\(\int \frac {(a+b \coth ^{-1}(c+d x))^3}{e+f x} \, dx\) [117]
Optimal result
Integrand size = 20, antiderivative size = 308 \[
\int \frac {\left (a+b \coth ^{-1}(c+d x)\right )^3}{e+f x} \, dx=-\frac {\left (a+b \coth ^{-1}(c+d x)\right )^3 \log \left (\frac {2}{1+c+d x}\right )}{f}+\frac {\left (a+b \coth ^{-1}(c+d x)\right )^3 \log \left (\frac {2 d (e+f x)}{(d e+f-c f) (1+c+d x)}\right )}{f}+\frac {3 b \left (a+b \coth ^{-1}(c+d x)\right )^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1+c+d x}\right )}{2 f}-\frac {3 b \left (a+b \coth ^{-1}(c+d x)\right )^2 \operatorname {PolyLog}\left (2,1-\frac {2 d (e+f x)}{(d e+f-c f) (1+c+d x)}\right )}{2 f}+\frac {3 b^2 \left (a+b \coth ^{-1}(c+d x)\right ) \operatorname {PolyLog}\left (3,1-\frac {2}{1+c+d x}\right )}{2 f}-\frac {3 b^2 \left (a+b \coth ^{-1}(c+d x)\right ) \operatorname {PolyLog}\left (3,1-\frac {2 d (e+f x)}{(d e+f-c f) (1+c+d x)}\right )}{2 f}+\frac {3 b^3 \operatorname {PolyLog}\left (4,1-\frac {2}{1+c+d x}\right )}{4 f}-\frac {3 b^3 \operatorname {PolyLog}\left (4,1-\frac {2 d (e+f x)}{(d e+f-c f) (1+c+d x)}\right )}{4 f}
\]
[Out]
-(a+b*arccoth(d*x+c))^3*ln(2/(d*x+c+1))/f+(a+b*arccoth(d*x+c))^3*ln(2*d*(f*x+e)/(-c*f+d*e+f)/(d*x+c+1))/f+3/2*
b*(a+b*arccoth(d*x+c))^2*polylog(2,1-2/(d*x+c+1))/f-3/2*b*(a+b*arccoth(d*x+c))^2*polylog(2,1-2*d*(f*x+e)/(-c*f
+d*e+f)/(d*x+c+1))/f+3/2*b^2*(a+b*arccoth(d*x+c))*polylog(3,1-2/(d*x+c+1))/f-3/2*b^2*(a+b*arccoth(d*x+c))*poly
log(3,1-2*d*(f*x+e)/(-c*f+d*e+f)/(d*x+c+1))/f+3/4*b^3*polylog(4,1-2/(d*x+c+1))/f-3/4*b^3*polylog(4,1-2*d*(f*x+
e)/(-c*f+d*e+f)/(d*x+c+1))/f
Rubi [A] (verified)
Time = 0.16 (sec) , antiderivative size = 308, normalized size of antiderivative = 1.00, number
of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {6247, 6062}
\[
\int \frac {\left (a+b \coth ^{-1}(c+d x)\right )^3}{e+f x} \, dx=-\frac {3 b^2 \left (a+b \coth ^{-1}(c+d x)\right ) \operatorname {PolyLog}\left (3,1-\frac {2 d (e+f x)}{(d e-c f+f) (c+d x+1)}\right )}{2 f}+\frac {3 b^2 \operatorname {PolyLog}\left (3,1-\frac {2}{c+d x+1}\right ) \left (a+b \coth ^{-1}(c+d x)\right )}{2 f}-\frac {3 b \left (a+b \coth ^{-1}(c+d x)\right )^2 \operatorname {PolyLog}\left (2,1-\frac {2 d (e+f x)}{(d e-c f+f) (c+d x+1)}\right )}{2 f}+\frac {\left (a+b \coth ^{-1}(c+d x)\right )^3 \log \left (\frac {2 d (e+f x)}{(c+d x+1) (-c f+d e+f)}\right )}{f}+\frac {3 b \operatorname {PolyLog}\left (2,1-\frac {2}{c+d x+1}\right ) \left (a+b \coth ^{-1}(c+d x)\right )^2}{2 f}-\frac {\log \left (\frac {2}{c+d x+1}\right ) \left (a+b \coth ^{-1}(c+d x)\right )^3}{f}-\frac {3 b^3 \operatorname {PolyLog}\left (4,1-\frac {2 d (e+f x)}{(d e-c f+f) (c+d x+1)}\right )}{4 f}+\frac {3 b^3 \operatorname {PolyLog}\left (4,1-\frac {2}{c+d x+1}\right )}{4 f}
\]
[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 6062
Int[((a_.) + ArcCoth[(c_.)*(x_)]*(b_.))^3/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-(a + b*ArcCoth[c*x])^3)*(Lo
g[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]
Rule 6247
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]
Rubi steps \begin{align*}
\text {integral}& = \frac {\text {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 \operatorname {PolyLog}\left (2,1-\frac {2}{1+c+d x}\right )}{2 f}-\frac {3 b \left (a+b \coth ^{-1}(c+d x)\right )^2 \operatorname {PolyLog}\left (2,1-\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 ) \operatorname {PolyLog}\left (3,1-\frac {2}{1+c+d x}\right )}{2 f}-\frac {3 b^2 \left (a+b \coth ^{-1}(c+d x)\right ) \operatorname {PolyLog}\left (3,1-\frac {2 d (e+f x)}{(d e+f-c f) (1+c+d x)}\right )}{2 f}+\frac {3 b^3 \operatorname {PolyLog}\left (4,1-\frac {2}{1+c+d x}\right )}{4 f}-\frac {3 b^3 \operatorname {PolyLog}\left (4,1-\frac {2 d (e+f x)}{(d e+f-c f) (1+c+d x)}\right )}{4 f} \\
\end{align*}
Mathematica [F]
\[
\int \frac {\left (a+b \coth ^{-1}(c+d x)\right )^3}{e+f x} \, dx=\int \frac {\left (a+b \coth ^{-1}(c+d x)\right )^3}{e+f x} \, dx
\]
[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]
Maple [C] (warning: unable to verify)
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 12.70 (sec) , antiderivative size = 3250, normalized size of antiderivative =
10.55
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| method | result | size |
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| derivativedivides |
\(\text {Expression too large to display}\) |
\(3250\) |
| default |
\(\text {Expression too large to display}\) |
\(3250\) |
| parts |
\(\text {Expression too large to display}\) |
\(3425\) |
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[In]
int((a+b*arccoth(d*x+c))^3/(f*x+e),x,method=_RETURNVERBOSE)
[Out]
1/d*(a^3*d*ln(c*f-d*e-f*(d*x+c))/f-b^3*d*(-ln(c*f-d*e-f*(d*x+c))/f*arccoth(d*x+c)^3-3/f*(-1/3*arccoth(d*x+c)^3
*ln(f*c*((d*x+c+1)/(d*x+c-1)-1)+(1-(d*x+c+1)/(d*x+c-1))*e*d+(-(d*x+c+1)/(d*x+c-1)-1)*f)+1/6*I*Pi*csgn(I*(f*c*(
(d*x+c+1)/(d*x+c-1)-1)+(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*(f*c*((d*x+c+1)/(d*x+c-1)-1)+(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))-csgn(I*(f*c*((d*x+c+1)/(d*x+c-1)-1)+(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))-csgn(I*(f*c*((d*x+c+1)/(d*x+c-1)-1)+(1-(d*x+c+1)/(d*x+c-1))*e
*d+(-(d*x+c+1)/(d*x+c-1)-1)*f))*csgn(I*(f*c*((d*x+c+1)/(d*x+c-1)-1)+(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*(f*c*((d*x+c+1)/(d*x+c-1)-1)+(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)*arccoth(d*x+c)^3+1/3*arccoth(d*x+c)^3*ln((d*x+c+1)/(d*x+c-1)-
1)-1/3*arccoth(d*x+c)^3*ln(1+1/((d*x+c-1)/(d*x+c+1))^(1/2))-arccoth(d*x+c)^2*polylog(2,-1/((d*x+c-1)/(d*x+c+1)
)^(1/2))+2*arccoth(d*x+c)*polylog(3,-1/((d*x+c-1)/(d*x+c+1))^(1/2))-2*polylog(4,-1/((d*x+c-1)/(d*x+c+1))^(1/2)
)-1/3*arccoth(d*x+c)^3*ln(1-1/((d*x+c-1)/(d*x+c+1))^(1/2))-arccoth(d*x+c)^2*polylog(2,1/((d*x+c-1)/(d*x+c+1))^
(1/2))+2*arccoth(d*x+c)*polylog(3,1/((d*x+c-1)/(d*x+c+1))^(1/2))-2*polylog(4,1/((d*x+c-1)/(d*x+c+1))^(1/2))+1/
3*f*c/(c*f-d*e-f)*arccoth(d*x+c)^3*ln(1-(d*x+c+1)*(c*f-d*e-f)/(c*f-d*e+f)/(d*x+c-1))+1/2*f*c/(c*f-d*e-f)*arcco
th(d*x+c)^2*polylog(2,(d*x+c+1)*(c*f-d*e-f)/(c*f-d*e+f)/(d*x+c-1))-1/2*f*c/(c*f-d*e-f)*arccoth(d*x+c)*polylog(
3,(d*x+c+1)*(c*f-d*e-f)/(c*f-d*e+f)/(d*x+c-1))+1/4*f*c/(c*f-d*e-f)*polylog(4,(d*x+c+1)*(c*f-d*e-f)/(c*f-d*e+f)
/(d*x+c-1))-1/3*f/(c*f-d*e-f)*arccoth(d*x+c)^3*ln(1-(d*x+c+1)*(c*f-d*e-f)/(c*f-d*e+f)/(d*x+c-1))-1/2*f/(c*f-d*
e-f)*arccoth(d*x+c)^2*polylog(2,(d*x+c+1)*(c*f-d*e-f)/(c*f-d*e+f)/(d*x+c-1))+1/2*f/(c*f-d*e-f)*arccoth(d*x+c)*
polylog(3,(d*x+c+1)*(c*f-d*e-f)/(c*f-d*e+f)/(d*x+c-1))-1/4*f/(c*f-d*e-f)*polylog(4,(d*x+c+1)*(c*f-d*e-f)/(c*f-
d*e+f)/(d*x+c-1))-1/3*e*d/(c*f-d*e-f)*arccoth(d*x+c)^3*ln(1-(d*x+c+1)*(c*f-d*e-f)/(c*f-d*e+f)/(d*x+c-1))-1/2*e
*d/(c*f-d*e-f)*arccoth(d*x+c)^2*polylog(2,(d*x+c+1)*(c*f-d*e-f)/(c*f-d*e+f)/(d*x+c-1))+1/2*e*d/(c*f-d*e-f)*arc
coth(d*x+c)*polylog(3,(d*x+c+1)*(c*f-d*e-f)/(c*f-d*e+f)/(d*x+c-1))-1/4*e*d/(c*f-d*e-f)*polylog(4,(d*x+c+1)*(c*
f-d*e-f)/(c*f-d*e+f)/(d*x+c-1))))-3*a*b^2*d*(-ln(c*f-d*e-f*(d*x+c))/f*arccoth(d*x+c)^2-2/f*(-1/2*arccoth(d*x+c
)^2*ln(f*c*((d*x+c+1)/(d*x+c-1)-1)+(1-(d*x+c+1)/(d*x+c-1))*e*d+(-(d*x+c+1)/(d*x+c-1)-1)*f)+1/4*I*Pi*csgn(I*(f*
c*((d*x+c+1)/(d*x+c-1)-1)+(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))*(cs
gn(I*(f*c*((d*x+c+1)/(d*x+c-1)-1)+(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))-csgn(I*(f*c*((d*x+c+1)/(d*x+c-1)-1)+(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))-csgn(I*(f*c*((d*x+c+1)/(d*x+c-1)-1)+(1-(d*x+c+1)/(d*x+c-1)
)*e*d+(-(d*x+c+1)/(d*x+c-1)-1)*f))*csgn(I*(f*c*((d*x+c+1)/(d*x+c-1)-1)+(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*(f*c*((d*x+c+1)/(d*x+c-1)-1)+(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)*arccoth(d*x+c)^2+1/2*arccoth(d*x+c)^2*ln((d*x+c+1)/(d*x+c-
1)-1)-1/2*arccoth(d*x+c)^2*ln(1-1/((d*x+c-1)/(d*x+c+1))^(1/2))-arccoth(d*x+c)*polylog(2,1/((d*x+c-1)/(d*x+c+1)
)^(1/2))+polylog(3,1/((d*x+c-1)/(d*x+c+1))^(1/2))-1/2*arccoth(d*x+c)^2*ln(1+1/((d*x+c-1)/(d*x+c+1))^(1/2))-arc
coth(d*x+c)*polylog(2,-1/((d*x+c-1)/(d*x+c+1))^(1/2))+polylog(3,-1/((d*x+c-1)/(d*x+c+1))^(1/2))+1/2*f*c/(c*f-d
*e-f)*arccoth(d*x+c)^2*ln(1-(d*x+c+1)*(c*f-d*e-f)/(c*f-d*e+f)/(d*x+c-1))+1/2*f*c/(c*f-d*e-f)*arccoth(d*x+c)*po
lylog(2,(d*x+c+1)*(c*f-d*e-f)/(c*f-d*e+f)/(d*x+c-1))-1/4*f*c/(c*f-d*e-f)*polylog(3,(d*x+c+1)*(c*f-d*e-f)/(c*f-
d*e+f)/(d*x+c-1))-1/2*f/(c*f-d*e-f)*arccoth(d*x+c)^2*ln(1-(d*x+c+1)*(c*f-d*e-f)/(c*f-d*e+f)/(d*x+c-1))-1/2*f/(
c*f-d*e-f)*arccoth(d*x+c)*polylog(2,(d*x+c+1)*(c*f-d*e-f)/(c*f-d*e+f)/(d*x+c-1))+1/4*f/(c*f-d*e-f)*polylog(3,(
d*x+c+1)*(c*f-d*e-f)/(c*f-d*e+f)/(d*x+c-1))-1/2*e*d/(c*f-d*e-f)*arccoth(d*x+c)^2*ln(1-(d*x+c+1)*(c*f-d*e-f)/(c
*f-d*e+f)/(d*x+c-1))-1/2*e*d/(c*f-d*e-f)*arccoth(d*x+c)*polylog(2,(d*x+c+1)*(c*f-d*e-f)/(c*f-d*e+f)/(d*x+c-1))
+1/4*e*d/(c*f-d*e-f)*polylog(3,(d*x+c+1)*(c*f-d*e-f)/(c*f-d*e+f)/(d*x+c-1))))-3*b*a^2*d*(-ln(c*f-d*e-f*(d*x+c)
)/f*arccoth(d*x+c)+1/f^2*(-1/2*f*(dilog((-f*(d*x+c)+f)/(-c*f+d*e+f))+ln(c*f-d*e-f*(d*x+c))*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(c*f-d*e-f*(d*x+c))*ln((-f*(d*x+c)-f)/(-c*f+d*e-f)))
)))
Fricas [F]
\[
\int \frac {\left (a+b \coth ^{-1}(c+d x)\right )^3}{e+f x} \, dx=\int { \frac {{\left (b \operatorname {arcoth}\left (d x + c\right ) + a\right )}^{3}}{f x + e} \,d x }
\]
[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)
Sympy [F]
\[
\int \frac {\left (a+b \coth ^{-1}(c+d x)\right )^3}{e+f x} \, dx=\int \frac {\left (a + b \operatorname {acoth}{\left (c + d x \right )}\right )^{3}}{e + f x}\, dx
\]
[In]
integrate((a+b*acoth(d*x+c))**3/(f*x+e),x)
[Out]
Integral((a + b*acoth(c + d*x))**3/(e + f*x), x)
Maxima [F]
\[
\int \frac {\left (a+b \coth ^{-1}(c+d x)\right )^3}{e+f x} \, dx=\int { \frac {{\left (b \operatorname {arcoth}\left (d x + c\right ) + a\right )}^{3}}{f x + e} \,d x }
\]
[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)
Giac [F]
\[
\int \frac {\left (a+b \coth ^{-1}(c+d x)\right )^3}{e+f x} \, dx=\int { \frac {{\left (b \operatorname {arcoth}\left (d x + c\right ) + a\right )}^{3}}{f x + e} \,d x }
\]
[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)
Mupad [F(-1)]
Timed out. \[
\int \frac {\left (a+b \coth ^{-1}(c+d x)\right )^3}{e+f x} \, dx=\int \frac {{\left (a+b\,\mathrm {acoth}\left (c+d\,x\right )\right )}^3}{e+f\,x} \,d x
\]
[In]
int((a + b*acoth(c + d*x))^3/(e + f*x),x)
[Out]
int((a + b*acoth(c + d*x))^3/(e + f*x), x)