3.2.23 \(\int \frac {\cot (c+d x)}{(a+i a \tan (c+d x))^{3/2}} \, dx\) [123]

3.2.23.1 Optimal result

Integrand size = 24, antiderivative size = 132 \[ \int \frac {\cot (c+d x)}{(a+i a \tan (c+d x))^{3/2}} \, dx=-\frac {2 \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {a}}\right )}{a^{3/2} d}+\frac {\text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{2 \sqrt {2} a^{3/2} d}+\frac {1}{3 d (a+i a \tan (c+d x))^{3/2}}+\frac {3}{2 a d \sqrt {a+i a \tan (c+d x)}} \]

output

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

3.2.23.2 Mathematica [A] (verified)

Time = 1.12 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.92 \[ \int \frac {\cot (c+d x)}{(a+i a \tan (c+d x))^{3/2}} \, dx=\frac {-\frac {24 \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {a}}\right )}{a^{3/2}}+\frac {3 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{a^{3/2}}+\frac {4}{(a+i a \tan (c+d x))^{3/2}}+\frac {18}{a \sqrt {a+i a \tan (c+d x)}}}{12 d} \]

input

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

output

((-24*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/Sqrt[a]])/a^(3/2) + (3*Sqrt[2]*Ar 
cTanh[Sqrt[a + I*a*Tan[c + d*x]]/(Sqrt[2]*Sqrt[a])])/a^(3/2) + 4/(a + I*a* 
Tan[c + d*x])^(3/2) + 18/(a*Sqrt[a + I*a*Tan[c + d*x]]))/(12*d)
 

3.2.23.3 Rubi [A] (verified)

Time = 0.93 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.07, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.583, Rules used = {3042, 4042, 27, 3042, 4079, 27, 3042, 4083, 3042, 3961, 219, 4082, 73, 221}

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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4042

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4079

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4083

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3961

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

\(\Big \downarrow \) 219

\(\displaystyle \frac {\frac {4 a \int \frac {(a-i a \tan (c+d x)) \sqrt {i \tan (c+d x) a+a}}{\tan (c+d x)}dx+\frac {\sqrt {2} a^{5/2} \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{d}}{2 a^2}+\frac {3 a}{d \sqrt {a+i a \tan (c+d x)}}}{2 a^2}+\frac {1}{3 d (a+i a \tan (c+d x))^{3/2}}\)

\(\Big \downarrow \) 4082

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

\(\Big \downarrow \) 73

\(\displaystyle \frac {\frac {\frac {\sqrt {2} a^{5/2} \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{d}-\frac {8 i a^2 \int \frac {1}{i-\frac {i (i \tan (c+d x) a+a)}{a}}d\sqrt {i \tan (c+d x) a+a}}{d}}{2 a^2}+\frac {3 a}{d \sqrt {a+i a \tan (c+d x)}}}{2 a^2}+\frac {1}{3 d (a+i a \tan (c+d x))^{3/2}}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {\frac {\frac {\sqrt {2} a^{5/2} \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{d}-\frac {8 a^{5/2} \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {a}}\right )}{d}}{2 a^2}+\frac {3 a}{d \sqrt {a+i a \tan (c+d x)}}}{2 a^2}+\frac {1}{3 d (a+i a \tan (c+d x))^{3/2}}\)

input

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

output

1/(3*d*(a + I*a*Tan[c + d*x])^(3/2)) + (((-8*a^(5/2)*ArcTanh[Sqrt[a + I*a* 
Tan[c + d*x]]/Sqrt[a]])/d + (Sqrt[2]*a^(5/2)*ArcTanh[Sqrt[a + I*a*Tan[c + 
d*x]]/(Sqrt[2]*Sqrt[a])])/d)/(2*a^2) + (3*a)/(d*Sqrt[a + I*a*Tan[c + d*x]] 
))/(2*a^2)
 

3.2.23.3.1 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 73

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ 
{p = Denominator[m]}, Simp[p/b   Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + 
 d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt 
Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL 
inearQ[a, b, c, d, m, n, x]
 

rule 219

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

rule 221

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

rule 3042

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

rule 3961

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

rule 4042

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

rule 4079

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*A + b*B)*(a + b*Tan[e + f*x])^m*((c + d*Tan[e + f*x])^(n + 1)/(2*f*m*( 
b*c - a*d))), x] + Simp[1/(2*a*m*(b*c - a*d))   Int[(a + b*Tan[e + f*x])^(m 
 + 1)*(c + d*Tan[e + f*x])^n*Simp[A*(b*c*m - a*d*(2*m + n + 1)) + B*(a*c*m 
- b*d*(n + 1)) + d*(A*b - a*B)*(m + n + 1)*Tan[e + f*x], x], x], x] /; Free 
Q[{a, b, c, d, e, f, A, B, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] 
 && LtQ[m, 0] &&  !GtQ[n, 0]
 

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 4083

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

3.2.23.4 Maple [A] (verified)

Time = 1.17 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.78

method	result	size
derivativedivides	\(\frac {2 a^{2} \left (-\frac {\operatorname {arctanh}\left (\frac {\sqrt {a +i a \tan \left (d x +c \right )}}{\sqrt {a}}\right )}{a^{\frac {7}{2}}}+\frac {\sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a +i a \tan \left (d x +c \right )}\, \sqrt {2}}{2 \sqrt {a}}\right )}{8 a^{\frac {7}{2}}}+\frac {3}{4 a^{3} \sqrt {a +i a \tan \left (d x +c \right )}}+\frac {1}{6 a^{2} \left (a +i a \tan \left (d x +c \right )\right )^{\frac {3}{2}}}\right )}{d}\)	\(103\)
default	\(\frac {2 a^{2} \left (-\frac {\operatorname {arctanh}\left (\frac {\sqrt {a +i a \tan \left (d x +c \right )}}{\sqrt {a}}\right )}{a^{\frac {7}{2}}}+\frac {\sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a +i a \tan \left (d x +c \right )}\, \sqrt {2}}{2 \sqrt {a}}\right )}{8 a^{\frac {7}{2}}}+\frac {3}{4 a^{3} \sqrt {a +i a \tan \left (d x +c \right )}}+\frac {1}{6 a^{2} \left (a +i a \tan \left (d x +c \right )\right )^{\frac {3}{2}}}\right )}{d}\)	\(103\)

input

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

output

2/d*a^2*(-1/a^(7/2)*arctanh((a+I*a*tan(d*x+c))^(1/2)/a^(1/2))+1/8/a^(7/2)* 
2^(1/2)*arctanh(1/2*(a+I*a*tan(d*x+c))^(1/2)*2^(1/2)/a^(1/2))+3/4/a^3/(a+I 
*a*tan(d*x+c))^(1/2)+1/6/a^2/(a+I*a*tan(d*x+c))^(3/2))
 

3.2.23.5 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. 500 vs. \(2 (99) = 198\).

Time = 0.25 (sec) , antiderivative size = 500, normalized size of antiderivative = 3.79 \[ \int \frac {\cot (c+d x)}{(a+i a \tan (c+d x))^{3/2}} \, dx=\frac {{\left (3 \, \sqrt {\frac {1}{2}} a^{2} d \sqrt {\frac {1}{a^{3} d^{2}}} e^{\left (3 i \, d x + 3 i \, c\right )} \log \left (4 \, {\left (\sqrt {2} \sqrt {\frac {1}{2}} {\left (a^{2} d e^{\left (2 i \, d x + 2 i \, c\right )} + a^{2} d\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {\frac {1}{a^{3} d^{2}}} + a e^{\left (i \, d x + i \, c\right )}\right )} e^{\left (-i \, d x - i \, c\right )}\right ) - 3 \, \sqrt {\frac {1}{2}} a^{2} d \sqrt {\frac {1}{a^{3} d^{2}}} e^{\left (3 i \, d x + 3 i \, c\right )} \log \left (-4 \, {\left (\sqrt {2} \sqrt {\frac {1}{2}} {\left (a^{2} d e^{\left (2 i \, d x + 2 i \, c\right )} + a^{2} d\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {\frac {1}{a^{3} d^{2}}} - a e^{\left (i \, d x + i \, c\right )}\right )} e^{\left (-i \, d x - i \, c\right )}\right ) - 6 \, a^{2} d \sqrt {\frac {1}{a^{3} d^{2}}} e^{\left (3 i \, d x + 3 i \, c\right )} \log \left (16 \, {\left (3 \, a^{2} e^{\left (2 i \, d x + 2 i \, c\right )} + 2 \, \sqrt {2} {\left (a^{3} d e^{\left (3 i \, d x + 3 i \, c\right )} + a^{3} d e^{\left (i \, d x + i \, c\right )}\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {\frac {1}{a^{3} d^{2}}} + a^{2}\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}\right ) + 6 \, a^{2} d \sqrt {\frac {1}{a^{3} d^{2}}} e^{\left (3 i \, d x + 3 i \, c\right )} \log \left (16 \, {\left (3 \, a^{2} e^{\left (2 i \, d x + 2 i \, c\right )} - 2 \, \sqrt {2} {\left (a^{3} d e^{\left (3 i \, d x + 3 i \, c\right )} + a^{3} d e^{\left (i \, d x + i \, c\right )}\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {\frac {1}{a^{3} d^{2}}} + a^{2}\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}\right ) + \sqrt {2} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} {\left (10 \, e^{\left (4 i \, d x + 4 i \, c\right )} + 11 \, e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right )}\right )} e^{\left (-3 i \, d x - 3 i \, c\right )}}{12 \, a^{2} d} \]

input

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

output

1/12*(3*sqrt(1/2)*a^2*d*sqrt(1/(a^3*d^2))*e^(3*I*d*x + 3*I*c)*log(4*(sqrt( 
2)*sqrt(1/2)*(a^2*d*e^(2*I*d*x + 2*I*c) + a^2*d)*sqrt(a/(e^(2*I*d*x + 2*I* 
c) + 1))*sqrt(1/(a^3*d^2)) + a*e^(I*d*x + I*c))*e^(-I*d*x - I*c)) - 3*sqrt 
(1/2)*a^2*d*sqrt(1/(a^3*d^2))*e^(3*I*d*x + 3*I*c)*log(-4*(sqrt(2)*sqrt(1/2 
)*(a^2*d*e^(2*I*d*x + 2*I*c) + a^2*d)*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1))*sq 
rt(1/(a^3*d^2)) - a*e^(I*d*x + I*c))*e^(-I*d*x - I*c)) - 6*a^2*d*sqrt(1/(a 
^3*d^2))*e^(3*I*d*x + 3*I*c)*log(16*(3*a^2*e^(2*I*d*x + 2*I*c) + 2*sqrt(2) 
*(a^3*d*e^(3*I*d*x + 3*I*c) + a^3*d*e^(I*d*x + I*c))*sqrt(a/(e^(2*I*d*x + 
2*I*c) + 1))*sqrt(1/(a^3*d^2)) + a^2)*e^(-2*I*d*x - 2*I*c)) + 6*a^2*d*sqrt 
(1/(a^3*d^2))*e^(3*I*d*x + 3*I*c)*log(16*(3*a^2*e^(2*I*d*x + 2*I*c) - 2*sq 
rt(2)*(a^3*d*e^(3*I*d*x + 3*I*c) + a^3*d*e^(I*d*x + I*c))*sqrt(a/(e^(2*I*d 
*x + 2*I*c) + 1))*sqrt(1/(a^3*d^2)) + a^2)*e^(-2*I*d*x - 2*I*c)) + sqrt(2) 
*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1))*(10*e^(4*I*d*x + 4*I*c) + 11*e^(2*I*d*x 
 + 2*I*c) + 1))*e^(-3*I*d*x - 3*I*c)/(a^2*d)
 

3.2.23.6 Sympy [F]

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

input

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

output

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

3.2.23.7 Maxima [A] (verification not implemented)

Time = 0.36 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.05 \[ \int \frac {\cot (c+d x)}{(a+i a \tan (c+d x))^{3/2}} \, dx=-\frac {\frac {3 \, \sqrt {2} \log \left (-\frac {\sqrt {2} \sqrt {a} - \sqrt {i \, a \tan \left (d x + c\right ) + a}}{\sqrt {2} \sqrt {a} + \sqrt {i \, a \tan \left (d x + c\right ) + a}}\right )}{a^{\frac {3}{2}}} - \frac {24 \, \log \left (\frac {\sqrt {i \, a \tan \left (d x + c\right ) + a} - \sqrt {a}}{\sqrt {i \, a \tan \left (d x + c\right ) + a} + \sqrt {a}}\right )}{a^{\frac {3}{2}}} - \frac {4 \, {\left (9 i \, a \tan \left (d x + c\right ) + 11 \, a\right )}}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {3}{2}} a}}{24 \, d} \]

input

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

output

-1/24*(3*sqrt(2)*log(-(sqrt(2)*sqrt(a) - sqrt(I*a*tan(d*x + c) + a))/(sqrt 
(2)*sqrt(a) + sqrt(I*a*tan(d*x + c) + a)))/a^(3/2) - 24*log((sqrt(I*a*tan( 
d*x + c) + a) - sqrt(a))/(sqrt(I*a*tan(d*x + c) + a) + sqrt(a)))/a^(3/2) - 
 4*(9*I*a*tan(d*x + c) + 11*a)/((I*a*tan(d*x + c) + a)^(3/2)*a))/d
 

3.2.23.8 Giac [F]

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

input

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

output

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

3.2.23.9 Mupad [B] (verification not implemented)

Time = 4.42 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.83 \[ \int \frac {\cot (c+d x)}{(a+i a \tan (c+d x))^{3/2}} \, dx=-\frac {2\,\mathrm {atanh}\left (\frac {a\,\sqrt {a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}}}{\sqrt {a^3}}\right )}{d\,\sqrt {a^3}}+\frac {\frac {3\,\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}{2\,a}+\frac {1}{3}}{d\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{3/2}}+\frac {\sqrt {2}\,\mathrm {atanh}\left (\frac {\sqrt {2}\,a\,\sqrt {a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}}}{2\,\sqrt {a^3}}\right )}{4\,d\,\sqrt {a^3}} \]

input

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

output

((3*(a + a*tan(c + d*x)*1i))/(2*a) + 1/3)/(d*(a + a*tan(c + d*x)*1i)^(3/2) 
) - (2*atanh((a*(a + a*tan(c + d*x)*1i)^(1/2))/(a^3)^(1/2)))/(d*(a^3)^(1/2 
)) + (2^(1/2)*atanh((2^(1/2)*a*(a + a*tan(c + d*x)*1i)^(1/2))/(2*(a^3)^(1/ 
2))))/(4*d*(a^3)^(1/2))