\(\int \frac {\cot ^{\frac {3}{2}}(c+d x)}{(a+i a \tan (c+d x))^{5/2}} \, dx\) [782]

Optimal result

Integrand size = 28, antiderivative size = 219 \[ \int \frac {\cot ^{\frac {3}{2}}(c+d x)}{(a+i a \tan (c+d x))^{5/2}} \, dx=\frac {\left (\frac {1}{8}+\frac {i}{8}\right ) \text {arctanh}\left (\frac {(1+i) \sqrt {a} \sqrt {\tan (c+d x)}}{\sqrt {a+i a \tan (c+d x)}}\right ) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}}{a^{5/2} d}+\frac {\sqrt {\cot (c+d x)}}{5 d (a+i a \tan (c+d x))^{5/2}}+\frac {17 \sqrt {\cot (c+d x)}}{30 a d (a+i a \tan (c+d x))^{3/2}}+\frac {151 \sqrt {\cot (c+d x)}}{60 a^2 d \sqrt {a+i a \tan (c+d x)}}-\frac {317 \sqrt {\cot (c+d x)} \sqrt {a+i a \tan (c+d x)}}{60 a^3 d} \] Output:

(1/8+1/8*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^(5/2)/d+1/5*cot(d*x+c)^(1/2)/d/(a+I 
*a*tan(d*x+c))^(5/2)+17/30*cot(d*x+c)^(1/2)/a/d/(a+I*a*tan(d*x+c))^(3/2)+1 
51/60*cot(d*x+c)^(1/2)/a^2/d/(a+I*a*tan(d*x+c))^(1/2)-317/60*cot(d*x+c)^(1 
/2)*(a+I*a*tan(d*x+c))^(1/2)/a^3/d
 

Mathematica [A] (verified)

Time = 1.14 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.68 \[ \int \frac {\cot ^{\frac {3}{2}}(c+d x)}{(a+i a \tan (c+d x))^{5/2}} \, dx=\frac {\frac {15 i \sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {i a \tan (c+d x)}}{\sqrt {a+i a \tan (c+d x)}}\right )}{\sqrt {i a \tan (c+d x)}}+\frac {634 i+1600 \cot (c+d x)-1230 i \cot ^2(c+d x)-240 \cot ^3(c+d x)}{(i+\cot (c+d x))^2 \sqrt {a+i a \tan (c+d x)}}}{120 a^2 d \sqrt {\cot (c+d x)}} \] Input:

Integrate[Cot[c + d*x]^(3/2)/(a + I*a*Tan[c + d*x])^(5/2),x]
 

Output:

(((15*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]] + (634*I + 1600*Cot[c + d*x] - (1230*I 
)*Cot[c + d*x]^2 - 240*Cot[c + d*x]^3)/((I + Cot[c + d*x])^2*Sqrt[a + I*a* 
Tan[c + d*x]]))/(120*a^2*d*Sqrt[Cot[c + d*x]])
 

Rubi [A] (verified)

Time = 1.32 (sec) , antiderivative size = 234, normalized size of antiderivative = 1.07, number of steps used = 18, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.607, Rules used = {3042, 4729, 3042, 4042, 27, 3042, 4079, 27, 3042, 4079, 27, 3042, 4081, 27, 3042, 4027, 218}

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.

\(\displaystyle \int \frac {\cot ^{\frac {3}{2}}(c+d x)}{(a+i a \tan (c+d x))^{5/2}} \, dx\)

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4729

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4042

\(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {\int \frac {11 a-6 i a \tan (c+d x)}{2 \tan ^{\frac {3}{2}}(c+d x) (i \tan (c+d x) a+a)^{3/2}}dx}{5 a^2}+\frac {1}{5 d \sqrt {\tan (c+d x)} (a+i a \tan (c+d x))^{5/2}}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {\int \frac {11 a-6 i a \tan (c+d x)}{\tan ^{\frac {3}{2}}(c+d x) (i \tan (c+d x) a+a)^{3/2}}dx}{10 a^2}+\frac {1}{5 d \sqrt {\tan (c+d x)} (a+i a \tan (c+d x))^{5/2}}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {\int \frac {11 a-6 i a \tan (c+d x)}{\tan (c+d x)^{3/2} (i \tan (c+d x) a+a)^{3/2}}dx}{10 a^2}+\frac {1}{5 d \sqrt {\tan (c+d x)} (a+i a \tan (c+d x))^{5/2}}\right )\)

\(\Big \downarrow \) 4079

\(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {\frac {\int \frac {83 a^2-68 i a^2 \tan (c+d x)}{2 \tan ^{\frac {3}{2}}(c+d x) \sqrt {i \tan (c+d x) a+a}}dx}{3 a^2}+\frac {17 a}{3 d \sqrt {\tan (c+d x)} (a+i a \tan (c+d x))^{3/2}}}{10 a^2}+\frac {1}{5 d \sqrt {\tan (c+d x)} (a+i a \tan (c+d x))^{5/2}}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {\frac {\int \frac {83 a^2-68 i a^2 \tan (c+d x)}{\tan ^{\frac {3}{2}}(c+d x) \sqrt {i \tan (c+d x) a+a}}dx}{6 a^2}+\frac {17 a}{3 d \sqrt {\tan (c+d x)} (a+i a \tan (c+d x))^{3/2}}}{10 a^2}+\frac {1}{5 d \sqrt {\tan (c+d x)} (a+i a \tan (c+d x))^{5/2}}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {\frac {\int \frac {83 a^2-68 i a^2 \tan (c+d x)}{\tan (c+d x)^{3/2} \sqrt {i \tan (c+d x) a+a}}dx}{6 a^2}+\frac {17 a}{3 d \sqrt {\tan (c+d x)} (a+i a \tan (c+d x))^{3/2}}}{10 a^2}+\frac {1}{5 d \sqrt {\tan (c+d x)} (a+i a \tan (c+d x))^{5/2}}\right )\)

\(\Big \downarrow \) 4079

\(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {\frac {\frac {\int \frac {\sqrt {i \tan (c+d x) a+a} \left (317 a^3-302 i a^3 \tan (c+d x)\right )}{2 \tan ^{\frac {3}{2}}(c+d x)}dx}{a^2}+\frac {151 a^2}{d \sqrt {\tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}}{6 a^2}+\frac {17 a}{3 d \sqrt {\tan (c+d x)} (a+i a \tan (c+d x))^{3/2}}}{10 a^2}+\frac {1}{5 d \sqrt {\tan (c+d x)} (a+i a \tan (c+d x))^{5/2}}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {\frac {\frac {\int \frac {\sqrt {i \tan (c+d x) a+a} \left (317 a^3-302 i a^3 \tan (c+d x)\right )}{\tan ^{\frac {3}{2}}(c+d x)}dx}{2 a^2}+\frac {151 a^2}{d \sqrt {\tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}}{6 a^2}+\frac {17 a}{3 d \sqrt {\tan (c+d x)} (a+i a \tan (c+d x))^{3/2}}}{10 a^2}+\frac {1}{5 d \sqrt {\tan (c+d x)} (a+i a \tan (c+d x))^{5/2}}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {\frac {\frac {\int \frac {\sqrt {i \tan (c+d x) a+a} \left (317 a^3-302 i a^3 \tan (c+d x)\right )}{\tan (c+d x)^{3/2}}dx}{2 a^2}+\frac {151 a^2}{d \sqrt {\tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}}{6 a^2}+\frac {17 a}{3 d \sqrt {\tan (c+d x)} (a+i a \tan (c+d x))^{3/2}}}{10 a^2}+\frac {1}{5 d \sqrt {\tan (c+d x)} (a+i a \tan (c+d x))^{5/2}}\right )\)

\(\Big \downarrow \) 4081

\(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {\frac {\frac {\frac {2 \int \frac {15 i a^4 \sqrt {i \tan (c+d x) a+a}}{2 \sqrt {\tan (c+d x)}}dx}{a}-\frac {634 a^3 \sqrt {a+i a \tan (c+d x)}}{d \sqrt {\tan (c+d x)}}}{2 a^2}+\frac {151 a^2}{d \sqrt {\tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}}{6 a^2}+\frac {17 a}{3 d \sqrt {\tan (c+d x)} (a+i a \tan (c+d x))^{3/2}}}{10 a^2}+\frac {1}{5 d \sqrt {\tan (c+d x)} (a+i a \tan (c+d x))^{5/2}}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {\frac {\frac {15 i a^3 \int \frac {\sqrt {i \tan (c+d x) a+a}}{\sqrt {\tan (c+d x)}}dx-\frac {634 a^3 \sqrt {a+i a \tan (c+d x)}}{d \sqrt {\tan (c+d x)}}}{2 a^2}+\frac {151 a^2}{d \sqrt {\tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}}{6 a^2}+\frac {17 a}{3 d \sqrt {\tan (c+d x)} (a+i a \tan (c+d x))^{3/2}}}{10 a^2}+\frac {1}{5 d \sqrt {\tan (c+d x)} (a+i a \tan (c+d x))^{5/2}}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {\frac {\frac {15 i a^3 \int \frac {\sqrt {i \tan (c+d x) a+a}}{\sqrt {\tan (c+d x)}}dx-\frac {634 a^3 \sqrt {a+i a \tan (c+d x)}}{d \sqrt {\tan (c+d x)}}}{2 a^2}+\frac {151 a^2}{d \sqrt {\tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}}{6 a^2}+\frac {17 a}{3 d \sqrt {\tan (c+d x)} (a+i a \tan (c+d x))^{3/2}}}{10 a^2}+\frac {1}{5 d \sqrt {\tan (c+d x)} (a+i a \tan (c+d x))^{5/2}}\right )\)

\(\Big \downarrow \) 4027

\(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {\frac {\frac {\frac {30 a^5 \int \frac {1}{-\frac {2 \tan (c+d x) a^2}{i \tan (c+d x) a+a}-i a}d\frac {\sqrt {\tan (c+d x)}}{\sqrt {i \tan (c+d x) a+a}}}{d}-\frac {634 a^3 \sqrt {a+i a \tan (c+d x)}}{d \sqrt {\tan (c+d x)}}}{2 a^2}+\frac {151 a^2}{d \sqrt {\tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}}{6 a^2}+\frac {17 a}{3 d \sqrt {\tan (c+d x)} (a+i a \tan (c+d x))^{3/2}}}{10 a^2}+\frac {1}{5 d \sqrt {\tan (c+d x)} (a+i a \tan (c+d x))^{5/2}}\right )\)

\(\Big \downarrow \) 218

\(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {\frac {\frac {151 a^2}{d \sqrt {\tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}+\frac {\frac {(15+15 i) a^{7/2} \text {arctanh}\left (\frac {(1+i) \sqrt {a} \sqrt {\tan (c+d x)}}{\sqrt {a+i a \tan (c+d x)}}\right )}{d}-\frac {634 a^3 \sqrt {a+i a \tan (c+d x)}}{d \sqrt {\tan (c+d x)}}}{2 a^2}}{6 a^2}+\frac {17 a}{3 d \sqrt {\tan (c+d x)} (a+i a \tan (c+d x))^{3/2}}}{10 a^2}+\frac {1}{5 d \sqrt {\tan (c+d x)} (a+i a \tan (c+d x))^{5/2}}\right )\)

Input:

Int[Cot[c + d*x]^(3/2)/(a + I*a*Tan[c + d*x])^(5/2),x]
 

Output:

Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]]*(1/(5*d*Sqrt[Tan[c + d*x]]*(a + I*a* 
Tan[c + d*x])^(5/2)) + ((17*a)/(3*d*Sqrt[Tan[c + d*x]]*(a + I*a*Tan[c + d* 
x])^(3/2)) + ((151*a^2)/(d*Sqrt[Tan[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]]) 
+ (((15 + 15*I)*a^(7/2)*ArcTanh[((1 + I)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[ 
a + I*a*Tan[c + d*x]]])/d - (634*a^3*Sqrt[a + I*a*Tan[c + d*x]])/(d*Sqrt[T 
an[c + d*x]]))/(2*a^2))/(6*a^2))/(10*a^2))
 

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 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 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 4081

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

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] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 627 vs. \(2 (174 ) = 348\).

Time = 0.44 (sec) , antiderivative size = 628, normalized size of antiderivative = 2.87

method	result	size
derivativedivides	\(-\frac {\left (\frac {1}{\tan \left (d x +c \right )}\right )^{\frac {3}{2}} \tan \left (d x +c \right ) \sqrt {a \left (1+i \tan \left (d x +c \right )\right )}\, \left (1268 \sqrt {-i a}\, \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}\, \tan \left (d x +c \right )^{4}+60 i \sqrt {2}\, \ln \left (-\frac {-2 \sqrt {2}\, \sqrt {-i a}\, \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}+i a -3 a \tan \left (d x +c \right )}{\tan \left (d x +c \right )+i}\right ) a \tan \left (d x +c \right )^{4}-15 \sqrt {2}\, \ln \left (-\frac {-2 \sqrt {2}\, \sqrt {-i a}\, \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}+i a -3 a \tan \left (d x +c \right )}{\tan \left (d x +c \right )+i}\right ) a \tan \left (d x +c \right )^{5}-5660 \sqrt {-i a}\, \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}\, \tan \left (d x +c \right )^{2}-60 i \sqrt {2}\, \ln \left (-\frac {-2 \sqrt {2}\, \sqrt {-i a}\, \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}+i a -3 a \tan \left (d x +c \right )}{\tan \left (d x +c \right )+i}\right ) a \tan \left (d x +c \right )^{2}-4468 i \sqrt {-i a}\, \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}\, \tan \left (d x +c \right )^{3}+90 \sqrt {2}\, \ln \left (-\frac {-2 \sqrt {2}\, \sqrt {-i a}\, \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}+i a -3 a \tan \left (d x +c \right )}{\tan \left (d x +c \right )+i}\right ) a \tan \left (d x +c \right )^{3}+2940 i \sqrt {-i a}\, \tan \left (d x +c \right ) \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}-15 \sqrt {2}\, \ln \left (-\frac {-2 \sqrt {2}\, \sqrt {-i a}\, \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}+i a -3 a \tan \left (d x +c \right )}{\tan \left (d x +c \right )+i}\right ) a \tan \left (d x +c \right )+480 \sqrt {-i a}\, \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}\right )}{240 d \,a^{3} \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}\, \left (-\tan \left (d x +c \right )+i\right )^{4} \sqrt {-i a}}\)	\(628\)
default	\(-\frac {\left (\frac {1}{\tan \left (d x +c \right )}\right )^{\frac {3}{2}} \tan \left (d x +c \right ) \sqrt {a \left (1+i \tan \left (d x +c \right )\right )}\, \left (1268 \sqrt {-i a}\, \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}\, \tan \left (d x +c \right )^{4}+60 i \sqrt {2}\, \ln \left (-\frac {-2 \sqrt {2}\, \sqrt {-i a}\, \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}+i a -3 a \tan \left (d x +c \right )}{\tan \left (d x +c \right )+i}\right ) a \tan \left (d x +c \right )^{4}-15 \sqrt {2}\, \ln \left (-\frac {-2 \sqrt {2}\, \sqrt {-i a}\, \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}+i a -3 a \tan \left (d x +c \right )}{\tan \left (d x +c \right )+i}\right ) a \tan \left (d x +c \right )^{5}-5660 \sqrt {-i a}\, \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}\, \tan \left (d x +c \right )^{2}-60 i \sqrt {2}\, \ln \left (-\frac {-2 \sqrt {2}\, \sqrt {-i a}\, \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}+i a -3 a \tan \left (d x +c \right )}{\tan \left (d x +c \right )+i}\right ) a \tan \left (d x +c \right )^{2}-4468 i \sqrt {-i a}\, \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}\, \tan \left (d x +c \right )^{3}+90 \sqrt {2}\, \ln \left (-\frac {-2 \sqrt {2}\, \sqrt {-i a}\, \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}+i a -3 a \tan \left (d x +c \right )}{\tan \left (d x +c \right )+i}\right ) a \tan \left (d x +c \right )^{3}+2940 i \sqrt {-i a}\, \tan \left (d x +c \right ) \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}-15 \sqrt {2}\, \ln \left (-\frac {-2 \sqrt {2}\, \sqrt {-i a}\, \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}+i a -3 a \tan \left (d x +c \right )}{\tan \left (d x +c \right )+i}\right ) a \tan \left (d x +c \right )+480 \sqrt {-i a}\, \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}\right )}{240 d \,a^{3} \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}\, \left (-\tan \left (d x +c \right )+i\right )^{4} \sqrt {-i a}}\)	\(628\)

Input:

int(cot(d*x+c)^(3/2)/(a+I*a*tan(d*x+c))^(5/2),x,method=_RETURNVERBOSE)
 

Output:

-1/240/d*(1/tan(d*x+c))^(3/2)*tan(d*x+c)*(a*(1+I*tan(d*x+c)))^(1/2)/a^3*(1 
268*(-I*a)^(1/2)*(a*tan(d*x+c)*(1+I*tan(d*x+c)))^(1/2)*tan(d*x+c)^4+60*I*2 
^(1/2)*ln(-(-2*2^(1/2)*(-I*a)^(1/2)*(a*tan(d*x+c)*(1+I*tan(d*x+c)))^(1/2)+ 
I*a-3*a*tan(d*x+c))/(tan(d*x+c)+I))*a*tan(d*x+c)^4-15*2^(1/2)*ln(-(-2*2^(1 
/2)*(-I*a)^(1/2)*(a*tan(d*x+c)*(1+I*tan(d*x+c)))^(1/2)+I*a-3*a*tan(d*x+c)) 
/(tan(d*x+c)+I))*a*tan(d*x+c)^5-5660*(-I*a)^(1/2)*(a*tan(d*x+c)*(1+I*tan(d 
*x+c)))^(1/2)*tan(d*x+c)^2-60*I*2^(1/2)*ln(-(-2*2^(1/2)*(-I*a)^(1/2)*(a*ta 
n(d*x+c)*(1+I*tan(d*x+c)))^(1/2)+I*a-3*a*tan(d*x+c))/(tan(d*x+c)+I))*a*tan 
(d*x+c)^2-4468*I*(-I*a)^(1/2)*(a*tan(d*x+c)*(1+I*tan(d*x+c)))^(1/2)*tan(d* 
x+c)^3+90*2^(1/2)*ln(-(-2*2^(1/2)*(-I*a)^(1/2)*(a*tan(d*x+c)*(1+I*tan(d*x+ 
c)))^(1/2)+I*a-3*a*tan(d*x+c))/(tan(d*x+c)+I))*a*tan(d*x+c)^3+2940*I*(-I*a 
)^(1/2)*(a*tan(d*x+c)*(1+I*tan(d*x+c)))^(1/2)*tan(d*x+c)-15*2^(1/2)*ln(-(- 
2*2^(1/2)*(-I*a)^(1/2)*(a*tan(d*x+c)*(1+I*tan(d*x+c)))^(1/2)+I*a-3*a*tan(d 
*x+c))/(tan(d*x+c)+I))*a*tan(d*x+c)+480*(-I*a)^(1/2)*(a*tan(d*x+c)*(1+I*ta 
n(d*x+c)))^(1/2))/(a*tan(d*x+c)*(1+I*tan(d*x+c)))^(1/2)/(-tan(d*x+c)+I)^4/ 
(-I*a)^(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. 368 vs. \(2 (165) = 330\).

Time = 0.09 (sec) , antiderivative size = 368, normalized size of antiderivative = 1.68 \[ \int \frac {\cot ^{\frac {3}{2}}(c+d x)}{(a+i a \tan (c+d x))^{5/2}} \, dx=\frac {{\left (30 \, a^{3} d \sqrt {\frac {i}{8 \, a^{5} d^{2}}} e^{\left (5 i \, d x + 5 i \, c\right )} \log \left (4 \, {\left (2 \, \sqrt {2} {\left (a^{3} d e^{\left (2 i \, d x + 2 i \, c\right )} - a^{3} d\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {\frac {i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} - 1}} \sqrt {\frac {i}{8 \, a^{5} d^{2}}} + i \, a e^{\left (i \, d x + i \, c\right )}\right )} e^{\left (-i \, d x - i \, c\right )}\right ) - 30 \, a^{3} d \sqrt {\frac {i}{8 \, a^{5} d^{2}}} e^{\left (5 i \, d x + 5 i \, c\right )} \log \left (-4 \, {\left (2 \, \sqrt {2} {\left (a^{3} d e^{\left (2 i \, d x + 2 i \, c\right )} - a^{3} d\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {\frac {i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} - 1}} \sqrt {\frac {i}{8 \, a^{5} d^{2}}} - i \, a e^{\left (i \, d x + i \, c\right )}\right )} e^{\left (-i \, d x - i \, c\right )}\right ) - \sqrt {2} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {\frac {i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} - 1}} {\left (463 \, e^{\left (6 i \, d x + 6 i \, c\right )} - 194 \, e^{\left (4 i \, d x + 4 i \, c\right )} - 26 \, e^{\left (2 i \, d x + 2 i \, c\right )} - 3\right )}\right )} e^{\left (-5 i \, d x - 5 i \, c\right )}}{120 \, a^{3} d} \] Input:

integrate(cot(d*x+c)^(3/2)/(a+I*a*tan(d*x+c))^(5/2),x, algorithm="fricas")
 

Output:

1/120*(30*a^3*d*sqrt(1/8*I/(a^5*d^2))*e^(5*I*d*x + 5*I*c)*log(4*(2*sqrt(2) 
*(a^3*d*e^(2*I*d*x + 2*I*c) - a^3*d)*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))*sqrt(1/8*I/(a^5*d 
^2)) + I*a*e^(I*d*x + I*c))*e^(-I*d*x - I*c)) - 30*a^3*d*sqrt(1/8*I/(a^5*d 
^2))*e^(5*I*d*x + 5*I*c)*log(-4*(2*sqrt(2)*(a^3*d*e^(2*I*d*x + 2*I*c) - a^ 
3*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(1/8*I/(a^5*d^2)) - I*a*e^(I*d*x + I*c))*e^(- 
I*d*x - I*c)) - 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))*(463*e^(6*I*d*x + 6*I*c) - 194 
*e^(4*I*d*x + 4*I*c) - 26*e^(2*I*d*x + 2*I*c) - 3))*e^(-5*I*d*x - 5*I*c)/( 
a^3*d)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\cot ^{\frac {3}{2}}(c+d x)}{(a+i a \tan (c+d x))^{5/2}} \, dx=\text {Timed out} \] Input:

integrate(cot(d*x+c)**(3/2)/(a+I*a*tan(d*x+c))**(5/2),x)
 

Output:

Timed out
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {\cot ^{\frac {3}{2}}(c+d x)}{(a+i a \tan (c+d x))^{5/2}} \, dx=\text {Exception raised: RuntimeError} \] Input:

integrate(cot(d*x+c)^(3/2)/(a+I*a*tan(d*x+c))^(5/2),x, algorithm="maxima")
 

Output:

Exception raised: RuntimeError >> ECL says: THROW: The catch RAT-ERR is un 
defined.
 

Giac [F(-2)]

Exception generated. \[ \int \frac {\cot ^{\frac {3}{2}}(c+d x)}{(a+i a \tan (c+d x))^{5/2}} \, dx=\text {Exception raised: TypeError} \] Input:

integrate(cot(d*x+c)^(3/2)/(a+I*a*tan(d*x+c))^(5/2),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. \[ \int \frac {\cot ^{\frac {3}{2}}(c+d x)}{(a+i a \tan (c+d x))^{5/2}} \, dx=\int \frac {{\mathrm {cot}\left (c+d\,x\right )}^{3/2}}{{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{5/2}} \,d x \] Input:

int(cot(c + d*x)^(3/2)/(a + a*tan(c + d*x)*1i)^(5/2),x)
 

Output:

int(cot(c + d*x)^(3/2)/(a + a*tan(c + d*x)*1i)^(5/2), x)
 

Reduce [F]

\[ \int \frac {\cot ^{\frac {3}{2}}(c+d x)}{(a+i a \tan (c+d x))^{5/2}} \, dx=-\frac {\int \frac {\sqrt {\cot \left (d x +c \right )}\, \cot \left (d x +c \right )}{\sqrt {\tan \left (d x +c \right ) i +1}\, \tan \left (d x +c \right )^{2}-2 \sqrt {\tan \left (d x +c \right ) i +1}\, \tan \left (d x +c \right ) i -\sqrt {\tan \left (d x +c \right ) i +1}}d x}{\sqrt {a}\, a^{2}} \] Input:

int(cot(d*x+c)^(3/2)/(a+I*a*tan(d*x+c))^(5/2),x)
 

Output:

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