\(\int \frac {\cot ^{\frac {3}{2}}(c+d x)}{a+i a \tan (c+d x)} \, dx\) [735]

Optimal result

Integrand size = 26, antiderivative size = 177 \[ \int \frac {\cot ^{\frac {3}{2}}(c+d x)}{a+i a \tan (c+d x)} \, dx=-\frac {\left (\frac {5}{4}+\frac {3 i}{4}\right ) \arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} a d}+\frac {\left (\frac {5}{4}+\frac {3 i}{4}\right ) \arctan \left (1+\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} a d}+\frac {\left (\frac {5}{4}-\frac {3 i}{4}\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {\cot (c+d x)}}{1+\cot (c+d x)}\right )}{\sqrt {2} a d}-\frac {5 \sqrt {\cot (c+d x)}}{2 a d}+\frac {\cot ^{\frac {3}{2}}(c+d x)}{2 d (i a+a \cot (c+d x))} \] Output:

(5/8+3/8*I)*arctan(-1+2^(1/2)*cot(d*x+c)^(1/2))*2^(1/2)/a/d+(5/8+3/8*I)*ar 
ctan(1+2^(1/2)*cot(d*x+c)^(1/2))*2^(1/2)/a/d+(5/8-3/8*I)*arctanh(2^(1/2)*c 
ot(d*x+c)^(1/2)/(1+cot(d*x+c)))*2^(1/2)/a/d-5/2*cot(d*x+c)^(1/2)/a/d+1/2*c 
ot(d*x+c)^(3/2)/d/(I*a+a*cot(d*x+c))
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.

Time = 0.43 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.62 \[ \int \frac {\cot ^{\frac {3}{2}}(c+d x)}{a+i a \tan (c+d x)} \, dx=\frac {\cot ^{\frac {3}{2}}(c+d x) \left (-\cot ^2(c+d x)+\cot (c+d x) (i+\cot (c+d x)) \operatorname {Hypergeometric2F1}\left (-\frac {5}{4},1,-\frac {1}{4},-\tan ^2(c+d x)\right )+(1-i \cot (c+d x)) \operatorname {Hypergeometric2F1}\left (-\frac {3}{4},1,\frac {1}{4},-\tan ^2(c+d x)\right )\right )}{2 a d (i+\cot (c+d x))} \] Input:

Integrate[Cot[c + d*x]^(3/2)/(a + I*a*Tan[c + d*x]),x]
 

Output:

(Cot[c + d*x]^(3/2)*(-Cot[c + d*x]^2 + Cot[c + d*x]*(I + Cot[c + d*x])*Hyp 
ergeometric2F1[-5/4, 1, -1/4, -Tan[c + d*x]^2] + (1 - I*Cot[c + d*x])*Hype 
rgeometric2F1[-3/4, 1, 1/4, -Tan[c + d*x]^2]))/(2*a*d*(I + Cot[c + d*x]))
 

Rubi [A] (verified)

Time = 0.76 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.15, number of steps used = 20, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.731, Rules used = {3042, 4156, 3042, 4033, 27, 3042, 4011, 3042, 4017, 25, 27, 1482, 1476, 1082, 217, 1479, 25, 27, 1103}

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.

\(\displaystyle \int \frac {\cot ^{\frac {3}{2}}(c+d x)}{a+i a \tan (c+d x)} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {\cot (c+d x)^{3/2}}{a+i a \tan (c+d x)}dx\)

\(\Big \downarrow \) 4156

\(\displaystyle \int \frac {\cot ^{\frac {5}{2}}(c+d x)}{a \cot (c+d x)+i a}dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {\left (-\tan \left (c+d x+\frac {\pi }{2}\right )\right )^{5/2}}{-a \tan \left (c+d x+\frac {\pi }{2}\right )+i a}dx\)

\(\Big \downarrow \) 4033

\(\displaystyle \frac {\cot ^{\frac {3}{2}}(c+d x)}{2 d (a \cot (c+d x)+i a)}-\frac {\int \frac {1}{2} \sqrt {\cot (c+d x)} (3 i a-5 a \cot (c+d x))dx}{2 a^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\cot ^{\frac {3}{2}}(c+d x)}{2 d (a \cot (c+d x)+i a)}-\frac {\int \sqrt {\cot (c+d x)} (3 i a-5 a \cot (c+d x))dx}{4 a^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\cot ^{\frac {3}{2}}(c+d x)}{2 d (a \cot (c+d x)+i a)}-\frac {\int \sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (5 \tan \left (c+d x+\frac {\pi }{2}\right ) a+3 i a\right )dx}{4 a^2}\)

\(\Big \downarrow \) 4011

\(\displaystyle \frac {\cot ^{\frac {3}{2}}(c+d x)}{2 d (a \cot (c+d x)+i a)}-\frac {\frac {10 a \sqrt {\cot (c+d x)}}{d}+\int \frac {3 i \cot (c+d x) a+5 a}{\sqrt {\cot (c+d x)}}dx}{4 a^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\cot ^{\frac {3}{2}}(c+d x)}{2 d (a \cot (c+d x)+i a)}-\frac {\frac {10 a \sqrt {\cot (c+d x)}}{d}+\int \frac {5 a-3 i a \tan \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )}}dx}{4 a^2}\)

\(\Big \downarrow \) 4017

\(\displaystyle \frac {\cot ^{\frac {3}{2}}(c+d x)}{2 d (a \cot (c+d x)+i a)}-\frac {\frac {10 a \sqrt {\cot (c+d x)}}{d}+\frac {2 \int -\frac {a (3 i \cot (c+d x)+5)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}}{d}}{4 a^2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\cot ^{\frac {3}{2}}(c+d x)}{2 d (a \cot (c+d x)+i a)}-\frac {\frac {10 a \sqrt {\cot (c+d x)}}{d}-\frac {2 \int \frac {a (3 i \cot (c+d x)+5)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}}{d}}{4 a^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\cot ^{\frac {3}{2}}(c+d x)}{2 d (a \cot (c+d x)+i a)}-\frac {\frac {10 a \sqrt {\cot (c+d x)}}{d}-\frac {2 a \int \frac {3 i \cot (c+d x)+5}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}}{d}}{4 a^2}\)

\(\Big \downarrow \) 1482

\(\displaystyle \frac {\cot ^{\frac {3}{2}}(c+d x)}{2 d (a \cot (c+d x)+i a)}-\frac {\frac {10 a \sqrt {\cot (c+d x)}}{d}-\frac {2 a \left (\left (\frac {5}{2}-\frac {3 i}{2}\right ) \int \frac {1-\cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}+\left (\frac {5}{2}+\frac {3 i}{2}\right ) \int \frac {\cot (c+d x)+1}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}\right )}{d}}{4 a^2}\)

\(\Big \downarrow \) 1476

\(\displaystyle \frac {\cot ^{\frac {3}{2}}(c+d x)}{2 d (a \cot (c+d x)+i a)}-\frac {\frac {10 a \sqrt {\cot (c+d x)}}{d}-\frac {2 a \left (\left (\frac {5}{2}-\frac {3 i}{2}\right ) \int \frac {1-\cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}+\left (\frac {5}{2}+\frac {3 i}{2}\right ) \left (\frac {1}{2} \int \frac {1}{\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}+\frac {1}{2} \int \frac {1}{\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}\right )\right )}{d}}{4 a^2}\)

\(\Big \downarrow \) 1082

\(\displaystyle \frac {\cot ^{\frac {3}{2}}(c+d x)}{2 d (a \cot (c+d x)+i a)}-\frac {\frac {10 a \sqrt {\cot (c+d x)}}{d}-\frac {2 a \left (\left (\frac {5}{2}-\frac {3 i}{2}\right ) \int \frac {1-\cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}+\left (\frac {5}{2}+\frac {3 i}{2}\right ) \left (\frac {\int \frac {1}{-\cot (c+d x)-1}d\left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}-\frac {\int \frac {1}{-\cot (c+d x)-1}d\left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}\right )\right )}{d}}{4 a^2}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {\cot ^{\frac {3}{2}}(c+d x)}{2 d (a \cot (c+d x)+i a)}-\frac {\frac {10 a \sqrt {\cot (c+d x)}}{d}-\frac {2 a \left (\left (\frac {5}{2}-\frac {3 i}{2}\right ) \int \frac {1-\cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}+\left (\frac {5}{2}+\frac {3 i}{2}\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d}}{4 a^2}\)

\(\Big \downarrow \) 1479

\(\displaystyle \frac {\cot ^{\frac {3}{2}}(c+d x)}{2 d (a \cot (c+d x)+i a)}-\frac {\frac {10 a \sqrt {\cot (c+d x)}}{d}-\frac {2 a \left (\left (\frac {5}{2}-\frac {3 i}{2}\right ) \left (-\frac {\int -\frac {\sqrt {2}-2 \sqrt {\cot (c+d x)}}{\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}\right )+\left (\frac {5}{2}+\frac {3 i}{2}\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d}}{4 a^2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\cot ^{\frac {3}{2}}(c+d x)}{2 d (a \cot (c+d x)+i a)}-\frac {\frac {10 a \sqrt {\cot (c+d x)}}{d}-\frac {2 a \left (\left (\frac {5}{2}-\frac {3 i}{2}\right ) \left (\frac {\int \frac {\sqrt {2}-2 \sqrt {\cot (c+d x)}}{\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}\right )+\left (\frac {5}{2}+\frac {3 i}{2}\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d}}{4 a^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\cot ^{\frac {3}{2}}(c+d x)}{2 d (a \cot (c+d x)+i a)}-\frac {\frac {10 a \sqrt {\cot (c+d x)}}{d}-\frac {2 a \left (\left (\frac {5}{2}-\frac {3 i}{2}\right ) \left (\frac {\int \frac {\sqrt {2}-2 \sqrt {\cot (c+d x)}}{\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}+\frac {1}{2} \int \frac {\sqrt {2} \sqrt {\cot (c+d x)}+1}{\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}\right )+\left (\frac {5}{2}+\frac {3 i}{2}\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d}}{4 a^2}\)

\(\Big \downarrow \) 1103

\(\displaystyle \frac {\cot ^{\frac {3}{2}}(c+d x)}{2 d (a \cot (c+d x)+i a)}-\frac {\frac {10 a \sqrt {\cot (c+d x)}}{d}-\frac {2 a \left (\left (\frac {5}{2}+\frac {3 i}{2}\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )+\left (\frac {5}{2}-\frac {3 i}{2}\right ) \left (\frac {\log \left (\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2}}\right )\right )}{d}}{4 a^2}\)

Input:

Int[Cot[c + d*x]^(3/2)/(a + I*a*Tan[c + d*x]),x]
 

Output:

Cot[c + d*x]^(3/2)/(2*d*(I*a + a*Cot[c + d*x])) - ((10*a*Sqrt[Cot[c + d*x] 
])/d - (2*a*((5/2 + (3*I)/2)*(-(ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]]/Sqr 
t[2]) + ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]]/Sqrt[2]) + (5/2 - (3*I)/2)* 
(-1/2*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]]/Sqrt[2] + Log[1 + 
 Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]]/(2*Sqrt[2]))))/d)/(4*a^2)
 

Defintions of rubi rules used

rule 25

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

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 217

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( 
-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & 
& (LtQ[a, 0] || LtQ[b, 0])
 

rule 1082

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S 
implify[a*(c/b^2)]}, Simp[-2/b   Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b 
)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /; Fre 
eQ[{a, b, c}, x]
 

rule 1103

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S 
imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, 
e}, x] && EqQ[2*c*d - b*e, 0]
 

rule 1476

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 
2*(d/e), 2]}, Simp[e/(2*c)   Int[1/Simp[d/e + q*x + x^2, x], x], x] + Simp[ 
e/(2*c)   Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e}, x] 
 && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]
 

rule 1479

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 
-2*(d/e), 2]}, Simp[e/(2*c*q)   Int[(q - 2*x)/Simp[d/e + q*x - x^2, x], x], 
 x] + Simp[e/(2*c*q)   Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /; F 
reeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]
 

rule 1482

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 
a*c, 2]}, Simp[(d*q + a*e)/(2*a*c)   Int[(q + c*x^2)/(a + c*x^4), x], x] + 
Simp[(d*q - a*e)/(2*a*c)   Int[(q - c*x^2)/(a + c*x^4), x], x]] /; FreeQ[{a 
, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && NeQ[c*d^2 - a*e^2, 0] && NegQ[(- 
a)*c]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 4011

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + 
 (f_.)*(x_)]), x_Symbol] :> Simp[d*((a + b*Tan[e + f*x])^m/(f*m)), x] + Int 
[(a + b*Tan[e + f*x])^(m - 1)*Simp[a*c - b*d + (b*c + a*d)*Tan[e + f*x], x] 
, x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 
 0] && GtQ[m, 0]
 

rule 4017

Int[((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)])/Sqrt[(b_.)*tan[(e_.) + (f_.)*(x_ 
)]], x_Symbol] :> Simp[2/f   Subst[Int[(b*c + d*x^2)/(b^2 + x^4), x], x, Sq 
rt[b*Tan[e + f*x]]], x] /; FreeQ[{b, c, d, e, f}, x] && NeQ[c^2 - d^2, 0] & 
& NeQ[c^2 + d^2, 0]
 

rule 4033

Int[((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)/((a_) + (b_.)*tan[(e_.) + 
(f_.)*(x_)]), x_Symbol] :> Simp[(b*c - a*d)*((c + d*Tan[e + f*x])^(n - 1)/( 
2*a*f*(a + b*Tan[e + f*x]))), x] + Simp[1/(2*a^2)   Int[(c + d*Tan[e + f*x] 
)^(n - 2)*Simp[a*c^2 + a*d^2*(n - 1) - b*c*d*n - d*(a*c*(n - 2) + b*d*n)*Ta 
n[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] 
 && EqQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && GtQ[n, 1]
 

rule 4156

Int[(cot[(e_.) + (f_.)*(x_)]*(d_.))^(m_)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x 
_)]^(n_.))^(p_.), x_Symbol] :> Simp[d^(n*p)   Int[(d*Cot[e + f*x])^(m - n*p 
)*(b + a*Cot[e + f*x]^n)^p, x], x] /; FreeQ[{a, b, d, e, f, m, n, p}, x] && 
  !IntegerQ[m] && IntegersQ[n, p]
 

Maple [A] (verified)

Time = 0.51 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.66

method	result	size
derivativedivides	\(\frac {-2 \sqrt {\cot \left (d x +c \right )}-\frac {i \arctan \left (\frac {2 \sqrt {\cot \left (d x +c \right )}}{\sqrt {2}-i \sqrt {2}}\right )}{\sqrt {2}-i \sqrt {2}}-\frac {i \sqrt {\cot \left (d x +c \right )}}{2 \left (i+\cot \left (d x +c \right )\right )}+\frac {4 i \arctan \left (\frac {2 \sqrt {\cot \left (d x +c \right )}}{i \sqrt {2}+\sqrt {2}}\right )}{i \sqrt {2}+\sqrt {2}}}{d a}\)	\(117\)
default	\(\frac {-2 \sqrt {\cot \left (d x +c \right )}-\frac {i \arctan \left (\frac {2 \sqrt {\cot \left (d x +c \right )}}{\sqrt {2}-i \sqrt {2}}\right )}{\sqrt {2}-i \sqrt {2}}-\frac {i \sqrt {\cot \left (d x +c \right )}}{2 \left (i+\cot \left (d x +c \right )\right )}+\frac {4 i \arctan \left (\frac {2 \sqrt {\cot \left (d x +c \right )}}{i \sqrt {2}+\sqrt {2}}\right )}{i \sqrt {2}+\sqrt {2}}}{d a}\)	\(117\)

Input:

int(cot(d*x+c)^(3/2)/(a+I*a*tan(d*x+c)),x,method=_RETURNVERBOSE)
 

Output:

1/d/a*(-2*cot(d*x+c)^(1/2)-I/(2^(1/2)-I*2^(1/2))*arctan(2*cot(d*x+c)^(1/2) 
/(2^(1/2)-I*2^(1/2)))-1/2*I*cot(d*x+c)^(1/2)/(I+cot(d*x+c))+4*I/(I*2^(1/2) 
+2^(1/2))*arctan(2*cot(d*x+c)^(1/2)/(I*2^(1/2)+2^(1/2))))
 

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. 469 vs. \(2 (130) = 260\).

Time = 0.12 (sec) , antiderivative size = 469, normalized size of antiderivative = 2.65 \[ \int \frac {\cot ^{\frac {3}{2}}(c+d x)}{a+i a \tan (c+d x)} \, dx=\frac {{\left (a d \sqrt {\frac {i}{4 \, a^{2} d^{2}}} e^{\left (2 i \, d x + 2 i \, c\right )} \log \left (2 \, {\left (2 \, {\left (a d e^{\left (2 i \, d x + 2 i \, c\right )} - a d\right )} \sqrt {\frac {i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} - 1}} \sqrt {\frac {i}{4 \, a^{2} d^{2}}} + i \, e^{\left (2 i \, d x + 2 i \, c\right )}\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}\right ) - a d \sqrt {\frac {i}{4 \, a^{2} d^{2}}} e^{\left (2 i \, d x + 2 i \, c\right )} \log \left (-2 \, {\left (2 \, {\left (a d e^{\left (2 i \, d x + 2 i \, c\right )} - a d\right )} \sqrt {\frac {i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} - 1}} \sqrt {\frac {i}{4 \, a^{2} d^{2}}} - i \, e^{\left (2 i \, d x + 2 i \, c\right )}\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}\right ) + a d \sqrt {-\frac {4 i}{a^{2} d^{2}}} e^{\left (2 i \, d x + 2 i \, c\right )} \log \left (\frac {{\left ({\left (a d e^{\left (2 i \, d x + 2 i \, c\right )} - a d\right )} \sqrt {\frac {i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} - 1}} \sqrt {-\frac {4 i}{a^{2} d^{2}}} + 2 i\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{a d}\right ) - a d \sqrt {-\frac {4 i}{a^{2} d^{2}}} e^{\left (2 i \, d x + 2 i \, c\right )} \log \left (-\frac {{\left ({\left (a d e^{\left (2 i \, d x + 2 i \, c\right )} - a d\right )} \sqrt {\frac {i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} - 1}} \sqrt {-\frac {4 i}{a^{2} d^{2}}} - 2 i\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{a d}\right ) - \sqrt {\frac {i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} - 1}} {\left (9 \, e^{\left (2 i \, d x + 2 i \, c\right )} - 1\right )}\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{4 \, a d} \] Input:

integrate(cot(d*x+c)^(3/2)/(a+I*a*tan(d*x+c)),x, algorithm="fricas")
 

Output:

1/4*(a*d*sqrt(1/4*I/(a^2*d^2))*e^(2*I*d*x + 2*I*c)*log(2*(2*(a*d*e^(2*I*d* 
x + 2*I*c) - a*d)*sqrt((I*e^(2*I*d*x + 2*I*c) + I)/(e^(2*I*d*x + 2*I*c) - 
1))*sqrt(1/4*I/(a^2*d^2)) + I*e^(2*I*d*x + 2*I*c))*e^(-2*I*d*x - 2*I*c)) - 
 a*d*sqrt(1/4*I/(a^2*d^2))*e^(2*I*d*x + 2*I*c)*log(-2*(2*(a*d*e^(2*I*d*x + 
 2*I*c) - a*d)*sqrt((I*e^(2*I*d*x + 2*I*c) + I)/(e^(2*I*d*x + 2*I*c) - 1)) 
*sqrt(1/4*I/(a^2*d^2)) - I*e^(2*I*d*x + 2*I*c))*e^(-2*I*d*x - 2*I*c)) + a* 
d*sqrt(-4*I/(a^2*d^2))*e^(2*I*d*x + 2*I*c)*log(((a*d*e^(2*I*d*x + 2*I*c) - 
 a*d)*sqrt((I*e^(2*I*d*x + 2*I*c) + I)/(e^(2*I*d*x + 2*I*c) - 1))*sqrt(-4* 
I/(a^2*d^2)) + 2*I)*e^(-2*I*d*x - 2*I*c)/(a*d)) - a*d*sqrt(-4*I/(a^2*d^2)) 
*e^(2*I*d*x + 2*I*c)*log(-((a*d*e^(2*I*d*x + 2*I*c) - a*d)*sqrt((I*e^(2*I* 
d*x + 2*I*c) + I)/(e^(2*I*d*x + 2*I*c) - 1))*sqrt(-4*I/(a^2*d^2)) - 2*I)*e 
^(-2*I*d*x - 2*I*c)/(a*d)) - sqrt((I*e^(2*I*d*x + 2*I*c) + I)/(e^(2*I*d*x 
+ 2*I*c) - 1))*(9*e^(2*I*d*x + 2*I*c) - 1))*e^(-2*I*d*x - 2*I*c)/(a*d)
                                                                                    
                                                                                    
 

Sympy [F]

\[ \int \frac {\cot ^{\frac {3}{2}}(c+d x)}{a+i a \tan (c+d x)} \, dx=- \frac {i \int \frac {\cot ^{\frac {3}{2}}{\left (c + d x \right )}}{\tan {\left (c + d x \right )} - i}\, dx}{a} \] Input:

integrate(cot(d*x+c)**(3/2)/(a+I*a*tan(d*x+c)),x)
 

Output:

-I*Integral(cot(c + d*x)**(3/2)/(tan(c + d*x) - I), x)/a
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {\cot ^{\frac {3}{2}}(c+d x)}{a+i a \tan (c+d x)} \, dx=\text {Exception raised: RuntimeError} \] Input:

integrate(cot(d*x+c)^(3/2)/(a+I*a*tan(d*x+c)),x, algorithm="maxima")
 

Output:

Exception raised: RuntimeError >> ECL says: expt: undefined: 0 to a negati 
ve exponent.
 

Giac [A] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.45 \[ \int \frac {\cot ^{\frac {3}{2}}(c+d x)}{a+i a \tan (c+d x)} \, dx=\frac {i \, {\left (\left (4 i - 4\right ) \, \sqrt {2} \arctan \left (\left (\frac {1}{2} i + \frac {1}{2}\right ) \, \sqrt {2} \sqrt {\tan \left (d x + c\right )}\right ) + \left (i + 1\right ) \, \sqrt {2} \arctan \left (-\left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {2} \sqrt {\tan \left (d x + c\right )}\right ) - \frac {2 \, {\left (-5 i \, \tan \left (d x + c\right ) - 4\right )}}{\tan \left (d x + c\right )^{\frac {3}{2}} - i \, \sqrt {\tan \left (d x + c\right )}}\right )}}{4 \, a d} \] Input:

integrate(cot(d*x+c)^(3/2)/(a+I*a*tan(d*x+c)),x, algorithm="giac")
 

Output:

1/4*I*((4*I - 4)*sqrt(2)*arctan((1/2*I + 1/2)*sqrt(2)*sqrt(tan(d*x + c))) 
+ (I + 1)*sqrt(2)*arctan(-(1/2*I - 1/2)*sqrt(2)*sqrt(tan(d*x + c))) - 2*(- 
5*I*tan(d*x + c) - 4)/(tan(d*x + c)^(3/2) - I*sqrt(tan(d*x + c))))/(a*d)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\cot ^{\frac {3}{2}}(c+d x)}{a+i a \tan (c+d x)} \, dx=\int \frac {{\mathrm {cot}\left (c+d\,x\right )}^{3/2}}{a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}} \,d x \] Input:

int(cot(c + d*x)^(3/2)/(a + a*tan(c + d*x)*1i),x)
 

Output:

int(cot(c + d*x)^(3/2)/(a + a*tan(c + d*x)*1i), x)
 

Reduce [F]

\[ \int \frac {\cot ^{\frac {3}{2}}(c+d x)}{a+i a \tan (c+d x)} \, dx=\frac {\int \frac {\sqrt {\cot \left (d x +c \right )}\, \cot \left (d x +c \right )}{\tan \left (d x +c \right ) i +1}d x}{a} \] Input:

int(cot(d*x+c)^(3/2)/(a+I*a*tan(d*x+c)),x)
 

Output:

int((sqrt(cot(c + d*x))*cot(c + d*x))/(tan(c + d*x)*i + 1),x)/a