\(\int \frac {a+b \cot ^{-1}(c+d x)}{e+f x} \, dx\) [20]

Optimal result

Integrand size = 18, antiderivative size = 162 \[ \int \frac {a+b \cot ^{-1}(c+d x)}{e+f x} \, dx=-\frac {\left (a+b \cot ^{-1}(c+d x)\right ) \log \left (\frac {2}{1-i (c+d x)}\right )}{f}+\frac {\left (a+b \cot ^{-1}(c+d x)\right ) \log \left (\frac {2 d (e+f x)}{(d e+(i-c) f) (1-i (c+d x))}\right )}{f}-\frac {i b \operatorname {PolyLog}\left (2,1-\frac {2}{1-i (c+d x)}\right )}{2 f}+\frac {i b \operatorname {PolyLog}\left (2,1-\frac {2 d (e+f x)}{(d e+(i-c) f) (1-i (c+d x))}\right )}{2 f} \] Output:

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

Mathematica [B] (verified)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(336\) vs. \(2(162)=324\).

Time = 0.30 (sec) , antiderivative size = 336, normalized size of antiderivative = 2.07 \[ \int \frac {a+b \cot ^{-1}(c+d x)}{e+f x} \, dx=\frac {a \log (e+f x)+b \left (\left (\cot ^{-1}(c+d x)+\arctan (c+d x)\right ) \log (e+f x)+\arctan (c+d x) \left (\log \left (\frac {1}{\sqrt {1+(c+d x)^2}}\right )-\log \left (\sin \left (\arctan \left (\frac {d e-c f}{f}\right )+\arctan (c+d x)\right )\right )\right )+\frac {1}{2} \left (\frac {1}{4} i (\pi -2 \arctan (c+d x))^2+i \left (\arctan \left (\frac {d e-c f}{f}\right )+\arctan (c+d x)\right )^2-(\pi -2 \arctan (c+d x)) \log \left (1+e^{-2 i \arctan (c+d x)}\right )-2 \left (\arctan \left (\frac {d e-c f}{f}\right )+\arctan (c+d x)\right ) \log \left (1-e^{2 i \left (\arctan \left (\frac {d e-c f}{f}\right )+\arctan (c+d x)\right )}\right )+(\pi -2 \arctan (c+d x)) \log \left (\frac {2}{\sqrt {1+(c+d x)^2}}\right )+2 \left (\arctan \left (\frac {d e-c f}{f}\right )+\arctan (c+d x)\right ) \log \left (2 \sin \left (\arctan \left (\frac {d e-c f}{f}\right )+\arctan (c+d x)\right )\right )+i \operatorname {PolyLog}\left (2,-e^{-2 i \arctan (c+d x)}\right )+i \operatorname {PolyLog}\left (2,e^{2 i \left (\arctan \left (\frac {d e-c f}{f}\right )+\arctan (c+d x)\right )}\right )\right )\right )}{f} \] Input:

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

Output:

(a*Log[e + f*x] + b*((ArcCot[c + d*x] + ArcTan[c + d*x])*Log[e + f*x] + Ar 
cTan[c + d*x]*(Log[1/Sqrt[1 + (c + d*x)^2]] - Log[Sin[ArcTan[(d*e - c*f)/f 
] + ArcTan[c + d*x]]]) + ((I/4)*(Pi - 2*ArcTan[c + d*x])^2 + I*(ArcTan[(d* 
e - c*f)/f] + ArcTan[c + d*x])^2 - (Pi - 2*ArcTan[c + d*x])*Log[1 + E^((-2 
*I)*ArcTan[c + d*x])] - 2*(ArcTan[(d*e - c*f)/f] + ArcTan[c + d*x])*Log[1 
- E^((2*I)*(ArcTan[(d*e - c*f)/f] + ArcTan[c + d*x]))] + (Pi - 2*ArcTan[c 
+ d*x])*Log[2/Sqrt[1 + (c + d*x)^2]] + 2*(ArcTan[(d*e - c*f)/f] + ArcTan[c 
 + d*x])*Log[2*Sin[ArcTan[(d*e - c*f)/f] + ArcTan[c + d*x]]] + I*PolyLog[2 
, -E^((-2*I)*ArcTan[c + d*x])] + I*PolyLog[2, E^((2*I)*(ArcTan[(d*e - c*f) 
/f] + ArcTan[c + d*x]))])/2))/f
 

Rubi [A] (verified)

Time = 0.54 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.11, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {5571, 27, 5382, 2849, 2752, 2897}

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.

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

\(\Big \downarrow \) 5571

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 5382

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

\(\Big \downarrow \) 2849

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

\(\Big \downarrow \) 2752

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

\(\Big \downarrow \) 2897

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

Input:

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

Output:

-(((a + b*ArcCot[c + d*x])*Log[2/(1 - I*(c + d*x))])/f) + ((a + b*ArcCot[c 
 + d*x])*Log[(2*(d*e - c*f + f*(c + d*x)))/((d*e + I*f - c*f)*(1 - I*(c + 
d*x)))])/f - ((I/2)*b*PolyLog[2, 1 - 2/(1 - I*(c + d*x))])/f + ((I/2)*b*Po 
lyLog[2, 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 2752

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-e^(-1))*PolyLo 
g[2, 1 - c*x], x] /; FreeQ[{c, d, e}, x] && EqQ[e + c*d, 0]
 

rule 2849

Int[Log[(c_.)/((d_) + (e_.)*(x_))]/((f_) + (g_.)*(x_)^2), x_Symbol] :> Simp 
[-e/g   Subst[Int[Log[2*d*x]/(1 - 2*d*x), x], x, 1/(d + e*x)], x] /; FreeQ[ 
{c, d, e, f, g}, x] && EqQ[c, 2*d] && EqQ[e^2*f + d^2*g, 0]
 

rule 2897

Int[Log[u_]*(Pq_)^(m_.), x_Symbol] :> With[{C = FullSimplify[Pq^m*((1 - u)/ 
D[u, x])]}, Simp[C*PolyLog[2, 1 - u], x] /; FreeQ[C, x]] /; IntegerQ[m] && 
PolyQ[Pq, x] && RationalFunctionQ[u, x] && LeQ[RationalFunctionExponents[u, 
 x][[2]], Expon[Pq, x]]
 

rule 5382

Int[((a_.) + ArcCot[(c_.)*(x_)]*(b_.))/((d_) + (e_.)*(x_)), x_Symbol] :> Si 
mp[(-(a + b*ArcCot[c*x]))*(Log[2/(1 - I*c*x)]/e), x] + (Simp[(a + b*ArcCot[ 
c*x])*(Log[2*c*((d + e*x)/((c*d + I*e)*(1 - I*c*x)))]/e), x] - Simp[b*(c/e) 
   Int[Log[2/(1 - I*c*x)]/(1 + c^2*x^2), x], x] + Simp[b*(c/e)   Int[Log[2* 
c*((d + e*x)/((c*d + I*e)*(1 - I*c*x)))]/(1 + c^2*x^2), x], 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 [A] (verified)

Time = 1.20 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.22

method	result	size
parts	\(\frac {a \ln \left (f x +e \right )}{f}+\frac {b \left (\frac {d \ln \left (f \left (d x +c \right )-c f +d e \right ) \operatorname {arccot}\left (d x +c \right )}{f}+d \left (-\frac {i \ln \left (f \left (d x +c \right )-c f +d e \right ) \left (\ln \left (\frac {i f -f \left (d x +c \right )}{-c f +d e +i f}\right )-\ln \left (\frac {i f +f \left (d x +c \right )}{c f -d e +i f}\right )\right )}{2 f}-\frac {i \left (\operatorname {dilog}\left (\frac {i f -f \left (d x +c \right )}{-c f +d e +i f}\right )-\operatorname {dilog}\left (\frac {i f +f \left (d x +c \right )}{c f -d e +i f}\right )\right )}{2 f}\right )\right )}{d}\)	\(197\)
derivativedivides	\(\frac {\frac {a d \ln \left (c f -d e -f \left (d x +c \right )\right )}{f}-b d \left (-\frac {\ln \left (c f -d e -f \left (d x +c \right )\right ) \operatorname {arccot}\left (d x +c \right )}{f}-\frac {i \ln \left (c f -d e -f \left (d x +c \right )\right ) \left (\ln \left (\frac {i f +f \left (d x +c \right )}{c f -d e +i f}\right )-\ln \left (\frac {i f -f \left (d x +c \right )}{-c f +d e +i f}\right )\right )}{2 f}-\frac {i \left (\operatorname {dilog}\left (\frac {i f +f \left (d x +c \right )}{c f -d e +i f}\right )-\operatorname {dilog}\left (\frac {i f -f \left (d x +c \right )}{-c f +d e +i f}\right )\right )}{2 f}\right )}{d}\)	\(211\)
default	\(\frac {\frac {a d \ln \left (c f -d e -f \left (d x +c \right )\right )}{f}-b d \left (-\frac {\ln \left (c f -d e -f \left (d x +c \right )\right ) \operatorname {arccot}\left (d x +c \right )}{f}-\frac {i \ln \left (c f -d e -f \left (d x +c \right )\right ) \left (\ln \left (\frac {i f +f \left (d x +c \right )}{c f -d e +i f}\right )-\ln \left (\frac {i f -f \left (d x +c \right )}{-c f +d e +i f}\right )\right )}{2 f}-\frac {i \left (\operatorname {dilog}\left (\frac {i f +f \left (d x +c \right )}{c f -d e +i f}\right )-\operatorname {dilog}\left (\frac {i f -f \left (d x +c \right )}{-c f +d e +i f}\right )\right )}{2 f}\right )}{d}\)	\(211\)
risch	\(-\frac {i b \operatorname {dilog}\left (\frac {i c f -i d e +\left (-i d x -i c +1\right ) f -f}{i c f -i d e -f}\right )}{2 f}-\frac {i b \ln \left (-i d x -i c +1\right ) \ln \left (\frac {i c f -i d e +\left (-i d x -i c +1\right ) f -f}{i c f -i d e -f}\right )}{2 f}+\frac {\ln \left (i c f -i d e +\left (-i d x -i c +1\right ) f -f \right ) b \pi }{2 f}+\frac {a \ln \left (i c f -i d e +\left (-i d x -i c +1\right ) f -f \right )}{f}+\frac {i b \operatorname {dilog}\left (\frac {-i c f +i d e +\left (i d x +i c +1\right ) f -f}{-i c f +i d e -f}\right )}{2 f}+\frac {i b \ln \left (i d x +i c +1\right ) \ln \left (\frac {-i c f +i d e +\left (i d x +i c +1\right ) f -f}{-i c f +i d e -f}\right )}{2 f}\)	\(302\)

Input:

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

Output:

a*ln(f*x+e)/f+b/d*(d*ln(f*(d*x+c)-c*f+d*e)/f*arccot(d*x+c)+d*(-1/2*I*ln(f* 
(d*x+c)-c*f+d*e)*(ln((I*f-f*(d*x+c))/(d*e+I*f-c*f))-ln((I*f+f*(d*x+c))/(c* 
f-d*e+I*f)))/f-1/2*I*(dilog((I*f-f*(d*x+c))/(d*e+I*f-c*f))-dilog((I*f+f*(d 
*x+c))/(c*f-d*e+I*f)))/f))
 

Fricas [F]

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

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

2*b*integrate(1/2*arctan2(1, d*x + c)/(f*x + e), x) + a*log(f*x + e)/f
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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