\(\int \cot ^{-1}(c+(1+i c) \tan (a+b x)) \, dx\) [164]

Optimal result

Integrand size = 17, antiderivative size = 85 \[ \int \cot ^{-1}(c+(1+i c) \tan (a+b x)) \, dx=\frac {b x^2}{2}+x \cot ^{-1}(c+(1+i c) \tan (a+b x))+\frac {1}{2} i x \log \left (1-i c e^{2 i a+2 i b x}\right )+\frac {\operatorname {PolyLog}\left (2,i c e^{2 i a+2 i b x}\right )}{4 b} \]

[Out]

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

Rubi [A] (verified)

Time = 0.09 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {5272, 2215, 2221, 2317, 2438} \[ \int \cot ^{-1}(c+(1+i c) \tan (a+b x)) \, dx=\frac {\operatorname {PolyLog}\left (2,i c e^{2 i a+2 i b x}\right )}{4 b}+\frac {1}{2} i x \log \left (1-i c e^{2 i a+2 i b x}\right )+x \cot ^{-1}(c+(1+i c) \tan (a+b x))+\frac {b x^2}{2} \]

[In]

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

[Out]

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

Rule 2215

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] - Dist[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 2221

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]
- Dist[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 2317

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[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 2438

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 5272

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

Rubi steps \begin{align*} \text {integral}& = x \cot ^{-1}(c+(1+i c) \tan (a+b x))+(i b) \int \frac {x}{i (1+i c)+c+c e^{2 i a+2 i b x}} \, dx \\ & = \frac {b x^2}{2}+x \cot ^{-1}(c+(1+i c) \tan (a+b x))-(b c) \int \frac {e^{2 i a+2 i b x} x}{i (1+i c)+c+c e^{2 i a+2 i 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 {1}{2} i \int \log \left (1+\frac {c e^{2 i a+2 i b x}}{i (1+i c)+c}\right ) \, 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 {\text {Subst}\left (\int \frac {\log \left (1+\frac {c x}{i (1+i c)+c}\right )}{x} \, dx,x,e^{2 i a+2 i b x}\right )}{4 b} \\ & = \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} \\ \end{align*}

Mathematica [B] (warning: unable to verify)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(967\) vs. \(2(85)=170\).

Time = 4.19 (sec) , antiderivative size = 967, normalized size of antiderivative = 11.38 \[ \int \cot ^{-1}(c+(1+i c) \tan (a+b x)) \, dx=x \cot ^{-1}(c+(1+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)+(1+i c) \sin (a+b x))}{2 c}\right ) \log (1-i \tan (b x))+\log \left (\frac {\sec (b x) ((1-i c) \cos (a+b x)+(-i+c) \sin (a+b x))}{2 \cos (a)-2 i \sin (a)}\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)+i (i+c) \sin (a)) (\cos (a+b x)-i \sin (a+b x))}{2 c}\right )+\operatorname {PolyLog}\left (2,\frac {1}{2} \sec (b x) ((1+i c) \cos (a)-(i+c) \sin (a)) (\cos (a+b x)+i \sin (a+b x))\right )\right ) \sec ^2(a+b x) (\cos (b x)+i \sin (b x)) (i \cos (b x)+\sin (b x)) ((1-i c) \cos (a+b x)+(-i+c) \sin (a+b x))}{((i+c) \cos (a+b x)+(1+i c) \sin (a+b x)) \left (2 b x-i \log \left (1-\frac {\sec (b x) ((-i+c) \cos (a)+i (i+c) \sin (a)) (\cos (a+b x)-i \sin (a+b x))}{2 c}\right )-i \log \left (1+\frac {1}{2} \sec (b x) ((-1-i c) \cos (a)+(i+c) \sin (a)) (\cos (a+b x)+i \sin (a+b x))\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)+(1+i c) \sin (a+b x)}+\frac {(i+c) (\log (1-i \tan (b x))-\log (1+i \tan (b x))) \sin (a+b x)}{(i+c) \cos (a+b x)+(1+i c) \sin (a+b x)}-2 i b x \tan (b x)+\log \left (1-\frac {\sec (b x) ((-i+c) \cos (a)+i (i+c) \sin (a)) (\cos (a+b x)-i \sin (a+b x))}{2 c}\right ) \tan (b x)-\log \left (1+\frac {1}{2} \sec (b x) ((-1-i c) \cos (a)+(i+c) \sin (a)) (\cos (a+b x)+i \sin (a+b x))\right ) \tan (b x)-\log (1-i \tan (b x)) \tan (b x)+\cos ^2(a) \log (1+i \tan (b x)) \tan (b x)+\log (1+i \tan (b x)) \sin ^2(a) \tan (b x)+\frac {\log \left (\frac {\sec (b x) ((1-i c) \cos (a+b x)+(-i+c) \sin (a+b x))}{2 \cos (a)-2 i \sin (a)}\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)+(1+i c) \sin (a+b x))}{2 c}\right ) \sec ^2(b x)}{i+\tan (b x)}\right ) (-i+\tan (a+b x)) (1-i c+(-i+c) \tan (a+b x))} \]

[In]

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

[Out]

x*ArcCot[c + (1 + I*c)*Tan[a + b*x]] - (I*x*((2*I)*b*x*Log[2*Cos[b*x]*(Cos[b*x] - I*Sin[b*x])] - Log[(Sec[b*x]
*(Cos[a] - I*Sin[a])*((I + c)*Cos[a + b*x] + (1 + I*c)*Sin[a + b*x]))/(2*c)]*Log[1 - I*Tan[b*x]] + Log[(Sec[b*
x]*((1 - I*c)*Cos[a + b*x] + (-I + c)*Sin[a + b*x]))/(2*Cos[a] - (2*I)*Sin[a])]*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] + I*(I + c)*Sin[a])*(Cos[a + b*x] - I*S
in[a + b*x]))/(2*c)] + PolyLog[2, (Sec[b*x]*((1 + I*c)*Cos[a] - (I + c)*Sin[a])*(Cos[a + b*x] + I*Sin[a + b*x]
))/2])*Sec[a + b*x]^2*(Cos[b*x] + I*Sin[b*x])*(I*Cos[b*x] + Sin[b*x])*((1 - I*c)*Cos[a + b*x] + (-I + c)*Sin[a
+ b*x]))/(((I + c)*Cos[a + b*x] + (1 + I*c)*Sin[a + b*x])*(2*b*x - I*Log[1 - (Sec[b*x]*((-I + c)*Cos[a] + I*(
I + c)*Sin[a])*(Cos[a + b*x] - I*Sin[a + b*x]))/(2*c)] - I*Log[1 + (Sec[b*x]*((-1 - I*c)*Cos[a] + (I + c)*Sin[
a])*(Cos[a + b*x] + I*Sin[a + b*x]))/2] - (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] + (1 + I*c)*Sin[a + b*x]) + ((I + c)*(Log[1 - I*Tan[b*x]] - Log[1 + I*Tan[b*x]])*Sin[a
+ b*x])/((I + c)*Cos[a + b*x] + (1 + I*c)*Sin[a + b*x]) - (2*I)*b*x*Tan[b*x] + Log[1 - (Sec[b*x]*((-I + c)*Co
s[a] + I*(I + c)*Sin[a])*(Cos[a + b*x] - I*Sin[a + b*x]))/(2*c)]*Tan[b*x] - Log[1 + (Sec[b*x]*((-1 - I*c)*Cos[
a] + (I + c)*Sin[a])*(Cos[a + b*x] + I*Sin[a + b*x]))/2]*Tan[b*x] - Log[1 - I*Tan[b*x]]*Tan[b*x] + Cos[a]^2*Lo
g[1 + I*Tan[b*x]]*Tan[b*x] + Log[1 + I*Tan[b*x]]*Sin[a]^2*Tan[b*x] + (Log[(Sec[b*x]*((1 - I*c)*Cos[a + b*x] +
(-I + c)*Sin[a + b*x]))/(2*Cos[a] - (2*I)*Sin[a])]*Sec[b*x]^2)/(-I + Tan[b*x]) - (Log[(Sec[b*x]*(Cos[a] - I*Si
n[a])*((I + c)*Cos[a + b*x] + (1 + I*c)*Sin[a + b*x]))/(2*c)]*Sec[b*x]^2)/(I + Tan[b*x]))*(-I + Tan[a + b*x])*
(1 - I*c + (-I + c)*Tan[a + b*x]))

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 561 vs. \(2 (69 ) = 138\).

Time = 2.06 (sec) , antiderivative size = 562, normalized size of antiderivative = 6.61

method	result	size
derivativedivides	\(\frac {\frac {\operatorname {arccot}\left (c +\left (i c +1\right ) \tan \left (b x +a \right )\right ) \ln \left (-c +\left (i c +1\right ) \tan \left (b x +a \right )+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 (-c +\left (i c +1\right ) \tan \left (b x +a \right )+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 (-c +\left (i c +1\right ) \tan \left (b x +a \right )+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+c +\left (i c +1\right ) \tan \left (b x +a \right )\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+c +\left (i c +1\right ) \tan \left (b x +a \right )\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+c +\left (i c +1\right ) \tan \left (b x +a \right )\right )}{2 i-2 c}+\left (i c +1\right )^{2} \left (\frac {-\frac {i \ln \left (-i+c +\left (i c +1\right ) \tan \left (b x +a \right )\right )^{2}}{4}+\frac {i \left (\operatorname {dilog}\left (-\frac {i \left (c +\left (i c +1\right ) \tan \left (b x +a \right )+i\right )}{2}\right )+\ln \left (-i+c +\left (i c +1\right ) \tan \left (b x +a \right )\right ) \ln \left (-\frac {i \left (c +\left (i c +1\right ) \tan \left (b x +a \right )+i\right )}{2}\right )\right )}{2}}{2 i-2 c}-\frac {\frac {i \left (\operatorname {dilog}\left (\frac {c +\left (i c +1\right ) \tan \left (b x +a \right )+i}{2 c}\right )+\ln \left (-c +\left (i c +1\right ) \tan \left (b x +a \right )+i\right ) \ln \left (\frac {c +\left (i c +1\right ) \tan \left (b x +a \right )+i}{2 c}\right )\right )}{2}-\frac {i \left (\operatorname {dilog}\left (\frac {-i+c +\left (i c +1\right ) \tan \left (b x +a \right )}{-2 i+2 c}\right )+\ln \left (-c +\left (i c +1\right ) \tan \left (b x +a \right )+i\right ) \ln \left (\frac {-i+c +\left (i c +1\right ) \tan \left (b x +a \right )}{-2 i+2 c}\right )\right )}{2}}{2 \left (i-c \right )}\right )}{b \left (i c +1\right )}\)	\(562\)
default	\(\frac {\frac {\operatorname {arccot}\left (c +\left (i c +1\right ) \tan \left (b x +a \right )\right ) \ln \left (-c +\left (i c +1\right ) \tan \left (b x +a \right )+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 (-c +\left (i c +1\right ) \tan \left (b x +a \right )+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 (-c +\left (i c +1\right ) \tan \left (b x +a \right )+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+c +\left (i c +1\right ) \tan \left (b x +a \right )\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+c +\left (i c +1\right ) \tan \left (b x +a \right )\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+c +\left (i c +1\right ) \tan \left (b x +a \right )\right )}{2 i-2 c}+\left (i c +1\right )^{2} \left (\frac {-\frac {i \ln \left (-i+c +\left (i c +1\right ) \tan \left (b x +a \right )\right )^{2}}{4}+\frac {i \left (\operatorname {dilog}\left (-\frac {i \left (c +\left (i c +1\right ) \tan \left (b x +a \right )+i\right )}{2}\right )+\ln \left (-i+c +\left (i c +1\right ) \tan \left (b x +a \right )\right ) \ln \left (-\frac {i \left (c +\left (i c +1\right ) \tan \left (b x +a \right )+i\right )}{2}\right )\right )}{2}}{2 i-2 c}-\frac {\frac {i \left (\operatorname {dilog}\left (\frac {c +\left (i c +1\right ) \tan \left (b x +a \right )+i}{2 c}\right )+\ln \left (-c +\left (i c +1\right ) \tan \left (b x +a \right )+i\right ) \ln \left (\frac {c +\left (i c +1\right ) \tan \left (b x +a \right )+i}{2 c}\right )\right )}{2}-\frac {i \left (\operatorname {dilog}\left (\frac {-i+c +\left (i c +1\right ) \tan \left (b x +a \right )}{-2 i+2 c}\right )+\ln \left (-c +\left (i c +1\right ) \tan \left (b x +a \right )+i\right ) \ln \left (\frac {-i+c +\left (i c +1\right ) \tan \left (b x +a \right )}{-2 i+2 c}\right )\right )}{2}}{2 \left (i-c \right )}\right )}{b \left (i c +1\right )}\)	\(562\)
risch	\(\text {Expression too large to display}\)	\(1244\)

[In]

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

[Out]

1/b/(I*c+1)*(arccot(c+(I*c+1)*tan(b*x+a))/(2*I-2*c)*ln(-c+(I*c+1)*tan(b*x+a)+I)*c^2-2*I*arccot(c+(I*c+1)*tan(b
*x+a))/(2*I-2*c)*ln(-c+(I*c+1)*tan(b*x+a)+I)*c-arccot(c+(I*c+1)*tan(b*x+a))/(2*I-2*c)*ln(-c+(I*c+1)*tan(b*x+a)
+I)-arccot(c+(I*c+1)*tan(b*x+a))/(2*I-2*c)*ln(-I+c+(I*c+1)*tan(b*x+a))*c^2+2*I*arccot(c+(I*c+1)*tan(b*x+a))/(2
*I-2*c)*ln(-I+c+(I*c+1)*tan(b*x+a))*c+arccot(c+(I*c+1)*tan(b*x+a))/(2*I-2*c)*ln(-I+c+(I*c+1)*tan(b*x+a))+(I*c+
1)^2*(1/2/(I-c)*(-1/4*I*ln(-I+c+(I*c+1)*tan(b*x+a))^2+1/2*I*(dilog(-1/2*I*(c+(I*c+1)*tan(b*x+a)+I))+ln(-I+c+(I
*c+1)*tan(b*x+a))*ln(-1/2*I*(c+(I*c+1)*tan(b*x+a)+I))))-1/2/(I-c)*(1/2*I*(dilog(1/2*(c+(I*c+1)*tan(b*x+a)+I)/c
)+ln(-c+(I*c+1)*tan(b*x+a)+I)*ln(1/2*(c+(I*c+1)*tan(b*x+a)+I)/c))-1/2*I*(dilog((-I+c+(I*c+1)*tan(b*x+a))/(-2*I
+2*c))+ln(-c+(I*c+1)*tan(b*x+a)+I)*ln((-I+c+(I*c+1)*tan(b*x+a))/(-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. 199 vs. \(2 (60) = 120\).

Time = 0.28 (sec) , antiderivative size = 199, normalized size of antiderivative = 2.34 \[ \int \cot ^{-1}(c+(1+i c) \tan (a+b x)) \, dx=\frac {b^{2} x^{2} - i \, b x \log \left (\frac {{\left (c e^{\left (2 i \, b x + 2 i \, a\right )} + i\right )} e^{\left (-2 i \, b x - 2 i \, a\right )}}{c - i}\right ) - a^{2} + {\left (i \, b x + i \, a\right )} \log \left (\frac {1}{2} \, \sqrt {4 i \, c} e^{\left (i \, b x + i \, a\right )} + 1\right ) + {\left (i \, b x + i \, a\right )} \log \left (-\frac {1}{2} \, \sqrt {4 i \, c} e^{\left (i \, b x + i \, a\right )} + 1\right ) - i \, a \log \left (\frac {2 \, c e^{\left (i \, b x + i \, a\right )} + i \, \sqrt {4 i \, c}}{2 \, c}\right ) - i \, a \log \left (\frac {2 \, c e^{\left (i \, b x + i \, a\right )} - i \, \sqrt {4 i \, c}}{2 \, c}\right ) + {\rm Li}_2\left (\frac {1}{2} \, \sqrt {4 i \, c} e^{\left (i \, b x + i \, a\right )}\right ) + {\rm Li}_2\left (-\frac {1}{2} \, \sqrt {4 i \, c} e^{\left (i \, b x + i \, a\right )}\right )}{2 \, b} \]

[In]

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

[Out]

1/2*(b^2*x^2 - I*b*x*log((c*e^(2*I*b*x + 2*I*a) + I)*e^(-2*I*b*x - 2*I*a)/(c - 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. \[ \int \cot ^{-1}(c+(1+i c) \tan (a+b x)) \, dx=\text {Exception raised: CoercionFailed} \]

[In]

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

[Out]

Exception raised: CoercionFailed >> Cannot convert _t0**2*I + 2*c*exp(2*I*a) - I*exp(2*I*a) of type <class 'sy
mpy.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. 455 vs. \(2 (60) = 120\).

Time = 0.29 (sec) , antiderivative size = 455, normalized size of antiderivative = 5.35 \[ \int \cot ^{-1}(c+(1+i c) \tan (a+b x)) \, dx=-\frac {{\left (-i \, c - 1\right )} {\left (\frac {4 i \, {\left (b x + a\right )} \log \left (-\frac {2 \, {\left (-i \, c^{2} + {\left (c^{2} - 2 i \, c - 1\right )} \tan \left (b x + a\right ) - 2 \, c + i\right )}}{2 i \, c^{2} - 2 \, {\left (c^{2} - 2 i \, c - 1\right )} \tan \left (b x + a\right ) + 2 i}\right )}{i \, c + 1} - \frac {i \, {\left (4 \, {\left (b x + a\right )} {\left (\log \left (-i \, c^{2} + {\left (c^{2} - 2 i \, c - 1\right )} \tan \left (b x + a\right ) - 2 \, c + i\right ) - \log \left (-i \, c^{2} + {\left (c^{2} - 2 i \, c - 1\right )} \tan \left (b x + a\right ) - i\right )\right )} + i \, \log \left (-i \, c^{2} + {\left (c^{2} - 2 i \, c - 1\right )} \tan \left (b x + a\right ) - 2 \, c + i\right )^{2} - 2 i \, \log \left (-i \, c^{2} + {\left (c^{2} - 2 i \, c - 1\right )} \tan \left (b x + a\right ) - i\right ) \log \left (-\frac {1}{2} \, {\left (c - i\right )} \tan \left (b x + a\right ) + \frac {1}{2} i \, c + \frac {1}{2}\right ) + 2 i \, \log \left (-i \, c^{2} + {\left (c^{2} - 2 i \, c - 1\right )} \tan \left (b x + a\right ) - i\right ) \log \left (-\frac {{\left (i \, c + 1\right )} \tan \left (b x + a\right ) + c + i}{2 \, c} + 1\right ) - 2 i \, \log \left (-i \, c^{2} + {\left (c^{2} - 2 i \, c - 1\right )} \tan \left (b x + a\right ) - 2 \, c + i\right ) \log \left (-\frac {1}{2} i \, \tan \left (b x + a\right ) + \frac {1}{2}\right ) - 2 i \, {\rm Li}_2\left (\frac {1}{2} \, {\left (c - i\right )} \tan \left (b x + a\right ) - \frac {1}{2} i \, c + \frac {1}{2}\right ) + 2 i \, {\rm Li}_2\left (\frac {{\left (i \, c + 1\right )} \tan \left (b x + a\right ) + c + i}{2 \, c}\right ) - 2 i \, {\rm Li}_2\left (\frac {1}{2} i \, \tan \left (b x + a\right ) + \frac {1}{2}\right )\right )}}{i \, c + 1}\right )} - 8 \, {\left (b x + a\right )} \operatorname {arccot}\left ({\left (i \, c + 1\right )} \tan \left (b x + a\right ) + c\right ) - \frac {4 \, {\left (b x + a\right )} {\left (c - i\right )} \log \left (-\frac {2 \, {\left (-i \, c^{2} + {\left (c^{2} - 2 i \, c - 1\right )} \tan \left (b x + a\right ) - 2 \, c + i\right )}}{2 i \, c^{2} - 2 \, {\left (c^{2} - 2 i \, c - 1\right )} \tan \left (b x + a\right ) + 2 i}\right )}{i \, c + 1}}{8 \, b} \]

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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