\(\int \frac {\sqrt {a+i a \tan (c+d x)}}{\cot ^{\frac {3}{2}}(c+d x)} \, dx\) [756]

Optimal result

Integrand size = 28, antiderivative size = 175 \[ \int \frac {\sqrt {a+i a \tan (c+d x)}}{\cot ^{\frac {3}{2}}(c+d x)} \, dx=-\frac {\sqrt [4]{-1} \sqrt {a} \arctan \left (\frac {(-1)^{3/4} \sqrt {a} \sqrt {\tan (c+d x)}}{\sqrt {a+i a \tan (c+d x)}}\right ) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}}{d}-\frac {(1-i) \sqrt {a} \text {arctanh}\left (\frac {(1+i) \sqrt {a} \sqrt {\tan (c+d x)}}{\sqrt {a+i a \tan (c+d x)}}\right ) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}}{d}+\frac {\sqrt {a+i a \tan (c+d x)}}{d \sqrt {\cot (c+d x)}} \] Output:

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

Mathematica [A] (verified)

Time = 0.70 (sec) , antiderivative size = 183, normalized size of antiderivative = 1.05 \[ \int \frac {\sqrt {a+i a \tan (c+d x)}}{\cot ^{\frac {3}{2}}(c+d x)} \, dx=\frac {\sqrt {\cot (c+d x)} \left (-(-1)^{3/4} a \text {arcsinh}\left (\sqrt [4]{-1} \sqrt {\tan (c+d x)}\right ) \sqrt {1+i \tan (c+d x)} \sqrt {\tan (c+d x)}+a (1+i \tan (c+d x)) \tan (c+d x)+i \sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {i a \tan (c+d x)}}{\sqrt {a+i a \tan (c+d x)}}\right ) \sqrt {i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)}\right )}{d \sqrt {a+i a \tan (c+d x)}} \] Input:

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

Output:

(Sqrt[Cot[c + d*x]]*(-((-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]]) + a*(1 + I*Tan[c + d*x])*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]]]*Sqrt[I*a*Tan[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]]))/(d*Sq 
rt[a + I*a*Tan[c + d*x]])
 

Rubi [A] (verified)

Time = 0.92 (sec) , antiderivative size = 164, normalized size of antiderivative = 0.94, 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, 4043, 27, 3042, 4038, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4729

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4043

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4038

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4027

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

\(\Big \downarrow \) 218

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

\(\Big \downarrow \) 4082

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

\(\Big \downarrow \) 65

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

\(\Big \downarrow \) 216

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

Input:

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

Output:

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

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 4038

Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(3/2)/Sqrt[(c_.) + (d_.)*tan[(e_ 
.) + (f_.)*(x_)]], x_Symbol] :> Simp[2*a   Int[Sqrt[a + b*Tan[e + f*x]]/Sqr 
t[c + d*Tan[e + f*x]], x], x] + Simp[b/a   Int[(b + a*Tan[e + f*x])*(Sqrt[a 
 + b*Tan[e + f*x]]/Sqrt[c + d*Tan[e + f*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]
 

rule 4043

Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + 
(f_.)*(x_)])^(n_), x_Symbol] :> Simp[d*(a + b*Tan[e + f*x])^m*((c + d*Tan[e 
 + f*x])^(n - 1)/(f*(m + n - 1))), x] - Simp[1/(a*(m + n - 1))   Int[(a + b 
*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^(n - 2)*Simp[d*(b*c*m + a*d*(-1 + n)) 
 - a*c^2*(m + n - 1) + d*(b*d*m - a*c*(m + 2*n - 2))*Tan[e + f*x], x], x], 
x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2 
, 0] && NeQ[c^2 + d^2, 0] && GtQ[n, 1] && NeQ[m + n - 1, 0] && (IntegerQ[n] 
 || 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 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. 839 vs. \(2 (140 ) = 280\).

Time = 0.96 (sec) , antiderivative size = 840, normalized size of antiderivative = 4.80

method	result	size
default	\(\text {Expression too large to display}\)	\(840\)

Input:

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

Output:

1/8/d*(a*(1+I*tan(d*x+c)))^(1/2)/(1/2*I*sin(d*x+c)-1/2+1/2*cos(d*x+c))/(-c 
sc(d*x+c)+cot(d*x+c))^(1/2)/cot(d*x+c)^(3/2)*(I*2^(1/2)*ln(-((-csc(d*x+c)+ 
cot(d*x+c))^(1/2)*2^(1/2)*sin(d*x+c)+sin(d*x+c)+cos(d*x+c)-1)/((-2*sin(1/2 
*d*x+1/2*c)^2*csc(d*x+c))^(1/2)*2^(1/2)*sin(d*x+c)-sin(d*x+c)-cos(d*x+c)+1 
))*(-2*cos(d*x+c)+2-sec(1/2*d*x+1/2*c)^2)+2^(1/2)*ln((-(-csc(d*x+c)+cot(d* 
x+c))^(1/2)*2^(1/2)*sin(d*x+c)+sin(d*x+c)+cos(d*x+c)-1)/((-2*sin(1/2*d*x+1 
/2*c)^2*csc(d*x+c))^(1/2)*2^(1/2)*sin(d*x+c)+sin(d*x+c)+cos(d*x+c)-1))*(-2 
*cos(d*x+c)+2-sec(1/2*d*x+1/2*c)^2)+I*2^(1/2)*arctan((-csc(d*x+c)+cot(d*x+ 
c))^(1/2)*2^(1/2)-1)*(-4*cos(d*x+c)+4-2*sec(1/2*d*x+1/2*c)^2)+2^(1/2)*arct 
an((-csc(d*x+c)+cot(d*x+c))^(1/2)*2^(1/2)-1)*(-4*cos(d*x+c)+4-2*sec(1/2*d* 
x+1/2*c)^2)+I*2^(1/2)*arctan((-csc(d*x+c)+cot(d*x+c))^(1/2)*2^(1/2)+1)*(-4 
*cos(d*x+c)+4-2*sec(1/2*d*x+1/2*c)^2)+2^(1/2)*arctan((-csc(d*x+c)+cot(d*x+ 
c))^(1/2)*2^(1/2)+1)*(-4*cos(d*x+c)+4-2*sec(1/2*d*x+1/2*c)^2)+I*ln((-csc(d 
*x+c)+cot(d*x+c))^(1/2)-1)*(-2*cos(d*x+c)+2-sec(1/2*d*x+1/2*c)^2)+ln((-csc 
(d*x+c)+cot(d*x+c))^(1/2)-1)*(2*cos(d*x+c)-2+sec(1/2*d*x+1/2*c)^2)+I*ln((- 
csc(d*x+c)+cot(d*x+c))^(1/2)+1)*(2*cos(d*x+c)-2+sec(1/2*d*x+1/2*c)^2)+ln(( 
-csc(d*x+c)+cot(d*x+c))^(1/2)+1)*(-2*cos(d*x+c)+2-sec(1/2*d*x+1/2*c)^2)+I* 
arctan((-csc(d*x+c)+cot(d*x+c))^(1/2))*(4*cos(d*x+c)-4+2*sec(1/2*d*x+1/2*c 
)^2)+arctan((-csc(d*x+c)+cot(d*x+c))^(1/2))*(4*cos(d*x+c)-4+2*sec(1/2*d*x+ 
1/2*c)^2)-4*cos(d*x+c)*(-csc(d*x+c)+cot(d*x+c))^(1/2)*tan(1/2*d*x+1/2*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. 610 vs. \(2 (133) = 266\).

Time = 0.12 (sec) , antiderivative size = 610, normalized size of antiderivative = 3.49 \[ \int \frac {\sqrt {a+i a \tan (c+d x)}}{\cot ^{\frac {3}{2}}(c+d x)} \, dx =\text {Too large to display} \] Input:

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

Output:

-1/4*(4*sqrt(2)*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))*(I*e^(3*I*d*x + 3*I*c) - I*e^(I*d*x + 
I*c)) + (d*e^(2*I*d*x + 2*I*c) + d)*sqrt(-8*I*a/d^2)*log((sqrt(2)*(I*d*e^( 
2*I*d*x + 2*I*c) - I*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(-8*I*a/d^2) + 4*I*a*e^(I* 
d*x + I*c))*e^(-I*d*x - I*c)) - (d*e^(2*I*d*x + 2*I*c) + d)*sqrt(-8*I*a/d^ 
2)*log((sqrt(2)*(-I*d*e^(2*I*d*x + 2*I*c) + I*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( 
-8*I*a/d^2) + 4*I*a*e^(I*d*x + I*c))*e^(-I*d*x - I*c)) - (d*e^(2*I*d*x + 2 
*I*c) + d)*sqrt(-I*a/d^2)*log(-16*(2*sqrt(2)*(a*d*e^(3*I*d*x + 3*I*c) - a* 
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(-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)) + (d*e^(2*I*d*x + 2*I*c) + d)*sqrt(-I 
*a/d^2)*log(16*(2*sqrt(2)*(a*d*e^(3*I*d*x + 3*I*c) - a*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(-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)))/(d*e^(2*I*d*x + 2*I*c) + d)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F(-2)]

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

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

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

Output:

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

Reduce [F]

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

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

Output:

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