\(\int \frac {\cos (c+d x)}{(a+i a \tan (c+d x))^{5/2}} \, dx\) [376]

Optimal result

Integrand size = 24, antiderivative size = 192 \[ \int \frac {\cos (c+d x)}{(a+i a \tan (c+d x))^{5/2}} \, dx=\frac {35 i \text {arctanh}\left (\frac {\sqrt {a} \sec (c+d x)}{\sqrt {2} \sqrt {a+i a \tan (c+d x)}}\right )}{128 \sqrt {2} a^{5/2} d}+\frac {i \cos (c+d x)}{6 d (a+i a \tan (c+d x))^{5/2}}+\frac {7 i \cos (c+d x)}{48 a d (a+i a \tan (c+d x))^{3/2}}+\frac {35 i \cos (c+d x)}{192 a^2 d \sqrt {a+i a \tan (c+d x)}}-\frac {35 i \cos (c+d x) \sqrt {a+i a \tan (c+d x)}}{128 a^3 d} \] Output:

35/256*I*arctanh(1/2*a^(1/2)*sec(d*x+c)*2^(1/2)/(a+I*a*tan(d*x+c))^(1/2))* 
2^(1/2)/a^(5/2)/d+1/6*I*cos(d*x+c)/d/(a+I*a*tan(d*x+c))^(5/2)+7/48*I*cos(d 
*x+c)/a/d/(a+I*a*tan(d*x+c))^(3/2)+35/192*I*cos(d*x+c)/a^2/d/(a+I*a*tan(d* 
x+c))^(1/2)-35/128*I*cos(d*x+c)*(a+I*a*tan(d*x+c))^(1/2)/a^3/d
 

Mathematica [A] (verified)

Time = 1.22 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.74 \[ \int \frac {\cos (c+d x)}{(a+i a \tan (c+d x))^{5/2}} \, dx=\frac {i \sec ^3(c+d x) \left (-125-105 e^{2 i (c+d x)} \sqrt {1+e^{2 i (c+d x)}} \text {arctanh}\left (\sqrt {1+e^{2 i (c+d x)}}\right )-85 \cos (2 (c+d x))+40 \cos (4 (c+d x))+7 i \sin (2 (c+d x))+56 i \sin (4 (c+d x))\right )}{768 a^2 d (-i+\tan (c+d x))^2 \sqrt {a+i a \tan (c+d x)}} \] Input:

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

Output:

((I/768)*Sec[c + d*x]^3*(-125 - 105*E^((2*I)*(c + d*x))*Sqrt[1 + E^((2*I)* 
(c + d*x))]*ArcTanh[Sqrt[1 + E^((2*I)*(c + d*x))]] - 85*Cos[2*(c + d*x)] + 
 40*Cos[4*(c + d*x)] + (7*I)*Sin[2*(c + d*x)] + (56*I)*Sin[4*(c + d*x)]))/ 
(a^2*d*(-I + Tan[c + d*x])^2*Sqrt[a + I*a*Tan[c + d*x]])
 

Rubi [A] (verified)

Time = 0.82 (sec) , antiderivative size = 203, normalized size of antiderivative = 1.06, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.458, Rules used = {3042, 3983, 3042, 3983, 3042, 3983, 3042, 3971, 3042, 3970, 219}

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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3983

\(\displaystyle \frac {7 \int \frac {\cos (c+d x)}{(i \tan (c+d x) a+a)^{3/2}}dx}{12 a}+\frac {i \cos (c+d x)}{6 d (a+i a \tan (c+d x))^{5/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {7 \int \frac {1}{\sec (c+d x) (i \tan (c+d x) a+a)^{3/2}}dx}{12 a}+\frac {i \cos (c+d x)}{6 d (a+i a \tan (c+d x))^{5/2}}\)

\(\Big \downarrow \) 3983

\(\displaystyle \frac {7 \left (\frac {5 \int \frac {\cos (c+d x)}{\sqrt {i \tan (c+d x) a+a}}dx}{8 a}+\frac {i \cos (c+d x)}{4 d (a+i a \tan (c+d x))^{3/2}}\right )}{12 a}+\frac {i \cos (c+d x)}{6 d (a+i a \tan (c+d x))^{5/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {7 \left (\frac {5 \int \frac {1}{\sec (c+d x) \sqrt {i \tan (c+d x) a+a}}dx}{8 a}+\frac {i \cos (c+d x)}{4 d (a+i a \tan (c+d x))^{3/2}}\right )}{12 a}+\frac {i \cos (c+d x)}{6 d (a+i a \tan (c+d x))^{5/2}}\)

\(\Big \downarrow \) 3983

\(\displaystyle \frac {7 \left (\frac {5 \left (\frac {3 \int \cos (c+d x) \sqrt {i \tan (c+d x) a+a}dx}{4 a}+\frac {i \cos (c+d x)}{2 d \sqrt {a+i a \tan (c+d x)}}\right )}{8 a}+\frac {i \cos (c+d x)}{4 d (a+i a \tan (c+d x))^{3/2}}\right )}{12 a}+\frac {i \cos (c+d x)}{6 d (a+i a \tan (c+d x))^{5/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {7 \left (\frac {5 \left (\frac {3 \int \frac {\sqrt {i \tan (c+d x) a+a}}{\sec (c+d x)}dx}{4 a}+\frac {i \cos (c+d x)}{2 d \sqrt {a+i a \tan (c+d x)}}\right )}{8 a}+\frac {i \cos (c+d x)}{4 d (a+i a \tan (c+d x))^{3/2}}\right )}{12 a}+\frac {i \cos (c+d x)}{6 d (a+i a \tan (c+d x))^{5/2}}\)

\(\Big \downarrow \) 3971

\(\displaystyle \frac {7 \left (\frac {5 \left (\frac {3 \left (\frac {1}{2} a \int \frac {\sec (c+d x)}{\sqrt {i \tan (c+d x) a+a}}dx-\frac {i \cos (c+d x) \sqrt {a+i a \tan (c+d x)}}{d}\right )}{4 a}+\frac {i \cos (c+d x)}{2 d \sqrt {a+i a \tan (c+d x)}}\right )}{8 a}+\frac {i \cos (c+d x)}{4 d (a+i a \tan (c+d x))^{3/2}}\right )}{12 a}+\frac {i \cos (c+d x)}{6 d (a+i a \tan (c+d x))^{5/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {7 \left (\frac {5 \left (\frac {3 \left (\frac {1}{2} a \int \frac {\sec (c+d x)}{\sqrt {i \tan (c+d x) a+a}}dx-\frac {i \cos (c+d x) \sqrt {a+i a \tan (c+d x)}}{d}\right )}{4 a}+\frac {i \cos (c+d x)}{2 d \sqrt {a+i a \tan (c+d x)}}\right )}{8 a}+\frac {i \cos (c+d x)}{4 d (a+i a \tan (c+d x))^{3/2}}\right )}{12 a}+\frac {i \cos (c+d x)}{6 d (a+i a \tan (c+d x))^{5/2}}\)

\(\Big \downarrow \) 3970

\(\displaystyle \frac {7 \left (\frac {5 \left (\frac {3 \left (\frac {i a \int \frac {1}{2-\frac {a \sec ^2(c+d x)}{i \tan (c+d x) a+a}}d\frac {\sec (c+d x)}{\sqrt {i \tan (c+d x) a+a}}}{d}-\frac {i \cos (c+d x) \sqrt {a+i a \tan (c+d x)}}{d}\right )}{4 a}+\frac {i \cos (c+d x)}{2 d \sqrt {a+i a \tan (c+d x)}}\right )}{8 a}+\frac {i \cos (c+d x)}{4 d (a+i a \tan (c+d x))^{3/2}}\right )}{12 a}+\frac {i \cos (c+d x)}{6 d (a+i a \tan (c+d x))^{5/2}}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {7 \left (\frac {5 \left (\frac {3 \left (\frac {i \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \sec (c+d x)}{\sqrt {2} \sqrt {a+i a \tan (c+d x)}}\right )}{\sqrt {2} d}-\frac {i \cos (c+d x) \sqrt {a+i a \tan (c+d x)}}{d}\right )}{4 a}+\frac {i \cos (c+d x)}{2 d \sqrt {a+i a \tan (c+d x)}}\right )}{8 a}+\frac {i \cos (c+d x)}{4 d (a+i a \tan (c+d x))^{3/2}}\right )}{12 a}+\frac {i \cos (c+d x)}{6 d (a+i a \tan (c+d x))^{5/2}}\)

Input:

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

Output:

((I/6)*Cos[c + d*x])/(d*(a + I*a*Tan[c + d*x])^(5/2)) + (7*(((I/4)*Cos[c + 
 d*x])/(d*(a + I*a*Tan[c + d*x])^(3/2)) + (5*(((I/2)*Cos[c + d*x])/(d*Sqrt 
[a + I*a*Tan[c + d*x]]) + (3*((I*Sqrt[a]*ArcTanh[(Sqrt[a]*Sec[c + d*x])/(S 
qrt[2]*Sqrt[a + I*a*Tan[c + d*x]])])/(Sqrt[2]*d) - (I*Cos[c + d*x]*Sqrt[a 
+ I*a*Tan[c + d*x]])/d))/(4*a)))/(8*a)))/(12*a)
 

Defintions of rubi rules used

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 3042

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

rule 3970

Int[sec[(e_.) + (f_.)*(x_)]/Sqrt[(a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]], x_S 
ymbol] :> Simp[-2*(a/(b*f))   Subst[Int[1/(2 - a*x^2), x], x, Sec[e + f*x]/ 
Sqrt[a + b*Tan[e + f*x]]], x] /; FreeQ[{a, b, e, f}, x] && EqQ[a^2 + b^2, 0 
]
 

rule 3971

Int[((d_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*( 
x_)])^(n_), x_Symbol] :> Simp[b*(d*Sec[e + f*x])^m*((a + b*Tan[e + f*x])^n/ 
(a*f*m)), x] + Simp[a/(2*d^2)   Int[(d*Sec[e + f*x])^(m + 2)*(a + b*Tan[e + 
 f*x])^(n - 1), x], x] /; FreeQ[{a, b, d, e, f}, x] && EqQ[a^2 + b^2, 0] && 
 EqQ[m/2 + n, 0] && GtQ[n, 0]
 

rule 3983

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

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. 432 vs. \(2 (155 ) = 310\).

Time = 9.64 (sec) , antiderivative size = 433, normalized size of antiderivative = 2.26

method	result	size
default	\(-\frac {i \operatorname {arctanh}\left (\frac {\left (i-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right ) \sqrt {2}}{2 \sqrt {\cot \left (d x +c \right )^{2}-2 \cot \left (d x +c \right ) \csc \left (d x +c \right )+\csc \left (d x +c \right )^{2}-1}}\right ) \left (420 \sin \left (d x +c \right )+210 \tan \left (d x +c \right )-105 \sec \left (d x +c \right ) \tan \left (d x +c \right )\right )+\operatorname {arctanh}\left (\frac {\left (i-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right ) \sqrt {2}}{2 \sqrt {\cot \left (d x +c \right )^{2}-2 \cot \left (d x +c \right ) \csc \left (d x +c \right )+\csc \left (d x +c \right )^{2}-1}}\right ) \left (420 \cos \left (d x +c \right )+210-315 \sec \left (d x +c \right )-105 \sec \left (d x +c \right )^{2}\right )+\sqrt {2}\, \sqrt {-\frac {2 \cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \left (-420 \sin \left (d x +c \right )+105 \sec \left (d x +c \right ) \tan \left (d x +c \right )\right )+i \sqrt {2}\, \sqrt {-\frac {2 \cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \left (420 \cos \left (d x +c \right )-315 \sec \left (d x +c \right )\right )+\tan \left (d x +c \right ) \left (-448 \cos \left (d x +c \right )^{2}+392 \cos \left (d x +c \right )+210\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}+i \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \left (320 \cos \left (d x +c \right )^{2}-520 \cos \left (d x +c \right )-490+140 \sec \left (d x +c \right )\right )}{768 d \left (1+i \tan \left (d x +c \right )\right )^{2} \left (\cos \left (d x +c \right )+1\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, a^{2} \sqrt {a \left (1+i \tan \left (d x +c \right )\right )}}\)	\(433\)

Input:

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

Output:

-1/768/d/(1+I*tan(d*x+c))^2/(cos(d*x+c)+1)/(-cos(d*x+c)/(cos(d*x+c)+1))^(1 
/2)/a^2/(a*(1+I*tan(d*x+c)))^(1/2)*(I*arctanh(1/2/(cot(d*x+c)^2-2*cot(d*x+ 
c)*csc(d*x+c)+csc(d*x+c)^2-1)^(1/2)*(I-cot(d*x+c)+csc(d*x+c))*2^(1/2))*(42 
0*sin(d*x+c)+210*tan(d*x+c)-105*sec(d*x+c)*tan(d*x+c))+arctanh(1/2/(cot(d* 
x+c)^2-2*cot(d*x+c)*csc(d*x+c)+csc(d*x+c)^2-1)^(1/2)*(I-cot(d*x+c)+csc(d*x 
+c))*2^(1/2))*(420*cos(d*x+c)+210-315*sec(d*x+c)-105*sec(d*x+c)^2)+2^(1/2) 
*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*(-420*sin(d*x+c)+105*sec(d*x+c)*tan( 
d*x+c))+I*2^(1/2)*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*(420*cos(d*x+c)-315 
*sec(d*x+c))+tan(d*x+c)*(-448*cos(d*x+c)^2+392*cos(d*x+c)+210)*(-cos(d*x+c 
)/(cos(d*x+c)+1))^(1/2)+I*(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*(320*cos(d*x+ 
c)^2-520*cos(d*x+c)-490+140*sec(d*x+c)))
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 289, normalized size of antiderivative = 1.51 \[ \int \frac {\cos (c+d x)}{(a+i a \tan (c+d x))^{5/2}} \, dx=\frac {{\left (-105 i \, \sqrt {\frac {1}{2}} a^{3} d \sqrt {\frac {1}{a^{5} d^{2}}} e^{\left (6 i \, d x + 6 i \, c\right )} \log \left (-\frac {35 \, {\left (\sqrt {2} \sqrt {\frac {1}{2}} {\left (i \, a^{2} d e^{\left (2 i \, d x + 2 i \, c\right )} + i \, a^{2} d\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {\frac {1}{a^{5} d^{2}}} - i\right )} e^{\left (-i \, d x - i \, c\right )}}{64 \, a^{2} d}\right ) + 105 i \, \sqrt {\frac {1}{2}} a^{3} d \sqrt {\frac {1}{a^{5} d^{2}}} e^{\left (6 i \, d x + 6 i \, c\right )} \log \left (-\frac {35 \, {\left (\sqrt {2} \sqrt {\frac {1}{2}} {\left (-i \, a^{2} d e^{\left (2 i \, d x + 2 i \, c\right )} - i \, a^{2} d\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {\frac {1}{a^{5} d^{2}}} - i\right )} e^{\left (-i \, d x - i \, c\right )}}{64 \, a^{2} d}\right ) + \sqrt {2} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} {\left (-48 i \, e^{\left (8 i \, d x + 8 i \, c\right )} + 39 i \, e^{\left (6 i \, d x + 6 i \, c\right )} + 125 i \, e^{\left (4 i \, d x + 4 i \, c\right )} + 46 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + 8 i\right )}\right )} e^{\left (-6 i \, d x - 6 i \, c\right )}}{768 \, a^{3} d} \] Input:

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

Output:

1/768*(-105*I*sqrt(1/2)*a^3*d*sqrt(1/(a^5*d^2))*e^(6*I*d*x + 6*I*c)*log(-3 
5/64*(sqrt(2)*sqrt(1/2)*(I*a^2*d*e^(2*I*d*x + 2*I*c) + I*a^2*d)*sqrt(a/(e^ 
(2*I*d*x + 2*I*c) + 1))*sqrt(1/(a^5*d^2)) - I)*e^(-I*d*x - I*c)/(a^2*d)) + 
 105*I*sqrt(1/2)*a^3*d*sqrt(1/(a^5*d^2))*e^(6*I*d*x + 6*I*c)*log(-35/64*(s 
qrt(2)*sqrt(1/2)*(-I*a^2*d*e^(2*I*d*x + 2*I*c) - I*a^2*d)*sqrt(a/(e^(2*I*d 
*x + 2*I*c) + 1))*sqrt(1/(a^5*d^2)) - I)*e^(-I*d*x - I*c)/(a^2*d)) + sqrt( 
2)*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1))*(-48*I*e^(8*I*d*x + 8*I*c) + 39*I*e^( 
6*I*d*x + 6*I*c) + 125*I*e^(4*I*d*x + 4*I*c) + 46*I*e^(2*I*d*x + 2*I*c) + 
8*I))*e^(-6*I*d*x - 6*I*c)/(a^3*d)
 

Sympy [F]

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

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

Output:

Integral(cos(c + d*x)/(I*a*(tan(c + d*x) - I))**(5/2), x)
 

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. 2297 vs. \(2 (145) = 290\).

Time = 0.33 (sec) , antiderivative size = 2297, normalized size of antiderivative = 11.96 \[ \int \frac {\cos (c+d x)}{(a+i a \tan (c+d x))^{5/2}} \, dx=\text {Too large to display} \] Input:

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

Output:

1/3072*(544*(cos(1/3*arctan2(sin(6*d*x + 6*c), cos(6*d*x + 6*c)))^2 + sin( 
1/3*arctan2(sin(6*d*x + 6*c), cos(6*d*x + 6*c)))^2 + 2*cos(1/3*arctan2(sin 
(6*d*x + 6*c), cos(6*d*x + 6*c))) + 1)^(3/4)*((-I*sqrt(2)*cos(6*d*x + 6*c) 
 - sqrt(2)*sin(6*d*x + 6*c))*cos(3/2*arctan2(sin(1/3*arctan2(sin(6*d*x + 6 
*c), cos(6*d*x + 6*c))), cos(1/3*arctan2(sin(6*d*x + 6*c), cos(6*d*x + 6*c 
))) + 1)) + (sqrt(2)*cos(6*d*x + 6*c) - I*sqrt(2)*sin(6*d*x + 6*c))*sin(3/ 
2*arctan2(sin(1/3*arctan2(sin(6*d*x + 6*c), cos(6*d*x + 6*c))), cos(1/3*ar 
ctan2(sin(6*d*x + 6*c), cos(6*d*x + 6*c))) + 1)))*sqrt(a) + 12*(cos(1/3*ar 
ctan2(sin(6*d*x + 6*c), cos(6*d*x + 6*c)))^2 + sin(1/3*arctan2(sin(6*d*x + 
 6*c), cos(6*d*x + 6*c)))^2 + 2*cos(1/3*arctan2(sin(6*d*x + 6*c), cos(6*d* 
x + 6*c))) + 1)^(1/4)*(29*((I*sqrt(2)*cos(6*d*x + 6*c) + sqrt(2)*sin(6*d*x 
 + 6*c))*cos(1/3*arctan2(sin(6*d*x + 6*c), cos(6*d*x + 6*c)))^2 + (I*sqrt( 
2)*cos(6*d*x + 6*c) + sqrt(2)*sin(6*d*x + 6*c))*sin(1/3*arctan2(sin(6*d*x 
+ 6*c), cos(6*d*x + 6*c)))^2 + 2*(I*sqrt(2)*cos(6*d*x + 6*c) + sqrt(2)*sin 
(6*d*x + 6*c))*cos(1/3*arctan2(sin(6*d*x + 6*c), cos(6*d*x + 6*c))) + I*sq 
rt(2)*cos(6*d*x + 6*c) + sqrt(2)*sin(6*d*x + 6*c))*cos(5/2*arctan2(sin(1/3 
*arctan2(sin(6*d*x + 6*c), cos(6*d*x + 6*c))), cos(1/3*arctan2(sin(6*d*x + 
 6*c), cos(6*d*x + 6*c))) + 1)) + (19*I*sqrt(2)*cos(6*d*x + 6*c) + 19*sqrt 
(2)*sin(6*d*x + 6*c) - 16*I*sqrt(2))*cos(1/2*arctan2(sin(1/3*arctan2(sin(6 
*d*x + 6*c), cos(6*d*x + 6*c))), cos(1/3*arctan2(sin(6*d*x + 6*c), cos(...
                                                                                    
                                                                                    
 

Giac [F(-2)]

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

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

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

Output:

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

Reduce [F]

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

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

Output:

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