dx}{f}-\frac {\left (a+b \coth ^{-1}(c+d x)\right )^3}{f (e+f x)}\)

\(\Big \downarrow \) 6672

\(\displaystyle \frac {3 b \int \frac {d \left (a+b \coth ^{-1}(c+d x)\right )^2}{\left (d \left (e-\frac {c f}{d}\right )+f (c+d x)\right ) \left (1-(c+d x)^2\right )}d(c+d x)}{f}-\frac {\left (a+b \coth ^{-1}(c+d x)\right )^3}{f (e+f x)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {3 b d \int \frac {\left (a+b \coth ^{-1}(c+d x)\right )^2}{(d e-c f+f (c+d x)) \left (1-(c+d x)^2\right )}d(c+d x)}{f}-\frac {\left (a+b \coth ^{-1}(c+d x)\right )^3}{f (e+f x)}\)

\(\Big \downarrow \) 7276

\(\displaystyle \frac {3 b d \int \left (-\frac {a^2}{(c+d x-1) (c+d x+1) (d e-c f+f (c+d x))}-\frac {2 b \coth ^{-1}(c+d x) a}{(c+d x-1) (c+d x+1) (d e-c f+f (c+d x))}-\frac {b^2 \coth ^{-1}(c+d x)^2}{(c+d x-1) (c+d x+1) (d e-c f+f (c+d x))}\right )d(c+d x)}{f}-\frac {\left (a+b \coth ^{-1}(c+d x)\right )^3}{f (e+f x)}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {3 b d \left (-\frac {\log (-c-d x+1) a^2}{2 (d e-c f+f)}+\frac {\log (c+d x+1) a^2}{2 (d e-(c+1) f)}-\frac {f \log (d e-c f+f (c+d x)) a^2}{(d e-c f+f) (d e-(c+1) f)}+\frac {b \coth ^{-1}(c+d x) \log \left (\frac {2}{-c-d x+1}\right ) a}{d e-c f+f}-\frac {b \coth ^{-1}(c+d x) \log \left (\frac {2}{c+d x+1}\right ) a}{d e-c f-f}+\frac {2 b f \coth ^{-1}(c+d x) \log \left (\frac {2}{c+d x+1}\right ) a}{(d e-c f+f) (d e-(c+1) f)}-\frac {2 b f \coth ^{-1}(c+d x) \log \left (\frac {2 (d e-c f+f (c+d x))}{(d e-c f+f) (c+d x+1)}\right ) a}{(d e-c f+f) (d e-(c+1) f)}+\frac {b \operatorname {PolyLog}\left (2,-\frac {c+d x+1}{-c-d x+1}\right ) a}{2 (d e-c f+f)}+\frac {b \operatorname {PolyLog}\left (2,1-\frac {2}{c+d x+1}\right ) a}{2 (d e-c f-f)}-\frac {b f \operatorname {PolyLog}\left (2,1-\frac {2}{c+d x+1}\right ) a}{(d e-c f+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 ) a}{(d e-c f+f) (d e-(c+1) f)}+\frac {b^2 \coth ^{-1}(c+d x)^2 \log \left (\frac {2}{-c-d x+1}\right )}{2 (d e-c f+f)}-\frac {b^2 \coth ^{-1}(c+d x)^2 \log \left (\frac {2}{c+d x+1}\right )}{2 (d e-c f-f)}+\frac {b^2 f \coth ^{-1}(c+d x)^2 \log \left (\frac {2}{c+d x+1}\right )}{(d e-c f+f) (d e-(c+1) f)}-\frac {b^2 f \coth ^{-1}(c+d x)^2 \log \left (\frac {2 (d e-c f+f (c+d x))}{(d e-c f+f) (c+d x+1)}\right )}{(d e-c f+f) (d e-(c+1) f)}+\frac {b^2 \coth ^{-1}(c+d x) \operatorname {PolyLog}\left (2,1-\frac {2}{-c-d x+1}\right )}{2 (d e-c f+f)}+\frac {b^2 \coth ^{-1}(c+d x) \operatorname {PolyLog}\left (2,1-\frac {2}{c+d x+1}\right )}{2 (d e-c f-f)}-\frac {b^2 f \coth ^{-1}(c+d x) \operatorname {PolyLog}\left (2,1-\frac {2}{c+d x+1}\right )}{(d e-c f+f) (d e-(c+1) f)}+\frac {b^2 f \coth ^{-1}(c+d x) \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 )}{(d e-c f+f) (d e-(c+1) f)}-\frac {b^2 \operatorname {PolyLog}\left (3,1-\frac {2}{-c-d x+1}\right )}{4 (d e-c f+f)}+\frac {b^2 \operatorname {PolyLog}\left (3,1-\frac {2}{c+d x+1}\right )}{4 (d e-c f-f)}-\frac {b^2 f \operatorname {PolyLog}\left (3,1-\frac {2}{c+d x+1}\right )}{2 (d e-c f+f) (d e-(c+1) f)}+\frac {b^2 f \operatorname {PolyLog}\left (3,1-\frac {2 (d e-c f+f (c+d x))}{(d e-c f+f) (c+d x+1)}\right )}{2 (d e-c f+f) (d e-(c+1) f)}\right )}{f}-\frac {\left (a+b \coth ^{-1}(c+d x)\right )^3}{f (e+f x)}\)

Input:

Int[(a + b*ArcCoth[c + d*x])^3/(e + f*x)^2,x]
 

Output:

-((a + b*ArcCoth[c + d*x])^3/(f*(e + f*x))) + (3*b*d*((a*b*ArcCoth[c + d*x 
]*Log[2/(1 - c - d*x)])/(d*e + f - c*f) + (b^2*ArcCoth[c + d*x]^2*Log[2/(1 
 - c - d*x)])/(2*(d*e + f - c*f)) - (a^2*Log[1 - c - d*x])/(2*(d*e + f - c 
*f)) - (a*b*ArcCoth[c + d*x]*Log[2/(1 + c + d*x)])/(d*e - f - c*f) + (2*a* 
b*f*ArcCoth[c + d*x]*Log[2/(1 + c + d*x)])/((d*e + f - c*f)*(d*e - (1 + c) 
*f)) - (b^2*ArcCoth[c + d*x]^2*Log[2/(1 + c + d*x)])/(2*(d*e - f - c*f)) + 
 (b^2*f*ArcCoth[c + d*x]^2*Log[2/(1 + c + d*x)])/((d*e + f - c*f)*(d*e - ( 
1 + c)*f)) + (a^2*Log[1 + c + d*x])/(2*(d*e - (1 + c)*f)) - (a^2*f*Log[d*e 
 - c*f + f*(c + d*x)])/((d*e + f - c*f)*(d*e - (1 + c)*f)) - (2*a*b*f*ArcC 
oth[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^2*f*ArcCoth[c + d*x]^2*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)) + (a*b*PolyLog[2, -((1 + c + d*x)/(1 - c - d*x) 
)])/(2*(d*e + f - c*f)) + (b^2*ArcCoth[c + d*x]*PolyLog[2, 1 - 2/(1 - c - 
d*x)])/(2*(d*e + f - c*f)) + (a*b*PolyLog[2, 1 - 2/(1 + c + d*x)])/(2*(d*e 
 - f - c*f)) - (a*b*f*PolyLog[2, 1 - 2/(1 + c + d*x)])/((d*e + f - c*f)*(d 
*e - (1 + c)*f)) + (b^2*ArcCoth[c + d*x]*PolyLog[2, 1 - 2/(1 + c + d*x)])/ 
(2*(d*e - f - c*f)) - (b^2*f*ArcCoth[c + d*x]*PolyLog[2, 1 - 2/(1 + c + d* 
x)])/((d*e + f - c*f)*(d*e - (1 + c)*f)) + (a*b*f*PolyLog[2, 1 - (2*(d*e - 
 c*f + f*(c + d*x)))/((d*e + f - c*f)*(1 + c + d*x))])/((d*e + f - c*f)...
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 6660

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]
 

rule 6672

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]
 

rule 7276

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]
 

rule 7292

Int[u_, x_Symbol] :> With[{v = NormalizeIntegrand[u, x]}, Int[v, x] /; v =! 
= u]
 

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 13.83 (sec) , antiderivative size = 4101, normalized size of antiderivative = 6.47

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(4101\)
default	\(\text {Expression too large to display}\)	\(4101\)
parts	\(\text {Expression too large to display}\)	\(4252\)

Input:

int((a+b*arccoth(d*x+c))^3/(f*x+e)^2,x,method=_RETURNVERBOSE)
 

Output:

1/d*(a^3*d^2/(c*f-d*e-f*(d*x+c))/f+b^3*d^2*(1/(c*f-d*e-f*(d*x+c))/f*arccot 
h(d*x+c)^3+3/f*(-1/(c*f-d*e-f)^2/(c*f-d*e+f)*f^2*c*arccoth(d*x+c)*polylog( 
2,(c*f-d*e-f)/(d*x+c-1)*(d*x+c+1)/(c*f-d*e+f))-1/2/(c*f-d*e-f)^2/(c*f-d*e+ 
f)*f*e*d*polylog(3,(c*f-d*e-f)/(d*x+c-1)*(d*x+c+1)/(c*f-d*e+f))-1/(c*f-d*e 
-f)^2/(c*f-d*e+f)*f^2*c*arccoth(d*x+c)^2*ln(1-(c*f-d*e-f)/(d*x+c-1)*(d*x+c 
+1)/(c*f-d*e+f))+1/2*I/(c*f-d*e-f)/(c*f-d*e+f)*Pi*csgn(I/((d*x+c-1)/(d*x+c 
+1))^(1/2))*csgn(I*(d*x+c+1)/(d*x+c-1))^2*c*f*arccoth(d*x+c)^2-1/4*I/(c*f- 
d*e-f)/(c*f-d*e+f)*Pi*csgn(I/((d*x+c+1)/(d*x+c-1)-1))*csgn(I/(d*x+c-1)*(d* 
x+c+1)/((d*x+c+1)/(d*x+c-1)-1))^2*d*e*arccoth(d*x+c)^2-1/4*I/(c*f-d*e-f)/( 
c*f-d*e+f)*Pi*csgn(I*(d*x+c+1)/(d*x+c-1))*csgn(I/(d*x+c-1)*(d*x+c+1)/((d*x 
+c+1)/(d*x+c-1)-1))^2*d*e*arccoth(d*x+c)^2+1/4*I/(c*f-d*e-f)/(c*f-d*e+f)*P 
i*csgn(I/((d*x+c+1)/(d*x+c-1)-1))*csgn(I*(d*x+c+1)/(d*x+c-1))*csgn(I/(d*x+ 
c-1)*(d*x+c+1)/((d*x+c+1)/(d*x+c-1)-1))*f*arccoth(d*x+c)^2+1/4*I/(c*f-d*e- 
f)/(c*f-d*e+f)*Pi*csgn(I/((d*x+c+1)/(d*x+c-1)-1))*csgn(I/(d*x+c-1)*(d*x+c+ 
1)/((d*x+c+1)/(d*x+c-1)-1))^2*c*f*arccoth(d*x+c)^2+1/4*I/(c*f-d*e-f)/(c*f- 
d*e+f)*Pi*csgn(I/((d*x+c-1)/(d*x+c+1))^(1/2))^2*csgn(I*(d*x+c+1)/(d*x+c-1) 
)*d*e*arccoth(d*x+c)^2-1/2*I/(c*f-d*e-f)/(c*f-d*e+f)*Pi*csgn(I/((d*x+c-1)/ 
(d*x+c+1))^(1/2))*csgn(I*(d*x+c+1)/(d*x+c-1))^2*d*e*arccoth(d*x+c)^2-arcco 
th(d*x+c)^2*f/(c*f-d*e-f)/(c*f-d*e+f)*ln(c*f-d*e-f*(d*x+c))-1/2/(c*f-d*e-f 
)^2/(c*f-d*e+f)*f^2*polylog(3,(c*f-d*e-f)/(d*x+c-1)*(d*x+c+1)/(c*f-d*e+...
 

Fricas [F]

\[ \int \frac {\left (a+b \coth ^{-1}(c+d x)\right )^3}{(e+f x)^2} \, dx=\int { \frac {{\left (b \operatorname {arcoth}\left (d x + c\right ) + a\right )}^{3}}{{\left (f x + e\right )}^{2}} \,d x } \] Input:

integrate((a+b*arccoth(d*x+c))^3/(f*x+e)^2,x, algorithm="fricas")
 

Output:

integral((b^3*arccoth(d*x + c)^3 + 3*a*b^2*arccoth(d*x + c)^2 + 3*a^2*b*ar 
ccoth(d*x + c) + a^3)/(f^2*x^2 + 2*e*f*x + e^2), x)
 

Sympy [F]

\[ \int \frac {\left (a+b \coth ^{-1}(c+d x)\right )^3}{(e+f x)^2} \, dx=\int \frac {\left (a + b \operatorname {acoth}{\left (c + d x \right )}\right )^{3}}{\left (e + f x\right )^{2}}\, dx \] Input:

integrate((a+b*acoth(d*x+c))**3/(f*x+e)**2,x)
 

Output:

Integral((a + b*acoth(c + d*x))**3/(e + f*x)**2, x)
 

Maxima [F]

\[ \int \frac {\left (a+b \coth ^{-1}(c+d x)\right )^3}{(e+f x)^2} \, dx=\int { \frac {{\left (b \operatorname {arcoth}\left (d x + c\right ) + a\right )}^{3}}{{\left (f x + e\right )}^{2}} \,d x } \] Input:

integrate((a+b*arccoth(d*x+c))^3/(f*x+e)^2,x, algorithm="maxima")
 

Output:

3/2*(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* 
arccoth(d*x + c)/(f^2*x + e*f))*a^2*b - a^3/(f^2*x + e*f) + 1/8*(((d^2*e*f 
 - c*d*f^2 + d*f^2)*b^3*x + (c*d*e*f - c^2*f^2 + d*e*f + f^2)*b^3)*log(d*x 
 + c + 1)^3 - 3*(2*(d^2*e^2 - 2*c*d*e*f + c^2*f^2 - f^2)*a*b^2 + ((d^2*e*f 
 - c*d*f^2 - d*f^2)*b^3*x + (c*d*e*f - c^2*f^2 - d*e*f + f^2)*b^3)*log(d*x 
 + c - 1))*log(d*x + c + 1)^2)/(d^2*e^3*f - 2*c*d*e^2*f^2 + c^2*e*f^3 - e* 
f^3 + (d^2*e^2*f^2 - 2*c*d*e*f^3 + c^2*f^4 - f^4)*x) + integrate(-1/8*(((d 
^2*e*f - c*d*f^2 + d*f^2)*b^3*x + (c*d*e*f - c^2*f^2 + d*e*f + f^2)*b^3)*l 
og(d*x + c - 1)^3 - 6*((d^2*e*f - c*d*f^2 + d*f^2)*a*b^2*x + (c*d*e*f - c^ 
2*f^2 + d*e*f + f^2)*a*b^2)*log(d*x + c - 1)^2 - 3*(4*(d^2*e*f - c*d*f^2 + 
 d*f^2)*a*b^2*x + 4*(d^2*e^2 - c*d*e*f + d*e*f)*a*b^2 + ((d^2*e*f - c*d*f^ 
2 + d*f^2)*b^3*x + (c*d*e*f - c^2*f^2 + d*e*f + f^2)*b^3)*log(d*x + c - 1) 
^2 + 2*(b^3*d^2*f^2*x^2 - 2*(c*d*e*f - c^2*f^2 + d*e*f + f^2)*a*b^2 + (c*d 
*e*f - d*e*f)*b^3 - (2*(d^2*e*f - c*d*f^2 + d*f^2)*a*b^2 - (d^2*e*f + c*d* 
f^2 - d*f^2)*b^3)*x)*log(d*x + c - 1))*log(d*x + c + 1))/(c*d*e^3*f - c^2* 
e^2*f^2 + d*e^3*f + e^2*f^2 + (d^2*e*f^3 - c*d*f^4 + d*f^4)*x^3 + (2*d^2*e 
^2*f^2 - c*d*e*f^3 - c^2*f^4 + 3*d*e*f^3 + f^4)*x^2 + (d^2*e^3*f + c*d*e^2 
*f^2 - 2*c^2*e*f^3 + 3*d*e^2*f^2 + 2*e*f^3)*x), x)
 

Giac [F]

\[ \int \frac {\left (a+b \coth ^{-1}(c+d x)\right )^3}{(e+f x)^2} \, dx=\int { \frac {{\left (b \operatorname {arcoth}\left (d x + c\right ) + a\right )}^{3}}{{\left (f x + e\right )}^{2}} \,d x } \] Input:

integrate((a+b*arccoth(d*x+c))^3/(f*x+e)^2,x, algorithm="giac")
 

Output:

integrate((b*arccoth(d*x + c) + a)^3/(f*x + e)^2, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b \coth ^{-1}(c+d x)\right )^3}{(e+f x)^2} \, dx=\int \frac {{\left (a+b\,\mathrm {acoth}\left (c+d\,x\right )\right )}^3}{{\left (e+f\,x\right )}^2} \,d x \] Input:

int((a + b*acoth(c + d*x))^3/(e + f*x)^2,x)
 

Output:

int((a + b*acoth(c + d*x))^3/(e + f*x)^2, x)
                                                                                    
                                                                                    
 

Reduce [F]

\[ \int \frac {\left (a+b \coth ^{-1}(c+d x)\right )^3}{(e+f x)^2} \, dx=\text {too large to display} \] Input:

int((a+b*acoth(d*x+c))^3/(f*x+e)^2,x)
 

Output:

( - 4*acoth(c + d*x)**3*b**3*c**6*e*f**5 + 16*acoth(c + d*x)**3*b**3*c**5* 
d*e**2*f**4 - 24*acoth(c + d*x)**3*b**3*c**4*d**2*e**3*f**3 - 4*acoth(c + 
d*x)**3*b**3*c**4*d**2*e**2*f**4*x + 8*acoth(c + d*x)**3*b**3*c**4*e*f**5 
+ 16*acoth(c + d*x)**3*b**3*c**3*d**3*e**4*f**2 + 16*acoth(c + d*x)**3*b** 
3*c**3*d**3*e**3*f**3*x - 24*acoth(c + d*x)**3*b**3*c**3*d*e**2*f**4 - 4*a 
coth(c + d*x)**3*b**3*c**2*d**4*e**5*f - 24*acoth(c + d*x)**3*b**3*c**2*d* 
*4*e**4*f**2*x + 28*acoth(c + d*x)**3*b**3*c**2*d**2*e**3*f**3 + 4*acoth(c 
 + d*x)**3*b**3*c**2*d**2*e**2*f**4*x - 4*acoth(c + d*x)**3*b**3*c**2*e*f* 
*5 + 16*acoth(c + d*x)**3*b**3*c*d**5*e**5*f*x - 16*acoth(c + d*x)**3*b**3 
*c*d**3*e**4*f**2 - 8*acoth(c + d*x)**3*b**3*c*d**3*e**3*f**3*x + 8*acoth( 
c + d*x)**3*b**3*c*d*e**2*f**4 - 4*acoth(c + d*x)**3*b**3*d**6*e**6*x + 4* 
acoth(c + d*x)**3*b**3*d**4*e**5*f + 4*acoth(c + d*x)**3*b**3*d**4*e**4*f* 
*2*x - 4*acoth(c + d*x)**3*b**3*d**2*e**3*f**3 - 12*acoth(c + d*x)**2*a*b* 
*2*c**6*e*f**5 + 48*acoth(c + d*x)**2*a*b**2*c**5*d*e**2*f**4 - 72*acoth(c 
 + d*x)**2*a*b**2*c**4*d**2*e**3*f**3 - 12*acoth(c + d*x)**2*a*b**2*c**4*d 
**2*e**2*f**4*x + 24*acoth(c + d*x)**2*a*b**2*c**4*e*f**5 + 48*acoth(c + d 
*x)**2*a*b**2*c**3*d**3*e**4*f**2 + 48*acoth(c + d*x)**2*a*b**2*c**3*d**3* 
e**3*f**3*x - 72*acoth(c + d*x)**2*a*b**2*c**3*d*e**2*f**4 - 12*acoth(c + 
d*x)**2*a*b**2*c**2*d**4*e**5*f - 72*acoth(c + d*x)**2*a*b**2*c**2*d**4*e* 
*4*f**2*x + 84*acoth(c + d*x)**2*a*b**2*c**2*d**2*e**3*f**3 + 12*acoth(...