\(\int \frac {(a+b \cot ^{-1}(c+d x))^2}{e+f x} \, dx\) [26]

Optimal result

Integrand size = 20, antiderivative size = 261 \[ \int \frac {\left (a+b \cot ^{-1}(c+d x)\right )^2}{e+f x} \, dx=-\frac {\left (a+b \cot ^{-1}(c+d x)\right )^2 \log \left (\frac {2}{1-i (c+d x)}\right )}{f}+\frac {\left (a+b \cot ^{-1}(c+d x)\right )^2 \log \left (\frac {2 d (e+f x)}{(d e+(i-c) f) (1-i (c+d x))}\right )}{f}-\frac {i b \left (a+b \cot ^{-1}(c+d x)\right ) \operatorname {PolyLog}\left (2,1-\frac {2}{1-i (c+d x)}\right )}{f}+\frac {i b \left (a+b \cot ^{-1}(c+d x)\right ) \operatorname {PolyLog}\left (2,1-\frac {2 d (e+f x)}{(d e+(i-c) f) (1-i (c+d x))}\right )}{f}-\frac {b^2 \operatorname {PolyLog}\left (3,1-\frac {2}{1-i (c+d x)}\right )}{2 f}+\frac {b^2 \operatorname {PolyLog}\left (3,1-\frac {2 d (e+f x)}{(d e+(i-c) f) (1-i (c+d x))}\right )}{2 f} \] Output:

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

Mathematica [F]

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

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

Output:

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

Rubi [A] (verified)

Time = 0.42 (sec) , antiderivative size = 288, 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, 5384}

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.

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

\(\Big \downarrow \) 5571

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 5384

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

Input:

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

Output:

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

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 5384

Int[((a_.) + ArcCot[(c_.)*(x_)]*(b_.))^2/((d_) + (e_.)*(x_)), x_Symbol] :> 
Simp[(-(a + b*ArcCot[c*x])^2)*(Log[2/(1 - I*c*x)]/e), x] + (Simp[(a + b*Arc 
Cot[c*x])^2*(Log[2*c*((d + e*x)/((c*d + I*e)*(1 - I*c*x)))]/e), x] - Simp[I 
*b*(a + b*ArcCot[c*x])*(PolyLog[2, 1 - 2/(1 - I*c*x)]/e), x] + Simp[I*b*(a 
+ b*ArcCot[c*x])*(PolyLog[2, 1 - 2*c*((d + e*x)/((c*d + I*e)*(1 - I*c*x)))] 
/e), x] - Simp[b^2*(PolyLog[3, 1 - 2/(1 - I*c*x)]/(2*e)), x] + Simp[b^2*(Po 
lyLog[3, 1 - 2*c*((d + e*x)/((c*d + I*e)*(1 - I*c*x)))]/(2*e)), x]) /; Free 
Q[{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 = 21.45 (sec) , antiderivative size = 1911, normalized size of antiderivative = 7.32

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(1911\)
default	\(\text {Expression too large to display}\)	\(1911\)
parts	\(\text {Expression too large to display}\)	\(2019\)

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

a^2*log(f*x + e)/f + integrate(1/16*(12*b^2*arctan2(1, d*x + c)^2 + b^2*lo 
g(d^2*x^2 + 2*c*d*x + c^2 + 1)^2 + 32*a*b*arctan2(1, d*x + c))/(f*x + e), 
x)
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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