\(\int \cot ^{-1}(c-(1-i c) \tan (a+b x)) \, dx\) [20]

Optimal result

Integrand size = 18, antiderivative size = 86 \[ \int \cot ^{-1}(c-(1-i c) \tan (a+b x)) \, dx=-\frac {b x^2}{2}+x \cot ^{-1}(c-(1-i c) \tan (a+b x))-\frac {1}{2} i x \log \left (1+i c e^{2 i a+2 i b x}\right )-\frac {\operatorname {PolyLog}\left (2,-i c e^{2 i a+2 i b x}\right )}{4 b} \] Output:

-1/2*b*x^2+x*arccot(c-(1-I*c)*tan(b*x+a))-1/2*I*x*ln(1+I*c*exp(2*I*a+2*I*b 
*x))-1/4*polylog(2,-I*c*exp(2*I*a+2*I*b*x))/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. \(847\) vs. \(2(86)=172\).

Time = 2.00 (sec) , antiderivative size = 847, normalized size of antiderivative = 9.85 \[ \int \cot ^{-1}(c-(1-i c) \tan (a+b x)) \, dx=x \cot ^{-1}(c+i (i+c) \tan (a+b x))-\frac {i x \left (-2 i b x \log (2 \cos (b x) (\cos (b x)-i \sin (b x)))+\log \left (\frac {\sec (b x) (\cos (a)-i \sin (a)) ((-i+c) \cos (a+b x)+i (i+c) \sin (a+b x))}{2 c}\right ) \log (1-i \tan (b x))-\log \left (\frac {1}{2} \sec (b x) (\cos (a)+i \sin (a)) ((1+i c) \cos (a+b x)-(i+c) \sin (a+b x))\right ) \log (1+i \tan (b x))+\operatorname {PolyLog}(2,-\cos (2 b x)+i \sin (2 b x))+\operatorname {PolyLog}\left (2,\frac {\sec (b x) ((i+c) \cos (a)+(1+i c) \sin (a)) (\cos (a+b x)-i \sin (a+b x))}{2 c}\right )-\operatorname {PolyLog}\left (2,\frac {1}{2} (\cos (a)+i \sin (a)) ((i+c) \cos (a)+(1+i c) \sin (a)) (-i+\tan (b x))\right )\right ) \sec (a+b x) (\cos (b x)+i \sin (b x)) (i \cos (b x)+\sin (b x))}{((-i+c) \cos (a+b x)+i (i+c) \sin (a+b x)) \left (-2 b x+i \log \left (1-\frac {\sec (b x) ((i+c) \cos (a)+(1+i c) \sin (a)) (\cos (a+b x)-i \sin (a+b x))}{2 c}\right )+\frac {i (i+c) \cos (a+b x) (\log (1-i \tan (b x))-\log (1+i \tan (b x)))}{(-i+c) \cos (a+b x)+i (i+c) \sin (a+b x)}+\frac {(1+i c) (\log (1-i \tan (b x))-\log (1+i \tan (b x))) \sin (a+b x)}{(-1-i c) \cos (a+b x)+(i+c) \sin (a+b x)}+2 i b x \tan (b x)-\log \left (1-\frac {\sec (b x) ((i+c) \cos (a)+(1+i c) \sin (a)) (\cos (a+b x)-i \sin (a+b x))}{2 c}\right ) \tan (b x)+\log (1-i \tan (b x)) \tan (b x)-\log (1+i \tan (b x)) \tan (b x)-\frac {\log \left (\frac {1}{2} \sec (b x) (\cos (a)+i \sin (a)) ((1+i c) \cos (a+b x)-(i+c) \sin (a+b x))\right ) \sec ^2(b x)}{-i+\tan (b x)}+\frac {\log \left (1-\frac {1}{2} (\cos (a)+i \sin (a)) ((i+c) \cos (a)+(1+i c) \sin (a)) (-i+\tan (b x))\right ) \sec ^2(b x)}{-i+\tan (b x)}+\frac {\log \left (\frac {\sec (b x) (\cos (a)-i \sin (a)) ((-i+c) \cos (a+b x)+i (i+c) \sin (a+b x))}{2 c}\right ) \sec ^2(b x)}{i+\tan (b x)}\right ) (-i+\tan (a+b x))} \] Input:

Integrate[ArcCot[c - (1 - I*c)*Tan[a + b*x]],x]
 

Output:

x*ArcCot[c + I*(I + c)*Tan[a + b*x]] - (I*x*((-2*I)*b*x*Log[2*Cos[b*x]*(Co 
s[b*x] - I*Sin[b*x])] + Log[(Sec[b*x]*(Cos[a] - I*Sin[a])*((-I + c)*Cos[a 
+ b*x] + I*(I + c)*Sin[a + b*x]))/(2*c)]*Log[1 - I*Tan[b*x]] - Log[(Sec[b* 
x]*(Cos[a] + I*Sin[a])*((1 + I*c)*Cos[a + b*x] - (I + c)*Sin[a + b*x]))/2] 
*Log[1 + I*Tan[b*x]] + PolyLog[2, -Cos[2*b*x] + I*Sin[2*b*x]] + PolyLog[2, 
 (Sec[b*x]*((I + c)*Cos[a] + (1 + I*c)*Sin[a])*(Cos[a + b*x] - I*Sin[a + b 
*x]))/(2*c)] - PolyLog[2, ((Cos[a] + I*Sin[a])*((I + c)*Cos[a] + (1 + I*c) 
*Sin[a])*(-I + Tan[b*x]))/2])*Sec[a + b*x]*(Cos[b*x] + I*Sin[b*x])*(I*Cos[ 
b*x] + Sin[b*x]))/(((-I + c)*Cos[a + b*x] + I*(I + c)*Sin[a + b*x])*(-2*b* 
x + I*Log[1 - (Sec[b*x]*((I + c)*Cos[a] + (1 + I*c)*Sin[a])*(Cos[a + b*x] 
- I*Sin[a + b*x]))/(2*c)] + (I*(I + c)*Cos[a + b*x]*(Log[1 - I*Tan[b*x]] - 
 Log[1 + I*Tan[b*x]]))/((-I + c)*Cos[a + b*x] + I*(I + c)*Sin[a + b*x]) + 
((1 + I*c)*(Log[1 - I*Tan[b*x]] - Log[1 + I*Tan[b*x]])*Sin[a + b*x])/((-1 
- I*c)*Cos[a + b*x] + (I + c)*Sin[a + b*x]) + (2*I)*b*x*Tan[b*x] - Log[1 - 
 (Sec[b*x]*((I + c)*Cos[a] + (1 + I*c)*Sin[a])*(Cos[a + b*x] - I*Sin[a + b 
*x]))/(2*c)]*Tan[b*x] + Log[1 - I*Tan[b*x]]*Tan[b*x] - Log[1 + I*Tan[b*x]] 
*Tan[b*x] - (Log[(Sec[b*x]*(Cos[a] + I*Sin[a])*((1 + I*c)*Cos[a + b*x] - ( 
I + c)*Sin[a + b*x]))/2]*Sec[b*x]^2)/(-I + Tan[b*x]) + (Log[1 - ((Cos[a] + 
 I*Sin[a])*((I + c)*Cos[a] + (1 + I*c)*Sin[a])*(-I + Tan[b*x]))/2]*Sec[b*x 
]^2)/(-I + Tan[b*x]) + (Log[(Sec[b*x]*(Cos[a] - I*Sin[a])*((-I + c)*Cos...
 

Rubi [A] (verified)

Time = 0.43 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.26, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {5687, 25, 2615, 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.

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

\(\Big \downarrow \) 5687

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2615

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

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

Input:

Int[ArcCot[c - (1 - I*c)*Tan[a + b*x]],x]
 

Output:

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

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 2615

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

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 5687

Int[ArcCot[(c_.) + (d_.)*Tan[(a_.) + (b_.)*(x_)]], x_Symbol] :> Simp[x*ArcC 
ot[c + d*Tan[a + b*x]], x] + Simp[I*b   Int[x/(c + I*d + c*E^(2*I*a + 2*I*b 
*x)), x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[(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. 594 vs. \(2 (70 ) = 140\).

Time = 2.44 (sec) , antiderivative size = 595, normalized size of antiderivative = 6.92

method	result	size
derivativedivides	\(\frac {-\frac {\operatorname {arccot}\left (c +\left (i c -1\right ) \tan \left (b x +a \right )\right ) \ln \left (-\left (i c -1\right ) \tan \left (b x +a \right )+c +i\right ) c^{2}}{2 i+2 c}-\frac {2 i \operatorname {arccot}\left (c +\left (i c -1\right ) \tan \left (b x +a \right )\right ) \ln \left (-\left (i c -1\right ) \tan \left (b x +a \right )+c +i\right ) c}{2 i+2 c}+\frac {\operatorname {arccot}\left (c +\left (i c -1\right ) \tan \left (b x +a \right )\right ) \ln \left (-\left (i c -1\right ) \tan \left (b x +a \right )+c +i\right )}{2 i+2 c}+\frac {\operatorname {arccot}\left (c +\left (i c -1\right ) \tan \left (b x +a \right )\right ) \ln \left (i+\left (i c -1\right ) \tan \left (b x +a \right )+c \right ) c^{2}}{2 i+2 c}+\frac {2 i \operatorname {arccot}\left (c +\left (i c -1\right ) \tan \left (b x +a \right )\right ) \ln \left (i+\left (i c -1\right ) \tan \left (b x +a \right )+c \right ) c}{2 i+2 c}-\frac {\operatorname {arccot}\left (c +\left (i c -1\right ) \tan \left (b x +a \right )\right ) \ln \left (i+\left (i c -1\right ) \tan \left (b x +a \right )+c \right )}{2 i+2 c}-\left (i c -1\right )^{2} \left (\frac {\frac {i \ln \left (i+\left (i c -1\right ) \tan \left (b x +a \right )+c \right )^{2}}{4}-\frac {i \left (\left (\ln \left (i+\left (i c -1\right ) \tan \left (b x +a \right )+c \right )-\ln \left (-\frac {i \left (i+\left (i c -1\right ) \tan \left (b x +a \right )+c \right )}{2}\right )\right ) \ln \left (-\frac {i \left (i-\left (i c -1\right ) \tan \left (b x +a \right )-c \right )}{2}\right )-\operatorname {dilog}\left (-\frac {i \left (i+\left (i c -1\right ) \tan \left (b x +a \right )+c \right )}{2}\right )\right )}{2}}{2 i+2 c}-\frac {-\frac {i \left (\operatorname {dilog}\left (-\frac {i-\left (i c -1\right ) \tan \left (b x +a \right )-c}{2 c}\right )+\ln \left (-\left (i c -1\right ) \tan \left (b x +a \right )+c +i\right ) \ln \left (-\frac {i-\left (i c -1\right ) \tan \left (b x +a \right )-c}{2 c}\right )\right )}{2}+\frac {i \left (\operatorname {dilog}\left (\frac {-i-\left (i c -1\right ) \tan \left (b x +a \right )-c}{-2 i-2 c}\right )+\ln \left (-\left (i c -1\right ) \tan \left (b x +a \right )+c +i\right ) \ln \left (\frac {-i-\left (i c -1\right ) \tan \left (b x +a \right )-c}{-2 i-2 c}\right )\right )}{2}}{2 \left (i+c \right )}\right )}{b \left (i c -1\right )}\)	\(595\)
default	\(\frac {-\frac {\operatorname {arccot}\left (c +\left (i c -1\right ) \tan \left (b x +a \right )\right ) \ln \left (-\left (i c -1\right ) \tan \left (b x +a \right )+c +i\right ) c^{2}}{2 i+2 c}-\frac {2 i \operatorname {arccot}\left (c +\left (i c -1\right ) \tan \left (b x +a \right )\right ) \ln \left (-\left (i c -1\right ) \tan \left (b x +a \right )+c +i\right ) c}{2 i+2 c}+\frac {\operatorname {arccot}\left (c +\left (i c -1\right ) \tan \left (b x +a \right )\right ) \ln \left (-\left (i c -1\right ) \tan \left (b x +a \right )+c +i\right )}{2 i+2 c}+\frac {\operatorname {arccot}\left (c +\left (i c -1\right ) \tan \left (b x +a \right )\right ) \ln \left (i+\left (i c -1\right ) \tan \left (b x +a \right )+c \right ) c^{2}}{2 i+2 c}+\frac {2 i \operatorname {arccot}\left (c +\left (i c -1\right ) \tan \left (b x +a \right )\right ) \ln \left (i+\left (i c -1\right ) \tan \left (b x +a \right )+c \right ) c}{2 i+2 c}-\frac {\operatorname {arccot}\left (c +\left (i c -1\right ) \tan \left (b x +a \right )\right ) \ln \left (i+\left (i c -1\right ) \tan \left (b x +a \right )+c \right )}{2 i+2 c}-\left (i c -1\right )^{2} \left (\frac {\frac {i \ln \left (i+\left (i c -1\right ) \tan \left (b x +a \right )+c \right )^{2}}{4}-\frac {i \left (\left (\ln \left (i+\left (i c -1\right ) \tan \left (b x +a \right )+c \right )-\ln \left (-\frac {i \left (i+\left (i c -1\right ) \tan \left (b x +a \right )+c \right )}{2}\right )\right ) \ln \left (-\frac {i \left (i-\left (i c -1\right ) \tan \left (b x +a \right )-c \right )}{2}\right )-\operatorname {dilog}\left (-\frac {i \left (i+\left (i c -1\right ) \tan \left (b x +a \right )+c \right )}{2}\right )\right )}{2}}{2 i+2 c}-\frac {-\frac {i \left (\operatorname {dilog}\left (-\frac {i-\left (i c -1\right ) \tan \left (b x +a \right )-c}{2 c}\right )+\ln \left (-\left (i c -1\right ) \tan \left (b x +a \right )+c +i\right ) \ln \left (-\frac {i-\left (i c -1\right ) \tan \left (b x +a \right )-c}{2 c}\right )\right )}{2}+\frac {i \left (\operatorname {dilog}\left (\frac {-i-\left (i c -1\right ) \tan \left (b x +a \right )-c}{-2 i-2 c}\right )+\ln \left (-\left (i c -1\right ) \tan \left (b x +a \right )+c +i\right ) \ln \left (\frac {-i-\left (i c -1\right ) \tan \left (b x +a \right )-c}{-2 i-2 c}\right )\right )}{2}}{2 \left (i+c \right )}\right )}{b \left (i c -1\right )}\)	\(595\)
risch	\(\text {Expression too large to display}\)	\(1244\)

Input:

int(arccot(c-(1-I*c)*tan(b*x+a)),x,method=_RETURNVERBOSE)
 

Output:

1/b/(I*c-1)*(-arccot(c+(I*c-1)*tan(b*x+a))/(2*I+2*c)*ln(-(I*c-1)*tan(b*x+a 
)+c+I)*c^2-2*I*arccot(c+(I*c-1)*tan(b*x+a))/(2*I+2*c)*ln(-(I*c-1)*tan(b*x+ 
a)+c+I)*c+arccot(c+(I*c-1)*tan(b*x+a))/(2*I+2*c)*ln(-(I*c-1)*tan(b*x+a)+c+ 
I)+arccot(c+(I*c-1)*tan(b*x+a))/(2*I+2*c)*ln(I+(I*c-1)*tan(b*x+a)+c)*c^2+2 
*I*arccot(c+(I*c-1)*tan(b*x+a))/(2*I+2*c)*ln(I+(I*c-1)*tan(b*x+a)+c)*c-arc 
cot(c+(I*c-1)*tan(b*x+a))/(2*I+2*c)*ln(I+(I*c-1)*tan(b*x+a)+c)-(I*c-1)^2*( 
1/2/(I+c)*(1/4*I*ln(I+(I*c-1)*tan(b*x+a)+c)^2-1/2*I*((ln(I+(I*c-1)*tan(b*x 
+a)+c)-ln(-1/2*I*(I+(I*c-1)*tan(b*x+a)+c)))*ln(-1/2*I*(I-(I*c-1)*tan(b*x+a 
)-c))-dilog(-1/2*I*(I+(I*c-1)*tan(b*x+a)+c))))-1/2/(I+c)*(-1/2*I*(dilog(-1 
/2*(I-(I*c-1)*tan(b*x+a)-c)/c)+ln(-(I*c-1)*tan(b*x+a)+c+I)*ln(-1/2*(I-(I*c 
-1)*tan(b*x+a)-c)/c))+1/2*I*(dilog((-I-(I*c-1)*tan(b*x+a)-c)/(-2*I-2*c))+l 
n(-(I*c-1)*tan(b*x+a)+c+I)*ln((-I-(I*c-1)*tan(b*x+a)-c)/(-2*I-2*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. 201 vs. \(2 (61) = 122\).

Time = 0.10 (sec) , antiderivative size = 201, normalized size of antiderivative = 2.34 \[ \int \cot ^{-1}(c-(1-i c) \tan (a+b x)) \, dx=-\frac {b^{2} x^{2} + i \, b x \log \left (\frac {{\left (c + i\right )} e^{\left (2 i \, b x + 2 i \, a\right )}}{c e^{\left (2 i \, b x + 2 i \, a\right )} - i}\right ) - a^{2} - {\left (-i \, b x - i \, a\right )} \log \left (\frac {1}{2} \, \sqrt {-4 i \, c} e^{\left (i \, b x + i \, a\right )} + 1\right ) - {\left (-i \, b x - i \, a\right )} \log \left (-\frac {1}{2} \, \sqrt {-4 i \, c} e^{\left (i \, b x + i \, a\right )} + 1\right ) - i \, a \log \left (\frac {2 \, c e^{\left (i \, b x + i \, a\right )} + i \, \sqrt {-4 i \, c}}{2 \, c}\right ) - i \, a \log \left (\frac {2 \, c e^{\left (i \, b x + i \, a\right )} - i \, \sqrt {-4 i \, c}}{2 \, c}\right ) + {\rm Li}_2\left (\frac {1}{2} \, \sqrt {-4 i \, c} e^{\left (i \, b x + i \, a\right )}\right ) + {\rm Li}_2\left (-\frac {1}{2} \, \sqrt {-4 i \, c} e^{\left (i \, b x + i \, a\right )}\right )}{2 \, b} \] Input:

integrate(arccot(c-(1-I*c)*tan(b*x+a)),x, algorithm="fricas")
 

Output:

-1/2*(b^2*x^2 + I*b*x*log((c + I)*e^(2*I*b*x + 2*I*a)/(c*e^(2*I*b*x + 2*I* 
a) - I)) - a^2 - (-I*b*x - I*a)*log(1/2*sqrt(-4*I*c)*e^(I*b*x + I*a) + 1) 
- (-I*b*x - I*a)*log(-1/2*sqrt(-4*I*c)*e^(I*b*x + I*a) + 1) - I*a*log(1/2* 
(2*c*e^(I*b*x + I*a) + I*sqrt(-4*I*c))/c) - I*a*log(1/2*(2*c*e^(I*b*x + I* 
a) - I*sqrt(-4*I*c))/c) + dilog(1/2*sqrt(-4*I*c)*e^(I*b*x + I*a)) + dilog( 
-1/2*sqrt(-4*I*c)*e^(I*b*x + I*a)))/b
 

Sympy [F(-2)]

Exception generated. \[ \int \cot ^{-1}(c-(1-i c) \tan (a+b x)) \, dx=\text {Exception raised: CoercionFailed} \] Input:

integrate(acot(c-(1-I*c)*tan(b*x+a)),x)
 

Output:

Exception raised: CoercionFailed >> Cannot convert 2*_t0**4*c**2*exp(4*I*a 
) + _t0**4*I*c*exp(4*I*a) - 3*_t0**2*I*c*exp(2*I*a) + _t0**2*exp(2*I*a) - 
1 of type <class 'sympy.core.add.Add'> to QQ_I[b,c,_t0,exp(I*a)]
 

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. 450 vs. \(2 (61) = 122\).

Time = 0.14 (sec) , antiderivative size = 450, normalized size of antiderivative = 5.23 \[ \int \cot ^{-1}(c-(1-i c) \tan (a+b x)) \, dx =\text {Too large to display} \] Input:

integrate(arccot(c-(1-I*c)*tan(b*x+a)),x, algorithm="maxima")
 

Output:

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

Giac [F]

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

integrate(arccot(c-(1-I*c)*tan(b*x+a)),x, algorithm="giac")
 

Output:

integrate(arccot(-(-I*c + 1)*tan(b*x + a) + c), x)
 

Mupad [F(-1)]

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

int(acot(c + tan(a + b*x)*(c*1i - 1)),x)
 

Output:

int(acot(c + tan(a + b*x)*(c*1i - 1)), x)
 

Reduce [F]

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

int(acot(c-(1-I*c)*tan(b*x+a)),x)
 

Output:

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