\(\int \frac {(a+b \cot ^{-1}(c+d x))^3}{e+f x} \, dx\) [31]

Optimal result

Integrand size = 20, antiderivative size = 372 \[ \int \frac {\left (a+b \cot ^{-1}(c+d x)\right )^3}{e+f x} \, dx=-\frac {\left (a+b \cot ^{-1}(c+d x)\right )^3 \log \left (\frac {2}{1-i (c+d x)}\right )}{f}+\frac {\left (a+b \cot ^{-1}(c+d x)\right )^3 \log \left (\frac {2 d (e+f x)}{(d e+(i-c) f) (1-i (c+d x))}\right )}{f}-\frac {3 i b \left (a+b \cot ^{-1}(c+d x)\right )^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1-i (c+d x)}\right )}{2 f}+\frac {3 i b \left (a+b \cot ^{-1}(c+d x)\right )^2 \operatorname {PolyLog}\left (2,1-\frac {2 d (e+f x)}{(d e+(i-c) f) (1-i (c+d x))}\right )}{2 f}-\frac {3 b^2 \left (a+b \cot ^{-1}(c+d x)\right ) \operatorname {PolyLog}\left (3,1-\frac {2}{1-i (c+d x)}\right )}{2 f}+\frac {3 b^2 \left (a+b \cot ^{-1}(c+d x)\right ) \operatorname {PolyLog}\left (3,1-\frac {2 d (e+f x)}{(d e+(i-c) f) (1-i (c+d x))}\right )}{2 f}+\frac {3 i b^3 \operatorname {PolyLog}\left (4,1-\frac {2}{1-i (c+d x)}\right )}{4 f}-\frac {3 i b^3 \operatorname {PolyLog}\left (4,1-\frac {2 d (e+f x)}{(d e+(i-c) f) (1-i (c+d x))}\right )}{4 f} \] Output:

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

Mathematica [F]

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

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

Output:

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

Rubi [A] (verified)

Time = 0.48 (sec) , antiderivative size = 408, normalized size of antiderivative = 1.10, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {5571, 27, 5386}

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.

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

\(\Big \downarrow \) 5571

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 5386

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

Input:

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

Output:

-(((a + b*ArcCot[c + d*x])^3*Log[2/(1 - I*(c + d*x))])/f) + ((a + b*ArcCot 
[c + d*x])^3*Log[(2*(d*e - c*f + f*(c + d*x)))/((d*e + I*f - c*f)*(1 - I*( 
c + d*x)))])/f - (((3*I)/2)*b*(a + b*ArcCot[c + d*x])^2*PolyLog[2, 1 - 2/( 
1 - I*(c + d*x))])/f + (((3*I)/2)*b*(a + b*ArcCot[c + d*x])^2*PolyLog[2, 1 
 - (2*(d*e - c*f + f*(c + d*x)))/((d*e + I*f - c*f)*(1 - I*(c + d*x)))])/f 
 - (3*b^2*(a + b*ArcCot[c + d*x])*PolyLog[3, 1 - 2/(1 - I*(c + d*x))])/(2* 
f) + (3*b^2*(a + b*ArcCot[c + d*x])*PolyLog[3, 1 - (2*(d*e - c*f + f*(c + 
d*x)))/((d*e + I*f - c*f)*(1 - I*(c + d*x)))])/(2*f) + (((3*I)/4)*b^3*Poly 
Log[4, 1 - 2/(1 - I*(c + d*x))])/f - (((3*I)/4)*b^3*PolyLog[4, 1 - (2*(d*e 
 - c*f + f*(c + d*x)))/((d*e + I*f - c*f)*(1 - I*(c + d*x)))])/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 5386

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

rule 5571

Int[((a_.) + ArcCot[(c_) + (d_.)*(x_)]*(b_.))^(p_.)*((e_.) + (f_.)*(x_))^(m 
_.), x_Symbol] :> Simp[1/d   Subst[Int[((d*e - c*f)/d + f*(x/d))^m*(a + b*A 
rcCot[x])^p, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && I 
GtQ[p, 0]
 

Maple [C] (warning: unable to verify)

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

Time = 14.32 (sec) , antiderivative size = 3903, normalized size of antiderivative = 10.49

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(3903\)
default	\(\text {Expression too large to display}\)	\(3903\)
parts	\(\text {Expression too large to display}\)	\(4133\)

Input:

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

Output:

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*arccot( 
d*x+c)^3-3/f*(-1/3*arccot(d*x+c)^3*ln(-I*f*(d*x+c+I)^2/(1+(d*x+c)^2)+c*f*( 
d*x+c+I)^2/(1+(d*x+c)^2)-d*e*(d*x+c+I)^2/(1+(d*x+c)^2)-I*f-c*f+d*e)-1/2*I* 
f/(-I*f+c*f-d*e)*arccot(d*x+c)*polylog(3,(d*e+I*f-c*f)/(-c*f+d*e-I*f)*(d*x 
+c+I)^2/(1+(d*x+c)^2))+1/3*arccot(d*x+c)^3*ln((d*x+c+I)^2/(1+(d*x+c)^2)-1) 
-1/3*arccot(d*x+c)^3*ln(1-(d*x+c+I)/(1+(d*x+c)^2)^(1/2))-1/2*I*c*f/(-I*f+c 
*f-d*e)*arccot(d*x+c)^2*polylog(2,(d*e+I*f-c*f)/(-c*f+d*e-I*f)*(d*x+c+I)^2 
/(1+(d*x+c)^2))-2*arccot(d*x+c)*polylog(3,(d*x+c+I)/(1+(d*x+c)^2)^(1/2))+I 
*arccot(d*x+c)^2*polylog(2,(d*x+c+I)/(1+(d*x+c)^2)^(1/2))-1/3*arccot(d*x+c 
)^3*ln(1+(d*x+c+I)/(1+(d*x+c)^2)^(1/2))+I*d*e*arccot(d*x+c)^2*polylog(2,(d 
*e+I*f-c*f)/(-c*f+d*e-I*f)*(d*x+c+I)^2/(1+(d*x+c)^2))/(-2*I*f+2*c*f-2*d*e) 
-2*arccot(d*x+c)*polylog(3,-(d*x+c+I)/(1+(d*x+c)^2)^(1/2))-2*I*polylog(4,- 
(d*x+c+I)/(1+(d*x+c)^2)^(1/2))+1/3*c*f/(-I*f+c*f-d*e)*arccot(d*x+c)^3*ln(1 
-(d*e+I*f-c*f)/(-c*f+d*e-I*f)*(d*x+c+I)^2/(1+(d*x+c)^2))+1/2*c*f/(-I*f+c*f 
-d*e)*arccot(d*x+c)*polylog(3,(d*e+I*f-c*f)/(-c*f+d*e-I*f)*(d*x+c+I)^2/(1+ 
(d*x+c)^2))-1/3*I*f/(-I*f+c*f-d*e)*arccot(d*x+c)^3*ln(1-(d*e+I*f-c*f)/(-c* 
f+d*e-I*f)*(d*x+c+I)^2/(1+(d*x+c)^2))+I*arccot(d*x+c)^2*polylog(2,-(d*x+c+ 
I)/(1+(d*x+c)^2)^(1/2))-I*d*e*polylog(4,(d*e+I*f-c*f)/(-c*f+d*e-I*f)*(d*x+ 
c+I)^2/(1+(d*x+c)^2))/(-4*I*f+4*c*f-4*d*e)+1/4*I*c*f/(-I*f+c*f-d*e)*polylo 
g(4,(d*e+I*f-c*f)/(-c*f+d*e-I*f)*(d*x+c+I)^2/(1+(d*x+c)^2))-1/2*f/(-I*f...
 

Fricas [F]

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

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

a^3*log(f*x + e)/f + integrate(1/32*(28*b^3*arctan2(1, d*x + c)^3 + 3*b^3* 
arctan2(1, d*x + c)*log(d^2*x^2 + 2*c*d*x + c^2 + 1)^2 + 96*a*b^2*arctan2( 
1, d*x + c)^2 + 96*a^2*b*arctan2(1, d*x + c))/(f*x + e), x)
 

Giac [F(-1)]

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

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

Output:

Timed out
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {\left (a+b \cot ^{-1}(c+d x)\right )^3}{e+f x} \, dx=\frac {3 \left (\int \frac {\mathit {acot} \left (d x +c \right )}{f x +e}d x \right ) a^{2} b f +\left (\int \frac {\mathit {acot} \left (d x +c \right )^{3}}{f x +e}d x \right ) b^{3} f +3 \left (\int \frac {\mathit {acot} \left (d x +c \right )^{2}}{f x +e}d x \right ) a \,b^{2} f +\mathrm {log}\left (f x +e \right ) a^{3}}{f} \] Input:

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

Output:

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