\(\int \cot ^{-1}(c+d \tan (a+b x)) \, dx\) [12]

Optimal result

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

x*arccot(c+d*tan(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
 

Mathematica [B] (warning: unable to verify)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(549\) vs. \(2(198)=396\).

Time = 0.53 (sec) , antiderivative size = 549, normalized size of antiderivative = 2.77 \[ \int \cot ^{-1}(c+d \tan (a+b x)) \, dx=x \cot ^{-1}(c+d \tan (a+b x))+\frac {x \left (4 a \sqrt {-d^2} \arctan (c+d \tan (a+b x))-i d \log (1-i \tan (a+b x)) \log \left (\frac {-c d+\sqrt {-d^2}-d^2 \tan (a+b x)}{-c d+i d^2+\sqrt {-d^2}}\right )+i d \log (1+i \tan (a+b x)) \log \left (\frac {c d-\sqrt {-d^2}+d^2 \tan (a+b x)}{c d+i d^2-\sqrt {-d^2}}\right )+i d \log (1-i \tan (a+b x)) \log \left (\frac {c d+\sqrt {-d^2}+d^2 \tan (a+b x)}{c d-i d^2+\sqrt {-d^2}}\right )-i d \log (1+i \tan (a+b x)) \log \left (\frac {c d+\sqrt {-d^2}+d^2 \tan (a+b x)}{c d+i d^2+\sqrt {-d^2}}\right )-i d \operatorname {PolyLog}\left (2,\frac {d^2 (1-i \tan (a+b x))}{i c d+d^2-i \sqrt {-d^2}}\right )+i d \operatorname {PolyLog}\left (2,\frac {d^2 (1-i \tan (a+b x))}{i c d+d^2+i \sqrt {-d^2}}\right )+i d \operatorname {PolyLog}\left (2,\frac {d^2 (1+i \tan (a+b x))}{-i c d+d^2+i \sqrt {-d^2}}\right )-i d \operatorname {PolyLog}\left (2,\frac {d^2 (1+i \tan (a+b x))}{d^2-i \left (c d+\sqrt {-d^2}\right )}\right )\right )}{2 \sqrt {-d^2} (2 a-i \log (1-i \tan (a+b x))+i \log (1+i \tan (a+b x)))} \] Input:

Integrate[ArcCot[c + d*Tan[a + b*x]],x]
 

Output:

x*ArcCot[c + d*Tan[a + b*x]] + (x*(4*a*Sqrt[-d^2]*ArcTan[c + d*Tan[a + b*x 
]] - I*d*Log[1 - I*Tan[a + b*x]]*Log[(-(c*d) + Sqrt[-d^2] - d^2*Tan[a + b* 
x])/(-(c*d) + I*d^2 + Sqrt[-d^2])] + I*d*Log[1 + I*Tan[a + b*x]]*Log[(c*d 
- Sqrt[-d^2] + d^2*Tan[a + b*x])/(c*d + I*d^2 - Sqrt[-d^2])] + I*d*Log[1 - 
 I*Tan[a + b*x]]*Log[(c*d + Sqrt[-d^2] + d^2*Tan[a + b*x])/(c*d - I*d^2 + 
Sqrt[-d^2])] - I*d*Log[1 + I*Tan[a + b*x]]*Log[(c*d + Sqrt[-d^2] + d^2*Tan 
[a + b*x])/(c*d + I*d^2 + Sqrt[-d^2])] - I*d*PolyLog[2, (d^2*(1 - I*Tan[a 
+ b*x]))/(I*c*d + d^2 - I*Sqrt[-d^2])] + I*d*PolyLog[2, (d^2*(1 - I*Tan[a 
+ b*x]))/(I*c*d + d^2 + I*Sqrt[-d^2])] + I*d*PolyLog[2, (d^2*(1 + I*Tan[a 
+ b*x]))/((-I)*c*d + d^2 + I*Sqrt[-d^2])] - I*d*PolyLog[2, (d^2*(1 + I*Tan 
[a + b*x]))/(d^2 - I*(c*d + Sqrt[-d^2]))]))/(2*Sqrt[-d^2]*(2*a - I*Log[1 - 
 I*Tan[a + b*x]] + I*Log[1 + I*Tan[a + b*x]]))
 

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 = {5691, 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.

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

\(\Big \downarrow \) 5691

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

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

Input:

Int[ArcCot[c + d*Tan[a + b*x]],x]
 

Output:

x*ArcCot[c + d*Tan[a + b*x]] + b*(1 + I*c + d)*(-1/2*(x*Log[1 + ((1 + I*c 
+ d)*E^((2*I)*a + (2*I)*b*x))/(1 + I*c - d)])/(b*(c - I*(1 + d))) + ((I/4) 
*PolyLog[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)*((x*Log[1 + ((c + I*(1 - d))*E^((2*I 
)*a + (2*I)*b*x))/(c + I*(1 + d))])/(2*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))))
 

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 5691

Int[ArcCot[(c_.) + (d_.)*Tan[(a_.) + (b_.)*(x_)]], x_Symbol] :> Simp[x*ArcC 
ot[c + d*Tan[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 
]
 

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1137 vs. \(2 (168 ) = 336\).

Time = 4.68 (sec) , antiderivative size = 1138, normalized size of antiderivative = 5.75

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(1138\)
default	\(\text {Expression too large to display}\)	\(1138\)
risch	\(\text {Expression too large to display}\)	\(4969\)

Input:

int(arccot(c+d*tan(b*x+a)),x,method=_RETURNVERBOSE)
 

Output:

1/b/d*(d*arctan(tan(b*x+a))*arccot(c+d*tan(b*x+a))+d^2*(-1/d*arctan(d*((c+ 
d*tan(b*x+a))/d-c/d)+c)*arctan(-(c+d*tan(b*x+a))/d+c/d)-1/d^2*(-1/2*I*d*ar 
ctan(d*((c+d*tan(b*x+a))/d-c/d)+c)*ln(1-(I*d+I+c)*(1+I*(d*((c+d*tan(b*x+a) 
)/d-c/d)+c))^2/((d*((c+d*tan(b*x+a))/d-c/d)+c)^2+1)/(-I*d+I-c))-1/2*d*arct 
an(d*((c+d*tan(b*x+a))/d-c/d)+c)^2-1/4*d*polylog(2,(I*d+I+c)*(1+I*(d*((c+d 
*tan(b*x+a))/d-c/d)+c))^2/((d*((c+d*tan(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*tan(b*x+a))/d-c/d)+c))^2/((d*((c+d 
*tan(b*x+a))/d-c/d)+c)^2+1)/(I*d+I-c))*arctan(d*((c+d*tan(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*tan(b*x+a))/d-c/d)+c))^2/ 
((d*((c+d*tan(b*x+a))/d-c/d)+c)^2+1)/(I*d+I-c))*arctan(d*((c+d*tan(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*tan(b* 
x+a))/d-c/d)+c))^2/((d*((c+d*tan(b*x+a))/d-c/d)+c)^2+1)/(I*d+I-c))*c*arcta 
n(d*((c+d*tan(b*x+a))/d-c/d)+c)+1/2*d^2*arctan(d*((c+d*tan(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*tan(b*x+a))/d-c/d 
)+c))^2/((d*((c+d*tan(b*x+a))/d-c/d)+c)^2+1)/(I*d+I-c))/(1+I*c+d)+1/2*d*ar 
ctan(d*((c+d*tan(b*x+a))/d-c/d)+c)^2/(1+I*c+d)+1/2*d/(c-I*d-I)*c*arctan(d* 
((c+d*tan(b*x+a))/d-c/d)+c)^2+1/4*d*polylog(2,(c-I*d+I)*(1+I*(d*((c+d*tan( 
b*x+a))/d-c/d)+c))^2/((d*((c+d*tan(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*tan(b*x+a))/d-c/d)+ 
c))^2/((d*((c+d*tan(b*x+a))/d-c/d)+c)^2+1)/(I*d+I-c))*c)))
 

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. 1101 vs. \(2 (141) = 282\).

Time = 0.15 (sec) , antiderivative size = 1101, normalized size of antiderivative = 5.56 \[ \int \cot ^{-1}(c+d \tan (a+b x)) \, dx=\text {Too large to display} \] Input:

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

Output:

1/8*(8*b*x*arccot(d*tan(b*x + a) + c) - 2*(-I*b*x - I*a)*log(-2*((I*c*d - 
d^2 + d)*tan(b*x + a)^2 - c^2 - I*c*d + (I*c^2 - 2*c*d - I*d^2 + I)*tan(b* 
x + a) + d - 1)/((c^2 + d^2 - 2*d + 1)*tan(b*x + a)^2 + c^2 + d^2 - 2*d + 
1)) - 2*(I*b*x + I*a)*log(-2*((I*c*d - d^2 - d)*tan(b*x + a)^2 - c^2 - I*c 
*d + (I*c^2 - 2*c*d - I*d^2 + I)*tan(b*x + a) - d - 1)/((c^2 + d^2 + 2*d + 
 1)*tan(b*x + a)^2 + c^2 + d^2 + 2*d + 1)) - 2*(I*b*x + I*a)*log(-2*((-I*c 
*d - d^2 + d)*tan(b*x + a)^2 - c^2 + I*c*d + (-I*c^2 - 2*c*d + I*d^2 - I)* 
tan(b*x + a) + d - 1)/((c^2 + d^2 - 2*d + 1)*tan(b*x + a)^2 + c^2 + d^2 - 
2*d + 1)) - 2*(-I*b*x - I*a)*log(-2*((-I*c*d - d^2 - d)*tan(b*x + a)^2 - c 
^2 + I*c*d + (-I*c^2 - 2*c*d + I*d^2 - I)*tan(b*x + a) - d - 1)/((c^2 + d^ 
2 + 2*d + 1)*tan(b*x + a)^2 + c^2 + d^2 + 2*d + 1)) - 2*I*a*log(((I*c*d + 
d^2 + d)*tan(b*x + a)^2 - c^2 + I*c*d + (I*c^2 + I*d^2 + 2*I*d + I)*tan(b* 
x + a) - d - 1)/(tan(b*x + a)^2 + 1)) + 2*I*a*log(((I*c*d + d^2 - d)*tan(b 
*x + a)^2 - c^2 + I*c*d + (I*c^2 + I*d^2 - 2*I*d + I)*tan(b*x + a) + d - 1 
)/(tan(b*x + a)^2 + 1)) - 2*I*a*log(((I*c*d - d^2 + d)*tan(b*x + a)^2 + c^ 
2 + I*c*d + (I*c^2 + I*d^2 - 2*I*d + I)*tan(b*x + a) - d + 1)/(tan(b*x + a 
)^2 + 1)) + 2*I*a*log(((I*c*d - d^2 - d)*tan(b*x + a)^2 + c^2 + I*c*d + (I 
*c^2 + I*d^2 + 2*I*d + I)*tan(b*x + a) + d + 1)/(tan(b*x + a)^2 + 1)) - di 
log(2*((I*c*d - d^2 + d)*tan(b*x + a)^2 - c^2 - I*c*d + (I*c^2 - 2*c*d - I 
*d^2 + I)*tan(b*x + a) + d - 1)/((c^2 + d^2 - 2*d + 1)*tan(b*x + a)^2 +...
                                                                                    
                                                                                    
 

Sympy [F]

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

integrate(acot(c+d*tan(b*x+a)),x)
 

Output:

Integral(acot(c + d*tan(a + b*x)), x)
 

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. 433 vs. \(2 (141) = 282\).

Time = 0.20 (sec) , antiderivative size = 433, normalized size of antiderivative = 2.19 \[ \int \cot ^{-1}(c+d \tan (a+b x)) \, dx=-\frac {d {\left (\frac {8 \, {\left (b x + a\right )} \arctan \left (\frac {d^{2} \tan \left (b x + a\right ) + c d}{d}\right )}{d} - \frac {4 \, {\left (b x + a\right )} \arctan \left (\frac {c d + {\left (d^{2} + d\right )} \tan \left (b x + a\right )}{c^{2} + d^{2} + 2 \, d + 1}, \frac {c d \tan \left (b x + a\right ) + c^{2} + d + 1}{c^{2} + d^{2} + 2 \, d + 1}\right ) - 4 \, {\left (b x + a\right )} \arctan \left (\frac {c d + {\left (d^{2} - d\right )} \tan \left (b x + a\right )}{c^{2} + d^{2} - 2 \, d + 1}, \frac {c d \tan \left (b x + a\right ) + c^{2} - d + 1}{c^{2} + d^{2} - 2 \, d + 1}\right ) + \log \left (\tan \left (b x + a\right )^{2} + 1\right ) \log \left (\frac {d^{2} \tan \left (b x + a\right )^{2} + 2 \, c d \tan \left (b x + a\right ) + c^{2} + 1}{c^{2} + d^{2} + 2 \, d + 1}\right ) - \log \left (\tan \left (b x + a\right )^{2} + 1\right ) \log \left (\frac {d^{2} \tan \left (b x + a\right )^{2} + 2 \, c d \tan \left (b x + a\right ) + c^{2} + 1}{c^{2} + d^{2} - 2 \, d + 1}\right ) + 2 \, {\rm Li}_2\left (-\frac {i \, d \tan \left (b x + a\right ) - d}{i \, c + d + 1}\right ) - 2 \, {\rm Li}_2\left (-\frac {i \, d \tan \left (b x + a\right ) - d}{i \, c + d - 1}\right ) + 2 \, {\rm Li}_2\left (\frac {i \, d \tan \left (b x + a\right ) + d}{-i \, c + d + 1}\right ) - 2 \, {\rm Li}_2\left (\frac {i \, d \tan \left (b x + a\right ) + d}{-i \, c + d - 1}\right )}{d}\right )} - 8 \, {\left (b x + a\right )} \operatorname {arccot}\left (d \tan \left (b x + a\right ) + c\right ) - 8 \, {\left (b x + a\right )} \arctan \left (\frac {d^{2} \tan \left (b x + a\right ) + c d}{d}\right )}{8 \, b} \] Input:

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

Output:

-1/8*(d*(8*(b*x + a)*arctan((d^2*tan(b*x + a) + c*d)/d)/d - (4*(b*x + a)*a 
rctan2((c*d + (d^2 + d)*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*(b*x + a)*arctan2((c*d + (d 
^2 - d)*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(tan(b*x + a)^2 + 1)*log((d^2*tan(b*x + a) 
^2 + 2*c*d*tan(b*x + a) + c^2 + 1)/(c^2 + d^2 + 2*d + 1)) - log(tan(b*x + 
a)^2 + 1)*log((d^2*tan(b*x + a)^2 + 2*c*d*tan(b*x + a) + c^2 + 1)/(c^2 + d 
^2 - 2*d + 1)) + 2*dilog(-(I*d*tan(b*x + a) - d)/(I*c + d + 1)) - 2*dilog( 
-(I*d*tan(b*x + a) - d)/(I*c + d - 1)) + 2*dilog((I*d*tan(b*x + a) + d)/(- 
I*c + d + 1)) - 2*dilog((I*d*tan(b*x + a) + d)/(-I*c + d - 1)))/d) - 8*(b* 
x + a)*arccot(d*tan(b*x + a) + c) - 8*(b*x + a)*arctan((d^2*tan(b*x + a) + 
 c*d)/d))/b
 

Giac [F]

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

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

Output:

integrate(arccot(d*tan(b*x + a) + c), x)
 

Mupad [F(-1)]

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

int(acot(c + d*tan(a + b*x)),x)
 

Output:

int(acot(c + d*tan(a + b*x)), x)
 

Reduce [F]

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

int(acot(c+d*tan(b*x+a)),x)
 

Output:

int(acot(tan(a + b*x)*d + c),x)