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

Optimal. Leaf size=308 \[ -\frac{3 b^2 \left (a+b \coth ^{-1}(c+d x)\right ) \text{PolyLog}\left (3,1-\frac{2 d (e+f x)}{(c+d x+1) (-c f+d e+f)}\right )}{2 f}+\frac{3 b^2 \text{PolyLog}\left (3,1-\frac{2}{c+d x+1}\right ) \left (a+b \coth ^{-1}(c+d x)\right )}{2 f}-\frac{3 b \left (a+b \coth ^{-1}(c+d x)\right )^2 \text{PolyLog}\left (2,1-\frac{2 d (e+f x)}{(c+d x+1) (-c f+d e+f)}\right )}{2 f}+\frac{3 b \text{PolyLog}\left (2,1-\frac{2}{c+d x+1}\right ) \left (a+b \coth ^{-1}(c+d x)\right )^2}{2 f}-\frac{3 b^3 \text{PolyLog}\left (4,1-\frac{2 d (e+f x)}{(c+d x+1) (-c f+d e+f)}\right )}{4 f}+\frac{3 b^3 \text{PolyLog}\left (4,1-\frac{2}{c+d x+1}\right )}{4 f}+\frac{\left (a+b \coth ^{-1}(c+d x)\right )^3 \log \left (\frac{2 d (e+f x)}{(c+d x+1) (-c f+d e+f)}\right )}{f}-\frac{\log \left (\frac{2}{c+d x+1}\right ) \left (a+b \coth ^{-1}(c+d x)\right )^3}{f} \]

[Out]

-(((a + b*ArcCoth[c + d*x])^3*Log[2/(1 + c + d*x)])/f) + ((a + b*ArcCoth[c + d*x])^3*Log[(2*d*(e + f*x))/((d*e
 + f - c*f)*(1 + c + d*x))])/f + (3*b*(a + b*ArcCoth[c + d*x])^2*PolyLog[2, 1 - 2/(1 + c + d*x)])/(2*f) - (3*b
*(a + b*ArcCoth[c + d*x])^2*PolyLog[2, 1 - (2*d*(e + f*x))/((d*e + f - c*f)*(1 + c + d*x))])/(2*f) + (3*b^2*(a
 + b*ArcCoth[c + d*x])*PolyLog[3, 1 - 2/(1 + c + d*x)])/(2*f) - (3*b^2*(a + b*ArcCoth[c + d*x])*PolyLog[3, 1 -
 (2*d*(e + f*x))/((d*e + f - c*f)*(1 + c + d*x))])/(2*f) + (3*b^3*PolyLog[4, 1 - 2/(1 + c + d*x)])/(4*f) - (3*
b^3*PolyLog[4, 1 - (2*d*(e + f*x))/((d*e + f - c*f)*(1 + c + d*x))])/(4*f)

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Rubi [A]  time = 0.187135, antiderivative size = 308, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {6112, 5925} \[ -\frac{3 b^2 \left (a+b \coth ^{-1}(c+d x)\right ) \text{PolyLog}\left (3,1-\frac{2 d (e+f x)}{(c+d x+1) (-c f+d e+f)}\right )}{2 f}+\frac{3 b^2 \text{PolyLog}\left (3,1-\frac{2}{c+d x+1}\right ) \left (a+b \coth ^{-1}(c+d x)\right )}{2 f}-\frac{3 b \left (a+b \coth ^{-1}(c+d x)\right )^2 \text{PolyLog}\left (2,1-\frac{2 d (e+f x)}{(c+d x+1) (-c f+d e+f)}\right )}{2 f}+\frac{3 b \text{PolyLog}\left (2,1-\frac{2}{c+d x+1}\right ) \left (a+b \coth ^{-1}(c+d x)\right )^2}{2 f}-\frac{3 b^3 \text{PolyLog}\left (4,1-\frac{2 d (e+f x)}{(c+d x+1) (-c f+d e+f)}\right )}{4 f}+\frac{3 b^3 \text{PolyLog}\left (4,1-\frac{2}{c+d x+1}\right )}{4 f}+\frac{\left (a+b \coth ^{-1}(c+d x)\right )^3 \log \left (\frac{2 d (e+f x)}{(c+d x+1) (-c f+d e+f)}\right )}{f}-\frac{\log \left (\frac{2}{c+d x+1}\right ) \left (a+b \coth ^{-1}(c+d x)\right )^3}{f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(((a + b*ArcCoth[c + d*x])^3*Log[2/(1 + c + d*x)])/f) + ((a + b*ArcCoth[c + d*x])^3*Log[(2*d*(e + f*x))/((d*e
 + f - c*f)*(1 + c + d*x))])/f + (3*b*(a + b*ArcCoth[c + d*x])^2*PolyLog[2, 1 - 2/(1 + c + d*x)])/(2*f) - (3*b
*(a + b*ArcCoth[c + d*x])^2*PolyLog[2, 1 - (2*d*(e + f*x))/((d*e + f - c*f)*(1 + c + d*x))])/(2*f) + (3*b^2*(a
 + b*ArcCoth[c + d*x])*PolyLog[3, 1 - 2/(1 + c + d*x)])/(2*f) - (3*b^2*(a + b*ArcCoth[c + d*x])*PolyLog[3, 1 -
 (2*d*(e + f*x))/((d*e + f - c*f)*(1 + c + d*x))])/(2*f) + (3*b^3*PolyLog[4, 1 - 2/(1 + c + d*x)])/(4*f) - (3*
b^3*PolyLog[4, 1 - (2*d*(e + f*x))/((d*e + f - c*f)*(1 + c + d*x))])/(4*f)

Rule 6112

Int[((a_.) + ArcCoth[(c_) + (d_.)*(x_)]*(b_.))^(p_.)*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Dist[1/d, Subst[
Int[((d*e - c*f)/d + (f*x)/d)^m*(a + b*ArcCoth[x])^p, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] &
& IGtQ[p, 0]

Rule 5925

Int[((a_.) + ArcCoth[(c_.)*(x_)]*(b_.))^3/((d_) + (e_.)*(x_)), x_Symbol] :> -Simp[((a + b*ArcCoth[c*x])^3*Log[
2/(1 + c*x)])/e, x] + (Simp[((a + b*ArcCoth[c*x])^3*Log[(2*c*(d + e*x))/((c*d + e)*(1 + c*x))])/e, x] + Simp[(
3*b*(a + b*ArcCoth[c*x])^2*PolyLog[2, 1 - 2/(1 + c*x)])/(2*e), x] - Simp[(3*b*(a + b*ArcCoth[c*x])^2*PolyLog[2
, 1 - (2*c*(d + e*x))/((c*d + e)*(1 + c*x))])/(2*e), x] + Simp[(3*b^2*(a + b*ArcCoth[c*x])*PolyLog[3, 1 - 2/(1
 + c*x)])/(2*e), x] - Simp[(3*b^2*(a + b*ArcCoth[c*x])*PolyLog[3, 1 - (2*c*(d + e*x))/((c*d + e)*(1 + c*x))])/
(2*e), x] + Simp[(3*b^3*PolyLog[4, 1 - 2/(1 + c*x)])/(4*e), x] - Simp[(3*b^3*PolyLog[4, 1 - (2*c*(d + e*x))/((
c*d + e)*(1 + c*x))])/(4*e), x]) /; FreeQ[{a, b, c, d, e}, x] && NeQ[c^2*d^2 - e^2, 0]

Rubi steps

\begin{align*} \int \frac{\left (a+b \coth ^{-1}(c+d x)\right )^3}{e+f x} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (a+b \coth ^{-1}(x)\right )^3}{\frac{d e-c f}{d}+\frac{f x}{d}} \, dx,x,c+d x\right )}{d}\\ &=-\frac{\left (a+b \coth ^{-1}(c+d x)\right )^3 \log \left (\frac{2}{1+c+d x}\right )}{f}+\frac{\left (a+b \coth ^{-1}(c+d x)\right )^3 \log \left (\frac{2 d (e+f x)}{(d e+f-c f) (1+c+d x)}\right )}{f}+\frac{3 b \left (a+b \coth ^{-1}(c+d x)\right )^2 \text{Li}_2\left (1-\frac{2}{1+c+d x}\right )}{2 f}-\frac{3 b \left (a+b \coth ^{-1}(c+d x)\right )^2 \text{Li}_2\left (1-\frac{2 d (e+f x)}{(d e+f-c f) (1+c+d x)}\right )}{2 f}+\frac{3 b^2 \left (a+b \coth ^{-1}(c+d x)\right ) \text{Li}_3\left (1-\frac{2}{1+c+d x}\right )}{2 f}-\frac{3 b^2 \left (a+b \coth ^{-1}(c+d x)\right ) \text{Li}_3\left (1-\frac{2 d (e+f x)}{(d e+f-c f) (1+c+d x)}\right )}{2 f}+\frac{3 b^3 \text{Li}_4\left (1-\frac{2}{1+c+d x}\right )}{4 f}-\frac{3 b^3 \text{Li}_4\left (1-\frac{2 d (e+f x)}{(d e+f-c f) (1+c+d x)}\right )}{4 f}\\ \end{align*}

Mathematica [F]  time = 22.4348, size = 0, normalized size = 0. \[ \int \frac{\left (a+b \coth ^{-1}(c+d x)\right )^3}{e+f x} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [C]  time = 0.71, size = 3796, normalized size = 12.3 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/2*a^2*b/f*dilog(((d*x+c)*f-f)/(c*f-d*e-f))-b^3/(c*f-d*e-f)*arccoth(d*x+c)^3*ln(1-(c*f-d*e-f)*(d*x+c+1)/(d*x+
c-1)/(c*f-d*e+f))+3/2*b^3/(c*f-d*e-f)*arccoth(d*x+c)*polylog(3,(c*f-d*e-f)*(d*x+c+1)/(d*x+c-1)/(c*f-d*e+f))-3*
b^3/f*arccoth(d*x+c)^2*polylog(2,1/((d*x+c-1)/(d*x+c+1))^(1/2))-b^3/f*arccoth(d*x+c)^3*ln(1-1/((d*x+c-1)/(d*x+
c+1))^(1/2))-3*b^3/f*arccoth(d*x+c)^2*polylog(2,-1/((d*x+c-1)/(d*x+c+1))^(1/2))+6*b^3/f*arccoth(d*x+c)*polylog
(3,-1/((d*x+c-1)/(d*x+c+1))^(1/2))+b^3*ln((d*x+c)*f-c*f+d*e)/f*arccoth(d*x+c)^3+b^3/f*arccoth(d*x+c)^3*ln((d*x
+c+1)/(d*x+c-1)-1)+6*b^3/f*arccoth(d*x+c)*polylog(3,1/((d*x+c-1)/(d*x+c+1))^(1/2))-b^3/f*arccoth(d*x+c)^3*ln(1
+1/((d*x+c-1)/(d*x+c+1))^(1/2))-b^3/f*arccoth(d*x+c)^3*ln(((d*x+c+1)/(d*x+c-1)-1)*c*f+(1-(d*x+c+1)/(d*x+c-1))*
e*d+(-(d*x+c+1)/(d*x+c-1)-1)*f)-3/2*b^3/(c*f-d*e-f)*arccoth(d*x+c)^2*polylog(2,(c*f-d*e-f)*(d*x+c+1)/(d*x+c-1)
/(c*f-d*e+f))+3/4*b^3*c/(c*f-d*e-f)*polylog(4,(c*f-d*e-f)*(d*x+c+1)/(d*x+c-1)/(c*f-d*e+f))+6*a*b^2/f*polylog(3
,1/((d*x+c-1)/(d*x+c+1))^(1/2))+6*a*b^2/f*polylog(3,-1/((d*x+c-1)/(d*x+c+1))^(1/2))+3/2*a*b^2/(c*f-d*e-f)*poly
log(3,(c*f-d*e-f)*(d*x+c+1)/(d*x+c-1)/(c*f-d*e+f))-3/2*a^2*b/f*dilog(((d*x+c)*f+f)/(c*f-d*e+f))+3/2*b^3*c/(c*f
-d*e-f)*arccoth(d*x+c)^2*polylog(2,(c*f-d*e-f)*(d*x+c+1)/(d*x+c-1)/(c*f-d*e+f))-6*a*b^2/f*arccoth(d*x+c)*polyl
og(2,-1/((d*x+c-1)/(d*x+c+1))^(1/2))-3*a*b^2/f*arccoth(d*x+c)^2*ln(1-1/((d*x+c-1)/(d*x+c+1))^(1/2))-6*a*b^2/f*
arccoth(d*x+c)*polylog(2,1/((d*x+c-1)/(d*x+c+1))^(1/2))-3*a*b^2/(c*f-d*e-f)*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))-3*a*b^2/(c*f-d*e-f)*arccoth(d*x+c)*polylog(2,(c*f-d*e-f)*(d*x+c+1)/(d*x+c-1)/
(c*f-d*e+f))+3*a*b^2/f*arccoth(d*x+c)^2*ln((d*x+c+1)/(d*x+c-1)-1)-3*a*b^2/f*arccoth(d*x+c)^2*ln(((d*x+c+1)/(d*
x+c-1)-1)*c*f+(1-(d*x+c+1)/(d*x+c-1))*e*d+(-(d*x+c+1)/(d*x+c-1)-1)*f)+b^3*c/(c*f-d*e-f)*arccoth(d*x+c)^3*ln(1-
(c*f-d*e-f)*(d*x+c+1)/(d*x+c-1)/(c*f-d*e+f))-3/2*b^3*c/(c*f-d*e-f)*arccoth(d*x+c)*polylog(3,(c*f-d*e-f)*(d*x+c
+1)/(d*x+c-1)/(c*f-d*e+f))+3*a^2*b*ln((d*x+c)*f-c*f+d*e)/f*arccoth(d*x+c)-I*b^3/f*Pi*arccoth(d*x+c)^3-3/2*a^2*
b/f*ln(((d*x+c)*f+f)/(c*f-d*e+f))*ln((d*x+c)*f-c*f+d*e)+3/2*a^2*b/f*ln(((d*x+c)*f-f)/(c*f-d*e-f))*ln((d*x+c)*f
-c*f+d*e)+3*a*b^2*ln((d*x+c)*f-c*f+d*e)/f*arccoth(d*x+c)^2-3*a*b^2/f*arccoth(d*x+c)^2*ln(1+1/((d*x+c-1)/(d*x+c
+1))^(1/2))-3/2*a*b^2*c/(c*f-d*e-f)*polylog(3,(c*f-d*e-f)*(d*x+c+1)/(d*x+c-1)/(c*f-d*e+f))-3*d*a*b^2/f*e/(c*f-
d*e-f)*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))-3*d*a*b^2/f*e/(c*f-d*e-f)*arccoth(d*
x+c)*polylog(2,(c*f-d*e-f)*(d*x+c+1)/(d*x+c-1)/(c*f-d*e+f))-3/2*I*a*b^2/f*Pi*arccoth(d*x+c)^2*csgn(I*(((d*x+c+
1)/(d*x+c-1)-1)*c*f+(1-(d*x+c+1)/(d*x+c-1))*e*d+(-(d*x+c+1)/(d*x+c-1)-1)*f)/((d*x+c+1)/(d*x+c-1)-1))^2*csgn(I/
((d*x+c+1)/(d*x+c-1)-1))-3/2*I*a*b^2/f*Pi*arccoth(d*x+c)^2*csgn(I*(((d*x+c+1)/(d*x+c-1)-1)*c*f+(1-(d*x+c+1)/(d
*x+c-1))*e*d+(-(d*x+c+1)/(d*x+c-1)-1)*f))*csgn(I*(((d*x+c+1)/(d*x+c-1)-1)*c*f+(1-(d*x+c+1)/(d*x+c-1))*e*d+(-(d
*x+c+1)/(d*x+c-1)-1)*f)/((d*x+c+1)/(d*x+c-1)-1))^2+1/2*I*b^3/f*Pi*arccoth(d*x+c)^3*csgn(I*(((d*x+c+1)/(d*x+c-1
)-1)*c*f+(1-(d*x+c+1)/(d*x+c-1))*e*d+(-(d*x+c+1)/(d*x+c-1)-1)*f))*csgn(I*(((d*x+c+1)/(d*x+c-1)-1)*c*f+(1-(d*x+
c+1)/(d*x+c-1))*e*d+(-(d*x+c+1)/(d*x+c-1)-1)*f)/((d*x+c+1)/(d*x+c-1)-1))*csgn(I/((d*x+c+1)/(d*x+c-1)-1))-6*b^3
/f*polylog(4,1/((d*x+c-1)/(d*x+c+1))^(1/2))+a^3*ln((d*x+c)*f-c*f+d*e)/f-3/4*b^3/(c*f-d*e-f)*polylog(4,(c*f-d*e
-f)*(d*x+c+1)/(d*x+c-1)/(c*f-d*e+f))-6*b^3/f*polylog(4,-1/((d*x+c-1)/(d*x+c+1))^(1/2))+3*a*b^2*c/(c*f-d*e-f)*a
rccoth(d*x+c)^2*ln(1-(c*f-d*e-f)*(d*x+c+1)/(d*x+c-1)/(c*f-d*e+f))+3*a*b^2*c/(c*f-d*e-f)*arccoth(d*x+c)*polylog
(2,(c*f-d*e-f)*(d*x+c+1)/(d*x+c-1)/(c*f-d*e+f))+I*b^3/f*Pi*arccoth(d*x+c)^3*csgn(I*(((d*x+c+1)/(d*x+c-1)-1)*c*
f+(1-(d*x+c+1)/(d*x+c-1))*e*d+(-(d*x+c+1)/(d*x+c-1)-1)*f)/((d*x+c+1)/(d*x+c-1)-1))^2-3/4*d*b^3/f*e/(c*f-d*e-f)
*polylog(4,(c*f-d*e-f)*(d*x+c+1)/(d*x+c-1)/(c*f-d*e+f))-3*I*a*b^2/f*Pi*arccoth(d*x+c)^2-1/2*I*b^3/f*Pi*arccoth
(d*x+c)^3*csgn(I*(((d*x+c+1)/(d*x+c-1)-1)*c*f+(1-(d*x+c+1)/(d*x+c-1))*e*d+(-(d*x+c+1)/(d*x+c-1)-1)*f)/((d*x+c+
1)/(d*x+c-1)-1))^3-3/2*I*a*b^2/f*Pi*arccoth(d*x+c)^2*csgn(I*(((d*x+c+1)/(d*x+c-1)-1)*c*f+(1-(d*x+c+1)/(d*x+c-1
))*e*d+(-(d*x+c+1)/(d*x+c-1)-1)*f)/((d*x+c+1)/(d*x+c-1)-1))^3-1/2*I*b^3/f*Pi*arccoth(d*x+c)^3*csgn(I*(((d*x+c+
1)/(d*x+c-1)-1)*c*f+(1-(d*x+c+1)/(d*x+c-1))*e*d+(-(d*x+c+1)/(d*x+c-1)-1)*f)/((d*x+c+1)/(d*x+c-1)-1))^2*csgn(I/
((d*x+c+1)/(d*x+c-1)-1))+3/2*d*a*b^2/f*e/(c*f-d*e-f)*polylog(3,(c*f-d*e-f)*(d*x+c+1)/(d*x+c-1)/(c*f-d*e+f))-3/
2*d*b^3/f*e/(c*f-d*e-f)*arccoth(d*x+c)^2*polylog(2,(c*f-d*e-f)*(d*x+c+1)/(d*x+c-1)/(c*f-d*e+f))-d*b^3/f*e/(c*f
-d*e-f)*arccoth(d*x+c)^3*ln(1-(c*f-d*e-f)*(d*x+c+1)/(d*x+c-1)/(c*f-d*e+f))+3/2*d*b^3/f*e/(c*f-d*e-f)*arccoth(d
*x+c)*polylog(3,(c*f-d*e-f)*(d*x+c+1)/(d*x+c-1)/(c*f-d*e+f))-1/2*I*b^3/f*Pi*arccoth(d*x+c)^3*csgn(I*(((d*x+c+1
)/(d*x+c-1)-1)*c*f+(1-(d*x+c+1)/(d*x+c-1))*e*d+(-(d*x+c+1)/(d*x+c-1)-1)*f))*csgn(I*(((d*x+c+1)/(d*x+c-1)-1)*c*
f+(1-(d*x+c+1)/(d*x+c-1))*e*d+(-(d*x+c+1)/(d*x+c-1)-1)*f)/((d*x+c+1)/(d*x+c-1)-1))^2+3*I*a*b^2/f*Pi*arccoth(d*
x+c)^2*csgn(I*(((d*x+c+1)/(d*x+c-1)-1)*c*f+(1-(d*x+c+1)/(d*x+c-1))*e*d+(-(d*x+c+1)/(d*x+c-1)-1)*f)/((d*x+c+1)/
(d*x+c-1)-1))^2+3/2*I*a*b^2/f*Pi*arccoth(d*x+c)^2*csgn(I*(((d*x+c+1)/(d*x+c-1)-1)*c*f+(1-(d*x+c+1)/(d*x+c-1))*
e*d+(-(d*x+c+1)/(d*x+c-1)-1)*f))*csgn(I*(((d*x+c+1)/(d*x+c-1)-1)*c*f+(1-(d*x+c+1)/(d*x+c-1))*e*d+(-(d*x+c+1)/(
d*x+c-1)-1)*f)/((d*x+c+1)/(d*x+c-1)-1))*csgn(I/((d*x+c+1)/(d*x+c-1)-1))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{a^{3} \log \left (f x + e\right )}{f} + \int \frac{b^{3}{\left (\log \left (\frac{1}{d x + c} + 1\right ) - \log \left (-\frac{1}{d x + c} + 1\right )\right )}^{3}}{8 \,{\left (f x + e\right )}} + \frac{3 \, a b^{2}{\left (\log \left (\frac{1}{d x + c} + 1\right ) - \log \left (-\frac{1}{d x + c} + 1\right )\right )}^{2}}{4 \,{\left (f x + e\right )}} + \frac{3 \, a^{2} b{\left (\log \left (\frac{1}{d x + c} + 1\right ) - \log \left (-\frac{1}{d x + c} + 1\right )\right )}}{2 \,{\left (f x + e\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{b^{3} \operatorname{arcoth}\left (d x + c\right )^{3} + 3 \, a b^{2} \operatorname{arcoth}\left (d x + c\right )^{2} + 3 \, a^{2} b \operatorname{arcoth}\left (d x + c\right ) + a^{3}}{f x + e}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \operatorname{arcoth}\left (d x + c\right ) + a\right )}^{3}}{f x + e}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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