\(\int \frac {1}{\cot ^{\frac {3}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}} \, dx\) [772]

Optimal result

Integrand size = 28, antiderivative size = 180 \[ \int \frac {1}{\cot ^{\frac {3}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}} \, dx=-\frac {2 \sqrt [4]{-1} \arctan \left (\frac {(-1)^{3/4} \sqrt {a} \sqrt {\tan (c+d x)}}{\sqrt {a+i a \tan (c+d x)}}\right ) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}}{\sqrt {a} d}-\frac {\left (\frac {1}{2}-\frac {i}{2}\right ) \text {arctanh}\left (\frac {(1+i) \sqrt {a} \sqrt {\tan (c+d x)}}{\sqrt {a+i a \tan (c+d x)}}\right ) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}}{\sqrt {a} d}-\frac {1}{d \sqrt {\cot (c+d x)} \sqrt {a+i a \tan (c+d x)}} \] Output:

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

Mathematica [A] (verified)

Time = 0.68 (sec) , antiderivative size = 178, normalized size of antiderivative = 0.99 \[ \int \frac {1}{\cot ^{\frac {3}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}} \, dx=\frac {\sqrt {\cot (c+d x)} \left (-4 (-1)^{3/4} a \text {arcsinh}\left (\sqrt [4]{-1} \sqrt {\tan (c+d x)}\right ) \sqrt {1+i \tan (c+d x)} \sqrt {\tan (c+d x)}-2 a \tan (c+d x)+i \sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {i a \tan (c+d x)}}{\sqrt {a+i a \tan (c+d x)}}\right ) \sqrt {i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)}\right )}{2 a d \sqrt {a+i a \tan (c+d x)}} \] Input:

Integrate[1/(Cot[c + d*x]^(3/2)*Sqrt[a + I*a*Tan[c + d*x]]),x]
 

Output:

(Sqrt[Cot[c + d*x]]*(-4*(-1)^(3/4)*a*ArcSinh[(-1)^(1/4)*Sqrt[Tan[c + d*x]] 
]*Sqrt[1 + I*Tan[c + d*x]]*Sqrt[Tan[c + d*x]] - 2*a*Tan[c + d*x] + I*Sqrt[ 
2]*ArcTanh[(Sqrt[2]*Sqrt[I*a*Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]]*Sq 
rt[I*a*Tan[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]]))/(2*a*d*Sqrt[a + I*a*Tan[ 
c + d*x]])
 

Rubi [A] (verified)

Time = 0.97 (sec) , antiderivative size = 165, normalized size of antiderivative = 0.92, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.464, Rules used = {3042, 4729, 3042, 4041, 27, 3042, 4084, 3042, 4027, 218, 4082, 65, 216}

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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4729

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4041

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4084

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4027

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

\(\Big \downarrow \) 218

\(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {2 \int \frac {(a-i a \tan (c+d x)) \sqrt {i \tan (c+d x) a+a}}{\sqrt {\tan (c+d x)}}dx-\frac {(1-i) a^{3/2} \text {arctanh}\left (\frac {(1+i) \sqrt {a} \sqrt {\tan (c+d x)}}{\sqrt {a+i a \tan (c+d x)}}\right )}{d}}{2 a^2}-\frac {\sqrt {\tan (c+d x)}}{d \sqrt {a+i a \tan (c+d x)}}\right )\)

\(\Big \downarrow \) 4082

\(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {\frac {2 a^2 \int \frac {1}{\sqrt {\tan (c+d x)} \sqrt {i \tan (c+d x) a+a}}d\tan (c+d x)}{d}-\frac {(1-i) a^{3/2} \text {arctanh}\left (\frac {(1+i) \sqrt {a} \sqrt {\tan (c+d x)}}{\sqrt {a+i a \tan (c+d x)}}\right )}{d}}{2 a^2}-\frac {\sqrt {\tan (c+d x)}}{d \sqrt {a+i a \tan (c+d x)}}\right )\)

\(\Big \downarrow \) 65

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

\(\Big \downarrow \) 216

\(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {-\frac {4 \sqrt [4]{-1} a^{3/2} \arctan \left (\frac {(-1)^{3/4} \sqrt {a} \sqrt {\tan (c+d x)}}{\sqrt {a+i a \tan (c+d x)}}\right )}{d}-\frac {(1-i) a^{3/2} \text {arctanh}\left (\frac {(1+i) \sqrt {a} \sqrt {\tan (c+d x)}}{\sqrt {a+i a \tan (c+d x)}}\right )}{d}}{2 a^2}-\frac {\sqrt {\tan (c+d x)}}{d \sqrt {a+i a \tan (c+d x)}}\right )\)

Input:

Int[1/(Cot[c + d*x]^(3/2)*Sqrt[a + I*a*Tan[c + d*x]]),x]
 

Output:

Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]]*(((-4*(-1)^(1/4)*a^(3/2)*ArcTan[((-1 
)^(3/4)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]])/d - ((1 - 
 I)*a^(3/2)*ArcTanh[((1 + I)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[ 
c + d*x]]])/d)/(2*a^2) - Sqrt[Tan[c + d*x]]/(d*Sqrt[a + I*a*Tan[c + d*x]]) 
)
 

Defintions of rubi rules used

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 65

Int[1/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[2   Sub 
st[Int[1/(b - d*x^2), x], x, Sqrt[b*x]/Sqrt[c + d*x]], x] /; FreeQ[{b, c, d 
}, x] &&  !GtQ[c, 0]
 

rule 216

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

rule 218

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

rule 3042

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

rule 4027

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

rule 4041

Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + 
(f_.)*(x_)])^(n_), x_Symbol] :> Simp[(-(b*c - a*d))*(a + b*Tan[e + f*x])^m* 
((c + d*Tan[e + f*x])^(n - 1)/(2*a*f*m)), x] + Simp[1/(2*a^2*m)   Int[(a + 
b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^(n - 2)*Simp[c*(a*c*m + b*d*(n 
 - 1)) - d*(b*c*m + a*d*(n - 1)) - d*(b*d*(m - n + 1) - a*c*(m + n - 1))*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] && LtQ[m, 0] && GtQ[n, 1] && (In 
tegerQ[m] || IntegersQ[2*m, 2*n])
 

rule 4082

Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*tan[(e_.) + 
(f_.)*(x_)])*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Sim 
p[b*(B/f)   Subst[Int[(a + b*x)^(m - 1)*(c + d*x)^n, x], x, Tan[e + f*x]], 
x] /; FreeQ[{a, b, c, d, e, f, A, B, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[ 
a^2 + b^2, 0] && EqQ[A*b + a*B, 0]
 

rule 4084

Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*tan[(e_.) + 
(f_.)*(x_)])*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Sim 
p[(A*b + a*B)/b   Int[(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^n, x], x] 
 - Simp[B/b   Int[(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^n*(a - b*Tan[ 
e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, A, B, m, n}, x] && NeQ[b*c - 
a*d, 0] && EqQ[a^2 + b^2, 0] && NeQ[A*b + a*B, 0]
 

rule 4729

Int[(cot[(a_.) + (b_.)*(x_)]*(c_.))^(m_.)*(u_), x_Symbol] :> Simp[(c*Cot[a 
+ b*x])^m*(c*Tan[a + b*x])^m   Int[ActivateTrig[u]/(c*Tan[a + b*x])^m, x], 
x] /; FreeQ[{a, b, c, m}, x] &&  !IntegerQ[m] && KnownTangentIntegrandQ[u, 
x]
 

Maple [B] (warning: unable to verify)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 429 vs. \(2 (141 ) = 282\).

Time = 2.14 (sec) , antiderivative size = 430, normalized size of antiderivative = 2.39

method	result	size
default	\(\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \left (i \sqrt {2}\, \left (\cos \left (d x +c \right )+1\right ) \arctan \left (\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {2}\, \sqrt {-\csc \left (d x +c \right )+\cot \left (d x +c \right )}\right )+i \sin \left (d x +c \right ) \ln \left (\sqrt {-\csc \left (d x +c \right )+\cot \left (d x +c \right )}+1\right )+i \sin \left (d x +c \right ) \ln \left (\sqrt {-\csc \left (d x +c \right )+\cot \left (d x +c \right )}+i\right )-i \sin \left (d x +c \right ) \ln \left (\sqrt {-\csc \left (d x +c \right )+\cot \left (d x +c \right )}-i\right )-i \sin \left (d x +c \right ) \ln \left (\sqrt {-\csc \left (d x +c \right )+\cot \left (d x +c \right )}-1\right )+i \left (\cos \left (d x +c \right )+1\right ) \sqrt {-\csc \left (d x +c \right )+\cot \left (d x +c \right )}-\sin \left (d x +c \right ) \sqrt {2}\, \arctan \left (\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {2}\, \sqrt {-\csc \left (d x +c \right )+\cot \left (d x +c \right )}\right )+\left (\cos \left (d x +c \right )+1\right ) \ln \left (\sqrt {-\csc \left (d x +c \right )+\cot \left (d x +c \right )}+1\right )+\left (\cos \left (d x +c \right )+1\right ) \ln \left (\sqrt {-\csc \left (d x +c \right )+\cot \left (d x +c \right )}+i\right )+\left (-\cos \left (d x +c \right )-1\right ) \ln \left (\sqrt {-\csc \left (d x +c \right )+\cot \left (d x +c \right )}-i\right )+\left (-\cos \left (d x +c \right )-1\right ) \ln \left (\sqrt {-\csc \left (d x +c \right )+\cot \left (d x +c \right )}-1\right )+\sqrt {-\csc \left (d x +c \right )+\cot \left (d x +c \right )}\, \left (-\cos \left (d x +c \right )-1\right )\right )}{d \left (\cos \left (d x +c \right )+1\right ) \sqrt {a \left (1+i \tan \left (d x +c \right )\right )}\, \sqrt {-\csc \left (d x +c \right )+\cot \left (d x +c \right )}\, \sqrt {\cot \left (d x +c \right )}}\)	\(430\)

Input:

int(1/cot(d*x+c)^(3/2)/(a+I*a*tan(d*x+c))^(1/2),x,method=_RETURNVERBOSE)
 

Output:

(1/2+1/2*I)/d*(I*2^(1/2)*(cos(d*x+c)+1)*arctan((1/2+1/2*I)*2^(1/2)*(-csc(d 
*x+c)+cot(d*x+c))^(1/2))+I*sin(d*x+c)*ln((-csc(d*x+c)+cot(d*x+c))^(1/2)+1) 
+I*sin(d*x+c)*ln((-csc(d*x+c)+cot(d*x+c))^(1/2)+I)-I*sin(d*x+c)*ln((-csc(d 
*x+c)+cot(d*x+c))^(1/2)-I)-I*sin(d*x+c)*ln((-csc(d*x+c)+cot(d*x+c))^(1/2)- 
1)+I*(cos(d*x+c)+1)*(-csc(d*x+c)+cot(d*x+c))^(1/2)-sin(d*x+c)*2^(1/2)*arct 
an((1/2+1/2*I)*2^(1/2)*(-csc(d*x+c)+cot(d*x+c))^(1/2))+(cos(d*x+c)+1)*ln(( 
-csc(d*x+c)+cot(d*x+c))^(1/2)+1)+(cos(d*x+c)+1)*ln((-csc(d*x+c)+cot(d*x+c) 
)^(1/2)+I)+(-cos(d*x+c)-1)*ln((-csc(d*x+c)+cot(d*x+c))^(1/2)-I)+(-cos(d*x+ 
c)-1)*ln((-csc(d*x+c)+cot(d*x+c))^(1/2)-1)+(-csc(d*x+c)+cot(d*x+c))^(1/2)* 
(-cos(d*x+c)-1))/(cos(d*x+c)+1)/(a*(1+I*tan(d*x+c)))^(1/2)/(-csc(d*x+c)+co 
t(d*x+c))^(1/2)/cot(d*x+c)^(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. 620 vs. \(2 (134) = 268\).

Time = 0.10 (sec) , antiderivative size = 620, normalized size of antiderivative = 3.44 \[ \int \frac {1}{\cot ^{\frac {3}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}} \, dx =\text {Too large to display} \] Input:

integrate(1/cot(d*x+c)^(3/2)/(a+I*a*tan(d*x+c))^(1/2),x, algorithm="fricas 
")
 

Output:

1/4*(a*d*sqrt(-2*I/(a*d^2))*e^(I*d*x + I*c)*log(-2*(sqrt(2)*(I*a*d*e^(2*I* 
d*x + 2*I*c) - I*a*d)*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1))*sqrt((I*e^(2*I*d*x 
 + 2*I*c) + I)/(e^(2*I*d*x + 2*I*c) - 1))*sqrt(-2*I/(a*d^2)) - 2*I*a*e^(I* 
d*x + I*c))*e^(-I*d*x - I*c)) - a*d*sqrt(-2*I/(a*d^2))*e^(I*d*x + I*c)*log 
(-2*(sqrt(2)*(-I*a*d*e^(2*I*d*x + 2*I*c) + I*a*d)*sqrt(a/(e^(2*I*d*x + 2*I 
*c) + 1))*sqrt((I*e^(2*I*d*x + 2*I*c) + I)/(e^(2*I*d*x + 2*I*c) - 1))*sqrt 
(-2*I/(a*d^2)) - 2*I*a*e^(I*d*x + I*c))*e^(-I*d*x - I*c)) + a*d*sqrt(-4*I/ 
(a*d^2))*e^(I*d*x + I*c)*log(-16*(sqrt(2)*(a^2*d*e^(3*I*d*x + 3*I*c) - a^2 
*d*e^(I*d*x + I*c))*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1))*sqrt((I*e^(2*I*d*x + 
 2*I*c) + I)/(e^(2*I*d*x + 2*I*c) - 1))*sqrt(-4*I/(a*d^2)) + 3*a^2*e^(2*I* 
d*x + 2*I*c) - a^2)*e^(-2*I*d*x - 2*I*c)) - a*d*sqrt(-4*I/(a*d^2))*e^(I*d* 
x + I*c)*log(16*(sqrt(2)*(a^2*d*e^(3*I*d*x + 3*I*c) - a^2*d*e^(I*d*x + I*c 
))*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1))*sqrt((I*e^(2*I*d*x + 2*I*c) + I)/(e^( 
2*I*d*x + 2*I*c) - 1))*sqrt(-4*I/(a*d^2)) - 3*a^2*e^(2*I*d*x + 2*I*c) + a^ 
2)*e^(-2*I*d*x - 2*I*c)) - 2*sqrt(2)*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1))*sqr 
t((I*e^(2*I*d*x + 2*I*c) + I)/(e^(2*I*d*x + 2*I*c) - 1))*(-I*e^(2*I*d*x + 
2*I*c) + I))*e^(-I*d*x - I*c)/(a*d)
 

Sympy [F]

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

integrate(1/cot(d*x+c)**(3/2)/(a+I*a*tan(d*x+c))**(1/2),x)
                                                                                    
                                                                                    
 

Output:

Integral(1/(sqrt(I*a*(tan(c + d*x) - I))*cot(c + d*x)**(3/2)), x)
 

Maxima [F]

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

integrate(1/cot(d*x+c)^(3/2)/(a+I*a*tan(d*x+c))^(1/2),x, algorithm="maxima 
")
 

Output:

integrate(1/(sqrt(I*a*tan(d*x + c) + a)*cot(d*x + c)^(3/2)), x)
 

Giac [F(-2)]

Exception generated. \[ \int \frac {1}{\cot ^{\frac {3}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}} \, dx=\text {Exception raised: TypeError} \] Input:

integrate(1/cot(d*x+c)^(3/2)/(a+I*a*tan(d*x+c))^(1/2),x, algorithm="giac")
 

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Error: Bad Argument TypeError: Bad 
Argument TypeDone
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(sqrt(a)*( - int((sqrt(tan(c + d*x)*i + 1)*sqrt(cot(c + d*x))*tan(c + d*x) 
)/(cot(c + d*x)**2*tan(c + d*x)**2 + cot(c + d*x)**2),x)*i + int((sqrt(tan 
(c + d*x)*i + 1)*sqrt(cot(c + d*x)))/(cot(c + d*x)**2*tan(c + d*x)**2 + co 
t(c + d*x)**2),x)))/a