\(\int \frac {1}{\cot ^{\frac {5}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}} \, dx\) [773]

Optimal result

Integrand size = 28, antiderivative size = 217 \[ \int \frac {1}{\cot ^{\frac {5}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}} \, dx=-\frac {(-1)^{3/4} \arctan \left (\frac {(-1)^{3/4} \sqrt {a} \sqrt {\tan (c+d x)}}{\sqrt {a+i a \tan (c+d x)}}\right ) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}}{\sqrt {a} d}+\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \text {arctanh}\left (\frac {(1+i) \sqrt {a} \sqrt {\tan (c+d x)}}{\sqrt {a+i a \tan (c+d x)}}\right ) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}}{\sqrt {a} d}-\frac {1}{d \cot ^{\frac {3}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}}-\frac {2 i \sqrt {a+i a \tan (c+d x)}}{a d \sqrt {\cot (c+d x)}} \] Output:

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

Mathematica [A] (verified)

Time = 0.73 (sec) , antiderivative size = 185, normalized size of antiderivative = 0.85 \[ \int \frac {1}{\cot ^{\frac {5}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}} \, dx=\frac {\sqrt {\cot (c+d x)} \left (2 \sqrt [4]{-1} a \text {arcsinh}\left (\sqrt [4]{-1} \sqrt {\tan (c+d x)}\right ) \sqrt {1+i \tan (c+d x)} \sqrt {\tan (c+d x)}+2 a \tan (c+d x) (-2 i+\tan (c+d x))+\sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {i a \tan (c+d x)}}{\sqrt {a+i a \tan (c+d x)}}\right ) \sqrt {i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)}\right )}{2 a d \sqrt {a+i a \tan (c+d x)}} \] Input:

Integrate[1/(Cot[c + d*x]^(5/2)*Sqrt[a + I*a*Tan[c + d*x]]),x]
 

Output:

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

Rubi [A] (verified)

Time = 1.26 (sec) , antiderivative size = 205, normalized size of antiderivative = 0.94, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.536, Rules used = {3042, 4729, 3042, 4041, 27, 3042, 4080, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4729

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4041

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4080

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4084

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4027

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

\(\Big \downarrow \) 218

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

\(\Big \downarrow \) 4082

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

\(\Big \downarrow \) 65

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

\(\Big \downarrow \) 216

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

Input:

Int[1/(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]]*(-(Tan[c + d*x]^(3/2)/(d*Sqrt[a + I* 
a*Tan[c + d*x]])) + (((-2*(-1)^(3/4)*a^(5/2)*ArcTan[((-1)^(3/4)*Sqrt[a]*Sq 
rt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]])/d + ((1 + I)*a^(5/2)*ArcTan 
h[((1 + I)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]])/d)/a - 
 ((4*I)*a*Sqrt[Tan[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]])/d)/(2*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 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 4080

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*(a + b*Tan[e + f*x])^m*((c + d*Tan[e + f*x])^n/(f*(m + n))), x] + Simp[ 
1/(a*(m + n))   Int[(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^(n - 1)*Sim 
p[a*A*c*(m + n) - B*(b*c*m + a*d*n) + (a*A*d*(m + n) - B*(b*d*m - a*c*n))*T 
an[e + f*x], 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] && 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 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. 876 vs. \(2 (172 ) = 344\).

Time = 2.43 (sec) , antiderivative size = 877, normalized size of antiderivative = 4.04

method	result	size
default	\(\text {Expression too large to display}\)	\(877\)

Input:

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

Output:

(1/32+1/32*I)/d*2^(1/2)*(I*(16*cos(1/2*d*x+1/2*c)^4-16*cos(1/2*d*x+1/2*c)^ 
2+4)*sin(1/2*d*x+1/2*c)*arctan((1/2+1/2*I)*2^(1/2)*(-csc(d*x+c)+cot(d*x+c) 
)^(1/2))+arctan((1/2+1/2*I)*2^(1/2)*(-csc(d*x+c)+cot(d*x+c))^(1/2))*(16*co 
s(1/2*d*x+1/2*c)^5-16*cos(1/2*d*x+1/2*c)^3+4*cos(1/2*d*x+1/2*c))+sin(1/2*d 
*x+1/2*c)*(4*cos(1/2*d*x+1/2*c)^4-4*cos(1/2*d*x+1/2*c)^2+1)*2^(1/2)*ln((-c 
sc(d*x+c)+cot(d*x+c))^(1/2)-I)+I*2^(1/2)*ln((-csc(d*x+c)+cot(d*x+c))^(1/2) 
-I)*(-4*cos(1/2*d*x+1/2*c)^5+4*cos(1/2*d*x+1/2*c)^3-cos(1/2*d*x+1/2*c))+si 
n(1/2*d*x+1/2*c)*(-4*cos(1/2*d*x+1/2*c)^4+4*cos(1/2*d*x+1/2*c)^2-1)*2^(1/2 
)*ln((-csc(d*x+c)+cot(d*x+c))^(1/2)+I)+I*2^(1/2)*ln((-csc(d*x+c)+cot(d*x+c 
))^(1/2)+I)*(4*cos(1/2*d*x+1/2*c)^5-4*cos(1/2*d*x+1/2*c)^3+cos(1/2*d*x+1/2 
*c))+sin(1/2*d*x+1/2*c)*(4*cos(1/2*d*x+1/2*c)^4-4*cos(1/2*d*x+1/2*c)^2+1)* 
2^(1/2)*ln((-csc(d*x+c)+cot(d*x+c))^(1/2)-1)+I*2^(1/2)*ln((-csc(d*x+c)+cot 
(d*x+c))^(1/2)-1)*(-4*cos(1/2*d*x+1/2*c)^5+4*cos(1/2*d*x+1/2*c)^3-cos(1/2* 
d*x+1/2*c))+sin(1/2*d*x+1/2*c)*(-4*cos(1/2*d*x+1/2*c)^4+4*cos(1/2*d*x+1/2* 
c)^2-1)*2^(1/2)*ln((-csc(d*x+c)+cot(d*x+c))^(1/2)+1)+I*2^(1/2)*ln((-csc(d* 
x+c)+cot(d*x+c))^(1/2)+1)*(4*cos(1/2*d*x+1/2*c)^5-4*cos(1/2*d*x+1/2*c)^3+c 
os(1/2*d*x+1/2*c))+I*2^(1/2)*(-csc(d*x+c)+cot(d*x+c))^(1/2)*((-8*cos(1/2*d 
*x+1/2*c)^2+4)*cos(1/2*d*x+1/2*c)^2*sin(1/2*d*x+1/2*c)-4*(2*cos(1/2*d*x+1/ 
2*c)^2-1)^2*cos(1/2*d*x+1/2*c))+2^(1/2)*(-csc(d*x+c)+cot(d*x+c))^(1/2)*((8 
*cos(1/2*d*x+1/2*c)^2-4)*cos(1/2*d*x+1/2*c)^2*sin(1/2*d*x+1/2*c)-4*(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. 703 vs. \(2 (163) = 326\).

Time = 0.09 (sec) , antiderivative size = 703, normalized size of antiderivative = 3.24 \[ \int \frac {1}{\cot ^{\frac {5}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}} \, dx =\text {Too large to display} \] Input:

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

Output:

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

Sympy [F]

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

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

Maxima [F]

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

integrate(1/cot(d*x+c)^(5/2)/(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)^(5/2)), x)
 

Giac [F(-2)]

Exception generated. \[ \int \frac {1}{\cot ^{\frac {5}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}} \, dx=\text {Exception raised: TypeError} \] Input:

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

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

Output:

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

Reduce [F]

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

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