\(\int \cot ^{\frac {5}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)} \, dx\) [752]

Optimal result

Integrand size = 28, antiderivative size = 140 \[ \int \cot ^{\frac {5}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)} \, dx=-\frac {(1-i) \sqrt {a} \text {arctanh}\left (\frac {(1+i) \sqrt {a} \sqrt {\tan (c+d x)}}{\sqrt {a+i a \tan (c+d x)}}\right ) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}}{d}-\frac {2 i \sqrt {\cot (c+d x)} \sqrt {a+i a \tan (c+d x)}}{3 d}-\frac {2 \cot ^{\frac {3}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}}{3 d} \] Output:

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

Mathematica [A] (verified)

Time = 0.48 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.79 \[ \int \cot ^{\frac {5}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)} \, dx=\frac {\sqrt {\cot (c+d x)} \left (3 i \sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {i a \tan (c+d x)}}{\sqrt {a+i a \tan (c+d x)}}\right ) \sqrt {i a \tan (c+d x)}-2 (i+\cot (c+d x)) \sqrt {a+i a \tan (c+d x)}\right )}{3 d} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.71 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.06, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.393, Rules used = {3042, 4729, 3042, 4044, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4729

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4044

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4081

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4027

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

\(\Big \downarrow \) 218

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

Input:

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

Output:

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

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 4044

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*(n + 1)*(c^2 + d^2))), x] - Simp[1/(a*(c^2 + d^2)*(n + 
1))   Int[(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^(n + 1)*Simp[b*d*m - 
a*c*(n + 1) + a*d*(m + n + 1)*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] && LtQ[n, -1] && (IntegerQ[n] || IntegersQ[2*m, 2*n])
 

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] (warning: unable to verify)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 622 vs. \(2 (112 ) = 224\).

Time = 2.26 (sec) , antiderivative size = 623, normalized size of antiderivative = 4.45

method	result	size
default	\(-\frac {\cot \left (d x +c \right )^{\frac {5}{2}} \sqrt {a \left (1+i \tan \left (d x +c \right )\right )}\, \left (-2 \left (2 \cos \left (d x +c \right )+1\right ) \sqrt {2}\, \tan \left (d x +c \right )^{2}-i \sqrt {2}\, \left (-4 \sin \left (d x +c \right )-2 \tan \left (d x +c \right )+2 \sec \left (d x +c \right ) \tan \left (d x +c \right )\right )+i \tan \left (d x +c \right )^{2} \left (-3 \cos \left (d x +c \right )-3\right ) \sqrt {\cot \left (d x +c \right )-\csc \left (d x +c \right )}\, \ln \left (-\frac {\sqrt {\cot \left (d x +c \right )-\csc \left (d x +c \right )}\, \sqrt {2}\, \sin \left (d x +c \right )+\sin \left (d x +c \right )+\cos \left (d x +c \right )-1}{\sqrt {-2 \csc \left (d x +c \right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}\, \sqrt {2}\, \sin \left (d x +c \right )-\sin \left (d x +c \right )-\cos \left (d x +c \right )+1}\right )+i \tan \left (d x +c \right )^{2} \left (-6 \cos \left (d x +c \right )-6\right ) \sqrt {\cot \left (d x +c \right )-\csc \left (d x +c \right )}\, \arctan \left (\sqrt {\cot \left (d x +c \right )-\csc \left (d x +c \right )}\, \sqrt {2}-1\right )+i \tan \left (d x +c \right )^{2} \left (-6 \cos \left (d x +c \right )-6\right ) \sqrt {\cot \left (d x +c \right )-\csc \left (d x +c \right )}\, \arctan \left (\sqrt {\cot \left (d x +c \right )-\csc \left (d x +c \right )}\, \sqrt {2}+1\right )+\tan \left (d x +c \right )^{2} \left (-3 \cos \left (d x +c \right )-3\right ) \sqrt {\cot \left (d x +c \right )-\csc \left (d x +c \right )}\, \ln \left (-\frac {\sqrt {\cot \left (d x +c \right )-\csc \left (d x +c \right )}\, \sqrt {2}\, \sin \left (d x +c \right )-\sin \left (d x +c \right )-\cos \left (d x +c \right )+1}{\sqrt {-2 \csc \left (d x +c \right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}\, \sqrt {2}\, \sin \left (d x +c \right )+\sin \left (d x +c \right )+\cos \left (d x +c \right )-1}\right )+\tan \left (d x +c \right )^{2} \left (-6 \cos \left (d x +c \right )-6\right ) \sqrt {\cot \left (d x +c \right )-\csc \left (d x +c \right )}\, \arctan \left (\sqrt {\cot \left (d x +c \right )-\csc \left (d x +c \right )}\, \sqrt {2}-1\right )+\tan \left (d x +c \right )^{2} \left (-6 \cos \left (d x +c \right )-6\right ) \sqrt {\cot \left (d x +c \right )-\csc \left (d x +c \right )}\, \arctan \left (\sqrt {\cot \left (d x +c \right )-\csc \left (d x +c \right )}\, \sqrt {2}+1\right )\right ) \sqrt {2}}{6 d \left (i \cos \left (d x +c \right )+i-\sin \left (d x +c \right )\right )}\)	\(623\)

Input:

int(cot(d*x+c)^(5/2)*(a+I*a*tan(d*x+c))^(1/2),x,method=_RETURNVERBOSE)
 

Output:

-1/6/d*cot(d*x+c)^(5/2)*(a*(1+I*tan(d*x+c)))^(1/2)/(I*cos(d*x+c)+I-sin(d*x 
+c))*(-2*(2*cos(d*x+c)+1)*2^(1/2)*tan(d*x+c)^2-I*2^(1/2)*(-4*sin(d*x+c)-2* 
tan(d*x+c)+2*sec(d*x+c)*tan(d*x+c))+I*tan(d*x+c)^2*(-3*cos(d*x+c)-3)*(cot( 
d*x+c)-csc(d*x+c))^(1/2)*ln(-((cot(d*x+c)-csc(d*x+c))^(1/2)*2^(1/2)*sin(d* 
x+c)+sin(d*x+c)+cos(d*x+c)-1)/((-2*csc(d*x+c)*sin(1/2*d*x+1/2*c)^2)^(1/2)* 
2^(1/2)*sin(d*x+c)-sin(d*x+c)-cos(d*x+c)+1))+I*tan(d*x+c)^2*(-6*cos(d*x+c) 
-6)*(cot(d*x+c)-csc(d*x+c))^(1/2)*arctan((cot(d*x+c)-csc(d*x+c))^(1/2)*2^( 
1/2)-1)+I*tan(d*x+c)^2*(-6*cos(d*x+c)-6)*(cot(d*x+c)-csc(d*x+c))^(1/2)*arc 
tan((cot(d*x+c)-csc(d*x+c))^(1/2)*2^(1/2)+1)+tan(d*x+c)^2*(-3*cos(d*x+c)-3 
)*(cot(d*x+c)-csc(d*x+c))^(1/2)*ln(-((cot(d*x+c)-csc(d*x+c))^(1/2)*2^(1/2) 
*sin(d*x+c)-sin(d*x+c)-cos(d*x+c)+1)/((-2*csc(d*x+c)*sin(1/2*d*x+1/2*c)^2) 
^(1/2)*2^(1/2)*sin(d*x+c)+sin(d*x+c)+cos(d*x+c)-1))+tan(d*x+c)^2*(-6*cos(d 
*x+c)-6)*(cot(d*x+c)-csc(d*x+c))^(1/2)*arctan((cot(d*x+c)-csc(d*x+c))^(1/2 
)*2^(1/2)-1)+tan(d*x+c)^2*(-6*cos(d*x+c)-6)*(cot(d*x+c)-csc(d*x+c))^(1/2)* 
arctan((cot(d*x+c)-csc(d*x+c))^(1/2)*2^(1/2)+1))*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. 326 vs. \(2 (104) = 208\).

Time = 0.11 (sec) , antiderivative size = 326, normalized size of antiderivative = 2.33 \[ \int \cot ^{\frac {5}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)} \, dx=\frac {-16 i \, \sqrt {2} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {\frac {i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} - 1}} e^{\left (3 i \, d x + 3 i \, c\right )} - 3 \, {\left (d e^{\left (2 i \, d x + 2 i \, c\right )} - d\right )} \sqrt {-\frac {8 i \, a}{d^{2}}} \log \left ({\left (\sqrt {2} {\left (i \, d e^{\left (2 i \, d x + 2 i \, c\right )} - i \, d\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {\frac {i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} - 1}} \sqrt {-\frac {8 i \, a}{d^{2}}} + 4 i \, a e^{\left (i \, d x + i \, c\right )}\right )} e^{\left (-i \, d x - i \, c\right )}\right ) + 3 \, {\left (d e^{\left (2 i \, d x + 2 i \, c\right )} - d\right )} \sqrt {-\frac {8 i \, a}{d^{2}}} \log \left ({\left (\sqrt {2} {\left (-i \, d e^{\left (2 i \, d x + 2 i \, c\right )} + i \, d\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {\frac {i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} - 1}} \sqrt {-\frac {8 i \, a}{d^{2}}} + 4 i \, a e^{\left (i \, d x + i \, c\right )}\right )} e^{\left (-i \, d x - i \, c\right )}\right )}{12 \, {\left (d e^{\left (2 i \, d x + 2 i \, c\right )} - d\right )}} \] Input:

integrate(cot(d*x+c)^(5/2)*(a+I*a*tan(d*x+c))^(1/2),x, algorithm="fricas")
 

Output:

1/12*(-16*I*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))*e^(3*I*d*x + 3*I*c) - 3*(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)) + 3*(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)
 

Sympy [F(-1)]

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

integrate(cot(d*x+c)**(5/2)*(a+I*a*tan(d*x+c))**(1/2),x)
 

Output:

Timed out
 

Maxima [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 961 vs. \(2 (104) = 208\).

Time = 0.43 (sec) , antiderivative size = 961, normalized size of antiderivative = 6.86 \[ \int \cot ^{\frac {5}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)} \, dx=\text {Too large to display} \] Input:

integrate(cot(d*x+c)^(5/2)*(a+I*a*tan(d*x+c))^(1/2),x, algorithm="maxima")
 

Output:

1/6*(2*sqrt(cos(2*d*x + 2*c)^2 + sin(2*d*x + 2*c)^2 - 2*cos(2*d*x + 2*c) + 
 1)*((-(3*I - 3)*cos(3*d*x + 3*c) - (I - 1)*cos(d*x + c) + (3*I + 3)*sin(3 
*d*x + 3*c) + (I + 1)*sin(d*x + c))*cos(3/2*arctan2(sin(2*d*x + 2*c), cos( 
2*d*x + 2*c) - 1)) + (-(3*I + 3)*cos(3*d*x + 3*c) - (I + 1)*cos(d*x + c) - 
 (3*I - 3)*sin(3*d*x + 3*c) - (I - 1)*sin(d*x + c))*sin(3/2*arctan2(sin(2* 
d*x + 2*c), cos(2*d*x + 2*c) - 1)))*sqrt(a) + 3*(2*(-(I + 1)*cos(2*d*x + 2 
*c)^2 - (I + 1)*sin(2*d*x + 2*c)^2 + (2*I + 2)*cos(2*d*x + 2*c) - I - 1)*a 
rctan2(2*(cos(2*d*x + 2*c)^2 + sin(2*d*x + 2*c)^2 - 2*cos(2*d*x + 2*c) + 1 
)^(1/4)*sin(1/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c) - 1)) + 2*sin(d 
*x + c), 2*(cos(2*d*x + 2*c)^2 + sin(2*d*x + 2*c)^2 - 2*cos(2*d*x + 2*c) + 
 1)^(1/4)*cos(1/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c) - 1)) + 2*cos 
(d*x + c)) + ((I - 1)*cos(2*d*x + 2*c)^2 + (I - 1)*sin(2*d*x + 2*c)^2 - (2 
*I - 2)*cos(2*d*x + 2*c) + I - 1)*log(4*cos(d*x + c)^2 + 4*sin(d*x + c)^2 
+ 4*sqrt(cos(2*d*x + 2*c)^2 + sin(2*d*x + 2*c)^2 - 2*cos(2*d*x + 2*c) + 1) 
*(cos(1/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c) - 1))^2 + sin(1/2*arc 
tan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c) - 1))^2) + 8*(cos(2*d*x + 2*c)^2 + 
 sin(2*d*x + 2*c)^2 - 2*cos(2*d*x + 2*c) + 1)^(1/4)*(cos(d*x + c)*cos(1/2* 
arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c) - 1)) + sin(d*x + c)*sin(1/2*ar 
ctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c) - 1)))))*(cos(2*d*x + 2*c)^2 + si 
n(2*d*x + 2*c)^2 - 2*cos(2*d*x + 2*c) + 1)^(1/4)*sqrt(a) + 2*(((-(I - 1...
 

Giac [F(-2)]

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

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

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

Output:

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

Reduce [F]

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

int(cot(d*x+c)^(5/2)*(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))*cot(c + d*x)**2,x)