\(\int \cos ^7(c+d x) (a+i a \tan (c+d x))^{5/2} \, dx\) [318]

Optimal result

Integrand size = 26, antiderivative size = 231 \[ \int \cos ^7(c+d x) (a+i a \tan (c+d x))^{5/2} \, dx=\frac {9 i a^{5/2} \text {arctanh}\left (\frac {\sqrt {a} \sec (c+d x)}{\sqrt {2} \sqrt {a+i a \tan (c+d x)}}\right )}{32 \sqrt {2} d}+\frac {3 i a^3 \cos (c+d x)}{16 d \sqrt {a+i a \tan (c+d x)}}-\frac {9 i a^2 \cos (c+d x) \sqrt {a+i a \tan (c+d x)}}{32 d}-\frac {3 i a^2 \cos ^3(c+d x) \sqrt {a+i a \tan (c+d x)}}{20 d}-\frac {9 i a \cos ^5(c+d x) (a+i a \tan (c+d x))^{3/2}}{70 d}-\frac {i \cos ^7(c+d x) (a+i a \tan (c+d x))^{5/2}}{7 d} \] Output:

9/64*I*a^(5/2)*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)/d+3/16*I*a^3*cos(d*x+c)/d/(a+I*a*tan(d*x+c))^(1/2)-9/32*I*a^ 
2*cos(d*x+c)*(a+I*a*tan(d*x+c))^(1/2)/d-3/20*I*a^2*cos(d*x+c)^3*(a+I*a*tan 
(d*x+c))^(1/2)/d-9/70*I*a*cos(d*x+c)^5*(a+I*a*tan(d*x+c))^(3/2)/d-1/7*I*co 
s(d*x+c)^7*(a+I*a*tan(d*x+c))^(5/2)/d
 

Mathematica [A] (verified)

Time = 1.69 (sec) , antiderivative size = 155, normalized size of antiderivative = 0.67 \[ \int \cos ^7(c+d x) (a+i a \tan (c+d x))^{5/2} \, dx=-\frac {i a^2 e^{-3 i (c+d x)} \left (-35+353 e^{2 i (c+d x)}+544 e^{4 i (c+d x)}+214 e^{6 i (c+d x)}+68 e^{8 i (c+d x)}+10 e^{10 i (c+d x)}-315 e^{2 i (c+d x)} \sqrt {1+e^{2 i (c+d x)}} \text {arctanh}\left (\sqrt {1+e^{2 i (c+d x)}}\right )\right ) \sqrt {a+i a \tan (c+d x)}}{2240 d} \] Input:

Integrate[Cos[c + d*x]^7*(a + I*a*Tan[c + d*x])^(5/2),x]
 

Output:

((-1/2240*I)*a^2*(-35 + 353*E^((2*I)*(c + d*x)) + 544*E^((4*I)*(c + d*x)) 
+ 214*E^((6*I)*(c + d*x)) + 68*E^((8*I)*(c + d*x)) + 10*E^((10*I)*(c + d*x 
)) - 315*E^((2*I)*(c + d*x))*Sqrt[1 + E^((2*I)*(c + d*x))]*ArcTanh[Sqrt[1 
+ E^((2*I)*(c + d*x))]])*Sqrt[a + I*a*Tan[c + d*x]])/(d*E^((3*I)*(c + d*x) 
))
 

Rubi [A] (verified)

Time = 1.04 (sec) , antiderivative size = 243, normalized size of antiderivative = 1.05, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3042, 3978, 3042, 3978, 3042, 3978, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3978

\(\displaystyle \frac {9}{14} a \int \cos ^5(c+d x) (i \tan (c+d x) a+a)^{3/2}dx-\frac {i \cos ^7(c+d x) (a+i a \tan (c+d x))^{5/2}}{7 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {9}{14} a \int \frac {(i \tan (c+d x) a+a)^{3/2}}{\sec (c+d x)^5}dx-\frac {i \cos ^7(c+d x) (a+i a \tan (c+d x))^{5/2}}{7 d}\)

\(\Big \downarrow \) 3978

\(\displaystyle \frac {9}{14} a \left (\frac {7}{10} a \int \cos ^3(c+d x) \sqrt {i \tan (c+d x) a+a}dx-\frac {i \cos ^5(c+d x) (a+i a \tan (c+d x))^{3/2}}{5 d}\right )-\frac {i \cos ^7(c+d x) (a+i a \tan (c+d x))^{5/2}}{7 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {9}{14} a \left (\frac {7}{10} a \int \frac {\sqrt {i \tan (c+d x) a+a}}{\sec (c+d x)^3}dx-\frac {i \cos ^5(c+d x) (a+i a \tan (c+d x))^{3/2}}{5 d}\right )-\frac {i \cos ^7(c+d x) (a+i a \tan (c+d x))^{5/2}}{7 d}\)

\(\Big \downarrow \) 3978

\(\displaystyle \frac {9}{14} a \left (\frac {7}{10} a \left (\frac {5}{6} a \int \frac {\cos (c+d x)}{\sqrt {i \tan (c+d x) a+a}}dx-\frac {i \cos ^3(c+d x) \sqrt {a+i a \tan (c+d x)}}{3 d}\right )-\frac {i \cos ^5(c+d x) (a+i a \tan (c+d x))^{3/2}}{5 d}\right )-\frac {i \cos ^7(c+d x) (a+i a \tan (c+d x))^{5/2}}{7 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {9}{14} a \left (\frac {7}{10} a \left (\frac {5}{6} a \int \frac {1}{\sec (c+d x) \sqrt {i \tan (c+d x) a+a}}dx-\frac {i \cos ^3(c+d x) \sqrt {a+i a \tan (c+d x)}}{3 d}\right )-\frac {i \cos ^5(c+d x) (a+i a \tan (c+d x))^{3/2}}{5 d}\right )-\frac {i \cos ^7(c+d x) (a+i a \tan (c+d x))^{5/2}}{7 d}\)

\(\Big \downarrow \) 3983

\(\displaystyle \frac {9}{14} a \left (\frac {7}{10} a \left (\frac {5}{6} a \left (\frac {3 \int \cos (c+d x) \sqrt {i \tan (c+d x) a+a}dx}{4 a}+\frac {i \cos (c+d x)}{2 d \sqrt {a+i a \tan (c+d x)}}\right )-\frac {i \cos ^3(c+d x) \sqrt {a+i a \tan (c+d x)}}{3 d}\right )-\frac {i \cos ^5(c+d x) (a+i a \tan (c+d x))^{3/2}}{5 d}\right )-\frac {i \cos ^7(c+d x) (a+i a \tan (c+d x))^{5/2}}{7 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {9}{14} a \left (\frac {7}{10} a \left (\frac {5}{6} a \left (\frac {3 \int \frac {\sqrt {i \tan (c+d x) a+a}}{\sec (c+d x)}dx}{4 a}+\frac {i \cos (c+d x)}{2 d \sqrt {a+i a \tan (c+d x)}}\right )-\frac {i \cos ^3(c+d x) \sqrt {a+i a \tan (c+d x)}}{3 d}\right )-\frac {i \cos ^5(c+d x) (a+i a \tan (c+d x))^{3/2}}{5 d}\right )-\frac {i \cos ^7(c+d x) (a+i a \tan (c+d x))^{5/2}}{7 d}\)

\(\Big \downarrow \) 3971

\(\displaystyle \frac {9}{14} a \left (\frac {7}{10} a \left (\frac {5}{6} a \left (\frac {3 \left (\frac {1}{2} a \int \frac {\sec (c+d x)}{\sqrt {i \tan (c+d x) a+a}}dx-\frac {i \cos (c+d x) \sqrt {a+i a \tan (c+d x)}}{d}\right )}{4 a}+\frac {i \cos (c+d x)}{2 d \sqrt {a+i a \tan (c+d x)}}\right )-\frac {i \cos ^3(c+d x) \sqrt {a+i a \tan (c+d x)}}{3 d}\right )-\frac {i \cos ^5(c+d x) (a+i a \tan (c+d x))^{3/2}}{5 d}\right )-\frac {i \cos ^7(c+d x) (a+i a \tan (c+d x))^{5/2}}{7 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {9}{14} a \left (\frac {7}{10} a \left (\frac {5}{6} a \left (\frac {3 \left (\frac {1}{2} a \int \frac {\sec (c+d x)}{\sqrt {i \tan (c+d x) a+a}}dx-\frac {i \cos (c+d x) \sqrt {a+i a \tan (c+d x)}}{d}\right )}{4 a}+\frac {i \cos (c+d x)}{2 d \sqrt {a+i a \tan (c+d x)}}\right )-\frac {i \cos ^3(c+d x) \sqrt {a+i a \tan (c+d x)}}{3 d}\right )-\frac {i \cos ^5(c+d x) (a+i a \tan (c+d x))^{3/2}}{5 d}\right )-\frac {i \cos ^7(c+d x) (a+i a \tan (c+d x))^{5/2}}{7 d}\)

\(\Big \downarrow \) 3970

\(\displaystyle \frac {9}{14} a \left (\frac {7}{10} a \left (\frac {5}{6} a \left (\frac {3 \left (\frac {i a \int \frac {1}{2-\frac {a \sec ^2(c+d x)}{i \tan (c+d x) a+a}}d\frac {\sec (c+d x)}{\sqrt {i \tan (c+d x) a+a}}}{d}-\frac {i \cos (c+d x) \sqrt {a+i a \tan (c+d x)}}{d}\right )}{4 a}+\frac {i \cos (c+d x)}{2 d \sqrt {a+i a \tan (c+d x)}}\right )-\frac {i \cos ^3(c+d x) \sqrt {a+i a \tan (c+d x)}}{3 d}\right )-\frac {i \cos ^5(c+d x) (a+i a \tan (c+d x))^{3/2}}{5 d}\right )-\frac {i \cos ^7(c+d x) (a+i a \tan (c+d x))^{5/2}}{7 d}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {9}{14} a \left (\frac {7}{10} a \left (\frac {5}{6} a \left (\frac {3 \left (\frac {i \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \sec (c+d x)}{\sqrt {2} \sqrt {a+i a \tan (c+d x)}}\right )}{\sqrt {2} d}-\frac {i \cos (c+d x) \sqrt {a+i a \tan (c+d x)}}{d}\right )}{4 a}+\frac {i \cos (c+d x)}{2 d \sqrt {a+i a \tan (c+d x)}}\right )-\frac {i \cos ^3(c+d x) \sqrt {a+i a \tan (c+d x)}}{3 d}\right )-\frac {i \cos ^5(c+d x) (a+i a \tan (c+d x))^{3/2}}{5 d}\right )-\frac {i \cos ^7(c+d x) (a+i a \tan (c+d x))^{5/2}}{7 d}\)

Input:

Int[Cos[c + d*x]^7*(a + I*a*Tan[c + d*x])^(5/2),x]
 

Output:

((-1/7*I)*Cos[c + d*x]^7*(a + I*a*Tan[c + d*x])^(5/2))/d + (9*a*(((-1/5*I) 
*Cos[c + d*x]^5*(a + I*a*Tan[c + d*x])^(3/2))/d + (7*a*(((-1/3*I)*Cos[c + 
d*x]^3*Sqrt[a + I*a*Tan[c + d*x]])/d + (5*a*(((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])/(Sq 
rt[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)))/6))/10))/14
 

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 3978

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*((m + n)/(m*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] && GtQ[n, 0] && LtQ[m, -1] && IntegersQ[2*m, 2*n]
 

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. 1342 vs. \(2 (188 ) = 376\).

Time = 3.42 (sec) , antiderivative size = 1343, normalized size of antiderivative = 5.81

\[\text {Expression too large to display}\]

Input:

int(cos(d*x+c)^7*(a+I*a*tan(d*x+c))^(5/2),x)
 

Output:

-1/2240/d*cos(d*x+c)^2*(I*cos(d*x+c)*(-400*cos(d*x+c)^4+2184*cos(d*x+c)^2- 
630)+(-1260*cos(d*x+c)^3-630*cos(d*x+c)^2+945*cos(d*x+c)+315)*(-cos(d*x+c) 
/(cos(d*x+c)+1))^(1/2)*tan(d*x+c)^2*arctanh(2^(1/2)*(cot(d*x+c)-csc(d*x+c) 
)/(cot(d*x+c)^2-2*cot(d*x+c)*csc(d*x+c)+csc(d*x+c)^2-1)^(1/2))+(2520*cos(d 
*x+c)^2+1260*cos(d*x+c)-630)*sin(d*x+c)*(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2) 
*tan(d*x+c)*arctanh(2^(1/2)*(cot(d*x+c)-csc(d*x+c))/(cot(d*x+c)^2-2*cot(d* 
x+c)*csc(d*x+c)+csc(d*x+c)^2-1)^(1/2))+I*(2520*cos(d*x+c)^2+1260*cos(d*x+c 
)-630)*sin(d*x+c)*(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*tan(d*x+c)*arctan(1/2 
*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*2^(1/2))+I*(-1260*cos(d*x+c)^2-630*c 
os(d*x+c)+315)*sin(d*x+c)*(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*arctanh(2^(1/ 
2)*(cot(d*x+c)-csc(d*x+c))/(cot(d*x+c)^2-2*cot(d*x+c)*csc(d*x+c)+csc(d*x+c 
)^2-1)^(1/2))+(1260*cos(d*x+c)^3+630*cos(d*x+c)^2-945*cos(d*x+c)-315)*(-co 
s(d*x+c)/(cos(d*x+c)+1))^(1/2)*arctanh(2^(1/2)*(cot(d*x+c)-csc(d*x+c))/(co 
t(d*x+c)^2-2*cot(d*x+c)*csc(d*x+c)+csc(d*x+c)^2-1)^(1/2))+(-1260*cos(d*x+c 
)^2-630*cos(d*x+c)+315)*sin(d*x+c)*(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*tan( 
d*x+c)^2*arctan(1/2*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*2^(1/2))+I*(-1260 
*cos(d*x+c)^3-630*cos(d*x+c)^2+945*cos(d*x+c)+315)*(-cos(d*x+c)/(cos(d*x+c 
)+1))^(1/2)*tan(d*x+c)^2*arctan(1/2*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*2 
^(1/2))+I*sin(d*x+c)*tan(d*x+c)*(400*cos(d*x+c)^4-2184*cos(d*x+c)^2+630)+( 
-2520*cos(d*x+c)^3-1260*cos(d*x+c)^2+1890*cos(d*x+c)+630)*(-cos(d*x+c)/...
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 300, normalized size of antiderivative = 1.30 \[ \int \cos ^7(c+d x) (a+i a \tan (c+d x))^{5/2} \, dx=-\frac {{\left (315 \, \sqrt {\frac {1}{2}} \sqrt {-\frac {a^{5}}{d^{2}}} d e^{\left (2 i \, d x + 2 i \, c\right )} \log \left (-\frac {9 \, {\left (-i \, a^{3} + \sqrt {2} \sqrt {\frac {1}{2}} \sqrt {-\frac {a^{5}}{d^{2}}} {\left (d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}}\right )} e^{\left (-i \, d x - i \, c\right )}}{16 \, d}\right ) - 315 \, \sqrt {\frac {1}{2}} \sqrt {-\frac {a^{5}}{d^{2}}} d e^{\left (2 i \, d x + 2 i \, c\right )} \log \left (-\frac {9 \, {\left (-i \, a^{3} - \sqrt {2} \sqrt {\frac {1}{2}} \sqrt {-\frac {a^{5}}{d^{2}}} {\left (d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}}\right )} e^{\left (-i \, d x - i \, c\right )}}{16 \, d}\right ) - \sqrt {2} {\left (-10 i \, a^{2} e^{\left (10 i \, d x + 10 i \, c\right )} - 68 i \, a^{2} e^{\left (8 i \, d x + 8 i \, c\right )} - 214 i \, a^{2} e^{\left (6 i \, d x + 6 i \, c\right )} - 544 i \, a^{2} e^{\left (4 i \, d x + 4 i \, c\right )} - 353 i \, a^{2} e^{\left (2 i \, d x + 2 i \, c\right )} + 35 i \, a^{2}\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}}\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{2240 \, d} \] Input:

integrate(cos(d*x+c)^7*(a+I*a*tan(d*x+c))^(5/2),x, algorithm="fricas")
 

Output:

-1/2240*(315*sqrt(1/2)*sqrt(-a^5/d^2)*d*e^(2*I*d*x + 2*I*c)*log(-9/16*(-I* 
a^3 + sqrt(2)*sqrt(1/2)*sqrt(-a^5/d^2)*(d*e^(2*I*d*x + 2*I*c) + d)*sqrt(a/ 
(e^(2*I*d*x + 2*I*c) + 1)))*e^(-I*d*x - I*c)/d) - 315*sqrt(1/2)*sqrt(-a^5/ 
d^2)*d*e^(2*I*d*x + 2*I*c)*log(-9/16*(-I*a^3 - sqrt(2)*sqrt(1/2)*sqrt(-a^5 
/d^2)*(d*e^(2*I*d*x + 2*I*c) + d)*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1)))*e^(-I 
*d*x - I*c)/d) - sqrt(2)*(-10*I*a^2*e^(10*I*d*x + 10*I*c) - 68*I*a^2*e^(8* 
I*d*x + 8*I*c) - 214*I*a^2*e^(6*I*d*x + 6*I*c) - 544*I*a^2*e^(4*I*d*x + 4* 
I*c) - 353*I*a^2*e^(2*I*d*x + 2*I*c) + 35*I*a^2)*sqrt(a/(e^(2*I*d*x + 2*I* 
c) + 1)))*e^(-2*I*d*x - 2*I*c)/d
 

Sympy [F(-1)]

Timed out. \[ \int \cos ^7(c+d x) (a+i a \tan (c+d x))^{5/2} \, dx=\text {Timed out} \] Input:

integrate(cos(d*x+c)**7*(a+I*a*tan(d*x+c))**(5/2),x)
 

Output:

Timed out
 

Maxima [F(-1)]

Timed out. \[ \int \cos ^7(c+d x) (a+i a \tan (c+d x))^{5/2} \, dx=\text {Timed out} \] Input:

integrate(cos(d*x+c)^7*(a+I*a*tan(d*x+c))^(5/2),x, algorithm="maxima")
 

Output:

Timed out
                                                                                    
                                                                                    
 

Giac [F(-2)]

Exception generated. \[ \int \cos ^7(c+d x) (a+i a \tan (c+d x))^{5/2} \, dx=\text {Exception raised: TypeError} \] Input:

integrate(cos(d*x+c)^7*(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. \[ \int \cos ^7(c+d x) (a+i a \tan (c+d x))^{5/2} \, dx=\int {\cos \left (c+d\,x\right )}^7\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{5/2} \,d x \] Input:

int(cos(c + d*x)^7*(a + a*tan(c + d*x)*1i)^(5/2),x)
 

Output:

int(cos(c + d*x)^7*(a + a*tan(c + d*x)*1i)^(5/2), x)
 

Reduce [F]

\[ \int \cos ^7(c+d x) (a+i a \tan (c+d x))^{5/2} \, dx=\sqrt {a}\, a^{2} \left (-\left (\int \sqrt {\tan \left (d x +c \right ) i +1}\, \cos \left (d x +c \right )^{7} \tan \left (d x +c \right )^{2}d x \right )+2 \left (\int \sqrt {\tan \left (d x +c \right ) i +1}\, \cos \left (d x +c \right )^{7} \tan \left (d x +c \right )d x \right ) i +\int \sqrt {\tan \left (d x +c \right ) i +1}\, \cos \left (d x +c \right )^{7}d x \right ) \] Input:

int(cos(d*x+c)^7*(a+I*a*tan(d*x+c))^(5/2),x)
 

Output:

sqrt(a)*a**2*( - int(sqrt(tan(c + d*x)*i + 1)*cos(c + d*x)**7*tan(c + d*x) 
**2,x) + 2*int(sqrt(tan(c + d*x)*i + 1)*cos(c + d*x)**7*tan(c + d*x),x)*i 
+ int(sqrt(tan(c + d*x)*i + 1)*cos(c + d*x)**7,x))