3.1.63 \(\int \arctan (c+d \cot (a+b x)) \, dx\) [63]

3.1.63.1 Optimal result

Integrand size = 11, antiderivative size = 198 \[ \int \arctan (c+d \cot (a+b x)) \, dx=x \arctan (c+d \cot (a+b x))+\frac {1}{2} i x \log \left (1-\frac {(1+i c-d) e^{2 i a+2 i b x}}{1+i c+d}\right )-\frac {1}{2} i x \log \left (1-\frac {(c+i (1+d)) e^{2 i a+2 i b x}}{c+i (1-d)}\right )+\frac {\operatorname {PolyLog}\left (2,\frac {(1+i c-d) e^{2 i a+2 i b x}}{1+i c+d}\right )}{4 b}-\frac {\operatorname {PolyLog}\left (2,\frac {(c+i (1+d)) e^{2 i a+2 i b x}}{c+i (1-d)}\right )}{4 b} \]

output

x*arctan(c+d*cot(b*x+a))+1/2*I*x*ln(1-(1+I*c-d)*exp(2*I*a+2*I*b*x)/(1+I*c+ 
d))-1/2*I*x*ln(1-(c+I*(1+d))*exp(2*I*a+2*I*b*x)/(c+I*(1-d)))+1/4*polylog(2 
,(1+I*c-d)*exp(2*I*a+2*I*b*x)/(1+I*c+d))/b-1/4*polylog(2,(c+I*(1+d))*exp(2 
*I*a+2*I*b*x)/(c+I*(1-d)))/b
 

3.1.63.2 Mathematica [B] (warning: unable to verify)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(1648\) vs. \(2(198)=396\).

Time = 21.13 (sec) , antiderivative size = 1648, normalized size of antiderivative = 8.32 \[ \int \arctan (c+d \cot (a+b x)) \, dx =\text {Too large to display} \]

input

Integrate[ArcTan[c + d*Cot[a + b*x]],x]
 

output

x*ArcTan[c + d*Cot[a + b*x]] + (d*(4*a*Sqrt[-d^2]*ArcTan[(c*d + Tan[a + b* 
x] + c^2*Tan[a + b*x])/d] + I*d*Log[1 + I*Tan[a + b*x]]*Log[(c*d - Sqrt[-d 
^2] + Tan[a + b*x] + c^2*Tan[a + b*x])/(I + I*c^2 + c*d - Sqrt[-d^2])] + I 
*d*Log[1 - I*Tan[a + b*x]]*Log[(c*d + Sqrt[-d^2] + Tan[a + b*x] + c^2*Tan[ 
a + b*x])/(-I - I*c^2 + c*d + Sqrt[-d^2])] - I*d*Log[1 + I*Tan[a + b*x]]*L 
og[(c*d + Sqrt[-d^2] + Tan[a + b*x] + c^2*Tan[a + b*x])/(I + I*c^2 + c*d + 
 Sqrt[-d^2])] - I*d*Log[1 - I*Tan[a + b*x]]*Log[(-(c*d) + Sqrt[-d^2] - (1 
+ c^2)*Tan[a + b*x])/(I + I*c^2 - c*d + Sqrt[-d^2])] - I*d*PolyLog[2, ((1 
+ c^2)*(1 - I*Tan[a + b*x]))/(1 + c^2 + I*c*d - I*Sqrt[-d^2])] + I*d*PolyL 
og[2, ((1 + c^2)*(1 - I*Tan[a + b*x]))/(1 + c^2 + I*c*d + I*Sqrt[-d^2])] - 
 I*d*PolyLog[2, ((1 + c^2)*(1 + I*Tan[a + b*x]))/(1 + c^2 - I*c*d - I*Sqrt 
[-d^2])] + I*d*PolyLog[2, ((1 + c^2)*(1 + I*Tan[a + b*x]))/(1 + c^2 - I*c* 
d + I*Sqrt[-d^2])])*((2*a)/(b*(-1 - c^2 - d^2 + Cos[2*(a + b*x)] + c^2*Cos 
[2*(a + b*x)] - d^2*Cos[2*(a + b*x)] - 2*c*d*Sin[2*(a + b*x)])) - (2*(a + 
b*x))/(b*(-1 - c^2 - d^2 + Cos[2*(a + b*x)] + c^2*Cos[2*(a + b*x)] - d^2*C 
os[2*(a + b*x)] - 2*c*d*Sin[2*(a + b*x)]))))/((d*Log[1 - ((1 + c^2)*(1 - I 
*Tan[a + b*x]))/(1 + c^2 + I*c*d - I*Sqrt[-d^2])]*Sec[a + b*x]^2)/(1 - I*T 
an[a + b*x]) - (d*Log[1 - ((1 + c^2)*(1 - I*Tan[a + b*x]))/(1 + c^2 + I*c* 
d + I*Sqrt[-d^2])]*Sec[a + b*x]^2)/(1 - I*Tan[a + b*x]) + (d*Log[(c*d + Sq 
rt[-d^2] + Tan[a + b*x] + c^2*Tan[a + b*x])/(-I - I*c^2 + c*d + Sqrt[-d...
 

3.1.63.3 Rubi [A] (verified)

Time = 0.68 (sec) , antiderivative size = 277, normalized size of antiderivative = 1.40, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {5692, 2620, 2715, 2838}

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.

\(\displaystyle \int \arctan (d \cot (a+b x)+c) \, dx\)

\(\Big \downarrow \) 5692

\(\displaystyle b (i c-d+1) \int \frac {e^{2 i a+2 i b x} x}{i c-(i c-d+1) e^{2 i a+2 i b x}+d+1}dx-b (-i c+d+1) \int \frac {e^{2 i a+2 i b x} x}{-i c-(-i c+d+1) e^{2 i a+2 i b x}-d+1}dx+x \arctan (d \cot (a+b x)+c)\)

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 2715

\(\displaystyle b (i c-d+1) \left (\frac {i \int e^{-2 i a-2 i b x} \log \left (1-\frac {(i c-d+1) e^{2 i a+2 i b x}}{i c+d+1}\right )de^{2 i a+2 i b x}}{4 b^2 (c-i (1-d))}+\frac {x \log \left (1-\frac {(i c-d+1) e^{2 i a+2 i b x}}{i c+d+1}\right )}{2 b (c-i (1-d))}\right )-b (-i c+d+1) \left (-\frac {i \int e^{-2 i a-2 i b x} \log \left (1-\frac {(c+i (d+1)) e^{2 i a+2 i b x}}{c+i (1-d)}\right )de^{2 i a+2 i b x}}{4 b^2 (c+i (d+1))}-\frac {x \log \left (1-\frac {(c+i (d+1)) e^{2 i a+2 i b x}}{c+i (1-d)}\right )}{2 b (c+i (d+1))}\right )+x \arctan (d \cot (a+b x)+c)\)

\(\Big \downarrow \) 2838

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

input

Int[ArcTan[c + d*Cot[a + b*x]],x]
 

output

x*ArcTan[c + d*Cot[a + b*x]] + b*(1 + I*c - d)*((x*Log[1 - ((1 + I*c - d)* 
E^((2*I)*a + (2*I)*b*x))/(1 + I*c + d)])/(2*b*(c - I*(1 - d))) - ((I/4)*Po 
lyLog[2, ((1 + I*c - d)*E^((2*I)*a + (2*I)*b*x))/(1 + I*c + d)])/(b^2*(c - 
 I*(1 - d)))) - b*(1 - I*c + d)*(-1/2*(x*Log[1 - ((c + I*(1 + d))*E^((2*I) 
*a + (2*I)*b*x))/(c + I*(1 - d))])/(b*(c + I*(1 + d))) + ((I/4)*PolyLog[2, 
 ((c + I*(1 + d))*E^((2*I)*a + (2*I)*b*x))/(c + I*(1 - d))])/(b^2*(c + I*( 
1 + d))))
 

3.1.63.3.1 Defintions of rubi rules used

rule 2620

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/ 
((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)), x_Symbol] :> Simp 
[((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x] - Si 
mp[d*(m/(b*f*g*n*Log[F]))   Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x 
)))^n/a)], x], x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]
 

rule 2715

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] 
:> Simp[1/(d*e*n*Log[F])   Subst[Int[Log[a + b*x]/x, x], x, (F^(e*(c + d*x) 
))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]
 

rule 2838

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

rule 5692

Int[ArcTan[(c_.) + Cot[(a_.) + (b_.)*(x_)]*(d_.)], x_Symbol] :> Simp[x*ArcT 
an[c + d*Cot[a + b*x]], x] + (Simp[b*(1 + I*c - d)   Int[x*(E^(2*I*a + 2*I* 
b*x)/(1 + I*c + d - (1 + I*c - d)*E^(2*I*a + 2*I*b*x))), x], x] - Simp[b*(1 
 - I*c + d)   Int[x*(E^(2*I*a + 2*I*b*x)/(1 - I*c - d - (1 - I*c + d)*E^(2* 
I*a + 2*I*b*x))), x], x]) /; FreeQ[{a, b, c, d}, x] && NeQ[(c + I*d)^2, -1]
 

3.1.63.4 Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1144 vs. \(2 (168 ) = 336\).

Time = 2.87 (sec) , antiderivative size = 1145, normalized size of antiderivative = 5.78

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(1145\)
default	\(\text {Expression too large to display}\)	\(1145\)
risch	\(\text {Expression too large to display}\)	\(4986\)

input

int(arctan(c+d*cot(b*x+a)),x,method=_RETURNVERBOSE)
 

output

1/b/d*(-d*(1/2*Pi-arccot(cot(b*x+a)))*arctan(c+d*cot(b*x+a))+d^2*(-1/d*arc 
tan(d*((c+d*cot(b*x+a))/d-c/d)+c)*arctan(-(c+d*cot(b*x+a))/d+c/d)-1/d^2*(- 
1/2*I*d*arctan(d*((c+d*cot(b*x+a))/d-c/d)+c)*ln(1-(I+c+I*d)*(1+I*(d*((c+d* 
cot(b*x+a))/d-c/d)+c))^2/((d*((c+d*cot(b*x+a))/d-c/d)+c)^2+1)/(-I*d+I-c))- 
1/2*d*arctan(d*((c+d*cot(b*x+a))/d-c/d)+c)^2-1/4*d*polylog(2,(I+c+I*d)*(1+ 
I*(d*((c+d*cot(b*x+a))/d-c/d)+c))^2/((d*((c+d*cot(b*x+a))/d-c/d)+c)^2+1)/( 
-I*d+I-c))+1/2*I*d^2*ln(1-(c-I*d+I)*(1+I*(d*((c+d*cot(b*x+a))/d-c/d)+c))^2 
/((d*((c+d*cot(b*x+a))/d-c/d)+c)^2+1)/(I*d+I-c))*arctan(d*((c+d*cot(b*x+a) 
)/d-c/d)+c)/(1+I*c+d)+1/2*I*d*ln(1-(c-I*d+I)*(1+I*(d*((c+d*cot(b*x+a))/d-c 
/d)+c))^2/((d*((c+d*cot(b*x+a))/d-c/d)+c)^2+1)/(I*d+I-c))*arctan(d*((c+d*c 
ot(b*x+a))/d-c/d)+c)/(1+I*c+d)+1/2*I*d/(c-I*d-I)*ln(1-(c-I*d+I)*(1+I*(d*(( 
c+d*cot(b*x+a))/d-c/d)+c))^2/((d*((c+d*cot(b*x+a))/d-c/d)+c)^2+1)/(I*d+I-c 
))*c*arctan(d*((c+d*cot(b*x+a))/d-c/d)+c)+1/2*d^2*arctan(d*((c+d*cot(b*x+a 
))/d-c/d)+c)^2/(1+I*c+d)+1/4*d^2*polylog(2,(c-I*d+I)*(1+I*(d*((c+d*cot(b*x 
+a))/d-c/d)+c))^2/((d*((c+d*cot(b*x+a))/d-c/d)+c)^2+1)/(I*d+I-c))/(1+I*c+d 
)+1/2*d*arctan(d*((c+d*cot(b*x+a))/d-c/d)+c)^2/(1+I*c+d)+1/2*d/(c-I*d-I)*c 
*arctan(d*((c+d*cot(b*x+a))/d-c/d)+c)^2+1/4*d*polylog(2,(c-I*d+I)*(1+I*(d* 
((c+d*cot(b*x+a))/d-c/d)+c))^2/((d*((c+d*cot(b*x+a))/d-c/d)+c)^2+1)/(I*d+I 
-c))/(1+I*c+d)+1/4*d/(c-I*d-I)*polylog(2,(c-I*d+I)*(1+I*(d*((c+d*cot(b*x+a 
))/d-c/d)+c))^2/((d*((c+d*cot(b*x+a))/d-c/d)+c)^2+1)/(I*d+I-c))*c)))
 

3.1.63.5 Fricas [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 965 vs. \(2 (140) = 280\).

Time = 0.40 (sec) , antiderivative size = 965, normalized size of antiderivative = 4.87 \[ \int \arctan (c+d \cot (a+b x)) \, dx=\text {Too large to display} \]

input

integrate(arctan(c+d*cot(b*x+a)),x, algorithm="fricas")
 

output

1/8*(8*b*x*arctan(d*cot(b*x + a) + c) - 2*I*a*log(1/2*c^2 + I*c*d - 1/2*d^ 
2 - 1/2*(c^2 + d^2 + 2*d + 1)*cos(2*b*x + 2*a) + 1/2*(I*c^2 + I*d^2 + 2*I* 
d + I)*sin(2*b*x + 2*a) + 1/2) + 2*I*a*log(1/2*c^2 + I*c*d - 1/2*d^2 - 1/2 
*(c^2 + d^2 - 2*d + 1)*cos(2*b*x + 2*a) + 1/2*(I*c^2 + I*d^2 - 2*I*d + I)* 
sin(2*b*x + 2*a) + 1/2) + 2*I*a*log(-1/2*c^2 + I*c*d + 1/2*d^2 + 1/2*(c^2 
+ d^2 + 2*d + 1)*cos(2*b*x + 2*a) + 1/2*(I*c^2 + I*d^2 + 2*I*d + I)*sin(2* 
b*x + 2*a) - 1/2) - 2*I*a*log(-1/2*c^2 + I*c*d + 1/2*d^2 + 1/2*(c^2 + d^2 
- 2*d + 1)*cos(2*b*x + 2*a) + 1/2*(I*c^2 + I*d^2 - 2*I*d + I)*sin(2*b*x + 
2*a) - 1/2) - 2*(-I*b*x - I*a)*log((c^2 + d^2 - (c^2 + 2*I*c*d - d^2 + 1)* 
cos(2*b*x + 2*a) + (-I*c^2 + 2*c*d + I*d^2 - I)*sin(2*b*x + 2*a) + 2*d + 1 
)/(c^2 + d^2 + 2*d + 1)) - 2*(I*b*x + I*a)*log((c^2 + d^2 - (c^2 - 2*I*c*d 
 - d^2 + 1)*cos(2*b*x + 2*a) + (I*c^2 + 2*c*d - I*d^2 + I)*sin(2*b*x + 2*a 
) + 2*d + 1)/(c^2 + d^2 + 2*d + 1)) - 2*(I*b*x + I*a)*log((c^2 + d^2 - (c^ 
2 + 2*I*c*d - d^2 + 1)*cos(2*b*x + 2*a) + (-I*c^2 + 2*c*d + I*d^2 - I)*sin 
(2*b*x + 2*a) - 2*d + 1)/(c^2 + d^2 - 2*d + 1)) - 2*(-I*b*x - I*a)*log((c^ 
2 + d^2 - (c^2 - 2*I*c*d - d^2 + 1)*cos(2*b*x + 2*a) + (I*c^2 + 2*c*d - I* 
d^2 + I)*sin(2*b*x + 2*a) - 2*d + 1)/(c^2 + d^2 - 2*d + 1)) + dilog(-(c^2 
+ d^2 - (c^2 + 2*I*c*d - d^2 + 1)*cos(2*b*x + 2*a) + (-I*c^2 + 2*c*d + I*d 
^2 - I)*sin(2*b*x + 2*a) + 2*d + 1)/(c^2 + d^2 + 2*d + 1) + 1) + dilog(-(c 
^2 + d^2 - (c^2 - 2*I*c*d - d^2 + 1)*cos(2*b*x + 2*a) + (I*c^2 + 2*c*d ...
 

3.1.63.6 Sympy [F]

\[ \int \arctan (c+d \cot (a+b x)) \, dx=\int \operatorname {atan}{\left (c + d \cot {\left (a + b x \right )} \right )}\, dx \]

input

integrate(atan(c+d*cot(b*x+a)),x)
 

output

Integral(atan(c + d*cot(a + b*x)), x)
 

3.1.63.7 Maxima [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 532 vs. \(2 (140) = 280\).

Time = 0.34 (sec) , antiderivative size = 532, normalized size of antiderivative = 2.69 \[ \int \arctan (c+d \cot (a+b x)) \, dx=-\frac {d {\left (\frac {8 \, {\left (b x + a\right )} \arctan \left (\frac {c d + {\left (c^{2} + 1\right )} \tan \left (b x + a\right )}{d}\right )}{d} - \frac {8 \, {\left (b x + a\right )} \arctan \left (\frac {c d + {\left (c^{2} + 1\right )} \tan \left (b x + a\right )}{d}\right ) - 4 \, \arctan \left (\frac {c d + {\left (c^{2} + 1\right )} \tan \left (b x + a\right )}{d}\right ) \arctan \left (\frac {c d + {\left (c^{2} + d + 1\right )} \tan \left (b x + a\right )}{c^{2} + d^{2} + 2 \, d + 1}, -\frac {c d \tan \left (b x + a\right ) - c^{2} - d - 1}{c^{2} + d^{2} + 2 \, d + 1}\right ) + 4 \, \arctan \left (\frac {c d + {\left (c^{2} + 1\right )} \tan \left (b x + a\right )}{d}\right ) \arctan \left (-\frac {c d + {\left (c^{2} - d + 1\right )} \tan \left (b x + a\right )}{c^{2} + d^{2} - 2 \, d + 1}, -\frac {c d \tan \left (b x + a\right ) - c^{2} + d - 1}{c^{2} + d^{2} - 2 \, d + 1}\right ) - {\left (\log \left (\frac {{\left (c^{2} + 1\right )} \tan \left (b x + a\right )^{2} + c^{2} + 1}{c^{2} + d^{2} + 2 \, d + 1}\right ) - \log \left (\frac {{\left (c^{2} + 1\right )} \tan \left (b x + a\right )^{2} + c^{2} + 1}{c^{2} + d^{2} - 2 \, d + 1}\right )\right )} \log \left ({\left (c^{2} + 1\right )} d^{2} + 2 \, {\left (c^{3} + c\right )} d \tan \left (b x + a\right ) + {\left (c^{4} + 2 \, c^{2} + 1\right )} \tan \left (b x + a\right )^{2}\right ) - 2 \, {\rm Li}_2\left (\frac {{\left (i \, c - 1\right )} \tan \left (b x + a\right ) + i \, d}{c + i \, d + i}\right ) + 2 \, {\rm Li}_2\left (\frac {{\left (i \, c + 1\right )} \tan \left (b x + a\right ) + i \, d}{c + i \, d - i}\right ) + 2 \, {\rm Li}_2\left (-\frac {{\left (i \, c - 1\right )} \tan \left (b x + a\right ) + i \, d}{c - i \, d + i}\right ) - 2 \, {\rm Li}_2\left (-\frac {{\left (i \, c + 1\right )} \tan \left (b x + a\right ) + i \, d}{c - i \, d - i}\right )}{d}\right )} - 8 \, {\left (b x + a\right )} \arctan \left (c + \frac {d}{\tan \left (b x + a\right )}\right ) - 8 \, {\left (b x + a\right )} \arctan \left (\frac {c d + {\left (c^{2} + 1\right )} \tan \left (b x + a\right )}{d}\right )}{8 \, b} \]

input

integrate(arctan(c+d*cot(b*x+a)),x, algorithm="maxima")
 

output

-1/8*(d*(8*(b*x + a)*arctan((c*d + (c^2 + 1)*tan(b*x + a))/d)/d - (8*(b*x 
+ a)*arctan((c*d + (c^2 + 1)*tan(b*x + a))/d) - 4*arctan((c*d + (c^2 + 1)* 
tan(b*x + a))/d)*arctan2((c*d + (c^2 + d + 1)*tan(b*x + a))/(c^2 + d^2 + 2 
*d + 1), -(c*d*tan(b*x + a) - c^2 - d - 1)/(c^2 + d^2 + 2*d + 1)) + 4*arct 
an((c*d + (c^2 + 1)*tan(b*x + a))/d)*arctan2(-(c*d + (c^2 - d + 1)*tan(b*x 
 + a))/(c^2 + d^2 - 2*d + 1), -(c*d*tan(b*x + a) - c^2 + d - 1)/(c^2 + d^2 
 - 2*d + 1)) - (log(((c^2 + 1)*tan(b*x + a)^2 + c^2 + 1)/(c^2 + d^2 + 2*d 
+ 1)) - log(((c^2 + 1)*tan(b*x + a)^2 + c^2 + 1)/(c^2 + d^2 - 2*d + 1)))*l 
og((c^2 + 1)*d^2 + 2*(c^3 + c)*d*tan(b*x + a) + (c^4 + 2*c^2 + 1)*tan(b*x 
+ a)^2) - 2*dilog(((I*c - 1)*tan(b*x + a) + I*d)/(c + I*d + I)) + 2*dilog( 
((I*c + 1)*tan(b*x + a) + I*d)/(c + I*d - I)) + 2*dilog(-((I*c - 1)*tan(b* 
x + a) + I*d)/(c - I*d + I)) - 2*dilog(-((I*c + 1)*tan(b*x + a) + I*d)/(c 
- I*d - I)))/d) - 8*(b*x + a)*arctan(c + d/tan(b*x + a)) - 8*(b*x + a)*arc 
tan((c*d + (c^2 + 1)*tan(b*x + a))/d))/b
 

3.1.63.8 Giac [F]

\[ \int \arctan (c+d \cot (a+b x)) \, dx=\int { \arctan \left (d \cot \left (b x + a\right ) + c\right ) \,d x } \]

input

integrate(arctan(c+d*cot(b*x+a)),x, algorithm="giac")
 

output

integrate(arctan(d*cot(b*x + a) + c), x)
 

3.1.63.9 Mupad [F(-1)]

Timed out. \[ \int \arctan (c+d \cot (a+b x)) \, dx=\int \mathrm {atan}\left (c+d\,\mathrm {cot}\left (a+b\,x\right )\right ) \,d x \]

input

int(atan(c + d*cot(a + b*x)),x)
 

output

int(atan(c + d*cot(a + b*x)), x)